Independent solution

How to solve this Linear Combinations of Random Variables question

Setup

Setup

Let u be the units assigned to the first asset, leaving 10-u units in the second asset.

S=uX+(10−u)YS=uX+(10-u)Y

Model

Model

Independence removes the covariance term, while each allocation coefficient is squared in the variance.

Var⁡(S)=30u2+20(10−u)2\operatorname{Var}(S)=30u^2+20(10-u)^2
Var⁡(S)=50u2−400u+2000\operatorname{Var}(S)=50u^2-400u+2000

Compute

Compute

Differentiate the convex quadratic and solve its first-order condition.

dduVar⁡(S)=100u−400\frac{d}{du}\operatorname{Var}(S)=100u-400
100u−400=0⟹u=4100u-400=0\quad\Longrightarrow\quad u=4
d2du2Var⁡(S)=100>0\frac{d^2}{du^2}\operatorname{Var}(S)=100>0

Answer

Answer

Four units should be assigned to the first asset.

4(C)\boxed{4\quad\text{(C)}}