THE COST OF CAPITAL & DISCOUNT RATES
As was illustrated in the Capital Investment handout, financing can be separated from the determination of value by reflecting the mix of debt and equity through the use of a weighted average cost of capital (WACC) rather than incorporating interest and principal payments into the cash flow calculations. This approach is most useful in valuing a new project or an acquisition where the implicit assumption is that the expansion will be financed in the same manner as the firm as a whole. It also allows for the separation of decision-making responsibilities: the manager of a division only needs to know what the appropriate cost of capital is in order to determine which projects are desirable, while corporate headquarters can decide how it wants to finance its operations. The use of the weighted average cost of capital as a hurdle rate, of course, assumes that the projects evaluated by it are of average risk; i.e., the same risk as the firm as a whole since the WACC is risk-adjusted for the firm as a whole.
Beninga & Sarig rely heavily on the need to establish the fact that the WACC is constant, regardless of the amount of debt financing used. In this manner, it is possible for them to find an “industry” cost of capital to use as their discount rate. It also allows them to avoid the circularity implicit in calculating a weighted average cost of capital for a specific firm.
All of the values in this calculation are based upon market values since the cost of capital is an opportunity cost. The values of the debt and equity are based upon market values, not book values. The market values represent the true economic values rather than the historic book values. Think of the equity in your home. Lenders are willing to lend based upon the market value of the home, not the historic price you paid for it (although you will probably have to pay for an appraisal to prove the market value). The same is true with companies.
The circularity of reasoning here is as follows: If you already know the market value of the debt and the equity of the firm, what is the purposes of calculating the WACC? It can’t be to value the firm because you already know that. The only true application in this case would be for valuing a new project or the acquisition of another company with the same risk and financed in the same manner. Remember that Beninga & Sarig are valuing the firm as a whole (Debt & Equity plus any other claims on it). From that total firm value, the value of the debt will be subtracted in order to arrive at the value of the equity. The alternative is to value the equity directly, but this can be more difficult.
The Cost of Debt
In calculating the true cost of debt, all associated costs should be considered as well. The result is an effective rate of interest to the company that is higher than the yield to maturity to the lender. In this manner, both the expected bankruptcy costs, which are reflected in the coupon rate of interest, and the agency costs, which are associated costs of the debt to the company, result in a rising effective cost of debt to the company as it employs more leverage in its capital structure.
Consider the following loan that a firm took out to finance an acquisition:
Face amount:$ 3,850,000
Interest Rate:10.406%
Processing Fee:$ 60,000
Monitoring Fee:$ 24,000 per year
Audit Fee:$ 25,000 per year
Principal & Interest Payments:$ 1,026,172
Net Loan Proceeds = $ 3,850,000 - $ 60,000 = $ 3,790,000
Annual Cost = $ 1,026,172 + $ 24,000 + $ 25,000 = $ 1,075,172
Year 0 Year 1 Year 2 Year 3 Year 4 Year 5
CFs3,790,000(1,075,172)(1,075,172)(1,075,172)(1,075,172)(1,075,172)
IRR = 12.91%
The effective rate of interest on the loan is 12.91% to the company. Note that since the annual audit fee is for accountants to audit the books and does not go to the bank, the annual proceeds to the bank amount to only $1,050,172 and the effective rate of interest that the bank is receiving is only 11.95%. It should also be pointed out that there are costs related to manpower that have not been included in this calculation. The company had to hire a comptroller to help the CFO who had become overburdened with work responsibilities. A lot of the additional work was related to managing the five additional manufacturing facilities that came with the acquisition. Some of it, however, was directly a result of having to report to the bank on a monthly basis, not only the consolidated financial results, but also the quick ratio and debt service coverage ratio to show that they met restrictive covenants of the loan. An allocation of some of the cost of the comptroller would be justified and would raise the effective cost of the debt to the company. In any event, you get the picture.
The Cost of Equity
The more difficult capital component cost to estimate is the required return on equity. The two most commonly used approaches to estimating stockholder required rates of return are the discounted cash flow, or Gordon Growth Model, and the risk-premium model. The Gordon Model, (or Dividend Valuation Model), is based upon an assumed constant rate of growth in dividends:
The first term is the dividend yield, while g is constant rate of growth in dividends, or the capital gain yield. The assumptions inherent in the use of the Gordon Model will be delved into in more detail when we look at Price-Earnings ratios.
The second approach is to calculate an appropriate risk-premium that is required in addition to the risk-free rate of interest.
The risk-free rate of interest is a function of the rate of inflation and the "real" rate of interest. The risk-premium is dependent upon the business risk and financial risk. As previously seen, the financial risk that results from the use of debt simply magnifies the business risk that is associated with a project or company.
The Capital Asset Pricing Model is the most well-known of the risk-premium models. The CAPM assumes that the only relevant risk is the systematic risk and, therefore, the appropriate risk premium is a multiple of the market's risk premium.
The difficulties of implementing the CAPM include obtaining a valid estimate of β as well as estimating the market's expected rate of return. Typically, the latter problem is circumvented by using historical averages of the market's realized risk premium, although the appropriate measure of "the market" is debatable. Estimating β, on the other hand, is difficult even for a firm that is publicly traded, and impossible for a firm that is privately-held.
Another approach to calculating the required rate of return on equity, called the “build-up method”, is to separate the overall risk-premium into a risk-premium for debt of the company and a risk-premium for the equity of the company. In this manner, the debt's risk-premium can be objectively observed by the rate of interest that the market is charging on the firm's bonds (or the bank's effective interest rate on loans if the firm is privately-held). To this is added an equity risk-premium based upon historical stock (total) risk-premiums in comparison to historical debt risk-premiums.
The most widely referenced source for historical returns used in calculating risk-premiums is Ibbotson Associates' Stocks, Bonds, Bills, and Inflation annual Yearbooks. This source tracks the market performance of a variety of financial instruments from 1926 to the present. Their data include the market performances of government treasury bills, treasury notes, treasury bonds, inflation rates, company returns by decile, large company stocks, small company stocks and corporate bonds. Using data from the 2001 Yearbook, the average (geometric) annual return for corporate bonds can be calculated from the Yearbook's total return index for the past 75 years as
Corp. Large Small
t-billsBonds Stocks Stocks
Cumulative Wealth Index 200820.509115.1542049.4489548.944
Cumulative Wealth Index 1925 1.000 1.000 1.000 1.000
Geometric Average Yield = 3.71% 5.89% 9.62% 11.67%
Corporate Bond Premium = 5.89% - 3.71% = 2.18%
Equity Risk Premium = 9.62% - 5.89% = 3.73%
Small Firm Risk Premium = 11.67% - 9.62% = 2.05%
Consider now a company that has a cost of debt of 8%. Then, adding in the equity risk-premium of 3.73% yields a cost of equity of 11.73%. If the firm is relatively small, another 2.05% might be added to adjust for firm size to yield a required rate of return of 13.78%. Note that this would only truly be consistent with publicly traded small stocks if the risk-free rate of interest was 5.82%; i.e., the risk-premium for the debt was still 2.18%.
Suppose instead that the required return on equity is being estimated for a privately-held firm. The argument could be made that the historic risk-premiums for publicly traded debt and equity are not relevant, but that provides no insight into what an appropriate required rate of return might be. Empirical research has shown, and U.S. courts have acknowledged, that a discount is appropriate in the valuation of privately-held securities. A discount in value is equivalent to an increase in rate of return. Thus, it is possible to utilize the preceding methodology as a means of estimating the required rate of return for a small, privately-held firm as well.
Assume a company's debt carries an 11% rate of interest. If the risk-free rate is 5%, then the debt risk-premium is 6%. Compared to the historic average risk-premium of 2.18% for large, publicly-traded corporate bonds, the company's debt risk-premium is 2.75 times the historic average (6%/2.18% = 2.75). Even disregarding the public vs. private nature of the debt, this is not inconsistent with the risk of the debt being 2.75 times that of a large public company. That is, such a risk-premium would be justified if the βdebt of the company were 2.75 times that of the average large corporation. Assuming that the beta of the equity of was also 2.75 times that of a large, publicly traded company, then the appropriate equity risk-premium would be 10.29%, or 2.75 times the historic average of 3.74%. Adding this to the cost of the debt would yield a required return on equity of 21.29% for the firm.
While 21.29% appears to be fairly high, it is not inconsistent with average realized rates of return on private equity; in fact, actual “required” rates of return in the range of 25% - 35% are not uncommon in proforma analysis by private equity, especially for troubled firms. The increased risk characteristics of the firm could be a result of a higher asset beta, higher leverage, or a combination of the two, as well as the risk of being privately-held (i.e., variability of obtainable prices for the securities). Thus, the resulting estimate of the required rate of return on equity is still consistent with financial theory. (It should be noted that no small firm premium should be included in this case since the debt and equity premiums calculated presumably would reflect this factor.)
Since the calculation of the cost of equity is just an estimate of what investors require, it is useful to use multiple approaches to generate an idea of what is a reasonable range of estimates. The final figure chosen must then be based upon the various results of different approaches combined with informed judgment. As an example, consider the impact of the turmoil that existed in the financial markets during 2009 when t-bill rates were close to zero percent for most of the year. Using the CAPM to calculate the cost of equity often resulted in estimates that were less than a company’s cost of debt. Clearly, the CAPM failed during this time period.
A Note on Use of Historic Averages
Arguments have been put forth that using past, realized returns does not reflect investor expectations. For example, large company stock investors did not expect to lose 37% of their equity investments in 2008, or an average of 2.19% per year for the five years 2004 - 2008. Conversely, they in all likelihood did not expect to earn 37%, or an average of 28.56%per year for the five years 1995 – 1999. Beninga and Sarig argue that historic averages, in the long-run, should be the same as investor expectations, however. If the long-run historic averages were different, then incentives would arise for risk-arbitrage in securities which would create the market forces that would result in market prices yielding historic averages consistent with expectations.
The other question that often arises over the use of historic averages is in regard to the appropriate period of time to consider for calculating the averages. It is generally considered to be more appropriate to use longer periods of time, rather than shorter periods, in order to capture the overall cyclical effects of the economy. The advantage of a longer time period also overcomes the inherent error of an historic estimate that can occur from choice of time periods, as the following graph depicts.
As may be observed, a short period of time can result in grossly underestimating (red line) or overestimating (blue line) the trend. A longer time period, even when measuring from a boom period (crest) to a recession (trough) results in a more representative estimate (gold line) of the true trend (dotted line).