Chapter 2 Some comments one Assignment 6

2.1 Exercise 5.9

Suppose that there are just two scenarios \(\omega_1\) and \(\omega_2\) and consider two risky securities with returns \(K_1\) and \(K_2\). Show that \(K_1 = aK_2 + b\) for some numbers \(a \neq 0\) and \(b\), and deduce that \(\rho_{12} = 1\) or −1.

First, recall that \(\rho_{12}= \frac{Cov(K_1,K_2)}{\sigma_1\sigma_2}\). In order to check the value of \(\rho_{12}\), we can first check the numerator and then check the denominator.

• Numerator:

\[ Cov(K_1,K_2) = Cov(aK_2+b,K_2) = aCov(K_2,K_2)= aVar(K_2)=a\sigma_2^2 \]

This holds because:

\[ Var(x) = a\\ Var(bx) = b^2a\\ Var(x+b) = a\\ Cov(x,y) = a\\ Cov(bx,by) = b*b*a=b^2a\\ Cov(bx,y) = ba \]

• Denominator: we want to find the relation between \(\sigma_1\) and \(\sigma_2\), i.e., we want to know the function \(f(x)\), such that \(\sigma_1^2 = f(\sigma_2^2)\).

\[ \sigma_1^2 = Var(K_1) = Var(aK_2+b) = a^2Var(K_2) = a^2\sigma_2^2\\ \Longrightarrow \sigma_1 = |a|\sigma_2\\ \]

These together give us:

\[ \rho_{12}=\frac{a\sigma_2^2}{|a|\sigma_2^2}=\frac{a}{|a|} = 1 \text{ or } -1 \]