# 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:

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