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Deriving a Stock’s Expected Return from CAPM and Equal-Weighted Market Data

Article Quant Q&A · Author: J. Seg
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Summary

The document outlines how to infer Stock Z’s expected return under CAPM when the market portfolio is composed of three stocks with equal weights. It proposes first calculating market variance from the component variances and covariances, then finding each stock’s beta as its covariance with the market divided by market variance. The correlation with the market can be derived by expanding the market return as the weighted sum of the three stock returns.

Once the betas for X and Y are known, their CAPM equations form two equations in the risk-free rate and expected market return. Solving those gives the market return, which can then be used with the market portfolio weights to infer Z’s expected return. The document gives a conceptual calculation path rather than numerical inputs or a worked result, and it assumes CAPM holds and the required return and covariance data are available.

Key ideas

  • Market variance for an equal-weighted portfolio depends on the variances and covariances of all component stocks.
  • A stock’s beta is its covariance with the market divided by market variance.
  • The covariance of a stock with the market can be expanded using the market portfolio’s weighted return.
  • CAPM equations for two stocks can be used to solve for the risk-free rate and expected market return.
  • The expected return of the remaining stock can then be inferred from the market portfolio relationship.

Tags

EquitiesStatisticsPortfolio construction
Full text
# Expected Return on Stock


# Expected Return on Stock












Suppose we have the following information on stocks $X$, $Y$, and $Z$:







Assume that the CAPM holds and that the market portfolio consists of the above three stocks weighted equally. Find the expected return of Stock Z.

Attempt: We can first get $\sigma_M$ by using the formula for the variance of a three-asset portfolio. Then, from there, we can solve for the $\beta$ for each stock using $\beta=\frac{\text{Cov}(R_i,R_M)}{\sigma_M^2}$. However, I'm not sure how to compute for the correlation between the stock return and the market return.

## Answer by JeanGuillaume (score 1)

https://quant.stackexchange.com/a/47582

To compute the correlation between the stock return (let us say $R_X$) and the market return $R_M$, you just write:

$\rho_{R_X,R_M} = \frac{Cov(R_X,R_M) }{\sigma_{R_X}\sigma_{R_M}} = \frac{Cov(R_X, \frac{1}{3}R_X + \frac{1}{3}R_Y + \frac{1}{3}R_Z ) }{\sigma_{R_X}\sigma_{R_M}} $

Once you get the market variance and the $\beta$, you just write the CAPM formula for $X$ and $Y$:

$E[R_X] = R_{rf} + \beta_{X,M}(E[R_M] - R_{rf})$

$E[R_Y] = R_{rf} + \beta_{Y,M}(E[R_M] - R_{rf})$

And then you go two équations with two unknows variables ($R_{rf}$ and $E[R_M]$). You solve it and finally you easily get $E[R_Z]$ from $E[R_M]$

Shown 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.