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Unformatted text preview: Financial Data Analysis Professor S. Kou, Department of IEOR, Columbia University Lecture 2. Estimation Risk and Portfolio Selection: Empirical Bayesian Estimators and the Black-Litterman Bayesian model Although the mean variance analysis is a major breakthrough in &nance, relatively few people in practice use it directly. Some main problems are (1) Sometimes it is hard to estimate the covariance matrix C , although it is a big problem if the number of assets involved is small. (2) The result is very sensitive to the mean vector R , which is very di cult to be estimated. (3) The portfolio weights are sensitive to estimation errors in C and R . (4) The whole theory is based on the quadratic utility function, which itself is problematic. The literature is quite rich on the estimation risk topic: Jobson and Ko- rkie (1980, 1981), Frost and Savarino (1986, 1988), Jorion (1986), Michaud (1989, 2000), Best and Grauer (1991), Black and Litterman (1992), Scherer (2004); see a survey paper by Brandt (2005) and the book by Meucci (2005). There are two main approaches to handle the di culties. First, there are empirical Bayesian estimation procedures, which are the ones that we shall discuss next. Later in this lecture we shall consider a popular Bayesian approach to do portfolio optimization, the Black-Litterman (1992) model, which is used by Goodman Sachs, . A key di/erence between the two ap- proaches is that the empirical Bayesian approaches only depend on historical data, while Bayesian approaches rely on both data and subjective views. 1 Problems with the empirical means and covari- ances Suppose that we observe a vector of N stock returns X 1 , X 2 , ..., X n during n periods. The mean vector is R 1 & N , and covariance is C N & N . One way to estimate the mean vector R is to simply use the sample mean vector ^ R ; ^ R = 1 n n X i =1 X i : Now assuming that X 1 , X 2 , ..., X n are i.i.d. random vector with the mul- tivariate normal distribution. Then S 2 = n X i =1 ( X i & ^ R ) > ( X i & ^ R ) has a Wishart distribution with E & S 2 = ( n & 1) C : 1 To estimate C one can use the sample covariance ^ C = 1 n & 1 S 2 : The commend in Splus for ^ C is var . Recall that in the presence of the risk-free rate, the vector of optimal mean variance e cient portfolio weights is given by w = & C & 1 ( R & r 1 ) To get an unbiased estimator for the portfolio weights we shall use an unbiased estimator ^ w = & ( n & N & 2) & S 2 & 1 ( ^ R & r 1 ) = & n & N & 2 n & 1 ^ C & 1 ( ^ R & r 1 ) : To show this estimator is unbiased we use the fact that S 2 and ^ R are inde- pendent, and & S 2 & 1 has an inverse Wishart distribution with mean C & 1 n & N & 2 : Indeed, E [ ^ w ] = & ( n & N & 2) E h & S 2 & 1 i E ( ^ R & r 1 ) = & ( n & N & 2) C & 1 n & N & 2 ( R & r 1 ) = & C & 1 ( R & r 1 ) = w: Kan and Zhou (2007) show that the above unbiased portfolios per- form better (highest expected out-of-sample performance) respect to the...
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This note was uploaded on 10/16/2010 for the course IEOR 4709 taught by Professor Stevenkou during the Fall '10 term at Columbia.
- Fall '10
- Financial Engineering