ABSTRACT. involved therein. This paper has been motivated by the desire to meet the challenge of statistical estimation. A new estimator for

Transcription

1 A Shorter-Length Confidence-Interval Estimator (CIE) for Sharpe-Ratio Using a Multiplier k* to the Usual Bootstrap-Resample CIE and Computational Intelligence Chandra Shekhar Bhatnagar 1, Chandrashekhar.Bhatnagar@sta.uwi.edu Ashok Sahai 2, sahai.ashok@gmail.com Viswanadham Bulusu 3, viswanadhambulusu1970@gmail.com ABSTRACT The population value of the Sharpe (1966) performance measure for portfolio i is defined as i rf i for i = 1, 2,..., n. It is simply the mean excess return over the standard deviation of the excess returns for the portfolio. The sample estimate of any of these portfolio measures is challenging not only because of the dynamic nature of this measure but also because of the statistical estimation issue involved therein. This paper has been motivated by the desire to meet the challenge of statistical estimation. A new estimator for Sharpe s ratio is formulated using the optimal value defined as k*, which is used as a multiplier for the usual sample-counterpart estimator. This formulation has been pivotal to an efficient Confidence-Interval estimator (CIE). Computational Intelligence has been deployed for the optimal mixing [using the optimal value of a design parameter, θ] of the proposed estimator and usual sample-counterpart estimator with the aim of achieving the shortest length of the proposed bootstrap-resample Confidence-Interval estimator (PBCIE) for Sharpe s ratio. We have been successful in achieving a shorter length through the proposed Bootstrapresample Confidence Interval Estimator [PBCIE] for Sharpe s ratio vis-à-vis the Usual Bootstrap-resample Confidence Interval Estimator [UBCIE], without paying the usual cost in terms of a greater Coverage Error. An empirical simulation study has been used to bring forth the potential gain through a more efficient estimation in the context, with the limitation of the normal distribution being followed by the population for the portfolios. The empirical simulation study is modeled around the values of parameters in the study by Vinod and Morey (1999) Key Words & Phrases: Bootstrap resampling; Computational intelligence; Coverage error & the length of the confidence interval estimation; Empirical simulation study. 1 Department of Management Studies, The University of the West Indies,St. Augustine, Trinidad 2 Department of Mathematics and Statistics, The University of the West Indies, St. Augustine, Trinidad 3 Principal, Aurora s Degree & PG College, Chikkadpally; Hydearbad, India. 1

2 I. Introduction The Sharpe s Ratio (1966, 1994) is one of the standard tools in evaluating the performance of portfolios. It is intuitive and lends itself easily to computation and interpretation. It represents the excess return per unit of standard deviation of return and can be easily looked up as the slope of the capital allocation line (CAL). Since the Sharpe s ratio gives us the price of risk, it follows that the ratio is widely used in comparing portfolio performance and identifying superior portfolios. Using Roy s (1952) argument to use the reward to- risk ratio in appraising portfolio (or any strategy s) performance, the Sharpe s ratio modifies Markowitz s (1952) efficient frontier of risky assets by incorporating a proxy risk-free rate (or the rate offered by a benchamrk portfolio resulting in what we understand as the Information Ratio). From there, it facilitates the task of identifying the optimum portfolio that is tangent to the the efficient frontier, having a property of maximizing the slope of CAL. Returns are often not normal (Goetzmann, Ingersoll, Spiegel and Welch (2002)), and comparable over time (Sharpe (1994), Lo (2002)) and since they serve as the first data admission in computing Sharpe s ratio, there is bound to be an estimation error in the ratio itself. The recognition of non-normality as a contributing factor in estimation errors has always nagged the researchers, more so since Michaud (1989) examined and questioned the optimality of the optimized portfolio in the Markowitz mean-variance paradigm. Further, due to the presence of a random denominator in the definition of the ratio, the sampling distribution of the Sharpe ratio is somewhat difficult to determine, even when using data from large samples (Christie (2007)). Lo (2002) set the ball rolling with his work on the statistical distribution of the Sharpe ratio under various scenarios for the return distribution. Earlier, Jobson and Korkie (1981) did consider a test of the difference between Sharpe ratios under the assumption of multivariate normality, but found that their test lacked power. We tackle the challenge of statistical estimation by attempting to propose a new estimator for Sharpe s ratio to serve as an efficient Confidence-Interval estimator (CIE). We draw from the work of Vinod and Morey (2001) who proposed a modified version of the Sharpe ratio, called the Double Sharpe ratio, to take into account estimation risk. We use bootstrap sampling and computational intelligence with the aim of achieving the shortest length proposed bootstrap-resample Confidence-Interval estimator of the Sharpe s Ratio. The procedure followed is similar but not limited to Sahai and Strepnek (2011) In the next section, we specify the Sharpe s ratio and discuss our proposed bootstrap confidence interval (CI) estimator for the ratio. Section III illustrates the relative gain from our proposed estimator. Section IV concludes the paper. 2

3 II. The Proposed Bootstrap Confidence Interval Estimator for Sharpe s Ratio Let us say, there is an investment position, P, invested partially in a risky portfolio and a risk-free asset. If w represents the weight on the risky portfolio i, µ represents the expected return, and r f is the risk-free rate of return, the expecte return of position P can be described as: w (1 w) r (1) P i f We can re-write (1) as follows: w wr r, or P i f f r w[ r ] (2) P f i f Since the risk of combined position P, as measured by the standard deviation of expected return, would emanate from only the risky portfolio, we can write the standard deviation of position P as: P w, which further implies that; i P w (3) i Substituting (3) in (2) yields the capital allocation line, CAL; r r (4) P P f i f i From (4), it can be seen that r i i f is the slope of the CAL, also known as the Sharpe s ratio, Sr i. The population value of the Sharpe s performance measure for portfolio i is simply the mean excess return over the standard deviation of the excess returns for the portfolio. Therefore, for any portfolio in general, we have the population value of the Sharpe s ratio as r i i f. In the aforesaid μ is the population mean, r f is the riskfree rate of return with zero variance and σ i is the portfolio s standard deviation. It is commonly seen in the literature, that the sample counterparts that estimate these population parameters, namely the sample mean, and the sample standard deviation are calculated from a random sample (x 1, x 2, x 3,, x n ) as follows: 3

4 x n i1 n x i, s n 2 i1 x i x n 1 2, and s s 2 This leads to an applied point estimate for Sharpe s ratio of; Srˆ x s As indicated earlier, one problem with the Sharpe s ratio is that its denominator is random, as it is computed using a data sample of returns on a given history and not the whole population of returns. So, it is difficult to evaluate its risk estimation. Vinod and Morey (2001) proposed a modified version of the Sharpe s ratio, called the Double Sharpe ratio, to take into account the estimation risk. This ratio is defined as Srˆ DSrˆ, where Srˆ is the standard deviation of the Sharpe ratio estimate, or the estimation risk. They do not use the sample (x 1, x 2, x 3,, x n ) directly in the calculation of their Double Sharpe Ratio. Instead, to calculate this standard deviation they use bootstrap methodology to generate 999 resamples from the original returns. We note here that the sampling error is seminal to the estimation risk or the estimation error. Vinod and Morey (2001) s Double Sharpe ratio takes care of this estimation error implicitly, just like the Sharpe s ratio. Srˆ Our proposed improvement of Vinod and Morey (2001) s Double Sharpe Ratio also uses 999 bootstrap resamples. These resamples have been used to calculate the bootstrap mean of these estimates and hence Confidence Interval Estimate (CIE) of Sharpe s ratio. The improved Confidence Interval Estimate (CIE) of Sharpe s ratio constructed and proposed in this paper consists in estimation of both the numerators & the denominators of the ratio. For doing so, our first target is to find an efficient estimator of the inverse of the normal standard deviation 1. For that we first prove the following result: 4

5 Lemma: For a random sample (x 1, x 2, x 3,, x n ) from a normal population N (μ, σ), * 1 k. is the Minimum Mean-Square-Error Estimator (MMSE) of 1 ; wherein; 2 n. 1 ( n 1) 2 k* ( n 3) 2 Proof: It can easily be checked that; k * 1 1 E. s 1 E s 2 Rest follows from the well-known fact that (n-1). s 2 ~ 2 n-1. Q.E.D. (5) We can handle the efficient estimation of numerator Sr ˆ of the Double Sharpe ratio, by possibly a more efficient estimator using bootstrap resamples. This new estimator will take care of the estimator error, rather less explicitly. To illustrate our proceedings comprehensibly we, for the ready comparative reference, borrow the study by Vinod and Morey (1999) and to keep things in perspective as we proceed forward, we reproduce a selection from their work: the resampling for the Sharpe measure is done with replacement of the original excess returns themselves for j =1,2,, J or 999 times. Thus, we calculate 999 Sharpe measures from the original excess return series. The choice of the odd number 999 is convenient, since the rank-ordered 25-th and 975-th values of estimated Sharpe ratios arranged from the smallest to the largest, yield a useful 95% confidence interval. It is from these 999 Sharpe measures that we calculate ( Sˆ r ). As an illustration, we have calculated the Sharpe and Double Sharpe ratios for the 30 largest growth mutual funds. (as of January 1998 in terms of overall assets managed). 5

6 Their work reports the excess mean monthly returns, the standard deviation of the excess monthly returns, the Sharpe ratio, the mean and standard deviation of the bootstrapped Sharpe ratios, the lower [0.025] and upper [0.975] confidence values of the bootstrapped Sharpe value, the 95% confidence interval width, and the Double Sharpe ratio. They mention that in their case, the sampling distribution that represents the estimation risk is non-normal with positive skewness. This is why the means of the bootstrapped Sharpe ratios are always slightly higher than the point estimates of Sharpe ratios, which ignore the estimation risk altogether. In line with their work, we compute the Usual Bootstrap-resample Confidence-Interval Estimator [UBCIE] of Sharpe s ratio using the the lower [0.025] and upper [0.975] confidence values of the bootstrapped Sharpe value, say LCV & UCV respectively. Our Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE] of Sharpe s ratio has two phases. In the first phase, we propose the Modified Bootstrap-resample Confidence-Interval Estimator [MBCIE] of Sharpe s ratio by modifying the LCV & UCV with k*, as below: MLCV = k*. LCV (6) and, MUCV = k*. UCV (7) In the second phase, we consider the Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE] of Sharpe s ratio as the one with the Lower & Upper Confidence Values, as below: PLCV = LCV + θ.(lcv - k*.lcv) (8) and PUCV = UCV + θ.(ucv - k*.ucv) (9) The parameter θ is the Design Parameter in the Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE] of Sharpe-Ratio proposed to be used Optimally to design the proposed PBCIE is such a manner as to lower the its length without paying the usual price in terms of the Coverage Error with the CIE. This has been achieved by using the Computational Intelligence to determine that Optimal Value of the design-parameter θ. The extensive simulational computational results lead to this sought-after Optimal- Value of the design-parameter θ to be

7 III. Empirical Study Illustrating the Relative Gain through our Proposed CI Estimator The empirical simulation study is carried out using a Matlab 2010b code for various illustrative values of the sample sizes ~ 11, 21, 31, 41, 51, 71, 85 & 101. The parent population is taken to be normal with the population Sharpe ratio Sr as 0.20 [around the values of the case-study in Vinod and Morey (1999)]. The various values of population standard deviation (likewise in that study) are taken as σ = 3.25, 3.75, 4.25, 4.75, 5.25, 5.75, 6.25, 6.75 & 7.25 for illustration. The number of replications in the simulation is taken to be quite large ~ In each of these (4444) replications, for a typical value-combination of n & σ, we let the computer generate the psuedo-random sample from the relevant Normal Distribution with the value of σ, and that of the Sharpe Ratio (Sr = 0.2). That sample is then used for the 999 bootstrap-resamples leading to the Lower & Upper values of the various (95%) CIs; LCV & UCV for Usual Bootstrap-resample Confidence-Interval Estimator [UBCIE] for Sharpe s ratio & PLCV & PUCV for Proposed Bootstrap-resample Confidence- Interval Estimator [PBCIE] for Sharpe s ratio. Thence, from all these 4444 replications, we calculate for these CIs {[UBCIE] & [PBCIE]}, the following performance-icons values: Coverage Probability ~ The relative frequency of the actual value being inside the relevant CI. Coverage Error ~ Abs. ( Coverage Probability) for the relevant CI. Length ~ average of the values Abs.(Lower Value of CI Upper value of the CI) for the relevant CI. Left Bias ~ average of the values Abs.(Lower Value of CI Actual Value of SR (0.2)) {iff the actual value is less than the lower value of the CI} for the relevant CI. Right Bias ~ average of the values Abs.(Upper Value of CI Actual Value of SR (0.2)) {iff the actual value is more than the upper value of the CI} for the relevant CI. Left Bias - Right Bias Relative Bias = Abs. Left Bias + Right Bias The resultant values of these Features of CIs are tabulated in Tables A.1 through A.9, in the appendix. As apparent from the tabulated values in the appendix, our Proposed Bootstrap-resample Confidence-Interval Estimator [PBCIE] of Sharpe s ratio is consistently shorter in its length than the Usual Bootstrap-resample 7

8 Confidence-Interval Estimator [UBCIE] of Sharpe s ratio. Nevertheless, we have evidently manipulated [through our optimal choice of the value of the design-parameter θ in our PBCIE of Sharpe s ratio using Computational Intelligence ] for the said Optimal mixing of UBCIE of Sharpe s ratio & the Modified Bootstrap-resample Confidence-Interval Estimator [MBCIE] of Sharpe s ratio. IV. Conclusion The Sharpe's ratio is one of the widely prevalent metrics for evaluating and comparing portfolio performance. It relates the reward of investing with its inherent risk. However, in a world where financial data does not strictly conform to desired statistical attributes, the ratio s reliability may be questioned due to statistical estimation error. This study seeks to and achieves an improvement in so far as a more robust statistical estimation is evident by using bootstrap resampling and computational intelligence. The simulation results demonstrate a marked improvement in terms of a shorter length of the proposed bootstrap-resample Confidence-Interval estimator (PBCIE) for Sharpe s ratio, while keeping the coverage error in check. This has been achieved by using k* as the multiplier for the usual sample counterpart estimator for the Sharpe s ratio of the population. Using computational intelligence, a unique design parameter, θ, has been arrived at for an optimal mixing of the proposed estimator and usual sample-counterpart estimator. References: Goetzmann, William N, Jonathan E Ingersoll, Matthew I Spiegel, and Ivo Welch, 2002, Sharpening Sharpe Ratios, Yale Working Paper H.D. Vinod and M.R. Morey, 1999, A Double Sharpe Ratio, Fordham University, New York H.M. Markowitz, 1952, Portfolio Selection, Journal of Finance, 7(1), Jobson, J.D. and B.M. Korkie, 1981, "Performance Hypothesis Testing with the Sharpe and Treynor Measures," Journal of Finance, 36, No. 4, Lo, Andrew W, 2002, The Statistics of Sharpe Ratios, Financial Analysts Journal, Michaud, Richard O, 1989, The Markowitz Optimization Enigma: Is 'Optimized' Optimal?, Financial Analysts Journal,

9 Roy, Arthur D., 1952, Safety First and Holding of Assets, Econometricia, Sahai, Ashok and Strepnek, Grant. H, 2011, An Estimation Error Corrected Sharpe Ratio using Bootstrap Resampling, Journal of Applied Finance and Banking, Vol.1, No.2, S. Christie, 2007, Beware the Sharpe Ratio, Macquarie University Applied Finance Centre, Sydney Vinod, H. D. and M. R. Morey, 2001 A Double Sharpe Ratio in Advances in Investment Analysis and Portfolio Management, Vol. 8. W.F. Sharpe, 1966, Mutual Fund Performance, Journal of Business, 39, W.F. Sharpe, 1994, The Sharpe Ratio, Journal of Portfolio Management, 21(1),

10 APPENDIX SIMULATION RESULTS [4444REPLICATIONS] FOR 95% CI : THE PROPOSED VIS-A-VIS THE USUAL BOOTSTRAP CI FOR Sr USING k* AND COMPUTATIONAL INTELLIGENCE Table A.1 n = 11, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

11 Table A.2 n = 21, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

12 Table A.3 n = 31, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

13 Table A.4 n = 41, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

14 Table A.5 n = 51, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

15 Table A.6 n = 61, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

16 Table A.7 n = 71, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

17 Table A.8 n = 85, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator

18 Table A.9 n = 101, k * = σ = % CI Estimator for SR Coverage Probability Coverage Error Length Left Bias Right Bias Relative Bias UBCIEstimator PBCIEstimator σ = UBCIEstimator PBCIEstimator σ = 4.25 UBCIEstimator PBCIEstimator σ = 4.75 UBCIEstimator PBCIEstimator σ = 5.25 UBCIEstimator PBCIEstimator σ = 5.75 UBCIEstimator PUBCIEstimator σ = 6.25 UBCIEstimator PBCIEstimator σ = 6.75 UBCIEstimator PBCIEstimator σ = 7.25 UBCIEstimator PBCIEstimator