< PreviousGuidance from Energy Capital Corp. v. U.S. While the Supreme Court has not issued a decision on discount rates for calculating the present value of lost profits, a federal appellate court provided guidance in the Energy Capital decision. 20 In Energy Capital, the trial court applied a risk-free rate to discount future lost profits to present value. The government argued that this was incorrect, asserting that “the discount rate represents the return an investor would require in order to risk investing capital in a particular venture and that such a rate must incorporate any risk that cash flows would not be realized.” 21 The appellate court’s response was, “It depends.” We do not hold that in every case a risk-adjusted discount rate is required. Rather, we merely hold that the appropriate discount rate is a question of fact. In a case where lost profits have been awarded, each party may present evidence regarding the value of those lost profits, including an appropriate discount rate. 22 The appellate court explained its reasoning as follows: Energy Capital argues that once the Court of Federal Claims determined that its profits were reasonably certain, no further consideration of risk was appropriate, because risk already had been considered in determining whether there would be any profits. We disagree. … Therefore, the fact that the trial court has determined that profits were reasonably certain does not mean that risk should play no role in valuing the stream of anticipated profits. In other words, by finding that Energy Capital’s lost profits were reasonably certain, the trial court determined that the probability that the … venture would be successful was high enough that a determination of profits would not be unduly speculative. The determination of the amount of those profits, however, could still be affected by the level of riskiness inherent in the venture. Energy Capital argues that the sole purpose in discounting is to account for the time value of money. Again, we disagree. When calculating the value of an 20 Energy Capital Corp. v. United States, 302 F.3d 1314 (Fed. Cir. 2002). 21 Ibid., 1330. 22 Ibid., 1333. 23 Ibid. 24 See, e.g., Sonoma Apartment Associates v. United States, 302 Fed. Cl. 90 (2017); Elk v. United States, 87 Fed. Cl. 70 (2009); Franconia Associates v. United States, 61 Fed. Cl. 718 (2004); PSKS, Inc. v. Leegin Creative Leather Products, Inc., 171 Fed. Appx. 464 (5th Circuit, 2006). 25 Michael A. Crain, “Discounting Lost Profits in Damage Measurements,” in The Comprehensive Guide to Economic Damages, 7th ed., vol. 1, eds. Jimmy S. Pappas, William Scally, and Steven M. Veenema (Portland, OR: Business Valuation Resources, 2023), 283–86. 26 Ibid., 284–85. anticipated cash flow stream pursuant to the DCF [discounted cash flow] method, the discount rate performs two functions: (i) it accounts for the time value of money; and (ii) it adjusts the value of the cash flow stream to account for risk. 23 Lost Profits, Financial Literature, and the Courts Energy Capital held that the discount rate was a question of fact to be argued by each side in the litigation. Other federal and state decisions have held this same position. 24 Because the appropriate discount rate is a question of fact and the facts of each litigated dispute are different, the assessment of the appropriate discount rate is different for each case. Therefore, the expert assigned to estimate future lost profits and discount them to present value must identify the best method for determining the appropriate discount rate for that specific case. A review of financial literature shows three popular categories for determining the discount rate for calculating the present value of future lost profits: 1. A rate of return on a safe investment (determined as a matter of law) 2. The injured party’s rate of return when investing the award (determined as a matter of fact) 3. The rate of return commensurate with the risk in receiving the lost profits (determined as a matter of fact) 25 The rate of return on safe investments has typically meant the rate of return on U.S. Treasury securities. The rate of return when investing the award covers a broad spectrum of options, including: • Rate of return on a conservative investment • Rate of return on an investment portfolio • The injured firm’s cost of debt • The injured firm’s weighted average cost of capital (WACC) • The injured firm’s cost of equity • Rate of return from an investment similar to the injured firm if the business was destroyed 26 10 The Value Examiner Litigation ConsultingThe rate of return commensurate with the risk in receiving the lost profits ties into the modeling approach discussed by Dunn and Harry. “Some of the legal cases that discuss the appropriate discount rate relate it to a rate of return that is commensurate with the risk that the injured firm would have incurred if it had actually received the lost profits (but for the injury). Some call this rate the ‘risk-adjusted’ discount rate. This term is somewhat misleading because the investment rate of return … also includes risk (i.e., risk from the relevant investment).” 27 Many experts prefer to apply the rate of return when investing the award. Financial literature supports applying the cost of equity or a WACC as the discount rate for lost profits. The most important distinction about the lost profits methodology is that it considers all information available up to the date of the ex post analysis, 28 which is typically close to the trial date and may be many years after the alleged legal violation. By this date, there may be additional information that can be used to produce an improved estimate of loss or a refined but-for set of projections, particularly for the time period between the date of harm and the ex post date. While not all uncertainty can be eliminated, it is likely that some of the uncertainty can be eliminated between the date of harm and the ex post date, given that the intervening events are observed in hindsight. 27 Ibid., 285. 28 Most lost profits analyses use an ex post method, discounting future lost profits to the date of trial. Applying an ex ante approach would discount lost profits to the date of harm. Because ex post is most commonly used, this discussion focuses on that approach. 29 Everett P. Harry, III, and Jeffrey H. Kinrich, Lost Profits Damages: Principles, Methods, and Applications, 2nd ed. (Ventnor City, NJ: Valuation Products and Services, 2022), 547. 30 Ibid., 557. The lost profits and business valuation methodologies generally produce different discount rates if those rates are determined as of different dates (e.g., ex ante versus ex post ). “The lost profits methodology discount rate should reflect the state of the markets as of the trial date, if the damages analysis is performed ex post . In some cases this will result in a lower discount rate as of the trial date; in some cases it will result in a higher discount rate.” 29 Discount Rates as a Question of Fact When determining the discount rate for a lost profits analysis, Harry and Kinrich note: The discount rate should account for two general types of risk beyond the risk-free rate or time-value-of-money component of the total discount rate. First, dispersion risk exists to the extent that a but-for outcome could vary from the expected stream of cash flows (i.e., the probability weighted average of the potential but-for outcomes). Greater potential volatility supports a higher discount rate. Second, plaintiff’s projection of cash flows may represent a hoped-for or success outcome in excess of the expected result. If so, additional unsystematic risk premia may be appropriately included in the discount rate. 30 11 May | June 2024 A Professional Development Journal for the Consulting DisciplinesA further review of financial literature provides examples of varying discount rates. This review also shows that experts have applied discount rates based on ordinary business risk or company-specific risk more often than the risk-free or risk-adjusted approaches. Regardless, decisions show that a wide range of discount rates have been found appropriate based on the facts of a case. A sampling of these cases is included below, separated into four categories: risk-free, risk- adjusted, ordinary business risk, and company-specific risk. Risk-Free Rates As explained in The Comprehensive Guide to Economic Damages , Courts generally agree that, when the parties fail to present evidence regarding the proper discount rate, it is appropriate for the court to use a risk-free rate. … In other cases, courts have also approved the use of a risk-free rate when both parties presented evidence supporting their choice of a discount rate. … Trial courts have often declined to exclude expert testimony that uses a risk-free rate. Similarly, when juries base their verdicts on such testimony, appellate courts have been reluctant to overturn the verdicts. In BGB Pet Supply, Inc. v. Nutro Products, Inc. , for example, the U.S. Court of Appeals for the 6th Circuit upheld a jury verdict derived from the plaintiff’s expert’s use of a 5% discount rate, which the court noted was slightly below the yield on one-year Treasury securities at the time of trial. Although the defendant argued for a 20% discount rate, the court explained that “the jury could reasonably accept [the expert’s] testimony in favor of a 5% discount rate.” 31 31 Robert M. Lloyd, “Discounting Damages: Case Law,” in The Comprehensive Guide to Economic Damages, 7th ed., vol. 1, eds. Jimmy S. Pappas, William Scally, and Steven M. Veenema (Portland, OR: Business Valuation Resources, 2023), 292; BGB Pet Supply, Inc. v. Nutro Products, Inc., 124 F.3d 196 (6th Cir. 1997). 32 Robert L. Dunn, Recovery of Damages for Lost Profits, 6th ed. (Westport, CT: Lawpress, 2005), 546. 33 Robert L. Dunn, Recovery of Damages for Lost Profits, 6th ed., September 2023 Supplement, eds. Sharon C. Rutberg and Wendy Janis Malkin (Westport, CT: Lawpress, 2023), 486. 34 Harry and Kinrich, Lost Profits Damages: Principles, Methods, and Applications, 557. Risk-free rate cases included the following: • Purina Mills, LLC v. Less, 295 F. Supp. 2d 1017 (N.D. Iowa 2003). “Rates used to discount future damages were the corresponding Treasury Bond rates ranging from 1.24% (1 year) to 3.17% (five years).” 32 • Board of Supervisors of Louisiana State University v. Gerson , 260 So. 3d 634 (La. App. 2018). The appellate court upheld an award of future lost profits “based on business owner’s income history, subject to discount rate of 2.5%.” 33 Risk-Adjusted Rates The Energy Capital opinion noted that the discount rate included the risk-free rate adjusted for the value of the cash flow stream to account for risk. It went on to say that not every case requires a risk-adjusted rate and the rate depends on the facts of the case. Many courts expect experts to include some form of risk adjustment when discounting future lost profits. The risk-adjusted rate category may also include “asset risk less than ordinary business risk” discount rates. This “impaired opportunity is a subset of plaintiff’s overall business. The related lost profits may reflect more defined and constrained risks than for the overall business. As such, the appropriate discount rate needs to be tailored to the particular lost opportunity. That is, the analyst should not assume that a general company WACC or hurdle rate is applicable, since the appropriate risk-sensitive discount rate may be lower than the company’s consolidated WACC.” 34 Courts generally agree that, when the parties fail to present evidence regarding the proper discount rate, it is appropriate for the court to use a risk-free rate. In other cases, courts have also approved the use of a risk-free rate when both parties presented evidence supporting their choice of a discount rate. 12 The Value Examiner Litigation ConsultingThe following cases exemplify such risk-adjusted rates: • Franconia Associates v. United States , 61 Fed. Cl. 718 (2004). “11% discount rate approved for projected cash flow from rent-regulated real property and 14% discount rate approved for projected cash flow from unregulated real property.” 35 • Soaring Wind Energy, LLC v. Catic USA, Inc. , 946 F.3d 742 (5th Cir. 2020) (Delaware law). “Upholding arbitration panel’s award of $62.9 million in lost profits where LLC members invested in projects other than LLC’s, breaching LLC agreement; calculation of damages was based on panel’s finding that defendants expected minimum 15% return on their investments and then applying discount rate to determine present value of lost profits.” 36 • Orozco v. WPV San Jose, LLC , 36 Cal. App. 5th 375 (2019). In this case, a California appellate court affirmed a jury award of $676,967 in lost profits. The case involved a fraudulent misrepresentation, by the lessor of commercial space, that it would not lease space in the same building to a restaurant that competed with the plaintiff’s restaurant. The award was based on testimony by plaintiff’s expert regarding the net present value of anticipated profits over a 10-year lease period. The plaintiff’s expert applied a discount rate of between 5 percent and 12 percent to account for risk factors, while the defense expert applied a discount rate of 26 percent. The jury award was close to, but more than, the defense’s projection of lost profits. The court stated that the “jury’s determination of riskiness (as embodied in its selection of an appropriate discount rate) represents a factual determination that we do not disturb on appeal.” 37 Ordinary Business Risk Rates Discount rates reflecting ordinary business risk are commonly based on the business’s WACC. This can be seen in the following cases: • NCMIC Finance Corp. v. Artino, 638 F. Supp. 2d 1042 (S.D. Iowa 2009). “Profits lost by leasing company as a result of former employee diverting leases to competitor discounted over 5 years as lease profits are received over a 5-year period at a 7.44% discount rate, which was the plaintiff’s weighted cost of capital and funding.” 38 35 Robert L. Dunn, Recovery of Damages for Lost Profits, September 2023 Supplement, 485. 36 Ibid., 484. 37 Ibid., 489. 38 Ibid., 485. 39 Ibid., 486. 40 Ibid., 485 (decision reversed on other grounds). 41 Harry and Kinrich, Lost Profits Damages: Principles, Methods, and Applications, 558. 42 Dunn, Recovery of Damages for Lost Profits, September 2023 Supplement, 486. 43 Ibid., 484–85. • SJW Property Commerce, Inc. v. Southwest Pinnacle Properties, Inc. , 328 S.W.3d 121 (Tex. App. 2010). “Not error to permit expert testimony to a 6.25% discount rate for plaintiff’s cost of capital to calculate damages for lost profits arising from a 1-year delay in receiving rental revenue.” 39 • Wye Oak Technology, Inc. v. Republic of Iraq , 2019 WL 4044046 (D.D.C. 2019). In this case, a discount rate of 30.2 percent was approved. The expert used the build-up method by totaling discount rates for three components. The discount rate was based on “discounted cash flow methodology and representing equity risk, small firm size, and industry risk components for work done by U.S. contractor under military equipment contracts with government of Iraq.” 40 Company-Specific Risk Rates The final level of discount rates reflects company-specific risk. “The company-specific risk premium (CRSP) is a discount rate adjustment (often positive but possibly negative) to account for the company’s risk variance from the industry’s average enterprise. Typically, the damages analyst increases the discount rate for the company’s unsystematic risk, which, by definition, exceeds the systematic risk for the relevant industry.” 41 The following cases reflect the inclusion of company-specific risk in discount rates: • Fairmont Supply Co. v. Hooks Industrial, Inc., 177 S.W.3d 529 (Tex. App. 2005). “On claim for damages for breach of requirements contract, plaintiff’s expert testified to a discount rate of 33%; defendant’s expert testified to a discount rate of 36%; jury verdict was within the evidence.” 42 • Spectrum Sciences & Software, Inc. v. United States , 98 Fed. Cl. 8 (2011). “Court fixed 16% discount rate in breach of contract action, derived by adding risk-free rate, equity risk premium, negative industry risk premium, size premium, and key customer dependence factor.” 43 The range of discount rates reflects the diversity of facts, financial conditions, and future loss periods being considered. These decisions provide examples of experts applying different discount rates under differing circumstances. The final arbiter of these calculations is the trier-of-fact, who 13 May | June 2024 A Professional Development Journal for the Consulting Disciplinesmay accept, reject, or adjust an expert’s discount rate for calculating the present value of future lost profits. A Texas case 44 reinforces the importance of the trier-of-fact in making the final decision: In Knox v. Taylor , the defendants sought to reverse a jury verdict on the ground that it was not supported by the evidence. One of their many grounds was that the plaintiffs’ expert had used a risk-free rate to discount lost profits in a business that was not risk-free. In contrast, the defendants’ expert had used 30%. The court held that the use of a risk-free rate was not grounds for reversal, because given the fact that the jury’s award was only a fraction of the amount the plaintiff[s’] expert estimated, the jury may consider any alleged flaws in his testimony. 45 Conclusion Financial experts regularly receive assignments asking them to estimate lost profits for all or a part of a business allegedly harmed by another person or business. In conducting their analyses, experts may determine that lost profits will occur in the future. In that case, projected future lost profits must be discounted to present value. One of the first decisions made by an expert is whether to model the projected cash flow of the injured business. 44 Knox v. Taylor, 992 S.W.2d 40 (Tex. Ct. App. 1999). 45 Robert M. Lloyd, “Discounting Damages: Case Law,” 300. Next, the expert must determine which of several approaches to use to calculate an appropriate discount rate, selecting the approach that best fits the facts of the case. The plaintiff and defendant may present opposing opinions as to the appropriate discount rate. The reasoning for each expert’s discount rate will assist the trier-of-fact in assessing not only which rate should be applied but also the reliability of each expert’s overall analysis. In the end, the trier-of-fact may accept a rate, reject it, or choose its own based on the trial testimony. The cases discussed above provide a range of accepted discount rates, stretching from 1.24 percent to 36 percent. If nothing else, these results show that determining a discount rate for lost profits is not a one-size-fits-all endeavor. The most commonly used source for the discount rate in lost profits matters is the injured business’s WACC. However, financial experts should recognize that different plaintiffs will have different financial circumstances, which will affect the resulting discount rate. Many factors play a role, including the period of loss, the injured firm’s profit history and financial condition, the market served, and overall economic conditions. That is why appropriate discount rates range from risk-free to company-specific and why experts have a broad spectrum of approaches for assessing the appropriate rate. Allyn Needham, PhD, CEA, is a partner at Shipp Needham Economic Analysis, LLC, a Fort Worth- based litigation support, consulting expert services, and economic research firm. Prior to joining Shipp Needham, he worked in the banking, finance, and insurance industries. As an expert, he has testified on matters relating to commercial damages, personal damages, business bankruptcy, and business valuation. Dr. Needham has published articles on financial and forensic economics, and provided continuing education presentations at professional economic, vocational rehabilitation, and bar association meetings. Email: aneedham@shippneedham.com. 14 The Value Examiner Litigation ConsultingThe Authority in Matters of Value ® Credentialing Databases / Software Continuing Professional Education NACVA.com | NACVA1@NACVA.com | (800) 677-2009 NACVA’s Certified Valuation Analyst ® (CVA ® ) and Master Analyst in Financial Forensics ® (MAFF ® ) designations are the only valuation and financial forensic credentials accredited by the National Commission for Certifying Agencies ® (NCCA ® ), the accreditation body of the Institute for Credentialing Excellence™ (ICE™). The CVA designation is also accredited by the ANSI National Accreditation Board ® (ANAB ® )This article demonstrates the importance of granular assumptions and modeling decisions when calculating size premiums. We explain these modeling choices in detail and present a series of size premiums under different sets of assumptions. All size premiums reflect a beta adjustment in the spirit of the modified capital asset pricing model, which is popularly used by valuation practitioners to model the cost of equity. We assess size premiums across size deciles based on market capitalization. While baseline size premiums reflect common practice, we find that small, reasonable deviations from common practice can result in material differences in size premiums. We offer alternative size premiums along with guidance indicating when they may be appropriate. 1 Much of the discussion in this section is motivated by Derek Zweig and Adam Luke, “Size and Capitalization Adjustments for Market-Based Pricing Multiples,” The Value Examiner, July/August 2022. 2 E.g., Jack L. Treynor, “Market Value, Time, and Risk” (unpublished manuscript, August 8, 1961), http://doi.org/10.2139/ssrn.2600356; William F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance 19, no. 3 (September 1964), https://doi.org/10.1111/j.1540-6261.1964.tb02865.x; John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics 47, no. 1 (February 1965), https://doi.org/10.2307/1924119; John Lintner, “Security Prices, Risk, and Maximal Gains From Diversification,” Journal of Finance 20, no. 4 (December 1965), https://doi.org/10.2307/2977249; Jan Mossin, “Equilibrium in a Capital Asset Market,” Econometrica 34, no. 4 (October 1966), https://www.jstor.org/stable/1910098?origin=JSTOR-pdf; Fischer Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business 45, no. 3 (July 1972), http://doi.org/10.1086/295472. 1. Background The ideal methodology to value an asset is to observe the prices paid for that asset in the open market. However, when valuing shares of a privately held company, no such marketplace exists. Valuation theory thus includes consideration of three commonly accepted approaches to value: the market approach, the income approach, and the asset approach. Each approach contains various valuation methods. The choice of which approaches and methods to use depends on the specific facts and circumstances of the valuation. Under the income approach, the value of a company is estimated based on the discounted value of projected future cash flows. This requires one to estimate both the growth rates of free cash flows and the risk of realizing those cash flows. To estimate risk, practitioners often build up the cost of capital by starting with a risk-free rate and adding “premiums” to account for additional risks. Similarly, the capital asset pricing model (CAPM) is a popular model for estimating the cost of equity. Due to the CAPM’s empirical shortcomings, it is common to append a size premium to better reflect the cost of equity for securities observed in the market. The size premium is meant to capture additional risk an investor is exposed to by owning equity in a company of a particular size. As discussed below, small companies are traditionally thought of as higher risk, implying that the size premium should be larger. In this article, we will study the theoretical support for this size premium, discuss its critiques, and ultimately estimate the size premium ourselves, examining whether these estimates are robust to a host of alternative modeling choices. 2. Concept Overview 1 Pioneering work from several authors 2 led to early adoption of the CAPM as the leading approach to explain the cost of 16 The Value Examiner Valuation 16 The Value Examiner Investigating the Sensitivity of the Size Premium By Derek Zweig, CFA, FRM, Timothy Sumner, and Adam Luke, PhDequity. Under several restrictive assumptions, 3 the CAPM states that the cost of equity (r i ) for security i is calculated using Equation 1: r i = r f + β i (r m − r f ) (1) Here, the cost of equity 4 is a function of the risk-free rate (r f ) , the excess return of the market (r m − r f ) , and security i ’s systematic risk coefficient ( β i ). β i = 0 indicates that security i has zero systematic risk, while β i = 1 indicates that security i ’s systematic risk is identical to that of the market as a whole. The market return (r m ) is often estimated using a broad equity index, such as the S&P 500. While this approach to estimating the cost of equity is straightforward, decades of empirical work suggest that the CAPM consistently underestimates equity returns. 5 Banz 6 was the first to note that the degree of underestimation seemed to coincide with different measures of company size. Banz used the simple linear functional form: r i = γ 0 + γ 1 β 1 + γ 2 ϕ i − ϕ m ϕ m (2) where r i is the equity return for security i , γ 0 is the risk- free rate (i.e., the return on a zero-beta portfolio), γ 1 is the expected market risk premium (r m − γ 0 ) , ϕ i is the market value of equity of security i , ϕ m is the average market value of equity for the market as a whole, and γ 2 is the constant measuring the contribution of ϕ i to the expected equity return. If there is no size premium, then γ 2 = 0 , and Equation 2 would reduce to traditional CAPM. 3 Investors: (a) have identical and complete information, (b) have homogeneous expectations, (c) are rational and risk-averse, (d) maximize utility, (e) diversify investments, and (f) are price takers. Additionally, all assets are assumed perfectly divisible and liquid, trading frictions (taxes and transaction costs) are assumed not to exist, and borrowing and lending is assumed to be available in unlimited amounts at the risk-free rate. 4 The cost of equity may be interpreted as the return one should expect to receive for investing in an equity security with a comparable risk profile. 5 E.g., S. Basu, “Investment Performance of Common Stocks in Relation to Their Price-Earnings Ratios: A Test of the Efficient Market Hypothesis,” Journal of Finance 32, no. 3 (June 1977 ), https://doi.org/10.2307/2326304; Rolf W. Banz, “The Relationship Between Return and Market Value of Common Stocks," Journal of Financial Economics 9, no. 1 (March 1981), https://doi. org/10.1016/0304-405X(81)90018-0; R. Ostermark, “Portfolio Efficiency of APT and CAPM in Two Scandinavian Stock Exchanges,” Omega 18, no. 4 (1990), https://doi.org/10.1016/0305- 0483(90)90034-7; Eugene F. Fama and Kenneth R. French, “The Capital Asset Pricing Model: Theory and Evidence,” Journal of Economic Perspectives 18, no. 3 (Summer 2004), https://doi. org/10.1257/0895330042162430; Kapil Choudhary and Sakshi Choudhary, “Testing Capital Asset Pricing Model: Empirical Evidences from Indian Equity Market,” Eurasian Journal of Business and Economics 3, no. 6 (November 2010), https://www.ejbe.org/index.php/EJBE/article/view/44; Mayur Agrawal, Mohapatra Debabrata, and Ilya Pollak, “Empirical Evidence Against CAPM: Relating Alphas and Returns to Betas,” in 2011 IEEE International Conference on Acoustics, Speech and Signal Processing, https://doi.org/10.1109/ICASSP.2011.5947662; Yasmeen et al., “The Capital Asset Pricing Model: Empirical Evidence from Pakistan” (MPRA Paper 41961, University Library of Munich, Germany, 2012). Note that these studies cover both developed and emerging markets. 6 Banz, “The Relationship Between Return and Market Value of Common Stocks.” 7 Ibid., 3. 8 For a more detailed discussion of these critiques, see Zweig and Luke, “Size and Capitalization Adjustments for Market-Based Pricing Multiples.” 9 E.g., Clifford Asness et al., “Size Matters, If You Control Your Junk,” Journal of Financial Economics 129, no. 3 (September 2018), https://doi.org/10.1016/j.jfineco.2018.05.006; Stefano Ciliberti et al., “The 'Size Premium' in Equity Markets: Where Is the Risk?,” Journal of Portfolio Management 45, no. 5 (July 2019), https://doi.org/10.3905/jpm.2019.1.086.; Roger Grabowski, “The Size Effect Continues to Be Relevant When Estimating the Cost of Capital,” Business Valuation Review 37, no. 3 (Fall 2018), https://www.kroll.com/-/media/assets/pdfs/publications/ valuation/grabowski-size-effect-and-cost-of-capital.pdf; Kewei Hou and Mathijs A. van Dijk, “Resurrecting the Size Effect: Firm Size, Profitability Shocks, and Expected Stock Returns," Review of Financial Studies 32, no. 7 (July 2019), https://doi.org/10.1093/rfs/hhy104. 10 Shannon P. Pratt and Roger J. Grabowski, Cost of Capital: Applications and Examples, 5th ed. (Hoboken, NJ: John Wiley & Sons, Inc., 2014). 11 For a more detailed discussion of the theoretical properties thought to support the existence of a size premium, see Zweig and Luke, “Size and Capitalization Adjustments.” Banz found that, on average, smaller firms have higher risk- adjusted returns than larger firms. This size premium was also found to be nonlinear, accelerating in magnitude as companies get increasingly small. A number of critiques followed, with a host of alternative explanations arising for the observed size premium. As Banz himself notes, “It is not known whether size per se is responsible for the [size premium] or whether size is just a proxy for one or more true unknown factors correlated with size.” 7 These critiques emphasize data artifacts such as bid-ask bounce and delisting bias, sampling nuances, time- dependence of the size treatment, and illiquidity. 8 While each critique is legitimate, and possibly devastating, recent empirical work 9 has attempted to refine the size premium calculation methodology to avoid these measurement pitfalls. If the premium is in fact valid, what practical and theoretical explanations account for its existence? From a theoretical standpoint, there are several properties of small stocks that make their risks fundamentally different from those of large stocks. 10 Sparsity of investment data and analyst coverage, protection of market share, resource base and access to capital markets, management depth, and customer base fragmentation are just a few of the properties thought to justify the theoretical existence of a size premium. 11 Each of the theoretical and practical differences between small and large firms should work, in theory, to increase the required return on an investment in small stocks. Of course, the size premium is not predictive and cannot be relied upon for any particular holding period. In other [ ] 17 May | June 2024 A Professional Development Journal for the Consulting Disciplineswords, the size premium tends to be cyclical, even prior to the 1980s. There are many historical periods in which small stocks significantly underperformed large stocks. If this were not true, there would not be an increased risk associated with small firms. The cyclicality should be a manifestation of the size risk. Thus, it is not surprising that the size premium fluctuates across time, only stabilizing over longer time horizons. 12 The empirical importance of the size premium has led to the modified CAPM as an improvement on the CAPM in measuring cost of equity, represented by Equation 3: r i = r f + β i (r m − r f ) + r i s (3) Here, r i s represents the size premium for security i . Notice that r i s is treated as a residual; that is, all return in excess of the CAPM-expected return is attributable to the size premium. Rearranging Equation 3 and taking expectation, the size premium for security i can be stated simply as: E(r i s ) = E(r i − r f ) − β i E(r m − r f )( 4) Valuation practitioners tend to rely on Equation 4 for the estimation of size premiums, which are then inserted into the modified CAPM (or a similar build-up model) to estimate the cost of equity for a private company. Accordingly, the baseline methodology employed to calculate the size premium in subsequent sections reflects that common practice. We apply “robustness” tests to common practice to understand how small methodological deviations impact results. 3. Methodology In this section, we describe a methodology for calculating size premiums that is consistent with common practice in the valuation industry. Broadly speaking, we estimate E(r i s ) from Equation 4. While the size premium, r s , will vary from one security to another, our goal is to separate securities into groups and present r i s in a way that can be generalized for use in valuation of a private company that belongs to group i . In other words, we are interested in the expected value of r i s for all groups i (defined below). To estimate the size premium, we must review a number of modeling choices in our analysis: 12 Grabowski, “The Size Effect Continues to Be Relevant.” 1. Population of securities . This determines which securities should be included in the analysis. 2. Segmentation of securities . This is a choice regarding how to break the population of securities into subgroups. Included in this decision is a selection of the metric by which companies are grouped. 3. Aggregation method. This describes how the returns for securities in each subgroup are condensed into a single return measure for the subgroup. 4. Estimation window . The period of time over which data is collected and analyzed. 5. Return period . The frequency with which returns are sampled over the estimation window. 6. Return averaging method . The averaging method used to represent expected return. 7. Risk-free rate . The return one should expect to receive without any risk of default. One must choose a risk-free instrument as a proxy for this theoretical return. 8. Systematic risk coefficient (beta) . One must select a method for calculating the systematic risk coefficient, an important variable in the modified CAPM. 9. Equity risk premium. A function of the risk-free return and the market index return. One must select a market index and make decisions regarding the units of the returns. Note that some of these modeling choices may include multiple modeling decisions. While theory allows some modeling choices to be eliminated, it is often unclear whether one choice is better than another. Accordingly, one may test more than one modeling choice to allow users to decide for themselves how best to use published size premiums. This highlights the importance of the robustness tests discussed in Section 5. The following subsections expand upon each modeling choice, describing the justification for our baseline approach as well as possible robustness tests. 3.1. Modeling Choices Below, we discuss each modeling choice in detail and identify the “baseline” modeling choice, which is meant to reflect common practice in estimating size premiums. (See Table 1.) We use non-baseline modeling choices as robustness tests to depict the sensitivity of the published size premiums to baseline modeling choices. 18 The Value Examiner ValuationTable 1: Baseline Modeling Summary Modeling ChoiceSelections Population of SecuritiesU.S. Exchange-Traded Equities Segmentation of Securities Size (Market Capitalization) Deciles Aggregation MethodMarket Capitalization Weighting Estimation Window1927–2022 Return PeriodAnnual Return Averaging Method Arithmetic Average Risk-Free Rate20-Year Treasury Systematic Risk Coefficient OLS Beta, Monthly Returns Equity Risk Premium S&P 500 Index; Nominal Returns; Pre-tax; Supply-Side 3.1.1. Population of Securities To properly estimate size premiums, it is necessary to perform a separate analysis in each jurisdiction. For our purposes, the population of equity securities is limited to exchange-traded public companies in the U.S. (excluding American depository receipts). To avoid delisting bias, delisted securities are included so long as pricing information is available following delisting. If a delisted security can no longer be traded, it is considered worthless. 3.1.2. Segmentation of Securities To assess the impact of size on return, we must “bucket” securities according to size. To the extent the impact of size on returns is nonlinear, a more granular bucketing provides a more precise estimate of the size premium. At one extreme, one could assign each security to its own bucket, though the analysis and results would quickly become unwieldy. On the opposite extreme, using only one bucket would fail to capture the effects of size. Academics and practitioners have become comfortable over time using 10 buckets. The population of equity securities are bucketed into deciles based on size, with the minimum, maximum, and weighted average size used to describe each decile. Breaking the population of securities into deciles allows nonlinearities to appear in the results without becoming unwieldy. We apply decile bucketing in our analysis, rebalancing deciles for changes in size on an 19 May | June 2024 A Professional Development Journal for the Consulting DisciplinesNext >