CAPM Beta, Idiosyncratic Risk, and Large Return Residuals
Summary
The document addresses two CAPM questions: how equal betas relate to total risk when assets have different regression R-squared values, and how to interpret a historical average return far above the model estimate. Under the CAPM framing described, beta scales the market risk premium, while R-squared indicates how much return variation the market regression explains. If two companies have the same beta but different R-squared values, the lower-R-squared company has more residual, idiosyncratic risk; this does not mean CAPM predicts greater total risk from beta alone.
The second example reports a beta of 1.41, an R-squared of 0.27, an estimated return of 11%, and a historical average of 60% with 70% volatility. The answer treats the gap as an unusually large apparent alpha and suggests checking for calculation errors or inconsistent inputs. It does not identify a specific failed CAPM assumption; it cautions that the sample may be short and asks whether returns are annualized.
Key ideas
- Equal beta implies equal CAPM compensation for systematic risk, not equal total volatility.
- A lower regression R-squared indicates that more return variation is left unexplained by the market factor.
- A large gap between a CAPM estimate and historical average appears as an alpha relative to the model.
- Check the return period, annualization, and calculations before attributing a large residual to a failed assumption.
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Full text
# Two specific questions about CAPM's assumptions and implications # Two specific questions about CAPM's assumptions and implications I have two questions about the CAPM model: the first is theoretical while the second is related to observed market data. First question: let's say we have company A and company B and we want to estimate their expected returns within the CAPM model. In this model the percentage of systematic risk is given by the $R^2$ of the regression of an investment's returns and market portfolio returns. Let's say that company A has 30% systematic risk while company B has 80% systematic risk. Then CAPM tells us that the Beta of an investment (market returns coefficient in the CAPM regression) is to be used as a scaling factor of the market portfolio expected extra-return in order to obtain the investment expected extra-return. Let's now suppose that the two companies have the same Beta. What are the implications of this problem in the CAPM model? Is the model suggesting that Company A has a much greater total risk than Company B? Second question: if we use real data to estimate Betas and we obtain the following results: - Beta = 1.41 - $R^2$=0.27 - CAPM estimated expected return = 11% but the historical average return for the same period is 60% (with a volatility of 70%). What would most likely be the CAPM assumption that did not hold? ## Answer by phdstudent (score 3) https://quant.stackexchange.com/a/74083 - If the two companies have indeed the same beta, then company A will have more total risk. Or in other words company A has more idiosyncratic risk (i.e. the risk that is not explained by the model); - If the historical average return is 60% (is this yearly?) and the CAPM implies an expected return of 11% for that company, that means that the company has a massive alpha (of 49%). There is likely either an error or an inconsistency in your calculations since that alpha seems massively large (unless the time-series is short).
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