A Basic Introduction to the Methodology Used to Determine a Discount Rate

A Basic Introduction to the Methodology Used to Determine a Discount Rate By Dubravka Tosic, Ph.D. The term discount rate is one of the most fundamental, widely used terms in finance and economics. Whether one is dealing with a business valuation or litigation involving allegations of lost profit or lost compensation, the financial and economic analysis performed typically involves the critical process of determining an appropriate discount rate. A discount rate is used to determine the present value of a stream of economic benefits expected to be generated in the future. Examples of such economic benefits are a stream of future cash flows for a business, asset, or an investment, or a stream of projected earnings for an individual. The present value of a stream of future cash flows must be determined for at least one simple reason, the time value of money. The concept of the time value of money answers questions such as whether $1,000 today is worth more, less or the same as $1,000 one year from today, and why? The time value of money reflects the simple idea that everyone would prefer to have one dollar today compared to receiving one dollar at a later date. According to this view, deferring receipt of money to a later date requires a return, i.e. interest. As such, the value today of money to be received in the future is less than the face value to be received in the future; i.e., a future receipt is discounted to arrive at today s value (or present value ). The present value of a future economic benefit can be calculated by using the following basic formula: PV FV 1 r t t Where: FV t is the future value of a cash payout a time t, r is the discount rate, and t is the length of time between the present and the future payout. 1

For example, if one anticipates receiving $100,000 five years from now from a business venture, the present value of this amount is calculated as: PV = $100,000 / (1 + r) 5 Where r is the discount rate to be used. If one assumes that a proper discount rate to use is a risk-free rate of 0.25%, the calculations would be: PV = $100,000 / (1 + 0.25) 5 = $88,385.43 Based on this calculation, $100,000 five years from now is worth $88,385.43 today. Another way to interpret this calculation is that an investment of $88,385.43 today, at an interest rate of 0.25%, will be valued at $100,000 in five years (assuming simple interest). This example illustrates that the discount rate is just the interest rate used to determine the present value of a stream of future cash flows. Net present value ( NPV ) is a similar concept to present value. NPV is basically the present value of a stream of cash flows received at expected time intervals (e.g. received over several years). In calculating the present value of a stream of cash flows, NPV considers both inflows and outflows. When the NPV of a cash flow is positive, the economic benefit is thought to have a positive present value. Therefore, the determination of the present value of a cash flow in essence determines a discounted cash flow. Hence, one of the fundamental ways of determining value is to first estimate or forecast a future cash flow which is then discounted with an appropriate discount rate to convert this forecasted cash flow to present value. The basic formula for NPV is: NPV C 0 C 1 C2 1 r CT... 1 r 2 1 r T Where: C0, C1, C2..CT are the projected or forecasted cash flows at time 0, 1, 2 T, and r is the discount rate to be used in the calculation. 2

A common type of NPV analysis is called a discounted cash flow model ( DCF ). A DCF analysis is generally one method used to value an income producing asset or business. Chart 1 (below) shows the impact varied discount rates have on the present value of $5,000,000 to be received five years from today. Chart 1 Present Value of $5 Million Using Different Discount Rates Present Value $5,000,000 $4,500,000 $4,000,000 $3,500,000 $3,000,000 $2,500,000 $2,000,000 $1,500,000 $1,000,000 $500,000 $0 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% Discount Rate Present Value From Chart 1 one can see that by applying a 2% discount rate, a cash flow of $5 million received five years from now has a present value of about $4.5 million. On the other hand, applying a discount rate of 30% reduces the present value of the $5 million substantially to approximately $1.3 million. Since a present value of $4.5 million is drastically different from a present value of $1.3 million, one can reach drastically different conclusions about the value of a forecasted economic benefit. Chart 1 demonstrates several key points: 1. There is an inverse relationship between the present value and the discount rate used. The higher the discount rate used, the lower the present value, and the lower the discount rate used, the higher the present value. 2. The present value of an economic benefit is sensitive to the discount rate chosen. For example, using a 12% discount rate above yields a present value of $2.8 million, while choosing a 14% discount rate yields a present value of almost $2.6 million. A 2% difference in the discount rate chosen results in over a $200,000 difference in present value. 3

3. The relationship between the discount rate and the present value is non-linear. The impact of changing the discount rate by two percentage points is much greater at lower discount rates than at higher discount rates. For example, the change in present value is much greater when moving from a 4% to a 6% discount rate, than from a 28% to a 30% discount rate. One can observe this by looking at the slope of the curve in Chart 1 the slope is steeper at lower discount rates than at higher discount rates. While simple in concept, determining the appropriate discount rate can be very complex, and is critically important to the reliability of the analysis. Many academic studies have been written on the proper determination of a discount rate, and economic and financial experts typically spend a considerable amount of time reviewing and scrutinizing discount rates used by other experts. It is not uncommon that a dispute between two experts arises in connection with the definition of an appropriate discount rate. In determining an appropriate discount rate, one may consider various factors including: 1. Pure interest compensation for deferring consuming or spending an amount today. 2. Inflation compensation for the possibility that money in the future will not buy the same goods that could be purchased today. 3. Risk compensation for the risk that the money will not be received in a year. 4. Reinvestment risk compensation for the possibility that an investment will not generate the same return in a year. 5. Variability of returns compensation for the variability that may occur when the return varies over a specific time period. In addition to the time value of money, the perceived risk of a future cash flow should be adequately reflected in the discount rate. In economics and finance, risk refers to the likelihood that one will receive a return on an investment that is different from what was expected. This includes not only the bad outcomes, where one receives less than expected, but also the good outcomes, where one receives more than expected. Typically, a positive relationship exists between risk and reward, where the greater expected reward comes only with exposure to greater risk. Adjustments for this perceived risk of future cash flows are made through the selection of an appropriate discount rate. The more 4

uncertain or riskier future cash flows are perceived to be, the higher the rate of return an investor will require, meaning that a higher discount rate would be required, leading to a lower present value. Conversely, the lower the perceived risk to an investment, the lower the required rate of return will be, leading to a higher present value. This implies that the discount rate for investments with risky cash flows should be higher than the discount rate for investments with risk free cash flows. Additionally, since a discount rate should consider the time value of money and the riskiness of the forecasted cash flows, it typically represents a company s cost of capital or an investor s required rate of return. Determining the present value of a forecast of future cash flows to a business requires an understanding of earnings generated by the business as well as returns for both debt and equity investors. Therefore, the cost of capital of a business is typically considered to be a combination of debt costs and equity costs, weighted by the relative proportion of each type of capital invested in the company. A methodology used to determine an appropriate discount rate is to calculate the Weighted Average Cost of Capital ( WACC ). WACC is the discount rate used to evaluate the NPV of stream of future cash flows. WACC is calculated as the weighted cost of the various components of a company s capital structure, typically consisting of debt and equity. The WACC equation can be expressed as: Where: E = market value of the company s equity V = total value of the company s equity and debt Re = the company s cost of equity D = the market value of the company s debt Rd = the company s cost of debt Tc = the company s corporate tax rate A balance sheet would show the company s capital structure and the value of assets owned by the company, both debt and equity. If a company s balance sheet shows $40 million in assets, of which $24 million is in debt and $16 million is in equity, then the debt to equity ratio is 24-16 (or 3-2). As a consequence, debt capital represents 60% of the company s assets, and equity capital represents 40% of 5

the company s assets. The WACC calculation, therefore, would weight debt capital by 60% and equity capital by 40%. In order to calculate WACC, one must first calculate the cost of debt capital, and then the cost of equity capital: 1. Cost of Debt The cost of debt may be calculated in various ways, and may be calculated by considering various types of debt. Most often, the cost of debt corresponds to the interest rate a company is paying on all of its debt, such as long-term bank debt, corporate bonds, leases, preferred stock, warrants, etc. Companies with higher risk will usually have a higher cost of debt. The cost of debt may be adjusted by the corporation s tax rate, to take into account that a reduction in taxable income also reduces the tax obligation. This adjustment is often referred to as tax-effecting the interest rate and involves the interest rate shield ( ITS ). While DCF analysis attempts to capture taxes and other financing effects in a WACC, other analysis such as the Adjusted Present Value ( APV ) analysis does not. 2. Cost of Equity The cost of equity can be considered the rate of return required by a company s common stockholders. If shareholders do not receive the return they expect on their investment, they may be inclined to sell their shares. Therefore, a company will want to ensure that it returns what its investors desire, through share appreciation and dividends. All methodologies used to calculate the cost of equity have three basic components in common: 5a. The risk-free rate This interest rate typically corresponds to what an investor expects to receive from an investment with zero risk. A U.S. Government bond is an example of such an investment instrument. In calculations, one would use the yield on a long-term U.S. Government security to approximate the risk-free rate. 5b. Beta or the Industry risk premium This measure quantifies a company s risk relative to the overall market, typically represented by the S&P 500. A company with a Beta greater than 1 would be considered riskier than the market, and a company with a Beta less than 1 would be considered less risky than the market. 5c. The equity risk premium This estimate is probably the most debated underlying figure used in the calculation of the cost of equity. Equity risk premium can be calculated using 6

a variety of approaches, and typically represents the additional return on stocks as compared to bonds. Although there are several leading methodologies used to estimate the cost of equity, two of the more prominent and widely used approaches will be discussed below: the Build-up Methodology and the Capital Asset Pricing Model (CAPM). a. Build-Up Methodology Ibbotson Associates is usually credited for developing the build-up method. According to this method, the cost of equity equals: D = R f + P e + P s + P i + P c Where: R f is the risk-free rate P e is the Equity risk premium P s is the Company size premium P i Industry risk premium P c Company specific risk premium By using the build-up method, one starts with the risk-free rate, and builds on top of it by adding additional risk premiums that reflect the compilation of all perceived risks. Risk-free rate (R f ) As described above, one may use the yield on a long-term U.S. Government security, such as the 20 year U.S. Treasury bond. Equity risk premium (P e ) Reflects the uncertainty as to the amount and timing of dividend distributions and gains realized from public company stock appreciation. An accepted procedure for estimating the equity risk premium ( ERP ) is to calculate the historical average of the difference between annual equity market returns and returns on risk-free securities. Various publications, such as the Ibbotson SBBI Classic Yearbook: Market Results for Stocks, Bonds, Bills and Inflation, publish these differences, delineated by various sizes of companies. 7

Company size premium (P s ) Financial studies indicate that smaller company size is associated with higher investment risk. As such, investors demand higher returns for securities issued by smaller companies. In further developing the build-up process, a size premium may be added to the risk-free rate and equity premium, especially if the analysis involves a small company. Industry risk premium (P i ) Certain industries are associated with higher risk. As such, investors demand higher returns for securities issued by companies in such industries. For such industries, adding an industry risk premium to the sum of the risk-free rate and the equity risk premium rate may therefore be appropriate. Company specific risk premium Certain companies, because of their unique characteristics such as unstable earnings or high leverage, may be associated with higher risk. Such risks would require higher returns. As such, adding a company specific risk premium to all other risk premiums already taken into account may be appropriate. The quantification of such company specific risks is often challenging and based on subjective criteria, and may or may not be based on adequate data and replicable calculations. As such, in a dispute or litigation setting, there may be considerable disagreement especially when it comes to this type of a risk premium. Chart 2 below provides an example of the build-up method, assuming all types of perceived risk are considered and used to determine an estimated cost of equity: Chart 2 Build-Up Method Risk-free rate 0.25% Equity risk premium 4.70% Company size premium 1.50% Systematic risk (uncontrollable) Industry risk premium 2.00% Company specific risk 3.30% premium Unsystematic risk (controllable) Discount rate 11.75% Total Risk 8

The sum of the risk-free rate, equity risk premium and size premium are often referred to as systematic or uncontrollable risk, as these risks are associated with general market movements. The industry risk premium and the company specific premium are referred to as unsystematic or controllable risk as these risks are associated with a specific industry or specific company. And, through diversification, an investor may be able to control such risks. b. Capital Asset Pricing Model (CAPM) CAPM is the preferred method used by finance and economics professionals for determining an appropriate discount rate. CAPM was initially developed by William Sharpe in 1964, a financial economist and Nobel Prize winner in Economics, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. The general formula for CAPM is: CAPM compares market returns for a particular company or industry to the market as a whole. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (r f ) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (Beta) that compares the returns of the asset to the market over a period of time and to the market premium (r m -r f ). The CAPM asserts that the expected return of a security (r a ) equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat the required return of an investor, then the investment may not be undertaken. For example, if one was analyzing a company in the retail industry, one may conclude that this company and industry are less consistent (or the company s stock price may be more volatile) than the overall market, and therefore have a Beta greater than 1. The CAPM method will take a riskfree rate and then add the equity return premium (ERP), adjusted by the retail industry s Beta and the company s Betas. 9

Despite the existence of more sophisticated and complex approaches to asset pricing (e.g. Arbitrage Pricing theory and Merton's Portfolio Problem), CAPM still remains popular due to its simplicity and usefulness. Once the cost of debt, the cost of equity, and the company s proportion of debt capital and equity capital are calculated, the WACC can then be determined. Using the previous example, where: - Debt capital represents 60% of a company total capital, - Equity capital represents 40% of the company s total capital, - Assuming the cost of debt is 6% (adjusted for the corporate tax rate), and - Cost of equity is 11.75% (as in Chart 2): WACC = (60% x 6%) + (40% x 11.75%) = 8.3% As one can see, WACC is the average of the discount rate for debt capital and the discount rate for equity capital, where the average is weighted by the relative proportions of debt capital and equity capital to total capital. In some calculations, it may be most appropriate to calculate not one discount rate for an entire period of analysis, but a separate discount rate for each year of analysis. In the calculation of the present value of projected lost profits, it may be that lost profits vary from year to year, and the discount rate varies from year to year. In this situation, the discounting is typically prepared for each year separately. Further, based on particular facts of the analysis, the financial and economic expert may determine that an appropriate discount rate would be based only on a specific cost of debt or based only on the cost of equity. Special Cases In certain calculations of the present value of future cash flows, financial and economic experts do not undertake the above process to calculate an appropriate discount rate. Due to legal precedent, there are special cases where the expert must use a risk-free rate to calculate the present value of a stream of future cash flows. For example, the calculation of the present value of economic damages in wrongful termination, employment discrimination, personal injury and wrongful death matters generally 10

involves the use of a risk-free rate as the discount rate. This methodology is based on the legal precedent and the seminal decisions made by the U.S. Supreme Court in Kelly v. Chesapeake & Ohio Railway Co. (1916) and Jones & Laughlin v. Pfeifer (1983), followed by several decisions from Federal Circuit courts. In these types of calculations, economic experts typically then adjust for certain perceived risks by making adjustments to projected future earnings (i.e., future forecasted cash flow ) themselves, and not the chosen risk-free discount rate. For example, projected future earnings are typically adjusted for the risk that the Plaintiff would not be alive, in the labor force, and employed each and every future year in the analysis. Projected earnings are adjusted for such risks by using publicly available data on the probability of life, probability of labor force participation, and probability of employment. There may be additional risk factors that projected earnings are adjusted for, that take into account particular characteristics of the Plaintiff, the company, or the occupation and industry employed in. Despite having to use risk-free rates to discount future estimated compensation to present value, views among economic and financial experts as to what risk-free rate is the most appropriate tend to vary. For example, some believe that the risk-free rate should be based on a historical average, while others believe that only current risk-free rates should be used. Conclusion Determining a proper discount rate is one of the most critical processes in financial and economic analysis. Using a discount rate that is too high will calculate a present value that would be unrealistically low. Similarly, using a discount rate that is too low will produce a value that is unrealistically high. Ultimately, inappropriate conclusions can be made based on such calculations, leading to decisions that may have negative and sometimes long-term consequences. Using reasonable, transparent assumptions and replicable data may allow a financial and economic analyst or expert to prepare appropriate and defensible analyses. Additionally, preparing sensitivity analysis where one varies specific assumptions in the calculations to observe their impact on the final result may contribute to a better understanding of the data, the assumptions and their impact, and the overall analysis. Such sensitivity analysis may uncover potential issues with the data being used and/or the assumptions made, and can contribute to a higher quality analysis. 11

References: Brealey, Richard, Stewart Myers, and Franklin Allen, Principles of Corporate Finance, 11 th Edition, McGraw-Hill Higher Education, January 2013. Damodaran, Aswath, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance, 2 nd Edition, Wiley, 2006. Damodaran, Aswath, Applied Corporate Finance: A User's Manual, Third Edition, Wiley, 2010. Fisher, Franklin M. and R. Craig Romaine, Janis Joplin s Yearbook and the Theory of Damages, Journal of Accounting, Auditing and Finance, Volume 5, Numbers ½, Winter/Spring 1990, pp. 145-157. Weil, Roman L., Peter B. Frank, Christian W. Hughes, and Michael J. Wagner, Litigation Services Handbook: The Role of the Financial Expert, Fourth Edition, Wiley, December 2006. 12