Sharpe ratio

Systematic Trading: A Unique New Method for Designing Trading and Investing Systems
by Robert Carver
Published 13 Sep 2015

How long do you need to decide if a rule has a positive Sharpe ratio (SR) and is worth keeping? To answer this I generated more random trading rule daily returns, this time assuming an underlying positive Sharpe ratio, which again I wouldn’t know in advance. As more trading history is generated I can estimate the SR each year and the average so far. At the same time I look at the distribution of those annual Sharpe ratios.49 This allows me to get 49. For a more technical discussion of this issue, see Andrew Lo’s paper ‘The Statistics of Sharpe ratios’ in Financial Analysts Journal 58:4, July/August 2002. 60 Chapter Three.

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Same volatility: If all assets had the same expected standard deviation of returns. This is always the case for the volatility standardised assets we’re using. 2. Same Sharpe ratio: If all assets had the same expected Sharpe ratio (SR). 3. Same correlation: If all assets had the same expected co-movement of returns. If these assumptions aren’t correct, then what should your portfolio look like?61 What kind of portfolio should we have with... 1. Same Sharpe ratio and correlation: Equal weights. 2. Significantly different Sharpe ratio (SR): Larger weights for assets that are expected to have higher SR, smaller for low SR. 3. Significantly different correlation: Larger weights for highly diversifying assets which have lower correlations to other assets, and smaller for less diversifying assets.

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The economic strategists have belatedly decided that bonds are now the way to go and they have shifted to underweight in all equities. Your previous Sharpe ratio adjustments are now inverted. 240 Chapter Fourteen. Asset Allocating Investor BOND INSTRUMENT WEIGHT ADJUSTMENTS US bonds Euro bonds UK bonds EM bonds Inflation bonds 6.67% 6.67% 6.67% 10.00% 10.00% Expected Sharpe ratio 0.6 0.6 0.6 0.6 0.6 Average Sharpe ratio 0.3 0.3 0.3 0.3 0.3 Outperformance 0.3 0.3 0.3 0.3 0.3 Sharpe ratio adjustment* 1.17 1.17 1.17 1.17 1.17 Renormalised instrument weight** 8.07% 8.07% 8.07% 12.11% 12.11% Instrument weight * From table 12 (page 86) column B. ** After adjustment weights add up to 96.6%.

Portfolio Design: A Modern Approach to Asset Allocation
by R. Marston
Published 29 Mar 2011

7 To assess the return on stocks versus the return on bonds, the first step is to adjust each return for risk. The Sharpe ratio adjusts each return by first subtracting the risk-free rate, rF , then dividing by the standard deviation. If rj is the return on asset j and σ j is its standard deviation, then the Sharpe ratio for asset j is [rj − rF ]/sj To calculate the Sharpe ratio, we use the arithmetic returns and standard deviations in Table 2.1. It’s possible to define a Sharpe ratio using geometric returns, but only if the standard deviation is defined over a similar long horizon.8 Using the returns in Table 1, the Sharpe ratios are defined by Stocks : [0.113 − 0.047]/0.146 = 0.45 Bonds : [0.063 − 0.047]/0.095 = 0.17 The Sharpe ratio for stocks is more than twice the size of the ratio for bonds.

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Suppose that an investment advisor has measured the return on a portfolio in terms of its Sharpe ratio and wants to compare it with some benchmark. (In a later chapter, the performance of the Yale portfolio is compared with the benchmark of university portfolios as a whole). The advisor could simply compare the Sharpe ratios of the portfolio and benchmark. Sharpe ratios rather than alphas are appropriate because it is the total return and total risk that is being assessed. Alpha∗ provides a way of comparing the Sharpe ratios by measuring the excess return earned by the portfolio relative to its benchmark.

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Viewed from this perspective, bonds are not very attractive. How much difference does it make if we invest in medium-term bonds or if we study a longer period of time? If the medium-term bond is used instead of the long-term bond, the Sharpe ratio is 0.29 for the period since 1951, still much less than the Sharpe ratio for stocks. If stocks and bonds are compared over the entire period since 1926, the Sharpe ratio for stocks is 0.40, while that of long-term bonds is 0.22 and medium-term bonds is 0.32. So in both cases, the risk-adjusted return on stocks is substantially higher than that of bonds. Of course, no investor must choose between stocks and bonds.

Advances in Financial Machine Learning
by Marcos Lopez de Prado
Published 2 Feb 2018

You backtest this strategy using the WF method, and the Sharpe ratio is 1.5. You then repeat the backtest on the reversed series and achieve a Sharpe ratio of –1.5. What would be the mathematical grounds for disregarding the second result, if any? You develop a mean-reverting strategy on a futures contract. Your WF backtest achieves a Sharpe ratio of 1.5. You increase the length of the warm-up period, and the Sharpe ratio drops to 0.7. You go ahead and present only the result with the higher Sharpe ratio, arguing that a strategy with a shorter warm-up is more realistic. Is this selection bias? Your strategy achieves a Sharpe ratio of 1.5 on a WF backtest, but a Sharpe ratio of 0.7 on a CV backtest.

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In this section, we account for the risks involved in achieving those results. 14.7.1 The Sharpe Ratio Suppose that a strategy's excess returns (in excess of the risk-free rate), {rt}t = 1, …, T, are IID Gaussian with mean μ and variance σ2. The Sharpe ratio (SR) is defined as The purpose of SR is to evaluate the skills of a particular strategy or investor. Since μ, σ are usually unknown, the true SR value cannot be known for certain. The inevitable consequence is that Sharpe ratio calculations may be the subject of substantial estimation errors. 14.7.2 The Probabilistic Sharpe Ratio The probabilistic Sharpe ratio (PSR) provides an adjusted estimate of SR, by removing the inflationary effect caused by short series with skewed and/or fat-tailed returns.

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The Sharpe ratio is a function of precision rather than accuracy, because passing on an opportunity (a negative) is not rewarded or punished directly (although too many negatives may lead to a small n, which will depress the Sharpe ratio toward zero). For example, for p = .55, , and achieving an annualized Sharpe ratio of 2 requires 396 bets per year. Snippet 15.1 verifies this result experimentally. Figure 15.1 plots the Sharpe ratio as a function of precision, for various betting frequencies. Figure 15.1 The relation between precision (x-axis) and sharpe ratio (y-axis) for various bet frequencies (n) Snippet 15.1 Targeting a Sharpe ratio as a function of the number of bets Solving for 0 ≤ p ≤ 1, we obtain , with solution This equation makes explicit the trade-off between precision (p) and frequency (n) for a given Sharpe ratio (θ).

Quantitative Trading: How to Build Your Own Algorithmic Trading Business
by Ernie Chan
Published 17 Nov 2008

Everyone agrees on what the riskfree rate is, but each trader can use a different market index to come up with their own favorite information ratio, rendering comparison difficult. (Actually, there are some subtleties in calculating the Sharpe ratio related to whether and how to subtract the risk-free rate, how to annualize your Sharpe ratio for ease of comparison, and so on. I will cover these subtleties in the next chapter, which will also contain an example on how to compute the Sharpe ratio for a dollar-neutral and a long-only strategy.) If the Sharpe ratio is such a nice performance measure across different strategies, you may wonder why it is not quoted more often instead of returns. In fact, when a colleague and I went to SAC Capital Advisors (assets under management: $14 billion) to pitch a strategy, their then head of risk management said to us: “Well, a high Sharpe ratio is certainly nice, but if you can get a higher return instead, we can all go buy bigger houses with our bonuses!”

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Maximum drawdown duration Equity in $ A drawdown 3x104 2x104 Maximum drawdown 1x104 2000 2001 2002 2003 Time FIGURE 2.1 Drawdown, Maximum Drawdown, and Maximum Drawdown Duration P1: JYS c02 JWBK321-Chan September 24, 2008 13:47 Fishing for Ideas Printer: Yet to come 21 As a rule of thumb, any strategy that has a Sharpe ratio of less than 1 is not suitable as a stand-alone strategy. For a strategy that achieves profitability almost every month, its (annualized) Sharpe ratio is typically greater than 2. For a strategy that is profitable almost every day, its Sharpe ratio is usually greater than 3. I will show you how to calculate Sharpe ratios for various strategies in Examples 3.4, 3.6, and 3.7 in the next chapter. How Deep and Long Is the Drawdown? A strategy suffers a drawdown whenever it has lost money recently.

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Most (but not all) of these problems associated with comparing returns can be avoided by quoting Sharpe ratio and drawdown instead as the standard performance measures. I introduced the concepts of the Sharpe ratio, maximum drawdown, and maximum drawdown duration in Chapter 2. Here, I will just note a number of subtleties associated with calculating the Sharpe ratio, and give some computational examples in both Excel and MATLAB. There is one subtlety that often confounds even seasoned portfolio managers when they calculate Sharpe ratios: should we or shouldn’t we subtract the risk-free rate from the returns of a dollarneutral portfolio?

Market Sense and Nonsense
by Jack D. Schwager
Published 5 Oct 2012

Although most investors would clearly prefer the return profile of Manager B, the Sharpe ratio decisively indicates the reverse ranking. The potential for a mismatch between Sharpe ratio rankings and investor preferences has led to the creation of other return/risk measures that seek to address the flaws of the Sharpe ratio. Before we review some of these alternative measures, we first consider the question: What are the implications of a negative Sharpe ratio? Although it is commonplace to see negative Sharpe ratios reported for managers whose returns are less than the risk-free return, negative Sharpe ratios are absolutely meaningless. When the Sharpe ratio is positive, greater volatility (as measured by the standard deviation), a negative characteristic, will reduce the Sharpe ratio, as it logically should.

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Despite the fact that Manager B is much worse than Manager A in terms of both return and volatility, Manager B has a higher (less negative) Sharpe ratio. This preposterous result is a direct consequence of higher volatility resulting in higher (less negative) Sharpe ratios when the Sharpe ratio is in negative territory. What should be done with negative Sharpe ratios? Ignore them.5 They are always worthless and frequently misleading. Table 8.3 A Comparison of Two Managers with Negative Sharpe Ratios Sortino Ratio The Sortino ratio addresses both the problems previously cited for the Sharpe ratio. First, it uses the compounded return, which is representative of the actual realized return over any period of time, instead of the arithmetic return.

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Because it distinguishes between upside and downside deviations, the Sortino ratio probably comes closer to reflecting investor preferences than does the Sharpe ratio and, in this sense, may be a better tool for comparing managers. But the Sortino ratio should be compared only with other Sortino ratios and never with Sharpe ratios. Symmetric Downside-Risk Sharpe Ratio The symmetric downside-risk (SDR) Sharpe ratio, which was introduced by William T. Ziemba,6 is similar in intent and construction to the Sortino ratio, but makes a critical adjustment to remove the inherent upward bias in the Sortino ratio vis-à-vis the Sharpe ratio. The SDR Sharpe ratio is defined as the compound return minus the risk-free return divided by the downside deviation.

Understanding Asset Allocation: An Intuitive Approach to Maximizing Your Portfolio
by Victor A. Canto
Published 2 Jan 2005

Value of $1 Return Top 2,919.50 30.5% Second 365.57 21.7% Third 92.48 16.3% Fourth/Median 31.89 12.2% Fifth 11.44 8.4% Sixth 1.75 1.9% Seventh 0.24 –4.7% Strategic Asset Allocation Based On… Period Sharpe Ratio $72.59 15.35% Yearly Sharpe Ratio $34.39 12.5% Market Weights $28.76 11.8% Source: MSCI, Research Insight, and Ibbotson Associates 112 UNDERSTANDING ASSET ALLOCATION Table 6.5 Risk measurements: 1975–2004. CAPM Beta Jensen’s Alpha T-Statistics Sharpe Ratio Small Cap 1 5.64% 2.07 0.65 Large Cap 1 0.00% Growth 1.06 –1.46% 1.87 0.43 Value 0.93 0.81% 1.67 0.60 International 0.62 –1.27% 0.04 0.29 0.53 Strategic Asset Allocation Based On… Period Sharpe Ratio 0.66 0.03% 24.09 0.72 Yearly Sharpe Ratio 0.52 0.02% 24.58 0.63 Market Weights 0.54 0.1% 27 0.57 Comparing the Historical- and Market-Based Allocations As I pointed out in Chapter 1, “In Search of the Upside,” financial economics developments over the past three decades provide us with the necessary tools to develop risk-adjusted returns in a rigorous and systematic way.

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Ideally, one would then search for those asset classes that would add alpha (that is, excess returns to a portfolio) without increasing beta (that is, the risk of the portfolio in relation to the benchmark). Now, let’s apply the Sharpe ratio. Once more, the Sharpe ratio divides a portfolio’s excess returns (returns less risk less Treasury bill returns) by its volatility. In effect, the Sharpe ratio treats each asset class as a separate portfolio, focusing on the standard deviations that measure total risk. If a portfolio in question represents an individual’s entire investment, then volatility matters and the Sharpe ratio is a fitting comparison tool. As such, the Sharpe ratio provides an appropriate way to compare and evaluate the size, style, and location choices within our SAA portfolio.

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This splits the index into two mutually exclusive groups designed to track two of the predominant investment styles in the U.S. equity market.”4 These styles are value and growth, a distinction William Sharpe found valid.5 The Sharpe ratio, named after William Sharpe (the 1990 Nobel Prize in Economics winner), divides a portfolio’s excess return (return less riskless T-bill return) by its volatility. In effect, the Sharpe ratio treats each asset class as a separate portfolio, focusing on the standard deviations that measure total risk. If the portfolio in question represents the entire investment of an individual, volatility matters—and the Sharpe ratio is an appropriate comparison Chapter 2 The Case for Cyclical Asset Allocation 21 tool. As such, the Sharpe ratio provides an apt way to compare and evaluate the size, style, location, and balance of portfolios.

Commodity Trading Advisors: Risk, Performance Analysis, and Selection
by Greg N. Gregoriou , Vassilios Karavas , François-Serge Lhabitant and Fabrice Douglas Rouah
Published 23 Sep 2004

CHAPTER 22 Risk-Adjusted Returns of CTAs: Using the Modified Sharpe Ratio Robert Christopherson and Greg N. Gregoriou any institutional investors use the traditional Sharpe ratio to examine the risk-adjusted performance of CTAs. However, this could pose problems due to the nonnormal returns of this alternative asset class. A modified VaR and modified Sharpe ratio solves the problem and can provide a superior tool for correctly measuring risk-adjusted performance. Here we rank 30 CTAs according to the Sharpe and modified Sharpe ratio and find that larger CTAs possess high modified Sharpe ratios. M INTRODUCTION The assessment of portfolio performance is fundamental for both investors and funds managers, as well as commodity trading advisors (CTAs).

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As a result of the recurring frequency of down markets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool and is considered a mainstream technique to estimate a CTA’s exposure to market risk. 377 378 PROGRAM EVALUATION, SELECTION, AND RETURNS With the large acceptance of VaR and, specifically, the modified VaR as a relevant risk management tool, a more suitable portfolio performance measure for CTAs can be formulated in term of the modified Sharpe ratio.1 Using the traditional Sharpe ratio to rank CTAs will underestimate the tail risk and overestimate performance. Distributions that are highly skewed will experience greater-than-average risk underestimation. The greater the distribution is from normal, the greater is the risk underestimation. In this chapter we rank 30 CTAs according to the Sharpe ratio and modified Sharpe ratio. Our results indicate that the modified Sharpe ratio is more accurate when examining nonnormal returns. Nonnormality of returns is present in the majority of CTA subtype classifications.

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The returns are usually positively skewed (the only exception is the arbitrage strategy) and their distributions tend to have fat tails, as evidenced by the large values for kurtosis. When risk and returns are considered together through the Sharpe ratio,3 the discretionary funds emerge with the highest Sharpe ratio (0.54) followed by other technical funds (with 0.38). Fundamental funds offer a Sharpe ratio of only 0.19. Correlation Analysis Table 4.3 reports the correlation coefficients between the various strategies for the January 1985 to December 2002 period. It indicates that the CTA 3The Sharpe ratio is the ratio of the excess return over the standard deviation. We use a risk-free rate of 5 percent for this calculation. 55 1.00 −0.18 0.41 0.25 0.12 0.26 0.98 0.68 0.93 0.73 0.14 0.81 Discret 0.41 0.20 1.00 0.14 0.13 0.18 0.27 0.16 0.42 0.27 0.00 0.32 Arb −0.18 1.00 0.20 −0.02 0.08 0.05 −0.21 −0.18 −0.13 −0.05 0.24 −0.01 0.25 −0.02 0.14 1.00 0.01 0.08 0.22 0.17 0.22 0.20 −0.02 0.12 Funda 0.12 0.08 0.13 0.01 1.00 0.62 0.12 −0.01 0.03 0.11 0.02 0.12 Option 0.26 0.05 0.18 0.08 0.62 1.00 0.25 0.09 0.14 0.13 0.01 0.29 Stock 0.98 −0.21 0.27 0.22 0.12 0.25 1.00 0.70 0.89 0.71 0.18 0.79 System 0.68 −0.18 0.16 0.17 −0.01 0.09 0.70 1.00 0.56 0.56 0.12 0.56 Teccur 0.93 −0.13 0.42 0.22 0.03 0.14 0.89 0.56 1.00 0.66 0.05 0.73 Tecdiv 0.73 −0.05 0.27 0.20 0.11 0.13 0.71 0.56 0.66 1.00 0.10 0.50 Tecfin 0.14 0.24 0.00 −0.02 0.02 0.01 0.18 0.12 0.05 0.10 1.00 0.09 Tecoth 0.81 −0.01 0.32 0.12 0.12 0.29 0.79 0.56 0.73 0.50 0.9 1.00 Nocat AllCTA = CTA Global Index; Arb = arbitrage; Discret = discretionary; Funda = fundamental; Stock = stock index; System = systematic funds; Teccur = technical currency; Tecdiv = technical diversified; Tecfin = technical financial/metals; Tecoth = other technical; Nocat = no category.

High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems
by Irene Aldridge
Published 1 Dec 2009

If returns of the trading strategy can be assumed to be normal, Jobson and Korkie (1981) showed that the error in Sharpe ratio estimation is normally distributed with mean 0 and standard deviation s = [(1/T)(1 + 0.5SR2 )]1/2 For a 90 percent confidence level, the claimed Sharpe ratio should be at least 1.645 times greater than the standard deviation of the Sharpe ratio errors, s. As a result, the minimum number of evaluation periods used for Sharpe ratio verification is Tmin = (1.6452 /SR2 )(1 + 0.5SR2 ) The Sharpe ratio SR used in the calculation of Tmin , however, should correspond to the frequency of estimation periods. If the annual Sharpe ratio claimed for a trading strategy is 2, and it is computed based on the basis of monthly data, then the corresponding monthly Sharpe ratio SR is 2/(12)0.5 = 0.5774.

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If the annual Sharpe ratio claimed for a trading strategy is 2, and it is computed based on the basis of monthly data, then the corresponding monthly Sharpe ratio SR is 2/(12)0.5 = 0.5774. On the other hand, if the claimed Sharpe ratio is computed based on daily data, then the corresponding Sharpe ratio SR 60 HIGH-FREQUENCY TRADING TABLE 5.2 Minimum Trading Strategy Performance Evaluation Times Required for Verification of Reported Sharpe Ratios Claimed Annualized Sharpe Ratio No. of Months Required (Monthly Performance Data) No. of Months Required (Daily Performance Data) 0.5 1.0 1.5 2.0 2.5 3.0 4.0 130.95 33.75 15.75 9.45 6.53 4.95 3.38 129.65 32.45 14.45 8.15 5.23 3.65 2.07 is 2/(250)0.5 = 0.1054.

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The profitability of a trading strategy is often measured by Sharpe ratios, a risk-adjusted return metric first proposed by Sharpe (1966). As Table 7.2 shows, maximum Sharpe ratios increase with increases in trading frequencies. From March 11, 2009, through March 22, 2009, the maximum possible annualized Sharpe ratio for EUR/USD trading strategies with daily position rebalancing was 37.3, while EUR/USD trading strategies that held positions for 10 seconds could potentially score Sharpe ratios well over the 5,000 mark. The maximum possible intra-day Sharpe ratio is computed as a sample period’s average range divided by the sample period’s standard deviation of the range, adjusted by square root of the number of observations in a year: SR = E[Range]
× (# Intra-day Periods) × (# Trading Days in a Year) σ [Range] (7.1) Note that high-frequency strategies normally do not carry overnight positions and, therefore, do not incur the overnight carry cost often proxied by the risk-free rate in Sharpe ratios of longer-term investments.

Market Risk Analysis, Quantitative Methods in Finance
by Carol Alexander
Published 2 Jan 2007

Weak stochastic dominance therefore indicates that any rational investor should prefer B to A. However, let us compare the Sharpe ratios of the two investments. We calculate the mean and standard deviation of the excess returns and divide the former by the latter. The results are shown in Table I.6.5. Hence, according to the Sharpe ratio, A should be preferred to B! Introduction to Portfolio Theory 259 Table I.6.5 Sharpe ratio and weak stochastic dominance Portfolio Expected excess return Standard deviation Sharpe ratio A B 8.0% 9.80% 0.8165 10.0% 13.56% 0.7372 The Sharpe ratio does not respect even weak stochastic dominance, and the example given above can be extended to other RAPMs derived in the CAPM framework.

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If the autocorrelation in excess returns is positive then (I.6.63) is greater than the square root of h, so the denominator in the Sharpe ratio will increase and the Sharpe ratio will be reduced. Conversely, if the autocorrelation is negative the Sharpe ratio will increase. Example I.6.12: Adjusting a Sharpe ratio for autocorrelation Ex post estimates of the mean and standard deviation of the excess returns on a portfolio, based on a historical sample of daily data, are 0.05% and 0.75%, respectively. Estimate the Sharpe ratio under the assumption that the daily excess returns are i.i.d. and that there are 250 trading days per year.

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Taking this positive autocorrelation into account will have the effect of reducing the Sharpe ratio that is estimated from reported returns. I.6.5.4 Adjusting the Sharpe Ratio for Higher Moments The use of the Sharpe ratio is limited to investments where returns are normally distributed and investors have a minimal type of risk aversion to variance alone, as if their utility function is exponential. Extensions of the Sharpe ratio have successfully widened its application to non-normally distributed returns but its extension to different types of utility function is more problematic. Another adjustment to the Sharpe ratio assumes investors are averse not only to a high volatility but also to negative skewness and to positive excess kurtosis.

Expected Returns: An Investor's Guide to Harvesting Market Rewards
by Antti Ilmanen
Published 4 Apr 2011

• I show empirically that many strategies with the best Sharpe ratios since the 1980s are of the second type. Their covariance with MU is high: they tend to lose money in bad times. This feature can largely explain their high Sharpe ratios. It is harder to explain the performance of government bonds since the 1980s as an equilibrium outcome, given the combination of a high Sharpe ratio and a wonderful diversification/hedging role. As noted, the high in-sample Sharpe ratios of Treasuries over that time frame likely reflect windfall gains from unanticipated yield declines. Note that the Sharpe ratios of Treasuries over periods ending in the early 1980s were very poor, reflecting windfall losses from unanticipated yield increases—a mirror image of the situation from the early 1980s to the present

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[3] If we make predictions of one-year returns, we may be able to make statistically significant inferences about excess returns using 20 years of historical data; whether such short-term forecasts are economically significant is a topic of debate. The Sharpe ratio is closely related to statistical significance. It is the ratio of mean excess return over its volatility, while statistical signifcance uses the same two variables plus sample size. (The standard error of a mean is σ/√N where σ is standard deviation and N is sample size. The denominator of a Sharpe ratio only includes σ.) Given a 20-year sample period, an asset’s annual excess return is statistically significantly above zero if the Sharpe ratio exceeds 0.44 (i.e., the critical value of a two-sided 5% significance level is 1.95 and 1.9/√20 ≈ 0.44).

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[2] Long data histories are necessary to make statistically significant inferences of one-year returns, even assuming constant expected returns and no structural changes (see Note 3 in Chapter 2). Assessing the statistical significance of multi-year horizon returns would require much longer data histories. [3] Sharpe ratio and relative return investors: The information ratio is the benchmark-oriented relative return manager’s counterpart to an absolute return manager’s Sharpe ratio. Information ratio is the mean excess return of a portfolio vs. its benchmark, divided by the tracking error (i.e., volatility of that excess return). Sharpe ratio is sometimes computed as the ratio of total return and its volatility, which is plain wrong; the return on a money market asset should be subtracted from total returns.

Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined
by Lasse Heje Pedersen
Published 12 Apr 2015

WARREN BUFFETT: THE ULTIMATE VALUE AND QUALITY INVESTOR Warren Buffett has become one of the world’s richest people based on his investment success over the past half century. How large a Sharpe ratio does it take to become the richest person in the world? Most investors guess that Warren Buffett must have realized a Sharpe ratio well north of 1 or even 2, perhaps based on Sharpe ratios promised by aggressive fund managers. The truth is that Buffett’s firm Berkshire Hathaway has delivered a Sharpe ratio of 0.76 from 1976 to 2011. While this is lower than some might have expected, it is nevertheless an extremely impressive number. Buffett’s Sharpe ratio is double that of the overall stock market over the same time period, which means that Buffett has delivered twice as much return per unit of risk.

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The standard deviation σ can often be estimated with more precision: It is the square root of the variance σ2, which is estimated as the squared deviations around the arithmetic average, 2.4. TIME HORIZONS AND ANNUALIZING PERFORMANCE MEASURES Performance measures depend on the horizon over which they are measured. For instance, table 2.1 shows that a strategy that has an annual Sharpe ratio of 1 has very different Sharpe ratios if measured over other time horizons, such as a Sharpe ratio of 2 over a four-year period and a mere 0.06 over a trading day. Hence, when we talk about performance measures, we need to be clear about the horizon. Furthermore, when we compare the performance of two different strategies or hedge funds, we need to make sure that the performance measures are calculated over the same time horizon.

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This portfolio does make money because it exploits the fact that, while safe and risky stocks have similar average returns, the safe stocks have significantly higher Sharpe ratios. This portfolio exploits the differences in Sharpe ratios by leveraging the safe shorts and deleveraging the risky ones so that both the long and short sides of the portfolio have a beta of 1. The portfolio is called a “betting against beta” (BAB) factor. The BAB factor for U.S. stocks has realized a Sharpe ratio of 0.78, as seen in figure 9.4. As also seen in the figure, the BAB factor has had positive performance in most global stock markets as well as in the credit markets, bond markets, and futures markets.

Finding Alphas: A Quantitative Approach to Building Trading Strategies
by Igor Tulchinsky
Published 30 Sep 2019

Suppose that a researcher is looking to identify at least one two-­year-­ long backtesting period with an annualized Sharpe ratio higher than 1. If he tries enough strategy configurations, he will eventually find one even if the strategies are actually random, with an expected out-­of-­sample Sharpe ratio of 0. By trying a large enough number of strategy configurations, a backtest can always be fitted to any desired performance for a fixed sample length. A signal can be defined as a strategy configuration whose last M days’ daily PnL Sharpe ratio was higher than S. In Table 9.1, a minimal Sharpe requirement runs across the top and the number of random simulations within which one can expect to see a signal satisfying the requirement runs down the left column.

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Note: as the number of alphas being tested out of sample increases, the out-of-sample test becomes more biased. An alpha can perform randomly well due to luck. Out-of-sample performance at the single-­ alpha level is inadequate when many alphas are tested. Increase the in-sample Sharpe ratio requirement: A higher Sharpe ratio reduces the risk of overfitting. If possible, it is better to test the model on a wider universe, where it should have a higher Sharpe, following the fundamental law of Sharpe ratios: the information ratio equals the information coefficient times the square root of breadth (Grinold and Kahn 1999). In the real world, unfortunately, there are Backtest – Signal or Overfitting?

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After each operation, the per-­dollar return was lower and the Sharpe ratio was higher. The Sharpe ratio was highest after simultaneous neutralization of the three factors. In summary, finding alphas is a constantly evolving process in a competitive market. Some alphas may become less powerful over the years. Because of the risks involved, it is wise to avoid high loadings of risk factors in our portfolios. As predicted by the AMH, innovation is the key to survival. Table 13.1 Example of factor neutralization on an alpha Annualized return Annualized volatility Sharpe ratio Original alpha 16.8% 10.9% 1.55 Neutralized momentum factor 13.3% 7.4% 1.79 Neutralized size factor 14.4% 8.1% 1.77 Neutralized value factor 14.6% 8.1% 1.81 Neutralized all three factors 13.4% 7.3% 1.84 14 Risk and Drawdowns By Hammad Khan and Rebecca Lehman Finding alphas is all about returns over risk.

The Missing Billionaires: A Guide to Better Financial Decisions
by Victor Haghani and James White
Published 27 Aug 2023

To do this, we ran a simulation in which half the time the expected excess return of equities was 1% and the other half it was 9%, which roughly matches the spread of expected excess returns experienced in the past 120 years.13 We found that dynamically scaling the exposure to equities over many simulated histories delivers a roughly 30% average improvement in the Sharpe ratio versus a static strategy. Against this backdrop, the historical experience of the past 120 years appears to be just a little bit worse than we'd expect. The simulation also suggests that over a shorter horizon of 40 years, the dynamic asset allocation has an 85% probability of generating a higher return and a 65% chance of resulting in a higher Sharpe ratio than a static weight strategy. Improvement in Sharpe Ratio Is a Twofer Since 1900, the dynamic strategy has generated a Sharpe ratio about 30% higher than that of the static strategy.

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And for the risk‐adjusted excess return of the optimal portfolio, we get: So, for our Base Case investor in a two‐asset world where the Sharpe ratio of the risky asset is 0.3, the risk‐adjusted excess return on his optimal portfolio will be 2.25%. This formula conveys the message that, with the idealized assumptions under which it holds, an investment with a higher Sharpe ratio generates a higher RAR, and the relationship is turbo‐charged in that doubling the Sharpe ratio quadruples the portfolio's risk‐adjusted excess return. But as we'll see in a moment, in real‐world situations, we'll need to know more than the Sharpe ratio of different investments to properly rank their attractiveness.

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Thirty years later, in a 1994 paper cheekily titled “The Sharpe Ratio,” Bill Sharpe himself conceded that his original term never gained popularity and gave his tacit approval to the term “Sharpe ratio,” which we still use today. Though the original name wasn't a hit, the concept was: a measure originally coined in the very specific context of a theoretical model came to be the de facto standard amongst academics and investors for measuring the risk/reward quality of a wide variety of real‐world investments.c Today, use of the Sharpe ratio in both language and practice is ubiquitous—and naturally, critiques of its use are nearly as abundant.

Trading Risk: Enhanced Profitability Through Risk Control
by Kenneth L. Grant
Published 1 Sep 2004

As it turns out, with a quick sleight of algebraic hand, we can manipulate the Sharpe Ratio equation to determine both what a given level of volatility will produce in terms of returns and (more important for our purposes here) what level of volatility is consistent with a given performance target. In order to understand this critical concept as thoroughly as its importance demands, 112 TRADING RISK let us take the steps involved one at a time. Begin by resurrecting the basic Sharpe Ratio equation: Sharpe Ratio  (Return  Risk-Free Rate)
Portfolio Volatility The Sharpe Ratio can be thought of as a scorecard of your performance as a portfolio manager from a risk-adjusted-return perspective.

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For example, consider two accounts: one that returned 25% in one year, with the first generating a standard deviation of 10%, and one that returned 20%. Under the Sharpe Ratio measurement, the first portfolio would be the better performer, because it ostensibly operated under a reduced risk profile while producing the same overall performance. There are many variations to the Sharpe Ratio calculation, but all of them attempt to capture the following concept: Sharp Ratio  (Return  Risk-Free Return)/Standard Deviation of Return 66 TRADING RISK Note that the right side of the equation can be expressed in terms of dollars or percentages, as long as the same convention applies to each side.

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In fact, my best advice to you (though it be blasphemy if uttered in the church of the pointy-headed) is to go ahead and calculate the standard deviations, without worrying about the validity of the normality assumption. When you do, try to write those lowercase sigmas with flourish and flair aplenty. As Sly Stone said at Woodstock, it’ll do you no harm. Sharpe Ratio Now that we have (more or less) thoroughly explored the concepts of both mean return and associated standard deviation, we can unite these concepts into what has become the industry standard for calculating riskadjusted return: the Sharpe Ratio. This ratio is designed to normalize returns against their associated volatility, the idea being that a unit of return can be judged qualitatively in inverse proportion to the amount of volatility required to produce it.

Cryptoassets: The Innovative Investor's Guide to Bitcoin and Beyond: The Innovative Investor's Guide to Bitcoin and Beyond
by Chris Burniske and Jack Tatar
Published 19 Oct 2017

Taking on a higher level of risk has no benefit in this light, and if a portfolio is unwisely constructed, investors can end up taking on more risk than they’re compensated for. Sharpe Ratio Similar to the concepts behind MPT, the Sharpe ratio was also created by a Nobel Prize winner, William F. Sharpe. The Sharpe ratio differs from the standard deviation of returns in that it calibrates returns per the unit of risk taken. The ratio divides the average expected return of an asset (minus the risk-free rate) by its standard deviation of returns. For example, if the expected return is 8 percent, and the standard deviation of returns is 5 percent, then its Sharpe ratio is 1.6. The higher the Sharpe ratio, the better an asset is compensating an investor for the associated risk.

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The higher the Sharpe ratio, the better an asset is compensating an investor for the associated risk. An asset with a negative Sharpe ratio is punishing the investor with negative returns and volatility. Importantly, absolute returns are only half the story for the Sharpe ratio. An asset with lower absolute returns can have a higher Sharpe ratio than a high-flying asset that experiences extreme volatility. For example, consider an equity asset that has an expected return of 12 percent with a volatility of 10 percent, versus a bond with an expected return of 5 percent but volatility of 3 percent. The former has a Sharpe ratio of 1.2 while the latter of 1.67 (assuming a risk-free rate of 0 percent).

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Currently, cryptoassets often have much higher volatility than other assets, and the Sharpe ratio enables us to understand this volatility in terms of the returns reaped. It’s still important to consider volatility outside of the Sharpe ratio in the context of the investor’s time horizon. While some volatile assets will have excellent Sharpe ratios over long time periods, those investments may not be appropriate for someone needing to place a down payment on a house three months from now. In comparing bitcoin to the FANG stocks, we observed that bitcoin had the highest volatility but also the highest returns by far. Interestingly, its Sharpe ratio was not just the highest but significantly so.

Risk Management in Trading
by Davis Edwards
Published 10 Jul 2014

Richard, the head of a trading desk, is examining the possibility of incorporating a new trading strategy into the trading desk operations. He has four possible strategies for which he calculated Sharpe Ratios and correlation with existing strategies. Which is the best strategy for Richard? A. Sharpe Ratio −0.5, correlation = 0 B. Sharpe Ratio −0.5, correlation = 1.0 C. Sharpe Ratio = +0.5, correlation = 0 D. Sharpe Ratio = +0.5, correlation = 1.0 Chang, a trader at a hedge fund, is examining two trading strategies. Strategy A has a 2.0 Sharpe ratio, Strategy B has a −0.1 Sharpe Ratio, and the strategies have a −1.0 correlation. What is the best combination of strategies? A. Only Strategy A B.

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Richard, the head of a trading desk, is examining the possibility of incorporating a new trading strategy into the trading desk operations. He has four possible strategies for which he calculated Sharpe Ratios and correlation with existing strategies. Which is the best strategy for Richard? A. Sharpe Ratio −0.5, correlation = 0 B. Sharpe Ratio −0.5, correlation = 1.0 C. Sharpe Ratio = +0.5, correlation = 0 D. Sharpe Ratio = +0.5, correlation = 1.0 Correct Answer: C Explanation: Since volatility can’t be negative, a negative Sharpe Ratio indicates a negative expected return. As a result, a strategy with a negative Sharpe Ratio is expected to lose money, and unless it offers a very high diversification benefit, would generally not merit investment.

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The simplest metric is called a Sharpe Ratio. The Sharpe Ratio measures average excess returns (returns above the risk-free rate) divided by the standard deviation of returns. (See Equation 4.1, Sharpe Ratio.) Sharpe Ratio = P * Average (Excess Daily Return) P * StdDev (Excess Daily Returns) (4.1) where P Excess Daily Return The time period adjustment for the Sharpe Ratio The average daily return minus an appropriately scaled risk free rate. For example: Excess Daily Return = daily return − risk free rate 109 Backtesting and Trade Forensics Most commonly, Sharpe Ratios are calculated based on daily returns and then annualized.

Investing Amid Low Expected Returns: Making the Most When Markets Offer the Least
by Antti Ilmanen
Published 24 Feb 2022

See also Naive Trading costs; Trading costs average market impact cost estimates, 229f minimization, 228f Covid-19 crisis, impact, 22, 73 Cowles data value, usage, 111 Credit carry strategy, 130 Credit excess return, 76f Credit market size, 75 Credit performance statistics, 78t Credit premium (premia), 74–81 Credit strategies, 125 Cross-country returns, 245 Cross-sectional approaches, time-series approaches (contrast), 182 Cross-sectional momentum, 117, 118, 182 Cross-sectional stock selection strategies, application, 121 Crowding concerns, 158–160 Cube, The, 197f Cumulative excess returns, 71f, 82f Cumulative return, 129f Currency carry, 125–126 Currency-hedged investor, function, 55 Cyclically-adjusted earnings yield (CAEY), 18, 59–60 aggregate measure, 113 averages, comparison, 65 judgement, in-sample signals (usage), 236 level, reduction, 68 quintile buckets, 238f scatter plot/time series, 237f usage, 174, 236 D Data mining, usage, 170–171 Defensive factor, 153 Defensive stock selection, low beta/quality basis, 132–133 Defensive strategies, 131–137 Defensive US equity strategies, performance, 135f Defined-benefit (DB) pension plan (D-B plans), 28–30 glide paths, adoption/changes, 32, 32f low expected return challenge, impact, 42–43 US public/corporate DB pensions, A-B evolving asset allocation, 38f Defined contribution (DC) pension, 16, 28 Defined contribution (DC) savers, low expected return challenge (impact), 44–45 di Bartolomeo, Dan, 160 Dilution effects, variation, 64 Dimson, Elroy, 36 Directional commodity index trading, 83 Direct real estate, listed real estate (comparison), 95 Discounting, 164, 170 Discount rate effect, 17–21, 17f Discretionary investing, systematic investing (contrast), 142b Disinflation, secular developments, 71 Disposition effect, 167 Diversification, 187–189, 191–194 A-B value, 190f usage, 191f Dividend discount model (DDM), 59, 174 Dividends-base Gordon model, 174 Dividends per share (DPS), 59, 60, 64–65 Dividend yield-based stock selection strategy, 130 Dividend yield differential, 127–128 Downward-sloping glide path, justification, 30–31 Dragon risk, 196 Drawdown control rules, 209–210 Dry powder, absorption challenge symptom, 100 Dynamic risk control strategies, 209–210 E Earnings per share (EPS), 59–62 Economic rationale, global change (impact), 169 Economic recessions (bad times measure), 154 Ellis, Charley, 36 Emerging market bonds, growth, 75 Empirical evidence, usage, 170–171 Employee satisfaction, evidence, 223 Endowment Model (Yale Model), 13, 37, 41 Endowments, 29, 45 Environmental, Social, and Governance (ESG) investing, 11, 26, 42, 205, 219–223 considerations, 6 ESG-Sharpe ratio frontier, stylized example, 222f framework, 220f preferences, 196 strategies, 168 themes, examples, 223t Equations, usage, 173–180 Equilibrium real yields/growth, comparison, 21 Equities beta, 101 bonds, shift, 35–36 contrarian timing, 240–242 equity-bond allocation, optimization, 201f equity-like risk-taking, 47 housing, performance comparison, 90–93 market beta, A-B scatterplot multi-asset average return, 154f money management process, 40 returns, GDP growth (weak empirical relationship), 67b valuation, level, 68 Equity allocation (stylized glide path), 30f, 32f Equity market, 168 correlations, evolution, 130f decline (bad times measure), 154 empirically decomposed equity market return, 66t ownership shares, evolution, 35f strategy Sharpe Ratios, predictor basis, 241f tail performance, 123f Equity premia, 55–69, 58f Estimation errors (MVO), 204–206 Excess returns, 9f, 70–71, 71f, 81, 82f Exchange-traded funds (ETFs), 34, 40, 83, 141 Exotic beta, 106 Expected real return (60/40 stock/bond portfolio), 19f Expected real return/expected inflation, US cash rate split, 54f Expected return, 62–64, 176, 197, 235 Expected Returns (Ilmanen), 10, 12, 75 Extrapolative strategies, 117–124 Ezra, Don, 36 F Factor, 8, 107 coefficients, 182 diversification/timing, usage, 242f factor-based investing, 47 momentum, 122 Fallen angels, 79–80, 102 Falling next-decade expectations, 20f Fama, Eugene, 107, 109, 111, 140, 152, 176, 178 Fama-French factor models, 140 Fama-MacBeth factor-mimicking portfolios, 183 Fear of missing out (FOMO), 246 Fees, 96, 225, 230–232 Five-factor model (Fama/French), 107 Fixed income (FI), 201 funds, positive credit beta tilt, 80 Flow data, understanding, 161b Forward-looking analysis, 65 Forward-looking equity premia (real yields), 59–62 Forward-looking indicators, 96 Forward-looking real equity return, 68f Forward-looking returns, 59 Forward-looking Treasury yields, decomposition, 71–72 French, Kenneth, 99, 107, 109, 111, 114, 140, 176, 230 Front-end opportunities, 80 Fundamental law of active management (FLAM), 178–180, 188, 193 Funding ratio (FR), 28, 31–32, 42 evolution, 43f Future excess returns, 238f G GAAP earnings (replacement), operating earnings (usage), 62 GARCH models, 215 Generic risk premium (1/price effect), 126 Geometric mean (GM), 81–82 Glide paths, adoption/changes, 32, 32f Global asset allocation, contrarian strategies (usage), 114 Global bond yield decline, convergence (relationship), 23f Global equities average compound/premia, 57f cumulative excess returns, comparison, 71f diversification, home bias (contrast), 188 drawdowns, 155 Global Financial Crisis (GFC), 4, 21, 38, 208 buyout index, drawdown, 99 volatility, 124 Global government bonds/global equities, cumulative excess returns (comparison), 71f Globalization, disinflationary impact, 53 Global Japanification, 21 Global market portfolio, 39b–40b Goetzmann, Will, 111 Gold, examination/history, 83, 85f Goobey, Ross, 36 Gordon growth model, 59, 62 Governance, problems, 248 Government bond index returns, 75 Gradualism, usage, 169 Great Depression, 36, 58, 208 Great Inflation, 53 Grinold, Richard, 178 Gross domestic product (GDP), 67b, 154 GSCI index, 83 H Hedge fund index cumulative excess return over cash, 144f Hedge funds, 144, 144f, 146f, 147, 149 industry excess-of-cash return, decomposition, 148t High yield (HY) bonds, 74, 76 High yield (HY) corporate bonds, characteristics, 76–77 High-yield default rates, 79f High yield (HY) market, OAS, 77 Historical average, 56 excess returns, 75–77, 81–83 return, 180 Historical bond yields, excess returns (relationship), 70–71 Historical equity premium, 56–59 Historical realized premium, 56 Historical returns, sources (understanding), 65 Home bias global equity diversification, contrast, 188 reduction, 37 Horizon, 165f, 181–182 Housel, Morgan, 248 Housing, 90–93 NCREIF commercial real estate index, real return (decomposition), 100f US housing, arithmetic mean returns (decomposition), 92f I Idiosyncratic momentum, 121 Idiosyncratic security risk, diversification, 109 Idiosyncratic volatility factor (IVOL), 134, 136 Illiquid alternatives, 36–37, 88–101, 203 Illiquid assets, 24 class, performance/risk statistics, 91t global wealth share, 89b–90b premia, 156 Illiquidity premia, 87, 95, 101, 146–147 correlations, 108t smoothing service, impact, 98–99, 99f Illiquidity proxy, A-D scatterplot multi-asset Sharpe ratio, 156f Impatience, 164–167 Incomes, equations, 174–175 Income strategies, 124–131 Index funds, market share loss, 40 Individual pension saver, 28 Inflation, 54 average inflation, real cash return (relationship), 53f GFC, impact, 21 hedging premium, 83, 85–86 increase, 215, 250 inflation-indexed government bonds, liquidity, 101–102 sensitivities, 85f Information ratio (IR), 80 Informed traders, 102 In-sample signals, usage, 236 Institutional investors asset management fees, estimates, 231f procyclicality, 246 Intangibles, B/P ratio, 112–113 Interest rate risk, 101 Internal rates of return, basis, 89, 99 Intertemporal CAPM, 176 Inverted Shiller CAPE, 174 Investable global market, history, 38 Investable indices, absence, 9 Investable, limit (broadening), 38 Investing, 34, 42, 47, 168, 243 active investing, passive investing (contrast), 140–142 principles, 250–251 superstars, examination, 147, 149, 149f systematic investing, discretionary investing (contrast), 142b Investment, 6–8, 11–15 models, taxonomy, 41f opportunities, 83, 247t problem/solution, identification, 30–31 returns, decadal perspective, 24–26 risk, management techniques, 209–210 success, forecasting skills (impact), 206 Investment-grade (IG) bonds, 74, 79 Investors, 27–33, 42–45 institutional investors, asset management fees (estimates), 231f low expected returns, impact, 249–250 patience, enhancement process, 167–169 portfolio construction process, 177–178 returns demand, 175–177 risk level, determination, 249–250 trading costs, differences, 226 value, addition process, 178–180 Irrational mispricing, rational reward, 152–153 K Kahneman, Daniel, 164 Kelly criterion, 216 Keynes, John Maynard, 99 L Large US endowments, asset allocation (evolution), 35f Last-decade performance, flatness, 119 Law of small numbers, 164–165 Leibowitz, Marty, 12, 36 Leverage, 156, 197 Leveraged buyout funds, impact, 95–96 Leverage risk management, 210 Liability matching, 206 long-duration bonds, demand, 73–74 Lifecycle investing, theories, 30 Line-item thinking, 168 Liquid alternatives, 106 Liquid ARP, 201 Liquid asset class, 118 premia, 51, 156 Liquid assets, 24, 156 Liquidity, 101, 102 Liquidity risk premium proxy, 101 Liquid public assets, 101–102 Listed real estate, direct real estate (comparison), 95 Long horizons, 181–182, 197 return, 236–237 Long-run return sources, 151–153 conviction/patiences, sustaining, 163 pyramid, 52f Long/short alternative risk premia variants, 8 Long-short value strategy, risks/pitfalls, 114–115 Long-shot bias, 136 Long/shot sides, valuation, 116f Long-term corporat bond returns, 75 Long-term per-share dividend/earnings growth, lag, 63 Long-term reversal patterns, 158f Lottery preferences, 156 Low expected return challenge, 15, 42–45 institutional answers, 47f investor responses, 27, 45–47 storyline, change, 79–80 Low expected returns, 15, 17–21 impact, 249–250 Serenity Prayer, relationship, 3–6 Low-for-longer policy rates, 55 Low-risk strategies, 131–137 LTREV Factors, 114 M Machine learning, 170, 183b–184b Macroeconomic exposures, 197–198, 200 Macroeconomic sensitivities, 199f Madoff, Bernie (returns), 167 Manager-specific alpha, 9 Market, 40 inefficiency, 143 liquidity, 87 moves, expectations, 126 risk, management, 214–216 yields, anchor, 19 Market beta, 231 exposures, problems, 114–115 Market risk premia (MRP), 146 Mark-to-market pricing, 89, 98 Marshmallow tests, 164 Maverick risk, 194, 210 Max expected utility, equation, 177 Mean reversion, appearance, 19 Mean-variance optimization (MVO), 177–178, 200–206 Medium-term expected returns, tactical timing, 235 Model errors (MVO), 204 Modern Portfolio Theory (MPT), 36, 202b Momentum equity market trail performance, 123f extrapolative strategy, 117–124 monthly/quarterly momentum, 158f opportunities, 247 patterns, 245f style premia, per-decade/century-long SRs, 122f US momentum-based stock selection strategies, cumulative performance, 119f Money losses, 158–160 management process, 40–42 Multi-asset portfolio, variation, 190 Multidimensionality, meaning, 88 Multi-factor strategies, trading (increase), 227 Multi-metric composite, 112 Multiyear procyclic investor flows/mean-reverting returns (connection), 167 Multiyear return chasing, 244–246 Mutual funds, 34, 41f N NCREIF commercial real estate index, real return (decomposition), 100f Net dilution, 64 Net rental yields, overstatement, 93 Net return, maximization, 228f Net total payout yield, components, 64f Net total yield (NTY), 63 Next-decades excess returns, scatter plot/time series, 237f NIPA profits, usage, 240 Non-investment-grade bonds, 74 Non-profit-seeking market participants, presence, 126 Non-standard preferences, 157 Normal backwardation, 86 Normal cash rate, perception, 52 Norway Model, 13, 40, 41 Notional diversification, impact, 188 O Objective long-run return expectations, 61f Off-the-run bond, 102 One-factor alphas, measurement, 140 On-the-run bond, 102 OOS forecasts, 239 Optimal portfolios, information (variation), 205t Option-adjusted spread (OAS), 77 Organizational commitment, increase, 168 Outcome bias, Serenity Prayer (contrast), 5–6 Out-of-the-money (OTM) index, 212 short-dated deep-OTM index, rolling, 213 Overdiversification/diworsification, danger, 193 Overtrading, 243 P Passive investing, active investing (contrast), 140–141 Passive money management, 40 Path-dependence, role, 43 Patience adversity, conviction (sustaining), 164–169 impatience, causes, 164–166 investor patience, enhancement process, 167–169 problems, 167 sustaining, 163 Pension fund balance sheet, terminology, 28–29 Pension plans, problems, 46 Per-decade equity premia, 57–58, 58f Performance metrics, IRR basis, 89 Peso problem, 167 Portfolios, 197f carbon emissions measure, 224 choice, risky assets (impact), 203f construction, 187, 195–196 insurance strategies, 209–210 investor construction, 177–178 macroeconomic exposures, 197–198, 200 optimal portfolios, information (variation), 205t portfolio-level MVO, usage, 201–203 rebalancing strategies, 114 return/risk assumptions, 201t return sources, classification, 146–147 review, broadness/infrequency, 168 top-down decisions, 195–200 total portfolio, illiquid share, 196–197 volatility, 214–216 Predictive techniques, 173, 180–183 Price-scaled predictors, usage, 240 Private assets, 88–101 classes, valuation/expected return data (absence), 101 performance, 99–100 Private credit, 75, 88 Private equity (PE) A-B ex-post/ex-ante edge (weakening), 97f burnout, returns, 95–96, 98 funds, returns (earning), 232 illiquid alternative asset class, 88 managers, fee pressure, 96 Private market returns, IRR basis, 90 Probability distribution, reshaping, 209 Proprietary alpha, 51, 140 Prospect theory (Kahneman-Tversky), 157 Prudent Man Rule, 36 Public flows, importance, 162 Public pension plans, impact, 46 Pure alpha, 232 Put strategies, trend strategies (contrast), 210–213 Q Quality-adjusted house price appreciation, 93 Quality-minus-junk (QMJ), 132, 134, 136 performance, 134f Quality strategies, 131–137 R Radical Uncertainty (Kay-King), 216 Rational risk premia, 155–158 Real asset returns, 100 Real cash rates, 53, 55 Real cash return, average inflation (relationship), 53f Real estate illiquid alternative asset class, 88 returns, 94 Real estate investment trusts (REITs), 95 Real income growth, 94 Realized asset class return, decadal perspective, 25f Realized asset return, windfall gains, 17f Realized nominal returns, 46 Realized real return (60/40 stock/bond portfolio), 19f Real short rates, gold price history (relationship), 85f Real yields (forward-looking equity premia), 59–62 Rearview-mirror expectations, 17–21 Rebalancing, 114, 192 Regression, 180, 182–183 Replacement rate (75%), annual savings rate (requirement), 44f Responsible investing, framework, 220f Rethinking the Equity Risk Premium (Ilmanen), 56 Retirement, savings (calculation), 30 Returns active returns, alpha (relationship), 139–146 constant-varying expected returns, time-varying expected returns (contrast), 180–181 cumulative excess returns, comparison, 71f cumulative return, 129f enhancement, risk management (impact), 216b–217b equation, 174 ESG investing, impact, 221–223 excess returns, historical bond yields (relationship), 70–71 expected real returns, 19f, 54f expected return, equation, 176 fair split, 232–233 future excess returns, 238f generation process, 174–175 historical average excess returns, 75–77 historical performance, 111–112 improvement, 8–9 investor demands, 175–177 medium-term expected returns, tactical timing, 235 multiyear return chasing, 244–246 net return, maximization, 228f private equity (burnout) returns, 95–96, 98 rolling relative return, 166f smooth returns, 167–169 Reversal patterns, 227, 245f Risk, 208–210 adjusted active return, subtraction (equation), 179 aversion, increase, 160 level, investor determination, 249–250 management, 187, 207 perception, 197 rational reward, 152–153 risk-based explanations, 110, 126 risk-mitigating strategies, 213 risk-mitigating strategies, performance, 213t risk-neutral hypotheses, 128 risk-reward trade-off, 45 Risk-adjusted return (information ratio), 80 patience, enhancement, 168–169 Riskless arbitrage, 102 Riskless bonds, yields, 16 Riskless cash return, 52–55, 153–155, 153f Riskless long-term yields, historical perspective, 21–24 Riskless rate, 16 Risk parity 60/40 stock/bond portfolio, contrast, 188–189 investing, 179 Risk premia, 16, 153–158 annual excess returns/Sharpe ratios, 9f bond risk premium, 69–74 generic risk premium (1/price effect), 126 Rolling relative return, 166f Roll, Richard, 178 S Safe assets, demand, 73–74 Samonov, Mikhail, 111 Samuelson's dictum, 145 Saving glut hypothesis, 23 Saving/investment plan, setup, 29 Scheidel, Walter, 154 Seasonal strategies, 103 Secular low expected return, challenge, 15 Selection biases, 170, 223 Serenity Prayer, 3–6 Sharpe ratio (SR), 9f, 120f, 133 A-D scatterplot multi-asset Sharpe ratio, 156f ESG-Sharpe ratio frontier, stylized example, 222f increase, 120, 191f, 242f long-run Sharpe ratios, 112t predictor basis, 241f underperformance, frequency, 165f Sharpe, William, 175 Short horizons equity, 241 long horizons, contrast, 181–182 Short-selling constraints, 152 Short-term bill yields, history, 70, 70f Short-term reversal, 117 Simple expected real return, 7f Size premium, 107b Skew, A-D scatterplot multi-asset Sharpe ratio, 156f Skewness preferences, 155 Slippage, 228 Small numbers, law, 164–165 Smart beta factor investing, 8 Smart money, 121 Smoothing service, impact, 98–99, 99f Smoothness preference, 99 Smooth returns, 167–169 Soros, George (examination), 149f Sorting methods, regression methods (contrast), 182–183 Sovereign wealth fund, perception, 33 Stable-minus-risky (SMR) factor, 134, 136 Stable-minus-risky market-neutral (SMRMN), 134, 136 Standard & Poor's 500 (S&P500) Berkshire Hathway, rolling relative return (contrast), 166f returns, 24, 76 St.

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Under standard assumptions (no non-normality or selection bias), statistical significance at the 95% confidence level requires a t-statistic near 2.0. Since the t-statistic is a product of realized Sharpe ratio and time period (the square root of years), a performance history with a Sharpe ratio of 0.4 will be significantly different from zero (with 95% confidence) after about 25 years (0.4 * √25 = 2), while a Sharpe ratio of 0.7 will be significantly different from zero after roughly 8 years (0.7 * √8 ≈ 2). 7 There are other entirely good reasons for changing allocations after losses. Investors don't have the same transparency as managers into the drivers of performance.

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, 1920–2020 Figure 8.2 A-B Scatterplot multi-asset average return on (A) volatility, (B)... Figures 8.3 A-D Scatterplot multi-asset Sharpe Ratio on (A) bad-times averag... Figure 8.4 Monthly and Quarterly Momentum and Long-Term Reversal Patterns fo... Chapter 9 Figure 9.1 Frequency of Underperformance, for a Given Horizon and Sharpe Rat... Figure 9.2 Rolling Relative Return of Berkshire Hathaway vs. S&P500, Jan 197... Chapter 11 Figure 11.1 Volatility as a Function of the Number of Assets and the Correla... Figure 11.2 A-B Value of Diversification Across Asset Classes and Across Sty... Figure 11.3 Sharpe Ratio Boosting Through Diversification, 1926–2020 Chapter 12 Figure 12.1 The Cube: Asset Class, Strategy Style, and Macro Factor Perspect...

Red-Blooded Risk: The Secret History of Wall Street
by Aaron Brown and Eric Kim
Published 10 Oct 2011

A strategy with an annualized Sharpe ratio of 2 will make more than the risk-free rate about 39 years out of 40. However, it’s hard to find Sharpe ratios near or above 1 in high-capacity, liquid strategies that are inexpensive to run. You don’t need a Sharpe ratio near or at 1 to get rich. For a Kelly investor, the long-term growth of capital above the risk-free rate is approximately equal to the Sharpe ratio squared (it’s actually always higher than this, substantially so for high Sharpe ratios, but that doesn’t affect the points I want to make). A Sharpe ratio of 1 means growing at 100 percent per year—that is, doubling your capital.

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A Sharpe ratio of 0.1 produces a 1 percent excess return; a Sharpe ratio of 0.2 gives about 4 percent; a Sharpe ratio of 0.5 gives about 25 percent. Since the high Sharpe ratio strategies have limited capacity but grow so quickly that initial investment is almost irrelevant, they are appropriate for people investing their own money. Sharpe ratios around 0.5 make good hedge fund strategies. The lower Sharpe ratios are useful for large institutions with cheap capital and high risk tolerance. Card counters gravitated to very high Sharpe ratio strategies. When I say these have limited capacity, I don’t mean there are always opportunities to invest a small amount in them.

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It’s still important to distinguish the difference, however, because the effect that the new generation of quants had on events was quite different from the effect from the old generation. Sharpe Ratios and Wealth Returning to the old school, the two main branches found different kinds of market opportunities, distinguished by Sharpe ratio. We’re going to get a bit mathematical again, but you don’t need the numbers to follow the argument. The Sharpe ratio of a strategy is defined as the return of the strategy minus what you could make investing the same capital in risk-free instruments, divided by the standard deviation of the return. It is a measure of risk-adjusted return. A strategy with an annualized Sharpe ratio of 1 will make more than the risk-free rate about five years out of six.

The New Science of Asset Allocation: Risk Management in a Multi-Asset World
by Thomas Schneeweis , Garry B. Crowder and Hossein Kazemi
Published 8 Mar 2010

The Sharpe Ratio has other well-known shortcomings, including: ■ ■ ■ In periods of historical negative returns, the strict Sharpe comparisons have little value. The Sharpe Ratio should be based on expected return and risk; however, in practice, actual performance over a particular period of time is often used. In periods of negative mean return, an asset may have a lower negative return as well as a lower standard deviation and yet report a lower Sharpe Ratio (e.g., more negative) than an alternative asset with a greater negative return and with a higher relative standard deviation. Gaming the Sharpe Ratio. A manager with a high Sharpe Ratio will get a close look from institutional investors even if the absolute returns are less than stellar.

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The higher the ratio, the more favorable the assumed risk-return characteristics of the investment. The Sharpe Ratio is computed as: Si = (Ri − Rf ) σi where R̄i is the estimated mean rate of return of the asset, Rf is the risk-free rate of return, and σi is the estimated standard deviation. This measure can be taken to show return obtained per unit of risk. While the Sharpe Ratio does offer the ability to rank assets with different return and risk (measured as standard deviation), its use may be limited to comparing portfolios that may realistically be viewed as alternatives to one another. First, the Sharpe Ratio has little to say about the relative return to risk of individual securities.

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There is simply too much randomness in the price movement of individual securities to make the Sharpe Ratio of any real use at the individual asset level. Moreover, the Measuring Risk 27 Sharpe Ratio does not take into account that the individual assets may themselves be used to create a portfolio. As discussed previously, the CAPM purports that the expected return of a security stems more from the covariance of the security with the market portfolio than from the stand-alone risk of the individual asset. The Sharpe Ratio has other well-known shortcomings, including: ■ ■ ■ In periods of historical negative returns, the strict Sharpe comparisons have little value.

Unknown Market Wizards: The Best Traders You've Never Heard Of
by Jack D. Schwager
Published 2 Nov 2020

Sall also provides a perfect example of the deficiency of the Sharpe ratio. His Sharpe ratio is 1.43, an excellent level, but not a particularly exceptional one. The Sharpe ratio’s primary pitfall is that its risk component (the standard deviation) penalizes large gains equivalently to large losses. For a trader like Sall, who has many spectacularly large gains, this penalty is severe. Sall’s adjusted13 Sortino ratio, a statistic that only penalizes downside volatility, is 12 times as large as his Sharpe ratio. This extreme imbalance between the two ratios is extraordinary. Most traders have an adjusted Sortino ratio to Sharpe ratio closer to 1:1.

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Therefore, it is still necessary to check return alone. The Sharpe Ratio The Sharpe ratio is the most widely used risk-adjusted return measure. The Sharpe ratio is defined as the average excess return divided by the standard deviation. Excess return is the return above the risk-free return (e.g., T-bill rate). For example, if the average return is 8% per year and the T-bill rate is 3%, the excess return would be 5%. The standard deviation is a measure of the variability of return. In essence, the Sharpe ratio is the average excess return normalized by the volatility of returns. There are two basic problems with the Sharpe ratio: The return measure is based on average rather than compounded return.

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The average annual compounded return of -13.4%, however, would reflect the reality (86.6% × 86.6% = 75%). The Sharpe ratio does not distinguish between upside and downside volatility. The risk measure inherent in the Sharpe ratio—the standard deviation—does not reflect the way most investors perceive risk. Traders and investors care about loss, not volatility. They are averse to downside volatility but actually like upside volatility. I have yet to meet an investor who complained because his manager made too much money in a month. The standard deviation, and by inference the Sharpe ratio, however, is indifferent between upside and downside volatility. This characteristic of the Sharpe ratio can result in rankings that would contradict most people’s perceptions and preferences.25 Sortino Ratio The Sortino ratio addresses both the problems cited for the Sharpe ratio.

The Volatility Smile
by Emanuel Derman,Michael B.Miller
Published 6 Sep 2016

Define a new variable 𝜆, the ratio of a security’s excess return to its volatility, so that 𝜆≡ 𝜇−r 𝜎 (2.2) The variable 𝜆 is the well-known Sharpe ratio. Now, for the portfolio of a risky security and riskless bonds in Equation 2.1, the Sharpe ratio is 𝜆P ≡ 𝜇P − r w𝜇 + (1 − w) r − r w (𝜇 − r) 𝜇 − r = = = ≡𝜆 𝜎P w𝜎 w𝜎 𝜎 (2.3) The Sharpe ratio of the portfolio is equal to the Sharpe ratio of the risky security. Diluting a portfolio by investing part of the portfolio in riskless bonds has no effect on the Sharpe ratio.6 Now consider another uncorrelated stock S′ that has the same volatility w𝜎 as the portfolio P. It has the same numerical risk as portfolio P consisting of S and a riskless bond, but, since it is a separate source of risk, uncorrelated with the behavior of S, both risks are unavoidable.

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A portfolio manager can always increase expected returns by taking more risk (by diluting the portfolio less or by borrowing more). In order to generate a higher Sharpe ratio, however, a portfolio manager must either increase excess returns without increasing risk, keep excess returns the same while lowering risk, or both increase excess returns and lower risk. All other things being equal, rational investors prefer investments with higher Sharpe ratios. Asset managers often employ leverage, and one additional feature of the Sharpe ratio as a measure of performance is that, assuming you can borrow at the riskless rate, the Sharpe ratio is invariant under changes in leverage. If a portfolio manager borrows the full value of her original portfolio with characteristics (𝜇, 𝜎) to invest twice as much in the same portfolio, her expected return increases to 2𝜇 − r, twice as much from the portfolio less the interest r paid on the loan.

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The excess return of the leveraged portfolio therefore doubles to 2𝜇 − 2r. But the portfolio also has double the volatility, so its Sharpe ratio remains the same. It is fitting that a measure of fund performance should not increase when the fund simply borrows more money to invest. Note that the Sharpe ratio is not dimensionless. When calculating the Sharpe ratio, we typically use annualized numbers. The average return is then the average return per year, and volatility is calculated as the square root of the standard deviation of returns per year, so that the dimension of 𝜆 is (year)−1/2 . The Sharpe ratio therefore depends on the units of time used to calculate the returns.

Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life
by Emanuel Derman
Published 13 Oct 2011

According to the EMM the test is the value of your portfolio’s Sharpe ratio. If your portfolio contains securities whose realized Sharpe ratios turn out to be greater than the market’s average Sharpe ratio, you have been smart. Of course, only if the EMM weren’t quite right could you hope to find exceptional Sharpe ratios. The theory of efficient markets has become so much a part of accepted market lore that professional money managers measure and report their Sharpe ratios with the hope of demonstrating their talent. Judging Investments by Their Sharpe Ratios Suppose that over the past 10 years a security A has produced an average annual return μA = 15%, with high returns of 20% and low returns of 10%, so that the annual volatility σA is 5%, corresponding to return fluctuations of ±5 percentage points about the mean.

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The market will be in equilibrium when the price of the option and the price of the stock are such that each security’s Sharpe ratio—each security’s expected excess return per unit of its risk—is the same for both, so that investors will have no reason to prefer one security over the other as a route to taking on Apple risk. Until that is the case, investors will preferentially buy the more efficient security and sell the less efficient one until their prices adjust to demand and they eventually provide the same risk premium. That’s equilibrium. By equating the Sharpe ratio of the stock and the Sharpe ratio of the option, Black and Scholes were able to derive and, a few years later, solve an equation for the model value of the call option.

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Therefore a stock with half its volatility, that is, 10%, must provide half as much expected return, that is, 2.5% Adding this to the riskless bond’s return of 2%, you should expect a return of 4.5% on a stock with volatility of 10%—if the EMM is correct. The Sharpe Ratio According to the Law of Proportionality of Risk and Return, a risk must provide an excess return . Put differently, this means that for any security the ratio of excess return to risk is always the same, since each is proportional to the weight w. Finance theorists like to write the Law of Proportionality of Risk and Return entirely in Greek symbols to make it seem oracular: This states that the ratio of excess return to risk is a “universal” constant, denoted by the Greek letter X (lambda), which is referred to as the Sharpe ratio, after William Sharpe, who first began to make use of this concept in the mid-1960s.

Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors
by Wesley R. Gray and Tobias E. Carlisle
Published 29 Nov 2012

By investigating the price metrics along these lines, we seek to find any problems not uncovered by the raw or adjusted performance measures. William Sharpe created the Sharpe ratio in 1966, intending it to be used to measure the risk-adjusted performance of mutual funds.5 Sharpe was interested in the extent to which managers took on extra risk to generate additional return. He wanted to find some measure that would adjust the return for the risk taken to generate it. He created the Sharpe ratio, which does this by examining the historical relationship between excess return—the return in excess of the risk-free rate—and volatility, which stands in for risk. The higher the Sharpe ratio, the more return is generated for each additional unit of volatility, and the better the price metric.

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Table 7.3 sets out the Sharpe and Sortino ratios and historical drawdown risk metrics for the market capitalization-weighted value decile portfolios of each price ratio. Table 7.3 shows that the enterprise multiples have the top risk-adjusted performance, whether we examine the results using the Sharpe ratio or the Sortino ratio. The enterprise multiple (EBIT variation) monthly Sharpe ratio of 0.58 is the highest, and its monthly Sortino ratio of 0.89 is also the highest. This means the enterprise multiple (EBIT variation) metric offers the best risk/reward ratio, whether we define risk as volatility (Sharpe ratio) or just downside volatility (Sortino ratio). The EBITDA variation also stands out with favorable Sharpe and Sortino ratios of 0.53 and 0.82, respectively.

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The higher the Sharpe ratio, the more return is generated for each additional unit of volatility, and the better the price metric. The Sortino ratio, like the Sharpe ratio, measures risk-adjusted return. The difference is that the Sortino ratio only measures downside volatility, while the Sharpe ratio measures both upside and downside volatility. The Sortino ratio doesn't adjust return for upside volatility, only for downside volatility, which we wish to avoid. The Sortino ratio also measures excess returns in excess of a minimum acceptable return. We use 5 percent per year as the minimum acceptable return in our analysis. The Sortino ratio therefore measures the excess return over a minimum acceptable return per unit of downside risk.

Beyond Diversification: What Every Investor Needs to Know About Asset Allocation
by Sebastien Page
Published 4 Nov 2020

Stefan Hubrich points out that there is another way to think about how managed volatility may increase Sharpe ratios in certain market environments: think of time diversification as similar to cross-asset diversification. Suppose we invest in five different stocks with the same Sharpe ratios but very different volatility levels. If we assume the stocks are uncorrelated, we should allocate equal risk (not equal value weights) to get the Sharpe ratio–maximizing portfolio. The same logic applies through time; the realized variance of the portfolio is basically the sum of the point-in-time variances. To get the highest Sharpe ratio through time, we should allocate equal risk to each period.

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Fat tails matter because asset allocators must forecast exposure to loss when they construct portfolios. For example, I’m always surprised at how many professional investors, who know the impact of fat tails on exposure to loss, still ignore them when they discuss Sharpe ratios (defined as excess return over cash, divided by volatility). Clearly, we should not use Sharpe ratios to compare strategies that sell optionality (like most alternative investments) with traditional asset classes, investment strategies, and products. But I continue to hear such comparisons everywhere. In early 2019, I attended a conference on risk premiums in New York.

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I was young, starting my career, and I was quite excited to see the strategy beat the pants off the MSCI World Index, with very limited look-ahead bias. (In hindsight, exceptional backtest performance is never that exceptional. A client once told me that he had never seen a backtest that didn’t work. He added that the only people who can consistently generate Sharpe ratios of 3.0 or above were quants running backtests, plus Bernie Madoff. Even without explicit look-ahead bias, researchers benefit from years of published research on what works and what doesn’t, which is itself an implicit look-ahead bias.) The last and hardest building block to forecast is valuation change.

Beyond the Random Walk: A Guide to Stock Market Anomalies and Low Risk Investing
by Vijay Singal
Published 15 Jun 2004

However, it is important to take risk into account because industry-momentum-based trading strategies are clearly riskier than holding a 91 92 Beyond the Random Walk broader market portfolio. The Sharpe ratio is used to compare the risk-adjusted returns.3 Sharpe ratios are reported below based on annual returns for the best-case scenarios, an average return of 4 percent for short-term Treasury bills during 1997–2001, and standard deviations (reported in parentheses) in Table 5.4. 25-week estimation period and 5-week holding period 5-week estimation period and 5-week holding period 5-week estimation period and 1-week holding period S&P 500 holding return 0.91 0.76 1.05 0.49 Since higher Sharpe ratios indicate superior investment, the best results are for the one-week holding period, with a ratio of 1.05.

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If an optimal currency portfolio, consisting of the German mark, Japanese yen, Swiss franc, and the British pound with the U.S. dollar as the risk-free asset, is formed, then it has been found that the portfolio would have generated an average excess return of 2.79 percent per year over the period November 1989 through June 1999. The Sharpe ratio for this portfolio is 0.69 compared with a Sharpe ratio of 0.53 for the U.S. Treasury index, 0.49 for an unhedged global Treasury index, 0.80 for a hedged global Treasury index, and 0.95 for the S&P 500. The Sharpe ratio for the S&P 500 is unusually high because of the high returns earned by stocks during this period. Overall, the evidence suggests that holding currencies can be a superior form of investment than several other forms of investment, after accounting for risk, even during the 1990s.

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Beyond the Random Walk Table 11.6 Bias in Currency Forward Rates The annual return over the period January 2000 to June 2002 is 15.6 percent, slightly better than the 13.4 percent return in Table 11.5. The excess return is 11.5 percent with a standard deviation of 10.7 percent and a Sharpe ratio of 1.07. The excess return is impressive, as is the risk-adjusted return as measured by the Sharpe ratio. Overall, implementation of the trading strategy reveals annual excess returns of 9.3 percent and 11.5 percent. The strategy seems to be eminently successful with Sharpe ratios of 0.85 and 1.07. Qualifications The evidence presented in this chapter and the trading strategy recommendations based on the evidence rely on past data.

Trend Following: How Great Traders Make Millions in Up or Down Markets
by Michael W. Covel
Published 19 Mar 2007

So stop taking about who I work for and start justifying the industry wide Sharpe Ratio of 0.09 to your invstors [sic]. You have been stealing investor money for too long, 2-20 for trend following really?????” Performance data for trend following traders, month-by-month performance, is there for all to see (Appendix B and Iasg.com). If those numbers are considered “stealing” to Bhardwaj, I can’t convince him to see another light. After word Further, Bhardwaj thinks the Sharpe ratio is an appropriate measure of trend following traders. It is not (see Chapter 3, “Performance Data”). Trend following trader David Harding has written on the Sharpe ratio: “The Sharpe ratio appears at first blush to reward returns (good) and penalize risks (bad).

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His previous job being a vice president of a disaster like AIG can’t look good on a resume. He’s probably lucky to be working at all.” Another reader responded: “The Sharpe ratio of CTAs [trend following traders] does not need to be ‘explained.’ Most investors want the investment to be profitable. The Sharpe ratio does not measure ‘risk’ as it is commonly understood. Even Sharpe, in his original paper, wrote about ‘variability’ not ‘risk.’ The Sharpe ratio punishes a spike up in the same way as a spike down. Many managers have strategies that produce a good average profit over a long period, but require the investors to accept some gyration in the mean time.

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Unlike some who dismiss trend following, these folks will get it since they are willing to learn something new. “Ahhh, now I get it. The buy and hold crowd justifies their 50% drawdowns and 0% 10 year total return by the Sharpe ratio, of course! I can hear the funds on the phone with their investors now, ‘Yes, you’ve lost half your money in the last year and you have made nothing in 10 years but don’t worry, the Sharpe ratio was good!’ Is this what Vanguard means by ‘sensible buy and hold’? If the Sharpe ratio is good then it must be sensible, returns and drawdowns be damned!” Bhardwaj is a pawn of the mutual fund industry. The mutual fund industry spends millions through lobbying in Washington and propaganda (i.e ‘academic research’) to keep trend following traders from advertising their performance.

Learn Algorithmic Trading
by Sebastien Donadio
Published 7 Nov 2019

We will use a week as the time horizon for our trading strategy: last_week = 0 weekly_pnls = [] weekly_losses = [] for i in range(0, num_days): if i - last_week >= 5: pnl_change = pnl[i] - pnl[last_week] weekly_pnls.append(pnl_change) if pnl_change < 0: weekly_losses.append(pnl_change) last_week = i from statistics import stdev, mean sharpe_ratio = mean(weekly_pnls) / stdev(weekly_pnls) sortino_ratio = mean(weekly_pnls) / stdev(weekly_losses) print('Sharpe ratio:', sharpe_ratio) print('Sortino ratio:', sortino_ratio) The preceding code will return the following output: Sharpe ratio: 0.09494748065583607 Sortino ratio: 0.11925614548156238 Here, we can see that the Sharpe ratio and the Sortino ratio are close to each other, which is what we expect since both are risk-adjusted return metrics. The Sortino ratio is slightly higher than the Sharpe ratio, which also makes sense since, by definition, the Sortino ratio does not consider large increases in PnLs as being contributions to the drawdown/risk for the trading strategy, indicating that the Sharpe ratio was, in fact, penalizing some large +ve jumps in PnLs.

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There are some very large profits and losses for some weeks, but they are very rare, which is also within the expectations of what the distribution should look like. Sharpe ratio Sharpe ratio is a very commonly used performance and risk metric that's used in the industry to measure and compare the performance of algorithmic trading strategies. Sharpe ratio is defined as the ratio of average PnL over a period of time and the PnL standard deviation over the same period. The benefit of the Sharpe ratio is that it captures the profitability of a trading strategy while also accounting for the risk by using the volatility of the returns.

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Models with negative values usually indicate issues in the training data or process and cannot be used: from sklearn.metrics import mean_squared_error, r2_score # The mean squared error print("Mean squared error: %.2f" % mean_squared_error(Y_train, ols.predict(X_train))) # Explained variance score: 1 is perfect prediction print('Variance score: %.2f' % r2_score(Y_train, ols.predict(X_train))) # The mean squared error print("Mean squared error: %.2f" % mean_squared_error(Y_test, ols.predict(X_test))) # Explained variance score: 1 is perfect prediction print('Variance score: %.2f' % r2_score(Y_test, ols.predict(X_test))) This code will return the following output: Mean squared error: 27.36 Variance score: 0.00 Mean squared error: 103.50 Variance score: -0.01 Finally, as shown in the code, let's use it to predict prices and calculate strategy returns: goog_data['Predicted_Signal'] = ols.predict(X) goog_data['GOOG_Returns'] = np.log(goog_data['Close'] / goog_data['Close'].shift(1)) def calculate_return(df, split_value, symbol): cum_goog_return = df[split_value:]['%s_Returns' % symbol].cumsum() * 100 df['Strategy_Returns'] = df['%s_Returns' % symbol] * df['Predicted_Signal'].shift(1) return cum_goog_return def calculate_strategy_return(df, split_value, symbol): cum_strategy_return = df[split_value:]['Strategy_Returns'].cumsum() * 100 return cum_strategy_return cum_goog_return = calculate_return(goog_data, split_value=len(X_train), symbol='GOOG') cum_strategy_return = calculate_strategy_return(goog_data, split_value=len(X_train), symbol='GOOG') def plot_chart(cum_symbol_return, cum_strategy_return, symbol): plt.figure(figsize=(10, 5)) plt.plot(cum_symbol_return, label='%s Returns' % symbol) plt.plot(cum_strategy_return, label='Strategy Returns') plt.legend() plot_chart(cum_goog_return, cum_strategy_return, symbol='GOOG') def sharpe_ratio(symbol_returns, strategy_returns): strategy_std = strategy_returns.std() sharpe = (strategy_returns - symbol_returns) / strategy_std return sharpe.mean() print(sharpe_ratio(cum_strategy_return, cum_goog_return)) This code will return the following output: 2.083840359081768 Let's now have a look at the graphical representation that is derived from the code: Here, we can observe that the simple linear regression model using only the two features, Open-Close and High-Low, returns positive returns.

Deep Value
by Tobias E. Carlisle
Published 19 Aug 2014

Figure 10.5 shows the compound annual growth rates for each of the portfolios in each decade. 200 DEEP VALUE TABLE 10.2 Performance Statistics By Decade for All Value Stocks, Deep Value Stocks, and Glamour Value Stocks (1951 to 2013) Deep Value Portfolios 1951–1959 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio 1960–1969 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio 1970–1979 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio 1980–1989 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio 1990–1999 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio 2000–2013 Compound Annual Growth Rate Arithmetic Average Return Standard Deviation Sharpe Ratio Glamour Value Portfolios All Value Portfolios 15.74% 15.82% 20.40% 0.85 12.30% 14.34% 23.63% 0.61 14.39% 15.82% 21.47% 0.74 19.24% 18.16% 15.42% 1.18 5.90% 4.65% 9.89% 0.47 12.71% 11.40% 14.38% 0.79 8.27% 9.44% 7.86% 1.20 9.13% 11.12% 14.35% 0.77 8.82% 10.28% 11.30% 0.91 14.92% 15.21% 11.69% 1.30 13.10% 13.14% 13.91% 0.95 14.18% 14.18% 12.55% 1.13 23.95% 22.91% 10.99% 2.08 6.92% 5.09% 12.56% 0.41 15.52% 14.00% 14.68% 0.95 12.97% 14.92% 12.97% 1.15 8.00% 9.73% 11.41% 0.85 10.69% 12.32% 12.27% 1.00 Data Source: Eyquem Investment Management LLC, Compustat.

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TABLE 5.1â•… Average Annualized Five-Year Performance Statistics by Capitalization for Price-to-Book Value Quintiles (1980 to 2013) Glamour 1 All Large Cap Small Cap Annual Return Standard Deviation Sharpe Ratio Annual Return Standard Deviation Sharpe Ratio Annual Return Standard Deviation Sharpe Ratio 8.80% 19.03% 0.21 9.85% 19.77% 0.25 8.23% 19.21% 0.17 2 10.82% 18.24% 0.33 11.62% 18.90% 0.36 10.46% 18.61% 0.30 3 11.32% 17.45% 0.37 11.78% 17.27% 0.40 11.22% 17.88% 0.35 4 11.89% 16.08% 0.43 11.65% 16.12% 0.42 12.14% 16.67% 0.43 Value Value Premium 5 (5-1) 13.48% 16.44% 0.52 13.36% 16.45% 0.52 13.41% 16.96% 0.50 4.68% 3.51% 5.18% 93 94 DEEP VALUE lowest returns with the worst Sharpe ratio, and the value quintile earns the highest returns with the best Sharpe ratio. The quintiles also perform in almost perfect inverse rank order from glamour to value (column 5 outperforms column 4 and so on).

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The standard deviation of returns to the Magic Formula over the full period was 16.93 percent, against 17.28 percent for the earnings yield, and $100,000,000 $10,000,000 S&P500 TR Magic Formula Earnings Yield Return on Capital $1,000,000 $100,000 $10,000 1965 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 $1,000 FIGURE 4.2â•… Logarithmic Chart of Magic Formula (Market Weight), Earnings Yield, Return on Capital, and S&P 500 (Total Return) Performance (1974 to 2011) Source: Eyquem Investment Management LLC. 61 The Acquirer’s Multiple TABLE 4.1 Performance Statistics for Magic Formula (Market Weight), Earnings Yield, Return on Capital, and S&P 500 (Total Return) (1974 to 2011) CAGR (%) Standard Deviation (%) Downside Deviation (%) Sharpe Ratio Sortino Ratio (MAR = 5%) Rolling 5-Year Outperformance (%) Rolling 10-Year Outperformance (%) Correlation Magic Formula Earnings Yield Return on Capital S&P 500 TR 13.94 16.93 12.02 0.55 0.80 15.95 17.28 11.88 0.64 0.96 10.37 17.04 11.35 0.35 0.56 10.46 15.84 11.16 0.37 0.56 — 15.11 84.38 80.10 — — 11.28 0.927 89.91 0.806 96.44 0.872 Data Source: Wesley Gray and Tobias Carlisle.

The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal
by Ludwig B. Chincarini
Published 29 Jul 2012

It was as if Meriwether and his 140 employees had a magic money tree. TABLE 2.1 LTCM Returns versus Standard Asset Classes The Sharpe ratio is an important, risk-adjusted tool for comparing the performances of different investments or portfolio managers. The higher the ratio, the better the portfolio manager. (See Box 2.1.) Table 2.1 shows that, before 1998, the LTCM fund had a Sharpe ratio five times that of the standard returns of U.S. Treasury bills and bonds. Box 2.1 The Sharpe Ratio The Sharpe ratio measures the return of a portfolio minus the risk-free rate divided by the portfolio’s standard deviation. It is a risk-adjusted return measure that assists in comparing different portfolios or investments, even in the presence of leverage.

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It’s important, but it’s not enough to judge who’s a better golfer. Distance matters, but so does accuracy. How often do drives land in the fairway? The Sharpe ratio is essentially drive distance divided by the variation of the drive distance from the center of the fairway. It lets us compare two golfers. It probably goes without saying that Phil Mickelson had a lower variation than the average amateur, and a much higher Sharpe ratio. He is clearly the better golfer. The LTCM portfolio, consisting mainly of fixed-income instruments, had a Sharpe ratio almost double the S&P 500 during a period in which the United States had one of the strongest bull markets in history.

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The new firm’s risk was about half that of LTCM, and its returns were between a third and a half of its predecessor’s. LTCM’s Sharpe ratio was 2.54 during its first three years, much higher than JWMP’s 1.63. JWMP existed from December 1999 to April 2009. It performed reasonably well during its first eight years, which led up to the 2008 financial crisis. Before 2008, the fund’s average annual return was 8.55%, with a standard deviation of 3.81%. The fund’s Sharpe ratio was 1.33. In comparison, the S&P 500 had a Sharpe ratio of 0.00 over the same period. The fund’s best monthly return was 3.64%; its worst monthly return was −2.99%.

Capital Ideas Evolving
by Peter L. Bernstein
Published 3 May 2007

The result is the gold at the end of the investor’s rainbow: a high Sharpe Ratio. The Sharpe Ratio is a measure of return relative to risk. Specifically, the Sharpe Ratio is the ratio of a portfolio’s realized return, minus the return on a riskless asset, divided by the volatility of that return. Higher is always better than lower—a bigger bang for the buck. Litterman pays close attention to the Sharpe Ratios of the portfolios under his supervision. “We still lose money all the time, between the good months. Nothing is easy. Nevertheless, we have been creating a Sharpe Ratio of 1.0 for ten years running, and that’s huge.

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In that case, they are behaving as though the ratio of alpha they expect to earn relative to the volatility of that active risk will work out to be only 0.01 percent to 0.05 percent. That is a razor-thin number. Litterman goes on to explain further: “If, for example, you expect a Sharpe Ratio of only 0.25, which is the approximate Sharpe Ratio of the equity market, then you should allocate your risk equally between beta and active risk. If you expect a Sharpe Ratio above 0.5—a return as high as one-half the volatility you experience—then clearly you want active risk to be the dominant risk in your portfolio.” Conservative investors holding a diversified portfolio half in equities and half in fixed-income can expect positive returns in the long run, but need to be realistic.

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The focus on risk management, however, does not mean that Litterman believes in any way that low risk is preferable to high risk. Control is what matters. Higher risk at a given Sharpe Ratio means more return, because the ratio is return divided by risk; holding the ratio constant, taking increased risk should lead to higher returns. Litterman is fascinated that so many investors are averse to taking active risk. Given a basic exposure to the equity markets—that is, beta—he believes there is an optimal amount of active risk associated with whatever Sharpe Ratio you assume. Most investors take too little active risk, with 90 percent or more of their total risk coming from beta—from the volatility of the market itself.

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues
by Alain Ruttiens
Published 24 Apr 2013

When it is the case, above performance measures cannot hold, in particular the Sharpe ratio and its variants. The Sortino Ratio The Sortino ratio is built in a similar way as the Sharpe ratio, except that, instead of dividing by the standard deviation of a series of past returns, the divisor is the downside semi-standard deviation σd, that is, the standard deviation of the past negative returns. This makes sense when the portfolio is involving asymmetric instruments like options (cf. Chapter 4, Sec-tion 4.3.7): Example. On the same data as for the previous Sharpe ratio example in Section 14.1.3, that is a fund invested in the S&P 500, data of year 2009, with a p.a. standard deviation of negative returns (downside semi-standard deviation) of 19.76%, the result is instead of 0.59 for the usual Sharpe ratio.

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Hence, it makes sense to consider that in a risky investment (r, σ) it should be the excess return only, that is, r − rf, that pays for the supported risk. Hence the Sharpe ratio: Practically speaking, for a given period of past data leading to r and σ measures, the rf rate must be of a non-defaultable government bill or bond of maturity coinciding with the same period of time as used for r and σ. The data for r and rf being usually expressed on a p.a. basis, σ must also be computed on a p.a. basis. Example. For a fund passively invested in the S&P 500 in 2009, the computed return and risk were 17.96% p.a. and 27.04% p.a. respectively (based on daily closing prices). The corresponding 12-month T-Bill was 2.004%. The Sharpe ratio is Note that in the fund industry, it is hard to achieve a Sharpe ratio above 1, which may be viewed as a reference level.

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It thus comes to the ratio of the Jensen's α and the Tracking Error, on a p.a. basis: This ratio gives an idea of the importance of the excess return obtained by a fund, considering the undergone excess risk, hence, a kind of “excess” Sharpe ratio. Global Example of Calculation of These Ratios Maximum Draw Down (MDD); Calmar Ratio The MDD is the highest loss over a given period. For a given series of periodic portfolio performances, it can refer to the worst periodic performance, but better on a cumulative level (if successive periodic performances are negative). Example. Consider the previous example related to the Sharpe ratio, of a fund invested in the S&P 500 and disclosing monthly performance (NAV): the corresponding prices data during 2009, but on a monthly basis, show a MDD of (−) 21.11% = (931.8 − 735.09)/931.8, during Jan + Feb 09 – see Figure 14.3.

Frequently Asked Questions in Quantitative Finance
by Paul Wilmott
Published 3 Jan 2007

The more sensible measures make allowance for the risk that has been taken, since a high return with low risk is much better than a high return with a lot of risk. Sharpe Ratio The Sharpe ratio is probably the most important non-trivial risk-adjusted performance measure. It is calculated as where µ is the return on the strategy over some specified period, r is the risk-free rate over that period and σ is the standard deviation of returns. The Sharpe ratio will be quoted in annualized terms. A high Sharpe ratio is intended to be a sign of a good strategy. If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of the risk-free rate.

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If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of the risk-free rate. In the expected return versus risk diagram of Modern Portfolio Theory the Sharpe ratio is the slope of the line joining each investment to the risk-free investment. Choosing the portfolio that maximizes the Sharpe ratio will give you the Market Portfolio. We also know from the Central Limit Theorem that if you have many different investments all that matters is the mean and the standard deviation. So as long as the CLT is valid the Sharpe ratio makes sense. The Sharpe ratio has been criticized for attaching equal weight to upside ‘risk’ as downside risk since the standard deviation incorporates both in its calculation.

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This may be important if returns are very skewed. Modigliani-Modigliani Measure The Modigliani-Modigliani or M2 measure is a simple linear transformation of the Sharpe ratio:M2 = r + ν × Sharpe where ν is the standard deviation of returns of the relevant benchmark. This is easily interpreted as the return you would expect from your portfolio if it were (de)leveraged to have the same volatility as the benchmark. Sortino Ratio The Sortino ratio is calculated in the same way as the Sharpe ratio except that it uses the square root of the semi-variance as the denominator measuring risk. The semi variance is measured in the same way as the variance except that all data points with positive return are replaced with zero, or with some target value.

Triumph of the Optimists: 101 Years of Global Investment Returns
by Elroy Dimson , Paul Marsh and Mike Staunton
Published 3 Feb 2002

Equivalently, we could just have compared the two Sharpe ratios. The excess return for the United States is 1.0672/1.00875 – 1 = 5.79 percent, so the US Sharpe ratio is 5.79/20.16 = 0.287. For the world, the excess return (relative to US treasury bills) is 1.0579/1.00875 – 1 = 4.87 percent, so the Sharpe ratio is 4.87/17.04 = 0.286. This leads to the same conclusion, namely, that over the twentieth century, US citizens who invested in the world equity portfolio would have achieved almost exactly the same reward-to-risk ratio as those who restricted themselves to US equities. These Sharpe ratios are shown in Figure 8-4, which also Figure 8-4: Comparative reward-to-risk ratios for US citizens investing in world versus US equities 0.5 Sharpe ratios from perspective of US investor 0.440 World United States World ex-US 0.4 0.3 0.286 0.422 0.407 0.287 0.218 0.187 0.2 0.151 0.1 0.049 0.0 1900–2000 1900–1949 1950–2000 113 Chapter 8: International investment gives the comparative ratios for the two halves of the twentieth century.

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For each country, we compute a relative Sharpe ratio by dividing the Sharpe ratio for an investment in the world index by the equivalent ratio for domestic investment. For example, for the United States, the relative Sharpe ratio based on the data from Figure 8-4 was 0.286/0.287 = 1.00 for 1900– 2000, and 0.440/0.422 = 1.04 for 1950–2000. A relative ratio of one indicates that the world index had the same reward to risk ratio as domestic investment; a ratio above one, that the world index dominated; and a ratio below unity, that domestic investment gave the highest reward for risk. Figure 8-5 shows the relative Sharpe ratios for each country.

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To assess this, we measure the reward per unit of risk using the Sharpe ratio. The latter is defined as the excess return on a portfolio over a given period, divided by the standard deviation of the portfolio’s returns. The excess return is the actual return, less the interest rate, typically taken as the treasury bill rate. The Sharpe ratio is defined in terms of excess Triumph of the Optimists: 101 Years of Global Investment Returns 112 returns in recognition of the fact that investors can blend an investment in equities with lending or borrowing at the interest rate to achieve any desired level of risk. To illustrate use of the Sharpe ratio, we compare the risk/return trade-off for US equities versus the world index.

Python for Finance
by Yuxing Yan
Published 24 Apr 2014

Our major function would start from Step 3 as shown in the following code: # Step 3: generate a return matrix (annul return) n=len(ticker) # number of stocks x2=ret_annual(ticker[0],begdate,enddate) for i in range(1,n): x_=ret_annual(ticker[i],begdate,enddate) x2=pd.merge(x2,x_,left_index=True,right_index=True) # using scipy array format R = sp.array(x2) print('Efficient porfolio (mean-variance) :ticker used') print(ticker) [ 216 ] Chapter 8 print('Sharpe ratio for an equal-weighted portfolio') equal_w=sp.ones(n, dtype=float) * 1.0 /n print(equal_w) print(sharpe(R,equal_w)) # for n stocks, we could only choose n-1 weights w0= sp.ones(n-1, dtype=float) * 1.0 /n w1 = fmin(negative_sharpe_n_minus_1_stock,w0) final_w = sp.append(w1, 1 - sum(w1)) final_sharpe = sharpe(R,final_w) print ('Optimal weights are ') print (final_w) print ('final Sharpe ratio is ') print(final_sharpe) From the following output, we know that if we use a naïve equal-weighted strategy, the Sharpe ratio is 0.63. However, the Sharpe ratio for our optimal portfolio is 0.67: Constructing an efficient frontier with n stocks Constructing an efficient frontier is always one of the most difficult tasks for finance instructors since the task involves matrix manipulation and a constrained optimization procedure.

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.] >>> The output shows that the function value is 3, and it is achieved by assigning x as 0. [ 214 ] Chapter 8 Constructing an optimal portfolio In finance, we are dealing with the trade-off between risk and return. One of the widely used criteria is the Sharpe ratio, which is defined as follows: 6KDUSH ( 5  5I VS (27) The following program would maximize the Sharpe ratio by changing the weights of the stock in the portfolio. We have several steps in the program: the input area is very simple, just several tickers in addition to the beginning and ending dates. Then, we define four functions: converting daily returns into annual ones, estimate a portfolio variance, estimate the Sharpe ratio, and estimate the nth weight when n-1 weights are given: from matplotlib.finance import quotes_historical_yahoo import numpy as np import pandas as pd import scipy as sp from scipy.optimize import fmin # Step 1: input area ticker=('IBM','WMT','C') # tickers begdate=(1990,1,1) # beginning date enddate=(2012,12,31) # ending date rf=0.0003 # annual risk-free rate In the second part of the program, we define a few functions: download data from Yahoo!

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Why is it claimed that the sn.npv() function from SciPY() is really a Present Value (PV) function? 22. Design a true NPV function using all cash flows, including today's cash flow. 23. The Sharpe ratio is used to measure the trade-off between risk and return: Sharpe = R − Rf σ Here, R is the expected returns for an individual security, and R f is the expected risk-free rate. σ is the volatility, that is, standard deviation of the return on the underlying security. Estimate Sharpe ratios for IBM, DELL, Citi, and W-Mart by using their latest five-year monthly data. [ 122 ] Visual Finance via Matplotlib Graphs and other visual representations have become more important in explaining many complex financial concepts, trading strategies, and formulae.

Optimization Methods in Finance
by Gerard Cornuejols and Reha Tutuncu
Published 2 Jan 2006

Now more commonly known as the Sharpe measure, or Sharpe ratio, this quantity measures the expected return per unit of risk (standard deviation) for a zero-investment strategy. The portfolio that maximizes the Sharpe ratio is found by solving the following problem: maxx µT x−rf (xT Qx)1/2 Ax = b Cx ≥ d. (5.3) In this form, this problem is not easy to solve. Although it has a nice, polyhedral feasible region, its objective function is somewhat complicated, and worse, is possibly non-concave. Therefore, (5.3) is not a convex optimization problem. The standard strategy to find the portfolio maximizing the Sharpe ratio, often called the optimal risky portfolio, is the following: First, one traces out the efficient frontier on a two dimensional return vs. standard deviation graph.

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We present these conditions next. xR is an optimal solution of problem (5.1) if and only if there exists λR ∈ <, γE ∈ <m , and γI ∈ <p satisfying the following conditions: QxR − λR µ − AT γE − C T γI = 0, µT xR ≥ R, AxR = b, CxR ≥ d, λR ≥ 0, λR (µT xR − R) = 0, γI ≥ 0, γIT (CxR − d) = 0. 5.2 (5.2) Maximizing the Sharpe Ratio Consider the setting in the previous subsection. Let us define the function σ(R) : [Rmin , Rmax ] → < as σ(R) := (xTR QxR )1/2 , where xR denotes the unique solution of problem (5.1). Since we assumed that Q is positive definite, it is easily shown that the function σ(R) is strictly convex in its domain. As mentioned before, the efficient frontier is the graph E = {(R, σ(R)) : R ∈ [Rmin , Rmax ]}. 5.2. MAXIMIZING THE SHARPE RATIO 61 We now consider a riskless asset whose expected return is rf ≥ 0. We will assume that rf < Rmin , which is natural since the portfolio xmin has a positive risk associated with it while the riskless asset does not.

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Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 3 4 4 5 6 7 8 9 11 11 . . . . . . . . 13 13 14 17 18 18 21 24 27 . . . . . 29 29 30 31 34 36 iv CONTENTS 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Quadratic Programming: Theory and Algorithms 4.1 The Quadratic Programming Problem . . . . . . . 4.2 Optimality Conditions . . . . . . . . . . . . . . . . 4.3 Interior-Point Methods . . . . . . . . . . . . . . . . 4.4 The Central Path . . . . . . . . . . . . . . . . . . . 4.5 Interior-Point Methods . . . . . . . . . . . . . . . . 4.5.1 Path-Following Algorithms . . . . . . . . . . 4.5.2 Centered Newton directions . . . . . . . . . 4.5.3 Neighborhoods of the Central Path . . . . . 4.5.4 A Long-Step Path-Following Algorithm . . . 4.5.5 Starting from an Infeasible Point . . . . . . 4.6 QP software . . . . . . . . . . . . . . . . . . . . . . 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 5 QP 5.1 5.2 5.3 5.4 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models and Tools in Finance Mean-Variance Optimization . . . . . . . . . . . . . . . . Maximizing the Sharpe Ratio . . . . . . . . . . . . . . . Returns-Based Style Analysis . . . . . . . . . . . . . . . Recovering Risk-Neural Probabilities from Options Prices Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Stochastic Programming Models 6.1 Introduction to Stochastic Programming . 6.2 Two Stage Problems with Recourse . . . . 6.3 Multi Stage Problems . . . . . . . . . . . . 6.4 Stochastic Programming Models and Tools 6.4.1 Asset/Liability Management . . . . 6.4.2 Corporate Debt Management . . . . . . . . . . . . . . . . . . . . . in Finance . . . . . . . . . . . . 7 Robust Optimization Models and Tools in Finance 7.1 Introduction to Robust Optimization . . . . . . . . . 7.2 Model Robustness . . . . . . . . . . . . . . . . . . . . 7.2.1 Robust Multi-Period Portfolio Selection . . . . 7.3 Solution Robustness . . . . . . . . . . . . . . . . . . 7.3.1 Robust Portfolio Selection . . . . . . . . . . . 7.3.2 Robust Asset Allocation: A Case Study . . . 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . 43 43 44 45 48 49 49 50 53 55 56 56 57 . . . . . 59 59 60 63 65 68 . . . . . . 71 71 72 74 76 76 78 . . . . . . . 83 83 83 84 88 88 90 92 CONTENTS 8 Conic Optimization 8.1 Conic Optimization Models and Tools in Finance 8.1.1 Minimum Risk Arbitrage . . . . . . . . . . 8.1.2 Approximating Covariance Matrices . . . . 8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . .

Inside the House of Money: Top Hedge Fund Traders on Profiting in a Global Market
by Steven Drobny
Published 31 Mar 2006

If repeated year in and year out, the expected average gain of 30 percent (0.50 × 0.60) minus the expected average loss of 20 percent (0.50 × 0.40) would produce an average annual return of 10 percent. (See Figure A.1.) Big Bet Performance Analysis Hedge fund managers are often judged by their Sharpe ratios, which are calculated as the fund’s return minus the risk-free rate divided by the volatility of returns. The Sharpe ratio is also known as the reward-tovolatility ratio and provides a sense of the quality of the managers’ returns per unit of risk. It allows for some element of comparison across managers. Hedge funds often strive for a Sharpe ratio of at least 1.0 such that their 70% 60% Probability 50% 40% 30% 20% 10% 0% –50% –40% –30% –20% –10% 0% Return FIGURE A.1 Hypothetical Big Bet Portfolio 10% 20% 30% 40% 50% WHY GLOBAL MACRO IS THE WAY TO GO 345 performance is commensurate with the amount of risk assumed.

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Hedge funds often strive for a Sharpe ratio of at least 1.0 such that their 70% 60% Probability 50% 40% 30% 20% 10% 0% –50% –40% –30% –20% –10% 0% Return FIGURE A.1 Hypothetical Big Bet Portfolio 10% 20% 30% 40% 50% WHY GLOBAL MACRO IS THE WAY TO GO 345 performance is commensurate with the amount of risk assumed. Hedge funds that produce a Sharpe ratio well over 1.0 attract investor interest while those with a Sharpe ratio well below 1.0 do not. In the example of the single big bet, with a volatility of approximately 50 percent, the strategy delivers a Sharpe ratio of 0.1 assuming a risk-free rate of 5 percent [(10 – 5)/50 = 0.1]. While investors would probably be pleased with a 50 percent return in the good years, it is unlikely that they would tolerate the negative 50 percent years and remain invested long enough to witness a 10 percent average annual return.

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Multibet Performance Analysis Checking the performance of the multibet approach, we find the strategy had a volatility of about 20 percent, resulting in a Sharpe ratio of 0.25.While this figure is not particularly impressive, it represents a drastic improvement to the 0.1 Sharpe ratio produced by the single big bet approach. Both the big bet strategy and the multibet strategy produced average annual returns of 10 percent, but the multibet approach produced much less volatility and thus a higher Sharpe ratio. In sum, the smoother ride of the diversified portfolio might have compelled investors to stick around. HOW TO PRODUCE SUPERIOR RISK-ADJUSTED RETURNS By simply increasing the number of independent bets from one to five per year, the same better-than-average hedge fund manager increased the qual- WHY GLOBAL MACRO IS THE WAY TO GO 347 ity of returns considerably.

The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett
by Jack (edited By) Guinan
Published 27 Jul 2009

Related Terms: • Capital Structure • Equity • Retained Earnings • Common Stock • Preferred Stock Sharpe Ratio What Does Sharpe Ratio Mean? A ratio developed by Nobel laureate William F. Sharpe that is used to measure risk-adjusted performance. rp − rf The Sharpe ratio is calculated by sub= tracting the risk-free rate, such as that σp of the 10-year U.S. Treasury bond, from Where : the rate of return of a portfolio and rp = Expected portfolio return then dividing the result by the stanrf = Rissk free rate dard deviation of the portfolio returns. σp = Portfolio standard deviation Investopedia explains Sharpe Ratio The Sharpe ratio indicates whether a portfolio’s returns are due to smart investment decisions or are a result of excess risk.

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σp = Portfolio standard deviation Investopedia explains Sharpe Ratio The Sharpe ratio indicates whether a portfolio’s returns are due to smart investment decisions or are a result of excess risk. This measurement is very useful because although one portfolio or fund can reap higher returns than its peers, it is a good investment only if those higher returns are not a result of taking on too much additional risk. The greater a portfolio’s Sharpe ratio is, the better its risk-adjusted performance has been. A variation of the Sharpe ratio is the Sortino ratio, which removes the effects of upward price movements on standard deviation to measure only return against downward price volatility. 270 The Investopedia Guide to Wall Speak Related Terms: • Portfolio • Risk-Free Rate of Return • Total Return • Risk • Standard Deviation Short (or Short Position) What Does Short (or Short Position) Mean?

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The excess return The Investopedia Guide to Wall Speak 9 of the fund relative to the return of the benchmark index is a fund’s alpha. (2) The abnormal rate of return on a security or portfolio in excess of what would be predicted by an equilibrium model such as the capital asset pricing model (CAPM). Investopedia explains Alpha (1) Alpha is one of five technical risk measures that are used in modern portfolio theory (MPT); the others are beta, standard deviation, R-squared, and the Sharpe ratio. These indicators help investors determine the risk-reward profile of a mutual fund. Simply stated, alpha often is considered to represent the value that a portfolio manager adds to or subtracts from a fund’s return. A positive alpha of 1.0 means the fund has outperformed its benchmark index by 1%.

Finance and the Good Society
by Robert J. Shiller
Published 1 Jan 2012

The manager can take home high management fees for all the years that the risks do not show up, and then walk away when the catastrophe finally comes. For years, nance students have been taught to use the Sharpe ratio to evaluate whether a portfolio manager is really beating the market. The Sharpe ratio, named after Stanford University nance professor William Sharpe, is the average excess return over the historical life of the manager’s portfolio above the return of the market of all possible investments as a whole divided by the standard deviation of the return over the historical life of the manager’s portfolio. A high Sharpe ratio is taken as a sign of a good investment manager. If the manager is outperforming the market consistently, then the numerator of the ratio should be large.

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If the manager is outperforming the market consistently, then the numerator of the ratio should be large. But if the manager is taking signi cant risks to achieve a high return relative to the market, that will show up in the denominator as high variability in the manager’s portfolio return, and thus bring down the Sharpe ratio. But the Sharpe ratio is not necessarily a reliable indicator of a manager’s performance, as the risks do not necessarily show up in a high standard deviation of returns for the portfolio over most of its life. If there is no news about the risks, then prices will not change, until the catastrophe comes.

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Investment companies can legally engage in such shenanigans if they disclose them. The private investment company Integral Investment Management, managed by former biologist Conrad Seghers, advertised, according to a Wall Street Journal story, an extremely high Sharpe ratio but disclosed that it was pursuing some unusual derivatives activities.17 According to Goetzmann and his co-authors, Integral was coming close to the optimal Sharpe ratio manipulation because of massive sales of out-of-the-money puts on U.S. equity indices and a short call position implicit in the hedge fund fees. The manipulation worked, and Integral managed to persuade the Art Institute of Chicago to invest $43 million of its endowment in Integral and related funds.

The Power of Passive Investing: More Wealth With Less Work
by Richard A. Ferri
Published 4 Nov 2010

Ironically, rather than using his own beta formula as the denominator in the equation, Sharpe used a portfolio’s standard deviation of return. Perhaps this was because Treynor beat him to the punch. Sharpe’s formula became known as the Sharpe Ratio. An interesting 1966 paper published by Sharpe in the Journal of Business evaluated the performance of 34 mutual funds over a period from 1954–1963 using the Sharpe ratio; the Treynor Ratio; and a third factor, fund expenses.12 Sharpe’s intent was to compare the three methods and perhaps determine which was better at determining skill among mutual fund managers. Sharpe found sufficient evidence that all three ratios had some predictability for selecting funds relative to each other, although no one method isolated funds that consistently outperformed the market as measured by the DJIA (Sharpe doesn’t disclose why he chose this limited market indicator when the more comprehensive S&P 500 existed).

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Here are the results: The market as measured by the DJIA was less than 11 active funds and better than the remaining 23 funds. Basically, there was one winning fund for every two losing funds, a win-loss ratio of 1 to 2. The Sharpe Ratio for the Dow was 0.67 while the average ratio for the 34 funds was only 0.63. This means the Dow had a better return per unit of risk than the average mutual fund in the study. The Treynor Ratio returned results similar to the Sharpe Ratio. Sharpe also tested fees as a predictor of return. He makes this important observation about fees near the conclusion of the paper: While it may be dangerous to generalize the results found during one ten-year period, it appears that the average mutual fund selects a portfolio at least as good as the Dow-Jones Industrials, but that the results actually obtained by the holder of mutual fund shares (after the costs associated with the operation of the fund have deducted) fall somewhat short of those from the Dow-Jones portfolio.

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These funds can experience higher share-price volatility than diversified funds because sector funds are subject to issues specific to a given sector. Securities and Exchange Commission (SEC) The federal government agency that regulates mutual funds, registered investment advisors, the stock and bond markets, and broker-dealers. The SEC was established by the Securities Exchange Act of 1934. Sharpe ratio A measure of risk-adjusted return. To calculate a Sharpe ratio, an asset’s excess return (its return in excess of the return generated by risk-free assets such as Treasury bills) is divided by the asset’s standard deviation. It can be calculated compared to a benchmark or an index. short sale The sale of a security or option contract that is not owned by the seller, usually to take advantage of an expected drop in the price of the security or option.

Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals
by David Aronson
Published 1 Nov 2006

However, there are many other performance measures that might be used: the Sharpe ratio,30 the profit factor,31 the mean return divided by the Ulcer Index,32 and so forth. It should be pointed out that the sampling distributions of these alternative performance statistics would be different from the sampling distribution of the mean. It should also be pointed out that the methods used in this book to generate the sampling distribution of the mean may be of limited value in generating sampling distributions for performance statistics with elongated right tails. This can occur with performance statistics that involve ratios such as the Sharpe ratio, the mean-return-to-Ulcer-Index ratio, and the profit factor.

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There are data suggesting that trend following in stocks is indeed a less rewarding enterprise.103 Lars Kestner compared the performance of trend-following systems for a portfolio of futures and a portfolio of stocks. The futures considered were 29 commodities in 8 different sectors.104 The stocks were represented by 31 large-cap stocks in 9 different industry sectors,105 and 3 stock indices. Risk-adjusted performance (Sharpe ratio) was computed for 5 different trend-following systems,106 over the period January 1, 1990 through December 31, 2001. The Sharpe ratio averaged over the five trend-following systems in futures was 384 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS .604 versus .046 in stocks. These results support the notion that futures trend followers are earning a risk premium that is not available to trend followers in stocks.

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A superior way to calculate the profit factor to transform it to have a natural zero point is to take the log of the ratio. The Ulcer Index is an alternative and possibly superior measure of risk that considers the magnitude of equity retracements, which are not directly considered by the Sharpe ratio. The standard deviation, the risk measure employed by the Sharpe ratio, does not take into account the sequence of winning and losing periods. For a definition of the Ulcer Index, see P.G. Martin and B.B. McCann, The Investor’s Guide to Fidelity Funds (New York: John Wiley & Sons, 1989), 75–79. A similar concept, the return to retracement ratio, is described by J.D.

Hedge Fund Market Wizards
by Jack D. Schwager
Published 24 Apr 2012

For the near nine-year period since its inception, Cornwall Capital has realized an average annual compounded net return of 40 percent (52 percent gross).2 The annualized standard deviation has been relatively high at 32 percent (37 percent gross). Cornwall’s Sharpe ratio of 1.12 (1.23 gross) represents very good performance based on this widely used return/risk measure, but greatly understates the true return/risk performance of the fund. Cornwall is the poster child for the inadequacy of the Sharpe ratio if applied to managers with non-normal return distributions. The crux of the problem is that the Sharpe ratio uses volatility as the proxy for risk. Because of the asymmetric design of its trades, Cornwall’s volatility consists mostly of upside volatility.

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Since its inception in 2004, Benedict’s fund has realized an average annualized compounded net return of 11.5 percent (19.3 percent gross). If this return does not sound sufficiently impressive, keep in mind that it was achieved with an extremely low annualized volatility of 5.8 percent and, even more impressive, a maximum drawdown of less than 5 percent. Benedict’s return/risk numbers are exemplary. His Sharpe ratio is very high at 1.5. The Sharpe ratio, however, understates Benedict’s performance because this statistic does not distinguish between upside and downside volatility, and in Benedict’s case, most of the limited volatility is on the upside. Benedict’s Gain to Pain ratio is an extremely high 3.4. (See Appendix A for an explanation of the Gain to Pain ratio.)

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If you are rigorous about acknowledging what that new data is telling you, you can really get somewhere. It may take a while. If you are trading the system, and it is not performing in line with expectations over some reasonable time frame, look for overfit and hindsight errors. If you are expecting a Sharpe ratio above 1, and you are getting a Sharpe ratio under 0.3, it means that you have made one or more important hindsight errors, or badly misjudged trading costs. I was using the data up to a year before the current date as the training data set, the final year data as the validation data set, and the ongoing realtime data as the test.

Safe Haven: Investing for Financial Storms
by Mark Spitznagel
Published 9 Aug 2021

In their world, it sounds reasonable, and it's a comfortable story, asserted but not proved: You've got to take more risk to make higher returns; sleeping well comes at a cost. No guts, no glory. To make matters worse, academics furthered this idea by positing that investing and risk mitigation are about lowering or calibrating a portfolio's volatility relative to its average return—the risk‐adjusted return or dreaded Sharpe ratio—unwittingly at the expense of the growth rate of wealth. They thus claim an intellectually dishonest victory based on their own theoretical scoreboard. It is a solution in search of a problem, and a bad idea. (It's even a big reason for our great dilemma.) I don't really believe that most investors even have this bad idea.

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Yet, curiously, the tools of modern finance would insist that leverage is basically always good when the expected arithmetic returns are positive like this. After all, leverage simply raises the arithmetic average, or expected return, and it doesn't affect the ratio of the average return to the standard deviation of those returns. (A plot of the Sharpe ratios overlaid on the previous chart would simply be a horizontal line—more free money for all as leverage increases, meanwhile ending wealth plunges.) And less risk would always only mean less return. What a disaster those superficial, pseudoscientific tools are! Leverage can indeed kill the golden goose.

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Imagine William Tell—the fourteenth‐century Swiss folk hero and expert marksman who led the Swiss Confederacy's rebellion against their Austrian oppressors—forced to shoot the apple off his son's head. He can miss, but only in one direction, and the stray arrow matters at least as much as his median arrow. He aims to minimize his exposure to bad luck. William Tell's shot requires both precision and accuracy—aim small, miss small. (Contrast this with maximizing a Sharpe ratio by increasing precision at the expense of accuracy.) Those remaining stray arrows are a fair criticism of the aggressive Kelly betting strategy, as it still has an abundance of risk. That 5th percentile outcome of 0.3 means a 70% loss—so a lot of bad things can still happen between the 5th and 50th percentile of our new shot grouping.

The Invisible Hands: Top Hedge Fund Traders on Bubbles, Crashes, and Real Money
by Steven Drobny
Published 18 Mar 2010

Basically, our risk process mimics our investment process in that we work down from macro factor diversification to micro factor risk, allowing us to integrate the equity and commodity strategies that comprise our overall portfolio. Sharpe Ratio The Sharpe, or reward-to-variability, ratio is a measure of the excess return (or risk premium) per unit of risk in an investment asset or a trading strategy. The Sharpe ratio is used to measure the return of an asset relative to the level of risk taken. When comparing two assets, an investor can compare the expected returns E[R] against the relative benchmarks with return Rf. The asset with the higher Sharpe ratio gives more return for the same risk. How do you deal with the closet dollar exposure?

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So I would like to see a combination of people whose thinking is logical, consistent, and sounds as if they would be alpha extracting over time (see box). Then I would want the track record to verify it. Alpha Versus Beta In modern portfolio theory (MPT), there are five basic statistical measurements: beta, alpha, standard deviation (volatility), R-squared (correlation), and the Sharpe ratio (return/risk). Beta measures both the correlation and volatility of a fund or security to a benchmark. For example, if a fund has a beta of 2.0 in relation to the S&P 500, the fund’s returns are on average double those of the S&P. If a fund has a beta of -0.5, the fund’s returns are on average half those of the S&P, and in the opposite direction.

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If I am 60 percent long commodities, I am really just a currency fund in drag because commodities have a high correlation to the U.S. dollar at the moment. In this sense, I am essentially running a 50 percent dollar short position. Much of what we do seeks to mitigate unintended consequences, enabling us to do more with less from a volatility standpoint, which should increase our Sharpe ratio over time (see box on page 244). Our risk system functions as a toolkit with two discrete parts, one of which is defensive, the other offensive. The defensive component tries to understand macro factor exposures, throwing up flags when unintended exposures creep into the portfolio. The offensive component is where we look at data for different rates of change in core factors, to see if the macro matrix is beginning to cross-correlate.

The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution
by Gregory Zuckerman
Published 5 Nov 2019

As a result, Kepler’s portfolio was market neutral, or reasonably immune to the stock market’s moves. Frey’s models usually just focused on whether relationships between clusters of stocks returned to their historic norms—a reversion-to-the-mean strategy. Constructing a portfolio of these investments figured to dampen the fund’s volatility, giving it a high Sharpe ratio. Named after economist William F. Sharpe, the Sharpe ratio is a commonly used measure of returns that incorporates a portfolio’s risk. A high Sharpe suggests a strong and stable historic performance. Kepler’s hedge fund, eventually renamed Nova, generated middling results that frustrated clients, a few of whom bolted. The fund was subsumed into Medallion while Frey continued his efforts, usually without tremendous success.

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By combining signals from these new markets with Medallion’s existing predictive algorithms in one main trading system, something remarkable seemed to happen. The correlations of Medallion’s trades to the overall market dropped, smoothing out returns and making them less connected to key financial markets. Investment professionals generally judge a portfolio’s risk by its Sharpe ratio, which measures returns in relation to volatility; the higher one’s Sharpe, the better. For most of the 1990s, Medallion had a strong Sharpe ratio of about 2.0, double the level of the S&P 500. But adding foreign-market algorithms and improving Medallion’s trading techniques sent its Sharpe soaring to about 6.0 in early 2003, about twice the ratio of the largest quant firms and a figure suggesting there was nearly no risk of the fund losing money over a whole year.

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That year, Medallion soared 99 percent, even after it charged clients 20 percent of their gains and 5 percent of the money invested with Simons. The firm now managed nearly $4 billion. Over the previous decade, Medallion and its 140 employees had enjoyed a better performance than funds managed by George Soros, Julian Robertson, Paul Tudor Jones, and other investing giants. Just as impressive, Medallion had recorded a Sharpe ratio of 2.5 in its most recent five-year period, suggesting the fund’s gains came with low volatility and risk compared with those of many competitors. Letting his guard down, Simons consented to an interview with Hal Lux, a writer at Institutional Investor magazine. Over coffee in his New York office, and later while sipping gin and tonics at Renaissance’s Long Island headquarters, Simons expressed confidence his gains would continue.

Nerds on Wall Street: Math, Machines and Wired Markets
by David J. Leinweber
Published 31 Dec 2008

Prognostications here include: an increase in the complexity of derivative and structured products driven by the demands of alpha-seeking strategies; some products’ requirement of willingness to commit capital in innovative ways; and increased trading interest in risk classes, over individual securities. *The Sharpe ratio is a measure of management skill that adjusts pure alpha (value added) by the variability of that value added. Details of the Sharpe ratio can be found at http:// en.wikipedia.org/wiki/Sharpe_ratio. Algorithm Wars 81 Both articles forecast an increasingly risk-centric view of trading. IBM opines, “As the industry matures, many traditional activities will come under increasing pressure and new value engines will emerge.

…

In Chapter 6, the last of this part, “Stupid Data Miner Tricks,” we see how with the right mix of hubris, stupidity, and CPU cycles, it is possible to do some real damage to your financial health. In investing, as in the bomb squad, knowing what not to do is extremely worthwhile. *The Sharpe ratio is a measure of management skill that adjusts pure alpha (value added) by the variability of that value added. The others (Jensen & Treynor) are refinements based on characteristics of the portfolio, such as beta. They are less commonly used. Details are here http://en.wikipedia. org/wiki/Sharpe_ratio. Chapter 4 Where Does Alpha Come From? Life Is Alpha. The Rest Is Details. —POPULAR T-SHIRT AT HEDGE FUND EVENTS T here was a time not too long ago when, if you posed the question “Where does alpha come from?”

…

Financial models never capture every aspect of market participants’ motivations. Varied outcomes likely. Simple games like tic-tac-toe can be modeled exactly. One action always leads to another. This is clearly not the case in trading. Performance feedback and reinforcement. Performance measurement is natural for trading agents. For alpha-seeking algos, metrics like the Sharpe ratio* fit. Pure execution algos use implementation cost or VWAP shortfall. Layered behaviors. Agents should have default behaviors that complete their tasks and avoid errors. Basic behavior is at the lower layers, more sophisticated behavior above. Some of these agents will be programs, and some will be people.

Derivatives Markets
by David Goldenberg
Published 2 Mar 2016

It is shown in mathematical finance that there are equivalent ways to ensure the existence of an EMM for the discounted underlying price process (no-arbitrage). One important necessary and sufficient condition is the existence of a ‘Market Price of Risk’ (MPR), which is also called the Sharpe Ratio. We mentioned this important concept briefly as a relative risk measure in Chapter 15. Under the appropriate technical assumptions, EMMs are in one-to-one correspondence with Sharpe ratios. This is because the market price of risk for security i, (MPRi)=(μi–r)/σi. is the generator of EMMs. In terms of risk-neutral probabilities and the actual probabilities, the mechanism of the transformation from pi to is Girsanov’s Theorem, provided the assumptions underlying Girsanov actually hold.

…

We can further simplify the [Risk Premia Cancellation Condition] by substituting Δ into it, which says, after a little algebra, that, or, Note that, just as , by exactly the same argument, and . This equilibrium condition, [Risk Premia Cancellation Condition, Returns], between the risk premium on the option and the risk premium on the underlying stock says that, in order for the hedge portfolio to be riskless, the Sharpe ratios of the option and the stock must be equal. The Sharpe ratio is a standard portfolio risk measure defined as risk-premium to standard deviation and was introduced in Chapter 15. We have also called it the Market Price of Risk (MPR). As we have just demonstrated, [Risk Premia Cancellation Condition, Returns] is a consequence of the fact that it is possible to replicate the call option in the (BOPM, N=1).

…

Even from the point of view of a risk-averse investor, we know that if the stock is assumed to be risk neutral then the option must also be risk neutral, because different risks cannot co-exist in the riskless hedge portfolio (see Chapter 17). If they did, there would be no way to cancel them out, as they must in order to generate a riskless hedge. By risk is meant relative risk, which is also called the ‘Sharpe ratio’. Once again, what do we know? We know that the no-arbitrage, replicable option price, C0, is given under the risk-neutralized stock price measure (pr,1–pr). That is, Evaluating Er(C1(ω)|C0) we obtain, Therefore, or, since pr=p′, We can also get the same result using the state price representation of C0 as, This says that the current no-arbitrage option price is the weighted sum of its payoffs times the no-arbitrage prices of the primitive AD securities.

Alpha Trader
by Brent Donnelly
Published 11 May 2021

Many studies cover the slow decline of trend following performance. This table from the Winton study63 shows the decline in trend following Sharpe ratios over time: 10-year gross Sharpe ratio. Fast: Weekly, Medium: 6-week, Slow: 13-week This does not show there are now zero profitable trend following strategies, but clearly the edge has dropped over time. This is not a surprise given the proliferation of academic literature on the topic in the 1990s and 2000s. As a strategy becomes well known, its Sharpe ratio grinds towards zero. Asymmetrical information and lack of transparency 1980 to 1996 Before the internet, markets were opaque and it was hard for those outside financial institutions to get an accurate view of prices.

…

This is the optimal business model for a manager that wants to protect the bank but also allow for a bit of upside if trading conditions are optimal. This same business model is employed by many pod-based hedge funds. If you have 100 traders all working with very tight stops but some of those traders are still able to generate 20% returns in a year, the overall returns and Sharpe ratio of the hedge fund will be outstanding. Some traders detest this business model because it makes them nervous. Taking risk without losing money sounds like an oxymoron. It really isn’t. It just means that you need to behave like a call option. Take as much risk as you want when you are profitable, but play extremely tight when you are not.

…

This is where self-awareness is important. If you are biased to inaction, you need to be less patient! Since I am always biased to action and overtrading, I need to be more patient. Think about where you lie on that spectrum. An interesting (though not very surprising) pattern that emerges from my innate lack of patience is that my Sharpe ratio is higher when I am in the red vs. when I am deep in the black. If I’m in the red for the year, or even for the month, my main goal is to avoid going deeper into the red. This is the zero bound effect I describe earlier, in the section on Courage. Everyone has a point of discomfort as they near their stop loss, and your goal is to stay away from that point of discomfort so that you are not weak when a huge opportunity or great trade idea emerges.

Personal Investing: The Missing Manual
by Bonnie Biafore , Amy E. Buttell and Carol Fabbri
Published 24 May 2010

If the new manager is an unknown, you may want to sell your shares—perhaps reinvesting the money in the former manager’s new fund. Keep in mind, you’ll have to pay taxes on any capital gains if you sell your shares. 94 Chapter 5 The Sharpe ratio helps you figure out whether a fund manager’s returns are due to great investment decisions or to an inordinate fondness for risk. It is a ratio of the return you earn to the risk you take, that is, how much the fund returns for the risk taken. The higher the Sharpe ratio, the better a fund’s risk-adjusted returns. To find a fund’s Sharpe ratio, click the Ratings & Risks tab on a Morningstar fund web page. Finding Funds Funds are investor-friendly investments, but finding the right funds isn’t a slam-dunk.

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See wages Savingforcollege.com website, 206 savings accounts, 2 saving toward financial goals, 31–33, 37–38, 43 The Second-Grader Portfolio, 168 second home, 35 SEC (Securities and Exchange Commission) website, 161 Section 529 college prepaid plan, 67, 200, 202–203 Section 529 college savings plan, 67, 200–202, 204–208 sector, diversification by, 162 secured bonds, 131 Securities and Exchange Commission (SEC) website, 161 senior bonds, 132 SEP-IRAs, 66, 187 shareholders’ equity. See equity share price of bonds, 139 shares of stock. See stock Sharpe ratio, 95 Shift+clicking, how to, 7 short-term bonds, 134 short-term capital gains, 63 short-term successes, desire for, 45 simple interest, 18 SIMPLE IRAs, 187 small businesses. See also companies retirement plans for, 187 SEP-IRAs for, 66 starting, loan for, 40 small-cap stocks, 80 SmartMoney’s One Asset Allocation System, 165 Social Security, 28–29, 191–193 benefits calculator for, 28 benefits from, based on retirement age, 25, 29 benefits from, delaying, 192–193 eligibility for, 28, 192 full retirement age, 29, 192 history of, 24 how long it will be around, 29 life expectancy and, 193 Wage Index website, 16 working while receiving benefits from, 193 speculative bonds, 132, 159 spending, guidelines for, 43–44, 194–196 spread, of bond prices, 139 Stafford loans, 210 stepped-up basis, for stock inheritances, 68 stock, 56–58, 103.

Madoff Talks: Uncovering the Untold Story Behind the Most Notorious Ponzi Scheme in History
by Jim Campbell
Published 26 Apr 2021

I knew it was the largest fraud in history. I didn’t really understand how victims thought back then. They didn’t know. Here’s a guy, he’s not hitting home runs. He’s hitting doubles. They didn’t know what volatility was. There was like 4 percent volatility or less. They didn’t know what a Sharpe ratio* was. The Sharpe ratio was three. They didn’t know that a three Sharpe ratio over a long period is a sure sign of a fraud. They weren’t trained in that. They had made their money somewhere else.”45 The Madoff monthly customer financial statements Norma Hill furnished to me were a sight to behold. (See Hill’s Madoff statement in Figure 7.1.)

…

† “Margin debit” refers to the loan balance in a margin account. It is the total owed by the customer to a broker for funds advanced to purchase securities. * The Hadassah Foundation is a charitable organization that invests in social change to empower girls and women in Israel and the United States. * The “Sharpe ratio” is the return earned in excess of the risk-free rate, per unit of volatility or total risk. 8 THE MADOFF FAMILY Did They Know? Bernie Madoff: “Andy and Catherine. I’m so sorry for everything. Dad”1 (From prison, apologizing in a one-sentence letter. See Figure 8.1.) Ruth Madoff: “What’s a Ponzi scheme?”

…

.; Madoff, Peter; Madoff, Ruth Alpern) Madoff Technologies LLC, 196 “Madoff Tops Charts; Skeptics Ask How” (Ocrant), 114 Madoff Victims Coalition, 181, 208 Manning Rule, 43–44 Manzke, Sandra, 192 MARHedge, 114, 120 Market bias, 118–119 Market-making and proprietary business (MM&PT), xix, 9, 82, 144–146, 233 Markopolos, Harry, xvi–xviii, 103, 109, 111, 112f, 113–114, 117–119, 121, 122, 124–126, 129–131, 153, 157, 160, 165, 204, 278–281, 291, 293 “May Day,” 33 Maya, Isaac, 87 McGraw Hill, 218 McKenna, Lawrence, 179 McMahon, Robert, 83, 91, 221 “Merc” (New York Mercantile Exchange), 119 Merkin, Ezra, xvii, 22, 161–162, 168 Merrill Lynch, 33, 45, 52, 121, 128, 280 Mitterrand, François, 58 MM&PT (see Market-making and proprietary business) Money laundering, 144–147, 151, 290–292 Montauk, Long Island, 20, 25 Morgan Stanley, 45, 75, 144, 145, 233, 273 MSIL (Madoff Securities International Limited), 20, 146 Mukasey, Marc, 17, 19–20, 80–82 Mukasey, Michael, 80 Municipal bonds, 64 Muntner, Sylvia, 30, 254 Naked options (naked shorts), 60, 125, 158, 160 NASD (see National Association of Securities Dealers) NASDAQ (National Association of Securities Dealers Automated Quotation System), 29, 32, 46, 117, 121, 131, 209, 252, 280, 293 Nasi, William, 88 National Association of Securities Dealers (NASD), 46, 52, 66, 108, 109, 134, 135, 137, 286 (See also Financial Industry Regulatory Authority [FINRA]; NASDAQ) National Securities Clearing Corporation (NSCC), 108 Nee, John, 133 NERO (Northeast Regional Office) (SEC), 113–115 Net Investment Method (NIM), 185–187 Net losers, 180 Net winners, 180 New Age Funds, 184 New Deal, 282 New Jersey, 203 New Times Securities Services, Inc., 184 New York City Ballet, 55 New York Mercantile Exchange (NYMEX, “Merc”), 119 New York Mets, 166–168, 197 New York Post, 239 New York State, 203, 220, 266 (See also Southern District of New York [SDNY]) New York State Attorney General, 44 New York Stock Exchange (NYSE), xxiv, 29, 31–33, 37, 39, 45, 46, 118, 134, 267 New York Times, xx NIM (Net Investment Method), 185–187 Noel, Alix, 156 Noel, Ariane, 156 Noel, Corina, 156 Noel, Lisina, 156 Noel, Marisa, 156 Noel, Monica, 156 Noel, Walter, 156, 168 Northeast Regional Office (NERO) (SEC), 113–115 NSCC (National Securities Clearing Corporation), 108 NYMEX (New York Mercantile Exchange), 119 NYSE (see New York Stock Exchange) OCC (see Options Clearing Corporation) Ocrant, Michael, 114, 120–121, 130, 263, 287 Office of Economic Analysis (OEA), 109 Office of Market Intelligence (OMI), 280 Office of the Whistleblower (SEC), 280, 281 Ogilvy and Mather, 189 O’Hara, Jerome “Jerry,” 82, 90–91, 94–95, 148, 152 Old Greenwich, Conn., xxi, 238–239 Oldenburg, Claes, 269 Optimal (hedge fund), 14, 16, 23 Options Clearing Corporation (OCC), 104, 108, 155 Ostrow, William, 129–134 Palm Beach, Fla., 12–13, 20, 23, 25, 156, 167 Palm Beach Country Club, xvii, 13, 20, 57 Paris, France, 20 Paulson, Hank, 181–182 Pension plans, 98 Perelman, Ron, 161 Perez, George, 82, 91, 94–96, 95f, 148, 152 Personal expenses, payment of, 245–246, 245f Pettitt, Brian, 22 PharmaSciences, 55 Phelan, John, 33, 37 Picard, Irving, xx, xxvi appointment of, 179 billing rate of, 142 BLM on, 167, 168, 194, 201–202, 236 in BLMIS offices, 152 and BLMIS “piggy bank,” 245–247 and Stanley Chais, 194–196 and Sonja Cohn, 161 fees received by, 47, 180, 184, 284–285 and JPMorgan Chase, 168 Andrew Madoff on, 241 Ruth Madoff sued by, 188, 234–235 and Ezra Merkin, 162 mission of, as Trustee, 179 perception of, as bully, 210 Ponzi scheme victims sued by, 189–190 recovery rate of, 180, 185 resilience of, 286 successes of, 190–193 tax fraud ignored by, 290 team of, 197–200 Picower, Barbara, 55, 56, 98 Picower, Emily, 55 Picower, Jeffry, xxiv, 51, 53–57, 60, 62–64, 85, 97–100, 142, 190–191, 193–196, 200, 201, 260, 269, 273, 274, 289–290 Picower Foundation, 55 Picower Institute for Medical Research, 55 Piedrahita, Andrés, 156 Pink sheets, 32 P&Ls (see Profit and loss reports) Pomerantz, Steven, 163–166, 256 Ponzi schemes, xvii, xxiv, 184 Pottruck, David, 21 Price Waterhouse (PW), 66, 110, 193, 277 Prime brokerages, 122 Pritzker family, 273 Private placement memoranda (PPMs), 157 Profit and loss reports (P&Ls), 44–45, 82, 146–147, 231–232 Promissory notes, 62 “PRT62V” programs, 93 Put options, 72 Putnam, 184 PW (see Price Waterhouse) QUANT hedge fund, 75–76 Quants, 41, 163 Queens College, 238 Rampart Investment Management, xv–xvii, 117–118, 120, 123, 126, 291 Random number generators, 92 RBS (Royal Bank of Scotland), 107 Reagan, Ronald, 274–275 Regulatory reform, need for systemic, 277–278 Regulatory reports, falsified, 150–151 Renaissance Technologies, 75, 115–116 Riopelle, Roland, 86, 88, 99, 262–263, 290 Riordan, Erin, 152 “Riskless arbitrage,” 35–36 Roberts, Paul, xix, 12, 46–47, 67, 87, 88–89, 94, 96, 98, 141–142, 151–153, 171, 172, 230, 269, 293 Robertson, Julian, 42 Robinhood, 39 Rosa Mexicano restaurant, 6 Rothko, Mark, 162–163 Round Hill Club, 156 RP/EQ, 146, 147, 232 “RuAnn Family Plan,” 87 Rule 2860, 109 Russia, 195, 291–292 San Marino, Calif., 156 Sandell, Laurie, xx SAR (Suspicious Activity Report), 172, 174 Schumer, Chuck, 210 Schwab (see Charles Schwab & Company) Securities and Exchange Commission (SEC), xvii, xviii, xxv and arrest of BLM, 8–10 basic questions not asked by, 103–105 BLM and incompetency of, 222, 293 BLM on respect of, for BLMIS, 36 BLMIS IT specialists and false reporting to, 94–95 BLMIS’s dealings with, 37, 42, 83, 90, 91, 157–159, 221 BLM’s claim to be on “short list” for chairman of, 131 BLM’s parents’ registration with, 243 Boston District Office of, 111, 114 broker-dealer vs. investment-advisory arms of, 105–106 and Stanley Chais, 195 dysfunction at, 137–138 earlier investigations of, 96–97, 277–278 exploitation of incompetence of, by BLM, 105–106 failure of, to detect BLM’s Ponzi scheme, 103–117 falsified reports filed with, 150–151 and FBI, 152 and FINRA, 136, 287 Securities and Exchange Commission (SEC) first exoneration of BLM by, 61–67 “job” of, 129 made-up “rule” of, 88 Harry Markopolos’ complaint to, 111, 113–114, 124 New York Regional Office of, 133, 226 Northeast Regional Office of, 113–115 Office of Economic Analysis of, 109 organizational silos at, 154 oversight needed for, 286 Ponzi scheme victims and failure of, 214 and Rampart Investment Management, 118 real trading never verified by, 103–104, 106–109 reforming the, 278–282 and Robinhood, 39 and SIPC, 181, 187, 283 and Sterling Equities, 166 2005 failed examination by, 129–134 utter failure of, as regulator, 294 whistleblower tips/revelations ignored by, 109–117 Securities Industry Automation Corporation (SIAC), 108 Securities Investor Protection Act (SIPA), 184, 186, 284–286 Securities Investor Protection Corporation (SIPC): and fees charged by Trustees, 184, 185, 188 GAO’s warning to, 278 goalposts moved by, 183–184 inadequate customer protection fund of, xxvi mission of, 142, 179 and Net Investment Method, 185–186 Hank Paulson and, 181–182 “protection” of investors offered by, 179–181 recovery rate of, 180 reforming the, 282–286 settlements made by, 196–197 top ten “bad guys” targeted by, 190–193 (See also Picard, Irving) Self-regulatory organizations (SROs), xxviii, 135, 136, 286 Serious Organised Crime Agency (SOCA), 174, 291 Settlements (made by SIPC), 196–197 “703 account,” xix, 10, 13, 15–16, 18, 23, 25, 78, 82–84, 87–88, 90, 97, 104, 109, 144–146, 152 Shapiro, Carl, 22, 23, 53, 56–57, 59, 64, 87, 97, 191, 193–196, 201, 273 Shapiro, Mary, 280–281 Shapiro, Ruth, 56 Sharpe ratio, 204 Sheehan, David, 185, 197 Shtup file, 94, 149 SIAC (Securities Industry Automation Corporation), 108 Sibley, Lee, 15, 153 Simons, Jim, 75–76, 115–116 SIPA (see Securities Investor Protection Act) SIPC (see Securities Investor Protection Corporation) Smith Barney, 45 SOCA (Serious Organised Crime Agency), 174, 291 Sodi, Marco, 156 Soft Screw (Oldenburg sculpture), 269 Solomon, Elaine, 7 Sorkin, Ira “Ike,” xxiii, 14, 16, 18–19, 61, 63, 193, 223–224, 226, 234, 240–241, 269 Soros, George, 42 Southern District of New York (SDNY), 196, 220, 226, 290 S&P 100 Index, 72, 92, 107, 149, 165, 167, 173 S&P 100 OEX, 77 S&P 500 Index, 33, 97, 284 SPCL programs, 92 Spear, Leeds & Kellogg, 45–46 Specialists, trading, 31 Split strike conversion (SSC) investment strategy, 71–77, 86, 92, 97, 100, 103, 108, 111, 116, 118–119, 132, 148, 149, 155, 157, 158, 162–166, 165f, 191, 202, 228, 257–258, 264, 276 Squillari, Eleanor, xviii–xix, 3–9, 11–13, 21, 23–24, 77, 89, 90, 160–161, 203, 220, 223, 225, 228, 229, 246, 255, 258, 259, 290, 293 Squillari, Sabrina, 24 SROs (see Self-regulatory organizations (SROs)) Stampfli, Josh, 37–41, 43–45, 146–147, 221–222 “STDTRADE” file, 92 Sterling Equities, 166–168 “STMTPro” program, 93 Stock options, 72 Strike price, 72 Structured notes, 118 Suh, Simona, 107–109 Suspicious Activity Report (SAR), 172, 174 Swaps, 121 Switzerland, 291 Synthetic structured products, 172–174 TARP bailout, 182 Tax fraud, 82, 88, 90, 266–267, 289–290 Tax shelters, 53, 54 Teicher, Victor, 162 Thema International Funds, 160, 161, 192 Tibbs, Susan, 109 Toub, Philip Jamchid, 156 “TRADE17” program, 92 “TRADE1701” program, 92 Transparency, trading, 32, 46, 76, 174 Treasury bills (T-bills), 64, 73–74, 98, 118, 119, 164, 173, 174, 206, 273 Treasury bonds, 22, 100, 146, 165 Tremont Fund, 173, 192 Trump administration, 282 Trust, 275–276 Tucker, Jeffrey, 22, 156, 168 Tufts University, xvii UBS (Union Bank of Switzerland), 107 Underwriters, IPO, 52 University of Alabama, 30 University of California, Berkeley, 163 US Congress, 184, 207, 210 US Department of Justice (DOJ), 44, 154, 175, 192, 196, 220, 234 US Department of Labor, 98 US Department of the Treasury, 182, 282 Vanderhonval, Bill, 23 Vanguard, 37, 39, 184 Vanity Fair, 156 Vanity Fair Corporation, 56 Victims of the Ponzi scheme, 203–214 Ambrosino family, 207–211 average age of, 185 BLM’s seeming lack of remorse for, 211–213 and Congress’ failure, 207 and the European banks, 201 Willard Foxton Jr., 207–208 hardship cases, 199–200 Norma Hill, 204, 205f, 206–207 and the IRS, 202–203 Jews as, 204 lesson learned by, 214 recoveries of, 213–214, 274 and SEC’s failure, 214 SIPC and, 179–202 Eleanor Squillari on, 203 Vienna, Austria, 161, 208 Vijayvergiya, Amit, 122, 159 Villehuchet, René-Thierry Magon de La, xv–xvii, 21, 119, 125, 155, 270 Volume Weighted Average Price (VWAP), 36, 148 Walker, Genevievette, 115 Walker, Richard, 110 Wall Street Journal, 45, 63 Walmart, 210 Ward, Grant, 113 Waters, Maxine, 210 Weinstein, Sheryl, 255–256 Weiss, Paul, 226 West, Deborah, 196 Wharton School of Business, 81, 221 Whistleblowers and others suspecting misconduct, 109–124 A&B feeder fund tip (1992), 110–111 anonymous informants (Oct. 2005), 116 Frank Casey, 103, 123–129 Neil Chelo, 122 “concerned citizen” (Dec. 2006), 116–117 hedge fund whistleblower (May 2003), 114–115 Harry Markopolos, 111–114, 117–119, 124–126 Ocrant and Arvedlund articles, 120–121 Renaissance Technologies internal emails (Apr. 2004), 115–116 Wilpon, Fred, 163, 166–168, 197 The Wizard of Lies (Henriques), 60 Yang, Chan, 173 Yelsey, Neil, 45, 147, 231 Yeshiva University, 161 Zames, Matt, 172–173 ABOUT THE AUTHOR Jim Campbell is the host of the nationally syndicated radio show Business Talk with Jim Campbell and his crime show: Forensic Talk with Jim Campbell.

More Money Than God: Hedge Funds and the Making of a New Elite
by Sebastian Mallaby
Published 9 Jun 2010

Thanks to the Nobel laureate William Sharpe, we have a way of testing whether this was so: If you divide the Tiger cubs’ returns by their volatility, you get a Sharpe ratio of 1.42—that is, a risk-adjusted return that is superior by far to any of the benchmarks. For instance, Hennessee’s general hedge-fund index had a Sharpe ratio of just 0.59. The comparison makes it difficult to resist the conclusion that the Tiger cubs learned something from Robertson. Let’s try to resist a little longer. There are ways for hedge funds to game the Sharpe ratio by behaving like undercapitalized insurance companies.2 For example, a fund can sell options that insure against extreme swings in the market.

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For example, Mark Wehrly, Farallon’s general counsel, reports that Farallon borrows about $25 for every $100 in equity. Mark Wehrly, interview with the author, July 25, 2008. 20. Robert Howard and Andre F. Perold, “Farallon Capital Management: Risk Arbitrage” (Harvard Business School case study 9-299-020, November 17, 1999). According to this HBS study, the Sharpe ratios for two Farallon funds between 1990 and 1997 were 1.38 and 1. 75. The S&P 500 had a Sharpe ratio of 0.50. 21. Enrique Boilini, who led Farallon’s investment in Alpargatas, recalls that Gabic, a similar textile company, did not attract the interest of a foreign hedge fund, with the result that its factories were liquidated and all its workers lost their jobs.

…

They used very little leverage, which in the wake of Long-Term’s blowup was a selling point in itself; partly as a result, their returns were almost miraculously steady.19 Farallon’s consistency was legendary: Between 1990 and 1997, there was not a single month in which the fund lost money. As a result, Farallon’s Sharpe ratio, a measure of returns adjusted for risk, was roughly three times higher than that of the broad stock market, making it an overwhelmingly attractive place for endowments to park savings.20 Even during the height of the dot-com madness, Steyer sailed along serenely. He did not ride the bubble like Stan Druckenmiller.

The Bogleheads' Guide to Investing
by Taylor Larimore , Michael Leboeuf and Mel Lindauer
Published 1 Jan 2006

Contrary to what these investors might believe, the article reported that the increased returns were actually found to be small or even nonexistent when compared with the additional risk (as measured by the volatility) taken on by those investors who didn't rebalance. In addition, the study showed that portfolios that were never rebalanced had the lowest Sharpe ratios of all the rebalancing methods studied. Since the Sharpe ratio measures the additional return an investor receives for taking on more risk, this lower ratio indicates that investors who didn't rebalance were not being compensated for the additional risk they were taking. This study's results came to the same conclusion as did Jack Bogle when he previously reported on the results from his 25-year rebalancing study in his classic 1993 book Bogle on Mutual Funds.

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One of their most important studies was to determine which of eleven common predictors of future mutual fund performance really worked. These predictors were Morningstar ratings; past performance; expenses; turnover; manager tenure; net sales; asset size; alpha; beta; standard deviation (SD); and the Sharpe ratio. Their study's conclusion: The expense ratio is the only reliable predictor of future mutual fund performance. In another study, Standard and Poor's examined all diversified U.S. stock funds in nine different Morningstar style-box categories. The study, reported in the September 2003 issue of Kiplinger magazine, divided the funds in each Style Box into two groups: funds with above-average costs, and funds with below-average costs.

…

Rollover: A tax-free transfer of assets from one retirement plan to another. Roth IRA: A tax-favored retirement plan. Contributions are not deductible, but earnings are tax-free during accumulation and also when withdrawn. Sector/specialty fund: A mutual fund that invests in a narrow segment of the market, such as health, technology, utilities, or real estate. Sharpe ratio: A measure of risk-adjusted performance developed by Nobel Laureate William Sharpe. Spousal IRA: An IRA established for a nonworking spouse. Standard deviation: A statistical measure of volatility. Taxable account: An account in which the securities are subject to annual federal taxes. Tax-deferred accounts: An account in which federal income taxes are deferred until withdrawn.

The Fund: Ray Dalio, Bridgewater Associates, and the Unraveling of a Wall Street Legend
by Rob Copeland
Published 7 Nov 2023

Dalio would have full access to the resources of Tudor Investment Corporation, and they would work together to develop what could become an investment fund. The goal wasn’t to make the most money possible in the shortest time, but to design a system that produced steady and sustainable gains, at the lowest possible risk. The ultimate barometer would be the program’s Sharpe ratio, a decades-old calculation of a portfolio’s returns in relation to volatility, or how much it swung. The higher the better. A strong Sharpe ratio was 2.0, though a ratio of 3.0 or even higher wasn’t out of the question for top investors. Dalio agglomerated his findings from years of writing newsletters. He distilled them all into what was essentially an “if this, then that” trading method.

…

He could only make changes by conducting deep research and changing the rules themselves based on historical data, as opposed to reacting one-off to any momentary market move. After several months of tweaking the blueprint, Dalio was satisfied. He brought it back to Jones, who handed it off to his team to analyze. The verdict: Dalio’s approach produced a Sharpe ratio of less than 1.0. Jones’s Southern accent did little to mask his palpable frustration: “What the hell am I supposed to do with this?” he asked Dalio. Jones had given a friend a fair shot and been rewarded with a design that failed the most important criterion. Jones would later tell an associate that he had judged Dalio to be unquestionably intelligent, but not much of an investment manager—certainly not up to the levels of Tudor Investment Corporation.

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the recession ended: Tim Sablik, “Recession of 1981–82,” Federal Reserve History, November 22, 2013. “What Is a Jeweler?”: Cassidy, “Mastering the Machine.” the best thing: Lawrence Delevingne and Michelle Celarier, “Ray Dalio’s Radical Truth,” Absolute Return, March 2, 2011. a matrix: Hilda Ochoa-Brillembourg, author interview. Sharpe ratio: Gregory Zuckerman, The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution (Portfolio, 2019). “What the hell”: A representative for Jones said this was “not his style or voice.” protect the retirement savings: Hilda Ochoa-Brillembourg, author interview. railroad bond prices: Robert McGough, “Fair Wind or Foul?

No One Would Listen: A True Financial Thriller
by Harry Markopolos
Published 1 Mar 2010

.; Vice-President and global head of equity derivatives research, Goldman Sachs (NY), ; One New York Plaza, New York, NY 10004; 24. Red Flag # 28: BM’s Sharpe Ratio of 2.55 (Attachment 1: Fairfield Sentry Ltd. Performance Data) is UNBELIEVABLY HIGH compared to the Sharpe Ratios experienced by the rest of the hedge fund industry. The SEC should obtain industry hedge fund rankings and see exactly how outstanding Fairfield Sentry Ltd.’s Sharpe Ratio is. Look at the hedge fund rankings for Fairfield Sentry Ltd. and see how their performance numbers compare to the rest of the industry. Then ask yourself how this is possible and why hasn’t the world come to acknowledge BM as the world’s best hedge fund manager?

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Among all the funds on the database in that same period, the Madoff/ Fairfield Sentry fund would place at number 16 if ranked by its absolute cumulative returns. Among 423 funds reporting returns over the last five years, most with less money and shorter track records, Fairfield Sentry would be ranked at 240 on an absolute return basis and come in number 10 if measured by risk-adjusted return as defined by its Sharpe ratio. What is striking to most observers is not so much the annual returns—which, though considered somewhat high for the strategy, could be attributed to the firm’s market making and trade execution capabilities—but the ability to provide such smooth returns with so little volatility. The best known entity using a similar strategy, a publicly traded mutual fund dating from 1978 called Gateway, has experienced far greater volatility and lower returns during the same period.

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The capital overseen by Madoff through Fairfield Sentry has a cumulative compound net return of 397.5 percent. Compared with the 41 funds in the Zurich database that reported for the same historical period, from July 1989 to February 2001, it would rank as the best performing fund for the period on a risk-adjusted basis, with a Sharpe ratio of 3.4 and a standard deviation of 3.0 percent. (Ranked strictly by standard deviation, the Fairfield Sentry funds would come in at number three, behind two other market neutral funds.) Questions Abound Bernard Madoff, the principal and founder of the firm, who is widely known as Bernie, is quick to note that one reason so few might recognize Madoff Securities as a hedge fund manager is because the firm makes no claim to being one.

Principles of Corporate Finance
by Richard A. Brealey , Stewart C. Myers and Franklin Allen
Published 15 Feb 2014

Start on the vertical axis at rf and draw the steepest line you can to the curved red line of efficient portfolios. That line will be tangent to the red line. The efficient portfolio at the tangency point is better than all the others. Notice that it offers the highest ratio of risk premium to standard deviation. This ratio of the risk premium to the standard deviation is called the Sharpe ratio: Investors track Sharpe ratios to measure the risk-adjusted performance of investment managers. (Take a look at the mini-case at the end of this chapter.) We can now separate the investor’s job into two stages. First, the best portfolio of common stocks must be selected—S in our example. Second, this portfolio must be blended with borrowing or lending to obtain an exposure to risk that suits the particular investor’s taste.

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Does that mean that Q’s price is too high or that R’s price is too low? 2. Portfolio risk and return For each of the following pairs of investments, state which would always be preferred by a rational investor (assuming that these are the only investments available to the investor): 3. Sharpe ratio Use the long-term data on security returns in Sections 7-1 and 7-2 to calculate the historical level of the Sharpe ratio of the market portfolio. 4. Efficient portfolios Figure 8.11 below purports to show the range of attainable combinations of expected return and standard deviation. a. Which diagram is incorrectly drawn and why? b. Which is the efficient set of portfolios?

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How would your answer change if the correlation coefficient were 0 or –.5? c. Is Mr. Scrooge’s portfolio better or worse than one invested entirely in share A, or is it not possible to say? 13. Sharpe ratio Look back at Problem 3 in Chapter 7. The risk-free interest rate in each of these years was as follows: a. Calculate the average return and standard deviation of returns for Ms. Sauros’s portfolio and for the market. Use these figures to calculate the Sharpe ratio for the portfolio and the market. On this measure did Ms. Sauros perform better or worse than the market? b. Now calculate the average return that you could have earned over this period if you had held a combination of the market and a risk-free loan.

In Pursuit of the Perfect Portfolio: The Stories, Voices, and Key Insights of the Pioneers Who Shaped the Way We Invest
by Andrew W. Lo and Stephen R. Foerster
Published 16 Aug 2021

At Wells Fargo he was involved in the creation of index funds, a visionary idea to develop the now-ubiquitous products that replicate overall market portfolios such as the S&P 500 index. During this time, Sharpe also developed a simple reward-to-variability measure, the now-famous Sharpe ratio. Mathematically speaking, the Sharpe ratio is the return on a stock or portfolio in excess of a risk-free return, divided by the standard deviation of the return. This simple measure is used extensively today to measure investment performance. In 1980, Sharpe was honored with his election as president of the American Finance Association.

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Mossin cited and critiqued Sharpe’s paper, noting that their main conclusions were consistent with one another but that Sharpe’s “lack of precision in the specification of equilibrium conditions leaves parts of his arguments somewhat indefinite.”54 Mossin discussed Sharpe’s “so-called ‘market line,’ ” or capital market line, but perhaps in a manner fitting a mathematically rigorous academic journal, he didn’t present a graph of the line itself. Mossin discussed the “price of risk,” the return-to-risk trade-off akin to the now-famous Sharpe ratio, but critiqued it as an unfortunate term, instead using the term “the price of risk reduction,” making the analogy that we would “certainly hesitate to use the term ‘price of garbage’ for a city sanitation fee.” Academics can be a critical bunch, and there are often differences of opinion with respect to the appropriate level of mathematical rigor in economic analysis.

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The communication and focus are on retirement income, the amount of an inflation-protected annuity an investor is able to purchase, rather than the current account balance. As Merton stated, “The secret sauce of Managed DC is that if you are willing to agree on a goal, say, for example, $58,000 per year, protected against inflation in retirement, and my competitors and I start with the same Sharpe ratio, but I use dynamic strategies based on the goal versus a 70:30 portfolio, then I promise you I’ll beat them. Focusing on the goal is like having 20 percent more assets.”82 Merton notes that if the typical individual in the past worked for forty years, retired at age sixty-five, and lived until seventy-five, he or she had to support fifty years of consumption on forty years of work, thus needing to save roughly 25 percent of his or her income.83 However, if that worker today lives for twenty years past retirement until age eighty-five, in order to support sixty years of consumption on forty years of work, the individual needs to save roughly 33 percent of his or her income.

Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy
by Cathy O'Neil
Published 5 Sep 2016

This is a result of the way we define a trader’s prowess, namely by his “Sharpe ratio,” which is calculated as the profits he generates divided by the risks in his portfolio. This ratio is crucial to a trader’s career, his annual bonus, his very sense of being. If you disembody those traders and consider them as a set of algorithms, those algorithms are relentlessly focused on optimizing the Sharpe ratio. Ideally, it will climb, or at least never fall too low. So if one of the risk reports on credit default swaps bumped up the risk calculation on one of a trader’s key holdings, his Sharpe ratio would tumble. This could cost him hundreds of thousands of dollars when it came time to calculate his year-end bonus.

New Market Wizards: Conversations With America's Top Traders
by Jack D. Schwager
Published 28 Jan 1994

I sincerely believe that the person who has the best daily Sharpe ratio at the end of the year is the best trader. [The Sharpe ratio is a statistical performance measure that normalizes return by risk, with the variability of returns being used to measure risk. Thus, for example, assume Trader A and Trader B managed identical-sized funds and made all the same trades, but Trader A always entered orders for double the number of contracts as Trader B. In this case, Trader A would realize double the percentage return, but because risk would also double, the Sharpe ratio would be the same for both traders.* Normally, the Sharpe ratio is measured using monthly data.

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In the area of The Silence of the Turtles / 139 futures traders, one reference source I used was the quarterly summary provided by Managed Accounts Reports. This report summarizes the performance of a large number of commodity trading advisors (CTAs), providing a single synopsis sheet for each advisor. At the bottom of each sheet is a summary table with key statistics, such as average annual percentage return, largest drawdown, Sharpe ratio (a return/risk measure), percentage of winning months, and the probabilities of witnessing a 50 percent, 30 percent, and 20 percent loss with the given CTA. To be objective, I flipped through the pages, glancing only at the tables (not the names at the top of the sheets) and checking off the names of those advisors whose exceptional performance seemed to jump off the page.

Extreme Money: Masters of the Universe and the Cult of Risk
by Satyajit Das
Published 14 Oct 2011

Assume risk-free government securities yield 5 percent and hedge fund A has returns of 20 percent with a volatility of 10 percent while hedge fund B has returns of 15 percent and a volatility of 5 percent. The Sharpe ratios are respectively: Hedge Fund A = [20%—5%] / 10% = 1.5 Hedge Fund B = [15%—5%] / 5% = 2.0 Despite lower absolute performance, B provides investors with greater returns relative to risk. Sharpe ratios are ex post (based on actual risk) rather than ex ante (expected risk). Actual returns should be compared to expected risk at the time the position was taken. Insufficient attention is paid to the asymmetry of hedge fund returns, which do not follow the familiar bell-shaped normal distribution.

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, 301 Hasset, Kevin, 97 Havel, Václav, 359 Hawala, 22, 24 Hawkin, Greg, 248 Hawking, Stephen, 126 Hayek, Frederick, 103 HE (home equity), 181 Heathrow Airport, 161 Hederman, Abbot Mark Patrick, 361 Hedge Fund Alley, 239 hedge funds, 73, 77, 80, 260 Alfred Winslow Jones, 240 Amaranth, 250, 252 Centaurus Energy, 319 clientele, 247-248, 250 compensation, 314 fees, 245 formula for, 239 Fortress, 318 George Soros, 240 Hedgestock, 252, 261-262 Hyman Minsky, 260-262 leverage, 254 markets, 241 Porsche, 257-260 returns, 243-244, 255-257 Sharpe ratios, 246-247 strategies, 241-243 structure of CDOs, 195 Hedgestock, 261-262 hedge funds, 252 hedging, 235 aspect of Black-Scholes model, 122 derivatives, 216-217 Heine, Heinrich, 38, 64 Heisenberg, Werner, 101 Heller, Walter, 129 Hellman, Lillian, 350 HELOC (home equity line of credit), 181 Hennessy, Peter, 278 Heritage Foundation, 350 Herodotus, 74 Hertz, 155 Hewlett-Packard (HP), 122 Heyman, William, 270 Hickman, W.

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See also options profits, 121 risk, 124 intellectual property rights, securitization of, 168 interest rates cutting of, 340-341 lowering of, 348 International Accounting Standards Board, 289 International Grain Council, 334 International Institute of Finance (IIF), 289 International Monetary Fund (IMF), 96 international reply coupons (IRCs), 33 Internet bubble (1990s), 54 stocks, 58 InterNorth, 55 interviews on financial TV shows, 94 invention of money, 24-25 investment banks, 57, 309 leveraged buyouts (LBOs), 147-148 percentage of jobs in, 313 separation from commercial banks, 66 investments alternative, 252 exotic products, 73-74 hedge funds Amaranth, 250-252 clientele, 247-250 fees, 245 Hedgestock, 252 markets, 241 returns, 243-244 Sharpe ratios, 246-247 strategies, 241-243 incentives, 348 IO (interest only) bonds, 178 Ireland, 83, 344 Irish Times, The, 356 Iron Chef, The, 168 Irving, John, 29 Ising model, 204 It’s A Wonderful Life, 65, 180 Italy, derivatives, 215-216 ITT Corporation, 60 J Jackson, Marjorie, 156 Jackson, Michael, 21 Jackson, Tony, 363 James, Oliver, 274 Japan debt, 357 financialization, 38-39 housewife traders, 40-41 lost decades, 357 retirement, 49-50 six sigma, 60 Jefferson County, Alabama, 211-214 Jefferson, Thomas, 91 Jenkins, Simon, 302 Jenson, Michael, 120, 138-141 Jiabao, Wen, 86-87 Jian, Ma, 295 Jintao, Hu, 363 jobbers, 53 jobs certifications, 309-310 finance, 307-308 Jobs, Steve, 164 John F.

The Little Book of Common Sense Investing: The Only Way to Guarantee Your Fair Share of Stock Market Returns
by John C. Bogle
Published 1 Jan 2007

EXHIBIT 16.1 “Smart Beta” Returns: 10-Year Period Ended December 31, 2016 Fundamental Index Fund Dividend Index Fund S&P 500 Index Fund Annual return 7.6% 6.6% 6.9% Risk (standard veviation) 17.7 15.1 15.3 Sharpe ratio* 0.39 0.38 0.40 Correlation with S&P 500 Index 0.97 0.97 1.00 *A measure of risk-adjusted return. You’ll note that the fundamental index fund earned higher returns while assuming higher risk than the S&P 500 fund. The dividend index, on the other hand, earned lower returns and carried lower risk. But when we calculate the risk-adjusted Sharpe ratio, the S&P 500 Index fund wins in both comparisons. The similarity of returns and risks in all three funds should not be surprising.

Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets
by Nassim Nicholas Taleb
Published 1 Jan 2001

The dull presenter was actually comparing “Sharpe ratios,” i.e., returns scaled by their standard deviations (both annualized), named after the financial economist William Sharpe, but the concept has been commonly used in statistics and called “coefficient of variation.” (Sharpe introduced the concept in the context of the normative theory of asset pricing to compute the expected portfolio returns given some risk profile, not as a statistical device.) Not counting the survivorship bias, over a given twelve-month period, assuming (very generously) the Gaussian distribution, the “Sharpe ratio” differences for two uncorrelated managers would exceed 1.8 with close to 50% probability.

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Not counting the survivorship bias, over a given twelve-month period, assuming (very generously) the Gaussian distribution, the “Sharpe ratio” differences for two uncorrelated managers would exceed 1.8 with close to 50% probability. The speaker was discussing “Sharpe ratio” differences of around .15! Even assuming a five-year observation window, something very rare with hedge fund managers, things do not get much better. Value of the seat: Even then, by some attribution bias, traders tend to believe that their income is due to their skills, not the “seat,” or the “franchise” (i.e., the value of the order flow).The seat has a value as the New York Stock Exchange specialist “book” is worth quite large sums: See Hilton (2003). See also Taleb (1997) for a discussion of the time and place advantage.

Trend Commandments: Trading for Exceptional Returns
by Michael W. Covel
Published 14 Jun 2011

You will never produce the mythologically consistent returns that many believe to exist. When trend followers hit home runs from the likes of Barings Bank, Long-Term Capital Management, and the 2008 market crash, they are targeting unknowable extreme occurrences that happen to occur with a probability greater than expected. The Sharpe ratio is oversold. It can give a false sense of precision and lead people to make predictions unwisely.2 Those occurrences are fat tails—in statistician speak. Trend following’s nature of riding a trend to the end when it bends, and then cutting losses very fast, puts you in a position to benefit when the next unexpected flood rolls in.

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See also trend following manipulation of monetary policy, 181-183 Marcus, Michael, 15 Longstreet, Roy, 239 market crashes, trend following during, 97-98 losers and winners market gurus, 147-148 crowd mentality, 117-120 during market crashes, 97-98 Efficient-Markets Hypothesis, 101-102 market predictors, 165-167 contradictions in predictions, 175-178 trend following versus, 194 hatred of trend following, 109-110 market price, importance of, 51-52 unexpected events, 91 market theories zero-sum game, 95 losses avoiding averaging, 79 271 fundamental analysis, 33-35 technical analysis, 35-36 markets drawdowns, 69-70 change in, 45 exit strategies, 75-76 entering, 73 what to trade, 65-67 272 Index McCain, John, 182 media, market predictions by, 177 Merton, Robert, 101 Miller, Merton, 101 origins of trend following, 221-231 Ostgaard, Stig, 221 P Paulson, John, 109 misleading information, spreading, 169-170 Pelley, Scott, 175 monetary policy, government manipulation of, 181-183 performance statistics for trend following, 15-23 money, capitalism and, 113-115 Picasso, Pablo, 88 money management, importance of, 61-62 players, types of, 205 morality of trend following, 109-110 Popoff, Peter, 166 pliability, 30 moving average, defined, 13 portfolio diversity, example of, 65-67 Mulvaney, Paul, 22 position sizing, 61-62 Munger, Charlie, 157 Prechter, Robert, 225 N–O predictions about market, 165-167 Nasdaq market crash (1973-1974), 151 net worth of trend following traders, 15 New Blueprints for Gains in Stocks and Grains (Dunnigan), 230 avoiding, 47-48 contradictions in, 175-178 trend following versus, 194 predictive technical analysis, 35 Nicklaus, Jack, 120 presidents, approval ratings based on economy, 181-182 Nikkei 225 stock index, 151 price action Obama, Barack, 182 optimism in trend following, 201-202 entering markets, 73 importance of, 51-52 profit targets, avoiding, 75-76 Index Profits in the Stock Market (Gartley), 226 Prospect Theory, 118 prudence, 30 Pujols, Albert, 138-139 Q–R quants, defined, 12 quarterly performance reports, 105-106 273 S S&P 500, defined, 13 Sack, Brian, 183 Schabacker, Richard W., 226 Schmidt, Eric, 47 Scholes, Myron, 101 scientific method in trend following, 134-135 Seidler, Howard, 20 Ramsey, Dave, 91 Seinfeld (television program), 161 Rand, Ayn, 113 selecting trading systems, 59-62 reactive technical analysis, 36 self-regulation, 124 “Reminiscences of a Stock Operator” (Lefèvre), 223 self-reliance, 30 repeatable alpha, defined, 12 Seykota, Ed, 15, 62 Rhea, Robert, 225 Shanks, Tom, 21 Ricardo, David, 223 Sharpe ratio, 137 risk assessment, 55-56 sheep mentality, 117-120 risk management, 61-62 short, defined, 12 Robbins, Anthony, 208 Simons, Jim, 135 Robertson, Pat, 166 The Simpsons (television program), 110 robustness of trend following, 85 Rogers, Jim, 23 Roosevelt, Franklin D., 114 Rosenberg, Michael, 201 rules of trend following, 201-202 Serling, Rod, 26 skill versus luck in trend following, 189-190 Smith, Vernon, 26 Social Security, 181 Sokol, David, 158 Soros, George, 189 274 Index speculation qualities of, 30 role of, 29-30 Speilberg, Steven, 208 spreading misleading information, 169-170 statistics in trend following, 137-140 Stone, Oliver, 29 story, fundamental analysis as, 33-35 Studies in Tape Reading (Wyckoff), 226 sunk costs, 118 Sunrise Capital, 5 drawdowns statistics, 70 performance statistics, 22 Technical Traders Guide to Computer Analysis of the Futures Markets (Lebeau), 60 Ten Years in Wall Street (Fowler), 224 three-phase systems, 60 Trader (documentary), 143 traders, investors versus, 29-30 trading gold, 173 trading systems, selecting, 59-62 “Trading With the Trend” (Dunnigan), 229 Transtrend, 5, 15 Trend Commandments (Covel) content of, 6 people written for, 9 trend following Swensen, David, 187 advantages of, 235-236 systematic global macro, defined, 12 compared to black box, 187 systematic trend following.

Statistical Arbitrage: Algorithmic Trading Insights and Techniques
by Andrew Pole
Published 14 Sep 2007

Suppose we pick only those trades that are profitable round trip. Daily profit variation will still, typically, be substantial. Experiments with real data using a popcorn process model show that the proportion of winning days can be as low as 52 percent for a strategy with 75 percent winning bets and a Sharpe ratio over 2. Reversion defined as any movement from today’s price in the direction of the center of the price distribution includes overshoot cases. The scenario characterizes as reversionary movement a price greater than the median that moves to any lower price—including to any price lower than the median, the so-called overshoot.

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See also Return decline catastrophe process, 194–198 catastrophe process forecasts, 198–200 catastrophe process theoretical interpretation, 205–209 Cuscore statistics and, 200–205, 211–221 risk management and, 209–211 trend change identification, 200–205 Revealed reversion, see Expected revealed reversion Reverse bets, 11 Reversion, law of, 67–89, 113–114, 139–140 first-order serial correlation and, 77–82 inhomogeneous variances and, 74–77 interstock volatility and, 67, 99–112, 164–165 looking several days ahead and, 87–89 nonconstant distributions and, 82–84 in nonstationary process, 136–137 serial correlation, 138–139 75 percent rule and, 68–74 in stationary random process, 114–136 temporal dynamics and, 91–98 U.S. bond futures and, 85–87 Reynders Gray, 26 Risk arbitrage, competition and, 160–161 Risk control, 26–32 event correlations, 31–32 forecast variance, 26–28 market exposure, 29–30 market impact, 30–31 229 Index Risks scenarios, 141–154 catastrophe process and, 209–211 correlation during loss episodes, 151–154 event risk, 142–145 new risk factors, 145–148 redemption tension, 148–150 Regulation Fair Disclosure (FD), 150–151 Royal Dutch Shell (RD)–British Petroleum (BP) spread, 46–47 S&P (Standard & Poor’s): S&P 500, 28 futures and exposure, 21 Sample distribution, 123 Santayana, George, 5n2 SARS (severe acute respiratory syndrome), 175 Securities and Exchange Commission (SEC), 3, 150–151 Seismology analogy, 200n1 September 11 terrorist attacks, 175 Sequentially structured variances, 136–137 Sequentially unstructured variances, 137 Serial correlation, 138–139 75 percent rule, 68–74, 117 first-order serial correlation and, 77–82 inhomogeneous variances and, 74–77, 136–137 looking several days ahead and, 87–89 nonconstant distributions and, 82–84 U.S. bond futures and, 85–87 Severe acute respiratory syndrome (SARS), 175 Shackleton, E. H., 113 Sharpe ratio, 116 Shaw (D.E.), 3, 189 Shell, see Royal Dutch Shell (RD)–British Petroleum (BP) spread Sinusoid, 19–20, 170 Spatial model analogy, 200n1 Specialists, 3, 156–157 Speer, Leeds & Kellog, 189 Spitzer, Elliot, 176, 180 Spread margins, 16–18. See also specific companies Standard & Poor’s (S&P): S&P 500, 28 futures and exposure, 21 Standard deviations, 16–18 Stationarity, 49, 84–85 Stationary random process, reversion in, 114–136 amount of reversion, 118–135 frequency of moves, 117 movements from other than median, 135–136 Statistical arbitrage, 1–7, 9–10 Stochastic resonance, 20, 50, 58–59, 169, 204 Stochastic volatility, 50–51 Stock split, 13n1 Stop loss, 39 Structural change, return decline and, 179–180 Structural models, 37–66 accuracy issues, 59–61 classical time series models, 47–52 doubling and, 81–83 exponentially weighted moving average, 40–47 factor model, 53–58, 63–66 stochastic resonance, 58–59 Stuart, Alan, 63 Student t distribution, 75, 124–126, 201 Sunamerica, Inc.

Fed Up!: Success, Excess and Crisis Through the Eyes of a Hedge Fund Macro Trader
by Colin Lancaster
Published 3 May 2021

Information is more readily available to virtually anyone, central banks have changed the markets, and the markets themselves have continued to evolve. Systematic strategies and the rise of the quants have made the markets more difficult. To help combat some of these factors, many firms started to embrace other, higher Sharpe-ratio strategies in lieu of the more traditional macro investing. In some ways, the classic macro approach has been a dying breed these past ten years. But still, there’s a lot of smart guys trying to earn a living doing this stuff and a crazy cast of characters of brokers and researchers who support their efforts

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But this is because he has seen people make mistakes, big mistakes, that cost them millions, even their lives. Back before the ‘08 crisis, before the tide went out, he made some of the greatest calls possible. One time, we were shown a deal from a guy in Los Angeles. It looked like a sure thing. A three-Sharpe-ratio business looking to grow. The returns were literally too good to be true. It originated in Minneapolis with the Petters scams.3 A guy was pitching one of the funds as an investment opportunity. But the Rabbi was on it and told us to stay away. When the tide went out, it was left lying on the beach.

Python for Data Analysis
by Wes McKinney
Published 30 Dec 2011

Finance: In [105]: import pandas.io.data as web In [106]: data = web.get_data_yahoo('SPY', '2006-01-01') In [107]: data Out[107]: <class 'pandas.core.frame.DataFrame'> DatetimeIndex: 1655 entries, 2006-01-03 00:00:00 to 2012-07-27 00:00:00 Data columns: Open 1655 non-null values High 1655 non-null values Low 1655 non-null values Close 1655 non-null values Volume 1655 non-null values Adj Close 1655 non-null values dtypes: float64(5), int64(1) Now, we’ll compute daily returns and a function for transforming the returns into a trend signal formed from a lagged moving sum: px = data['Adj Close'] returns = px.pct_change() def to_index(rets): index = (1 + rets).cumprod() first_loc = max(index.notnull().argmax() - 1, 0) index.values[first_loc] = 1 return index def trend_signal(rets, lookback, lag): signal = pd.rolling_sum(rets, lookback, min_periods=lookback - 5) return signal.shift(lag) Using this function, we can (naively) create and test a trading strategy that trades this momentum signal every Friday: In [109]: signal = trend_signal(returns, 100, 3) In [110]: trade_friday = signal.resample('W-FRI').resample('B', fill_method='ffill') In [111]: trade_rets = trade_friday.shift(1) * returns We can then convert the strategy returns to a return index and plot them (see Figure 11-1): In [112]: to_index(trade_rets).plot() Figure 11-1. SPY momentum strategy return index Suppose you wanted to decompose the strategy performance into more and less volatile periods of trading. Trailing one-year annualized standard deviation is a simple measure of volatility, and we can compute Sharpe ratios to assess the reward-to-risk ratio in various volatility regimes: vol = pd.rolling_std(returns, 250, min_periods=200) * np.sqrt(250) def sharpe(rets, ann=250): return rets.mean() / rets.std() * np.sqrt(ann) Now, dividing vol into quartiles with qcut and aggregating with sharpe we obtain: In [114]: trade_rets.groupby(pd.qcut(vol, 4)).agg(sharpe) Out[114]: [0.0955, 0.16] 0.490051 (0.16, 0.188] 0.482788 (0.188, 0.231] -0.731199 (0.231, 0.457] 0.570500 These results show that the strategy performed the best during the period when the volatility was the highest.

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First, I’ll load historical prices for a portfolio of financial and technology stocks: names = ['AAPL', 'GOOG', 'MSFT', 'DELL', 'GS', 'MS', 'BAC', 'C'] def get_px(stock, start, end): return web.get_data_yahoo(stock, start, end)['Adj Close'] px = DataFrame({n: get_px(n, '1/1/2009', '6/1/2012') for n in names}) We can easily plot the cumulative returns of each stock (see Figure 11-2): In [117]: px = px.asfreq('B').fillna(method='pad') In [118]: rets = px.pct_change() In [119]: ((1 + rets).cumprod() - 1).plot() For the portfolio construction, we’ll compute momentum over a certain lookback, then rank in descending order and standardize: def calc_mom(price, lookback, lag): mom_ret = price.shift(lag).pct_change(lookback) ranks = mom_ret.rank(axis=1, ascending=False) demeaned = ranks - ranks.mean(axis=1) return demeaned / demeaned.std(axis=1) With this transform function in hand, we can set up a strategy backtesting function that computes a portfolio for a particular lookback and holding period (days between trading), returning the overall Sharpe ratio: compound = lambda x : (1 + x).prod() - 1 daily_sr = lambda x: x.mean() / x.std() def strat_sr(prices, lb, hold): # Compute portfolio weights freq = '%dB' % hold port = calc_mom(prices, lb, lag=1) daily_rets = prices.pct_change() # Compute portfolio returns port = port.shift(1).resample(freq, how='first') returns = daily_rets.resample(freq, how=compound) port_rets = (port * returns).sum(axis=1) return daily_sr(port_rets) * np.sqrt(252 / hold) Figure 11-2.

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Cumulative returns for each of the stocks When called with the prices and a parameter combination, this function returns a scalar value: In [122]: strat_sr(px, 70, 30) Out[122]: 0.27421582756800583 From there, you can evaluate the strat_sr function over a grid of parameters, storing them as you go in a defaultdict and finally putting the results in a DataFrame: from collections import defaultdict lookbacks = range(20, 90, 5) holdings = range(20, 90, 5) dd = defaultdict(dict) for lb in lookbacks: for hold in holdings: dd[lb][hold] = strat_sr(px, lb, hold) ddf = DataFrame(dd) ddf.index.name = 'Holding Period' ddf.columns.name = 'Lookback Period' To visualize the results and get an idea of what’s going on, here is a function that uses matplotlib to produce a heatmap with some adornments: import matplotlib.pyplot as plt def heatmap(df, cmap=plt.cm.gray_r): fig = plt.figure() ax = fig.add_subplot(111) axim = ax.imshow(df.values, cmap=cmap, interpolation='nearest') ax.set_xlabel(df.columns.name) ax.set_xticks(np.arange(len(df.columns))) ax.set_xticklabels(list(df.columns)) ax.set_ylabel(df.index.name) ax.set_yticks(np.arange(len(df.index))) ax.set_yticklabels(list(df.index)) plt.colorbar(axim) Calling this function on the backtest results, we get Figure 11-3: In [125]: heatmap(ddf) Figure 11-3. Heatmap of momentum strategy Sharpe ratio (higher is better) over various lookbacks and holding periods Future Contract Rolling A future is an ubiquitous form of derivative contract; it is an agreement to take delivery of a certain asset (such as oil, gold, or shares of the FTSE 100 index) on a particular date. In practice, modeling and trading futures contracts on equities, currencies, commodities, bonds, and other asset classes is complicated by the time-limited nature of each contract.

A Man for All Markets
by Edward O. Thorp
Published 15 Nov 2016

The unlevered annualized return for XYZ before fees, at 18.21 percent, is more than double that of the S&P; the riskiness, as measured by the standard deviation, is 6.68 percent. The ratio of (annualized) return to risk for XYZ at 2.73 is more than five times that of the S&P. Estimating 5 percent as the average three-month T-bill rate over the period, the corresponding Sharpe ratios are 0.18 for the S&P versus 1.98 for XYZ. The graph in Appendix E, XYZ Performance Comparison, displays two major “epochs.” The first, from August 12, 1992, to early October 1998, shows a steady increase. The second epoch, from then until September 13, 2002, has a higher rate of return, including a remarkable six-month spurt just after the collapse (after four years) of the large hedge fund called, ironically, Long-Term Capital Management.

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For updated Consumer Price Index numbers and for month-by-month values, go to www.bls.gov/cpi or do the usual Google search. Appendix B * * * HISTORICAL RETURNS Table 10: Historical Returns on Asset Classes, 1926–2013 Series Compound Annual Return* Average Annual Return** Standard Deviation Real (after inflation) Compound Annual Return* Sharpe Ratio† Large Company Stocks 10.1% 12.1% 20.2% 6.9% 0.43 Small Company Stocks 12.3% 16.9% 32.3% 9.1% 0.41 Long-Term Corporate Bonds 6.0% 6.3% 8.4% 2.9% 0.33 Long-Term Government Bonds 5.5% 5.9% 9.8% 2.4% 0.24 Intermediate-Term Government Bonds 5.3% 5.4% 5.7% 2.3% 0.33 US Treasury Bills 3.5% 3.5% 3.1% 0.5% ——— Inflation 3.0% 3.0% 4.1% ——— ——— * Geometric Mean ** Arithmetic Mean † Arithmetic From: Ibbotson, Stocks, Bonds, Bills and Inflation, Yearbook, Morningstar, 2014.

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of the impact Market impact refers to the fact that “market orders” to buy are, on average, filled at or above the last previous price and “market orders” to sell tend to be filled at or below the last previous price. gained 9 percent The accounting period with an odd length of five months arose here for PNP because the fiscal year end for PNP changed in 1987 from October 31 to December 31. statistics confirmed Common metrics include the Sharpe ratio, the Sortino ratio, the distribution of drawdowns, and the MAR ratio (annual return divided by maximum drawdown). See, for instance, the three-part series by William Ziemba in Wilmott magazine: “The Great Investors,” March, May and July 2006. or losing quarters For comparison, the S&P 500 was down in eleven of the thirty-two full quarters and small company stocks lost in thirteen.

Concentrated Investing
by Allen C. Benello
Published 7 Dec 2016

Table 4.5â•… Performance Statistics for Concentrated Combination Value Portfolios, Rolling Annual Rebalance (1963 to 2015) Number of Stocks 300 200 100 50 40 30 25 20 15 10 5 1 Return (%) 17.2 17.9 18.6 19.5 19.8 20.1 20.4 20.4 20.7 20.6 21.6 22.8 Sharpe ratio 0.73 0.78 0.81 0.84 0.84 0.84 0.85 0.84 0.83 0.80 0.78 0.63 104 Concentrated Investing Figure 4.3â•… Concentrated Combination Value Portfolios, Rolling Annual Rebalance (1963 to 2015) Source: O’Shaughnessy Asset Management, LLC. O’Shaughnessy found the one-stock portfolios generated the best raw returns at 22.8 percent per year compound. As the number of portfolio holdings swelled from one, the raw returns fell off in rank order. The 25-stock portfolios generated the best Sharpe ratio—a measure of risk-adjusted returns—at 0.85. As the portfolio holdings increased beyond 25 positions, the risk-adjusted returns diminished in rank order.

Python for Algorithmic Trading: From Idea to Cloud Deployment
by Yves Hilpisch
Published 8 Dec 2020

Under these assumptions, the daily values scale up to the yearly ones from before and one gets the following: One now has to maximize the following quantity to achieve maximum long-term wealth when investing in the stock: Using a Taylor series expansion, one finally arrives at the following: Or for infinitely many trading points in time (that is, for continuous trading), one arrives at the following: The optimal fraction then is given through the first order condition by the following expression: This represents the expected excess return of the stock over the risk-free rate divided by the variance of the returns. This expression looks similar to the Sharpe ratio but is different. A real-world example shall illustrate the application of the preceding formula and its role in leveraging equity deployed to trading strategies. The trading strategy under consideration is simply a passive long position in the S&P 500 index. To this end, base data is quickly retrieved and required statistics are easily derived: In [16]: raw = pd.read_csv('http://hilpisch.com/pyalgo_eikon_eod_data.csv', index_col=0, parse_dates=True) In [17]: symbol = '.SPX' In [18]: data = pd.DataFrame(raw[symbol]) In [19]: data['return'] = np.log(data / data.shift(1)) In [20]: data.dropna(inplace=True) In [21]: data.tail() Out[21]: .SPX return Date 2019-12-23 3224.01 0.000866 2019-12-24 3223.38 -0.000195 2019-12-27 3240.02 0.000034 2019-12-30 3221.29 -0.005798 2019-12-31 3230.78 0.002942 The statistical properties of the S&P 500 index over the period covered suggest an optimal fraction of about 4.5 to be invested in the long position in the index.

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Scatter plot of S&P 500 and VIX log returns with linear regression line Having financial time series data stored in a pandas DataFrame object makes the calculation of typical statistics straightforward: In [213]: ret = rets.mean() * 252 In [214]: ret Out[214]: SPX 0.104995 VIX -0.037526 dtype: float64 In [215]: vol = rets.std() * math.sqrt(252) In [216]: vol Out[216]: SPX 0.147902 VIX 1.229086 dtype: float64 In [217]: (ret - 0.01) / vol Out[217]: SPX 0.642279 VIX -0.038667 dtype: float64 Calculates the annualized mean return for the two indexes. Calculates the annualized standard deviation. Calculates the Sharpe ratio for a risk-free short rate of 1%. The maximum drawdown, which we only calculate for the S&P 500 index, is a bit more involved. For its calculation, we use the .cummax() method, which records the running, historical maximum of the time series up to a certain date. Consider the following code that generates the plot in Figure A-10: In [218]: plt.figure(figsize=(10, 6)) spxvix['SPX'].plot(label='S&P 500') spxvix['SPX'].cummax().plot(label='running maximum') plt.legend(loc=0); Instantiates a new figure object.

The Big Secret for the Small Investor: A New Route to Long-Term Investment Success
by Joel Greenblatt
Published 11 Apr 2011

Shameless tout: For a fuller explanation of the concepts behind earnings yield and return on capital, feel free to read The Little Book That Still Beats the Market (by Joel Greenblatt). 3. After modeled costs of trading and market impact. Data source: CompuStat point-in-time database. Value-weighted Index versus Russell 1000 during test period: Beta = 1.0; Upside capture = 116%; Downside capture = 89%; Sharpe ratio = .61 versus .27; Sortino = .91 versus .38; Information ratio = .85. 4. These could include variants of price-to-earnings, price-to-cash-flow, price-to-dividends, price-to-book, price-to-sales, or any number of other measures of potential cheapness. 5. See appendix. 6. Even after subtracting typical mutual fund expenses (1.25 percent per year), returns would have approximated 12.65 percent per year and turned $1 into $11.50 during the test period in a nontaxable account. 7.

All About Asset Allocation, Second Edition
by Richard Ferri
Published 11 Jul 2010

These funds can experience higher share-price volatility than diversified funds because sector funds are subject to issues specific to a given sector. Securities and Exchange Commission (SEC) The federal government agency that regulates mutual funds, registered investment advisors, the stock and bond markets, and broker-dealers. The SEC was established by the Securities Exchange Act of 1934. Sharpe Ratio A measure of risk-adjusted return. To calculate a Sharpe ratio, an asset’s excess returns (its return in excess of the return generated by risk-free Glossary 329 assets such as Treasury bills) is divided by the asset’s standard deviation. It can be calculated compared to a benchmark or an index. Short Sale The sale of a security or option contract that is not owned by the seller, usually to take advantage of an expected drop in the price of the security or option.

Keeping at It: The Quest for Sound Money and Good Government
by Paul Volcker and Christine Harper
Published 30 Oct 2018

He concluded with a strong warning: businesses, particularly financial businesses, that were not fully aware of and capable of using the new instruments of finance would be doomed to failure. I found myself sitting in the audience next to William Sharpe, a 1990 Nobel laureate in economics whose “Sharpe ratio” has become a widely accepted measure of risk-adjusted returns for fund managers. I nudged him and asked how much this new financial engineering contributed to economic growth, measured by GNP. “Nothing,” he whispered back to me. It was not the answer I anticipated. “So what does it do?” was my response.

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See also Woodrow Wilson School of Public and International Affairs Schultz, Fred, 104, 152 Schweitzer, Albert, 84 Schweitzer, Pierre-Paul, 73, 84 Scranton, Bill, 174 Sears Roebuck, 204 Securities and Exchange Commission (SEC) Citibank and, 99 international accounting standards and, 193–194, 196, 197 PCAOB and, 201 responsibilities, 230 Waste Management Inc. suit, 198 Seger, Martha, 142 Seidenstein, Tom, 195 Selig, Bud, 158 Senate Banking Committee, 114, 122, 130, 200, 213, 216–217 Senate Ethics Committee, 130 September 11 attacks, 235–236 Sevan, Benon, 186 Shapiro, Eli, 42 Shapiro, Harold, 160 Sharpe ratio, 206 Sharpe, William, 206 Shultz, George, 85 fixed rate exchange/end and, 68, 70, 71, 80, 81, 82 Friedman and, 33 Group of Five, 83n international monetary reform/IMF meeting and, 82–83, 84 leaving Nixon administration, 90 Nixon/administration and, 33, 68, 70, 71, 80, 81, 82 Pierre-Paul Schweitzer and, 84 Reagan/administration and, 33 Treasury and, xi, 80, 81, 82–83 Volcker and, 150 Whitehead and, 149 Shultze, Charlie, 106 Siemens AG bribes, 190–191 Silva Herzog, Jesús (“Chucho”), 131, 132, 133, 134 silver market, xii, 124 Simon, Bill, 90, 112 Singh, Natwar, 185 Skilling, Jeffrey, 197 Slade, Elliot, 154 Slaughter, Anne-Marie, 161 Smithies, Arthur, 23, 31 Smithsonian agreement, xi, 79–80, 82, 87 Solomon, Tony, 101 Latin American debt crisis and, 132, 133 Solow, Bob, 23 Soros, George, 156–157 Soviet Union collapse/views, 3 Spacek, Leonard, 198, 200 special drawing rights (SDRs), 57, 65–66, 84, 85 Sproul, Allan, 39, 95 Staats, Elmer, 122 “stagflation,” 99, 223 State Department Bretton Woods fixed exchange rates and, 51, 73 Camp David meetings (1971) and, 73 Stein, Herb, 71 Sternlight, Peter, 106 Stevenson, Adlai, 30, 210 stock market crash (1929), 37 Stockman, David, 114 Stokes, Donald, 159 Strong, Benjamin, 37 “subprime mortgages,” 206, 232 Sumita, Satoshi, 142 Summers, Larry, 211, 212 supply-side economics creator, 143n Swiss banks–Nazi victims investigation accounting firms, 178, 178n “Bergier Commission” and, 177, 179, 180 class-action lawsuits, 177, 177n as Independent Committee of Eminent Persons (ICEP) investigation, 177–181 Jewish community/priorities and, 176, 177–178 research/results, 178–181 Swiss banks/priorities and, 176, 177, 178, 178n Volcker and, 176–177 “Volcker Commission” and, xv, 177–181 World Jewish Congress and, 176, 177 “target zones” (exchange rates), 139, 143 Tariff Act (1930), 73 Tarullo, Daniel, 215 tax consulting, 201 Taylor, Bill, 155, 242 Teaneck, New Jersey, 7, 8, 10, 10n.

The City
by Tony Norfield

But if the investment goes wrong, the return on equity becomes a big negative, incorporating not only the losses from the operating business but also the extra drain on profits from the interest and debt repayments that still have to be made. As one might expect, particular calculations have been developed for this kind of economy to make an adjustment for this risk. In portfolio investment theory, or the mathematics of financial parasitism, investment returns are divided by the volatility of the returns when calculating a ‘Sharpe ratio’ on investment performance.5 Not surprisingly, financial companies have much higher leverage ratios than non-financial ones. Banks, especially, are in a good position to manage this because they can create loans and deposits very much larger than the equity capital that has been invested in bank operations.

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See also financial power; economic power price fixing 118, 120–1 privatisation 120 production 77, 78, 86 productive capital 90 productivity 69, 148–9, 150 profit and profitability 21–2, 77, 129–39 American rate of 153–9, 154, 157 amount 149, 150 comparing 136–9 and costs 149–50 drivers 135–6 equalising 137 and financial crisis 135 and financial returns 135 and fixed assets 137 global 156 and interest rates 156–7 investment location 152 and labour costs 155–6 leverage 130–6, 133, 134 measurement 153 moribund 159–60 and productivity 148–9, 150 rate of 130, 147–50 restoration of 152, 154–5 return on equity (RoE) 129–30 role of financial assets 137 sources 129, 139–47, 143 volatility 131 property bubbles 151 proprietary positions 79 quantitative easing 65, 152, 157–9 Radcliffe Report 35 Reagan, Ronald 65 real economy, the 5 reform agenda 214 regulation 9, 215–16 American 39–40 state 115–16 Regulation Q (US) 39, 48 regulatory arbitrage 42 rentiers 98 repo borrowing 46 return on assets 132 return on equity (RoE) 129–30, 132 returns, distribution of 102–3 revenue-earning assets, bank creation of 137, 137–8 Rich, Marc 122 risk 130 rivalry 28 Robin Hood tax 215–16 Rolling Stone 161 Rome, Treaty of 33, 57, 58, 69 Royal Bank of Scotland 4 Royal Dutch/Shell Group plc 3, 112–13, 220 Rubin, Robert 16 Russia 18, 93, 106, 107, 109, 113–14, 154, 171, 216, 222, 223, 224 SABMiller 121 Samsung 118, 120, 121–2 sanctions 113–14, 172, 216 San Francisco 37 Saudi Arabia 55–6, 109, 208, 220 Scotland, independence referendum 217–18 Second World War 26–8, 151 seigniorage 43, 163–6 Shanghai 18, 182, 222–3 shareholders, power 88 share issues 179 shares 139 value 86–7 share swaps 179, 199 Sharpe ratio 131 Shell 114 Shenzhen 18 Sherman Antitrust Act (US) 119 short-term money-market instruments 79, 144 Simon, William 55–6 Singapore 177–8, 179, 206, 209 Snowden, Edward 59 South Africa 18, 222, 224 South Korea 101, 120 Soviet Union 29, 30 Spain 65 speculation, madcap 135–6 stamp duty 48 Standard Chartered 225 Standard Oil 119 state, the and capitalism 111–15 and finance 115–17 and imperialism 119 imperialist 126 power 115, 126 sterling collapse, 1992 62 convertibility 31 devaluation, 1949 31 Sterling Area, the 31–3, 35, 58 stock exchange prices 182 stock market crash, 1987 xi, 68 stock swaps 179 Suez crisis 27–8, 60 supply-chain links 77 surplus labour 149 surplus value 90, 140 appropriation of 173–4 creation of 149 global system 161 and securities 144–6 transfer of 97 Sveriges Riksbank 116 Sweden 4, 9, 116 Swiss Bankers’ Association 176 Switzerland 4, 9, 165, 175–6, 178, 184 Taibbi, Matt 161 Taiwan 18 Tanzanian groundnut scheme 33 tariffs 114 tax and taxation 114, 215–16 tax avoidance 48 tax havens 2, 22, 71, 184, 193, 209, 211 Temasek 177 Thailand 101 Thatcher, Margaret xi, 63, 65, 69 Tokyo 49, 50 Toyota 121 trade gap 188 trade patterns 6, 60–1, 61 trading revenues 146–7 Treasury, the 48 UK Independence Party 219 unemployment 53 Unilever 112–13 United Kingdom anti-monopoly policies 119–21 appropriation of value 187 balance of payments 187, 188, 200, 203 bank assets and liabilities 4, 141–4, 143 bank borrowing 206–8 bank deficits 206–10 bank lending 208–10 bank leverage data 134 banking centre status 50 banks 4, 116, 134, 191–7, 192, 194, 196, 206–10, 214 borrowing 201–2, 204–5 China policy 225–6, 227 colonial exploitation 30–3 colonialism 30–1 credit rating 204–5 current account balance 188–90, 189, 190 current account deficit 200, 202, 211, 217 current account surplus 33, 34, 69 debt costs 204–5 decolonisation 30 direct investment returns 202–3, 203 domination 162 earnings from financial services 43–4 economic power 2 economic weakness 35 Empire protectionism 30, 33 equity holdings 102–3 equity market capitalisation and turnover 181, 182 and the EU 16–17, 21 European policy 53–4 FDI 107, 200, 205 financial account 197–200, 199 financial machine 22 financial market share 70–1, 71 financial operations 185–212 financial policy 14, 44–7, 65–70 financial position, 1950s 34–5 financial power 2, 3, 64–5 financial sector benefits 185 financial sector employment 186 financial sector tax revenues 186 financial services assets location 205–6, 207 financial services exports 174 financial services revenues 190–7, 192, 194, 196 financial wealth 103 foreign direct investments 3, 66 foreign exchange trading 109 foreign exchange turnover 193–5, 194 foreign investment income 189–90 freedom of action 63, 64 GDP 4, 106, 107, 155–6 Gowan’s analysis 11–12 Helleiner’s analysis 13–14, 70 Hilferding’s analysis 93, 94 imperialism 7–8, 186, 228 imperial strategy 59–65 inflows of foreign money-capital 69 international banking index 108 international banking position 191–2, 192 international banking share 70–1, 71 international financial revenues 10 international investment position 200–1, 201 investment income 200–4, 203 investment income, 1899–1913 98 invisible empire 7–8 joins EEC 34, 57–8 Lend-Lease debts 29, 36 liabilities 204–5, 206–7 military interventions 1–2 military spending 110 monetary policy 66 OPEC’s investment in 57 pension fund assets 103 quantitative easing 158–9 relationship with America 21, 27–8, 29–30, 36, 59, 73 relationship with Europe 62 return to gold standard 23–6 seigniorage 165–6 status 1–2, 3–4, 27–8, 30, 30–5, 52, 73–4, 110, 111, 196, 204, 210, 213, 214, 216–17, 227–8 tax havens 193 trade currency pricing 163 trade deficit 44, 188–9, 189, 211 trade gap 22 trade pattern 60–1, 61 wealth distribution 102–3 United Nations 3–4, 123 United Nations Security Council 109 United States of America 2, 205 aftermath of First World War 24 anti-monopoly policies 119–21 attitude to Bretton Woods 31 bank Eurodollar deposits 41–2 bank leverage data 132–3 bank regulation 36 banking system fragmented 40 banks 132–3, 193 capital flows from 38 capital markets 55 China policy 226, 227 Chinese challenge to 17–18 corporate rate of profit 153–5, 154 cost of living 155 credit bubble 156 credit restrictions 42 current account deficit 167–8 domestic market 28 domination 3, 7, 10, 11, 26–7, 35, 162, 183–4, 223 equity holdings 102 equity market capitalisation and turnover 181–2, 181 exorbitant privilege 166–9 FDI 107 Federal Reserve System 40 financial business split 37 and financial crises 12 financial market regulations 39–40 financial policy 11, 65–6, 67–8 financial power 6, 11–12, 14–15, 55, 170–3, 183 financial services exports 173–4 financial services revenues 190 financial system 36–40 foreign direct investments 3, 42 foreign exchange trading 108–9, 109 foreign exchange turnover 194, 194 foreign investment revenues 9–10 foreign investment stock position 169 foreign securities market 48 GDP 106, 107 gold reserves 39 gold standard abandoned 54 Gowan’s analysis 11–12 hegemony 21, 105 Helleiner’s analysis 12–14 Hilferding’s analysis 93 imperial advantage 164 imperialism 12, 14–15, 166–9 interbank money market 46 interest rates 168 international banking index 108 international banking position 192, 192 international banking share 50, 71, 71 international capital outflows 66 investment income 169 leverage ratios 131 military spending 109 millionaires 99 mortgage-backed securities 140 mortgage debt 56 Panitch and Gindin’s analysis 14–17 privileged position 22 quantitative easing 157–8 rate of profit 153–9, 154, 157 relationship with Britain 21, 27–8, 29–30, 36, 59, 73 relationship with Saudi Arabia 55–6 role in the world economy 12–14, 14–17 role of 111 seigniorage 163–5 status 105, 110, 111 UK-based lending 209–10 UK debt 29 working-class living standards 154–5 United Technologies 121 unsecured loans, interbank 46 US–China Economic and Security Review Commission 18 US Commerce Department 56 US dollar, the 6, 15, 30 Chinese foreign exchange reserves 167 domination 55, 164–5 exchange rate 68, 163 exorbitant privilege 166–9 financial power 170–3 global role 170–3 glut 38–9 gold standard abandoned 54 imperial advantage 164 privileged position 22 as reserve currency of choice 166 status 35, 108–9 UK colonies surplus 33 US Federal Reserve 14, 41, 42, 44, 116, 157–8, 157, 172–3, 185 US Treasury 14 value xi–xiii appropriation of 187 creation 76, 104, 150–1 extraction 77, 104 relations 90 surplus 90, 97, 144–5, 149 transfer of 164–5 Vietnam War 54, 105 Vodafone Group plc 3, 180 Volcker, Paul 156 Volkswagen 121 wages 77, 144, 148, 152, 155–6 Wal-Mart 77, 155 waste, elimination of 152 wealth concentration of 91–2 distribution 102–3 fictitious capital 88, 147 UK household 103 West, Admiral Lord 1–2, 3, 4 West African Marketing Boards 32 WhatsApp 91 Wilson, Harold 32 Wolf, Martin 214 working-class, living standards 154–5 working hours 38 World Bank 14, 27, 29, 73, 223 world economy 2, 9–10 American role 14–17 world hierarchy 105–11, 111 world monetary system 10 ‘zero hour’ contracts 152 zero sum games 145 Zuckerberg, Mark 91

The Elements of Choice: Why the Way We Decide Matters
by Eric J. Johnson
Published 12 Oct 2021

The first one might be good if you are doing calculations, but for gathering a general impression, remembering outcomes, and making inferences, it appears that the last two representations are better. Dan Goldstein from Microsoft Research, whom you met earlier when we discussed defaults, and William Sharpe, a Nobel Prize winner in economics, have tried to apply this idea to understanding financial outcomes. Those in finance might recognize “Sharpe ratios,” a measure of the financial returns of a stock relative to its risk, based on Sharpe’s research.20 There are numerical terms, like beta, that describe the volatility of an investment. These are useful for the technically trained, but Goldstein and Sharpe wanted to help ordinary people who are investing for retirement.

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Armstrong, 209–10 Robinhood, 282–91, 350n Roth, Al, 162–63, 165 rule of 72, 347n Russo, Jay, 194–95 Rutgers University, 138–39 Saddler, Claudette, 164–65, 168–69 Saddler, Radcliffe, 159–60, 161, 164–65, 168–69, 182–83 salt content labels, 256–59, 258 Salvation Army, 71–72 Sam Houston University, 188 Sanders, Bernie, 45–46 “Save the Last Dance for Me” (Bruine de Bruin), 206 scaling metrics, 260–63 Scheibehenne, Benjamin, 166–67 Schoar, Antoinette, 104–5 school choice, 159–66, 168–71, 180, 182–83, 212–13 Schwartz, Barry, 167 screening, 47–52, 164–65 dating sites, 47–52, 164–65 use of term, 47–48 Zappos, 278 search results, ordering of, 199–202 Securities and Exchange Commission (SEC), 288 self-control, 34–35 sequential presentations, 192, 203–5, 207–8 Sharpe, William, 293–94, 351n Sharpe ratios, 293–94 Shealy, Tripp, 247–48 Silva-Risso, Jorge, 68–69 simulated outcome plots, 294, 295 simultaneous presentations, 203–5, 208 Singapore and organ donation, 114 SiriusXM, 277 Siskel, Gene, 251 Skiles, Jeff, 23–24, 27–28 sleight of hand, 56 slotting allowances, 209–10 Slovic, Paul, 101–2 sludge, 18–19, 267 smaller-sooner outcome, 34–35, 37, 39 smart defaults, 153–54, 155, 266, 274 smart-enough defaults, 275 “Snoverkill,” 67 “Snowpocalypse,” 67 social justice, 318 Social Security benefits, 88–98, 310, 336–37n finding right box, 92–95 Retirement Estimator, 95–98 Soll, Jack, 224–29 “Song of Myself” (Whitman), 63 sort order, 213, 278–79 spaghetti plots, 295, 296, 297 Spain and organ donation, 110, 115–16, 338–39n speech recognition, 42 spread trade, 288 Stanford University, 3–4 Stango, Victor, 243 Starbucks, 152 Stash Financial, Inc., 284 status quo bias, 340n stem cell transplants, 107–11 stock options, 285–89 stopping points, 171–72, 196 stop signs, 259 storm tracks, 294–97, 296 straight line metrics, 240–44 “stupid human tricks,” 4 subliminal perception, 300–1 subscription services, 126–27, 267, 314–15 Sullenberger, Chesley “Sully,” 22–29, 44, 102, 332–33n load shedding, 23–24, 27, 28, 44, 61 Sunstein, Cass, 11, 95, 118, 121, 135, 308 supermarket shelves, 209–11 sustainability and targets, 248–50 Sydnor, Justin, 174–75 System 1 thinking, 319–20, 333n target date funds, 153–54, 275 targets, 245–50 taxes, 236–37 taxis and tipping, 12, 128–29, 323 Temple University, 35 Tenev, Vlad, 283 Tesla, 235, 284, 285–86 Tetlock, Phil, 352n Thaler, Richard, 11, 95, 118, 121, 139–40, 214–15, 216 “thumbs-up, thumbs-down” scale, 240, 251, 271 Tinder, 45–48, 276 Tinder Thumb, 46 tipping and taxis, 12, 128–29, 323 Todd, Peter, 166–67 Toyota RAV4, 225–26 traffic light colors, 251, 252, 256, 257, 258 Trump, Donald, 190, 261, 291–93 “tunnel vision,” 332–33n Twitter, 276, 295, 324 “tyranny of choice,” 166 Uber and tipping, 128–29, 323 Uber ratings, 238–39, 250 uncertainty, 291–97 Ungemach, Christoph, 234–35 United States map, 72–73, 335n University of Basel, 166–67 University of California, Riverside, 67 University of Chicago, 117 University of Iowa, 59–62 University of London, 305 University of Michigan, 50–51 University of Nebraska, 283, 289 University of Pennsylvania, 131–32, 306 unsubscribe links, 129 Ursu, Raluca, 200–2 US Airways Flight 1549, 21–29 user models, 266, 273–77 usury laws, 243 utility bills, 134–37, 347n Verizon, privacy settings, 315–16 Vicary, James H., 299–301 visuals, 208–11 Vogelsang, Timothy, 69 voter registration, 156–58 walking, 30, 122, 208 Wall, Dan, 35, 37 wallpaper backgrounds, 65–66 warning labels, 257, 259, 307 Watt, James, 255 weather, 66–69 car sales and, 68–69 climate change and, 66–68 Weber, Elke, 21, 44, 352n airplane tickets and taxes, 236 building sustainability, 247–48 decision by distortion, 194–95 default studies, 141–42 environmental regulations, 261–62 greenhouse gas ratings, 234–35 query theory, 75–76 website design, 5, 13–15 auto buying, 146–51 backgrounds, 65–66 health insurance exchanges, 13–15, 173–82, 279–81 Weill Cornell Medical College, 9–10 Wertlieb, Stacey, 116 Wharton School, 4–5, 141, 182 Whitman, Walt, 63 Whole Foods, 209–10 Wilson, Woodrow, 186–87, 195 wine tastings, 204, 206–7, 207 wireframe, 13–14 Wonderlic, 230 Woods, Erika, 44–48, 103–4 word associations, 60–63, 235–36 Yang, Sybil, 217–18 Yelp, 40, 265 Zappos, 278–79 Zaval, Lisa, 67 Zink, Sheldon, 116 Zinman, Jonathan, 243 Zoom, 155–56 “Zoom bombing,” 156 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z About the Author Eric J.

How I Invest My Money: Finance Experts Reveal How They Save, Spend, and Invest
by Brian Portnoy and Joshua Brown
Published 17 Nov 2020

I purchased a small amount of her ETF in my Roth IRA. I expect I will make an additional exception or two for products from people I admire. My experience as a financial advisor has taught me that the “why” is more important than the “how” when it comes to investing. My financial goals are much more important than the alpha or the Sharpe ratio of my portfolio. As long as I prioritize paying myself first and avoid unnecessary mistakes, I am confident that my investment portfolio will do its job. Leighann Miko Leighann Miko is a CERTIFIED FINANCIAL PLANNER™ and the founder of Equalis Financial, a Los Angeles-based independent fee-only firm providing financial planning, business management, and investment management services.

Capital Allocators: How the World’s Elite Money Managers Lead and Invest
by Ted Seides
Published 23 Mar 2021

Allocators will never be rewarded for the past success of a manager. Thoughtful CIOs scrutinize a track record and use creative analytics to focus on the strengths and weaknesses of the manager. Allocators weigh an array of statistics to measure returns relative to a benchmark, including time-weighted and dollar-weighted returns, Sharpe ratio (return per unit of risk, measured by standard deviation of returns), Sortino ratio (return per unit of downside risk), exposure-adjusted returns, and peer analysis. Each of these can be fine-tuned depending on the question a CIO seeks to answer. Comparing time-weighted and dollar-weighted returns can inform a CIO about the quality of the investor base and the manager’s messaging.

Mastering Private Equity
by Zeisberger, Claudia,Prahl, Michael,White, Bowen , Michael Prahl and Bowen White
Published 15 Jun 2017

The theoretical (risk) impact of including PE in an investor’s portfolio is well understood: adding private, unlisted companies to a traditional portfolio of public equity and fixed income will help diversify the overall portfolio exposure by reducing its volatility and the risk of large drawdowns, thereby increasing its Sharpe ratio. PE’s historically lower volatility of annualized returns, relative to those from a comparable public equity index, seems to fulfill the theory (see Exhibit 23.22). Of course, the argument needs to be carefully back-tested3 since the actual diversification effect (and returns achieved) will naturally depend on the investor’s existing portfolio, the choice of PE/venture capital (VC) funds it may be able to access and the correlation of those assets to its overall portfolio.

…

PE firms have neither the competitive edge, nor the agility to profit from macro trading. Currency movement is an unwanted and uncontrollable risk that contributes only to volatility of returns. A portfolio theory approach would suggest that the risk performance of an investment (as measured by the Sharpe ratio) can be improved if some of the volatility of returns can be eliminated at a suitably acceptable cost. What is deemed “acceptable” will depend upon the amount of volatility that is eliminated, but the current global trend towards ever-lower rates skews the calculation towards active currency hedging in an increasing number of cases: with interest rates in most countries at or close to zero, interest differentials have compressed, and the costs of hedging through forwards have fallen dramatically.

The Little Book of Hedge Funds
by Anthony Scaramucci
Published 30 Apr 2012

As history has proven, statistical and quantitative techniques have done far more harm than good to both capital markets and hedge fund investors (think LTCM). It’s hard to convince yourself that levering any investment strategy 100:1 is safe unless you are both egregiously arrogant and have developed such sophisticated models that nothing can go wrong. After all, what good is a Sharpe Ratio of 4 for three years when you lose 100 percent of your money in the fourth year? That said, the application of data analysis can be helpful when applied by thoughtful, humble minds. The operational due diligence process is focused on making sure the manager does not or cannot do anything completely stupid on the business side to blow up his business.

Damsel in Distressed: My Life in the Golden Age of Hedge Funds
by Dominique Mielle
Published 6 Sep 2021

It is a muscle that you exercise and stretch. Risk tolerance is no different from physical endurance. Everyone has some; it’s only a matter of degree. What matters is how you quantify and manage it, or in finance terms, the amount of return you garner per unit of risk, a concept aptly named the Sharpe ratio, conceived by the 1990 Nobel laureate in economic sciences, William F. Sharpe. Which brings me to the other class that remains seared in my mind. Simply called portfolio management, it was taught by Bill Sharpe himself, a pioneer in financial economics, whose Capital Asset Pricing Model revolutionized portfolio management theory.

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton
by Colin Read
Published 16 Jul 2012

In his research, he continued to look into ways in which theoretical concepts can be reduced to methodologies that can be applied by practitioners. For instance, he produced a discrete-time binomial option pricing procedure that offered a readily applicable procedure for BlackScholes securities pricing, which will be covered in the next part of this book. He also developed the Sharpe ratio, a measure of the risk of a mutual or index fund versus its reward. Sharpe continued to work to make financial concepts more democratic and more accessible. He helped develop Financial Engines, an Internetbased application to deliver investment advice online. 78 The Rise of the Quants Ever concerned about the practitioner’s side of finance, Sharpe began to consult with investment houses, first Merrill Lynch and then Wells Fargo.

A Primer for the Mathematics of Financial Engineering
by Dan Stefanica
Published 4 Apr 2008

For notation purposes, assume that asset 4 is the asset with uncorrelated return, i.e., Pi,4 = 0, for i = 1 : 3. Let Wi be the weight of asset i in the BFinding efficient portfolios is one of the fundamental problems answered by the modern portfolio theory of Markowitz and Sharpe; see Markowitz [17] and Sharpe [25] for seminal ~apers. Of all the efficient portfolio~, the portfolio with the higheflt Sharpe ratio E~l~?, l.e., the expected return above the rIsk free rate r f normalized by the standard deviation 0-( R) of the return, is called the market portfolio (or the tangency portfolio) and plays an important role in the Capital Asset Pricing Model (CAPM). 262 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD. portfolio, for i = 1 : 4.

Heads I Win, Tails I Win
by Spencer Jakab
Published 21 Jun 2016

A piece of boilerplate in fund brochures that people routinely skim over says, “Investors should carefully consider investment objectives, risks, charges, and expenses.” Costs are higher than people realize. The number you see when looking up a mutual fund is its expense ratio. That’s a good place to start but not the whole story. William Sharpe, a Nobel Prize–winning economist and the man behind the famous Sharpe ratio used to measure portfolio risk, recently tried to calculate the all-in costs of active funds. Some aren’t at all obvious. In 2013 the average expense ratio was 1.12 percent, but service charges also averaged 0.5 percent. Then there are the transaction costs within a fund—paying commissions to brokers to buy and sell stocks.

Monte Carlo Simulation and Finance
by Don L. McLeish
Published 1 Apr 2005

If the market portfolio m has standard deviation σm and mean ηm , then the line L is described by the relation η=r+ ηm − r σ. σm For any investment with mean return η and standard deviation of return σ to be competitive, it must lie on this efficient frontier, i.e. it must satisfy the relation η − r = β(ηm − r), where β = σ or equivalently σm (2.19) η−r (ηm − r) . = σ σm This is the most important result in the capital asset pricing model. The excess return of a stock η − r divided by its standard deviation σ is supposed constant, and is called the Sharpe ratio or the market price of risk. The constant β called the beta of the stock or portfolio and represents the change in the expected portfolio return for each unit change in the market. It is also the ratio of the standard deviations of return of the stock and the market. Values of β > 1 indicate a stock that is more variable than the market and tends to have higher positive and negative returns, whereas values of β < 1 are investments that are more conservative and less volatile than the market as a whole.

Broken Markets: How High Frequency Trading and Predatory Practices on Wall Street Are Destroying Investor Confidence and Your Portfolio
by Sal Arnuk and Joseph Saluzzi
Published 21 May 2012

This typically involves: • Large technological expenditures in hardware, software and data • Latency sensitivity (order generation and execution taking place in sub-second speeds) • High quantities of orders, each small in size • Short holding periods, measured in seconds versus hours, days, or longer • Starts and ends each day with virtually no net positions • Little human intervention Will Psomadelis, Head of Trading at Schroeder Investment Management, in a paper titled “High Frequency Trading - Credible Research Tells the Real Story,” found that HFT “returns are abnormally high, with Sharpe ratios often in the order of nine or double digits. Well-known names in the HFT space include GETCO, Infinium, and Optiver.”1 Although initial perceptions focused on how HFT has shrunken spreads and generated liquidity that investors could embrace, in recent years those perceptions have turned quite negative.

High-Frequency Trading
by David Easley , Marcos López de Prado and Maureen O'Hara
Published 28 Sep 2013

The most profitable holding-to-close strategy is when BadMax trades in at the first non-clustered execution (net Alpha 5.6bp). In this strategy, by only establishing positions at the first non-clustered execution, BadMax avoids all subsequent clustered executions where the price has already moved away. All the holdingto-close strategies, however, are associated with high risk and low Sharpe ratios. This chapter is based on two Goldman Sachs reports by the authors (“Information Leakage”, Street Smart no. 45, and “Do algo executions leak information?” (with Mark Gurliacci), Street Smart no. 46, both dated February 2, 2012). This material was prepared by the Goldman Sachs Execution Strategies Group and is not the product of Goldman Sachs Global Investment Research.

My Life as a Quant: Reflections on Physics and Finance
by Emanuel Derman
Published 1 Jan 2004

We lived in a campus dorm and luxuriated in our freedom from corporate life, running on the MIT track in the late afternoons and eating in Cambridge in the evenings. Myers's course focused on the Capital Asset Pricing Model, and I was captivated by the apparent similarity between financial theory and thermodynamics. I saw a perhaps-too-facile correspondence between heat and money, temperature and risk, and entropy and the Sharpe ratio, but have never since figured out how to exploit it. The course was brief and intense and required more work than we put into it. One of the lecturers was Terry Marsh, now a Professor at Berkeley and a founding partner of the financial software firm Quantal. At that time he was just beginning to make his reputation, and I was always happy to run into him years later at professional finance meetings or when I gave a seminar at the Haas business school at Berkeley.

The Physics of Wall Street: A Brief History of Predicting the Unpredictable
by James Owen Weatherall
Published 2 Jan 2013

“. . . a variety of computer programs known as genetic algorithms”: For more on genetic algorithms, see, for instance, Mitchell (1998). For Packard’s early contributions, see Packard (1988, 1990). “. . . over the firm’s first fifteen years . . .”: More specifically, this person told me that the company had a Sharpe ratio of 3. 7. Tyranny of the Dragon King “Didier Sornette looked at the data again”: The opening story, which plays out throughout the chapter, is a dramatization, but the basic details are correct. In late summer 1997, Sornette observed a pattern in U.S. financial data that he had previously argued could be used to predict financial crashes; he contacted his colleagues Olivier Ledoit and Anders Johansen and proceeded as described here.

Trillions: How a Band of Wall Street Renegades Invented the Index Fund and Changed Finance Forever
by Robin Wigglesworth
Published 11 Oct 2021

Chris Welles, “Who Is Barr Rosenberg? And What the Hell Is He Talking About?,” Institutional Investor, May 1978. 9. Narasimhan Jegadeesh and Sheridan Titman, “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” Journal of Finance, March 1993. 10. Robert Huebscher, “Sharpe Ratio Inventor: ‘When I Hear Smart Beta It Makes Me Sick,’ ” Business Insider, May 22, 2014. 11. Eugene Fama and Kenneth French, “The Cross-Section of Expected Stock Returns,” Journal of Finance, June 1992. 12. Robin Wigglesworth, “Can Factor Investing Kill Off the Hedge Fund?,” Financial Times, July 22, 2018. 13.

Nobody's Fool: Why We Get Taken in and What We Can Do About It
by Daniel Simons and Christopher Chabris
Published 10 Jul 2023

Roth (Alexandra Roth Book Project, 2010), 212–219. We obtained the same historical performance data to calculate the year-to-year volatility (standard deviation of returns) for De Vita’s sixteen funds and for the Madoff fund (as reported for the Fairfield Sentry feeder fund) from 1991 to 2007. The Sharpe ratio (return over volatility, in each case relative to a risk-free asset, typically US Treasury bills) was 3.02 for Madoff and typically below 0.5 for the non-Ponzi funds. 19. We transcribed the quotation from Agassi’s Unscriptd interview, “Andre Agassi Interview | Beat Boris Becker by Observing His Tongue,” YouTube [https://www.youtube.com/watch?

Hedgehogging
by Barton Biggs
Published 3 Jan 2005

The media loved it and published the names of all the supposedly smart, sophisticated individuals and institutions who had lost their money. Everybody was deeply embarrassed, and ever since the big institutions have been obsessed with risk analytics and throw around terms like stress-testing portfolios, value at risk (VAR), and Sharpe ratios. The funds of funds employ sophisticated quantitative analytics to add value by strategically allocating among the different hedge-fund classes.The hedge-fund universe is usually broken down into seven broad investment style classifications.These are event driven, fixed-income arbitrage, global convertible bond arbitrage, equity market-neutral, long/short equity, global macro, and commodity trading funds.

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street
by Justin Fox
Published 29 May 2009

He presented his conclusions to a room full of Yale trustees and alumni up from Wall Street. “I looked around that room and all I could see after my pitch was angry faces,” he said. “Yale discarded my recommendation completely.” Years later, the “Treynor ratio” became a much-used measure of investment manager performance. Even better known is the “Sharpe ratio,” a similar gauge—originally termed “reward-to-volatility”—that fellow CAPM pioneer Bill Sharpe introduced in a paper in 1966. Most famous of all is probably “alpha,” devised by Michael Jensen for his 1968 Chicago Ph.D. dissertation on mutual fund performance. Alpha is a portfolio’s performance minus the performance of a hypothetical benchmark portfolio of equivalent risk.

How I Became a Quant: Insights From 25 of Wall Street's Elite
by Richard R. Lindsey and Barry Schachter
Published 30 Jun 2007

The environment met all the CAPM assumptions: All participants had same time horizon and the same knowledge of the securities. The market contained a riskless security, the security return distributions were normal and all the available assets were tradable and held by the investors, and so on. Bossaerts expected the students to trade their portfolios to attain the highest Sharpe ratio, which he had set to be that of the capitalization weighted market. He assumed the students would drive their portfolios to the zero line, as shown in Figure 4.1. However, the results were all over the place when they traded using continuous market mechanisms like those used by the NYSE, Nasdaq, and London’s SETS (Stock Exchange Electronic Transfer Service).

Investment: A History
by Norton Reamer and Jesse Downing
Published 19 Feb 2016

That said, to provide some empirical insight on the matter of the more recent performance of these strategies, the aggregate risk and return figures from 1994 to 2011, assembled by KPMG and the Centre for Hedge Fund Research (table 8.1), show that relative value and event-driven funds have been the strongest performers on a risk-adjusted basis (as measured by their Sharpe ratios).25 By contrast, short bias funds have tended to have the least attractive risk-reward characteristics, returning just over 1 percent per year but table 8.1 Statistics for Hedge Fund Strategies equity emerging event hedge m a rkets driven cta a nd relative m a rket m acro va lue neutr a l short bi as Annualized Mean 10.58 9.60 10.32 8.39 8.23 5.73 1.04 Annualized Std 9.49 14.25 6.97 6.69 4.35 3.30 18.96 Annualized Sharpe 0.74 0.42 0.97 0.72 1.06 0.65 –0.13 Source: Rober t Mirsky, Anthony Cowell, and Andrew Baker, “The Value of the Hedge Fund Industr y to Investors, Markets, and the Broader Economy,” KPMG and the Centre for Hedge Fund Research, Imperial College, London, last modified April 2012, http://www.kpmg.com/KY/en /Documents/the-value-of-the-hedge-fund-industr y-par t-1.pdf, 11. 

Trade Your Way to Financial Freedom
by van K. Tharp
Published 1 Jan 1998

They go bankrupt (or at least lose their capital) within a year or two because they don’t understand what their edge is or they don’t know how to capitalize on it. In addition, they are seldom risking 0.5 percent of their total equity per trade. 2. Jack Schwager, The New Market Wizards (New York: HarperCollins, 1992). 3. While I don’t wish to reveal my proprietary indicator in this book, it is highly correlated with the Sharpe Ratio. Furthermore, our research shows that the higher the ranking with this indicator, the easier it is to use position sizing to meet your objectives. CHAPTER 14 Position Sizing—the Key to Meeting Your Objectives When I get a 30 percent profit, I take a third. When I get a 50 percent profit I take another third.