Minimizing Risk

The prices (in dollars) of two stocks are random variables XX and YY with E(X)=E(Y)=P.E(X) = E(Y) = P. Estimators of these prices have variances of 4 and 8 respectively. Let P^\hat{P} be an unbiased estimator of PP with

P^=aX+(1a)Y.\hat{P} = aX + (1 - a)Y.

For what value of aa is the variance of P^\hat{P} minimized?

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