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:

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