Minimizing Risk

Probability Level 3

The prices (in dollars) of two stocks are random variables X X and Y Y 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 P P with

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

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

2 3 \frac{2}{3} 1 2 \frac{1}{2} 0 1

Andrew Ellinor by Andrew Ellinor , 33, USA

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1 solution

Andrew Ellinor
Jan 18, 2016

Let's compute the variance of the estimator P ^ : \hat{P}:

Var ( P ^ ) = Var ( a X ) + Var [ ( 1 − a ) Y ] = a 2 Var ( X ) + ( 1 − a ) 2 Var ( Y ) = 4 a 2 + 8 ( 1 − a ) 2 = 12 a 2 − 16 a + 8. \begin{aligned} \text{Var}(\hat{P}) &= \text{Var}(aX) + \text{Var}[(1 - a)Y] \\ &= a^2\text{Var}(X) + (1 - a)^2\text{Var}(Y) \\ &= 4a^2 + 8(1 - a)^2 \\ &= 12a^2 - 16a + 8. \end{aligned}

This quadratic, as with any concave up quadratic, is minimized at x = − b 2 a x = -\dfrac{b}{2a} , which in this instance is 2 3 . \dfrac23.

11 Helpful 0 Interesting 2 Brilliant 0 Confused

how did you put in 4 an 8 in the place of Var[X] and Var[Y] ?

Hafiz muhammad Ibrahim jaffar - 11 months, 1 week ago

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