Bayesian portfolio selection: an empirical analysis of JSE-ALSI 2003-2010

3. METHODOLOGY AND MODELLING

a) Mean-VaR Portfolio Selection

The expected return on the portfolio by minimizing the variance of return will be like this:

Min 12XT?Xsubject to { XTi = 1 and XTu = up (1)

We assume in the model that the market has n assets, the rate of return of a single asset i is a random variable and the expected rate of return for the asset is estimated through the observed return of the asset at time t. Then ri =1Tt=1TRit (2)

The mean conditional VaR; Max {E (rp) - CVaRá (rp); this is the alternative (or supplement) to VaR and it capture the risk metric for continuous distributions.

For continuous distributions, CVaR, also known as the Mean Excess Loss, Mean Shortfall, and Tail VaR, is the conditional expected loss given that the loss exceeds VaR. That is, CVaR is given by

öá(x)= ?(1- á)?^2 ?_(f (x,y)>æ(x,á)) f (x, y)p(y)dy, (3)

The method to be used in this framework first is the Mean-Variance that determines the weight of the portfolio and the mean VaR weight in order to have the mean of return and the portfolio risk. The following step is looking to the budget risk where we have the mean of the portfolio return and the portfolio risk according to the Mean Conditional Value at Risk (MCVaR).

b) Bayesian portfolio selection

The Bayesian portfolio selection is coming from conditional probability distribution and it provides inferences about the distribution of returns which is unknown by applying available signal or information. The inference makes the probability statements about the generating process of return.

Bayesian theorem is used to determine what one's probability for the hypothesis should be, once the outcome D from the experiment is known. The probability of the hypothesis once the outcome from the experiment is known is called the posterior probability^2(*).

p(èA) = P(A,?)p(A)=pA?p(?)p(A) p(èA)?P(Aè)P(è) Posterior?Likelihood x Prior (4) The prior is normally distributed and the posterior probability is proportional to the likelihood of the observed data, multiplied by the prior probability, and is given by Bayes' theorem. As the investor with rational behavior will provide a predictive mean-variance p(Rt+1At) (5) The equation (4) is called the posterior distribution of future returns Let t denote the time, R the expected return and A the historical return data observed.

To predict we observe A, and compute p(A) the predictive distribution

P(A) = ?p(è)p(èA)dè (6). The regression model will be Yi=Xiß+?i where ?i~Normal(0,ói) (7).

Bayesian regression model of joint distribution of Y and X

P(X,Y)= P(XØ)P(YX,â,s2) (8) Where Ø is a priori or a set of information independent of (â,ó).

At this stage we will compare two methods of Black Litterman where the first one will consist on the covariance through the BLCOP in order to get the weight and the risk of the portfolio by using the historical returns and the other stage will be based on the Bayesian CAPM which generates the alpha and beta coefficient of each stock by using the Markov Chain Monte Carlo package in R software (MCMC pack), we applying the excess return observed and the alpha in addition with the beta which forms the product with the excess market estimate in order to obtain the excess return of each stock estimate and after that we use the covariance of this dataset according to BLCOP where we have the weight and risk of the portfolio.

* 2 Slides of Bayesian asset pricing