Part I: Why Bayesian Beta?

Evan Thomas

— 2 min read

I came across both the Blume and Vasicek beta adjustments while working on estimating a regulated utility’s Return on Equity (ROE). One of the standard methods used in this context is the Capital Asset Pricing Model (CAPM), which takes the form: $$ R_i = R_f + \beta_i (R_m - R_f) $$ Here, $R_i$ is the expected return of security $i$, $R_f$ is the risk-free rate, $R_m$ is the expected market return, and $\beta_i$ measures the security’s sensitivity to movements in the overall market. It has become common to adjust betas in order to account for mean reversion, and two such adjustments are: Blume (1971) and Vasicek (1973).

Blume proposed a simple regression-based mean reversion of beta toward the market average, typically shown as: $$\beta_{adj} = \frac{1}{3} + \frac{2}{3}\beta_{raw}$$

Vasicek’s Bayesian adjustment, by contrast, uses a precision-weighted average to adjust betas on an individual basis: $$\beta_i^{\text{adj}} = w_i \bar{\beta}+ (1 - w_i)\hat{\beta_i}, \quad w_i = \frac{s_i^2}{s_i^2 + {\sigma_{\beta}} ^ 2} $$

Here, ${s_i}^2​$ is the sampling variance of the individual firm’s beta (estimated with regression) and ${\sigma_{\beta}} ^ 2$ is the cross-sectional variance of true betas across firms (the prior variance).

I wanted to understand both methods from first principles. Blume’s method is relatively straightforward, using historical regressions of individual betas over time. But in reviewing the Vasicek paper, it became clear that the Bayesian Statistics and decision theory were outside of my comfort zone. If I was going to recommend a ROE derived from a Vasicek-adjusted beta, I wanted to understand the math, not just the high level idea of "updating beliefs."

As it turns out, the Vasicek paper leaves out most of the derivation to the reader. So I had no choice but to dig deeper. That brought me to the Normal-Normal Bayesian update, the foundation of Vasicek's beta adjustment.

The next post will walk through how the posterior distribution is derived from a normal prior and a normal likelihood, but will start from the beginning with Bayes' theorem.

But that's only the first step, we can build the posterior distribution, but it doesn't tell us what estimate to report. Vasicek chooses the posterior mean, and the posterior is arranged into a recognizable function form so that the mean can be simply extracted. Even still, Vasicek doesn't provide an explicit defense of why the mean is the right choice.

I have now learned that that answer lies in decision theory. Choosing an estimate from the posterior requires the notion of loss, or a cost of being wrong. The posterior mean turns out to be the optimal decision.

Deriving the posterior distribution will be the focus of Part II. Part III will introduce decision theory and the role of loss functions. Finally, in Part IV, we'll return to Vasicek's model and tie the full Bayesian logic together.

Further reading

  • Blume (1971) - On the Assessment of Risk
  • Vasicek (1973) A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas

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