The Sharpe Ratio: Statistics and Applications

Description

The Sharpe Ratio: Statistics and Applications is the most widely used metric for comparing the performance of financial assets. The Markowitz portfolio is the portfolio with the highest Sharpe ratio. The Sharpe Ratio: Statistics and Applications examines the statistical properties of the Sharpe ratio and Markowitz portfolio, both under the simplifying assumption of Gaussian returns, and asymptotically. Connections are drawn between the financial measures and classical statistics including Student's t, Hotelling's T^2 and the Hotelling-Lawley trace. The robustness of these statistics to heteroskedasticity, autocorrelation, fat tails and skew of returns are considered. The construction of portfolios to maximize the Sharpe is expanded from the usual static unconditional model to include subspace constraints, hedging out assets, and the use of conditioning information on both expected returns and risk. The Sharpe Ratio: Statistics and Applications is the most comprehensive treatment of the statistical properties of the Sharpe ratio and Markowitz portfolio ever published. Features: 1. Material on single asset problems, market timing, unconditional and conditional portfolio problems, hedged portfolios. 2. Inference via both Frequentist and Bayesian paradigms. 3. A comprehensive treatment of overoptimism and overfitting of trading strategies. 4. Advice on backtesting strategies. 5. Dozens of examples and hundreds of exercises for self study. The Sharpe Ratio: Statistics and Applications is an essential reference for the practicing quant strategist and the researcher alike, and an invaluable textbook for the student. Cover Half Title Title Page Copyright Page Dedication Contents Foreword Preface List of Figures List of Tables Symbols I. The Sharpe Ratio 1. The Sharpe Ratio and the Signal-Noise Ratio 1.1. Introduction 1.1.1. Which returns? 1.2. Units of Sharpe ratio 1.2.1. Probability of a loss 1.2.2. Interpretation of the Sharpe ratio 1.3. Historical perspective 1.4. Linear attribution models 1.4.1. Examples of linear attribution models 1.4.2. † Heteroskedasticity attribution models 1.5. † Should you maximize Sharpe ratio? 1.6. † Probability of a loss, revisited 1.7. † Drawdowns and the signal-noise ratio 1.7.1. Controlling drawdowns via the signal-noise ratio 1.8. Exercises 2. The Sharpe Ratio for Gaussian Returns 2.1. The non-central t-distribution 2.1.1. Distribution of the Sharpe ratio 2.1.2. Distribution of the ex-factor Sharpe ratio 2.2. † Density and distribution of the Sharpe ratio 2.2.1. † The density of the Sharpe ratio 2.2.2. The distribution and quantile of the Sharpe ratio 2.3. Moments of the Sharpe ratio 2.3.1. The Sharpe ratio is biased 2.3.2. Moments under up-sampling 2.3.3. Unbiased estimation and efficiency 2.4. † The lambda prime distribution 2.5. Frequentist inference on the signal-noise ratio 2.5.1. Confidence intervals 2.5.2. † Symmetric confidence intervals 2.5.3. Hypothesis tests 2.5.4. † Two one-sided and other intersection union tests 2.5.5. Hypothesis tests for the linear attribution model 2.5.6. † Confidence intervals, linear attribution model 2.5.7. Type I errors and false discovery rate 2.5.8. Power and sample size 2.5.9. † Frequentist prediction intervals 2.6. † Likelihoodist inference on the signal-noise ratio 2.6.1. Likelihood ratio test 2.7. † Bayesian inference on the signal-noise ratio 2.7.1. Ex-Factor signal-noise ratio 2.7.2. Credible intervals 2.7.3. Posterior prediction intervals 2.8. Exercises 3. The Sharpe Ratio for Other Returns 3.1. The Sharpe ratio for (non-i.i.d.) elliptical returns 3.1.1. Independent Gaussian returns 3.1.2. Homoskedastic i.i.d. Gaussian returns 3.1.3. Heteroskedastic independent Gaussian returns 3.1.4. Homoskedastic autocorrelated Gaussian returns 3.2. Asymptotic distribution of Sharpe ratio 3.2.1. Frequentist analysis 3.2.2. Scalar case 3.2.3. Asymptotic bias and variance of the Sharpe ratio 3.2.4. † Concentration inequalities 3.2.5. † Survivorship bias and finite moments 3.3. Asymptotic distribution of functions of Sharpe ratios 3.3.1. Frequentist analysis 3.3.2. Bayesian analysis 3.4. † The ex-factor Sharpe ratio for elliptical returns 3.4.1. Market term, constant expectation 3.4.2. Homoskedastic i.i.d. elliptical returns 3.4.3. Homoskedastic autocorrelated Gaussian returns 3.5. † Asymptotic distribution of ex-factor Sharpe ratio 3.5.1. Testing the ex-factor Sharpe ratio 3.5.2. Ex-factor Sharpe ratio prediction intervals 3.6. Sharpe ratio and non-normality, an empirical study 3.7. Exercises 4. Overoptimism 4.1. Overoptimism by selection 4.1.1. Multiple hypothesis testing 4.1.2. Multiple hypothesis testing, correlated returns 4.1.3. One-sided alternatives 4.1.4. Hansen's asymptotic correction 4.1.5. Conditional inference 4.1.6. † Subspace approximation 4.2. A post hoc test on the Sharpe ratio 4.3. Exercises II. Maximizing the Signal-Noise Ratio 5. Maximizing the Sharpe Ratio 5.1. As an optimization problem 5.2. Portfolio optimization 5.2.1. Basic risk constraint 5.2.2. Basic subspace constraint 5.2.3. Spanning and hedging 5.2.4. Inequality constraints 5.3. Conditional portfolio optimization 5.3.1. † Conditional heteroskedasticity models 5.3.2. Optimal conditional portfolios 5.3.3. Constrained optimal conditional portfolios 5.4. Exercises 6. Portfolio Inference for Gaussian Returns 6.1. Optimal sample portfolios 6.1.1. Optimal sample conditional portfolios 6.1.2. Hotelling's T2 6.1.3. Multivariate general linear hypothesis 6.1.4. † Distribution of summary statistics 6.2. Moments of summary statistics 6.2.1. Asymptotic moments 6.2.2. Unbiased estimation 6.3. Frequentist inference 6.3.1. Confidence intervals 6.3.2. Hypothesis tests 6.3.3. Power and sample size 6.4. Likelihoodist inference 6.4.1. Likelihood ratio test 6.4.2. Traditional likelihood analysis 6.5. † Bayesian inference 6.5.1. Credible intervals 6.5.2. Bayesian inference, conditional expectation 6.6. Exercises 7. Portfolio Inference for Other Returns 7.1. The second moment matrix 7.1.1. Distribution under Gaussian returns 7.2. Asymptotic distribution of the second moment 7.2.1. Asymptotic distribution, i.i.d. elliptical returns 7.2.2. Asymptotic distribution, matrix normal returns 7.2.3. Asymptotic squared maximal Sharpe ratio, heteroskedastic returns 7.2.4. Asymptotic squared maximal Sharpe ratio, autocorrelated returns 7.3. † Constrained problems 7.3.1. † Subspace constraint 7.3.2. † Hedging const 7.3.3. † Conditional heteroskedasticity 7.3.4. † Conditional expectation and heteroskedasticity 7.4. Frequentist inference via the second moment 7.4.1. Confidence intervals 7.4.2. Hypothesis tests 7.5. † Likelihoodist inference on the second moment 7.6. † Bayesian inference via the second moment 7.7. Exercises 8. Overoptimism and Overfitting 8.1. The "haircut" 8.1.1. Haircut of the Markowitz portfolio 8.1.2. † Attribution of error 8.2. A bound on achieved signal-noise ratio 8.2.1. Limits to diversification 8.2.2. † Choosing a stopped clock 8.3. Inference on achieved signal-noise ratio 8.3.1. Confidence intervals 8.3.2. Hypothesis testing 8.3.3. Strategy selection 8.3.4. † Cross validation 8.4. Exercises 9. Market Timing 9.1. Market timing with a single binary feature 9.1.1. Gain in power 9.1.2. Testing assumptions 9.2. Market timing with a discrete feature 9.2.1. Estimation and inference 9.3. Market timing with a continuous feature 9.3.1. Reasonable timing of signals 9.4. Nonparametric market timing 9.5. Exercises 10. † Backtesting 10.1. On backtesting 10.2. Exercises A. Prerequisites A.1. Linear algebra A.2. Probability distributions A.2.1. Univariate probability distributions A.2.2. Multivariate probability distributions A.2.3. Matrixvariate probability distributions A.3. Statistical practice A.4. † Matrix derivatives A.5. The central limit theorem and delta method A.6. Exercises References Index