# 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: $$\beta = (r_i - r_f) / (r_m - r_f)$$

About the second question:

analytically speaking the answer of Kenneth Chen is correct but let me add that this representation is valid in Single Index Model case and not necessarily in CAPM. In CAPM setting, strictly speaking, the is no way to answer for you question because CAPM say not enough about second moments. See here Difference between CAPM and single index model