littlebook The Little Book of Valuation

Discount Rates

            Cash flows that are riskier should be assessed a lower value than more stable cashflows, but how do we measure risk and reflect it in value? In conventional discounted cash flow valuation models, the discount rate becomes the vehicle for conveying our concerns about risk. We use higher discount rates on riskier cash flows and lower discount rates on safer cash flows. In this section, we will begin be contrasting how the risk in equity can vary from the risk in a business, and then consider the mechanics of estimating the cost of equity and capital.

Business Risk versus Equity Risk

            Before we delve into the details of risk measurement and discount rates, we should draw a contrast between two different ways of thinking about risk that relate back to the financial balance sheet that we presented in chapter 1. In the first, we think about the risk in a firm’s operations or assets, i.e., the risk in the business. In the second, we look at the risk in the equity investment in this business. Figure 2.5 captures the differences between the two measures:

Figure 2.5: Risk in Business versus Risk in Equity

As with any other aspect of the balance sheet, this one has to balance has well, with the weighted risk in the assets being equal to the weighted risk in the ingredients to capital – debt and equity. Note that the risk in the equity investment in a business is partly determined by the risk of the business the firm is in and partly by its choice on how much debt to use to fund that business. The equity in a safe business can be rendered risky, if the firm uses enough debt to fund that business.

            In discount rate terms, the risk in the equity in a business is measured with the cost of equity, whereas the risk in the business is captured in the cost of capital. The latter will be a weighted average of the cost of equity and the cost of debt, with the weights reflecting the proportional use of each source of funding.

Measuring Equity Risk and the Cost of Equity

            Measuring the risk in equity investments and converting that risk measure into a cost of equity is rendered difficult by two factors. The first is that equity has an implicit cost, which is unobservable, unlike debt, which comes with an explicit cost in the form of an interest rate. The second is that risk in the eyes of the beholder and different equity investors in the same business can have very different perceptions of risk in that business and demand different expected returns as a consequence.

The Diversified Marginal Investor

If there were only one equity investor in a company, estimating equity risk and the cost of equity would be a far simpler exercise. We would measure the risk of investing in equity in that company to the investor and assess a reasonable rate of return, given that risk. In a publicly traded company, we run into the practical problem that the equity investors number in the hundreds, if not the thousands, and that they not only vary in size, from small to large investors, but also in risk aversion. So, whose perspective should we take when measuring risk and cost of equity? In corporate finance and valuation, we develop the notion of the marginal investor, i.e., the investor most likely to influence the market price of publicly traded equity. The marginal investor in a publicly traded stock has to own enough stock in the company to make a difference and be willing to trade on that stock. The common theme shared by risk and return models in finance is that the marginal investor is diversified, and we measure the risk in an investment as the risk added to a diversified portfolio.  Put another way, it is only that portion of the risk in an investment that is attributable to the broader market or economy, and hence not diversifiable, that should be built into expected returns.

Models for Expected Return (Cost of Equity)

It is on the issue of how best to measure this non-diversifiable risk that the different risk and return models in finance part ways. Let use consider the alternatives:

Š      In the capital asset pricing model (CAPM), this risk is captured in the beta that we assign an asset/business, with that number carrying the burden of measuring exposure to all of the components of market risk. The expected return on an investment can then be specified as a function of three variables – the riskfree rate, the beta of the investment and the equity risk premium (the premium demanded for investing in the average risk investment):

Expected Return = Riskfree Rate + BetaInvestment (Equity Risk Premium)

The riskfree rate and equity risk premium are the same for all investments in a market but the beta will capture the market risk exposure of the investment; a beta of one represents an average risk investment, and betas above (below) one indicate investments that are riskier (safer) than the average risk investment in the market.

Š      In the arbitrage pricing and multi-factor models, we allow for multiple sources of non-diversifiable (or market) risk and estimate betas against each one. The expected return on an investment can be written as a function of the multiple betas (relative to each market risk factor) and the risk premium for that factor. If there are k factors in the model with bji and Risk Premiumj representing the beta and risk premium of factor j, the expected return on the investment can be written as:

Expected Return =

Note that the capital asset pricing model can be written as a special case of these multi-factor models, with a single factor (the market) replacing the multiple factors.

Š      The final class of models can be categorized as proxy models. In these models, we essentially give up on measuring risk directly and instead look at historical data for clues on what types of investments (stocks) have earned high returns in the past, and then use the common characteristic(s) that they share as a measure of risk. For instance, researchers have found that market capitalization and price to book ratios are correlated with returns; stocks with small market capitalization and low price to book ratios have historically earned higher returns than large market stocks with higher price to book ratios. Using the historical data, we can then estimate the expected return for a company, based on its market capitalization and price to book ratio.

Expected Return = a + b(Market Capitalization) + c (Price to Book Ratio)

Since we are no longer working within the confines of an economic model, it is not surprising that researchers keep finding new variables (trading volume, price momentum) that improve the predictive power of these models. The open question, though, is whether these variables are truly proxies for risk or indicators of market inefficiency. In effect, we may be explaining away the misvaluatiion of classes of stock by the market by using proxy models for risk.     

Estimation Issues

            With the CAPM and multi-factor models, the inputs that we need for the expected return are straightforward. We need to come up with a risk free rate and an equity risk premium (or premiums in the multi-factor models) to use across all investments. Once we have these market-wide estimates, we then have to measure the risk (beta or betas) in individual investments. In this section, we will lay out the broad principles that will govern these estimates but we will return in future chapters to the details of how best to make these estimates for different types of businesses:

Š      The riskfree rate is the expected return on an investment with guaranteed returns; in effect, you expected return is also your actual return. Since the return is guaranteed, there are two conditions that an investment has to meet to be riskfree. The first is that the entity making the guarantee has to have no default risk; this is why we use government securities to derive riskfree rates, a necessary though not always a sufficient condition. As we will see in chapter 6, there is default risk in many government securities that is priced into the expected return. The second is that the time horizon matters. A six-month treasury bill is not riskfree, if you are looking at a five-year time horizon, since we are exposed to reinvestment risk. In fact, even a 5-year treasury bond may not be riskless, since the coupons received every six months have to be reinvested. Clearly, getting a riskfree rate is not as simple as it looks at the outset.

Š      The equity risk premium is the premium that investors demand for investing in risky assets (or equities) as a class, relative to the riskfree rate. It will be a function not only of how much risk investors perceive in equities, as a class, but the risk aversion that they bring to the market. It also follows that the equity risk premium can change over time, as market risk and risk aversion both change. The conventional practice for estimating equity risk premiums is to use the historical risk premium, i.e., the premium investors have earned over long periods (say 75 years) investing in equities instead of riskfree (or close to riskfree) investments. In chapter 7, we will question the efficacy of this process and offer alternatives.

Š      To estimate the beta in the CAPM and betas in multi-factor models, we draw on statistical techniques and historical data. The standard approach for estimating the CAPM beta is to run a regression of returns on a stock against returns on a broad equity market index, with the slope capturing how much the stock moves, for any given market move. To estimate betas in the arbitrage pricing model, we use historical return data on stocks and factor analysis to extract both the number of factors in the models, as well as factor betas for individual companies. As a consequence, the beta estimates that we obtain will always be backward looking (since they are derived from past data) and noisy (they are statistical estimates, with standard errors). In addition, these approaches clearly will not work for investments that do not have a trading history (young companies, divisions of publicly traded companies). One solution is to replace the regression beta with a bottom-up beta, i.e., a beta that is based upon industry averages for the businesses that the firm is in, adjusted for differences in financial leverage.[1] Since industry averages are more precise than individual regression betas, and the weights on the businesses can reflect the current mix of a firm, bottom up betas generally offer better estimates for the future.

The Cost of Debt

            While equity investors receive residual cash flows and bear the bulk of the operating risk in most firms, lenders to the firm also face the risk that they will not receive their promised payments – interest expenses and principal repayments. It is to cover this default risk that lenders add a “default spread” to the riskless rate when they lend money to firms; the greater the perceived risk of default, the greater the default spread and the cost of debt. The other dimension on which debt and equity can vary is in their treatment for tax purposes, with cashflows to equity investors (dividends and stock buybacks) coming from after-tax cash flows, whereas interest payments are tax deductible.  In effect, the tax law provides a benefit to debt and lowers the cost of borrowing to businesses.

            To estimate the cost of debt for a firm, we need three components. The first is the riskfree rate, an input to the cost of equity as well. As a general rule, the riskfree rate used to estimate the cost of equity should be used to compute the cost of debt as well; if the cost of equity is based upon a long-term riskfree rate, as it often is, the cost of debt should be based upon the same rate. The second is the default spread and there are three approaches that are used, depending upon the firm being analyzed.

Š      If the firm has traded bonds outstanding, the current market interest rate on the bond (yield to maturity) is used as the cost of debt. This is appropriate only if the bond is liquid and is representative of the overall debt of the firm; even risky firms can issue safe bonds, backed up by the most secure assets of the firms.

Š      If the firm has a bond rating from an established ratings agency such as S&P or Moody’s, we can estimate a default spread based upon the rating. In September 2008, for instance, the default spread for BBB rated bonds was 2% and would have been used as the spread for any BBB rated company.

Š      If the firm is unrated and has debt outstanding (bank loans), we can estimate a “synthetic” rating for the firm, based upon its financial ratios. A simple, albeit effective approach for estimating the synthetic ratio is to base it entirely on the interest coverage ratio (EBIT/ Interest expense) of a firm; higher interest coverage ratios will yield higher ratings and lower interest coverage ratios.

The final input needed to estimate the cost of debt is the tax rate. Since interest expenses save you taxes at the margin, the tax rate that is relevant for this calculation is not the effective tax rate but the marginal tax rate. In the United States, where the federal corporate tax rate is 35% and state and local taxes add to this, the marginal tax rate for corporations in 2008 was close to 40%, much higher than the average effective tax rate, across companies, of 28%. The after-tax cost of debt for a firm is therefore:

After-tax cost of debt = (Riskfree Rate + Default Spread) (1- Marginal tax rate)

The after-tax cost of debt for most firms will be significantly lower than the cost of equity for two reasons. First, debt in a firm is generally less risky than its equity, leading to lower expected returns. Second, there is a tax saving associated with debt that does not exist with equity.

Debt Ratios and the Cost of Capital

            Once we have estimated the costs of debt and equity, we still have to assign weights for the two ingredients. To come up with this value, we could start with the mix of debt and equity that the firm uses right now. In making this estimate, the values that we should use are market values, rather than book values. For publicly traded firms, estimating the market value of equity is usually a trivial exercise, where we multiply the share price by the number of shares outstanding. Estimating the market value of debt is usually a more difficult exercise, since most firms have some debt that is not traded. Though many practitioners fall back on book value of debt as a proxy of market value, estimating the market value of debt is still a better practice.

            Once we have the current market value weights for debt and equity for use in the cost of capital, we have a follow up judgment to make in terms of whether these weights will change or remain stable. If we assume that they will change, we have to specify both what the right or target mix for the firm will be and how soon the change will occur.  In an acquisition, for instance, we can assume that the acquirer can replace the existing mix with the target mix instantaneously. As passive investors in publicly traded firms, we have to be more cautious, since we do not control how a firm funds its operations. In this case, we may adjust the debt ratio from the current mix to the target over time, with concurrent changes in the costs of debt, equity and capital. In fact, the last point about debt ratios and costs of capital changing over time is worth reemphasizing. As companies change over time, we should expect the cost of capital to change as well.



[1] The simplest and most widely used equation relating betas to debt to equity ratios is based on the assumption that debt provides a tax advantage and that the beta of debt is zero.

Beta for equity = Beta of business * (1+ (1- tax rate) (Debt/ Equity))

The beta for equity is a levered beta, whereas the beta of the business is titled an unlevered beta. Regression betas are equity betas and are thus levered – the debt to equity ratio over the regression period is embedded in the beta.