Weight minimising variance of a blend of two variables

Question

$X_1$ and $X_2$ are independent with variances $25$ and $100$ . For $c\in[0,1]$ let $Y=cX_1+(1-c)X_2$ . Which $c$ gives $Y$ the smallest variance?

Short answer

$c=4/5$ . $\operatorname{Var}(Y)=25c^2+100(1-c)^2$ ; setting the derivative $50c-200(1-c)$ to zero gives $c=\tfrac45$ . In general the optimal weight is inversely proportional to variance: $\tfrac{1/25}{1/25+1/100}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Expectation, Mean, and Variance
Independence
Conditional Expectation and Variance Revisited

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free