Estimating the cost of equity for a private company

In assessing the cost of equity for publicly traded firms, we looked at the risk of investments through the eyes of the marginal investors in these firms. With the added assumption that these investors were well diversified, we were able to define risk in terms of risk added on to a diversified portfolio or market risk. The beta (in the CAPM) and betas (in the multi-factor models) that measure this risk are usually estimated using historical stock prices. The absence of historical price information for private firm equity and the failure on the part of many private firm owners to diversify can create serious problems with estimating and using betas for these firms.

#### Approaches to Estimating Market Betas

The standard process of estimating the beta in the capital asset pricing model involves running a regression of stock returns against market returns. Multi-factor models use other statistical techniques, but they also require historical price information. In the absence of such information, as is the case with private firms, there are three ways in which we can estimate betas.

##### 1. Accounting Betas

While price information is not available for private firms, accounting earnings information is. We could regress changes in a private firmÕs accounting earnings against changes in earnings for an equity index (such as the S&P 500) to estimate an accounting beta.

D EarningsPrivate firm = a + b D EarningsS&P 500

The slope of the regression (b) is the accounting beta for the firm. Using operating earnings would yield an unlevered beta, whereas using net income would yield a levered or equity beta.

There are two significant limitations with this approach. The first is that private firms usually measure earnings only once a year, leading to regressions with few observations and limited statistical power. The second is that earnings are often smoothed out and subject to accounting judgments, leading to mismeasurement of accounting betas.

##### 2. Fundamental Betas

There have been attempts made by researchers to relate the betas of publicly traded firms to observable variables such as earnings growth, debt ratios and variance in earnings. Beaver, Kettler, and Scholes (1970) examined the relationship between betas and seven variables - dividend payout, asset growth, leverage, liquidity, asset size, earnings variability and the accounting beta. Rosenberg and Guy (1976) also attempted a similar analysis. The following is a regression that we ran relating the betas of NYSE and AMEX stocks in 1996 to four variables: coefficient of variation in operating income (CVOI), book debt/equity (D/E), historical growth in earnings (g) and the book value of total assets (TA).

Beta = 0.6507   + 0.25 CVOI   + 0.09 D/E + 0.54 g   - 0.000009 TA         R2=18%

where

CVOI    = Coefficient of Variation in Operating Income

= Standard Deviation in Operating Income/ Average Operating Income

We could measure each of these variables for a private firm and use these to estimate the beta for the firm. While this approach is simple, it is only as good as the underlying regression. The low R2 suggests that the beta estimates that emerge from it are likely to have large standard errors.

##### 3. Bottom-up Betas

When valuing publicly traded firms, we used the unlevered betas of the businesses that the firms operated in to estimate bottom-up betas Š the costs of equity were based upon these betas. We did so because of the low standard errors on these estimates (due to the averaging across large numbers of firms) and the forward looking nature of the estimates (because the business mix used to weight betas can be changed). We can estimate bottom-up betas for private firms and these betas have the same advantages that they do for publicly traded firms. Thus, the beta for a private steel firm can be estimated by looking at the average betas for publicly traded steel companies. Any differences in financial or even operating leverage can be adjusted for in the final estimate.

In making the adjustment of unlevered betas for financial leverage, we do run into a problem with private firms, since the debt to equity ratio that should be used is a market value ratio. While many analysts use the book value debt to equity ratio to substitute for the market ratio for private firms, we would suggest one of the following alternatives.

a. Assume that the private firmÕs market leverage will resemble the average for the industry. If this is the case, the levered beta for the private firm can be written as:

b private firm = bunlevered (1 + (1 - tax rate) (Industry Average Debt/Equity))

b. Use the private firmÕs target debt to equity ratio (if management is willing to specify such a target) or its optimal debt ratio (if one can be estimated) to estimate the beta.

b private firm = bunlevered (1 + (1 - tax rate) (Optimal Debt/Equity))

The adjustment for operating leverage is simpler and is based upon the proportion of the private firmÕs costs that are fixed. If this proportion is greater than is typical in the industry, the beta used for the private firm should be higher than the average for the industry.

Betas measure the risk added by an investment to a diversified portfolio. Consequently, they are best suited for firms where the marginal investor is diversified. With private firms, the owner is often the only investor and thus can be viewed as the marginal investor. Furthermore, in most private firms, the owner tends to have much of his or her wealth invested in the private business and does not have an opportunity to diversify. Consequently, it can be argued that betas will understate the exposure to market risk in these firms.

At the limit, if the owner has all of his or her wealth invested in the private business and is completely undiversified, that owner is exposed to all risk in the firm and it is not just the market risk (which is what the beta measures). There is a fairly simple adjustment that can allow us to bring in this non-diversifiable risk into the beta computation. To arrive at this adjustment, assume that the standard deviation in the private firmÕs equity value (which measures total risk) is sj and that the standard deviation in the market index is sm. If the correlation between the stock and the index is defined to be rjm, the market beta can be written as:

Market beta

To measure exposure to total risk (sj), we could divide the market beta by rjm. This would yield the following.

This is a relative standard deviation measure, where the standard deviation of the private firmÕs equity value is scaled against the market indexÕs standard deviation to yield what we will call a total beta.

Total Beta

The total beta will be higher than the market beta and will depend upon the correlation between the firm and the market Š the lower the correlation, the higher the total beta.

You might wonder how a total beta can be estimated for a private firm, where the absence of market prices seems to rule out the calculation of either a market beta or a correlation coefficient. Note though, that we were able to estimate the market beta of the sector by looking at publicly traded firms in the business. We can obtain the correlation coefficient by looking at the same sample and use it to estimate a total beta for a private firm.