5. Sharpe Model Function
We use Sharpe Model Function for measuring a portfolio performance, which is used as a Fitness Function for Genetic Algorithms.
According to Sharpe's model a satisfactory simplification would be to abandon the covariances of each security with other security and to substitute information on the relationship of each security of the market. In his terms, it is posible to consider the return of each security to be represented by the following equation:
where Ri is the return on security, ai and bi are parameters, ci is a random variable with an expected value of zero, and I is the level of some index, in our project a common stock price index. Sharpe's idea that the return on a security varies with its sensitivity to changes in the market(as measured by bi) implies something about the pricing of assets and the relationship of price to this sensitivity. This would lead to an equation of the form E(Ri)=ai+bi*E(Rm), where E(Rm)-the expected return of the market as a whole. Thus, the expected return and variance of any portfolio P are the simply
E(Rp) = ap + bp * E(Rm) and sp = bp2 * sm + S(ai*si) where ap= S(ai * ai), bp=S(ai*bi) and si - the variance of the error term of security i.
Using this model, and selecting portfolios with genetic algorithms, project present selected
Advantages of the market model: This model radicaly reduces both the number of variables needed to determine efficient portfolios and the calculation to find these variables. For each security to be considered, all that needed is ai,bi and sigma(i). These we calculated using regrssion by ploting actual past values of Ri and Rm against each other.