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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=yXb
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