by Hans Tallis Leave a Comment

A paper titled “Best Practice for Cost-of-Capital Estimates” (Levi and Welch, 2017) focuses on implementation recommendations for CAPM-based cost of capital estimates. The authors’ focus isn’t on finding the “right” cost of capital (which many take to be the expected return for the asset), rather they focus on methodological choices that produce the most stable values over time: namely, creating CoC estimates that best predict next period’s CoC.

(Beta can (via CAPM) forecast future returns with an R2 of ~5% — or less — but can forecast future beta with an R2 of ~50%.)

Given the empirical weaknesses of CAPM, this seems like a useful approach for “best practices” guidance. We’ll summarize the findings in this post. The full paper is an excellent read.

### Problems in Estimating Cost of Capital

Before we present the recipe, here’s a summary of the issues in estimating cost of capital that the authors address:

- CAPM is, by some measurements, used by 90% of management teams to determine cost of capital
- Notwithstanding, Jagannathan points out that most firms’ hurdle rates are notably higher than their CAPM-derived costs of capital.

- Beta estimation is subject to standard statistical errors. Specifically, low beta estimates are likely to understate systematic risk, and high estimates are likely to overstate risk and with credit hero training is possible to avoid financial risks.
- Beta changes over time: companies change asset mix, capital structure, etc. Moreover, as shown in Figure 2 of the Levi and Welch paper, sector betas change meaningfully over time.
- Most academic papers ignore these problems, and several proposed fixes have been nearly “forgotten”.
- Smaller firms have lower beta (averaging 0.5); larger average 0.9. (This might be due to index trading artifacts; see our earlier post on this.)
- The adjustments below may increase the ability of the CoC estimates to forecast actual returns, but these improvements are at best marginal (and suffer from CAPM’s fundamental weakness in this regard). The CoC adjustments were determined to maximize e.g. the ability of beta to forecast one-year future beta — which enhances the stability of CoC estimates. That said, these adjustments do not degrade the ability to forecast returns; as such, there is no reason not to use them.

### Recipe to Calculate Stable Equity Beta Estimates

- Calculate CAPM using historical prices covariance: 1-3 years’ history, daily prices.
- If beta is assumed to be stable, use a longer period.
- To estimate local betas for non-US markets, use one year’s history.

- “Shrink” the beta using the Vasicek adjustment, which biases the estimate toward the prior estimate (1.0, or sector beta if available and preferred) for less confident (i.e. higher standard error) estimates. The Vasicek adjustment improves the R2 for future beta forecasts from 50% to 57%.
- The formula is available in the Levi and Welch paper, and another treatment is given by Gray and Hall, 2013.) Assuming the raw beta was estimated by OLS regression, define c = SE(beta_raw)^2 / [ SE(beta_raw)^2 + SEE^2 ], where SE(beta_raw) is standard error for beta_raw, and SEE is the standard error of the estimate. Then Beta_Vasicek = (1-c) * Beta_prior + c * Beta_raw, where Beta_prior is our prior expectation for beta, usually 1.0 or alternately a sector average beta.
- An alternative is to include industry and firm size in the shrinkage calculation, per Karolyi (1992). However while Karolyi’s approach improves the predictive power of adjusted beta (on next period beta) over Vasicek’s, the improvement is not large.

- Shrink the beta
*a second time*: beta = 0.25 + 0.70 * (Beta_Vasicek). This improves the forecasting ability of the adjusted beta to 69%. (These coefficients were determined by a straightforward regression of forecasted beta on adjusted beta; the 0.25/0.70 values were from a regression the authors calculated in 2013. These values compare will with the Bloomberg/Blume beta adjustment of 0.33/0.67.)- Alternately, if covariances were measured on monthly (vs. daily) prices, shrink using coefficients of 0.50/0.50.
- For smaller firms, it may be better to shrink toward the small-firm average of 0.50, instead of the target of 0.95 that the above coefficients provide.

- For capital budgeting purposes, note that betas seem to mean revert over time. The paper suggests that, compared to 30% shrinkage for one-year beta forecasts, use 40% for ten-year beta forecasts.

Finally, note that “*Industry betas are practically useless for individual stocks*… the corporate finance textbook suggestion of using industry market betas `because they are less noisy’ is not justifiable.” Measured by the ability to forecast a firm’s future beta, peer average beta has an R2 of 0.30 vs. an R2 of 0.50 for the firm’s own beta.

Finally, a few author recommendations on market risk premium (based on historical returns):

- Prefer geometric to arithmetic averaging
- MRP to short-term bonds appears to be 5-7%; to long-term bonds appears to be 2-4%