The change in the price of Apples each month is normally distributed with a standard deviation of 10 cents. The change in the price of Oranges each month is normally distributed with a standard deviation of 20 cents. You can assume that there is only a single Apple price that all apples follow, and there is a single Orange price that all Oranges follow. These monthly price changes of Apples and Oranges have correlation of $0.6$. Is two apples and one orange more or less risky than owning four apples?
Lets call the random variable for apples $A$, and the rv for oranges, $O$.
$$Var(A_1 + A_2 + O) = Var(2A) + Var(O) + 2 Cov(2A,O) = 4Var(A) + Var(O) + 4Cov(A,O)$$
$$ = 4*0.1^2+0.2^2+4*0.6*0.2*0.1 = 0.128$$
and
$$Var(A_1 + A_2 + A_3 + A_4) = Var(4A) + 2 \sum_{1 \leq i \leq j \leq 4}Cov(A,A) $$ $$ = 16*0.1^2 + 2 \sum_{1 \leq i \leq j \leq 4}1*0.1*0.2 = 16*0.1^2 + 2*3!*1*0.1*0.1 = 0.28 $$
Is this correct? Im not sure whether the 4 apples should have correlation 1 as they are all from the same price series? Is there an easier way to spot the answer?
In the case of owning two apples and an orange, we have $$\operatorname{var}(2A+O)$$ $$=4\operatorname{var}(A)+\operatorname{var}(O)+4\operatorname{cov}(A, O)$$ $$=4\cdot 0.10^2+0.20^2+4\cdot 0.60\cdot 0.10\cdot 0.20$$ $$=0.04+0.04+0.048$$ $$=0.128$$ In the case of owning four apples, we have $$\operatorname{var}(4A)$$ $$=16\operatorname{var}(A)$$ $$=16\cdot 0.10^2$$ $$=0.16$$ Therefore, owning two apples and an orange is less risky than owning four apples.