CAPM Beta and the Market Share of Return Variance
Summary
The document asks how to infer a stock’s beta from return expectations and how to attribute its variance to market risk. One response writes returns as a market-linked component plus an idiosyncratic residual. Under the single-index assumptions that the residual is uncorrelated with the market, total variance separates into a beta-scaled market variance and residual variance. The market share is the systematic component divided by total stock variance; the response also identifies this share with squared correlation between the stock and market returns.
A key qualification in another reply is that this variance decomposition follows from a single-index model and is not guaranteed by CAPM alone, which does not specify enough about second moments. The question of determining beta also requires the stock’s expected return if using the expected-return equation; knowing only the risk-free rate and expected market return is insufficient. Thus the formulas rely on assumptions beyond the CAPM expected-return relation.
Key ideas
- Under a single-index model, stock return variance separates into systematic market variance and residual variance.
- The systematic share of variance is the beta-squared market variance divided by total stock variance.
- Under the stated assumptions, this share equals the squared correlation between stock and market returns.
- The variance decomposition is not implied by CAPM alone, and calculating beta from expected returns also requires the stock’s expected return.
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# Fundamental CAPM questions
# Fundamental CAPM questions
A couple questions about the CAPM model:
- If I only know the riskfree rate and expected market return, how do I solve for $\beta$ ?
- 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?
I searched this question through google, but I cannot find reasonable answer. Please help me! Thanks in advance for your help.
## Answer by Kenneth Chen (score 5)
https://quant.stackexchange.com/a/21425
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
$$\beta_i = \frac{\text{Cov}[r_i,r_M]}{\text{Var} [r_M]}$$
## Answer by Mark Joshi (score 0)
https://quant.stackexchange.com/a/21418
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.
## Answer by markowitz (score 0)
https://quant.stackexchange.com/a/45169
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 modelShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.