Regression of beta toward the mean

Beta measures how much a stock moves with the market. A beta of 1.00 means it moves with the market exactly.

When researchers tested CAPM, they checked whether a beta measured today still holds tomorrow. It often does not — and it drifts in a particular direction.

Portfolios with a high beta tended to see it fall over time. Portfolios with a low beta tended to see it rise. Both moved towards 1.00.

That is the regression of beta toward the mean. The mean here is the market's own beta of 1.00.

It matters for a simple reason. If you build a portfolio on a measured beta, the beta you measured is not the beta you will live with. A historical beta needs treating as a starting point, not a fact.

Where the finding comes from (section 16.10). Testing CAPM essentially revolved around testing the stability of beta as a measure of risk, and the relationship between beta and the realised rate of return. The workbook reports four key results, and this is the second of them:

• Beta is unstable for individual stocks but stable for portfolios. The larger the portfolio — the workbook's example is over 50 stocks — and the longer the period — over 26 weeks — the more stable the beta estimate.
• Betas tended to regress toward the mean. Specifically, high-beta portfolios tended to decline over time towards 1.00, whereas low-beta portfolios tended to increase over time towards 1.00.
• Sharpe and Cooper found a positive relationship between return and risk, although not completely linear.
• Black, Jensen and Scholes examined risk and return for portfolios of stocks and found a positive linear relationship between monthly excess return and portfolio beta.

Why the two findings belong together. Beta is stable enough to use at the portfolio level, but even there it does not stand still — it decays towards the market's own beta of 1.00. So a portfolio built to a target beta of 1.5 will, left alone, tend to become less aggressive; one built to 0.6 will tend to become less defensive.

What follows for a manager. A beta estimated from the past has to be maintained if it is to be held, which is one of the reasons beta is listed in Chapter 17 among the parameters a risk framework controls.

Illustrative figures applying the workbook's finding. A PMS runs two strategies, each of Rs 5 crore, built on measured three-year betas.

Now price what the drift costs the aggressive client. Suppose the market rises 10% in the fourth year and the risk-free rate is 6%.

• At the beta the client signed up for, 1.60, the strategy's expected excess return over the risk-free rate is 1.60 x 4% = 6.4%, or Rs 32,00,000 on Rs 5 crore over and above the risk-free return.
• At the drifted beta of 1.32, it is 1.32 x 4% = 5.28%, or Rs 26,40,000.

The gap of Rs 5,60,000 is not a manager mistake and not a market event. It is the portfolio quietly becoming something less aggressive than the mandate.

The defensive client has the opposite problem. Her beta has risen from 0.55 to 0.78, so in a 20% market fall she is now exposed to roughly 0.78 of it rather than 0.55 — on Rs 5 crore, that is about Rs 23,00,000 of extra downside she never asked for. Both clients need the beta re-measured and the portfolio brought back to mandate.

Chapter 16 (Introduction to Capital Market Theory), section 16.10 (Empirical test of CAPM), lists the four key results of the CAPM tests, of which this is one. The section is short and is examined as a list.

Expect a question asking in which direction betas were found to move (both high and low betas move towards 1.00), or one pairing the finding with the stability result — beta unstable for individual stocks, stable for portfolios, more stable for portfolios of over 50 stocks measured over more than 26 weeks. The named researchers, Sharpe and Cooper and Black, Jensen and Scholes, and what each found, are also fair game.

• Both directions move to 1.00. It is not only that high betas fall; low betas rise. A question offering just one half of the movement is wrong.
• 1.00 is the mean here — the market's own beta, not the average beta of the manager's own portfolio.
• This is a finding about portfolios, not single stocks. The workbook separately reports that beta is unstable for individual stocks, so the regression result is stated at the portfolio level.
• Over 50 stocks and over 26 weeks are the workbook's own figures for when a beta estimate becomes stable. They are easy marks and easy to forget.
• Black, Jensen and Scholes found a linear relationship; Sharpe and Cooper found a positive but not completely linear one. The pairing of name to finding is examinable.
• Regression toward the mean is not the same as mean reversion in prices. This is about the risk measure drifting, not about a stock price returning to a level.