Fixed: Why Personal Finance is Broken and How to Make it Work for Everyone
by John Y. Campbell and Tarun Ramadorai · 25 Jul 2025
standard deviation, a statistical measure of variability that captures the magnitude of typical fluctuations. To measure how successful an investment has been, economists use the Sharpe ratio, named after economics Nobel laureate William Sharpe and calculated as the ratio of the historical average return on the investment (over and above the risk
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might earn in a bank savings account or money market mutual fund) to the standard deviation of that return. An investor who earns a high Sharpe ratio is well rewarded for risk and should respond by investing more aggressively.10 So how can an investor get a high
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Sharpe ratio? The key is to diversify: to combine different risky investments that don’t move perfectly in lockstep but vary to some degree independently. Even if
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each investment has a low Sharpe ratio in isolation, when combined in a diversified portfolio containing many such investments, the random fluctuations in any given investment will tend to be offset by
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in the opposite direction in other investments, while the positive average returns offered by each investment will add up. The result will be a higher Sharpe ratio on the portfolio as a whole. Economists are notorious for the gloomy saying “there’s no such thing as a free lunch,” but here we
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below 20% per year.12 Since a randomly selected individual stock has an expected return equal to that of the market as a whole, the Sharpe ratio for the US market index is therefore over three times greater than that for a typical individual stock. One might hope to get a
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0.3 on the US market, corresponding to an average return of about 6% above the risk-free interest rate, but the Sharpe ratio would be less than 0.1 for an individual stock.13 In the past it was much easier for wealthy people to exploit diversification, because
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.” 10. A classic formula for optimal risk-taking says that the fraction of your wealth that you should invest in a risky asset is the Sharpe ratio on the asset divided by your risk aversion times the standard deviation of the risky asset return. Equivalently, the risk you should allow in your
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portfolio is the Sharpe ratio divided by your risk aversion. If you have the appetite for a textbook explanation complete with math, see John Campbell, Financial Decisions and Markets: A
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), 309n39 Securities and Exchange Commission (SEC) (US), 216 self-insurance, 295n34 SEP-IRA account, 297n7 shares, people with limited resources owning, 8 Sharpe, William, 131 Sharpe ratio, 292n10; diversification and, 131–132 Shiller, Robert, 55 Shipley, Simon, 4, 63, 263 shocks: coping with financial, 9; effects of systemic, 82 short-term borrowing
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of financial vulnerability in, 83, 84; real estate transaction costs in, 113; real returns on higher education in, 102–106, 103; refinancing in, 120–121; Sharpe ratio for stock market in, 132; shift from defined benefit to defined contribution pensions in, 154–155; student loan program in, 106–107; sufficiency of retirement
Systematic Trading: A Unique New Method for Designing Trading and Investing Systems
by Robert Carver · 13 Sep 2015
51 52 65 67 xiii Systematic Trading Chapter Four. Portfolio Allocation Chapter overview Optimising gone bad Saving optimisation from itself Making weights by hand Incorporating Sharpe ratios 69 70 70 75 77 85 Part Three. Framework 91 Chapter Five. Framework Overview 93 Chapter overview 93 A bad example 94 Why a modular
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Classifying trading styles The main ways in which you can classify rules into trading styles, and the important characteristics each style has. Achievable Sharpe ratios What are realistic Sharpe ratios to expect? 25 Systematic Trading What makes a good trading rule Staunch systems trader This section mostly relates to how trading rules are fitted
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this is another useful source of return. Skew and unlikely events premium The rational investor of classical financial theory only cares about an asset’s Sharpe ratio (SR) – it’s average returns adjusted for their standard deviation. This only makes sense if all assets have symmetrically distributed returns. But in practice
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gains with occasional large losses (as a seller of insurance does). This implies that the skew of each asset’s returns is different. CONCEPT: SHARPE RATIO (SR) The Sharpe ratio (SR) measures how profitable a trading strategy or holding an asset has been, or is expected to be. To calculate it you take returns
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losses – negative skew. Figure 5 and table 1 show the return distribution and statistics of two assets with different skews but the same Sharpe ratio. FIGURE 5: TWO ASSETS, SAME SHARPE RATIO, DIFFERENT SKEWS Which is better, positive or negative skew? It depends... 33 Systematic Trading TABLE 1: CHARACTERISTICS OF NEGATIVE VERSUS POSITIVE SKEW
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futures gives you highly positive skew, but as you are effectively purchasing insurance against unexpectedly high equity volatility it also tends to have a negative Sharpe ratio (SR). Similarly selling the futures gives a positive SR, but with an extremely negative skew. Each of these futures has skew around four times
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monitor trading activity in real time so you need very tight controls and good monitoring systems. All this means that realising the very high theoretical Sharpe ratio from super fast strategies is no picnic. The domain of high frequency trading mostly falls outside the scope of this book. Technical vs fundamental
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shows the average number of rules accepted from pools of arbitrary unprofitable rules, given different pool sizes (rows), which were tested to see if their Sharpe ratio exceeded a minimum cutoff (columns). 48. Many of the examples in this chapter will use imaginary daily returns of various arbitrary trading rule variations.
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These fake returns are randomly generated with the kind of characteristics I want: expected mean, standard deviation (from which we get the Sharpe ratio), and where relevant correlation with other variations. I then run these tests many times and report the average result, so the answer is not influenced
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note: These are the returns you’d get if you traded an equally weighted portfolio of instruments using the same trading rule. The improvement in Sharpe ratio (SR) comes from the diversification effect across instruments. The alternative method is to stitch consecutive series of instrument returns together. This gives a very
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should you fit them individually. In practice this is rarely necessary. Compare apples with apples I’ve assumed throughout this chapter that you will use Sharpe ratio (SR) to compare rules. But comparing a positively skewed rule and a negatively skewed alternative purely on SR is highly misleading, as negative skew
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looking at performance, but at behaviour such as trading speed and correlation with other variations. 3. Allocate forecast weights to each variation, taking uncertainty about Sharpe ratios into account. Poor rules will have lower weight, but are rarely entirely excluded. This process means that I don’t use performance to fit trading
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can help us understand what is going wrong. Making weights by hand How to use a simple method called handcrafting to get portfolio weights. Incorporating Sharpe ratios Using additional information about expected performance to improve handcrafted weights. Optimising gone bad Introducing optimisation Portfolio optimisation will find the set of asset weights which
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give the best expected risk adjusted returns, usually measured by Sharpe ratio. The inputs to this are the expected average returns, standard deviation of returns, and their correlation. The standard method for doing this was first
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standardisation I spoke about in chapter two, ‘Systematic Trading Rules’. By using this technique you simplify the problem and only need to use expected Sharpe ratios and correlations to work out your weights. Although you won’t be optimising the underlying positions in individual assets like equities or bonds in your
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of sample expanding window. 71 Systematic Trading The calculation is done using the classic Markowitz optimisation; I find the maximum risk adjusted return (e.g. Sharpe ratio) using the estimated means and correlations, and standard deviations (which are all identical because I’ve used volatility standardisation). I also don’t allow weights
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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
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data, and the past contains periods which were very different, then the optimal portfolios will be close to equal weights. But with significant differences in Sharpe ratios or correlations similar portfolios will crop up repeatedly, 75 Systematic Trading and the average will reflect that. The averaged weights naturally reflect the amount of
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system I outline in chapter fifteen for staunch systems traders. Using in sample handcrafting rather than rolling out of sample bootstrapping gave an insignificant advantage (Sharpe ratio of 0.54 rather than 0.52). In comparison in sample single period optimisation produced an unrealistically high SR of 0.84; although when
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for the group that matters. 85 Systematic Trading TABLE 12: HOW MUCH SHOULD YOU ADJUST HANDCRAFTED WEIGHTS BY IF YOU HAVE SOME INFORMATION ABOUT ASSET SHARPE RATIOS? Adjustment factor SR difference to average (A) With certainty e.g. costs (B) Without certainty, more than ten years’ data (C) Without certainty, less
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year bond 68% 53% 50% 58% S&P 500 equities 32% 27% 25% 24% NASDAQ equities 0% 20% 25% 18% When I bring in Sharpe ratio estimates, handcrafting up-weights better performing assets and produces similar results to bootstrapping, but does not result in extreme portfolios like single period optimisation. Once
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is much more work than using the bootstrapping method. 89 Systematic Trading TABLE 14: WITH HOW MANY PINCHES OF SALT SHOULD WE TREAT BACK-TESTED SHARPE RATIOS? Pessimism factor Single period optimisation, uses SR, in sample 25% Single period optimisation, uses SR, out of sample 75% Bootstrapping, uses SR, in sample
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‘Volatility targeting’, and chapter ten, ‘Position sizing’. You’ve seen before that the ratio of average return to standard deviation of returns is the Sharpe ratio. So expected Sharpe ratios make good forecasts, which makes creating trading rules very intuitive. Forecasts should have a consistent scale I said above that a forecast needs to
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proportional to return adjusted for standard deviation. So using an equal forecast for all instruments is equivalent to assuming all assets have the same expected Sharpe ratio. There are certain advantages to using this constant forecast approach within the systematic framework, rather than just buying a normal risk parity portfolio. For
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% 56% 1.0 100% 100% 2.0 200% 400% The table shows the expected annual return (as a percentage of trading capital), given different Sharpe ratio (SR) (rows) and using the optimal Kelly percentage volatility target. Expected return is SR multiplied by percentage volatility target. This finding is potentially dangerous when
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the recommended percentage volatility target for those who can backtest their dynamic trading systems, depending on the skew of returns (columns) and achievable back-tested Sharpe ratios (SR) (rows) after making adjustments to simulated results from table 14 on page 90. Optimal volatility target is calculated using Half-Kelly. For negative
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Avoid very low volatility instruments requiring insanely high leverage. • The recommended percentage volatility in tables 25 and 26, depending on the back-tested or expected Sharpe ratio and skew of your trading system, and the type of trader or investor you are. Percentage volatility target should remain unchanged. Trading capital The amount
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faster, and apparently superior, variation rather than a slower, cheaper and supposedly inferior alternative (see table 6 on page 64). The relative pre-cost Sharpe ratio you will achieve in practice is extremely uncertain, whilst expected costs can be predicted with much more accuracy. There is one further danger in blindly
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be getting consistently negative execution costs. In conclusion then, to day trade even the cheapest instruments you will need to achieve very high pre-cost Sharpe ratios, or have an execution strategy that consistently captures the spread. This isn’t impossible, but clearly only those with a proven record of achieving
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. Selecting, fitting and calibrating trading rules (staunch systems traders) If you decide to fit your trading rules then you must consider both pre-cost Sharpe ratio and costs. Remember that you have more certainty of costs than of pre-cost performance. With my preferred method of fitting you shouldn’t look
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likely costs to avoid overweighting trading rule variations which perform well but are also relatively expensive to trade. Setting the risk percentage When checking your Sharpe ratio to set the risk percentage, you should use after-cost performance. Setting the look-back for estimating price volatility After forecast changes this is the
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holding periods and you should do your own bootstrapping with realistic costs to check this. 194 Chapter Twelve. Speed and Size More conservative estimation of Sharpe ratio to set risk percentage This section is relevant to all readers Back in chapter nine, ‘Volatility targeting’, I said that you should have realistic expectations
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value of one instrument block in the currency of the instrument, as calculated in chapter ten, ‘Position Sizing’, (page 158). Standardised cost Annualised cost in Sharpe ratio (SR) per round trip (buy and sell): Twice the cost to trade one block divided by annualised instrument currency volatility (daily instrument currency volatility multiplied
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loss rule to give an average holding period so that turnover multiplied by standardised cost of your most expensive instrument is at most 0.08 Sharpe ratio units a year. Selecting trading rules You should exclude trading rules which are unaffordable for a given instrument given my recommended maximum for staunch
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systems traders of spending no more than 0.13 Sharpe ratio units a year on costs. This can be calculated for each instrument individually, or using the highest and most conservative instrument cost. Forecast weights
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for trading rules Bootstrap the weights with cost adjustment or if using handcrafting I recommend assuming the same pre-cost Sharpe ratio (SR). This means adjusting handcrafted weights depending on the SR cost versus average across trading rules, using ‘with certainty’ column A of table 13 (
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by more than 25%. Instrument weights for trading subsystems Bootstrap the weights with cost adjustment, or if handcrafting I recommend assuming the same after cost Sharpe ratio, implying no adjustment to instrument weights. 205 Systematic Trading Too much capital When simple cost calculations aren’t enough If you can’t always trade
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instrument blocks with a maximum forecast of 20. You should also check that the standardised costs of trading any potential instrument will be 0.01 Sharpe ratio units or less, for reasons explained in detail below. Finally you should avoid very low volatility instruments, for the numerous reasons enumerated throughout part
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constraints are expressed in risk adjusted proportions. An extension to the basic example will be to adjust instrument weights according to some expectations about asset Sharpe ratios. These aren’t produced by a systematic trading rule, but this setup is common in institutional investing when allocations need to be adjusted to reflect
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(page 86), column B ‘Without certainty’. 233 Systematic Trading Portfolio instrument position Subsystem position multiplied by instrument weight (from table 41 for the example), with Sharpe ratio adjustments if required, and by instrument diversification multiplier (1.61 in the example). Rounded target position Portfolio instrument position rounded to nearest instrument block (in
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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
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. Handcrafted optimisation A method of portfolio optimisation where you set weights by hand, grouping assets together and using only estimates of correlations (and optionally, Sharpe ratios). See page 78. Ideas first A type of fitting where you conceive an idea, choose a specific trading rule and then back-test the rule
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return over some time period divided by the standard deviation of returns over the same time period. In this book I normally use the annualised Sharpe ratio – annualised returns divided by annualised standard deviation. If we’re not using derivatives the ‘risk free’ interest rate should be deducted from annualised returns
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allocating investor, 234- 244 Diary of trading, for staunch systems trader, 255- 257 Diversification, 20, 42, 44, 73f, 104, 107, 165, 170, 206 and Sharpe ratios, 65f, 147, 165 of instruments rather than rules, 68 and forecasts, 113 Dow Jones stock index, 23 Education of a Speculator, 17 Einstein, 70 Elliot
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, 155, 182 Instrument subsystem position, 175, 233 Instrument weights, 166-167, 169, 173, 175, 189, 198, 201, 202, 203, 206, 215, 229, 253 and Sharpe ratios, 168 and asset allocating investors, 226 and crash of 2008, 244 Gambling, 15, 20 Gaussian normal distribution, 22, 32&f, 39, 111f, 113, 114, 139f
Trillions: How a Band of Wall Street Renegades Invented the Index Fund and Changed Finance Forever
by Robin Wigglesworth · 11 Oct 2021 · 432pp · 106,612 words
. 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
Capital Ideas Evolving
by Peter L. Bernstein · 3 May 2007
outperformed their benchmarks (usually market indexes) by significant margins, and the two that fell behind their benchmarks have been operational for a relatively short time. Sharpe ratios (total return divided by volatility) compare favorably in all cases. The five strategies in operation for the longest periods of time (and their dates of
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the opportunity to create alpha. 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
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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
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a Sharpe Ratio of 1.0 for ten years running, and that’s huge. In the market as a whole, the volatility is between three and five times
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as high as the expected return.” I asked Litterman whether he thought Goldman could sustain a Sharpe Ratio that high into the indefinite future. “Absolutely not,” he responded. “Neither can we create unlimited wealth. Something has to give. . . . New competitors show up every
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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
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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
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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
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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
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all long/short, little or no capital is involved. The primary metric you are allocating, then, is risk, and the Sharpe Ratio is the most useful guide in that process. The Sharpe Ratio measures the ratio of return over the risk-free rate to the volatility of return. That is the design. Execution is
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equilibrium as explained by, 215, 218 Goldman Sachs career by, 214 –215, 216–218 on managing transaction costs, 224 on market eff iciency, 244 on Sharpe Ratios of portfolios, 221–222 on shifting from individual assets to portfolio, 228–229 Lo, Andrew Adaptive Market Hypothesis notion of, 62– 64 Capital Ideas applications
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Taiwan stock market (1990s), 56–57 Leibowitz’s work with CREF, 200 –202 Litterman on combining long and short position of, 230 –232 Litterman on Sharpe Ratios of, 221–222 Litterman on shifting from individual assets to, 228–229 Markowitz’s theory on selecting, 100, 103, 105–109 Merton’s Country A
Trading Risk: Enhanced Profitability Through Risk Control
by Kenneth L. Grant · 1 Sep 2004
Time Spans Graphical Representation of Daily P/L Histogram of P/L Observations Statistics A Tribute to Sir Isaac Newton Average P/L Standard Deviation Sharpe Ratio Median P/L Percentage of Winning Days Performance Ratio, Average P/L, Winning Days versus Losing Days xiv 1 19 21 24 26 32 37
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Calculation in Portfolio Management Scenario Analysis Technical Analysis CHAPTER 5 Setting Appropriate Exposure Levels (Rule 1) Determining the Appropriate Ranges of Exposure Method 1: Inverted Sharpe Ratio Method 2: Managing Volatility as a Percentage of Trading Capital Drawdowns and Netting Risk Asymmetric Payoff Function CHAPTER 6 Adjusting Portfolio Exposure (Rule 2) Size
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in this chapter, we will formalize the calculation of riskadjusted return in a number of important ways, most notably by introducing such calculations as the Sharpe ratio and the Return over Maximum Drawdown (ROMAD) statistic. Standard Deviation Statistic for Nonnormal Distributions. As indicated earlier, the standard deviation statistic is one that, in
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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
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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
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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
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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
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the markets you trade. For all but a few specified situations, the appropriate rate of return is that of U.S. Treasury instruments. In the Sharpe Ratio calculation, the risk-free return is subtracted from the total return of the portfolio in order to isolate that portion of the performance that is
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outcome of this is that an individual who takes capital and invests it in Treasuries earns exactly the risk-free rate and, therefore, generates a Sharpe Ratio of precisely 0, while portfolios that fail to earn even this modest level of return actually have negative Sharpes. Therefore, it is only for performance
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above the minimum benchmark of government coupons—and as such is deemed to be associated with risk-bearing market activities—that the Sharpe Ratio begins to recognize a positive, risk-adjusted return. • Standard Deviation of Return. This is our old friend/nemesis, which we thought was beaten to death
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%, would have an annualized standard deviation of approximately $159,000, or 15.9%. It is necessary to achieve this normalization of time spans across the Sharpe Ratio equation in order to derive results that do not fall into the category of nonsensical. Note that the equation allows for adjustments that Understanding the
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Sharpe and are wondering if you should be proud or ashamed. As a rule of thumb, I believe that individuals should almost always strive for Sharpe Ratios, calculated under the methodology just described, equal to or in excess of 1.0. For example, if we assume a 5% risk-free rate and
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—(20% Return 5% Risk-Free Return)/15% Standard Deviation 1.0. It is certainly possible to achieve significant financial objectives over extended time spans with Sharpe Ratios below this benchmark; however, the attractiveness of the returns from a risk-adjusted perspective quite naturally diminishes accordingly. In these cases, the providers of capital
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deviation calculation; and I urge you to evaluate these situations with extreme care. This brings us to the final element of our discussion on the Sharpe Ratio, namely, its limitations, which are largely tied to the accuracy of the standard deviation calculation as a proxy for exposure and to the applicability of
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, portfolio managers who employ these strategies can enjoy stable returns with low volatility for extended periods of time—often ranging into years. The associated high Sharpe Ratio, however, disguises the fact that every now and again, an abrupt market move will cause a substantial loss in these portfolios. When this occurs, we
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exposures and the hazards associated with using historical returns as a proxy for future exposures. For these reasons, and a number of others, although the Sharpe Ratio remains an important benchmark for risk-adjusted return, it is best 68 TRADING RISK used in conjunction with analytics that don’t rely on standard
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800,000 700,000 100,000 600,000 Return Stats Total P/L $ Total Return(% Buy. Pwr.) Avg P/L (Daily $) Avg P/L (Daily %) Sharpe Ratio (Daily) 361,394 12.05% 5,647 0.16% 0.20 19,000 0.63% 297 0.00% 0.00 50,000 619,000 20
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appropriate exposure parameters, which is the essence of Chapter 5, is one that applies many now-recognizable elements of the statistical tool kit, including volatility, Sharpe Ratio, and a number of others. With the statistical tools firmly in place, aimed at both the overall portfolio (Chapter 3) and its various subcomponents (Chapter
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volatility of individual financial instruments and because, as we will see in the next several chapters, it can be used, through a modification of the Sharpe Ratio, as a means of sizing investment levels and volatilities to specific, risk-adjusted return targets. Setting of VaR confidence intervals becomes most relevant when the
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dual objective of reaching our goals without assuming undue exposure. Specifically, the model features a dual methodology that draws on such concepts as Portfolio Volatility, Sharpe Ratio, Targeted Return, and Maximum Acceptable Loss in order to identify the exposure level sufficient to reach our targets while minimizing both the probability and the
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magnitude of loss-threshold violations. Method 1: Inverted Sharpe Ratio We’ll start the process by focusing on the less intuitive task of developing a methodology for setting a sensible lower bound for portfolio exposure
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risk. Most famously, and most relevantly for our purposes, these concepts are embodied in the Sharpe Ratio calculation, which you may recall is a ratio of returns to their associated volatility. If we can estimate our Sharpe Ratio with relative accuracy, we have a pretty good notion of the kind of return we can
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in order to generate the level of performance we desire? 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
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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. It
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risk, which, as argued earlier, is a concept that is expressed most succinctly in the portfolio volatility statistic. Thus, once you have carefully analyzed your Sharpe Ratio and are comfortable that the number you have calculated is one that you can sustain, you can invert this equation to determine what level of
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risk assumption. Note that for the purposes of this discussion, we will designate this figure to be the “Sustainable Sharpe.” It differs from the “actual” Sharpe Ratio derived for selected time sequences by containing a qualitative overlay that bases the figure on a conservative assessment of what you are likely to achieve
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in the future, given the dynamic nature of market conditions. The “actual” Sharpe Ratio is based solely on your trading history. Our “Sustainable Sharpe” is one that uses the historical Sharpe (and perhaps other inputs) to derive a lower
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bound for what we can confidently achieve as a Sharpe Ratio on a going forward basis. The results of this algebraic maneuvering are as follows: 1. (Return Risk-Free Rate) Sustainable Sharpe Portfolio Volatility 2. Portfolio
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what they say about performance. In order to do so, however, we must take a careful look at our options for characterizing volatility. In the Sharpe Ratio calculation, volatility is represented as the standard deviation of portfolio returns. So, whatever statistic you are currently employing as your benchmark exposure measurement, it must
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on current portfolio characteristics. If you have access to a VaR calculation, it is therefore possible to substitute this figure into the denominator of the Sharpe Ratio, as long as you take care to scale down the confidence interval statistic to the one standard deviation level (for example, if you are using
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begs the question of what portfolio volatility is consistent with a 25% return target, again assuming the portfolio in question is able to sustain a Sharpe Ratio of 2.0. We can arrive at this answer quite simply using the second equation by inputting Return, Risk-Free Rate, and Sustainable Sharpe, and
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10% annualized volatility in order to produce returns of 25%. In considering these concepts, it is important to understand that the algebraic manipulations of the Sharpe Ratio that are embodied in the two equations are nothing more than rough approximations of the amount of volatility that is consistent with a given target
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on historical risk-adjusted performance. The algebraic results will never be entirely accurate because, in the first place, the figure you select as your Sustainable Sharpe Ratio will almost certainly deviate from both your historical performance and from your best guess as to what actual Sharpe you can produce in the future
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low single digits, say, 3%, it has very little chance of ever hitting the 25% objective (indeed, it can only do so if the actual Sharpe Ratio is more than triple the estimated Sustainable Sharpe). By the same token, if this account trades to an annualized volatility profile of, say, 30%, the
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of the target or the level of risk assumption, or some combination of the two. The point here is that by combining information about your Sharpe Ratio with expectations of return, you begin to create a picture of what type of risk levels are appropriate to these objectives; volatility levels significantly below
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potential inconsistencies between return targets and risk tolerances. Method 2: Managing Volatility as a Percentage of Trading Capital As you may have surmised, the Inverted Sharpe Ratio method is best adapted to determining the lower bound of exposure that is consistent with reaching your targeted objectives. The main insight that it will
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.10 Cumulative P/L 800,000 700,000 Return Stats Total P/L $ Total Return(% Buy. Pwr.) Average P/L (Daily $) Average P/L (Daily %) Sharpe Ratio (Daily) Daily P/L 150,000 100,000 600,000 50,000 500,000 0 300,000 Volatility Stats Std. Dev. (Daily P/L $) Std
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such a way as to render them consistent with both objectives and constraints. These concepts, which can be thought of as analogues to the Inverted Sharpe Ratio and Percent of Risk Capital methodologies described in Chapter 5, can be used contemporaneously to form a useful (and aesthetically pleasing, for those who like
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feel that the expected return on a given, single transaction ought to be somewhere around zero. By using the Sustainable Sharpe component of the Inverted Sharpe ratio methodology, by contrast, we are basing our return estimates on (1) portfolio-level data (which I have argued in the immediately preceding discussion is more
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our own performance. Moreover, as careful readers of Chapter 5 will recall, the Inverted Sharpe methodology does not call for the use of your actual Sharpe ratio in the setting of exposure parameters, but rather suggests you set this input at a comfort level that you can sustain across most, if not
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all, market conditions. Prudent portfolio managers will set their Sustainable Sharpe ratio inputs at levels below their actual Sharpes so as to render them entirely consistent with an approach that uses past performance data as a means
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–148 optionality, 148–153 position level volatility, 141–142 position size, 134–135 time horizon, 142–144 Exposure range determination: components of, 110–111 inverted Sharpe Ratio, 111–114, 248–250 volatility management, as trading capital percentage, 114–126 Fat tails, 116, 118 Federal Reserve Regulation T, 148 Financial statements, 185, 225
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, as influential factor, 87, 105 In-the-money option, 150 Intraday prices, 39 Intraday P/L, 42 Intraday trading, 162 Intrinsic value, 87, 150 Inverted Sharpe Ratio, 111–114, 248–250 Investment. See Risk Management Investment Investment plan: adjustments to, 218–219 at the margin performance, 236–237 expectations and, 234–236
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, 63 Risk transference, 241 Scenario analysis, 84, 104–106 Scientific method, 5–6 Self-directed traders, 123 Self-funded traders, 110 Serial correlation, 76–78 Sharpe Ratio, 65–68 equation, 65 Inverted, 111–114, 248–250 limitations, 67–68 Sustainable, 112, 249 Short put options, 153 Short selling, 148–149, 152–154
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correlations, 73–79 drawdown, 70–73 historical perspective, 53–56 257 median P/L, 68 percentage (%) of winning days, 68–69 performance ratio, 68–69 Sharpe Ratio, 64–68, 100 standard deviation, 57–66 winning days vs. losing days, 69–70 Stock index, benchmark, 73–74 Stock market crashes, impact of, 14
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Stop-out level, 20–21, 26–32, 118–119, 122–124, 190, 193, 227, 233–234 Strike price, 149–150 Support level, 107–108 Sustainable Sharpe Ratio, 112, 249–250 Target return(s): nominal, 20, 24–26 optimal, 20–24 Technical analysis, 77, 106–108 10% Rule, 116, 122–123, 249, 251
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 · 19 Oct 2017 · 416pp · 106,532 words
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
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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. An asset with a negative Sharpe ratio is punishing the investor with negative returns and volatility. Importantly, absolute returns are
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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
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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). The ratio provides a mathematical method to compare
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of bitcoin and the FANG stocks since Facebook’s IPO Data sourced from Bloomberg and CoinDesk SHARPE RATIO Absolute returns and volatility are important in their own right, but when they’re put together they yield the Sharpe ratio, which is an equally important metric for investors to consider. Remember that by dividing the
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absolute returns9 by the volatility, we can calibrate the returns for the risk taken. The higher the Sharpe ratio, the more the asset is compensating investors for the risk. This is an extremely important metric in the context of modern portfolio theory, because while
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an apples-to-apples comparison between cryptoassets and other traditional and alternative assets. 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
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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
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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. Bitcoin compensated investors twice as well for the risk they took than Facebook did and 40 percent
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better than Netflix, its closest contender (see Figure 7.14). Figure 7.14 Sharpe ratio of bitcoin and the FANG stocks since Facebook’s IPO Data sourced from Bloomberg and CoinDesk Bitcoin and the FANG four’s
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Sharpe ratio comparison clearly illustrates the importance of combining solid returns and low volatility. While Facebook’s annual returns were just shy of Amazon’s and better
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7.15, we once again see the relationship between risk and reward playing out as we view bitcoin’s Sharpe ratio every full year from 2011 through 2016. Figure 7.15 Bitcoin’s annual Sharpe ratios since the start of Mt. Gox Data sourced from CoinDesk The year 2014 was the only time bitcoin
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had a negative Sharpe ratio, when it lost 60 percent of its value from the start to the end of the year. Recall that 2014 was the year of bitcoin’
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, 2016 was bitcoin’s best risk-adjusted return year since 2013. Digging into the comparison between 2013 and 2016, it’s remarkable that 2013’s Sharpe ratio was only double that of 2016, even though bitcoin’s returns in 2013 were so much greater, as shown in Figure 7.16. Figure 7
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than that of 2016, it would be reasonable to expect bitcoin in 2013 to have had a Sharpe ratio many times greater than in 2016. However, this is where both daily volatility and the way the Sharpe ratio is calculated come into play.11 First, volatility in 2013 was triple that of 2016, which
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much lower returns but still have a risk-reward ratio within the same ballpark as 2013. Second, the Sharpe ratio is calculated using average weekly returns, not total capital appreciation over the year. The Sharpe ratio is also revealing when comparing bitcoin to the broader market indices of the S&P 500, the DJIA
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around for a long time and are relatively steady when compared with fast-moving tech names. Figure 7.17 shows a comparison of bitcoin’s Sharpe ratio to the aforementioned three broad market indices, using the same period that we used for comparing the absolute returns of these assets: July 19, 2010
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through January 3, 2017. Figure 7.17 Bitcoin’s Sharpe ratio compared to major U.S. stock indices since the start of Mt. Gox Data sourced from Bloomberg and CoinDesk Once again, this chart reveals how
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absolute returns are tempered by volatility when calculating the Sharpe ratio. Although bitcoin’s Sharpe ratio is roughly 60 percent higher than the three broad market indices, this is a far cry from its absolute returns, which were roughly
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20 times greater than the broad market indices on an annual basis during the same period. In Figure 7.18 we compare bitcoin’s Sharpe ratio in 2016 to that of the broad market indices. Because 2016 was bitcoin’s lowest year of volatility (in the range of a small- to
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mid-cap stock), it is the most appropriate period to compare it to equities. What’s most surprising is bitcoin’s Sharpe ratio in 2016 was almost as high as its overall Sharpe ratio since the launch of Mt. Gox, the first exchange that gave mainstream investors access to bitcoin (1.65 for 2016
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vs. 1.66 since Mt. Gox). Figure 7.18 Bitcoin’s Sharpe ratio compared to major U.S. stock indices in 2016 Data sourced from Bloomberg and CoinDesk Some people are apt to think that the best years
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to be a bitcoin investor are past. However, looking at the Sharpe Ratio, 2016 had risk-adjusted returns that were as good as those of an investor who bought bitcoin when the mainstream first had the opportunity to
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to provide superior absolute returns to improve the risk-reward ratio of the overall portfolio. Since the Sharpe ratio is returns divided by risk, if the risk gets smaller, then the denominator gets smaller, making the Sharpe ratio bigger. The returns don’t have to change at all. However, it is possible for an
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to bitcoin and volatility was 4 percent higher. In this case the volatility was worth it, because the bitcoin portfolio had a 22 percent greater Sharpe Ratio, offering more return for the risk taken (note that comparison calculations in the text were made using unrounded numbers, while the tables show rounded numbers
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reasonable to expect that even a 1 percent allocation to bitcoin would put a drag on the returns of the portfolio and also lower the Sharpe ratio. However, here is where the power of rebalancing and dollar cost averaging would have come into play. An investor would have endured one year of
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. More surprisingly, the portfolio with bitcoin would have had lower volatility! The power of diversification is becoming evident, and it leads to a marginally superior Sharpe ratio for the investor who held bitcoin as a 1 percent position in his or her portfolio during this period (see Figure 7.22). Figure 7
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portfolio with a 1 percent allocation of bitcoin would have been less volatile, while improving compound annual returns by 0.6 percent, ultimately yielding a Sharpe ratio 14 percent better. Operating in the wild, innovative investors would have experienced the joy of a golden asset that both decreased volatility and increased returns
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when added to their portfolio, providing a double boost to the Sharpe ratio. Figure 7.23 Comparative performance of a two-year portfolio with and without a 1 percent allocation of Bitcoin Data sourced from Bloomberg and CoinDesk
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over a series of days, weeks, or months. As these assets mature and their volatility decreases, recall that this can help boost the Sharpe ratio. Recall that since the Sharpe ratio is absolute returns13 divided by volatility, if volatility comes down, then the returns don’t have to be as stupendously good for the
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Sharpe ratio to still be a standout. MARKETPLACE BEHAVIOR: CORRELATIONS As an asset class is first emerging, it will be uncorrelated with the broader capital markets because
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of, 85, 87–90 Ponzi myth of, 158 prices of, 83–84, 94, 118, 127, 145–148 Satoshi and, 3–4, 7–9, 36, 173 Sharpe ratio, 97–98, 100 SMA for, 208 smart contracts on, 53–54 software as, 11 supply of, 36–38, 179 transactions and, 122, 204, 301n24 venture
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Employee Pension Plan Services, 18 by companies, 113, 216 decentralization of, 13 geography and, 220 OTC as, 216–218 Shapeshift, 228 Sharpe, William F., 73 Sharpe ratio, 73, 97–101 Silbert, Barry, 231 Silicon, 116 Silicon Valley, 247 the Silk Road, 22, 23 Silver, 166 Simons, James, 77 Simple mail transfer protocol
Market Risk Analysis, Quantitative Methods in Finance
by Carol Alexander · 2 Jan 2007 · 320pp · 33,385 words
I.6.5.1 CAPM RAPMs I.6.5.2 Making Decisions Using the Sharpe Ratio I.6.5.3 Adjusting the Sharpe Ratio for Autocorrelation I.6.5.4 Adjusting the Sharpe Ratio for Higher Moments I.6.5.5 Generalized Sharpe Ratio I.6.5.6 Kappa Indices, Omega and Sortino Ratio Summary and Conclusions 232
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returns) I.6.2 Two investments (utility of outcomes) I.6.3 Returns characteristics for two portfolios I.6.4 Two investments I.6.5 Sharpe ratio and weak stochastic dominance I.6.6 Returns on an actively managed fund and its benchmark I.6.7 Statistics on excess returns I.6
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.8 Sharpe ratios and adjusted Sharpe ratios I.6.9 Kappa indices 154 154 164 167 205 205 208 221 227 228 237 258 259 261 262 262 264 List of
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formula for minimum variance portfolio The Markowitz problem Minimum variance portfolio with many constraints The CML equation Stochastic dominance and the Sharpe ratio Adjusting a Sharpe ratio for autocorrelation Adjusted Sharpe ratio Computing a generalized Sharpe ratio Omega, Sortino and kappa indices 196 198 208 212 215 222 227 228 235 236 241 242 244 245 246 252
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that is generated. What are the characteristics of P&L that we desire? For trading strategies we may look for strategies that produce a high Sharpe ratio, or that maximize some other risk adjusted performance measure, as outlined in Section I.6.5. But for a pure hedging strategy we should seek
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rate to invest more in P than we have in our funds. Introduction to Portfolio Theory 251 The in (I.6.43) is called the Sharpe ratio. It follows that the expected return on any new portfolio that is a combination of the portfolio P and the risk free asset is on
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allocation is called the market portfolio, and we denote it M. Efficient portfolios with riskless lending and borrowing Expected Return M P Rf Slope is Sharpe ratio for portfolio P StDev Figure I.6.10 Market portfolio Some investors may choose to borrow at the risk free rate and others may choose
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CML is equal to the risk free rate of return, i.e. 0.05 in this example, and the slope of the CML is the Sharpe ratio (I.6.43) for the market portfolio, i.e. 01 − 005 = 025 = 02 Hence the equation of the CML is = 0
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This section describes the risk adjusted performance measures that are linked to the capital asset pricing model and its extensions. A fundamental RAPM is the Sharpe ratio, , which applies if the distributions to be compared have the same shape in the region of their expected values. For example, all the distributions that
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are being ranked should be normal, or they should all have a Student t distribution with the same degrees of freedom. The Sharpe ratio is relevant to a portfolio where there is the option to invest in a risk free asset. For instance, it can apply to the decision
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assume that there is unlimited risk free lending and borrowing at the risk free rate Rf . In Section I.6.4.1 we introduced the Sharpe ratio as the slope of the capital market line. It is the excess of the expected return on an asset (or more generally on an investment
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and Dimitriu (2005). Although they are popular, the classical RAPMs have only a limited range of application. For instance, implicit in the use of the Sharpe ratio is the assumption that investors’ preferences can be represented by the exponential utility and that returns are normally distributed. A logarithmic utility function satisfies the
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an exponential utility function applied to a non-normal probability distribution of returns does not lead to the Sharpe ratio either. I.6.5.2 Making Decisions Using the Sharpe Ratio The stochastic dominance axiom of utility was the fourth of the conditions stated in Section I.6.2.1 for the existence of
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should choose an investment that is weakly dominated by another. Yet we now show that the Sharpe ratio can fail to rank investments according to weak stochastic dominance. Example I.6.11: Stochastic dominance and the Sharpe ratio Consider two portfolios A and B. The distributions of their returns in excess of the risk
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free rate are given in Table I.6.4. Which is the better portfolio? Does the ranking of the portfolios by the Sharpe ratio agree with this preference? Table I.6.4 Two investments Probability 0.1 0.8 0.1 Excess return A Excess return B 20% 10
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excess return from B is 40%. 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
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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
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% 0.7372 The Sharpe ratio does not respect even weak stochastic dominance, and the example given above can
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framework. For this reason they are not good metrics to use as a basis for decisions about uncertain investments. I.6.5.3 Adjusting the Sharpe Ratio for Autocorrelation The risk and return in an RAPM are forecast ex ante using a model for the risk and the expected return. Often both
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far as investors may believe that historical information can tell us something about the future behaviour of financial assets. An ex post estimate of the Sharpe ratio may be taken as R ˆ = (I.6.61) s where R is the sample mean and s is the sample standard deviation of the excess
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.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
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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. Now suppose that daily
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returns have an autocorrelation of 0.2. What is the adjusted Sharpe ratio that accounts for this autocorrelation? Solution We have the ordinary Sharpe ratio, 005% × 250 125% = = 10541 √ 1186% 075% × 250 But calculating the adjustment (I.6.63) to
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.2 in the spreadsheet gives an annualizing factor of 19.35 instead of 250 = 1581 for the standard deviation. This gives an autocorrelation adjusted Sharpe ratio, SR = ASR1 = 125% 005% × 250 = = 08612 075% × 1935 1451% Clearly the adjustment to a
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Sharpe ratio for autocorrelation can be very significant. Hedge funds, for instance, tend to smooth their reported returns and in so doing can induce a strong positive
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in them. 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
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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
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the Sharpe ratio assumes investors are averse not only to a high volatility but also to negative skewness and to positive excess kurtosis. It is estimated as28 ASR2
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= ˆ + ˆ ˆ 2 k̂ ˆ 3 − 6 24 (I.6.64) where ˆ is the estimate of the ordinary Sharpe ratio, or of the Sharpe ratio that has been adjusted for autocorrelation as described above, and ˆ and k̂ are the estimated skewness and excess kurtosis of the returns. The adjustment
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will tend to lower the Sharpe ratio if there is negative skewness and positive excess kurtosis in the returns, which is often the case. 28 The factors 1/6 and 1/24
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Taylor series expansion (I.6.23). Pézier (2008a) derives this formula as an approximation to the generalized Sharpe ratio described in the next section. Introduction to Portfolio Theory 261 Example I.6.13: Adjusted Sharpe ratio The monthly returns and excess returns on an actively managed fund and its benchmark are given in Table
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I.6.6. Calculate the Sharpe ratios of both the fund and the benchmark and the adjustments for autocorrelation and higher moments described above. Table I.6.6 Returns on an actively
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% –0.742 1.808 –0.220 12.28% 4.21% 16.76% –0.573 1.754 –0.218 11.57% Now we compute the ordinary Sharpe ratio and its two adjustments based on (I.6.63) and (I.6.64). The results are shown in Table I.6.8. There is a
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on the fund and on the benchmark. Adjusting for this autocorrelation leads to a large increase in the Sharpe ratio, so ASR1 is greater than the ordinary SR. Table I.6.8 Sharpe ratios and adjusted Sharpe ratios SR ASR1 ASR2 Fund Benchmark 02747 03984 03884 02510 03634 03561 Both
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ASR1 . However, in this example the most significant adjustment to the Sharpe ratios stems from the autocorrelation in excess returns. I.6.5.5 Generalized Sharpe Ratio Another extension of the Sharpe ratio to the case where returns are not normally distributed is the generalized Sharpe ratio introduced by Hodges (1997). This assumes investors have an exponential utility
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serve to decrease his utility. The utility gained at that point defines the maximum expected utility, EU∗ , and we use this to define the generalized Sharpe ratio of the portfolio as GSR = −2 ln−EU∗ 1/2 (I.6.65) Introduction to Portfolio Theory 263 Although this performance measure has nice properties
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using the Excel Solver, and then we set the maximum CE equal to the maximum expected utility in (I.6.65) to find the generalized Sharpe ratio. Pézier (2008a) approximates the value of q that maximizes (I.6.66) as q∗ ≈ 2 and shows for this choice of q that the maximum
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1 3 1 3 (I.6.67) − + k EU∗ ≈ − exp − 2 6 24 But / = , the ordinary Sharpe ratio. Hence, 1 1 −2 ln −EU∗ ≈ 2 + 3 − k4 (I.6.68) 3 12 Note the generalized Sharpe ratio corresponding to the exponential utility is the square root of (I.6.68), i.e. for
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− k4 (I.6.69) 3 12 Thus, when the utility function is exponential and the returns are normally distributed the generalized Sharpe ratio is identical to the ordinary Sharpe ratio. Otherwise a negative skewness and high positive kurtosis will have the effect of reducing the GSR relative to the ordinary SR. Example I
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.6.14: Computing a generalized Sharpe ratio Assuming the investor has an exponential utility function, approximate the generalized Sharpe ratio for the fund and the benchmark of Example I.6.13. Solution Continuing the previous example, the spreadsheet computes
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for the benchmark it is 0.3560. Note that these are almost exactly the same as the results in Table I.6.8 for the Sharpe ratio adjusted for skewness and excess kurtosis. Indeed, Pézier (2008a) shows that (I.6.69) is approximately equal to (I.6.64). I.6.5.6
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adjusted performance measures to recommend allocations to investors. The classical risk adjusted performance measure is the Sharpe ratio. This measure is only appropriate under restrictive assumptions about portfolio returns characteristics and the investor’s preferences. However, the Sharpe ratio has a number of simple and approximate extensions: for instance, to accommodate autocorrelation, skewness and
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stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327–343. Hodges, S.D. (1997) A generalisation of the Sharpe ratio and its applications to valuation bounds and risk measures. Financial Options Research Centre Working Paper, University of Warwick. Jagannathan, R. and Wang, Z. (1996) The
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137–8 Assets, tradable 1 Asymptotic mean integrated square error 107 Asymptotic properties of OLS estimators 156 Autocorrelation 175–9, 184, 259–62 Autocorrelation adjusted Sharpe ratio 259–62 Autoregression 135 Auxiliary regression 177 Backtesting 183 Backward difference operator 19 Bandwidth, kernel 106–7 Bank 225 Barra model 181 Basic calculus 1
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–Markov theorem 157, 175, 184 Generalized extreme value (GEV) distribution 101–3 Generalized least squares (GLS) 178–9 Generalized Pareto distribution 101, 103–5 Generalized Sharpe ratio 262–3 General linear model, regression 161–2 Geometric Brownian motion 21–2 lognormal asset price distribution 213–14 SDE 134 stochastic process 141 time
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likelihood estimation) 72, 130–4, 141, 202–3 Modified duration 2 Modified Newton method 192–3 Moments probability distribution 78–83, 140 sample 82–3 Sharpe ratio 260–3 Monotonic function 13–14, 35 Monte Carlo simulation 129, 217–22 correlated simulation 220–2 empirical distribution 217–18 random numbers 217 time
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–7 Ranking investments 256 287 288 Index RAPMs (risk adjusted performance measures) 256–67 CAPM 257–8 kappa indices 263–5 omega statistic 263–5 Sharpe ratio 250–1, 252, 257–63, 267 Sortino ratio 263–5 Realization, random variable 75 Realized variance 182 Rebalancing of portfolio 17–18, 26, 248–9
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–7 systematic risk 181, 250, 252 Risk adjusted performance measure (RAPM) 256–67 CAPM 257–8, 266 kappa indices 263–5 omega statistic 263–5 Sharpe ratio 251, 252, 257–63, 267 Sortino ratio 263–5 Risk averse investor 248 Risk aversion coefficients 231–4, 237 Risk factor sensitivities 33 Risk free
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plot 112–13, 144–5 SDE (stochastic differential equation) 136 Security market line (SML) 253–4 Self-financing portfolio 18 Sensitivities 1–2, 33–4 Sharpe ratio 257–63, 267 autocorrelation adjusted 259–62 CML 251, 252 generalized 262–3 higher moment adjusted 260–2 making decision 258 stochastic dominance 258–9
Expected Returns: An Investor's Guide to Harvesting Market Rewards
by Antti Ilmanen · 4 Apr 2011 · 1,088pp · 228,743 words
Walk Hypothesis S Spot price, spot rate SBRP Survey-based Bond Risk Premium SDF Stochastic Discount Factor SMB Small Minus Big, size premium proxy SR Sharpe Ratio SWF Sovereign Wealth Fund TED Treasury–Eurodollar (deposit) rate spread in money markets TIPS Treasury Inflation-Protected Securities, real bonds UIP Uncovered Interest Parity (
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returns and various biases make standard statistical inferences inappropriate. I prefer to assess the consistency of a return series “in investment language” by reporting the Sharpe ratio (which is closely related to statistical significance) and by displaying cumulative performance graphs that I find particularly informative for experienced users. [8] Personally, I
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measured volatility and their correlations with other assets. Both positive illiquidity premia and the understatement of risk in the denominator of the Sharpe ratio calculation imply that we should observe higher Sharpe ratios for less liquid assets. Figure 2.2 confirms such a positive relation, based on my subjective estimates of illiquidity (with scale
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Band of America Merrill Lynch, S&P GSCI, MIT-CRE, FTSE, Global Property Research, UBS, NCREIF, Hedge Fund Research, Cambridge Associates. Figure 2.2. Scatterplotting Sharpe ratios on illiquidity estimates. Sources: Bloomberg, MSCI Barra, Ken French’s website, Citigroup, Barclays Capital, J.P. Morgan, Band of America Merrill Lynch, S&P GSCI
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relations can be highly sample-specific. Sources: Bloomberg, MSCI Barra, Ken French’s website, Citigroup, Dimson–Marsh–Staunton (2010). Figure 2.4. Many long-run Sharpe ratios plummeted in 2008. Sources: Bloomberg, MSCI Barra, Ken French’s website, Citigroup, Barclays Capital, MIT-CRE, S&P GSCI, Hedge Fund Research, Cambridge Associates,
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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
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. Moreover, when investors expect improving macroeconomic conditions they tend to expect higher equity returns and lower volatility; this result provides direct evidence of procyclical expected Sharpe ratios. Amromin and Sharpe (2009) argue that these results lend support to behavioral explanations related to representativeness: optimism regarding the macroeconomy translates (too) directly into
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return predictability reflects the high required equity risk premium during bad times when investors’ risk aversion is especially high. Many studies document countercyclical ex ante Sharpe ratios based on valuation ratios or yield curve steepness and then assume that such predictability reflects investors’ rational risk assessments and preferences. While I firmly believe
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an important real-world phenomenon, I also believe that psychological factors exacerbate boom–bust cycles. At least retail investors’ subjective return expectations exhibit procyclical expected Sharpe ratios in a manner consistent with behavioral sentiment stories. Risk. Looking at various risk indicators, the relation between equity market volatility and subsequent market returns is
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yield spread volatilities tend to be lower at short maturities, at least for highly rated debt. Together these features explain high ex ante Sharpe ratios—and lesser ex post Sharpe ratios. Only an exceptionally dramatic spread widening (or actual default) can make short-dated carry trades underperform ex post, and since the 1980s this
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high-grade spread curves, and (b) corresponding break-even spread-widening cushions. Source: Citigroup, own calculations. Why has this opportunity—with persistently high ex ante Sharpe ratios—not been arbitraged away? The above analysis ignores funding rate spreads that would make carry positions less attractive. Financing rates have limited impact for long
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overweighting positive-roll commodities, and extending futures maturity to longer (deferred) contracts from the most liquid front contracts. All approaches appear to boost returns and Sharpe ratios but they also involve higher trading costs (and, likely, fees). 11.4 HEDGE FUNDS 11.4.1 Introduction Hedge funds (HFs) are pools of
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on biases and risks Negative stories come in two forms: biases and risks. First, recall that all reporting to HF indices is voluntary. The high Sharpe ratios discussed earlier partly reflect various reporting biases (survivor, backfill, etc.) that almost surely result in overstated reported returns and understated risk measures. FoF numbers
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of HFR index style subsets between 1990 and 2009. Most styles earned double-digit returns and all styles except short-selling equities achieved a high Sharpe ratio. Various reporting biases, together with standard concerns about sample-specific findings, make it unclear whether this graph tells us anything useful about future returns.
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(The graph also shows stock market correlations which were positive for most style subsets. Short-sellers are the exception—this group’s low Sharpe ratio is consistent with its superior diversification ability.) Limited data hint at relative mean reversion tendencies across sectors: if a style is successful over multiple years
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expected returns—occasional tailwinds for each sector—but fund access and liquidity issues make tactical allocation difficult in practice. Figure 11.11. Average returns and Sharpe ratios of HFR hedge fund sector subindices, 1990–2009. Sources: Bloomberg, Hedge Fund Research. There is limited literature on time-varying expected returns for HFs.
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quality of HFs has deteriorated and that this too has reduced industry profitability. Yet others stress that institutional investors required lower volatility and accepted lower Sharpe ratios than traditional HF investors, making single-digit returns and volatilities the new norm.) The capacity question can be raised for HFs overall and for specific
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Value-vs.-growth cumulative outperformance since 1926 (U.S.)/1975 (non-U.S.). Sources: Ken French’s website, Bloomberg. Figure 12.2. Sixty-month rolling Sharpe ratios of value growth long–short portfolios. Sources: Ken French’s website, Bloomberg. The value premium has been especially consistent after the Great Depression. Indeed, the
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ante value indicator is available, multi-year average return is often a reasonable substitute. The consistency of the results is impressive: all strategies have positive Sharpe ratios, ranging between 0.1 and 0.9. Value strategies worked especially well for stock selection in Japan and for equity country allocation. Other studies
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after rising prices. The complementary nature (negative correlation) of value and momentum strategies means that combining the two can sharply reduce portfolio volatility and boost Sharpe ratios:• One of my favourite clichés is “when value and momentum clash, value almost always loses first.” This aphorism suggests patience when responding to value signals
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been embarrassingly successful for over 50 years. In the past 15 years they have fared even better in emerging markets. Despite recent losses, long-run Sharpe ratios for currency carry strategies are higher than for equities, fixed income, or credit. The embarrassment partly reflects the sheer naivety of these strategies. Academics
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the SR from 0.61 to 0.70. Dynamic portfolio construction. Scaling position sizes inversely by recent historical or option-implied volatility tends to improve Sharpe ratios. Using mean variance optimization in sizing carry positions is another potential approach, but its chief benefit is to take advantage of such volatility normalization.
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really bad times. This class of trading strategies performs especially well in prolonged good times. Persistent success in turn attracts return-chasing inflows, which improves Sharpe ratios further and results in overvaluation and overcrowding. Until that rare disaster materializes, many investors underestimate the risk and extrapolate from past performance (or hope that
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1.4. Figure 14.2. Cumulative excess returns of trend-following strategies, 1993–2009. Sources: Bloomberg, Brevan Howard, own calculations. Figure 14.3. Contrasting Sharpe ratios for trading strategies that buy past winners and sell recent losers based on recent and intermediate momentum (in four equity market contexts and in two
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indices provided historically attractive payoffs for two decades from the crash of October 1987 until the fall of 2008—with high but misleading long-run Sharpe ratios. Selling volatility on other asset classes has also earned reasonably consistent profits, whereas it is less clear that selling volatility on individual equities is
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even worse for the “pure” volatility arbitrage strategy than for covered-option-writing strategies. Over 1989–2009, the latter still display higher returns and Sharpe ratios than a simple long-equities strategy (first column), but this advantage comes with less appealing tail risk or higher moment exposures: more negative skewness and
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probability of positive returns so that they can accumulate large fees before a rare disaster materializes, especially since overstated smoothness of returns can boost the Sharpe ratio and attract larger AUM until the blowup occurs. Main challenges and main explanations The three best historical trading opportunities and thus the three main empirical
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any remaining opportunities difficult. Specifically, Broadie–Chernov–Johannes (2009) argue that options are often thought to be mispriced because the performance metrics that are used (Sharpe ratios and CAPM alphas) are ill-suited for option analysis, especially over short samples. After documenting the huge challenge for rational models—massively negative average returns
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of risk, depending on the covariance of losses with bad times. Since volatility is an insufficient risk measure, reasonable investors may accept a lower portfolio Sharpe ratio if they can trade it off against better diversification characteristics, more attractive asymmetry, higher liquidity, and lesser tail risks. These themes are covered through this
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in which a subset of investors, whom they call “lotto investors”, display a direct preference for positive skewness. These lotto investors are willing to sacrifice Sharpe ratio (mean variance efficiency) to get higher skewness. They are deliberately underdiversified because diversification reduces the skewness they seek. The authors document that a meaningful segment
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the evolution of a profitable investment strategy—say, FX carry or harvesting illiquidity premia. Figure 20.2 shows a stylized lifecycle of the ex ante Sharpe ratio. Once the strategy’s past success attracts widespread attention, it triggers profit-seeking inflows through extrapolation or learning. Inflows may temporarily boost profitability but growing
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• After an overview, this chapter drills into each path and also contains essays on related topics such as rebalancing and diversification return, looking beyond the Sharpe ratio, and time diversification. 28.1 INTRODUCTION: HOW CAN INVESTORS ENHANCE RETURNS? It is time to start drawing the investment lessons from the preceding 27 chapters
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window. Gains from risk parity investing would rise with a lower stock–bond correlation, lower relative volatility between bonds and stocks, and a higher relative Sharpe ratio between them. Broadening into several moderately correlated asset classes would improve diversification and the portfolio’s SR faster, even if the generally high correlations of
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(0.30) than does the strategy style composite (—0.05). Thanks to its superior diversification, the strategy style composite has a double Sharpe ratio compared with its constituents’ average Sharpe ratio (1.2 vs. 0.6). This simply reflects the fact that with four uncorrelated strategies, portfolio volatility is about half the constituents’
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the fundamental law of active management (see Section 28.2.4). In contrast, the asset class composite sees a much more modest volatility reduction and Sharpe ratio increase. This example focuses on diversification and deliberately steers away from average returns (which happen to be higher for the strategy style composite). Performance differences
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available at SSRN: http://ssrn.com/ abstract = 1650232. Amromin, Gene; and Steven A. Sharpe (2009), “Expectations of risk and return among household investors: Are their Sharpe ratios countercyclical?” working paper, available at SSRN: http://ssrn.com/abstract=1327134 Ang, Andrew; Turan G. Bali; and Nusret Cakici (2010), “The joint cross section of
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rates foreign exchange forward-looking indicators forward-looking measures generic proxy role historical returns long-horizon investors non-zero yield spreads real asset investing roll Sharpe ratios 2008 slide tactical forecasting cash, ERPC cash flow catastrophes see also “bad times” CAY see consumption/wealth ratio CCW see covered call writing CDOs
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facts FoFs leverage managers negative factors neglected risks outperformance performance assessment positive factors prospective returns replication strategies return biases returns 1984—2009 risks sector performance Sharpe ratios 2008 skilled investors tail risks voluntary reporting hedged global government bonds hedging currency HFR index HFs see hedge funds high-yield (HY) bonds hindsight
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tail risks riskless returns risk management arithmetic mean bearing more risk diversification return enhancing returns geometric mean monetizing risk reduction rebalancing risk adjustment risk reduction Sharpe ratios smart risk taking volatility targeting risk-neutral default probabilities risk preferences risk premia growth return predictability risk premium hypothesis volatility see also bond risk premium
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reversible sustainable ten trends securities see also TIPS selection bias see also data mining; overfitting self-attribution self-deception selling volatility sentiment indicators share buybacks Sharpe ratios (SRs) active investing BAB balancing stock—bond portfolios BRP commodity momentum CRP currency carry emerging markets feedback loops fixed income front-end trading hedge funds
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Treasuries spot prices spreads funding rate industry-adjusted value spread non-zero yield OASs “spread products” swap Treasury see also credit spreads 8, SRs see Sharpe Ratios stagflation stagnation standalone skewness state regulation static strategies status seeking Staunton, Mike stochastic discount factor (SDF) stock markets stocks balanced portfolios Chen three-factor model
The New Science of Asset Allocation: Risk Management in a Multi-Asset World
by Thomas Schneeweis, Garry B. Crowder and Hossein Kazemi · 8 Mar 2010 · 317pp · 106,130 words
Allocation in the Modern World Product Development: Yesterday, Today, and Tomorrow Notes CHAPTER 2 Measuring Risk What Is Risk? Traditional Approaches to Risk Measurement Classic Sharpe Ratio Other Measures of Risk Assessment Portfolio Risk Measures Other Measures of Portfolio Risk Measurement Value at Risk Notes CHAPTER 3 Alpha and Beta, and the
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more complete and dynamic process of risk estimation and return determination would more adequately describe the expected return and risk tradeoff. 7. For example, the Sharpe Ratio, defined as: Si = (Ri − Rf ) σi was meant to provide evidence of the relative benefit of two efficient risky portfolios on the capital market line
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and became the performance measurement vehicle of choice. Note that the Sharpe Ratio for an individual asset or portfolio merely provides evidence of the number of standard deviations the mean return of a portfolio/asset is from the
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overwhelms any individual approach if for the simple reason that there are too many individuals, each with their own unique set of investment concerns. CLASSIC SHARPE RATIO For much of this and the previous chapter we have emphasized the wide range of risks involved in asset allocation and security return estimation; however
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, for many, when the choice is between two (or more) assets, one way of ranking investments (the Sharpe Ratio) is based on simplifying risk into a single parameter (e.g., standard deviation). This ratio essentially divides the return of the security (after first subtracting
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price risk (standard deviation of return) of the security. 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
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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
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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. There is simply too much randomness in the price movement of individual
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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
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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. The
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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
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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
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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. Investment managers employ a number of tactics
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to improve their measured Sharpe Ratio. For most asset classes, increasing the time interval used to measure standard deviation will result in a lower estimate of volatility. For example, the annualized
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the annual return measure is derived by compounding the monthly returns, but the standard deviation estimate is calculated from the (not compounded) monthly returns, the Sharpe Ratio will be upwardly biased. Options change the return distribution. Rather than approximating a normal distribution, options produce skewed, kurtotic, or leptokurtotic return distributions, depending on
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years without paying off once, and a 50% chance of surviving five years. If the manager is lucky, this strategy will show a significantly higher Sharpe Ratio, as the premiums flow directly to the bottom line with no apparent increase in volatility. Strategies that involve taking on default risk, liquidity risk, or
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other forms of catastrophe risk have the same ability to report an upwardly biased Sharpe Ratio. 28 ■ THE NEW SCIENCE OF ASSET ALLOCATION Smoothing is also a source of potential bias.4 Some illiquid investments are priced using models, which can
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that understate monthly gains or losses, thereby reducing reported volatility. OTHER MEASURES OF RISK ASSESSMENT Security and Asset Risk Measurement One potential disadvantage of the Sharpe Ratio measure is that even if it is used to compare similar asset class portfolios it may not provide a reasonable basis for comparison when portfolios
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exist within a multi-asset class portfolio (e.g., commodities, stocks, bonds, private equity) since the Sharpe Ratio is based on a portfolio’s stand-alone variance, and not its covariance with other assets that are included in a multi-asset portfolio. Another
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measure is generally measured as: Ti = (E (Ri ) − Rf ) βi As a consequence, the Treynor measure addresses one of the drawbacks mentioned earlier regarding the Sharpe Ratio. The Treynor measure works well when adding assets to a multi-asset market portfolio as the betas of the assets can be used as a
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the risk, the strategy can be implemented directly using options. 5. See Treynor (1965). 6. For purposes of clarity, theoretical and empirical issues in the Sharpe Ratio were well known by most individuals from the very start. However, as is often pointed out it is better to light a candle than curse
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and that investors do not demand a premium for volatility. EXHIBIT 3.1 Alpha Determination: Alternative Risk-Adjusted Benchmarks Alpha Benchmark Model T-Bill CAPM Sharpe Ratio Multi-Factor Alpha Determination E(Ri) − Rf α = (E(Ri) − Rf) − β(E(Rm) − Rf) E (Rm ) − Rf × σ i ⎞⎟ α = ( E (Ri ) − Rf ) − ⎛⎜ ⎝ ⎠ σm
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based benchmark have already been discussed. It assumes that market risk is the only relevant source of risk. The Sharpe Ratio based model assumes in part that there exists a known market portfolio Sharpe Ratio (e.g., .70). In the case of multi-factor models, identifying the factors is the most serious problem. More
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(Rp ) − Rf ⎞ EBK = Rc − ⎢⎜ ⎟⎠ ( ρcp ) σ c + Rf ⎥ ⎝ σ p ⎦ ⎣ where E(Rc) = Break-even rate of return required for the asset to improve the Sharpe Ratio of alternative index p Rc = Rate of return on asset c Rf = Riskless rate of return E(Rp) = Rate of return on alternative index p
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. Some investment managers emphasize the non-correlation of their strategy with the S&P 500 and then turn around and offer a comparison of their Sharpe Ratio with that of the S&P 500 to indicate superior alpha performance. Even in this case, the comparison will not indicate its potential alpha benefit
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, active manager portfolios; nor does it provide an indication of whether another like investment will have produced a similar or even greater increase in the Sharpe Ratio of the newly constructed portfolio. In short, the ability of a manager to achieve alpha is based on the ability to achieve a return via
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or from the selection of assets with nonlinear payoffs relative to market, may deliver improved risk-adjusted returns. NOTES 1. There is extensive literature on Sharpe Ratios and alternative relative risk comparison measures (e.g., the Jensen and the Treynor indices). See Bodie, Kane, and Marcus (2008). There have also been additional
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2 - 0.500 1.000 MP 2 MP 3 MP 1 9.00% 8.80% 8.60% 8.40% 8.20% 8.00% 7.80% Sharpe Ratio Standard Dev. Max Drawdown 72 73 EXHIBIT 4.8 MP 1 MP 2 MP 3 1.00 0.80 0.60 0.40 0.20
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High Grade Corp. Russell 1000 Rusell 2000 MSCI EAFE MSCI Emerging Mkt HFRI Distressed HFRI Equity Long Short Total Weights Annual Return Annual Std Dev Sharpe Ratio Correlation: Original Port. Traditional Portfolio Portfolio with HFRI Distressed 5.0% 5.0% 5.0% 10.0% 5.0% 10.0% 40.0% 10.0
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competitive funds would be similar. PERFORMANCE ANALYTICS PROVIDE A COMPLETE MEANS TO DETERMINE BETTER PERFORMING MANAGERS No investment package is complete without the inclusion of Sharpe ratios, information ratios, beta comparisons, and so on. However, while the analytical comparison of like managers forms the basis of manager selection, one must be reminded
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that for comparison purposes, an asset’s Sharpe ratio tells us little as to the marginal risk that an asset adds to a portfolio, and beta estimation is fraught with error (e.g., which
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, 155 Security market line, 6 Seed capital, 153 Selection bias, 192 Semi-standard deviation, 97 Semi-variance, 30 Sensitivity, market, 74–82, 82–84, 89 Sharpe Ratio, 18, 26–28, 37, 43 Skewness, 29, 62, 97, 223 Slope of the yield curve, 101 Small minus big (SMB) factor, 45 Smoothing, 28, 175
Triumph of the Optimists: 101 Years of Global Investment Returns
by Elroy Dimson, Paul Marsh and Mike Staunton · 3 Feb 2002 · 353pp · 148,895 words
. These two indexes correspond to a policy of diversifying across all sixteen countries in proportion to their size. In section 8.3, we use the Sharpe ratio, which measures the reward per unit of risk, to analyze whether investors in the United States and elsewhere, would, with hindsight, have been better off
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-US assets. Did these lower returns more than offset the lower risk? 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
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. 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
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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. From 1900–2000, US real equity returns had a standard deviation
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the complexities of multiperiod investment (see Sharpe, 1994). 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
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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
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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
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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
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index in real, local currency terms from the perspective of investors from each of the other fifteen countries. 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
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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
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, 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. Countries are ranked from lowest to highest ratio over the period 1900–2000, and ratios are also shown for 1950–2000. Figure
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8-5: Ex post gains from holding the world equity portfolio relative to domestic investment Relative Sharpe ratio (Sharpe ratio world index/Sharpe ratio domestic equities) 2.5 2.4 1900–2000 2.3 1950–2000 2.0 1.9 1.8 1.8 1.5 1.4
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., 1964, Capital asset prices: A theory of market equilibrium under the condition of risk. Journal of Finance 19: 425–42 Sharpe, W.F., 1994, The Sharpe ratio. Journal of Portfolio Management 21(1): 49–58 Shiller, R.J., 1981, Do stock prices move too much to be justified by subsequent changes in
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Regulated businesses, 216–7 Regularities, see anomalies Regulation, 3, 18, 163, 186, 216, 217 Reid, K., 141 Reinganum, M., 131 Reward-to-risk ratio, see Sharpe ratio Repurchases, 143, 149, 158–63, 177, 191 Risager, O., 244 Risk, 54–62 see also currency risk, default risk, market risk, portfolio risk, risk premium
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–28, 35, 36, 37, 138, 188, 299 Sell-in-May, 135, 138, September 11th 2001, 47, 58, 117, 168, 177, 178, 213 Shares, see equities Sharpe ratio, 105, 111–4, 208 Sharpe, K.P., 239 Sharpe, W.F., xi, 105, 111, 112, 113, 145, 180 Shell, 29 Shiller, R.J., 84, 158
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