Why Portfolio Beta Cannot Be Inferred from Volatility Alone

Summary

The document presents a CAPM multiple choice problem asking for the beta of an equally weighted portfolio. It supplies the market volatility, each stock’s volatility, and the portfolio volatility, then considers decomposing each stock’s variance into market driven and residual components. The missing quantities are the individual betas and residual variances.

The question highlights an identification limit: the given volatilities do not specify how each stock’s returns co-move with the market, which is needed to determine beta. Nor do the supplied aggregate volatility figures alone pin down the residual risk structure or cross-stock relationships. The text offers no answer or supporting dataset, and asks whether further information is required; the stated inputs are insufficient to uniquely calculate portfolio beta without additional covariance or beta information.

Key ideas

• Beta depends on an asset’s covariance with the market, not on its volatility alone.
• The individual stock volatilities do not reveal their market sensitivities.
• Portfolio volatility alone does not determine the portfolio beta from the stated inputs.
• Additional covariance, beta, or equivalent factor exposure information is needed to solve the problem.

Tags

# Calculation of portfolio beta (CAPM)


# Calculation of portfolio beta (CAPM)












Let the market risk be $\sigma_m=28\%$. A portfolio consists of four stocks, all with the same weight ($w_i=0.25$ for all $i$). We also know that $\sigma_a=18\%,\sigma_b=36\%,\sigma_c=22\%,\sigma_d=17\%$ where $\sigma_i$ is the standard deviation of stock $i$ and the portfolio's risk is $\sigma_p=24\%$

Calculate the portfolio beta ?

This is a multiple choice question with choices:(A) 0.71 (B) 0.74 (C) 0.77 (D) 0.79 (E) 0.82

I tried using the formulas $\sigma_p^2=\beta_p^2\sigma_m^2+\sum_{i=1}^n{w_i^2\sigma_{\epsilon,i}^2}$ and $\sigma_i^2=\beta_i^2\sigma_m^2+\sigma_{\epsilon,i}^2$ but i get stuck because I know neither the individual betas nor the error terms' variances.

Am I missing something or is the given data insufficient? Any tips?

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