Question

The trade balance, B X-M where X Exports, M- imports Show, in details, that the variance of B V(X)V(M) -2Cov(X,M)

Answer 1

Let B=X-M

Var(B)= E[ (B – E(B)² ]     

= E(B²) - B² where B² is the expected value of E(B)

=E[ (X-$\small \mu$_X)² - 2(M-$\small \mu$_M)(X-$\small \mu$_X) + (M-$\small \mu$_M)² ]  

= Var (X) - 2Cov(X,M) + Var(M)

Since we know Var(X-Y)= Var(X) + Var(Y) - 2Cov(X,Y)

and  $\small \mu$_X= E(X) and  $\small \mu$_M=E(M)