# Fundamental CAPM questions

A couple questions about the CAPM model:

1. If I only know the riskfree rate and expected market return, how do I solve for $\beta$ ?

2. Given the stock's variance, how do I solve the percentage of it that is due to market risk and how do I interpret this?

Because you have CAPM therefore the following holds:

$$r_i = r_f + \beta_i (r_M - r_f) + \epsilon_i$$

where $r_i$ is the expected return of stock $i$, $r_f$ is the risk free return and $r_M$ is the expected market return, and $\epsilon$ is an idiosyncratic return adjustment or an error.

Now if you take the $\text{Var}[\cdot]$ operator over the equation above you should have.

$$\begin{split} \text{Var}[r_i] & =\text{Var} \left [ r_f + \beta_i (r_M - r_f) + \epsilon_i \right ] \\ & = \text{Var}[r_f] + \beta_i^2 \text{Var} [r_M - r_f] + \text{Var} [\epsilon_i] \\ & = 0 + \beta_i^2 \text{Var} [r_M] + \text{Var} [\epsilon_i] \\ & = \beta_i^2 \text{Var} [r_M] + \text{Var} [\epsilon_i] \end{split}$$

This is the relation you're looking for, it's a decomposition of variance. (Notice there's no covariance terms by the assumption of CAPM). It tells you that the variance of your stock return is two-fold. First, it comes from the systematic risk where you bear from market risk, namely $\beta_i^2 \text{Var} [r_M]$. Second, each stock has its idiosyncratic risk, which is $\text{Var} [\epsilon_i]$.

Now in this problem you have $\beta_i$, $\text{Var} [r_M]$ and $\text{Var}[r_i]$ given. What percentage of this variance is due to market risk you ask? That's just

$$\frac{\beta_i^2 \text{Var} [r_M]}{\text{Var}[r_i]}$$

Now if you want to be even more convenient. You can re-write the above relation as

$$\frac{\beta_i^2 \text{Var} [r_M]}{\text{Var}[r_i]} = \frac{\text{Cov}^2[r_i,r_M]}{\text{Var} [r_M] \text{Var} [r_i]} = \rho_i^2$$

This is because

$$\beta_i = \frac{\text{Cov}[r_i,r_M]}{\text{Var} [r_M]}$$

Hint:

write the return of stock B as

$R_B=\beta R_M + e$

where $e$ is uncorrelated to $R_M.$ Find a formula for its variance and substitute all the terms you know.

I'm not sure to understand correctly the first question but if so: you have to know the stock return also. In this case the answer is trivial: $$\beta = (r_i - r_f) / (r_m - r_f)$$