Sharpe Ratio
Feb 7, 2025
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores the Sharpe ratio, examining both its practical use and the risks of overfitting, selection bias, and estimation errors. It highlights statistical corrections for backtest inflation, algorithmic refinements for risk-adjusted returns, and portfolio construction techniques that balance theoretical improvements with implementation challenges.
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models
Joseph Mulligan, Antoine (Jack) Jacquier & Johannes Muhle-Karbe
We study how much the in-sample performance of trading strategies based on linear predictive models is reduced out-of-sample due to overfitting. More specifically, we compute the in-and out-of-sample means and variances of the corresponding PnLs and use these to derive a closed-form approximation for the corresponding Sharpe ratios. We find that the out-of-sample "replication ratio" diminishes for complex strategies with many assets based on many weak rather than a few strong trading signals, and increases when more training data is used. The substantial quantitative importance of these effects is illustrated with an empirical case study for commodity futures following the methodology of Gârleanu and Pedersen.
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits
Sabrina Khurshid, Mohammed Shahid Abdulla & Gourab Ghatak
Sharpe ratio (SR) is a critical parameter in characterizing financial time series as it jointly considers the reward and the volatility of any stock/portfolio through its mean and standard deviation. Deriving online algorithms for optimizing the SR is particularly challenging since even offline policies experience constant regret with respect to the best expert (Even-Dar et al., 2006). This paper focuses on optimizing the regularized square SR (RSSR) by considering two settings: regret minimization (RM) and best arm identification (BAI). In this regard, we propose a novel multiarmed bandit (MAB) algorithm for RM called UCB-RSSR for RSSR maximization. We derive a path-dependent concentration bound for the estimate of the RSSR. Based on that, we derive the regret guarantees of UCB-RSSR and show that it evolves as O(log n) for the two-armed bandit case played for a horizon n. We also consider algorithms for the fixed budget setting of the BAI problems, i.e., sequential halving and successive rejects, and propose SHSR and SuRSR algorithms. We derive the upper bound for the error probability of BAI algorithms. We demonstrate that UCB-RSSR outperforms the only other known SR optimizing bandit algorithm, U-UCB (Cassel et al., 2023). We also study the efficacy of proposed BAI algorithms for 6 different setups and discuss the cases where our proposed algorithms are suitable. Our research highlights that our proposed algorithms will find extensive applications in risk-aware portfolio management problems.
Marginal Sharpe Ratio
Dragan Sestovic
The Marginal Sharpe Ratio (MSR) of an investment strategy with respect to a total portfolio can be defined as the derivative of the total portfolio Sharpe ratio (SR) over the allocation weight. Defined in such a way, MSR takes into account not only the contribution of the new strategy to the portfolio expected returns, but also the expected change of the portfolio risk profile due to diversification. It is similar to the well known concept of marginal risk. In this paper we derive analytical expressions for MSR. This leads to very simple and intuitive formulas, such as for example, MSR = (σ/σ_p) SR - β SR_p. Here SR and σ refer to the SR and the standard deviation of returns for the individual strategy. SR_p and σ_p refer to the SR and the standard deviation of returns for the base portfolio. β refers to the beta coefficient of the linear regression of the strategy returns against the total portfolio returns. The formula can be used for quick, "back-of-envelope" calculations for appraisals of new investment opportunities taking into account the risk/return profile of a base portfolio. The use of the MSR formula is demonstrated with simple numerical experiments. * Any opinions, findings, and conclusion or recommendations expressed in this material are those of the author and do not necessarily reflect the view of the Abu Dhabi Investment Authority.
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks
Felix Kircher & Daniel Rösch
We consider the problem of maximizing the out-of-sample Sharpe ratio when portfolio weights have to be estimated. We apply an improved bootstrap-based estimator, and an approximative estimator derived from a Taylor series. In a simulation study and empirical analysis with 15 datasets the proposed estimators outperform the minimum variance and equally weighted portfolio strategies. Out-of-sample Sharpe ratios improve by 15 and 32 percent on average, respectively, in the empirical analysis. While effectively dealing with estimation risks, the estimators produce considerable amounts of turnover. Realized net Sharpe ratios improve by 40 percent on average when the effects of accruing transaction costs are incorporated ex-ante into estimation of portfolio weights. When adding a risk-free asset, net Sharpe ratios remain of the same magnitude and portfolio volatility does not exceed a predefined target level.
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality
David H. Bailey & Marcos Lopez de Prado
With the advent in recent years of large financial data sets, machine learning and high-performance computing, analysts can backtest millions (if not billions) of alternative investment strategies. Backtest optimizers search for combinations of parameters that maximize the simulated historical performance of a strategy, leading to backtest overfitting.
The problem of performance inflation extends beyond backtesting. More generally, researchers and investors tend to report only positive outcomes, a phenomenon known as selection bias. Not controlling for the number of trials involved in a particular discovery leads to over- optimistic performance expectations.
The Deflated Sharpe Ratio (DSR) corrects for two leading sources of performance inflation: Selection bias under multiple testing and non-Normally distributed returns. In doing so, DSR helps separate legitimate empirical findings from statistical flukes.
Computation of the marginal contribution of Sharpe ratio and other performance ratios
Eric Benhamou & Beatrice Guez
Computing incremental contribution of performance ratios like Sharpe, Treynor, Calmar or Sterling ratios is of paramount importance for asset managers. Leveraging Euler's homogeneous function theorem, we are able to prove that these performance ratios are indeed a linear combination of individual modified performance ratios. This allows not only deriving a condition for a new asset to provide incremental performance for the portfolio but also to identify the key drivers of these performance ratios. We provide various numerical examples of this performance ratio decomposition.
References
Computation of the marginal contribution of Sharpe ratio and other performance
ratios. April 2021. Benhamou, E. and Guez, B. Université Paris-Dauphine Research Paper
Forthcoming. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3824133
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and
Non-Normality. July 2014. Bailey, D.H. and López de Prado, M. Journal of Portfolio
Management, 40 (5), pp. 94-107. 2014 (40th Anniversary Special Issue). Available at SSRN: http://dx.doi.org/10.2139/ssrn.2460551
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models. January 2025. Mulligan, J.; Jacquier, A. and Muhle-Karbe, J. Available at SSRN: http://dx.doi.org/10.2139/ssrn.5086171
Marginal Sharpe Ratio. August 2024. Sestovic, D. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4916095
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits.
January 2025. Khurshid, S., Abdulla, M.S. and Ghatak, G. Machine Learning, 114 (32) Available at Springer: https://doi.org/10.1007/s10994-024-06680-2
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks.
December 2021. Kircher, F. and Rösch, D. Journal of Banking & Finance, 133 (1). Available at Elsevier: https://doi.org/10.1016/j.jbankfin.2021.106281
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores the Sharpe ratio, examining both its practical use and the risks of overfitting, selection bias, and estimation errors. It highlights statistical corrections for backtest inflation, algorithmic refinements for risk-adjusted returns, and portfolio construction techniques that balance theoretical improvements with implementation challenges.
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models
Joseph Mulligan, Antoine (Jack) Jacquier & Johannes Muhle-Karbe
We study how much the in-sample performance of trading strategies based on linear predictive models is reduced out-of-sample due to overfitting. More specifically, we compute the in-and out-of-sample means and variances of the corresponding PnLs and use these to derive a closed-form approximation for the corresponding Sharpe ratios. We find that the out-of-sample "replication ratio" diminishes for complex strategies with many assets based on many weak rather than a few strong trading signals, and increases when more training data is used. The substantial quantitative importance of these effects is illustrated with an empirical case study for commodity futures following the methodology of Gârleanu and Pedersen.
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits
Sabrina Khurshid, Mohammed Shahid Abdulla & Gourab Ghatak
Sharpe ratio (SR) is a critical parameter in characterizing financial time series as it jointly considers the reward and the volatility of any stock/portfolio through its mean and standard deviation. Deriving online algorithms for optimizing the SR is particularly challenging since even offline policies experience constant regret with respect to the best expert (Even-Dar et al., 2006). This paper focuses on optimizing the regularized square SR (RSSR) by considering two settings: regret minimization (RM) and best arm identification (BAI). In this regard, we propose a novel multiarmed bandit (MAB) algorithm for RM called UCB-RSSR for RSSR maximization. We derive a path-dependent concentration bound for the estimate of the RSSR. Based on that, we derive the regret guarantees of UCB-RSSR and show that it evolves as O(log n) for the two-armed bandit case played for a horizon n. We also consider algorithms for the fixed budget setting of the BAI problems, i.e., sequential halving and successive rejects, and propose SHSR and SuRSR algorithms. We derive the upper bound for the error probability of BAI algorithms. We demonstrate that UCB-RSSR outperforms the only other known SR optimizing bandit algorithm, U-UCB (Cassel et al., 2023). We also study the efficacy of proposed BAI algorithms for 6 different setups and discuss the cases where our proposed algorithms are suitable. Our research highlights that our proposed algorithms will find extensive applications in risk-aware portfolio management problems.
Marginal Sharpe Ratio
Dragan Sestovic
The Marginal Sharpe Ratio (MSR) of an investment strategy with respect to a total portfolio can be defined as the derivative of the total portfolio Sharpe ratio (SR) over the allocation weight. Defined in such a way, MSR takes into account not only the contribution of the new strategy to the portfolio expected returns, but also the expected change of the portfolio risk profile due to diversification. It is similar to the well known concept of marginal risk. In this paper we derive analytical expressions for MSR. This leads to very simple and intuitive formulas, such as for example, MSR = (σ/σ_p) SR - β SR_p. Here SR and σ refer to the SR and the standard deviation of returns for the individual strategy. SR_p and σ_p refer to the SR and the standard deviation of returns for the base portfolio. β refers to the beta coefficient of the linear regression of the strategy returns against the total portfolio returns. The formula can be used for quick, "back-of-envelope" calculations for appraisals of new investment opportunities taking into account the risk/return profile of a base portfolio. The use of the MSR formula is demonstrated with simple numerical experiments. * Any opinions, findings, and conclusion or recommendations expressed in this material are those of the author and do not necessarily reflect the view of the Abu Dhabi Investment Authority.
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks
Felix Kircher & Daniel Rösch
We consider the problem of maximizing the out-of-sample Sharpe ratio when portfolio weights have to be estimated. We apply an improved bootstrap-based estimator, and an approximative estimator derived from a Taylor series. In a simulation study and empirical analysis with 15 datasets the proposed estimators outperform the minimum variance and equally weighted portfolio strategies. Out-of-sample Sharpe ratios improve by 15 and 32 percent on average, respectively, in the empirical analysis. While effectively dealing with estimation risks, the estimators produce considerable amounts of turnover. Realized net Sharpe ratios improve by 40 percent on average when the effects of accruing transaction costs are incorporated ex-ante into estimation of portfolio weights. When adding a risk-free asset, net Sharpe ratios remain of the same magnitude and portfolio volatility does not exceed a predefined target level.
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality
David H. Bailey & Marcos Lopez de Prado
With the advent in recent years of large financial data sets, machine learning and high-performance computing, analysts can backtest millions (if not billions) of alternative investment strategies. Backtest optimizers search for combinations of parameters that maximize the simulated historical performance of a strategy, leading to backtest overfitting.
The problem of performance inflation extends beyond backtesting. More generally, researchers and investors tend to report only positive outcomes, a phenomenon known as selection bias. Not controlling for the number of trials involved in a particular discovery leads to over- optimistic performance expectations.
The Deflated Sharpe Ratio (DSR) corrects for two leading sources of performance inflation: Selection bias under multiple testing and non-Normally distributed returns. In doing so, DSR helps separate legitimate empirical findings from statistical flukes.
Computation of the marginal contribution of Sharpe ratio and other performance ratios
Eric Benhamou & Beatrice Guez
Computing incremental contribution of performance ratios like Sharpe, Treynor, Calmar or Sterling ratios is of paramount importance for asset managers. Leveraging Euler's homogeneous function theorem, we are able to prove that these performance ratios are indeed a linear combination of individual modified performance ratios. This allows not only deriving a condition for a new asset to provide incremental performance for the portfolio but also to identify the key drivers of these performance ratios. We provide various numerical examples of this performance ratio decomposition.
References
Computation of the marginal contribution of Sharpe ratio and other performance
ratios. April 2021. Benhamou, E. and Guez, B. Université Paris-Dauphine Research Paper
Forthcoming. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3824133
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and
Non-Normality. July 2014. Bailey, D.H. and López de Prado, M. Journal of Portfolio
Management, 40 (5), pp. 94-107. 2014 (40th Anniversary Special Issue). Available at SSRN: http://dx.doi.org/10.2139/ssrn.2460551
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models. January 2025. Mulligan, J.; Jacquier, A. and Muhle-Karbe, J. Available at SSRN: http://dx.doi.org/10.2139/ssrn.5086171
Marginal Sharpe Ratio. August 2024. Sestovic, D. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4916095
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits.
January 2025. Khurshid, S., Abdulla, M.S. and Ghatak, G. Machine Learning, 114 (32) Available at Springer: https://doi.org/10.1007/s10994-024-06680-2
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks.
December 2021. Kircher, F. and Rösch, D. Journal of Banking & Finance, 133 (1). Available at Elsevier: https://doi.org/10.1016/j.jbankfin.2021.106281
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition explores the Sharpe ratio, examining both its practical use and the risks of overfitting, selection bias, and estimation errors. It highlights statistical corrections for backtest inflation, algorithmic refinements for risk-adjusted returns, and portfolio construction techniques that balance theoretical improvements with implementation challenges.
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models
Joseph Mulligan, Antoine (Jack) Jacquier & Johannes Muhle-Karbe
We study how much the in-sample performance of trading strategies based on linear predictive models is reduced out-of-sample due to overfitting. More specifically, we compute the in-and out-of-sample means and variances of the corresponding PnLs and use these to derive a closed-form approximation for the corresponding Sharpe ratios. We find that the out-of-sample "replication ratio" diminishes for complex strategies with many assets based on many weak rather than a few strong trading signals, and increases when more training data is used. The substantial quantitative importance of these effects is illustrated with an empirical case study for commodity futures following the methodology of Gârleanu and Pedersen.
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits
Sabrina Khurshid, Mohammed Shahid Abdulla & Gourab Ghatak
Sharpe ratio (SR) is a critical parameter in characterizing financial time series as it jointly considers the reward and the volatility of any stock/portfolio through its mean and standard deviation. Deriving online algorithms for optimizing the SR is particularly challenging since even offline policies experience constant regret with respect to the best expert (Even-Dar et al., 2006). This paper focuses on optimizing the regularized square SR (RSSR) by considering two settings: regret minimization (RM) and best arm identification (BAI). In this regard, we propose a novel multiarmed bandit (MAB) algorithm for RM called UCB-RSSR for RSSR maximization. We derive a path-dependent concentration bound for the estimate of the RSSR. Based on that, we derive the regret guarantees of UCB-RSSR and show that it evolves as O(log n) for the two-armed bandit case played for a horizon n. We also consider algorithms for the fixed budget setting of the BAI problems, i.e., sequential halving and successive rejects, and propose SHSR and SuRSR algorithms. We derive the upper bound for the error probability of BAI algorithms. We demonstrate that UCB-RSSR outperforms the only other known SR optimizing bandit algorithm, U-UCB (Cassel et al., 2023). We also study the efficacy of proposed BAI algorithms for 6 different setups and discuss the cases where our proposed algorithms are suitable. Our research highlights that our proposed algorithms will find extensive applications in risk-aware portfolio management problems.
Marginal Sharpe Ratio
Dragan Sestovic
The Marginal Sharpe Ratio (MSR) of an investment strategy with respect to a total portfolio can be defined as the derivative of the total portfolio Sharpe ratio (SR) over the allocation weight. Defined in such a way, MSR takes into account not only the contribution of the new strategy to the portfolio expected returns, but also the expected change of the portfolio risk profile due to diversification. It is similar to the well known concept of marginal risk. In this paper we derive analytical expressions for MSR. This leads to very simple and intuitive formulas, such as for example, MSR = (σ/σ_p) SR - β SR_p. Here SR and σ refer to the SR and the standard deviation of returns for the individual strategy. SR_p and σ_p refer to the SR and the standard deviation of returns for the base portfolio. β refers to the beta coefficient of the linear regression of the strategy returns against the total portfolio returns. The formula can be used for quick, "back-of-envelope" calculations for appraisals of new investment opportunities taking into account the risk/return profile of a base portfolio. The use of the MSR formula is demonstrated with simple numerical experiments. * Any opinions, findings, and conclusion or recommendations expressed in this material are those of the author and do not necessarily reflect the view of the Abu Dhabi Investment Authority.
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks
Felix Kircher & Daniel Rösch
We consider the problem of maximizing the out-of-sample Sharpe ratio when portfolio weights have to be estimated. We apply an improved bootstrap-based estimator, and an approximative estimator derived from a Taylor series. In a simulation study and empirical analysis with 15 datasets the proposed estimators outperform the minimum variance and equally weighted portfolio strategies. Out-of-sample Sharpe ratios improve by 15 and 32 percent on average, respectively, in the empirical analysis. While effectively dealing with estimation risks, the estimators produce considerable amounts of turnover. Realized net Sharpe ratios improve by 40 percent on average when the effects of accruing transaction costs are incorporated ex-ante into estimation of portfolio weights. When adding a risk-free asset, net Sharpe ratios remain of the same magnitude and portfolio volatility does not exceed a predefined target level.
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality
David H. Bailey & Marcos Lopez de Prado
With the advent in recent years of large financial data sets, machine learning and high-performance computing, analysts can backtest millions (if not billions) of alternative investment strategies. Backtest optimizers search for combinations of parameters that maximize the simulated historical performance of a strategy, leading to backtest overfitting.
The problem of performance inflation extends beyond backtesting. More generally, researchers and investors tend to report only positive outcomes, a phenomenon known as selection bias. Not controlling for the number of trials involved in a particular discovery leads to over- optimistic performance expectations.
The Deflated Sharpe Ratio (DSR) corrects for two leading sources of performance inflation: Selection bias under multiple testing and non-Normally distributed returns. In doing so, DSR helps separate legitimate empirical findings from statistical flukes.
Computation of the marginal contribution of Sharpe ratio and other performance ratios
Eric Benhamou & Beatrice Guez
Computing incremental contribution of performance ratios like Sharpe, Treynor, Calmar or Sterling ratios is of paramount importance for asset managers. Leveraging Euler's homogeneous function theorem, we are able to prove that these performance ratios are indeed a linear combination of individual modified performance ratios. This allows not only deriving a condition for a new asset to provide incremental performance for the portfolio but also to identify the key drivers of these performance ratios. We provide various numerical examples of this performance ratio decomposition.
References
Computation of the marginal contribution of Sharpe ratio and other performance
ratios. April 2021. Benhamou, E. and Guez, B. Université Paris-Dauphine Research Paper
Forthcoming. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3824133
The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and
Non-Normality. July 2014. Bailey, D.H. and López de Prado, M. Journal of Portfolio
Management, 40 (5), pp. 94-107. 2014 (40th Anniversary Special Issue). Available at SSRN: http://dx.doi.org/10.2139/ssrn.2460551
In-Sample and Out-of-Sample Sharpe Ratios for Linear Predictive Models. January 2025. Mulligan, J.; Jacquier, A. and Muhle-Karbe, J. Available at SSRN: http://dx.doi.org/10.2139/ssrn.5086171
Marginal Sharpe Ratio. August 2024. Sestovic, D. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4916095
Optimizing sharpe ratio: risk-adjusted decision-making in multi-armed bandits.
January 2025. Khurshid, S., Abdulla, M.S. and Ghatak, G. Machine Learning, 114 (32) Available at Springer: https://doi.org/10.1007/s10994-024-06680-2
A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks.
December 2021. Kircher, F. and Rösch, D. Journal of Banking & Finance, 133 (1). Available at Elsevier: https://doi.org/10.1016/j.jbankfin.2021.106281
