< Previousannual basis (at the beginning of each calendar year). If a security starts trading mid-year, we use its first available estimate of size for ranking. Decile 1 represents the largest strata of securities, while Decile 10 represents the smallest. In conjunction with the choice of bucketing, one must also decide which metric to use as a proxy for size. There are many options—including market capitalization, enterprise value, total revenue, EBITDA, total assets, and trading volume—each of which is a viable candidate for representing size. Financial statement metrics carry some downsides in that accounting decisions can impact the size metric. The further down the income statement one travels, the more accruals come into play. Similarly, balance sheet items differ depending on financing choices. Different levels of capital intensity can cause one firm to appear larger or smaller than another. To avoid these pitfalls, it is common to define size via market capitalization when estimating size premiums, which is the baseline approach we take in this analysis. 3.1.3. Aggregation Method Once the population of securities is defined and the segmentation approach settled, one must decide how to aggregate return statistics up to the bucket (decile) level. The most intuitive approach is to take the weighted average return of all constituent securities in a decile to arrive at the decile-level return. However, one must decide which weighting scheme to apply. Popular choices include value weighting, price weighting, and equal weighting. Value weighting applies a weight to each constituent security return based on the security’s market capitalization, while price weighting bases weights on share price. Equal weighting gives all securities an equal weight. Each weighting method has a theoretical interpretation. Price weighting produces a bucket value proportional to the average stock price of constituent securities in the bucket. While this may be a useful property for tracking average sector performance, it places emphasis on arbitrary accounting decisions that feed into share volumes. Despite the use of price weighting by the popular Dow Jones Industrial Average index, price weighting does not have strong justification as a weighting scheme in this analysis. Market capitalization weighting and equal weighting can be thought of as opposing views on portfolio rebalancing. With market capitalization weighting, companies with larger market capitalization within a decile receive greater weight. As share prices, and hence market capitalizations, change, the weights are automatically updated. This weighting scheme is akin to a buy-and-hold strategy, where a portfolio is never rebalanced. Equal weighting assigns a weight of 1 − n to each of the n constituent securities in a decile. The size of constituent securities will change, reflecting returns, but growth of a security will not cause it to demand more weight in the return calculation for the decile. This weighting scheme is akin to a dynamic rebalancing strategy, where a portfolio is rebalanced after each return period. While both weighting schemes have theoretical justifications, rebalancing a portfolio causes one to incur trading fees and 20 The Value Examiner Valuationcommissions, and may have tax consequences. Rebalancing too frequently will cause returns to be lower after accounting for these costs. Though investors do periodically rebalance, so that their portfolio returns will not perfectly reflect market capitalization weighted returns, equal weighting takes an arbitrary stance on rebalancing frequency and arguably requires consideration of the impact of rebalancing costs on returns (especially when many constituent securities are included and the rebalancing frequency is high). Beyond rebalancing assumptions, obstacles like bid-ask bounce also pose an issue when aggregating the returns of equity securities with prices near zero (i.e., penny stocks). As prices jump from bid to ask, and vice versa, the upward movement from bid to ask will be larger in magnitude from a percentage standpoint than the downward movement from ask to bid. Giving an equal weight to these returns can cause an upward bias in the aggregated return. 13 Whichever weighting scheme one prefers, it is important for purposes of this analysis that the aggregation method for the deciles matches that of the index selected for calculating the equity risk premium (see Section 3.1.9). All things considered, market capitalization weighting is the most common approach to aggregating security returns. This weighting scheme provides more flexibility in selecting an index for calculating the equity risk premium and avoids some tricky obstacles. Market capitalization weighting is the baseline approach to return aggregation. We conduct a robustness test to determine the impact of using equal weighting for both decile and index construction on size premiums (see Section 5). 13 This is arguably most relevant only for the smallest strata of equities, as larger stocks tend to be more liquid and have smaller bid-ask spreads relative to the price of the securities. By focusing on exchange-traded securities, bid-ask bounce is not expected to have a material impact on aggregated returns, even in the smallest bucket. Further, different choices of return period can mitigate this impact further (see Section 3.1.5). 14 Roger G. Ibbotson and James P. Harrington, Stocks, Bonds, Bills, and Inflation (SBBI), 2021 Summary Edition (Charlottesville, VA: CFA Institute, 2020), 116, https://www.cfainstitute.org/-/ media/documents/book/rf-publication/2021/sbbi-summary-edition-2021.ashx. 3.1.4. Estimation Window Stock exchanges in the U.S. date back to the late eighteenth century, starting with the Philadelphia Stock Exchange in 1790 and followed shortly thereafter by the New York Stock Exchange in 1792. The emergence of stock exchanges in the U.S. coincides with the advent of corporations, a budding global capital market, and the dawn of the American industrial revolution. At the time, publicly traded corporations were few in number, thinly traded, and enjoyed only a primitive standardization of performance reporting. It was not until 20 years later that economically pivotal industrial corporations began to benefit from public equity markets. Incorporations accelerated over the remainder of the nineteenth century, defined by waves of mergers and acquisitions, industry consolidation, and expanding reach of antitrust laws. While equity market data was recorded over this period, it was of low quality. Data points were often missing, recorded at irregular intervals, highly subject to human error, and generally disorganized. There is some consensus that high quality market data became available around 1927 in the U.S. With relatively high-quality return data dating back to 1927, the question becomes how much of this data is relevant when pricing an equity today? It is common practice to take a long-run perspective when calculating size premiums; 14 that is, one should use all data available to determine expected decile returns, market index returns, and risk-free returns. Ideally, one would use return data that embodies all major future events that an investor might experience. Given that we do not know what the future holds, one might reasonably decide to include all high-quality historical data. Whichever weighting scheme one prefers, it is important for purposes of this analysis that the aggregation method for the deciles matches that of the index selected for calculating the equity risk premium. 21 May | June 2024 A Professional Development Journal for the Consulting DisciplinesThis ensures that we capture as many historical events as possible, which may inform us about future events. This is a compelling approach and is used when selecting the baseline estimation window for this analysis. There are, however, challenges with using such a long period of time. The goal of extracting size premiums from historical equity return data is to support cost-of-capital estimations used in valuations today. To this end, valuation analysts should be confident that the pricing regime in which the size premium is calculated matches the pricing regime today. If market pricing follows fundamentally different rules, the size premium calculated from a different regime, while historically interesting, may not be relevant to a modern valuation. Note that regimes are different from events. While economic changes are constantly occurring, many prove to be transitory (e.g., jumps). We may loosely classify these transitory changes as “events.” An event can have a large impact on observed data over a short period of time, but ultimately does not reflect a permanent change in economic behavior. “Regimes” represent behavioral changes that persist over time. The occurrence of an event in the distant past does not disqualify that past data from being relevant today. A regime change, on the other hand, might mean that conclusions drawn from data prior to the regime change lack external validity. 15 Andrew Ang and Allan G. Timmermann, “Regime Changes and Financial Markets” (Netspar Discussion Paper No. 06/2011-068, June 20, 2011), https://doi.org/10.2139/ssrn.1919497. 16 Like bid-ask bounce, lead-lag effects are most relevant for the smallest, illiquid equity securities. Focusing on exchange-traded securities should mitigate the impact of lead-lag effects, though strategic choice of return period is still important here. Regime changes may take many forms, often reflecting legal, political, or technological developments. In calculating size premiums, it is important to understand regimes for both equity and interest rate markets. It is well documented that both of these markets have experienced regime changes more than once in just the past several decades. 15 Rather than attempt to identify specific regime changes that have impacted equity markets, we bypass this challenge through the use of rolling and shrinking windows. We look at many possible estimation windows to determine the chosen estimation window’s impact in calculating the size premium (see Section 5). 3.1.5. Return Period Once we decide on an estimation window, we must choose a frequency with which returns are collected within the period. At the high frequency extreme, returns could be calculated intraday. Such a high frequency has its downsides. First, intraday return data is difficult to obtain, only recently available (in the context of an estimation window beginning in 1927), and often missing for many securities. Even when available, intraday trading is different for every security and from one day to the next, so the return intervals are irregular, posing a challenge for traditional statistical techniques. Phenomena like bid-ask bounce can also result in a biased analysis of return data, especially at the intraday frequency. Beyond these challenges, we must consider not just the level of returns but the relationship of decile returns relative to a market index. This means choosing a return period that appropriately accounts for lead-lag effects. In other words, to the extent there is a delay in the impact of changes in one time series on another, high frequency return periods will not reflect this. To avoid this issue, shorter return frequencies (i.e., longer return periods) may be preferred. 16 Many of these considerations apply at the daily, weekly, and even monthly return periods as well. Some researchers use the sum beta to ensure lead-lag effects are captured (see Section 3.1.8 for more on beta calculation). Nonetheless, the longer the return period, the more likely lead-lag effects are properly considered. At the opposite (low) frequency extreme, the return period is set equal to the estimation window. This provides only a 22 The Value Examiner Valuationsingle return data point for each decile and for the market index. Statistically, this allows very few inferences to be drawn from the data. The same can be said of multiyear return periods, as there are only 95 years between 1927 and 2022. The smaller the sample of returns, the higher the standard error when estimating average returns and the beta coefficient (see Sections 3.1.6 and 3.1.8 for more discussion on average returns and the beta coefficient, respectively). Consequently, the largest sample possible is desirable, which implies a preference for a shorter return period. To balance these conflicting considerations, monthly or annual return periods are most commonly chosen. These frequencies are most readily available and complete, capture return relationships, and provide large enough samples to draw robust statistical inferences. We use an annual return period in the baseline analysis. 3.1.6. Return Averaging Method Now that we have determined which estimation window to examine and the return period(s) to use, we must determine how to take expectation of return. To this end, there are two common approaches: arithmetic average and geometric average. The arithmetic average return, r (A) , across n return periods is calculated as follows: where r t is the return for return period t . In our parlance, the periods t = 1 through t = n represent the estimation window. The geometric average return r (G) across n return periods is: The difference between these two averaging methods can be thought of as the “compounding effect”; that is, geometric returns account for the compounding of return 17 Stephen Abbot, Understanding Analysis, 2nd ed. (New York: Springer, 2015), 61. Note that arithmetic and geometric averaging will only be equal in the special case where returns are constant for every return period in the estimation window. 18 George Casella and Roger L. Berger, Statistical Inference (Independence, KY: Cengage Learning, 2002). 19 See, e.g., Benoit Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Business 36, no. 4 (October 1963), https://doi.org/10.1086/294632; Eugene F. Fama, “The Behavior of Stock-Market Prices,” Journal of Business 38, no. 1 (January 1965), https://doi.org/10.1086/294743; Benoit Mandelbrot and Howard M. Taylor, “On the Distribution of Stock Price Differences,” Operations Research 15, no. 6 (November–December 1967), https://doi.org/10.1287/opre.15.6.1057; Andrew W. Lo and A. Craig MacKinley, “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test,” Review of Financial Studies 1, no. 1, (January 1988), https://doi.org/10.1093/rfs/1.1.41. 20 Nassim N. Taleb and George A. Martin, “The Illusion of Thin-Tails under Aggregation” (working paper, January 2012), https://www.scribd.com/doc/152200528/The-Illusion-of-Thin-Tails-Under- Aggregation-SSRN-Id1987562?doc_id=152200528&download=true&order=634953640. 21 Aswath Damodaran, “Equity Risk Premiums (ERP): Determinants, Estimation and Implications—The 2023 Edition” (NYU Stern School of Business, March 23, 2023), http://dx.doi.org/10.2139/ ssrn.4398884. from one return period to the next. Not only is the original investment receiving return, but returns are also generating returns from one period to the next. In this context, compounding has the paradoxical effect of reducing the average. Generally speaking, the geometric average will be less than the arithmetic average. 17 This is because, with the inclusion of compounding from one return period to the next, the average return can be lower but still achieve the same total return. Arithmetic averaging, on the other hand, does not consider the gains from compounding. Consistent with common practice, we use the arithmetic average in the baseline results of this analysis. The preference for using the arithmetic average stems from certain desirable statistical properties of the arithmetic average under a set of restrictive assumptions. 18 If equity returns follow a normal distribution (or some other well-behaved distribution for which the central limit theorem works quickly), and if returns are independent from one return period to the next, then the arithmetic average is the best (lowest variance), unbiased estimator of return for the next period. There is mounting evidence, however, that the arithmetic average is an imperfect choice for estimating a premium that is used to build up the cost of capital. It is well documented that equity returns may not follow a well behaved distribution, 19 even after various forms of aggregation. 20 Rather, equity returns are better fit to power-law distributions, wherein “extreme” returns dominate when measuring statistical moments (some of which may even be infinite). In such a case, the central limit theorem may be less than helpful in finite time, calling into question the claim that the arithmetic average is the best return estimator. Furthermore, there is empirical evidence that equities are negatively serially correlated, not independent. This means strong performance in one period tends to be followed by relatively poor performance in the next, and vice versa. In this context, arithmetic average returns may become biased upwards, resulting in overstated size premiums as compared to geometric averaging. 21 For example, if a stock price 23 May | June 2024 A Professional Development Journal for the Consulting Disciplinesincreases from $100 to $200 and then falls back to $100, the stock has not gained any value. However, this represents a 100 percent return followed by a −50 percent return, which results in a 25 percent arithmetic return (and a 0 percent geometric return). In this way, arithmetic averaging gives the appearance of higher returns to higher volatility stocks. Note also that valuation models tend to involve multiple periods over which a discount rate is applied, meaning the discount rate must be compounded when applied to cash flows in multiple future years. Compounding with a premium derived from arithmetic averages may be illogical. Geometric averages do not suffer from this logical inconsistency, as compounding is built into their calculation. To understand how the resulting size premiums are impacted by return averaging method, we restate our results using the geometric average in Section 5. 3.1.7. Risk-Free Rate The risk-free rate represents a rate of return that is free of default risk. Notice that this definition still allows for the possibility that the risk-free return carries some kind of risk, just not default risk. The risk-free rate has long been considered an abstraction, thought not to exist in its purest form. In practice, certain securities have been designated as the best risk-free proxies. It is common in the valuation industry to use sovereign borrowing rates; specifically, borrowing rates of developed sovereigns that borrow in their own currency. In the U.S., Treasury securities can be used to measure sovereign borrowing rates. However, there are some limitations to the use of sovereign rates as the risk-free analog. 22 First, sovereign securities often receive beneficial tax treatment that may impact the level of the rate. As tax rates change, so will the coupons on these securities. Additionally, there are no capital requirements imposed on banks for holding Treasury securities, while other low-risk instruments do require capital allocation. This may artificially increase the demand for Treasury securities relative to other low-risk instruments, suppressing sovereign borrowing rates to levels they might otherwise exceed. Lastly, despite the ability of sovereigns to always meet debt obligations denominated in 22 John Hull, Options, Futures, and Other Derivatives (London: Pearson, 2018). 23 Some have suggested it is best to use a short-duration instrument when proxying the risk-free rate to reduce price risk. This is sensible for a short-term holding period, but over a longer-term holding period one must roll investments in short-duration instruments, reintroducing price risk and suggesting an instrument with a duration similar to the holding period may be more appropriate. 24 Pitchbook, 2022 Annual U.S. PE Middle Market Report, March 14, 2023, https://pitchbook.com/news/reports/2022-annual-us-pe-middle-market-report#:~:text=Some%20firms%20are%20 still%20shelling,of%20sponsors%20shifting%20down%20market. their own currency, there has been a long history of sovereign defaults due to political factors. This political risk implies that default risk is not truly absent for sovereign securities. Any market instrument used to proxy risk-free returns is also subject to price risk; that is, as market returns change, these risk-free securities will experience unrealized capital gains and losses. Accordingly, the total return to a risk-free instrument includes both “default-free” cash flows and capital gains (or losses). The inclusion of returns attributable to price risk is inconsistent with the spirit of the risk-free return and, in turn, is ignored when benchmarking risk-free rates. As a result, capital gains on risk-free securities should be ignored, leaving only the rate of return from contractually promised cash flows. 23 For purposes of this analysis, we use Treasury securities as the proxy for a risk-free instrument. The choice remains as to which Treasury security to use; that is, which Treasury security maturity bucket represents an appropriate opportunity cost to an equity investment. Holding periods for publicly traded equities range from intraday for day traders to many decades for buy-and- hold investors (or even into perpetuity for bequeathed investments). By definition, there is no market for privately owned equity, making intraday holding periods nearly impossible. Consortia of private investors, known as private equity groups (PEGs), tend to hold private companies for five to seven years on average 24 (see Figure 1). Owner–operators of private companies may hold their equity much longer, perhaps their entire professional lives. Figure 1: Average Holding Period (Private Equity Groups) 24 The Value Examiner ValuationAll things considered, because cash flows tend to be discounted into perpetuity, it has become common practice to use the 20-year Treasury bond as the best proxy for the risk-free rate for private equity investments. We note, however, that arguments can be made for use of five-, 10-, or 30-year Treasury securities as the appropriate proxy for risk-free rates. Thirty-year Treasury bonds suffer from a similar problem as secured overnight financing, in that they have large gaps in available historical data. We use the coupon rate on 20-year Treasury bonds as the baseline risk-free rate in this analysis. And we use the coupon rate on five-year Treasury bonds in robustness testing to understand how the choice of the risk-free rate may impact size premiums (see Section 5). One last note on the risk-free rate, regarding the use of normalized rates instead of the current risk-free rate for each period. When building up the cost of capital using the modified CAPM, some practitioners use the risk-free rate commensurate with the valuation date. This is equivalent to changing r f to be a constant in Equation 4, the constant being equal to the current period risk-free return. Other practitioners feel that interest rates at the time of the valuation date may have deviated from their normal level due to policy rate setting by the Federal Open Market Committee, and will “normalize” the interest rate to its average level. The latter approach is consistent with a time-dependent risk-free rate in Equation 4. Our baseline analysis uses a normalized risk-free rate. 3.1.8. Systematic Risk Coefficient (Beta) Several methodologies can be used to calculate the beta. These methodologies are discussed here, though the ultimate goal of this analysis is to arrive at an externally valid size premium. That means the size premiums derived must be consistent with the broader approach used to build up the cost of equity. To match practitioner preferences, our baseline approach assumes beta is calculated using ordinary least squares (OLS) regression (the “OLS beta”) without any adjustments, which is how beta is often treated in the modified CAPM when building up the cost of equity. 25 May | June 2024 A Professional Development Journal for the Consulting DisciplinesBefore introducing variants to the OLS beta, we redefine the OLS beta here by restating Equation 1: r i − r f = β i (r m − r f ) where the subscript i indexes decile i and β is the OLS beta. The OLS beta is calculated by regressing the decile risk premium (r i − r f ) on the market risk premium (r m − r f ) without an intercept. The OLS solution takes the following form: β i = Cov(r i ,r m ) σ r m 2 (5) where Cov(r i ,r m ) is the covariance between decile and market returns and σ r m 2 is the variance of market returns. There are several variants of beta that have garnered interest in the academic valuation world. The variant considered in this article involves the Blume adjustment. 25 The Blume- adjusted beta, β BA , is meant to be forward looking and reflects an empirical observation that a stock’s beta tends to mean-revert to the market average over time. 26 The market average here refers to a beta equal to one. The Blume- adjusted beta is a weighted average of the mean-reverting level (i.e., 1) and the OLS beta: β BA = w 1 β + w 2 (1) (6) In Equation 6, β refers to the OLS beta from Equation 5 and weights w 1 and w 2 are arbitrary such that w 1 + w 2 = 1 . It is common to set w 1 = 0.67 and w 2 = 0.33. As mentioned above, practitioners must be consistent in their methodology when incorporating size premiums. Incorporating a size premium in the modified CAPM that is calculated using one methodology, but then calculating beta in the modified CAPM using a different methodology will result in a mis- specified cost of equity. Accordingly, we apply the OLS beta as the baseline methodology in our calculation of size premiums, though we apply the Blume-adjusted beta in robustness tests. Lastly, it should be noted that the estimation window and return period used when calculating the beta should be consistent with that discussed in Sections 3.1.4 and 3.1.5. It is common when building up the cost of capital 25 The sum beta and the Vasicek beta are other popular variants. The impact of using these variants is not explored in this paper. 26 Marshall E. Blume, “Betas and Their Regression Tendencies,” Journal of Finance 30, no. 3 (June 1975), https://doi.org/10.1111/j.1540-6261.1975.tb01850.x. 27 Diversified except for size. 28 Damodaran, “Equity Risk Premiums.” for a private company to limit the estimation window for calculating beta to the last three to five years. This approach emphasizes business cycle behavior, capturing systematic risk within the context of the current business cycle. A limited estimation window may also avoid regime changes for individual companies, as mergers and acquisitions can fundamentally change systematic risk. When dealing with diversified deciles, 27 idiosyncratic considerations are no longer expected to change systematic risk. While regime changes are still possible, the baseline assumption is that the estimation window and return period should match that from which average return premiums are calculated (again, see Sections 3.1.4 and 3.1.5). 3.1.9. Equity Risk Premium The equity risk premium (ERP) represents the excess return of the market over the risk-free return. In Equation 3, it is represented by the term ( r m − r f ). Theoretically, the ERP is meant to compensate investors for the risk inherent in an equity investment. Similar to the systematic risk coefficient discussed in Section 3.1.8, the ERP may be calculated in several ways. Popular methods include: 28 • Survey-based ERP • Market-implied ERP from discounted cash flow models, default spreads, or option pricing models • Historical ERP As before, it is important that the choice of ERP in calculating size premiums is consistent with the ERP used by practitioners when building up the cost of equity. Accordingly, the baseline methodology for calculating the ERP is to use the historical ERP. The historical method involves estimating the average return to the market index over an estimation window and subtracting the average risk-free return over the same estimation window. In selecting the return averaging method, return period, and estimation window, one must consistently apply the same modeling choice for both decile and index returns to ensure the ERP is expressed in the same terms as the decile premiums. 26 The Value Examiner ValuationWe discussed the choice of risk-free rate in Section 3.1.7. Now, we must choose an appropriate proxy for the market index. As explained in Section 3.1.3, market capitalization weighting is the baseline approach for return aggregation at the decile level, so the market index should also be market capitalization weighted. Some popular market-capitalization- weighted indices are listed in Table 2. Table 2: Market-Capitalization-Weighted Indices IndexConstituent CountInception FTSE 1001001984 NASDAQ Composite2,500+1971 NYSE Composite3601965 Russell 20002,0001984 S&P 500 29 5001923 Wilshire 50005,0001974 Ideally, the chosen market index will represent the market as a whole, having no size bias. This may suggest a preference for an index with more (and size-diverse) constituents, but this preference must be balanced against the need for a long history. Additionally, it should be noted that use of an index may introduce a survivorship bias, as firms whose performance declines enough will be removed from the index, biasing index returns upward. Though the selection criteria 30 for inclusion in the S&P 500 may bias the index returns towards larger equity securities, the S&P 500’s long history and relative diversification have made it a common choice for calculating size premiums. Our baseline approach uses the S&P 500 as the market index. Because we apply such a long estimation window (see Section 3.1.4), we must also decide whether to use nominal or real returns (i.e., whether or not to adjust for inflation). The important point here is that all returns (for deciles, the market index, and the risk-free instrument) are expressed the same way. Our baseline approach uses nominal returns and expresses all returns pre-tax so as to avoid complications regarding changing tax laws and different tax rates for different investors. Our last note on the ERP involves the forward-looking nature of the estimate. For the size premium to be useful as a component of the cost of capital in valuation engagements, it should be 29 The S&P 500 did not have 500 constituent companies until 1957. Upon its inception in 1923, the index had only 233 constituent companies. 30 S&P Global, S&P Dow Jones Indices, S&P U.S. Indices Methodology, May 2024, https://www.spglobal.com/spdji/en/documents/methodologies/methodology-sp-us-indices.pdf. 31 Damodaran, “Equity Risk Premiums”; Roger G. Ibbotson and Peng Chen, “Long-Run Stock Returns: Participating in the Real Economy,” Financial Analysts Journal 59, no. 1 (January–February 2003), http://www.jstor.org/stable/4480453; Elroy Dimson, Paul Marsh, and Mike Staunton, “Equity Premia Around the World” (London Business School, October 7, 2011), http://dx.doi. org/10.2139/ssrn.1940165. 32 Ibid. forward-looking. Accordingly, the ERP should also be forward- looking. This means understanding which portion of the ERP is sustainable and which portion should be left in the past. To this end, it has become common to decompose the historical ERP into parts: 31 • Growth in income return (i.e., dividends and buybacks) • Growth from inflation • Growth in real earnings • Growth from expansion (or contraction) of pricing multiples It has been argued that the first three of these components are sustainable under the current monetary regime; that is, one should expect income return growth, real earnings growth, and inflation to continue at roughly their historical averages. Continued expansion (or contraction) of pricing multiples, however, is not thought to be a sustainable phenomenon. Figure 2 depicts the pricing multiple of the market index over the full estimation window. Figure 2: S&P 500 Price-Earnings Ratio (Full Estimation Window) Depending on the methodology, estimation window, and securities market analyzed, research suggests that anywhere from 0.5 percent to 1.25 percent of the ERP is attributable to expansion in pricing multiples. 32 An ERP that includes the first three components listed above and excludes the last component is known as the “supply-side ERP.” To ensure a forward-looking estimate of the ERP, we use the supply- side ERP in our baseline approach. We include use of the 27 May | June 2024 A Professional Development Journal for the Consulting Disciplinesunadjusted historical ERP to calculate size premiums as a robustness test in Section 5. 4. Data The time series data used in this analysis consists primarily of return data for different groupings of securities. Three categories of return data required sourcing: • Decile return time series • Market index return time series • Risk-free rate return time series We sourced size-based decile return data from the Value Analytics proprietary historical return database. 33 This data repository includes monthly and annual total return time series for value-weighted and equal-weighted portfolios constructed based on size. Decile return data spans from 1927 to 2022. We also sourced market index total return time series data from the Value Analytics proprietary return database. Market index data spans from 1927 to 2022. We pieced together risk-free return data from a variety of sources, each relied upon for specific time periods. We sourced market yields on constant maturity U.S. Treasury securities spanning 1962 to 2022 from FRED. 34 We used the 33 https://www.valueanalytics.org. 34 “Market Yield on U.S. Treasury Securities at X-Year Constant Maturity, Quoted on an Investment Basis,” FRED Economic Data, Federal Reserve Bank of St. Louis, accessed October 31, 2023, https://fred.stlouisfed.org/categories/115. 35 Economic Report of the President Transmitted to the Congress, Together with the Annual Report of the Council of Economic Advisors (Washington, D.C.: Government Printing Office, February 2011), https://www.govinfo.gov/content/pkg/ERP-2011/pdf/ERP-2011.pdf. 36 Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1914–1941 (1943), https://fraser.stlouisfed.org/title/38. Economic Report of the President and the Annual Report of the Council of Economic Advisors to impute Treasury yield data from 1942 to 1962. 35 And for data prior to 1942, we relied on the Board of Governors of the Federal Reserve System’s Banking and Monetary Statistics 1914–1941. 36 5. Results Table 3 presents the size premiums for each decile, as of 2022, under the baseline modeling choices described in Table 1. The average market capitalization corresponding to each decile, as of 2022, is summarized in Table 4. The following robustness tests were conducted to assess the sensitivity of the published size premiums to the baseline modeling choices: • Historical ERP (Alt 1) • Equally weighted returns (Alt 2) • 40-year estimation window (Alt 3) • Geometric average returns (Alt 4) • Five-year Treasury rate as risk-free rate (Alt 5) • Blume-adjusted beta (Alt 6) Each modeling adjustment was applied in isolation, meaning the above adjustments were not made simultaneously. These results are also shown in Table 3. Table 3: Size Premiums (Baseline Modeling Choices) DecileBaselineAlt 1Alt 2Alt 3Alt 4Alt 5Alt 6 10.93%-0.26%0.51%1.48%0.99%0.96%0.83% 21.79%0.45%1.66%2.44%1.62%1.74%1.92% 32.32%0.92%2.03%1.94%2.05%2.25%2.55% 42.60%1.16%2.34%1.86%1.98%2.51%2.90% 52.90%1.39%2.84%1.09%2.26%2.77%3.29% 62.77%1.24%2.61%1.11%1.90%2.64%3.20% 73.31%1.74%3.38%-0.03%2.05%3.15%3.80% 83.36%1.72%3.68%1.03%1.92%3.17%3.95% 93.85%2.14%5.14%0.64%1.22%3.62%4.55% 105.10%3.37%11.91%1.30%1.74%4.87%5.84% 28 The Value Examiner ValuationTable 4: Average Market Capitalization ($ in millions) BucketMarket Capitalization Decile 1$159,132 Decile 2$27,868 Decile 3$12,441 Decile 4$6,893 Decile 5$4,333 Decile 6$2,998 Decile 7$1,839 Decile 8$1,053 Decile 9$496 Decile 10$108 Using the unadjusted historical ERP (Alt 1) implies an assumption that all components of equity returns from the estimation window will remain stable, on average, into the future. Specifically, the portion of the ERP attributable to pricing multiple expansion over the estimation window is now included. By increasing the ERP, the size premium declines for each decile relative to the baseline modeling choice of using the supply-side ERP. Using equally weighted returns (Alt 2), the largest firms in each decile have reduced influence over the associated size premiums. The size premiums more closely reflect that of the average firm in each decile. If valuing a firm whose size is not near the top of its decile, the valuation practitioner should consider using the size premiums from Alt 2. A shorter estimation window (Alt 3) emphasizes more recent market return behavior when estimating size premiums. Events in the distant past may convey more relevant information for highly mature industries whose operating environment has been unaffected by changing economic regimes. For budding or relatively new industries, a shorter estimation window may be appropriate. Using a shorter estimation window, size premiums are significantly smaller, especially for deciles representing smaller firms. Additionally, the size premiums are no longer remotely monotonic. This may call into question the existence of the size premium in more recent decades. The size premiums calculated using the geometric average (Alt 4) may be more appropriate for multi-period discounted 37 Specifically, the claim is that the decile 10 size premium is greater than the decile 1 size premium. This serves as the alternative hypothesis. Accordingly, the null hypothesis says the decile 10 size premium is less than or equal to the decile 1 size premium. The p-values represent the significance level with which we may reject the null hypothesis. Rejection of the null hypothesis supports the claim. It should be noted that the matched t-test with dependent samples assumes normality of the sample differences. cash flow models wherein the discount rate compounds from one period to the next, though it is also common to use the baseline results for this scenario. It is clear that using the geometric average instead of the arithmetic average has the effect of reducing size premiums for deciles representing smaller firms and eliminates the near- monotonicity of the premiums from the baseline results. This alternative specification may also call into question the existence of the size premium. The choice of risk-free rate should consider the holding period of the company being valued. As discussed in Section 3.1.7, PEGs tend to have much shorter holding periods than owner-operators. Using the five-year Treasury rate as the risk-free rate (Alt 5) implies a much shorter holding period than the 20-year Treasury rate. If the willing buyer pool consists primarily of financial buyers, rather than strategic buyers, a valuation practitioner should consider using the size premiums from Alt 5. The Blume-adjusted beta transforms the OLS beta to make it more forward-looking, assuming the OLS beta tends towards a value of one. If using the Blume-adjusted beta when building up the modified CAPM, valuation practitioners should consider using the size premiums from Alt 6. Since smaller deciles tend to have more cyclical OLS beta values that exceed one, the Blume adjustment serves to reduce the beta of smaller deciles and increase the associated size premium. As a final exercise, we tested the premiums for deciles 1 and 10 in Table 3 for significance using a matched t-test. Theory dictates that the size premium for decile 10 should be greater than the size premium for decile 1. Accordingly, we tested whether the difference between deciles 10 and 1 is greater than zero. This was done for the baseline methodology and all six alternative approaches. The p-values for this exercise are shown in Table 5. 37 Table 5: Significance Test Approachp-value Baseline0.078 Alt 10.108 Alt 20.003 Alt 30.879 Alt 40.458 Alt 50.092 Alt 60.044 29 May | June 2024 A Professional Development Journal for the Consulting DisciplinesNext >