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Using CAPM to Assess a Portfolio Manager’s Return

Article Quant Q&A · Author: Lost1
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Summary

The document evaluates a portfolio manager’s claim that a 10% loss was good performance in a year when the market fell 30%. With a beta of 0.2 and a risk-free rate of 5%, the CAPM expected return is calculated as −2%. Comparing the observed return with that benchmark gives a negative alpha, so the result is worse than expected under the model, despite being less negative than the market return.

That comparison is conditional on CAPM being an appropriate model. The discussion notes that CAPM assumptions may not hold in practice. It also distinguishes an expected return from the outcome in one realized year: without estimates of beta uncertainty and residual return variance, the confidence interval around actual performance cannot be assessed. The available information supports a point estimate comparison, but not a statistical judgment about whether the manager’s underperformance is significant.

Key ideas

  • CAPM estimates the expected portfolio return from the risk-free rate and the portfolio’s market beta.
  • The stated beta and market return imply an expected loss of 2% for the year.
  • A realized loss of 10% is below the CAPM estimate, even though it is smaller than the market loss.
  • A single year’s result cannot be judged for statistical significance without return and beta uncertainty estimates.
  • The conclusion depends on CAPM assumptions, which may not hold in practice.

Tags

StatisticsRisk managementPortfolio construction
Full text
# Question 1.18 from Hull's Financial Risk management CAPM


# Question 1.18 from Hull's Financial Risk management CAPM












A portfolio manager maintains an active portfolio with beta of 0.2. Risk-free rate is 5% The market return for a particular year is -30% The fund produced a result of -10%. He claimed the return was good given the circumstances - discuss.

So the expected return should be:

5% +0.2x(-30%-5%)=-2%

So the alpha is negative and this is bad performance.

Is there anything more that can be said about this apart from this naive calculation?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/14739

I think there is not too much to say. At first glance it looks good if the manager loses $10\%$ if the whole market loses $30\%$.

But plugging the beta and the risk-free rate into the CAPM formula we see that we would have expected a loss of $2\%$ only. So the $10\%$ are much worse than expected.

Note however that there are various reason's why CAPM just does not hold. But in the text book world, where CAPM holds perferctly. The manager simply did worse than expected.

## Answer by James (score 0)

https://quant.stackexchange.com/a/14749

There would be a lot more to say if we could take into account that the -2% is the $expected$ return, but the confidence interval for an actual (observed) return can be of any width, e.g. (-22%; 20%). However, we can't compute it here because the problem doesn't supply us with the variance of beta and the error term from the fitted CAPM.

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.