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# lec2 - Financial Data Analysis Professor S Kou Department...

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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

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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) ² S 2 ³ ± 1 ( ^ R ° r 1 ) = ° ± n ° N ° 2 n ° 1 ± ´ ^ C µ ± 1 ( ^ R ° r 1 ) : To show this estimator is unbiased we use the fact that S 2 and ^ R are inde- pendent, and ² S 2 ³ ± 1 has an inverse Wishart distribution with mean C ± 1 n ° N ° 2 : Indeed, E [ ^ w ] = ° ± ( n ° N ° 2) ± E h ² S 2 ³ ± 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 one based on an unbiased estimate of the covariance matrix (i.e. without the term n ± N ± 2 n ± 1 ), and that the MLE (obtained using ´ 1 n ± 1 S 2 µ ± 1 to estimate C ± 1 ). However, it is still an open problem to compute
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lec2 - Financial Data Analysis Professor S Kou Department...

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