Estimating and Forecasting Betas

Estimating and Forecasting Betas

Estimating and forecasting betas for stocks that are potential candidates for inclusion in a portfolio is a prerequisite to apply the single-index model. Analysts could be asked to provide subjective estimates of beta for a security or a portfolio. Conversely, estimates of future beta could be arrived at by estimating beta from past data and using this historical beta as an estimate of the future beta. 

There is evidence that historical betas provide useful information about future betas. Usually, we estimate the location of the line using regression analysis. Such estimates are subject to error. As such, the estimate may not be equal to the true value that existed in the period. Furthermore, the process is complicated by the fact that historical betas are not perfectly stationary over time. Researchers concluded that order bias, security-beta nonstationarity, and thin trading account for their regression tendency bias. We would expect changes as the fundamental characteristics of the firm change because, as a risk measure, it should be related to the capital structure of the firm and, thus, it should change as the capital structure changes; obviously, markets in which firms grow faster are more affected than others. Despite such errors, the most straightforward way to forecast Beta for a future period is to use an estimate obtained via regression analysis from a past period. Note that because portfolio betas are measured with less error (errors in estimating beta for individual securities are thought to cancel), and because betas on portfolios change less than betas on securities, historical betas on portfolios are better predictors of future betas than are historical betas on securities.

The value of estimated betas

When considering the task of estimating and forecasting betas, researchers (Elton, Gruber, and Urich, 1978) have compared the ability of the following models to forecast the correlation structure between securities: 

  1. the historical correlation matrix itself 
  2. forecasts of the correlation matrix prepared by estimating betas from the prior historical period
  3. forecasts of the correlation matrix prepared by estimating betas from the prior two periods and updating via the Blume technique 
  4. forecasts prepared as in the third model but where the updating is done via the Vasicek Bayesian technique 

One of the most striking results of the study was that the historical correlation matrix itself was the poorest of all techniques. In most cases, it was outperformed by all of the beta forecasting techniques at a statistically significant level. This indicates that a large part of the observed correlation structure between securities, not captured by the single- index model, represents random noise with respect to forecasting. The point to note is that the single-index model, developed to simplify the inputs to portfolio analysis and thought to lose information because of the simplification involved, actually does a better job of forecasting than the full set of historical data.

Relationship between Beta, Correlation and Relative Volatility

In the single index model, the equity beta is the product of the market correlation and the relative volatility of the portfolio with respect to the index or benchmark. The correlation is bounded above and below by +1 and −1 and the relative volatility is always positive. So the portfolio beta can be very large and negative if the portfolio is negatively correlated with the market, which happens especially when short positions are held. On the other hand, very high values of beta can be experienced for portfolios containing many risky stocks that are also highly correlated with the market.

Application: Estimating a 3 Years Monthly Beta

The last three years of monthly CLOSE prices are required:

  • The monthly prices should be the close price on the first trading day of each month (note that this is not necessarily the first day of each month).
  • The end date should be the first trading day of the month prior to the current month
  • The start date should be the first trading day of the month 36 months prior to the end date

The following Jupyter notebook produces the computation:

Other possible estimations for historical beta

The task of estimating and forecasting betas can e accomplished in different ways. Indeed, there are several different methods to calculate beta depending on the frequency of data (monthly, weekly, etc.), the sample period (2 years, 3, years, 5 years, etc.), and fundamental considerations (levered, unlevered, relevered, etc.). However, when considering stocks’ betas:

  • Yahoo compute a 3 years monthly beta including the average of the unfinished current month against the S&P 500 Index
  • Bloomberg compute a 2 years weekly beta including the average of the unfinished current month against the S&P 500 Index
  • Value Line compute a 5 years quarterly beta including the average of the unfinished current month against the NYSE Index

Finally, raw betas are often adjusted to account for beta’s mean reverting property according to different formulas and methods:

  • Bloomberg: Adjusted beta = (.67) * Raw beta + (.33) * 1.0
  • Generalized: Adjusted beta = (.75) * Raw beta + (.25) * 1.0

Note that betas computed for CAPM (not historical betas) involve selecting an industry segment or a set of similar firms, compute the unlevered beta, and apply the selected capital structure to obtain the levered beta for the model.

References

Elton, E., Gruber, M., Brown, S., Goetzmann, W. (2007). Modern Portfolio Theory and Investment Analysis, 11th edition.

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