Forensic & Valuation Services Practice Aid Discount Rates, Risk, and Uncertainty in Economic Damages Calculations Page 1
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Notice to Readers This publication is designed to provide illustrative information with respect to the subject matter covered. It does not establish standards or preferred practices. The material was prepared by the AICPA staff and volunteers and has not been considered or acted upon by AICPA senior technical committees or the AICPA board of directors and does not represent an official opinion or position of the AICPA. It is provided with the understanding that the AICPA staff and the publisher are not engaged in rendering any legal, accounting, or other professional service. If legal advice or other expert assistance is required, the services of a competent professional person should be sought. The AICPA staff and this publisher make no representations, warranties, or guarantees about, and assume no responsibility for, the content or application of the material contained herein and expressly disclaim all liability for any damages arising out of the use of, reference to, or reliance on such material. Acknowledgements The principal authors of this practice aid are Christian Tregillis, Scott Bouchner, David Duffus, and Greg Regan. In addition, a special note of gratitude is extended to Everett Harry, Jarit Loughmiller, and Jed Greene, who assisted in drafting or reviewing sections of this practice aid. Members of the 2010 2011 and 2011 2012 AICPA Forensic and Litigation Services (FLS) Committee and Forensic and Valuation Services (FVS) Executive Committee also provided information and advice to the authors and AICPA staff for this practice aid. AICPA Staff Jeannette Koger Director Member Specialization & Credentialing Elaine Leggett Technical Manager FVS Section Barbara Andrews Project Manager FVS Section Page 3
Chapter 1 Introduction Experts evaluate the time value of money in both valuation and economic damages analyses. Courts, however, have offered little guidance with respect to acceptable methods to identify and implement an appropriate discount rate to accomplish this objective particularly in the economic damages context. Moreover, the published opinions in this area are frequently fact-specific and are not always consistent (see chapter 7, "Example Case Law," in this practice aid for a case law summary). Consequently, experts are called upon to interpret and apply the case circumstances using an appropriate methodology. fn 1 Terms in this practice aid that are in italics are defined in the glossary. This practice aid was written with an expectation that the reader has a basic understanding of the concept of the time value of money. Building on this knowledge, this practice aid presents the bases for certain general approaches to account for risk in the development of a damages model. These approaches include the capital markets approach, which accounts for risk with the use of a risk-adjusted discount rate, and the expected cash flow approach, which by comparison places a greater emphasis on adjustments for risk directly in the cash flow model and on the use of a lower discount rate. With that in mind, an initial review of some of the applicable foundational elements and terminology provides a basis for further discussion. This initial review focuses on the similarities of an economic damages analysis to the valuation of business interests and intangible assets. Specifically, a damages analysis often requires the conversion of values from future time periods to present value, such as when an award in litigation is intended to compensate a plaintiff for future lost profits. The litigation context often requires different methods from those used in a valuation of business interests or intangible assets, depending on the facts of the particular matter. fn 2 This practice aid is not intended to provide an exhaustive discussion of the cost of capital or valuation theory. For a list of fn 1 For example, see Michael Crain, Bonnie Goldsmith, and Michael Wagner, "Response to One Man s Opinion," CPA Expert, Spring 2004, 4 6, "Case law recognizes that each case has its own set of unique facts and circumstances and requires the expert to consider these relevant facts and circumstances in calculating damages. Therefore, CPAs should determine the most appropriate methodology to use in the damages calculation under the specific facts and circumstances of the case." fn 2 Practitioners performing economic damages calculations are subject to Statement on Standards for Consulting Services No. 1, Consulting Services: Definitions and Standards (AICPA, Professional Standards, CS sec. 100). In addition, practitioners performing business valuations should be aware of the standards set forth in Statement on Standards for Valuation Services (SSVS) No. 1, Valuation of a Business, Business Ownership Interest, Security, or Intangible Asset (AICPA, Professional Standards, VS sec. 100). Although SSVS No. 1 does not cover lost profits calculations, those developmental standards may be applicable to other economic damages, including loss of business value. Page 4
additional reference materials and publications, see the appendix, "Resource and Reference List," in this practice aid. The Concept of Time Value of Money The concept of the time value of money defines the value of money at one point in time as compared to the value of the money, including the promise or expectation of such, at some other point in time. This concept is within the purview of financial experts, accountants, and economists (hereafter referred to as "experts"), all of whom may find themselves tasked with the conversion of values across time periods. For example, a financial instrument may be required to be reported on a balance sheet at its present value. Or a business may be valued as the sum of the present value of expected future cash flows to be generated by the business. These and other applications may also be seen in the context of legal disputes. For example, consider the event when a company has experienced lost profits as a result of a breach of contract. The lost profits may be from the past or the future. In a transactional context, the price paid for a business is likely to be influenced by the negotiations of a specific transaction, with a host of issues specific to the transaction s context affecting the negotiated value. Similarly, financial computations and values related to a dispute, such as in an award of economic damages, can yield different results given the potential for small but impactful inputs and assumptions even with seemingly similar fact sets. In such cases, courts are often tasked with awarding an exact amount to a plaintiff as of the date of judgment. Circumstances require the trier of fact, as well as experts and legal counsel, to evaluate the facts pertaining to the matter in question. For example, an expert may be engaged to opine about the measure of damages such as lost profits or lost business value, the magnitude of the lost income stream, the duration of the damage period, and the appropriate discount rate to apply to determine the present value of the damages. Effectively accounting for and articulating these subtleties can have a profound impact on the amount of a judgment in litigation. In the course of the expert s development of the bases for these opinions, the objective remains the same to make the plaintiff whole for any losses sustained as of result of the defendant s alleged misconduct. Page 5
Chapter 2 Overview Of Time Value Of Money Discounting and present value concepts are based on the idea that a dollar invested today will grow to be worth more in the future. A dollar invested today can earn interest at an interest rate and be worth more in the future, or a future dollar can be discounted with a discount rate to conclude the present value of that future dollar. This idea reflects the notion that there is some value to the use of money for a period of time, as well as the notion that its present value is affected by the risk that the future value will actually occur at the expected level. The calculation of the future value in one year of $1.00 at a 10 percent interest rate (or rate of return) is shown as follows: $1.00 (1 + 10%) 1 = $1.10 Present Value (1 + Interest Rate) number of compounding periods = Future Value The discounting of a future value for one year to a present value is illustrated as follows: $1.00 = $1.10 (1 + 10%) 1 Present Value = Future Value number of compounding periods (1 + Discount Rate) In this instance, the discount factor to convert the future value to a present value for one year at a 10 percent discount rate is 1/1.10, or 0.9091, when the present value is equal to the future value multiplied by the discount factor. fn 1 The formula for the conversion of the value of money in one period to the value of money in another period is affected by the number of compounding periods (time). In the previous examples, the 10 percent interest rate was applied to an investment period of one year. Similarly, the 10 percent discount rate was used to discount the $1.10 future value one period to a present value of $1.00. With an increased number of compounding perifn 1 The discount factor can also be referred to as the "present value factor." Additionally, some practitioners work with the reciprocal of the discount factor, such that it grows as the discounting increases, and the present value is the future value divided by this version of a discount factor. This is a functionally equivalent calculation. Page 6
ods, the amount of growth in the money value increases. The future value of $1.00, assuming a constant 10 percent interest rate for two years, is calculated as follows: $1.00 (1 + 10%) 2 = $1.21 Present Value (1 + Discount Rate) number of compounding periods = Future Value Figure 2-1 The future value example assumes that the original dollar is invested for two periods (years), with the same 10 percent interest rate applied for both years. Thus, the originally invested dollar grows to $1.10 after one year, and then it is compounded at 10 percent again in the second year. At the end of the second year, the original dollar is worth $1.10 1.10, or $1.21. In this present value calculation, the discount factor is 1.00/1.21, or 0.8264. In the event that different interest rates are used in different periods in the compounding process, the future value formula is similar, only slightly more complex. For example, assuming that 10 percent is the interest rate for year one and 20 percent is the interest rate for year two, then the future value calculation performed is as follows: $1.00 (1 + 10%) 1 (1 + 20%) 1 = $1.32 The impact of a higher discount rate is to reduce the present value of future amount of money more than a lower discount rate. fn 2 For example, a $1,000 stream of cash flow for five years will be computed to have a higher present value if the discount rate used to perform the calculation is lower, all else equal. Lower Discount Rate Higher Discount Rate Annual Cash Discount Discount Present Discount Discount Present Period Flow Rate Factor Value Rate Factor Value 1 $1,000 10% 0.9091 $909 20% 0.8333 $833 2 $1,000 10% 0.8264 $826 20% 0.6944 $694 3 $1,000 10% 0.7513 $751 20% 0.5787 $579 4 $1,000 10% 0.6830 $683 20% 0.4823 $482 5 $1,000 10% 0.6209 $621 20% 0.4019 $402 fn 2 For purposes of this practice aid, the terms income and cash flow are both used in describing the development of a present value analysis. In practice, factors such as additions to working capital may call for adjustments to income in order to accurately model value to an investor or owner of an asset. However, income metrics may function well in calculating economic damages in some situations and may be simpler for presentation purposes. Page 7
Total $5,000 $3,791 $2,991 As a result, the core inputs that define the conversion of value of a single cash flow from one period to another period are (1) the value in one period, (2) the interest or other rate(s) of return (which may differ period by period), and (3) the number of compounding periods. Use of a Mid-Period Convention The time value of money is also affected if the cash flow amounts are not scheduled to be received at the beginning or the end of a period. In figure 2-1, the Period 1 cash flow receives exactly one period of discounting because the cash flow amounts are assumed to be received as a lump sum at the end of each period. But, if the cash flow amounts were to be earned evenly throughout the period, then a mid-period convention would call for the amount to be discounted for one half of a period. For example, the Period 2 cash flow in the lower discount rate scenario would be discounted for 1.5 periods. In that case, the Period 2 cash flow would be worth $867 instead of $826, which assumes an end-of-period timing, calculated as follows: Nominal Rates and Real Rates $1,000 / (1 + 10%) 1.5 = $867 Discount rates and interest rates can be expressed in nominal terms or in real terms. A nominal interest rate represents an interest rate without an adjustment for inflation. In other words, a nominal interest rate is the stated interest rate. By comparison, a real interest rate represents the interest rate after an adjustment for inflation, such that the rate is calculated as follows: Nominal Interest Rate = (1 + Expected Rate of Inflation) (1 + Real Interest Rate) 1 In practice, when cash flow amounts are presented in nominal dollars, then nominal discount rates should generally be used. And, when cash flow amounts are presented in real dollars, real discount rates should generally be used. Cash Flows and Variability Discounting a future income or cash flow stream to present value is a fundamental procedure in asset valuation. fn 3 Rational buyers are assumed to be willing to pay no more for fn 3 Shannon Pratt, with Alina Niculita, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: McGraw Hill, 2008), 56. Page 8
Figure 2-2 the subject asset than the present value of what they expect to receive in the future. Similarly, rational sellers are expected to sell the subject asset for a price no less than the present value of what they would expect to receive in the future. Like a business or asset valuation performed for nonlitigation purposes, an economic damages computation often takes into consideration the same factors as those considered by buyers and sellers involved in a transaction. These factors include economic benefits to be received in the future and the inherent risks of realizing those benefits. The value to buyers decreases as the risk associated with the income stream increases. The definition of "risk," however, is not as simple as a percentage likelihood of achieving an expected or forecasted result. At times, the possible outcome alternatives are binary either (1) achieve the expected cash flow level or (2) completely fail. This binary outcome scenario is implied in figure 2-2. Promised Probability of Failure Cash Probability of Total Expected Period Cash Flow Delivery Flow Failure Cash Flow 1 $1,000 50% $0 50% $500 2 $1,000 50% $0 50% $500 3 $1,000 50% $0 50% $500 4 $1,000 50% $0 50% $500 5 $1,000 50% $0 50% $500 Total $5,000 $0 $2,500 Figure 2-3 In the previous example, there are only two possible results each year: $1,000 or $0. With a 50 percent likelihood of achievement applied uniformly, the probability weighted expected value is $2,500. There is no time value of money calculation applied in this example. The capital markets approach expresses risk in terms of the variability of results around the expected income (or cash flow) value. As depicted in figure 2-3, given investors aversion to risk and variability, a future value that is certain (the bar graph on the left side) typically is valued higher than a 50 percent chance of twice the same future value (the bar graph on the right side). Page 9
Figure 2-4 The bar graph on the left side of figure 2-3 represents a certain outcome, while the bar graph on the right side has two different possible outcomes. Investors typically pay more for the same expected value when the variability of the outcome is less than an alternative investment with the same expected value but more variability. A certainty equivalent is the amount that an investor would accept in exchange for the rights to an outcome that is uncertain. This concept is represented in an example of earnings per share (EPS) variation in figure 2-4. In figure 2-4, Distribution 1 includes possible outcomes ranging from a loss to a gain in excess of $10 per share and a mean expected earnings of $5 per share. However, given the range of possible outcomes and investors' risk aversion, the certainty equivalent is less than the expected EPS of $5 in Distribution 1. In Distribution 2, the range of possible outcomes is tighter, although the mean expected outcome is the same ($5 EPS). Also, there is a lower probability in Distribution 2 of outcomes with substantial divergence from the expected $5 EPS than in Distribution 1. In Page 10
Figure 2-5 the capital markets, although both of these company scenarios offer the same $5 expected EPS as a weighted average of the EPS outcomes, the Distribution 2 expectation of less relative variation causes this probability-weighted expected EPS to be worth more than that represented in Distribution 1. fn 4 The certainty equivalents of both EPS distributions in figure 2-4 are less than the mean expected EPS, with a greater penalty to the distribution with greater risk, as seen in the variability of outcomes. This observation is consistent with a 1916 opinion from the U.S. Supreme Court: "It is self-evident that a given sum of money in hand is worth more than the like sum of money payable in the future." fn 5 The relationship between the expected value of an uncertain outcome and its certainty equivalent is affected by an individual investor's risk tolerance. In other words, some investors may be more willing to tolerate a wide range of outcomes for an individual investment, given the investor s ability to diversify away some of the risk. Also, an individual investor may be willing to accept outcomes within a certain range. For example, Distribution 1, with a possible negative EPS, may be unacceptable to some investors. In practice, experts often look at the variability of expected outcomes, represented statistically by the standard deviation. The standard deviation measures how far from the mean the actual result is likely to be. In figure 2-4, the standard deviation of Distribution 1 is greater than the standard deviation of Distribution 2. The standard deviation increases as the range of outcomes increases or as the likelihood of outcomes far from the expected value increases ("heavier tails"). The standard deviation of an investment s expected economic income is typically an important input into the rate of return that investors expect from that investment. In another example, a biotech company may have a 10 percent probability of making $1 billion, a 40 percent probability of making $10 million, and a 50 percent probability of losing $100 million. Based on these hypothetical data, the expected value of the income of the biotech company may be estimated as presented in figure 2-5. Scenario Income Probability Expected 1 $1B 10.00% $100M 2 $10M 40.00% $4M 3 ($100M) 50.00% ($50M) fn 4 The illustrative distributions are bell-shaped; however, in practice distribution curves may be skewed. For example, a mortgage lender may have little opportunity to recover more than the note-related principal and interest, but it may face a variety of potential scenarios for recovering part, or none, of the subject principal or interest. fn 5 Chesapeake & Ohio Railway Co. v. Kelly, 241 U.S. 485, 489 (1916). See also 241 U.S. at 485, 490, 493. Page 11
Scenario Income Probability Expected Total Expected Income $54M In contrast, a United States Treasury bill with an expected payment of $54 million, given the government s de minimus risk of default, would yield the same expected income as the biotech company. However, the Treasury bill would be worth more than the example biotech company, even though both investments have the same expected value. fn 6 The valuation of the biotech company would typically involve a valuation adjustment (or penalty) for variability risk, in addition to the discount to reach the $54 million expected value from the $1 billion income from the success scenario. As discussed in this practice aid, the variability of the cash flow streams associated with these investment alternatives is manifested in differing rates of return. In an economic damages analysis, when two different possible outcomes have the same expected value but different ranges of outcomes, then this difference normally results in different present values. The Impact of Time The time to delivery can also affect the present value beyond just discounting for additional future periods at the selected discount rate. For example, if the potential economic return is extended farther into the future, investors may (1) perceive greater uncertainty and (2) conclude a lower present value than is explained by just discounting for additional future periods at the otherwise selected discount rate. This conclusion would be seen in a higher discount rate, as well as an increased number of discount periods. Additionally, the risk-free component will generally be greater simply due to the greater period of time to use the subject investment. For example, a 30-year Treasury bond typically yields more than a 3-month Treasury bill. In the context of an economic damages computation, the time during which economic harm is experienced is often referred to as the damage period. The damage period is defined not only by the judgment of the expert, but also by factors such as applicable law, any contractual arrangements between the disputing parties, the documentary case evidence, witness statements, and market or product research. The damage period may range from a relatively short time frame to, in some situations, perpetuity. Given the effect of the compounding of discount rates, the length of the damage period may be an important factor to consider in selecting a rate or rates for a present value determination. Some experts have used multiple debt instruments, with different maturities for cash flow amounts from different time periods. As an illustration, this may result in a fn 6 There is no time value component in this example. Page 12
1-year treasury rate for damages in one time range in a damages calculation and a 5-year treasury for another time range in the calculation. Others have used a rate from one security to be applied to the cash flow stream uniformly. As an example, an expert may use a 3-year security to discount all periods in a 5-year damage period. Each of these two general approaches has been accepted by courts. Summary of Factors Considered by Discount Rates In summary, the amount that a buyer is willing to pay for a business, an asset, or a stream of income, or that a plaintiff should be awarded to replace such an item, is a function of a number of factors, including the expected cash flow or income stream (taking into account the risk of achievement), fn 7 the degree of penalty for risk of variability in returns, the timing of the cash flow or income stream, and the risk-free rate of return. fn 7 In the context of litigation, considerable debate is focused on a plaintiff s "expected" results but for the defendant. Page 13