At a 4 percent annual discount rate, this cash flow series (again, with an undiscounted value of $6.15 million) has a present value of $3.83 million. A two-point increase in the discount rate to 6 percent lowers the present value by nearly $750,000 to about $3.1 million. Discounting these same cash flows at a 28 percent rate yields a present value of about $695,000, and at a 30 percent rate, the cash flows have a present value of about $639,000.
This shows three important aspects of discounting. First, there is an inverse relationship between the discount rate and present value, meaning the larger the discount rate, the smaller the present value. Second, present value is sensitive to small movements in the discount rate (e.g., in this series, moving the discount rate from 4 percent to about 11.2 percent cuts the present value of the cash flow stream by half). Third, the relationship between the discount rate and present value is nonlinear. A two-point change in the discount rate from 4 to 6 percent has a greater overall impact than a two-point change from 28 to 30 percent. It is no wonder that this aspect of business valuation is often the most highly controversial.
As we previously discussed, it is important to define the income stream to be discounted and apply a discount rate appropriate to that income stream. A business’s assets are funded either by debt or equity. Therefore, business operations can be viewed as generating two cash flow streams: the one a return to the debt holders (usually in the form of interest expense) and the other a return to equity holders (usually in the form of dividends, changes in the stock value, or both). The stream of returns to debt holders should be discounted at an appropriate cost of debt capital, and the stream of returns to equity holders should be discounted at an appropriate cost of equity capital. The two discount rates can then be combined and, when weighted by relative amounts of debt and equity capital, form a weighted average cost of capital (WACC). The previous installment in this series presented considerations for determining a company’s cost of debt. The remainder of this article will address the derivation of the cost of equity.
Cost is often thought of as a dollar value exchanged to acquire a good or an interest. Economists tend to measure cost as the value of the next best opportunity forgone (not chosen). This expands the concept of cost from monetary terms to include anything of value. Cost of equity capital is, therefore, the opportunity cost of an investor’s forgone return on some next best investment. What might that return have been? Of course, that depends on the investment. So, if the investor wants a low- or no-risk investment, he or she has to accept that the investment will likely produce very low returns. To be enticed to make a riskier investment, the investor must be convinced of the possibility of higher returns. How much higher those returns must be will depend on the investor’s estimate of their riskiness. The riskier the investment, the higher the investor’s required rate of return, all other things being equal. Therefore, in determining the required rate of return on an equity investment, an investor must understand the nature (basis, amount, growth, among others) of economic benefits to be received in the future as well as the risk of whether such estimates of economic benefits will come to fruition.
One way to estimate the required rate of return is to look at the actual rate of return on an identical investment. Although no two investments are really identical, some may be similar enough for a required rate of return to be inferred. Take two companies of similar size, in the same industry, serving similar customers in similar geographies, subject to essentially the same business and economic risks, and capitalized similarly. If they are both actively traded on a national exchange, the returns on their stock will likely be similar (assuming investors believe such similarities will persist in the future). The extent to which such consistency can occur in big, publicly traded companies will be greater than for small, privately held companies. Also, the term “privately held” almost ensures that access to return information will be limited (and more likely nonexistent).
Like most methods, this discount rate derivation method begins with a risk-free rate of return and then builds on that to yield a rate of return representative of the riskiness of the investment being considered. A risk-free investment is one generally considered to be default risk free, and it provides a risk-free return. Because most businesses are expected to operate for long periods of time, the yield on a 20-year U.S. Treasury—backed by the full faith and credit of the U.S. government—is often used as a risk-free rate (currently about 2.3 percent). To be enticed to make an investment in riskier equities markets, an investor would have to receive a greater rate of return; in other words, the risk-free rate plus a premium for the extra risk associated with equity investments. Investors estimate higher expected rates of return by taking a published risk-free rate of return and adding an equity risk premium (ERP), representing excess returns of equity markets (e.g., dividends and capital gains) over and above returns on risk-free securities. Nonetheless, because equity returns are generally considered riskier (or more volatile) than returns on risk-free securities, investors often require higher rates of return for investment in equity securities.
A widely accepted procedure for estimating ERP is a historical averaging of the difference between annual equity market returns and returns on risk-free securities. One ready source for this information is the Ibbotson Associates’ annual Ibbotson SBBI Valuation Yearbook: Market Results for Stocks, Bonds, Bills and Inflation, which publishes means of historical differences between various risk-free rates (including long-term (20-year) U.S. Treasury coupon bond yields) and 1-year stock market returns for a series of years from 1926 to the current year publication. Because the proxy for one-year stock market returns used for this comparison is often based on the Standard & Poor’s (S&P) 500 or a composite index from the New York Stock Exchange, the ERP so derived will calculate historical excess returns on a fully diversified portfolio of large company stocks over the risk-free rate. If the company being valued is smaller, an ERP based on the returns on the stocks of smaller companies is often more appropriate. For that reason, the SBBI data presents equity risk premium information in various size categories of business capitalization—large, mid, small, and micro—as well as a breakdown into deciles (largest to smallest) and sub-deciles of the smallest decile. An analysis of this data reveals that historical ERPs show an inverse relationship to company size (the smaller the company, the larger the ERP). This makes intuitive sense in that one would expect smaller companies to be riskier investments (requiring higher rates of return) than their larger, more established and diversified counterparts. So, to further the build-up process, we may add a size premium to the already combined risk-free rate of return and equity risk premium.
A few words of caution are in order. The fact that such historical information is readily available should not convey simplicity in determining the appropriate ERP to use in building a discount rate. Historical ERPs may or may not be relevant proxies for future expectations—something that has been seen more often since the 2007 economic crisis. Many valuation professionals believe it is more appropriate to use an expected, rather than a historical, ERP. In fact, there are four schools of thought on the derivation of an ERP: (1) historical ERPs; (2) consensus ERPs derived through surveys of financial economists and other academicians who study various economic fundamentals (e.g., earnings, dividends, productivity, and growth); (3) demand-side ERPs, derived from additional return demanded by investors to take on the risk of equity investments; and (4) supply-side ERPs, developed through analysis of company-supplied components of total return (e.g., company earnings). Each method will produce a different ERP estimate. (Recall that present value is sensitive to small movements in the discount rate, of which ERP is a crucial component.) A greater discussion of these four methods is beyond the scope of this article, but suffice it to say that the derivation of the ERP is not settled science.
These methods produce estimates of risk premiums for equity markets as a whole. A potential investor in a privately held business may believe it appropriate to build in risks specific to that business itself in determining a required rate of return. Additional company-specific risk factors often show up in business valuation reports as a subjective amount added to the rate built up from other market data. Such “fudge factors” (so described by Richard A. Brealey, Stewart C. Myers, and Franklin Allen in their oft-cited text Principles of Corporate Finance (McGraw Hill 2008)) are often the subject of considerable discussion in expert depositions because they are based on the valuation expert’s judgment and not on a replicable scientific methodology or empirical data source.
Another criticism of company-specific risk factors is that they can, theoretically, be “diversified away” (in fact, they are often referred to as diversifiable risks). The idea is that by holding a fully diversified portfolio of investments (which may include the company being valued), such diversification will reduce (or perhaps eliminate) the impact a negative, unforeseen event affecting the company could have on the portfolio returns as a whole. This concept has its basis in market portfolio theory and the early work of Harry Markowitz (a winner of the Nobel Prize and contemporary of Milton Friedman at the University of Chicago), which posits that the aim of portfolio diversification is to reduce the collective risk on the portfolio below the risk of any single asset held. Based on this theory, some valuation experts believe company-specific risks should be excluded per se. However, just because some company-specific risk could be diversified away does not mean that all investors hold the required level of diversification such that all company-specific risks are diversified away. In other words, diversification may not be an option; therefore, the elimination of specific-company risk may not be achievable. This is an interesting point of contention among experts and has been the subject of many scholarly treatises. For our purposes, suffice it to say experts should take care in developing defensible opinions.
Another name for company-specific risk is “unsystematic” risk, meaning such risk is idiosyncratic to the company itself and does not arise as a result of market-related movements. This is why it can be affected (i.e., reduced) through diversification. “Systematic” risk, conversely, does relate to market changes. For the same reasons that unsystematic risk can be controlled through diversification, systematic risk is, by its very nature, non-diversifiable. (Note we are discussing systematic risk and not systemic risk, which has become a popular term of late and which refers to the collapse of entire market systems as opposed to risk associated with an individual business or component of the market system.)
Capital Asset Pricing Model
Introduced by William Sharpe in 1964, and building on Markowitz’s earlier work, the capital asset pricing model (CAPM) “is a conceptual cornerstone of modern capital market theory” (Shannon Pratt & Alina Niculita, Valuing a Business 185 (2008)). It is based on the concept that an asset’s price can be more reliably predicted by analyzing the relationship between the asset and systematic market risk. CAPM analyzes market returns for a particular investment (e.g., stock) or in particular industries and how such returns relate to market returns as a whole. For instance, if one were valuing cash flows generated by an electric utility, the build-up method would add an overall market equity risk premium to a risk-free rate. CAPM goes a step further and establishes the relationship between changes in electric utility market returns and market returns as a whole. Certain industries, like electricity and gas, basic food services, and education, tend to operate fairly consistently, regardless of overall market volatility. In those cases, a 1 percent change in overall market return may correlate closely with a change of less than 1 percent in these specific markets. So, if one were to determine that, for every 1 percent change in market returns, there is a 0.7 percent change in electric utility market returns, one could theoretically improve on the build-up method by adding a risk-free rate plus 70 percent of overall market ERP. This 70 percent factor is known as the electric utility market’s “beta.” On the other hand, industries like retail, entertainment, hotel and gaming, and semiconductor equipment typically reflect a beta greater than 1, meaning a 1 percent change in overall market return correlates to a change of more than 1 percent in return for these industries. Therefore, CAPM can be differentiated from the build-up method in that it takes a risk-free rate and then adds an ERP, adjusted by that industry’s, or company’s, beta.
Fama-French Three-Factor Model
While CAPM relies on one variable, beta, to help determine expected rates of return on income producing assets, Eugene Fama and Kenneth French developed a model that relies on three factors to describe stock returns. The research of Fama and French found that value stocks (those with a high book-to-market ratio) consistently outperform growth stocks and that small capitalization stocks consistently outperform large capitalization stocks. Therefore, in addition to CAPM’s systematic risk factor, the Fama-French model adds two more variables to capture these influences. One variable, HML, which stands for “high minus low,” captures historical differences between the return on stocks with a high book-to-market value and the return on stocks with a low book-to-market ratio. Specifically, it is the average return on two value portfolios (one containing small companies and the other containing large companies) minus the average return on two growth portfolios (again one containing small companies and the other large). The other variable, SMB, which stands for “small minus big,” captures historical differences between small and large capitalization stocks. Specifically, SMB equals the average return on three small company portfolios (value, neutral, and growth stocks) minus the average return on three big company portfolios. The model analyzes returns for a particular asset or industry and determines how sensitive changes in such market returns are to all three factors (systematic risk, HML, and SMB).
Arbitrage Pricing Theory
While CAPM is a single variable model, and the Fama-French model has three factors against which returns are measured, arbitrage pricing theory (APT) measures sensitivity to any number of other risk factors. Risk factors considered in APT are typically manifested in unexpected movements in asset prices and are non-diversifiable. Such factors typically capture risks related to the following factors:
- Investor confidence—changes in investors’ willingness to undertake relatively risky investments. It is measured as the difference between the rate of return on relatively risky corporate bonds and the rate of return on government bonds, both with 20-year maturities;
- Investment time horizon—unanticipated changes in investors’ desired time payouts. It is measured as the difference between the return on 20-year government bonds and 30-day Treasury bills;
- Inflation—a combination of the unexpected components of short- and long-run inflation rates; and
- Business cycle risk—unanticipated changes in the level of real business activity. The expected values of a business activity index are computed at both the beginning and the end of a month. Then business cycle risk is calculated as the difference between the end-of-month value and the beginning-of-month value. (Shannon Pratt and Roger Grabowski present a fairly concise discussion of these factors in Cost of Capital: Applications and Examples (2010). They cite as their source a talk based on a paper, “A Practitioner’s Guide to Arbitrage Pricing Theory,” by Edwin Burmeister, Richard Roll, and Stephen A. Ross, written for the Research Foundation of the Institute of Chartered Financial Analysts, and an exhibit drawn from “Notes for Controlling Risks Using Arbitrage Pricing Techniques,” by Edwin Burmeister.)
In summary, Parts II and III of this series discuss some of the theory of discount rate development. Much of the mathematics involved in development and proof of these theories is rigorous, and I have purposely steered clear of this as such discussions often cloud what is otherwise accessible finance theory. Part II focused on determining the cost of debt, while Part III dealt with the cost of equity. Adding these two costs, appropriately weighted by an entity’s debt versus equity financing, can produce a single weighted average cost of capital (WACC) factor, which can be used to transform forecasts of future cash flows to a single present value. I hope that by presenting some of the theoretical complexities of discount rate development, the myriad of recognized approaches, and a variety of information sources, I have conveyed that this is an evolving science and that there is no cookie-cutter formula that can be reliably applied in every circumstance.