Beta Beta is a measure of systematic risk. It measures the responsiveness of a stock's expected return to changes in the value of the whole stock market. Let R p = return on a portfolio of n assets R i = return on asset i = 1, . .., n X i = proportion of the portfolio held in asset i (1) R p = X 1 R 1 + X 2 R 2 + . .. + X n R n The expected return on this portfolio equals (2) E(R p ) = E(X 1 R 1 ) + E(X 2 R 2 ) + . .. + E(X n R n ) A measure of the risk for this portfolio is the standard deviation of the portfolio's return or its squared value, the variance of the portfolio's return, s p 2 . (3) s p 2 = E[R p- E(R p )] 2 This simplifies to (4) s p 2 = X 1 s 1p + X 2 s 2p + . .. + X n s np , where s ip equals the covariance of the return on asset i with the portfolio's return. It is a measure of how the return on asset i changes with changes in the return of the whole portfolio. Equation (4) tells us that the contribution to risk of asset i to the portfolio is X i s ip . So, the proportion of risk contributed by asset i equals
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