value at risk Concepts

Commodity Trading Advisors: Risk, Performance Analysis, and Selection

by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant and Fabrice Douglas Rouah · 23 Sep 2004

-Enhancing Diversification 336 David Kuo Chuen Lee, Francis Koh, and Kok Fai Phoon CHAPTER 20 Incorporating CTAs into the Asset Allocation Process: A Mean-Modified Value at Risk Framework Maher Kooli 358 viii CONTENTS CHAPTER 21 ARMA Modeling of CTA Returns 367 Vassilios N. Karavas and L. Joe Moffitt CHAPTER 22 Risk-Adjusted

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stock and bond indices. Concerning extreme risks, the Edhec CTA Index is closer to the bond index than to the stock index with a modified value at risk (VaR) (also referred to as Cornish Fisher VaR4) of −6.52 percent as opposed to −13.49 4cf. Favre and Galeano (2002b) for more details on

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skill. Note that coskewness remains irrelevant if it can be diversified away, but skewness may have some signaling value. Additionally, the popularity of the related value at risk (VaR) measure11 and the common practice of reporting drawdown12 information for various alternative investments suggest that skewness may be important, whether in terms of investor utility

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option strategy. Using a two-step regression procedure, the authors document the asymmetric return stream associated with CTAs and then provide a method for calculating value at risk. The authors also examine a passive trend-following commodity index and find it to have a similar put optionlike return distribution. The authors also demonstrate

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how commodity trading programs can be combined with other hedge fund strategies to produce a return stream that has significantly lower value at risk parameters. Chapter 10 examines the relationships between various risk measures for CTAs. The relationships are extremely important in asset allocation. If two measures (e.g

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overview of the managed futures industry. We then measure the long-volatility exposure captured these strategies. Next we apply Monte Carlo simulation to estimate the value at risk for longvolatility strategies. Last, we demonstrate some practical risk management strategies that may be employed with managed futures. BRIEF REVIEW OF THE MANAGED FUTURES INDUSTRY

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long volatility exposure of a long put option as well as the premium payment when all performs well. Our next step is to provide some Value at Risk analysis. 194 RISK AND MANAGED FUTURES INVESTING Diversified Excess Returns 15.00% 10.00% 5.00% 0.00% –5.00% –10.00% –20.00% –15

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to simulate the returns to trend-following strategies for developing risk estimates. Specifically, we can run Monte Carlo simulations with our mimicking portfolios and estimate value at risk (VaR). Armed with these data, we can estimate the probability of the risk of loss associated with long volatility strategies. This is important to help us

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-month VaR for the Barclay Commodity Trading Index is −0.93 percent at a 1 percent confidence level and −0.69 percent at a 5 percent confidence level. This means that we can state with a 99 per- 196 RISK AND MANAGED FUTURES INVESTING TABLE 9.2 Monte Carlo Simulation of Value at Risk

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fund styles to minimize and manage volatility risk. 201 Measuring the Long Volatility Strategies of Managed Futures TABLE 9.3 Monte Carlo Value at Risk 1 Month VaR @ 1% Confidence Level 1 Month VaR @ 5% Confidence Level Maximum Loss Number of Simulations Merger Arbitrage Merger Arbitrage and Managed Futures −6.04000% −3.1500% −3.1400

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risk and return characteristics. Chapter 20 analyzes the risk and return benefits of CTAs, as an alternative investment class. Then it shows, using a modified Value at Risk as a more precise measure of risk, how CTAs can be integrated into existing investment strategies and how we can determine the optimal proportion of

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’s losses do not exceed a client’s comfort level. Risk Measures On a per-strategy basis, it is useful to examine each strategy’s: ■ ■ ■ ■ ■ Value at risk based on recent volatilities and correlations Worst-case loss during normal times Worst-case loss during well-defined eventful periods Incremental contribution to portfolio

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value at risk Incremental contribution to worst-case portfolio event risk The last two measures give an indication if the strategy is a risk reducer or risk enhancer.

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On a portfolio-wide basis, it is useful to examine the portfolio’s: ■ ■ ■ Value at risk based on recent volatilities and correlations Worst-case loss during normal times Worst-case loss during well-defined eventful periods Each measure should be compared

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Fall 1998 bond market debacle Aftermath of 9/11/01 attacks 286 TABLE 15.3 PROGRAM EVALUATION, SELECTION, AND RETURNS Strategy-Level Risk Measures Strategy Value at Risk Worst-Case Loss during Normal Times Worst-Case Loss during Eventful Period Deferred Reverse Soybean Crush Spread 2.78% −1.09% −1.42% Long Deferred

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clients who are investing in a nontraditional investment for diversification benefits. Therefore, in addition to examining a portfolio’s risk based on recent fluctuations using value at risk measures, a manager also should examine how the portfolio would have performed during the eventful times listed in Table 15.2. Tables 15.3 and

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Outright Short Deferred Wheat Spread Long Deferred Gasoline Outright Long Deferred Gasoline vs. Heating Oil Spread Long Deferred Hog Spread Incremental Contribution to a Portfolio Value at Risk Incremental Contribution to Worst-Case Portfolio a Event Risk 0.08% −0.24% 0.17% 0.19% 0.04% 0.02% 0.33% 0.81

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liquidity risk borne may be an important difference between hedge funds and CTAs. CHAPTER 20 Incorporating CTAs into the Asset Allocation Process: A Mean-Modified Value at Risk Framework Maher Kooli alue at risk has become a heavily used risk management tool, and an important approach for setting capital requirements for banks. In

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examine the effect of including a CTA in a traditional portfolio. Using a mean-modified value at risk framework, we examine the case of a Canadian pension fund and compute the optimal portfolio by minimizing the modified value at risk at a given confidence level. V INTRODUCTION For the individual or the institutional investor who is

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trading advisors manage client assets on a discretionary basis using global futures markets as an investment medium. 360 PROGRAM EVALUATION, SELECTION, AND RETURNS MEAN-MODIFIED VALUE AT RISK FRAMEWORK Investment decisions are made to achieve an optimal risk/return trade-off from the available opportunities. To meet this objective, the portfolio manager has

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faced on investments is the same as the perception to the upward potential. Thus, investors needed a more precise measure of downside risk. With the value at risk (VaR) approach, it is possible to measure the amount of portfolio wealth that can be lost over a given period of time with a certain probability

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(1952) has been criticized often due to its utilization of variance as a measure of risk exposure when examining the nonnormal returns of CTAs. The value at risk (VaR) measure for financial risk has become accepted as a better measure for investment firms, large banks, and pension funds. As a result of the recurring

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.cve.com. Christoffersen, P. (2003) Elements of Financial Risk Management. San Diego, CA: Academic Press. Chung, S. Y. (1999) “Portfolio Risk Measurement: A Review of Value at Risk.” Journal of Alternative Investments, Vol. 2, No. 1, pp. 34–42. Clark, P. K. (1973) “A Subordinated Stochastic Process Model with Finite Variance for Speculative

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Performance Using Loess Fit Regression.” Journal of Alternative Investments, Vol. 4, No. 4, pp. 8–24. Favre, L., and J.-A. Galeano. (2002b) “Mean-Modified Value-at-Risk Optimization with Hedge Funds.” Journal of Alternative Investments, Vol. 5, No. 2, pp. 21–25. Favre, L., and A. Singer. (2002) “The Difficulties in Measuring

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. 23, No. 2, pp. 389–416. Jorgensen, R. B. (2003) Individually Managed Accounts: An Investor’s Guide. New York: John Wiley & Sons. Jorion, P. (2001). Value at Risk: The New Benchmark for Managing Financial Risk, Second Edition. New York: McGraw-Hill. Karavas, V. N., and S. Siokos (2003) “The Hedge Fund Indices Universe

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Continuous-Time Model with a Knockout Feature.” Applied Mathematical Finance, Vol. 7, No. 2, pp. 115–125. Rockafellar, R. T., and S. Uryasev. (2001) “Conditional Value-at-Risk for General Loss Distributions.” Research Report, ISE Department, University of Florida, Gainesville, FL. Ross, M. (1999, September 20) “CBOT, IFB Trade Observations.” Farm Week, Bloomington

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volatility strategies, 183–202 demonstration of, 185–188 fitting regression line, 189–191 mimicking portfolios of strategies, 192–195 risk management using, 198–201 and value at risk, 195–198 Losses, investing with CTAs after, 45–47 Macroportfolio hedging, 286–287 Managed Account Reports, 51 Managed futures. See also Commodity trading advisors; specific

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continued and volume–market volatility relationship, 164–181 and volume–price volatility relationship, 161–164 Markowitz, Harry, xxv MAR (minimal accepted return), 86 Mean-modified value at risk framework, 358–366 Mean-variance sufficiency, 85–87 Medium-term CTAs, 80n1 Merger arbitrage, 198–201 Mimicking portfolios, 192–195 Minimal accepted return (MAR), 86

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, see Mount Lucas Management Index MMA (managed money accounts), 155 Modern portfolio theory, xxv Modified Sharpe ratio, 279, 377–384 Modified value at risk, 22–23, 360–361, 379. See also Meanmodified value at risk framework Moment-based efficiency measure, 295–297 MOM (manager of managers), 7 Monte Carlo simulation, 35–37, 195, 196 Mount Lucas

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Risk: for comparative indices, 22–23 CTA characteristics and level of, 43–44 downside, 221–224 interdependence of measures for, 203–219 423 mean-modified value at risk framework, 358–366 performance evaluation and measures of, 82–87 time diversification as hedge against, 385–398 time-varying, 84–85 Risk-adjusted returns, 377

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bonds market, 155–160, 163–165, 169–181 Tremont TASS, 7 Trend-following CTAs, 7, 80, 183–184, 244, 287–288 Treynor ratio, 82 Value at Risk (VaR), 358 mean-modified value at risk framework, 358–366 measuring, 378–379 for mimicking portfolios, 195 modified, 379 Variance ratios, 166–171 Volatility, see Long volatility strategies; Market volatility

Analysis of Financial Time Series

by Ruey S. Tsay · 14 Oct 2001

B. Approximation to Standard Normal Probability, 253 7. Extreme Values, Quantile Estimation, and Value at Risk 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8. 256 Value at Risk, 256 RiskMetrics, 259 An Econometric Approach to VaR Calculation, 262 Quantile Estimation, 267 Extreme Value Theory, 270 An Extreme Value Approach to

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this book include the combination of recent developments in financial econometrics in the econometric and statistical literature. The developments discussed include the timely topics of Value at Risk (VaR), highfrequency data analysis, and Markov Chain Monte Carlo (MCMC) methods. In particular, the book covers some recent results that are yet to appear in

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academic journals; see Chapter 6 on derivative pricing using jump diffusion with closed-form formulas, Chapter 7 on Value at Risk calculation using extreme value theory based on a nonhomogeneous two-dimensional Poisson process, and Chapter 9 on multivariate volatility models with time-varying correlations. MCMC

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tailed distributions, and their application 1 2 FINANCIAL TIME SERIES AND THEIR CHARACTERISTICS to financial risk management. In particular, it discusses various methods for calculating Value at Risk of a financial position. Chapter 8 focuses on multivariate time series analysis and simple multivariate models. It studies the lead-lag relationship between time series

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This volatility evolves over time. Volatility is also important in risk management. As discussed in Chapter 7, volatility modeling provides a simple approach to calculating value at risk of a financial position. Finally, modeling the volatility of a time series can improve the efficiency in parameter estimation and the accuracy in interval forecast

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(1) for all forecast horizons. This special IGARCH(1, 1) model is the volatility model used in RiskMetrics, which is an approach for calculating Value at Risk; see Chapter 7. 3.6 THE GARCH-M MODEL In finance, the return of a security may depend on its volatility. To model such a

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of Financial Time Series. Ruey S. Tsay Copyright  2002 John Wiley & Sons, Inc. ISBN: 0-471-41544-8 CHAPTER 7 Extreme Values, Quantile Estimation, and Value at Risk Extreme price movements in the financial markets are rare, but important. The stock market crash on Wall Street in October 1987 and other big financial

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seemingly large daily price movements in high-tech stocks have further generated discussions on market risk and margin setting for financial institutions. As a result, value at risk (VaR) has become a widely used measure of market risk in risk management. In this chapter, we discuss various methods for calculating

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IBM stock from July 3, 1962 to December 31, 1998 for 9190 observations. 7.1 VALUE AT RISK There are several types of risk in financial markets. Credit risk, liquidity risk, and market risk are three examples. Value at risk (VaR) is mainly concerned with market risk. It is a single estimate of the amount by which

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an institution’s position in a risk category could decline due to general market movements during a given holding 256 257 -0.2 log return -0.1 0.0 0.1 VALUE AT RISK 1970 1980

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index t. Denote the cumulative distribution function (CDF) of V () by F (x). We define the VaR of a long position over the time horizon with probability p as p = Pr[V () ≤ VaR] = F (VaR). (7.1) 258 VALUE AT RISK Since the holder of a long financial position suffers a loss when V () < 0, the

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), i=1 2 | F ) can be obtained recursively. Using r where Var(at+i | Ft ) = E(σt+i t t−1 = at−1 = σt−1 t−1 , we can rewrite the volatility equation of the IGARCH(1, 1) model 260 VALUE AT RISK in Eq. (7.2) as 2 2 σt2 = σt−1 + (1

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rule is invalid. Consider the simple model: r t = µ + at , at = σt t , 2 2 σt2 = ασt−1 + (1 − α)at−1 , µ = 0 262 VALUE AT RISK where { t } is a standard Gaussian white noise series. The assumption that µ = 0 holds for returns of many heavily traded stocks on the NYSE; see

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quantile used to calculate the 1-period horizon VaR at time index t is tv ( p)σ̂t (1) , r̂t (1) − √ v/(v − 2) where tv ( p) is the pth quantile of a Student-t distribution with v degrees of freedom. 264 VALUE AT RISK Example 7.3. Consider again the daily IBM

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write the -step ahead forecast error at the forecast origin h as eh () = rh+ − rh () = ah+ + ψ1 ah+−1 + · · · + ψ−1 ah+1 ; 266 VALUE AT RISK see Eq. (2.30) and the associated forecast error. The forecast error of the expected k-period return r̂h [k] is the sum of

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r(i) as the ith order statistic of the sample. In particular, r(1) is the sample minimum and r(n) the sample maximum. 268 VALUE AT RISK Assume that the returns are independent and identically distributed random variables that have a continuous distribution with probability density function (pdf) f (x) and CDF

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use to estimate the unknown parameters of the extreme value distribution. Clearly, the estimates obtained may depend on the choice of subperiod length n. 274 VALUE AT RISK 7.5.2.1 The Parametric Approach Two parametric approaches are available. They are the maximum likelihood and regression methods. Maximum likelihood method Assuming that

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returns of IBM stock. The sample period is from July 3, 1962 to December 31, 1998: (a) positive returns, and (b) negative returns. 278 VALUE AT RISK the scatter plots of the Hill estimator kh (q) against q. For both positive and negative extreme daily log returns, the estimator is stable except

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In financial applications, the case of kn = 0 is of major interest. (7.24) 280 VALUE AT RISK Part II For a given lower (or left tail) probability p∗ , the quantile rn∗ of Eq. (7.24) is the VaR based on the extreme value theory for the subperiod minima. The next step is to

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extreme value theory. 7.6.1 Discussion We have applied various methods of VaR calculation to the daily log returns of IBM stock for a long position of $10 million. Consider the VaR of the position for the 282 VALUE AT RISK next trading day. If the probability is 5%, which means that with probability

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be estimated by maximizing the logarithm of this likelihood function. Since the scale parameter α is nonnegative, we use ln(α) in the estimation. 288 VALUE AT RISK Table 7.3. Estimation Results of a Two-Dimensional Homogeneous Poisson Model for the Daily Negative Log Returns of IBM Stock From July 3, 1962

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that the three parameters k, α, and β are time-varying and are linear functions of the explanatory variables. Specifically, when explanatory variables xt 290 VALUE AT RISK are available, we assume that kt = γ0 + γ1 x1t + · · · + γv xvt ≡ γ0 + γ xt ln(αt ) = δ0 + δ1 x1t + · · · + δv xvt ≡ δ0 + δ xt (

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with mean 1; see also Smith (1999). We can then apply the QQ-plot to check the validity of the GPD assumption for excesses. 292 VALUE AT RISK Independence A simple way to check the independence assumption, after adjusting for the effects of explanatory variables, is to examine the sample autocorrelation functions of

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= rt − r̄ , r̄ = 1 9190 rt , 9190 t=1 where rt is the daily log return in percentages, and employ the following explanatory variables: 294 VALUE AT RISK 1. x1t : an indicator variable for October, November, and December. That is, x1t = 1 if t is in October, November, or December. This variable is

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Case I of Example 7.3. Again, as expected, the effect of extreme values (i.e., heavy tails) on VaR is more pronounced when the tail probability used is small. 296 VALUE AT RISK An advantage of using explanatory variables is that the parameters are adaptive to the change in market conditions. For example

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“d-geln.dat.” Suppose that you hold a long position on the stock valued at $1 million. Use the tail probability 0.05. Compute the value at risk of your position for 1-day horizon and 15-day horizon using the following methods: (a) The RiskMetrics method. (b) A Gaussian ARMA-GARCH

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Systems stock from 1991 to 1999 with 2275 observations. Suppose that you hold a long position of Cisco stock valued at $1 million. Compute the Value at Risk of your position for the next trading day using probability p = 0.01. (a) Use the RiskMetrics method. (b) Use a GARCH model with

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index from 1980 to 1999. All returns are in percentages and include dividend distributions. Assume that the tail probability of interest is 0.01. Calculate Value at Risk for the following financial positions for the first trading day of year 2000. (a) Long on Hewlett-Packard stock of $1 million and S&

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35, 502–516. Cox, D. R., and Hinkley, D. V. (1974), Theoretical Statistics, London: Chapman and Hall. Danielsson, J., and De Vries, C. G. (1997a), “Value at risk and extreme returns,” working paper, London School of Economics, London, U.K. Danielsson, J., and De Vries, C. G. (1997b), “Tail index and quantile estimation

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Haan, L. (1989), “On the estimation of extreme value index and large quantile estimation,” Annals of Statistics, 17, 1795–1832. 298 VALUE AT RISK Duffie, D., and Pan, J. (1997), “An overview of value at risk,” Journal of Derivatives, Spring, 7–48. Embrechts, P., Kuppelberg, C., and Mikosch, T. (1997), Modelling Extremal Events, Berlin: Springer

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), “The frequency distribution of the annual maximum (or minimum) of meteorological elements,” Quarterly Journal of the Royal Meteorological Society, 81, 158–171. Jorion, P. (1997), Value at Risk: The New Benchmark for Controlling Market Risk. The McGraw-Hill Company: Chicago. Koenker, R. W., and Bassett, G. W. (1978), “Regression quantiles,” Econometrica, 46,

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F. M. (1999a), “Optimal margin level in futures markets: Extreme price movements,” The Journal of Futures Markets, 19, 127–152. Longin, F. M. (1999b), “From value at risk to stress testing: The extreme value approach,” working paper, Centre for Economic Policy Research, London, UK. Pickands, J. (1975), “Statistical inference using extreme order statistics

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volatilities have many important financial applications. They play an important role in portfolio selection and asset allocation, and they can be used to compute the Value at Risk of a financial position consisting of multiple assets. Consider a multivariate return series {rt }. We adopt the same approach as the univariate case by

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common factor xt = 0.769r1t + 0.605r2t is treated as given. 9.5 APPLICATION We illustrate the application of multivariate volatility models by considering the Value at Risk (VaR) of a financial position with multiple assets. Suppose that an investor holds a long position in the stocks of Cisco Systems and Intel Corporation each

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January 2, 1991 to December 31, 1999 to build volatility models. The VaR is computed using the 1-step ahead forecasts at the end of data span and 5% critical values. 386 MULTIVARIATE VOLATILITY MODELS Let VaR1 be the value at risk for holding the position on Cisco Systems stock and VaR2 for holding

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345. Using these forecasts, we have VaR1 = $30504, VaR2 = $39512, and the overall value at risk VaR = $57648. The estimated VaR of the three approaches are similar. The univariate models give the lowest VaR, whereas the constant-correlation model produces the highest VaR. The range of the difference is about $1100. The time-varying volatility model seems

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the implications of the model and compute 1-step ahead volatility forecast at the forecast origin t = 888. 6. An investor is interested in daily Value at Risk of his position on holding long $0.5 million of Dell stock and $1 million of Cisco Systems stock. Use 5% critical values and

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had before. In some cases, the Bayesian solutions might be advantageous. For example, consider the calculation of Value at Risk in Chapter 7. A Bayesian solution can easily take into consideration the parameter uncertainty in VaR calculation. However, the approach requires intensive computation. Let θ be the vector of unknown parameters of an entertained

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of interest. The predictive distribution is more informative than a simple point forecast. It can be used, for instance, to obtain the quantiles needed in Value at Risk calculation. 10.10 OTHER APPLICATIONS The MCMC method is applicable to many other financial problems. For example, Zhang, Russell, and Tsay (2000) use it

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for IBM transaction data, and Eraker (2001) and Elerian, Chib and Shephard (2001) use it to estimate diffusion equations. The method is also useful in Value at Risk calculation because it provides a natural way to evaluate predictive distributions. The main question is not whether the methods can be used in most financial

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to obtain a predictive distribution for 1-step ahead volatility forecast at the forecast origin December 1999. Finally, use the predictive distribution to compute the Value at Risk of a long position worth $1 million with probability 0.01 for the next trading day. 6. Build a bivariate stochastic volatility model for the

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self-exciting, 131 smooth, 134 Threshold co-integration, 334 Time plot, 14 Transactions data, 181 Unit-root test, 60 Unit-root time series, 56 Value at Risk, 256, 385 VaR econometric approach, 262 homogeneous Poisson process, 288 inhomogeneous Poisson process, 289 RiskMetrics, 259 of a short position, 283 traditional extreme value, 279 Vector AR

How the City Really Works: The Definitive Guide to Money and Investing in London's Square Mile

by Alexander Davidson · 1 Apr 2008 · 368pp · 32,950 words

a company is to trade in derivatives, it must understand their value. Software data will calculate the value at risk, known as VAR, which is how much the company is willing to lose at any time. The VAR changes daily. Banks have thousands of loans on their books, both receiving and giving. They  68 HOW

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theory 175–76 UBS Warburg 103, 136 UK Listing Authority 44 Undertakings for Collective Investments in Transferable Securities (UCITS) 156 United Capital Asset Management 95 value at risk (VAR) virtual banks 20 virt-x 140 67–68 weighted-average cost of capital (WACC) see discounted cash ﬂow analysis wholesale banking 20 wholesale markets 78

The Mathematics of Banking and Finance

by Dennis W. Cox and Michael A. A. Cox · 30 Apr 2006 · 312pp · 35,664 words

Truncated Normal Distribution 249 249 249 249 251 252 252 254 255 256 257 260 28 Value at Risk 28.1 Introduction 28.2 Extreme Value Distributions 28.2.1 A worked example of value at risk 28.3 Calculating Value at Risk 261 261 262 262 264 29 Sensitivity Analysis 29.1 Introduction 29.2 The Application of

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). The functions are shown in Figures 27.11, 27.12 and 27.13 for σ = 2 and μ = 2, 4 and 6. 28 Value at Risk 28.1 INTRODUCTION Value at risk, or VaR is an attempt to estimate the greatest loss likely if a defined risk were to occur. For example, it could represent the loss

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, it not only allows the estimation of actual results, but also provides the associated confidence limits. 28.2.1 A worked example of value at risk The following describes the value at risk concept and provides a worked example. Consider the case of a portfolio worth $14 billion, where gold makes up 50% of the portfolio

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$, € and ¥ (i.e. $2.8 billion is held in dollars, $2.8 billion is held in euros and $1.4 billion is held in yen). Value at Risk 263 If the portfolio is managed in dollars, there are three currency-related risks: 1. The risk arising from movements in the dollar-denominated gold

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a normal distribution. VaR is calculated as: √ VaR = α x T x ⎛ ⎞⎛ ⎞ 1,400 2,800 7,000 0.00053 0.00004 0.00017 1,400 ⎝0.00004 0.00042 0.00012⎠ ⎝2,800 ⎠ = 461 = 2.33 0.00017 0.00012 0.00054 7,000 So this portfolio’s value at risk is $461 million

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exactly as before. α, the new column vector of position is: ⎛ ⎞ 2,240 x = ⎝ 4,480 ⎠ . 2,800 264 Mathematics of Banking The value at risk is calculated as before: √ VaR = α x T x ⎛ ⎞⎛ ⎞ 2,240 4,480 2,800 0.00053 0.00004 0.00017 2,240 ⎝0.00004 0.00042 0

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, a much lower level of accuracy is actually achieved – perhaps only 80%. 28.3 CALCULATING VALUE AT RISK Value at risk for a single position is calculated as: VaR = Sensitivity of position to change in market prices × Estimated change in price or VaR = Amount of the position × Volatility of the position = xσ where x is the position size

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asset prices as well as the risks in the individual instruments. This can be written as: VaR = VaR21 + VaR22 + 2ρ12 VaR1 VaR2 where VaR1 is the value at risk arising from the first risk factor, VaR2 is the value at risk arising from the second risk factor, and ρ12 is the correlation between movements in the two

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risk factors. Given the definition of VaR above, this can be written as: VaR = x12 σ12 + x22 σ22 + 2ρ12 x1 σ1 x2 σ2 Value at Risk 265 where σ1 and σ2 are the confidence level volatilities for the two risk factors (equivalently σ12 and

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regression 108–20 net present value (NPV) 228–9, 231–2 risk 227–34 upper/lower quartiles, concepts 39–41 valuations, options 58, 97–8 value at risk (VaR) calculation 264–5 concepts 261–5 examples 262–3 extreme value distributions 262–4 importance 261 variable costs, stock control 197–201 variables bar charts

Tools for Computational Finance

by Rüdiger Seydel · 2 Jan 2002 · 313pp · 34,042 words

52 Chapter 1 Modeling Tools for Financial Options x and a constant α > 0. A correct modeling of the tails is an integral basis for value at risk (VaR) calculations. For the risk aspect compare [BaN97], [Dowd98], [EKM97], [ArDEH99]. For distributions that match empirical data see [EK95], [Shi99], [BP00], [MRGS00], [BTT00]. Estimates of future

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question arises whether the risk-neutral valuation principle is obeyed. An important application of Monte Carlo methods is the calculation of risk indices such as value at risk, see the notes on Section 1.7/1.8. Monte Carlo methods are especially attractive for multifactor models with high dimension. The demands for accuracy

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in a macroeconomic model. Chaos, Solitons and Fractals 12 (2001) 805-822. J.L. Doob: Stochastic Processes. John Wiley, New York (1953). K. Dowd: Beyond Value at Risk: The New Science of Risk Management. Wiley & Sons, Chichester (1998). D. Duﬃe: Dynamic Asset Pricing Theory. Second Edition. Princeton University Press, Princeton (1996). E. Eberlein

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Truncation error 299 161, 172 Underlying 1–2, 5, 58 Uniform distribution 61–74, 77–79, 88, 90, 255 Upwind scheme 209, 226–234, 237 Value at Risk 52, 116 Value function 9 Van der Corput sequence 82–83, 86 Van Leer 235 Variance 14–15, 40–43, 51, 53, 78–79, 87

Trading Risk: Enhanced Profitability Through Risk Control

by Kenneth L. Grant · 1 Sep 2004

Putting It All Together CHAPTER 4 The Risk Components of an Individual Portfolio Historical Volatility Options Implied Volatility Correlation Value at Risk (VaR) Justification for VaR Calculations Types of VaR Calculations Testing VaR Accuracy Setting VaR Parameters Use of VaR Calculation in Portfolio Management Scenario Analysis Technical Analysis CHAPTER 5 Setting Appropriate Exposure Levels (Rule 1) Determining the

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of individual position exposure, 84 TRADING RISK it is also useful to measure the interplay between the two—a concept captured perhaps most rigorously through Value at Risk (VaR) methodologies, developed by the money management industry over the past couple of decades to add precision to their own risk-estimation efforts. In this chapter

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to gain a better The Risk Components of an Individual Portfolio 91 understanding of interactive pricing dynamics across the fullest range of available market conditions. VALUE AT RISK (VaR) Through the efforts of modern-day financial engineers, a new paradigm has emerged: It is now possible, nay, even fashionable, to combine the concepts

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exposure estimate. This work, most of which has been conducted over the past 15 or so years, is most broadly synthesized under the heading of Value at Risk, which is now thought of as the standard methodology for risk management in the financial services industry. The underlying objective is to aggregate all risks

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Portfolio 93 after the fact, the focus turned toward approaches that would predict this variable; and the result is the development and institutionalization of Value at Risk. Although VaR can be both mystifying and maddening to those whose actions are governed by it, the methodology represents a significant upgrade over its predecessor approaches under

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outcomes. Perhaps the best alternative at your disposal in this regard is the results of a Value at Risk (VaR) calculation, which have the advantage of being based 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

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risk adjustments on returns by using not the standard deviation of returns as your benchmark, but rather statistics produced by a Value at Risk (VaR) model. Recall from the earlier discussion on VaR that its precise intent is to predict the volatility of portfolios based on the way that their pricing patterns have shown them

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market value, 172–174 number of daily transactions, 170–171 number of positions, 174 portfolio diversification and, 145 risk exposure and, 83–84, 90–91 VaR, 178–179 volatility, 177–179 Credit spreads, 105 Cross-collateralization, 198 Cross correlation analysis, 75–76 Daily net change, 75 Daily transactions, number of,

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portfolio, risk components: correlation, 90–91 historical volatility, 84–88, 96–97 options implied volatility, 86–89 scenario analysis, 104–106 technical analysis, 106–108 value at risk (VaR), 91–104 Individual trades, risk components of: core transactions-level statistics, 161–168, 209, 211 correlation analysis, 168–181 influential factors, generally, 208–211 performance

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snapshot statistics, 160–161 transaction defined, 158–160 Two-sided market, 135–137, 140 Underlying markets, 117 Underlying price, 149–150 Unit impact ratio, 187 Value at Risk (VaR) calculation: accuracy testing, 98–99, 103 and correlation analysis, 178–179 implications of, generally, 84, 91–92 justification of, 92–94 parameter setting, 99–

Risk Management in Trading

by Davis Edwards · 10 Jul 2014

CHAPTER 2 Financial Markets 33 CHAPTER 3 Financial Mathematics 61 CHAPTER 4 Backtesting and Trade Forensics 95 CHAPTER 5 Mark to Market 121 CHAPTER 6 Value-at-Risk 141 CHAPTER 7 Hedging 177 CHAPTER 8 Options, Greeks, and Non-Linear Risks 199 CHAPTER 9 Credit Value Adjustments (CVA) 237 vii viii

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return divided by the volatility of returns. The reason that these measures are commonly used is because trading limits are typically determined by a value at risk (VAR) calculation. Since VAR measures volatility of returns, for traders to maximize their profits they need to maximize returns for a given level of volatility. In other

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small amount of money to initiate a trade. Instead, position limits are commonly based on a volatility‐based estimate of size called value‐at‐risk, abbreviated VAR. Trading desks typically have several VAR limits. The first limit, a soft limitt, indicates the target size of the trading portfolio. The second limit, a hard limitt,

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desks use these limits to ensure that traders are following trading rules set by the firm and to ensure that diversification is working properly. Value‐at‐risk was originally designed as a way to apply consistent size limits across any type of investment. It has been expanded since that time to estimate

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traders are only trading to get under risk limits. This makes Value‐at‐Risk both a helpful tool and a source of danger to traders. POSITION LIMITS Value‐at‐risk was first invented to describe the size of a risk. Trading desks will often use VAR to limit the size of trades and investments. Prior to

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compare sizes of investments. When first implemented, the goal of VAR was to have a quick way to answer the question “How much money does the firm have at risk” within 15 minutes of the market close. Since that time, value‐at‐risk has become a standard way to describe the size of

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evolved over time and is now used for a variety of purposes. In addition to describing the size of risk, VAR is commonly used for regulatory Value-at-Risk 143 purposes (to calculate required cash reserves) and by senior management (to estimate the worst‐case losses for trading/investment positions). As a result,

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an extremely rare move. Rare moves are not well described by typical behavior because they have different root causes than normal price moves. WHAT IS VALUE-AT-RISK? Value‐at‐risk uses a factor common to all financial instruments (daily changes in value caused by mark‐to‐market accounting) to establish an apples‐to‐ apples

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V@R to distinguish it from the mathematical abbreviation for variance, which is commonly abbreviated var. 144 RISK MANAGEMENT IN TRADING KEY CONCEPT: VALUE AT RISK (VAR) IS DEFINED MATHEMATICALLY Value‐at‐risk is typically defined as the maximum expected loss on a financial instrument, or a portfolio of financial instruments, over a given period of time

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5 percent chance of losing more than the VAR amount over the next day. ■ 99 percent 5‐Day VAR. The company has a 1 percent chance of losing more than the VAR amount over the next five trading days. As a measure of size, value‐at‐risk is a fundamental building block of risk management

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its name (this is called a parametric model). In addition, the frequency at which different moves would be observed could be graphed and described 145 Value-at-Risk +/− 2 Standard Deviations contain approximately 95.5% of samples +/− 3 Standard Deviations contain approximately 99.9% of samples −3.0 −2.8 −2.6

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occur due to a stock market crash. Extreme events are difficult to predict using VAR for several reasons. First, VAR is based on a common denominatorr that crosses commodity, Value-at-Risk 147 KEY CONCEPT: VALUE-AT-RISK IS A MEASURE OF SIZE Value‐at‐risk is way to describe the size of a trading position. It does this by

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into the specific risks facing a trading position. ■ ■ ■ ■ ■ Value‐at‐risk

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traders to make large bets in the riskiest assets. To solve these problems, value‐at‐risk describes size by typical changes in value (expected profits and losses) rather than in terms of money used to make the investment. VAR has largely replaced position limits and capital requirements based on the concept of number

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period of time. (See Equation 6.1, Discrete Returns.) Returnt = Pricet −1 Pricet −1 or Pricet − Pricet −1 Returnt = Pricet −1 (6.1) 149 Value-at-Risk where Returnt Pricet Pricet–1 Discrete Period Return. The percent change in value between the current period and the prior period Price. The price in

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. Because it is simpler to use, and the limitations of parametric VAR are relatively unimportant for the purpose of setting position limits and calculating capital requirements, parametric VAR is the more common form of value‐at‐risk calculation. Common assumptions associated with parametric VAR are that returns are independent, identically distributed (have the same volatility

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same for every financial instrument. As a result, the normal distribution is primarily defined by a single parameter, the standard deviation. This looks 151 Value-at-Risk Standard Deviation = σ Standard Deviation = 1.5 σ FIGURE 6.3 Normal Distributions with Different Standard Deviations like a bell curve whose width is determined

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= 15%/√12 = 4.330% ■ 1‐Month 99% = 4.330% * 2.326 = 10.072% (6.5) Confidence ■ 1‐Month 99% VAR = $200MM * 10.072% = $20.144MM Answer = $20.144MM Value-at-Risk 153 Approximately once every 100 months, Truman should expect to have a monthly loss greater than $20.144MM. Note: The 1.645

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calculation. Exponentially weighted returns use a decay factor, commonly called lambda (λ) that progressively decreases the weight of each sample as the samples get 157 Value-at-Risk further back in time. For example, the current day’s sample (t = 0) receives a 100 percent weighting. The earlier samples get progressively less

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values today and the previous sum is the addition of today’s value and the subtraction of whatever value is dropping out of the calculation. Value-at-Risk 159 methods is called generalized autoregressive conditionally hetroskedastic models (GARCH). The term GARCH comes from the following abbreviations: ■ ■ ■ Generalized (G). This term means that

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operational problems that exacerbate risky situations since a major use of VAR is to set position limits and calculate regulatory capital requirements. These are both calculations that can exacerbate financial crises because they can force portfolio liquidations. For example, if value‐at‐risk is being used to limit the size of trading positions,

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and exacerbate losses in periods of volatile markets. VAR is commonly used to limit the size of trading positions. When volatility suddenly spikes, the practical effect is that traders begin selling their positions into the market to get back under their trading 163 Value-at-Risk limits. This simultaneously reduces liquidity (most institutions

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a measure of uncertainty or risk. An investment might involve no risk or uncertainty and result in zero VAR. There can never be negative uncertainty. 165 Value-at-Risk VARIANCE/COVARIANCE MATRIX When more than two assets are combined, the formula to calculate the combined volatility becomes complicated. To simplify this calculation, the

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similar to historically observed returns. TABLE 6.2 S&P 500 VAR Type of Volatility Model 99% One‐Day VAR 95% One‐Day VAR Historical Simulation −3.75% −1.75% Parametric −3.00% −2.00% Value-at-Risk 169 KEY CONCEPT: NON-PARAMETRIC VAR Statistical sampling techniques can be used to extend historical simulation in a

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/2 28 6/ VAR Backtest 8 07 6 00 00 /2 6/ 28 6/ 28 /2 00 5 4 20.00% 15.00% 10.00% 5.00% 0.00% −5.00% −10.00% −15.00% −20.00% 171 Value-at-Risk THE MISUSE OF VAR A major problem with VAR is that VAR gets used for multiple

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2.0 2.2 2.5 2.7 2.9 5.00% Profit or Loss FIGURE 6.11 Expected Shortfall Value-at-Risk 173 KEY CONCEPT: EXPECTED SHORTFALL MEASURES SIZE Expected shortfall, like VAR, describes whether the size of an investment is large or small rather than whether it is good or bad.

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financial instruments like stocks and bonds. Risk managers have developed a variety of techniques to model this risk and fit it into the position limit (value‐at‐risk) framework used for other financial instruments. The most common technique used to monitor option risk is to break the option into several risk factors.

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handful of exposures, options portfolios can be concisely described to senior management. This also allows position limits, like value‐at‐risk, to be applied to options. In practice, option portfolios often have a value‐at‐risk limit as well as a limit on each of their Greeks. OPTIONS Options are financial derivatives that give their

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(things like price of the underlying, time to expiration, and volatility) affect a dependent variable (the value of the option). One measure of risk, value‐at‐risk (VAR), examines linear changes in value. This is only somewhat useful for options because linear approximations of curved lines break down if prices move too far

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to expiration. In addition, gamma is always a positive quantity—it always helps option buyers and hurts option sellers. Gamma is also important to value‐at‐risk (VAR) calculations. VAR estimates can incorporate both delta and gamma approximations to better predict how a portfolio will move with respect to a risk factor. In the

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contract if a default occurs at some pre‐specified point in the future (like a year). This calculation tends to work very much like a value‐at‐risk (VAR) calculation and is commonly used in risk management. CALCULATING A CREDIT VALUE ADJUSTMENT A CVA calculation has two main steps. First, it is necessary

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49–50 default, 239 delta, 202, 212–218 and moneyness, 218 as payoff approximation, 213 delta calculations, exceptions to, 216–217 delta/gamma approximations for value-at-risk, 223 INDEX derivative first, 83–85 mathematical, 88 second, 84–85 derivatives, 37–40 calculus, 87–89 Greeks as, 203–204 discrete risks, 23–

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160 moneyness, delta and, 218 monitoring risk, 27–28 Monte Carlo testing, 105–106 mortgage-backed securities, 50 N non-linear risk, 205 non-parametric value-at-risk, 167–169 normal distribution, 81–82 O obligor, 239 operational risk, 25 option contracts, 38 option risk, managing, 2, 11 options, 55–59, 199

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estimating volatility for, 153–161 PD. See probability of default PDF. See probability density function percent returns, 148–150 phi, 203, 232–234 Index portfolio value-at-risk, calculating, 161–164 position limits, 142–143 setting, 2, 11 potential future exposure, current exposure and, 260 pre-trade monitoring, 117–118 preferred stock,

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V validation, data, 96–97 value of options, 204–207 value-at-risk limits, in practice, 170 value-at-risk sensitivity, 162–163 value-at-risk as size measure, 147 defined, 143–147 misuse of, 171–173 non-parametric, 167–169 parametric, 150–161 zero and, 164 VAR. See value-at-risk variables, 62–63 variance, 69–70 variance/covariance matrix, 165

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–167 vega, 203, 230–232 time and, 232 volatile earnings, created via hedging, 187 volatility, 69–70, 201 estimating, 153–161 volume notation, 202 W wrong-way risk, 255 Z zero coupon bonds, 49 zero, value-at-risk and,

Optimization Methods in Finance

by Gerard Cornuejols and Reha Tutuncu · 2 Jan 2006 · 130pp · 11,880 words

Fundamental Theorem of 3.1.1 Replication . . . . . . . . . . . . . . . . . . . . 3.1.2 Risk-Neutral Probabilities . . . . . . . . . . . . 3.2 Arbitrage Detection Using Linear Programming . . . . 3.3 Risk Measures: Conditional Value-at-Risk . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 3 4 4 5 6 7 8 9 11 11 . . . . . . . . 13 13 14 17 18 18 21 24 27

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vulnerabilities of a company. Examples of these risk measures include portfolio variance as in the Markowitz MVO model, the Valueat-Risk (VaR) and the expected shortfall (also known as conditional VaR, or CVaR)). Furthermore, risk control techniques need to be developed and implemented to adapt to the rapid changes in the values

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, . . . , n − 1 3. The function C(Ki ) := S0i defined on the set {K1 , K2 , . . . , Kn } is a strictly convex function. 3.3 Risk Measures: Conditional Value-at-Risk Financial activities involve risk. Our stock or mutual fund holdings carry the risk of losing value due to market conditions. Even money invested in a

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which comes from quantitative risk measures that adequately reflect the vulnerabilities of a company. Perhaps the best-known risk measure is Value-at-Risk (VaR) developed by financial engineers at J.P. Morgan. VaR is a measure related to percentiles of loss distributions and represents the predicted maximum loss with a specified probability level (e

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.g., 95%) over a certain period of time (e.g., one day). Consider, for example, a random variable 3.3. RISK MEASURES: CONDITIONAL VALUE-AT-RISK 37 X that

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VaR value. This and other undesirable features of VaR led to the development of alternative risk measures. One well-known modification of VaR is obtained by computing the expected loss given that

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the loss exceeds VaR. This quantity is often called conditional Value-at-Risk or CVaR. There are several alternative names for this measure in the finance literature including Mean Expected Loss, Mean Shortfall

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) = VaRα (x), CVaRα (x) = (3.14) 3.3. RISK MEASURES: CONDITIONAL VALUE-AT-RISK 39 i.e., CVaR of a portfolio is always at least as big as its VaR. Consequently, portfolios with small CVaR also have small VaR. However, in general minimizing CVaR and VaR are not equivalent. Since the definition of CVaR involves the

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maximize the expected wealth at the end of the planning horizon. In practice, one might have a different objective. For example, in some cases, minimizing Value at Risk (VaR) might be more appropriate. Other priorities may dictate other objective functions. To address the issue of the most appropriate objective function, one must understand the

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. Pınar. Minimum risk arbitrage with risky financial contracts. Technical report, Bilkent University, Ankara, Turkey, 2001. [12] R. T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk. The Journal of Risk, 2:21–41, 2000. [13] W. F. Sharpe. Determining a fund’s effective asset mix. Investment Management Review, pages 59–69

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. Robust asset allocation. Technical report, Department of Mathematical Sciences, Carnegie Mellon University, August 2002. To appear in Annals of Operations Research. [17] S. Uryasev. Conditional value-at-risk: Optimization algorithms and applications. Financial Engineering News, 14:1–6, 2000.

Mathematics for Finance: An Introduction to Financial Engineering

by Marek Capinski and Tomasz Zastawniak · 6 Jul 2003

.1.1 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.1.2 Greek Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.2 Hedging Business Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.2.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 9.2.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.3 Speculating with Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.3.1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.3.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 10

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from unexpected future changes of various market variables such as commodity prices, interest rates or stock prices. We shall introduce a measure of risk called Value at Risk (VaR), which has recently become very popular. Derivative securities will be used to design portfolios with a view to reducing this kind of risk. Finally, we

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of risk, related to an intuitive understanding of risk as the size and likelihood of a possible loss. 202 Mathematics for Finance 9.2.1 Value at Risk Let us present the basic idea using a simple example. We buy a share of stock for S(0) = 100 dollars to sell it after

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a loss not exceeding this amount is 95%. This is referred to as Value at Risk at 95% conﬁdence level and denoted by VaR. (Other conﬁdence levels can also be used.) So, VaR is an amount such that P (100er − S(1) < VaR) = 95%. It should be noted that the majority of textbooks neglect the time

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time; exercise time; delivery time time step Greek parameter theta row matrix with all entries 1 portfolio value; forward contract value, futures contract value variance value at risk Greek parameter vega symmetric random walk; weights in a portfolio weights in a portfolio as a row matrix Wiener process, Brownian motion position in a

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money 21 310 trinomial tree model Mathematics for Finance 64 underlying 85, 147 undiversiﬁable risk 122 unit bond 39 value at risk 202 value of a portfolio 2 value of a strategy 76, 87 VaR 202 variable interest 255 Vasiček model 260 vega 197 vega neutral portfolio volatility 71 weights in a portfolio

Handbook of Modeling High-Frequency Data in Finance

by Frederi G. Viens, Maria C. Mariani and Ionut Florescu · 20 Dec 2011 · 443pp · 51,804 words

the standard deviation (StD) of portfolio return at a ﬁxed time horizon, but there are several other measures of risk in common use, such as value at risk (VaR), expected shortfall (ES), and others (see Artzner et al., 1999; Rockafellar and Uryasev, 2002). Each of these is a kind of measure of the width

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pdf of the returns distribution, but for simplicity and concreteness, we focus here on forecasting VaR; other kinds of risk forecasts will be similar. 7.3.1 VALUE AT RISK DEFINITION 7.12 Value at Risk (VaR). Given α ∈ (0, 1), the value at risk at conﬁdence level α for loss L of a security or a portfolio is deﬁned as

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Typical values for α are between 0.95 and 0.995. VaR can also be based on returns instead of losses, in which case α takes a small value such as 0.05 or 0.01. For example, intuitively, a 95% value at risk, VaR0.95 , is a level L such that a loss

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. EM-based maximum likelihood parameter estimation of multivariate generalized hyperbolic distributions with ﬁxed λ. Stat Comput 2004;14:67–77. Rockafellar R, Uryasev S. Conditional value-at-risk for general loss distributions. J Bank Finance 2002;26:1443–1471. Serﬂing RJ. Approximation theorems of mathematical statistics. New York: Wiley; 1980. Sharpe WF.

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North America dataset, 54 Conditional density function, 173 Conditional distribution, 29, 30 Conditional expected returns, 181 Conditional normal distribution, density of, 173 Conditional VaR, 188–189, 207. See also Value at risk (VaR) 423 Conditional variances, 203, 206, 208 of the GARCH(1,1) process, 180 Conﬁdence intervals, for forecasts, 187–188 Consecutive trades, 129

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Daily returns, 4, 14 Daily returns scenario, 215–216 Daily return/volatility, 211–212 Daily sampled indices, analysis of, 132–141 Daily VaR forecast, backtesting, 199–200. See also Value at risk (VaR) Index Data for NIG and VG model estimation, 18 statistical behavior of, 345 Data analysis methods, 122–128 truncated Lévy ﬂight

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, 185–186, 212 High-low frequency density, 210 High-low frequency method, 200, 212, 215–216 limits of, 195 Index High-low frequency VaR forecast, 186. See also Value at risk (VaR) High parameter values, 136 High trading activity, 42 Hilbert space, 387 Hillebrand, Eric, xiii, 75 HL estimator, 263. See also VaRHL Hölder

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, 64 Multinomial recombining tree algorithm, 221, 226 Multinomial tree approximation method, 97–115 Multiple timescale forecasts, 185–188 Multiscale method, 217 Multiscale VaR forecast backtest results, 202. See also Value at risk (VaR) Multiscale volume classiﬁcation, 33–35 Multistock automated trading system, 66 Multivariate normal distribution, 170 Multivariate normal mean–variance mixture distribution, 165–166

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after retirement, 321 Utility estimations, 287 Utility functions, 296, 299 of power type, 305 Utility loss, 290 Value at risk (VaR), 163, 165, 176. See also VaR entries Value function, 304, 307, 312, 313 for the constant coefﬁcients case, 318 VaR error, 201. See also Value at risk (VaR) VaR estimates, based on Monte Carlo simulation, 199 VaRFixed , 213, 214, 215

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The Big Short: Inside the Doomsday Machine

by Michael Lewis · 1 Nov 2009 · 265pp · 93,231 words

Nerds on Wall Street: Math, Machines and Wired Markets

by David J. Leinweber · 31 Dec 2008 · 402pp · 110,972 words

13 Bankers: The Wall Street Takeover and the Next Financial Meltdown

by Simon Johnson and James Kwak · 29 Mar 2010 · 430pp · 109,064 words

The Ascent of Money: A Financial History of the World

by Niall Ferguson · 13 Nov 2007 · 471pp · 124,585 words

What They Do With Your Money: How the Financial System Fails Us, and How to Fix It

by Stephen Davis, Jon Lukomnik and David Pitt-Watson · 30 Apr 2016 · 304pp · 80,965 words

Too big to fail: the inside story of how Wall Street and Washington fought to save the financial system from crisis--and themselves

by Andrew Ross Sorkin · 15 Oct 2009 · 351pp · 102,379 words

The Black Box Society: The Secret Algorithms That Control Money and Information

by Frank Pasquale · 17 Nov 2014 · 320pp · 87,853 words

The Global Money Markets

by Frank J. Fabozzi, Steven V. Mann and Moorad Choudhry · 14 Jul 2002

Profiting Without Producing: How Finance Exploits Us All

by Costas Lapavitsas · 14 Aug 2013 · 554pp · 158,687 words

Getting a Job in Hedge Funds: An Inside Look at How Funds Hire

by Adam Zoia and Aaron Finkel · 8 Feb 2008 · 192pp · 75,440 words

Big Mistakes: The Best Investors and Their Worst Investments

by Michael Batnick · 21 May 2018 · 198pp · 53,264 words

Obliquity: Why Our Goals Are Best Achieved Indirectly

by John Kay · 30 Apr 2010 · 237pp · 50,758 words

The End of Accounting and the Path Forward for Investors and Managers (Wiley Finance)

by Feng Gu · 26 Jun 2016

Planet Ponzi

by Mitch Feierstein · 2 Feb 2012 · 393pp · 115,263 words

Unfinished Business

by Tamim Bayoumi · 405pp · 109,114 words

Foolproof: Why Safety Can Be Dangerous and How Danger Makes Us Safe

by Greg Ip · 12 Oct 2015 · 309pp · 95,495 words

Crisis Economics: A Crash Course in the Future of Finance

by Nouriel Roubini and Stephen Mihm · 10 May 2010 · 491pp · 131,769 words

A Man for All Markets

by Edward O. Thorp · 15 Nov 2016 · 505pp · 142,118 words

Beyond Diversification: What Every Investor Needs to Know About Asset Allocation

by Sebastien Page · 4 Nov 2020 · 367pp · 97,136 words

The Tyranny of Nostalgia: Half a Century of British Economic Decline

by Russell Jones · 15 Jan 2023 · 463pp · 140,499 words

Hedgehogging

by Barton Biggs · 3 Jan 2005

How to Kick Ass on Wall Street

by Andy Kessler · 4 Jun 2012 · 77pp · 18,414 words

MacroWikinomics: Rebooting Business and the World

by Don Tapscott and Anthony D. Williams · 28 Sep 2010 · 552pp · 168,518 words

Investing Amid Low Expected Returns: Making the Most When Markets Offer the Least

by Antti Ilmanen · 24 Feb 2022