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Bootstrap With Example. The following papers give good and simple examples of the use of the bootstrap methods for statistical inference. 1] Vinod, H. D. and M. R. Morey (2000) “Confidence Intervals and Hypothesis Testing for the Sharpe and Treynor Performance Measures: A Bootstrap Approach,” in Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo and A. S. Weigend (Ed.s) Computational Finance 1999 MIT Press, Cambridge Mass., 2000. Chapter 3, pages 25-39. 2] Vinod, H. D. and M. R. Morey (2001) “A Double Sharpe Ratio” in Advances in Investment Analysis and Portfolio Management, Vol. 8, Lee, Cheng Few, (ed.), New York, 2001, JAI-Elsevier Science pp. 57–65. See Abstracts and Electronic copies of some of my recent papers are available on my author page at the SSRN Electronic Library at: http://ssrn.com/author=139267 this is the oldest in the list on top The point of this article is to show that besides risk associated with the ups and downs of the market, there is also some risk associated with the fact that the characteristics of the probability distribution of market returns is not known exactly, but estimated from random data. We suggest a new way of incorporating estimation risk with an additional denominator to the estimated Sharpe Ratio. We start with some notation from financial economics. Let ri,t represent the excess return from the i-th portfolio in period t, where i = 1, 2, ..., n. A random sample of T excess returns on the n portfolios is then illustrated by r ' t [ r1t , r2 t ,... rnt ], where t = 1, 2, ..., T and where rt is assumed to be multivariate normal random variable, with mean = { i }, i = 1, 2, ..., n and a covariance matrix = ( ij ) where i , j 1, 2,... n. It is well-known that the unbiased estimators of the (n x 1) mean vector and the (n x n) covariance matrix are respectively, r T 1 rt , T t 1 and S sij 1 T ( r r )( rt r )'. T 1 t 1 t (1) These two estimators are then used to form the estimators of the traditional Sharpe performance measure. The population value of the Sharpe (1966) performance measure for portfolio i is defined as Sh i i , i = 1, 2, ..., n. It is simply the mean excess return over the standard i deviation of the excess returns for the portfolio. The conventional sample-based point estimates of the Sharpe performance measure used in (1) are then ri for i = 1, 2, ..., n. Sh i si (2) The Sharpe ratio is defined in equation (2) as the ratio of the mean excess return to its standard deviation. We define the Double Sharpe ratio by DSh i Sh i , sSh i (3) where s sh i is the standard deviation of the Sharpe ratio estimate, or the estimation risk. As is clear in (3), the Double Sharpe penalizes a portfolio for higher estimation risk. Because of the presence of the random denominator si in the definition of (2), the Sharpe ratio does not permit an easy method for evaluating the estimation risk in the point estimate itself. This is because the small-sample distribution of the Sharpe measure is non-normal and hence the usual method based on the ratio of the statistic to its standard errors is biased and unreliable. Same problem holds for our double Sharpe Ratio. What is bootstrap methodology? Assume a statistic, , is based on a sample of size, T. In the bootstrap methodology, instead of assuming the shape of the sampling distribution of , one empirically approximates the entire sampling distribution of by investigating the variation of over a large number (say 999) of pseudo samples obtained by re-sampling the same data. For the re-sampling, a Monte-Carlo simulation is used on the available sample values. This is conducted by randomly drawing, with replacement. We create a large number (999) of resamples of size T from the original sample. Each resample has T elements, however any given resample could have some of the original data points represented more than once and some not at all. Note that, by construction, each element of the original sample has the same probability, (1/T), of being in a sample. The initial idea behind bootstrapping was that a relative frequency distribution of ’s calculated from the resamples can be a good approximation to its sampling distribution. This idea has since been extended to conditional models and one-step conditional moments.1 In this paper, 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 s sh i . BOOTSTRAP FOR THE REGRESSION PROBLEM y = X + with say T=50 observations estimate b by OLS and compute the residuals e=yXb For example, let b* denote resampled regression coefficients, b the original coefficients, and the unknown parameters. The extended bootstrap approximates the properties of (b- ) by the observable (b*b). See Davison and Hinkley (1997) for recent references and Vinod (1993) for references to earlier attempts. 1 y = Xb + reshuffled residuals (with replacement, i.e., with repetition) bootstrap method is to create 999 regression problems. Create 999 sets of y data (each with 50 observations) Now you have 999 regression problems. Estimate some 999 times and find b* estimates. Now the sampling distribution approximation is available by comparing b is approximated by b*b