Effect of Liquidity on Size Premium and its Implications for Financial Valuations

Effect of Liquidity on Size Premium and its Implications for Financial Valuations [** Working Title] Frank Torchio and Sunita Surana Preliminary Draft August 2013

I. Size Premiums and Fair Value Discounted Cash Flow ( DCF ) analysis is one of, if not, the key valuation method taught by academics and used by practitioners. A critical parameter of a DCF analysis is the weighted average cost of capital (used to discount expected cash flows) which comprises in part the cost of equity. The cost of equity is generally computed using a version of the capital asset pricing model ( CAPM ) 1, for which the equation is: Cost of Equity = Risk-free Rate + (Beta x Equity Risk Premium) (1) Many valuation practitioners generally consider it appropriate to include in the calculation of the cost of equity a premium based on the market capitalization of equity or size of the firm being valued. Empirical studies, most notably published in the Ibbotson SBBI Yearbooks ( Ibbotson SBBI ), have shown that the CAPM alone does not fully account for the higher historical returns earned by smaller companies. 2 These studies show that historical returns for small firms are systematically greater than the returns implied by their betas (betaadjusted returns) from the standard CAPM in (1). That is, the greater risk of smaller-sized firms is not fully accounted for in the standard beta calculations for these firms. To account for this size-related effect, one of the variations of the CAPM equation includes a size premium, defined as: Cost of Equity = Risk-free Rate + (Beta x Equity Risk Premium) + Size Premium (2) 1 The CAPM is the cornerstone of asset pricing theory and is widely used for the estimation of cost of capital. See, for example, Sharpe (1964) and Fama and French (2004). 2 See Ibbotson SBBI 2011Valuation Yearbook, pp. 87-90. Banz (1981) first presented evidence that smaller firms earned higher risk-adjusted returns. 2

The higher the size premium, the higher is the cost of equity, and consequently the lower is the DCF value, all else the same. Ibbotson SBBI has measured historic size premiums by constructing portfolios of traded stocks by size. The size premiums are computed as the average returns for each size portfolio less the average of the returns predicted by CAPM in (1) for the stocks in each portfolio. Ibbotson has constructed both size-quartile portfolios and size-decile portfolios. In 2001, Ibbotson refined its size analysis by dividing decile 10, the smallest stock decile, into 10a and 10b. 3 In 2010, Ibbotson further divided the 10th decile into four size categories: 10w, 10x, 10y, and 10z. 4 As can be seen in Table 1, the size premiums increase as the company size decreases. 3 Ibbotson SBBI 2001Valuation Yearbook, pp. 122-3. 4 Ibbotson SBBI 2010 Valuation Yearbook, p. 91. 3

Table 1 Size Premiums for Size-Quartile and Size-Decile Portfolios Quartile Groups Large Cap (1-2) Mid Cap (3-5) Low Cap (6-8) Micro Cap (9-10) Size Premium n/a 1.20% 1.98% 4.07% Decile Groups Market Capitalization of Largest Company in Decile ($ millions) Size Premium 1 314,623-0.38% 2 15,080 0.81% 3 6,794 1.01% 4 3,711 1.20% 5 2,509 1.81% 6 1,776 1.82% 7 1,212 1.88% 8 772 2.65% 9 478 2.94% 10w 236 3.99% 10x 179 4.96% 10y 143 9.15% 10z 86 12.06% Notes: Source of data is Ibbotson SBBI 2011 Valuation Yearbook. Since the data in our analysis covers the time period 1926-2010, for comparability purposes, throughout this paper we report the statistics from the 2011 Yearbook that uses data also from 1926-2010. Market capitalization in each decile is as of September 30, 2010. Figure 1 plots the size premiums for the ten deciles (diamonds shapes) and also shows the size premiums for categories 10w, 10x, 10y, and 10z (circle shapes) contained in the Ibbotson SBBI 2011 Yearbook. As can be seen in Figure 1, the increase in the size premium is approximately linear for decile 1 (-0.38%) through decile 9 (2.94%). But for the smallest size decile (decile 10), the premium increases substantially to 6.36%, which is above the linear trend line based on the size premiums for deciles 1 through 9. Within decile 10, the increase in size premiums is even more dramatic and ranges from 3.99% for size category 10w to 12.06% for the smallest size category, 10z. 4

Figure 1 Size Premiums for Size-Decile Portfolios and Sub-Groups of the Tenth Decile Source: Ibbotson SBBI 2011 Valuation Yearbook. The substantial size premiums measured for very small companies has been criticized and its appropriateness continues to be debated among practitioners and researchers. 5 One argument raised by critics concerns the effects from the lack of liquidity that disproportionately affects stocks of smaller sized companies. The argument is that the lack of liquidity for many small sized firms causes transactions costs of trading a share of stock to be greater, which in turn results in greater observed historic returns to properly compensate investors for holding these stocks relative to more liquid stocks. Ibbotson does not disagree that lower liquidity will contribute to the magnitude of the calculated size premiums. Ibbotson correctly responds that it is irrelevant whether or not the 5 For example, some argue that portfolios of smallest size companies contain a disproportionate number of financially distressed firms or contain the so called fallen angels. See Ibbotson SBBI 2011 Valuation Yearbook, pp. 94-102 for a comprehensive review of the criticisms and Ibbotson s responses. 5

computed size premium also reflects the lower liquidity in smaller-sized companies when computing a stock s fair market value; the return to equity used to compute fair market value should include the additional return required to compensate investors for holding less liquid stocks. 6 This is because these transactions costs are real, can be substantial, and will affect the prices paid for a share of stock. 7 For example, if an investor were contemplating a purchase of 10 shares of a privately held company, the investor would certainly take into account in the purchase price she pays the transaction costs she would have to incur in order to sell those shares at a later time. So, to the extent that the computed size premium includes the effects of less liquid stocks is largely irrelevant because it is generally appropriate that a fair market valuation reflect any illiquidity effect on the expected return. In certain circumstances, however, the purpose of valuation is not to assess the fair market value, but rather to analyze and compute fair value. For example, the appropriate measure of value in an appraisal for a merger is not fair market value but rather fair value. The key differences between fair market value and fair value are that fair value requires that there be: (1) no discount -- either direct or implied -- for the lack of liquidity; and (2) no discount for minority interest. 8 6 According to valuation literature, fair market value is generally defined as: the amount at which property would change hands between a willing seller and a willing buyer when neither is acting under compulsion and when both have reasonable knowledge of the relevant facts. See Pratt (2008), pp. 41-42. 7 Under the standard of fair market value, the focus must be on the specific property (ownership interest) being valued, basically as is, including control and marketability characteristics. Therefore, minority interests in closely held corporations or partnerships are valued to reflect lack of control and lack of marketability characteristics. See Pratt (2009), p. 10. 8 See Laro and Pratt (2011), pp. 12-13. 6

Therefore, there is general agreement that the fair value of even a completely illiquid, privately held stock in an appraisal should not reflect any discount to the valuation that would obtain for the same stock that traded in a completely liquid market. But, if a key valuation parameter -- the cost of equity -- includes a premium for low liquidity, then the valuation obtained will necessarily reflect a discount for illiquidity. Is the value obtained from such an analysis the fair value of the stock if that value reflects an implicit discount for illiquidity? Because fair value is a legal concept, the answer to this question is left to the courts and legal scholars. This paper, however, provides an economic context to assist in answering this question by quantifying the liquidity premium reflected in the size-decile premiums that are currently used by many practitioners. Because the Ibbotson SBBI Yearbooks are by far the most commonly cited and used source for size premiums, we use the methods suggested by Ibbotson for computing size premiums and for measuring liquidity. The rest of the paper is organized as follows. The next section discusses the relationship between liquidity and asset pricing. The third section describes the data. Liquidity premiums, without accounting for size, are calculated in the fourth section. The fifth section discusses the replication of size premiums contained the Ibbotson SBBI 2011 Yearbook. The sixth section shows the amount of liquidity premium subsumed in the calculation of size premiums for sizedecile portfolios and for the sub-groups of the tenth decile. Finally, the last section provides concluding remarks. II. Liquidity and Asset Pricing The marketability or liquidity of an asset refers to the degree to which it can be converted to cash quickly without incurring large transaction costs or price concessions. Financial theory reasons that liquidity affects asset prices because investors price securities according to their 7

returns net of trading costs, such as transactions costs and expected price concessions, and consequently investors require a greater return for higher expected costs of achieving liquidity, all else the same. Thus, given two assets with the same expected cash flows but with different liquidity, investors will pay less (demand a higher return) to hold the more illiquid asset. Over the last 15 years, liquidity has been the subject of considerable research in the financial literature. Many of these studies are discussed by Amihud, Mendelson, and Pedersen (2005) who review the literature concerning the effects of liquidity on asset prices. Thus, financial economists expect that asset and security prices will differ systematically depending on the marketability characteristics of the securities, all else equal. For example, restricted stock should be priced at a discount from the unrestricted stock s traded market price on a liquid exchange. Indeed, many empirical studies of restricted stock, of the relationship between stock returns and bid-ask spreads, of a company s block transactions, and of private sales of a company s stock prior to the company s initial public offering have confirmed that high transaction costs and other restrictions generally cause securities to be priced at significant discounts from the market prices of comparable (often otherwise identical) liquid securities. Table 2 summarizes the illiquidity discounts measured in several restricted stock studies. These studies report median illiquidity discounts for restricted stock of 9% to 45% and means of 13% to 42% based on hundreds of transactions. Because the restricted stock studies provide a direct measure of the costs of illiquidity, many experts and courts have chosen to rely on these studies as an empirical guide in selecting illiquidity discounts to apply to the computed value from a DCF analysis to arrive at a fair market value measure. 8

Pre-1990 Studies Long-Horizon Studies 1990-1997 Studies 1997-2007 Studies (one year holding period) Study Table 2 Studies of Restricted Stock Marketability Discounts Time Period Notes: [1] Discounts Involved in Purchases of Common Stock (1966-1969), Institutional Investor Study Report of the Securities and Exchange Commission, H.R. Doc. No. 64, Part 5, 92nd Congress, 1st Session, 1971, pp. 2444-56, cited in Pratt, Reilly and Schweihs (2000), pp. 396-398, 404. [2] Cited in Pratt, Reilly and Schweihs (2000), pp. 398-399, 404. [3] Cited in Pratt, Reilly and Schweihs (2000), pp. 399, 404. [4] Cited in Pratt, Reilly and Schweihs (2000), pp. 400, 404. [5] Cited in Pratt, Reilly and Schweihs (2000), pp. 400, 404. [6] Unpublished study, cited in Pratt, Reilly and Schweihs (2000), pp. 400, 404. [7] Discount for private placement of unregistered shares. Wruck also reports an average premium of 4.1% and median discount of 1.8% for 36 private placements of registered shares. [8] Discount for private placement of restricted shares. Hertzel and Smith also report an average discount of 15.6% for 88 private placements of non-restricted shares. [9] Oliver, R. & Meyers, R. Discounts Seen in Private Placements of Restricted Stock: The Management Planning, Inc., Long-Term Study (1980-1996). Chapter 5 in Reilly and Schweihs (2000), cited in Pratt, Reilly and Schweihs (2000), pp. 401-404. Median is approximate. [10] The 1980-1997 time period is assumed to match the previous study with additional transactions updated. [11] Unregisterd Shares. Bajaj et al. also report discounts for 37 private placements of registered shares of 14% (average) and 10% (median). [12] Study obtained from Columbia Financial Advisors, Inc. [13] Unpublished study, cited in Laro and Pratt (2005), p.289. [14] Unpublished study, cited in Laro and Pratt (2011), p. 287. [15] Unpublished study, cited in Laro and Pratt (2011), p. 287. 9 Sample Size Price Discount Median Group Average Price Discount Group Average SEC overall average (1971) 1 Jan 1966 - June 1969 398 n/a 25.8% Gelman (1972) 1968-1970 89 33.0% 33.0% Trout (1977) 2 1968-1972 60 n/a 33.5% Moroney (1973) 3 na 146 33.0% 35.6% Maher (1976) 4 1969-1973 n/a n/a 30.9% 35.4% 31.6% Pittock and Stryker (1983) 5 Oct 1978 - June 1982 28 45.0% n/a Willamette Management Associates 6 Jan 1981 - May 1984 33 31.2% n/a Wruck (1989) 7 July 1979 - Dec 1985 37 12.2% 13.5% Hertzel and Smith (1983) 8 Jan 1980 - May 1987 18 n/a 42.0% Silber (1991) 1981-1988 69 n/a 33.8% Hall and Polacek (1994) 1979 - Apr 1992 100+ n/a 23.0% Oliver and Meyers (2000) 9 Jan 1980 - Dec 1996 53 25.0% 27.1% 22.1% Robak and Hall (2001) 1980 - Apr 1997 230 20.1% 22.3% 24.1% Hall (2003) 10 1980-1997 238 21.3% n/a Hall and Polacek (1994) May 1991 - Apr 1992 17 n/a 21.0% Bajaj, Denis, Ferris and Sarin (2001) 11 Jan 1990 - Dec 1995 51 26.5% 28.1% 18.7% 22.1% Johnson (1999) 1991-1995 72 n/a 20.2% Finnerty (2003) Jan 1991 - Feb 1997 101 15.5% 20.1% Aschwald (2000) Jan 1996 - Apr 1997 23 14.0% 21.0% Aschwald (2000) Jan 1997 - Dec 1998 15 9.0% 13.0% Columbia Financial Advisors, Inc. 12 June 1997 - June 2000 32 11.7% 13.5% Average Columbia Financial Advisors, Inc. 12 Jan 1999 - June 2000 24 11.7% 13.7% 14.6% Hall (2003) 1997-2000 182 25.9% n/a 16.7% FMV Opinions 13 1997-2003 187 n/a 22.5% FMV Opinions 14 1997-2007 311 n/a 20.6% Post 2007 Studies FMV Opinions 15 (six month holding 2008 43 n/a n/a 12.6% 12.6% period)

The consensus in the liquidity literature is that theory and empirical evidence strongly support three findings. First, investors require returns that compensate for the level of illiquidity of an investment. 9 Second, stocks in publicly traded equity markets can have substantially different degrees of liquidity; liquidity is not a binary variable. That is, one cannot simply divide stocks into two categories of liquidity based solely on whether or not the stock is publicly traded. While stocks that are not publicly traded are generally characterized as illiquid, even among publicly traded stocks, there can be important and substantial differences in the degree of liquidity. 10 Third, illiquidity is correlated with size. That is, across all publicly traded stocks, more small stocks tend to have lower liquidity than do large stocks. 11 Practitioners and academics are in general agreement that the negative relationship between returns and liquidity is stronger for smaller stocks. In a recent study, Ibbotson et al. (2013) empirically studied the effect on returns from differing levels of liquidity across all size quartile portfolios of publicly traded stocks between 1972 and 2011. Their findings are presented in Table 3. Ibbotson et al. found that within each size quartile portfolio, low liquidity portfolios generally earned higher returns than the high liquidity portfolios. The authors, however, find that the size impact is quite inconsistent across various levels of liquidity. Specifically, while among low liquidity stocks, small-sized stock portfolios earned higher returns than the large stock portfolios, the opposite is true for high 9 The effect of liquidity on asset prices was first examined by Amihud and Mendelson (1986). See also Datar, Naik, and Radcliffe (1998) and Amihud, Mendelson, and Pedersen (2005). 10 For example, Amihud and Mendelson (1986) divide publicly traded stocks into seven liquidity groups that show significant variability. 11 See, for example, Amihud (2002) and Ibbotson et al. (2013). 10

liquidity stocks. Thus, based on Ibbotson et al. (2013), at high levels of liquidity, the effect of small size does not result in higher returns compared to the larger firms. Table 3 Annualized Arithmetic Return for Size and Liquidity Quartile Portfolios from Ibbotson et al. (2013) Quartile Low Liquidity Mid-Low Liquidity Mid-High Liquidity High Liquidity Micro Cap 17.92% 20.00% 15.40% 6.78% Small Cap 17.07% 16.82% 15.38% 9.89% Mid Cap 15.01% 15.34% 14.51% 11.66% Large Cap 12.83% 12.86% 12.81% 11.58% Source: Ibbotson, R.G., Chen, Z., Kim, D. Y.-J., and Hu, W.Y., 2013. Liquidity as an Investment Style. Financial Analysts Journal, 69(3), Table 2 (partial). Ibbotson et al. (2013) conclude: Therefore, size does not capture liquidity (i.e., the liquidity premium holds regardless of size group). Conversely, the size effect does not hold across all liquidity quartiles, especially in the highest-turnover quartile. The liquidity effect, however, is strongest among microcap stocks and declines from micro- to small- to mid- to large-cap stocks. The key finding implied by Ibbotson et al. (2013) is that for smaller-sized companies the historic returns are substantially different between low liquidity and high liquidity stocks (first row of Table 3). If the CAPM returns are not similarly different across the various liquidity groups, then it implies that a substantial portion of the measure of what is generally referred to as size premium subsumes the premium that compensates investors for holding low liquidity stocks. II.a Liquidity Premium Creates an Illiquidity Discount to the Stock s Valuation If the size premium subsumes a substantial component that compensates investors for holding less liquid stocks, then inclusion of such a size premium in the cost of equity will necessarily result in an illiquidity discount to the stock s fair value. There is no economic 11

distinction between a DCF in which the cost of capital is increased by a liquidity premium versus a DCF that uses a cost of capital with no liquidity premium, but to which an illiquidity discount is then applied. As discussed in the recent Ibbotson SBBI Yearbooks, the economic equivalence can be simply shown as follows. 12 First, the valuation from a simple DCF model can be approximated by the perpetuity model: V L = C / R where, V L = value of a liquid security; C = annual cash flow; and R = discount rate of the liquid security. (3) Assuming, C = $10 and R = 10%, gives V L = $10/10% or $100. Because less liquid stocks are expected to earn higher returns to compensate investors for the lack of liquidity (all else equal), the value of a less liquid stock can be expressed as: V I = C / (R + P) (4) where, V I = value of a less liquid security; and P = liquidity premium. Alternatively, the value of the less liquid stock can be computed by applying an illiquidity discount, D, to the value of the liquid stock, yielding the following equation: V L (1 D) = V I (5) Solving for D gives: D = P / (R + P). Assuming a 5% liquidity premium yields a 33.3% discount to the value of the liquid security. In other words, a liquidity premium added to the cost of equity can always be mathematically translated to a discount for the lack of liquidity. 12 See, for example, Ibbotson SBBI 2011 Valuation Yearbook, p. 101. 12

There are two inferences from this simple economic equivalence. First, if one is computing the fair value of a privately held stock, it is duplicative to apply a full illiquidity discount factor to a DCF valuation when the DCF uses a cost of equity that includes a liquidity premium. Second, if one is computing the fair value of a stock, the fair value computation will reflect an implicit discount for lack of liquidity if the cost of equity includes a liquidity premium. In the following sections we empirically investigate the magnitude of the liquidity premium. III. Data Description We use monthly common stock data from 1926 through 2010 compiled by the Center for Research in Security Prices ( CRSP ) at the University of Chicago Booth School of Business. All common stock traded on the New York Stock Exchange ( NYSE ), American Stock Exchange ( AMEX ) 13, and NASDAQ stock markets are used. From 1926 through 2010, over 3 million monthly-level observations are available. Monthly returns on the Standard & Poor s ( S&P ) 500 index and 30-day U.S. Treasury bill total return from 1926 through 2010 are also obtained from CRSP. Finally, long-term mean income return component of 20-year government bonds and long-term equity risk premia are obtained from Ibbotson SBBI 2011 Valuation Yearbook. For the analysis of adjustment to NASDAQ volume to account for the potential overcounting of traded volume, we use daily common stock data from 1990 through 2012 from CRSP. For this time period, over 40 million daily-level observations are available. 13 In October 2008, AMEX was acquired by NYSE Euronext. 13

III.a Adjustment to NASDAQ Volume Volumes in quote-driven dealer markets like NASDAQ were historically higher than order-driven markets like NYSE because public buyers and sellers traded though the intermediation of dealers on NASDAQ, leading to an over count of trades among public traders. On NYSE, public buyers and sellers mostly traded among themselves. The differing market structures caused higher volumes for NASDAQ stocks, all else equal. However, regulatory interventions by the SEC (e.g., the 1997 SEC mandated order handling rules at NASDAQ 14 ) as well as the rapid growth of electronic trading mechanisms have caused the trading patterns on different platforms to converge. Ibbotson et al. (2013) divide volumes of NASDAQ stocks based on an adjustment factor suggested by Anderson and Dyl (2005). Based on stocks switching from NASDAQ to NYSE from 1997 through 2002, Anderson and Dyl find that the mean daily volume declined an average of 24.7% and the median decrease was 37.9%. Further, based on the finding in Anderson and Dyl (2007) that the relative over-reporting of NASDAQ stocks has not lessened during 2003-2005 relative to the 1990-1996 time period, Ibbotson et al. (2013) apply the adjustment factor throughout their study period of 1972-2011. While Anderson and Dyl (2007) found no evidence that the over-reporting has lessened for NASDAQ stocks, using data from 1993-2010, Harris (2011) reports that over time volumes between NYSE and NASDAQ stocks have become more and more similar leading to the homogenization of US equity markets. In light of the differences in the findings in Harris (2011) and Anderson and Dyl (2007) and the vast changes in the stock trading landscape, we analyzed the volume of common stocks 14 See, for example, McInish, Van Ness, and Van Ness (1998). 14

switching from NASDAQ to NYSE starting in 1990 (since the regulatory changes likely to impact the over counting of traded volume and the rapid growth of electronic trading mechanisms occurred after 1990). We studied volume on sixty trading days before and after the switch (with day one being the day of the switch). For each company switching from NASDAQ to NYSE, average volume during 60 trading days before the switch is divided by average volume during 60 trading days after the switch. The ratio of the two volumes is shown in figure 2. The horizontal lines represent the median ratios for consecutive five year periods beginning in 1990. As is evident from the figure and consistent with Harris (2011), the ratio has been decreasing over time. 15

4 Figure 2 Average Volume on NASDAQ Before the Switch Divided by Average Volume on NYSE After the Switch 3 Median 2 2.07 1.75 1.38 1.32 1.21 1 0 1985 1990 1995 2000 2005 2010 2015 Note: In order to focus on presenting the medians, the figure shows ratios between 0 and 4. A few outliers (greater than 4) are not shown in the figure but are used in the computation of the median ratios. We use the median ratios over five year intervals to adjust NASDAQ volume. Specifically, the ratio of 2.07 is used to divide NASDAQ volume prior to 1994; 1.75 is used to divide NASDAQ volume between 1995 and 1999; 1.38 is used to divide NASDAQ volume between 2000 and 2004; 1.32 is used to divide NASDAQ volume between 2005 and 2009; and 1.21 is used to divide NASDAQ volume after 2009. 16

IV. Liquidity Premium We measure liquidity using average monthly share turnover in each quarter. 15 Share turnover is calculated as the volume of shares traded each month divided by the number of shares outstanding at each month-end. This measure of liquidity is similar to the one used in Ibbotson et al. (2013), except that whereas Ibbotson et al. (2013) create annual portfolios of stocks and hence measure liquidity annually, we create portfolios of stocks that are rebalanced quarterly in keeping with the size premium methodology in the Ibbotson SBBI Yearbooks, and hence, we update the liquidity measure each quarter. The steps we take to create the liquidity-based portfolios of stocks are described below. We first rank companies with primary listings on the NYSE based on their liquidity at the end of each quarter. As mentioned above, liquidity at the end of each quarter is measured as average monthly share turnover in that quarter. Second, based on these rankings, the companies are divided into two equally populated groups based on the median liquidity at the end of each quarter. Group H contains companies with liquidity greater than or equal to the median liquidity, or the high liquidity companies. Group L contains companies with liquidity lower than the median liquidity, or the low liquidity companies. The ranking of the stocks thus yields a liquidity cutoff demarcating the H and L groups at the end of each quarter. These liquidity cutoffs obtained from NYSE stocks are then used to assign common stocks listed on AMEX and Nasdaq to one of the two liquidity groups based on the end of quarter liquidity measure for the AMEX and Nasdaq stocks. 15 Turnover rates and bid-ask spreads are typically used as measures of liquidity. See Amihud et al. (2005). Another measure of liquidity is the price impact such as the ratio of absolute stock return to its dollar volume, averaged over some period (Amihud, 2002). 17

Third, the liquidity groupings are rebalanced quarterly. Each month, every company is assigned a liquidity group based on its liquidity categorization in the previous quarter. For example, the liquidity category that a company falls under as of the quarter ending in March is used to assign its liquidity group for April, May, and June of that year. Thus, each stock remains in the same liquidity group for each of the three months that follow its assignment to a liquidity group using the liquidity measure from the end of the previous quarter. This methodology is similar to the methodology used by CRSP and the Ibbotson SBBI Yearbooks for the creation of the size-decile portfolios (discussed further below). Fourth, we compute monthly portfolio returns as the average returns of the stocks in each liquidity group from 1926 to 2010. Annual portfolio returns are computed by compounding the monthly returns. 16 Fifth, to compute the risk-adjusted portfolio returns, we compute the beta for each liquidity portfolio using the single-factor CAPM model. Following the SBBI Yearbooks, we use the following single-factor regression equation to estimate each portfolio s systematic risk (commonly referred to in the literature as the portfolio s beta). where, (r l r f ) = α l + β l (r m r f ) + ε l (5) r l represents monthly return on portfolio l; r f represents 30-day U.S. treasury bill total return; and r m represents monthly return on the market, measured by the S&P 500 index. The slope of the regression, β l, in equation (5) measures the portfolio s sensitivity to variations in the market return, or its exposure to systematic risk. months. 16 Annual return is computed when monthly return data is available for each of the twelve 18

Using the beta for each liquidity portfolio, β l, a CAPM portfolio return (in excess of the riskless rate) is computed as the product of the estimated beta for that portfolio and the equity risk premium (the difference between the mean total return of the S&P 500 index and the mean income return component of 20-year government bonds for the time period 1926-2010). Finally, for each liquidity portfolio of stocks, we calculate the difference between the portfolio s actual average annual return in excess of the risk-free rate (measured by the mean income return component of 20-year government bonds for the time period 1926-2010) and the CAPM return also in excess of the risk-free rate. We refer to this difference as the liquidity premium. Table 4 presents the liquidity premiums for the H and L liquidity groups. We find that while the high liquidity stocks have a liquidity premium of less than 1%, the liquidity premium for the low liquidity stocks is over 7%. Table 4 Long-Term Returns in Excess of Estimated CAPM Returns for High and Low Liquidity Categories Liquidity Group Beta Actual Arithmetic Mean Return Actual Return in Excess of [1] [2] [3] [4] = [3] - 5.17% CAPM Return in Excess of [5] = β * (11.88% - 5.17%) Liquidity Premium (Return in Excess of CAPM Return) [6] = [4] - [5] H 1.36 14.95% 9.78% 9.09% 0.69% L 1.05 19.50% 14.33% 7.02% 7.31% Notes: Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). Using the same methodology described above, we also categorize the data into liquidity quartiles. The results are presented in Table 5. Again, we find that liquidity premium increases 19

as the stocks get less liquid. These findings suggest that the CAPM underestimation of returns is a function of stock liquidity. Table 5 Long-Term Returns in Excess of Estimated CAPM Returns for High, Mid-High, Mid-Low, and Low Liquidity Categories Liquidity Group Beta Actual Arithmetic Mean Return Actual Return in Excess of [1] [2] [3] [4] = [3] - 5.17% CAPM Return in Excess of [5] = β * (11.88% - 5.17%) Liquidity Premium (Return in Excess of CAPM Return) [6] = [4] - [5] H 1.40 13.03% 7.86% 9.42% -1.56% MH 1.30 17.20% 12.03% 8.73% 3.31% ML 1.17 18.90% 13.73% 7.87% 5.86% L 0.95 19.61% 14.44% 6.35% 8.10% Notes: Historical riskless rate is measured by the arithmetic mean income return component of the 20- year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). MH denotes mid-high and ML denotes mid-low. The analysis so far does not distinguish liquidity premiums for stocks stratified by their size. We next study how liquidity impacts the premiums for each of the size-decile portfolios and for the sub-groups of the tenth size-decile. But before we assess the impact of liquidity for these stock portfolios, we create size-decile portfolios in a manner similar to CRSP. V. Replication of Ibbotson Size Premiums Using the monthly stock data, we next create size-based portfolios of stocks in a manner similar to CRSP and used in the Ibbotson SBBI Yearbooks. This involves the following steps (that are similar to the steps discussed above to create the liquidity-based portfolios). 20

Using monthly data from 1926-2010, we first rank the companies with primary listings on the NYSE based on their market capitalizations at the end of each quarter. Market capitalization is calculated as the product of the closing price on the last trading date of the quarter and the shares outstanding. 17 Second, based on these rankings, the companies are divided into equally populated deciles (decile 1 contains the largest companies, and decile 10 the smallest). Thus, the ranking of the NYSE stocks yield size cutoffs for each decile, where the cutoffs are the highest and lowest market capitalizations within each size-decile. These decile cutoffs obtained from NYSE stocks are then used to assign common stocks listed on AMEX and Nasdaq to one of the size deciles based on the end of fiscal quarter market capitalization for the AMEX and Nasdaq stocks. Third, size-decile portfolios are constructed for the stocks based on the size-decile rankings. Each month, every company is assigned a portfolio based on its decile ranking in the previous quarter. For example, the decile ranking of a stock based on its market capitalization as of the quarter ending in March is used to determine the stock s size-decile for the months of April, May, and June. Thus, each stock remains in the same size-decile for each of three months that follow its assignment to a size-decile using the market capitalization from the end of the previous quarter. Fourth, monthly portfolio returns are computed as the weighted average returns of the stocks in each size-decile, using market capitalizations based on the shares outstanding and 17 In the calculation of the liquidity portfolios discussed above we calculated the average monthly share turnover over a quarter instead of just using the last month in the quarter in order to preserve observations in the analysis that otherwise would be lost due to missing volume data in the last months of the quarters. 21

closing price for the last trading day of the previous month as weights. Annual portfolio returns are computed by compounding the monthly returns. Fifth, to compute the risk-adjusted portfolio returns, we compute the beta for each sizedecile portfolio using the single-factor CAPM model. Again, following the SBBI Yearbooks, we use the following single-factor regression equation to estimate each portfolio s systematic risk. where, (r s r f ) = α s + β s (r m r f ) + ε s (6) r s represents monthly return on portfolio s; r f represents 30-day U.S. treasury bill total return; and r m represents monthly return on the market, measured by the S&P 500 index. The slope of the regression, β s, in equation (6) measures the portfolio s sensitivity to variations in the market return, or its exposure to systematic risk. Using the beta for each size-decile portfolio, β s, a CAPM portfolio return (in excess of the riskless rate) is computed as the product of the estimated beta for that portfolio and the equity risk premium. Finally, for each size-decile portfolio of stocks, we calculate the difference between the portfolio s actual average annual return in excess of the risk-free rate (measured by the mean income return component of 20-year government bonds for the time period 1926-2010) and the CAPM return also in excess of the risk-free rate. This difference is what is referred to as the size premium in the Ibbotson SBBI publications. Table 6 presents the size premiums by size-decile. As the table shows, our computed size premiums are very similar to the size premiums reported in the Ibbotson SBBI 2011 Yearbook. 22

Table 6 Comparison of Ibbotson SBBI and Torchio-Surana Long-Term Returns in Excess of Estimated CAPM Returns for Size-Decile Portfolios Actual Return in CAPM Return in Decile Beta Actual Arithmetic Mean Return Excess of Excess of Size Premium (Return in Excess of CAPM Return) [1] [2] [3] [4] = [2] - [3] [5] [6] [7] = [5] - [6] [8] = [5] - 5.17% [9] = [2] * (11.88%-5.17%) [10] = [8] - [9] [11] [12] = [10] - [11] TS SBBI TS - SBBI TS SBBI TS - SBBI TS TS TS SBBI TS - SBBI 1 0.92 0.91 0.01 10.87% 10.92% -0.05% 5.70% 6.15% -0.44% -0.38% -0.06% 2 1.03 1.03 0.00 13.02% 12.92% 0.10% 7.85% 6.89% 0.96% 0.81% 0.15% 3 1.10 1.10 0.00 13.44% 13.56% -0.12% 8.27% 7.40% 0.88% 1.01% -0.13% 4 1.13 1.12 0.01 14.04% 13.91% 0.13% 8.87% 7.57% 1.30% 1.20% 0.10% 5 1.17 1.16 0.01 14.66% 14.75% -0.09% 9.49% 7.82% 1.67% 1.81% -0.14% 6 1.19 1.19 0.00 14.93% 14.95% -0.02% 9.76% 7.97% 1.79% 1.82% -0.03% 7 1.23 1.24-0.01 15.24% 15.38% -0.14% 10.07% 8.25% 1.82% 1.88% -0.06% 8 1.30 1.30 0.00 16.20% 16.54% -0.34% 11.03% 8.70% 2.33% 2.65% -0.32% 9 1.34 1.35-0.01 17.01% 17.16% -0.15% 11.84% 8.99% 2.85% 2.94% -0.09% 10 1.40 1.41-0.01 21.49% 20.97% 0.52% 16.32% 9.40% 6.93% 6.36% 0.57% Notes: Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). As discussed above, Ibbotson divides the smallest size-decile, decile 10, into 4 size subgroups (10w, 10x, 10y, and 10z) using the same methodology that is used to construct the 10 size-decile portfolios. In Table 7, we replicate that analysis and show the computed size premiums for sub-size groups w, x, y, and z for decile 10 are quite close to the size premiums reported in the Ibbotson SBBI 2011 Yearbook. One reason for the differences between our estimates and those reported in the Ibbotson SBBI Yearbook is that Ibbotson SBBI uses its internal database of companies for the sub-groups of the tenth decile (based on email communication with Morningstar). Notwithstanding the differences, the results are wholly consistent with that from Ibbotson. 23

Table 7 Comparison of Ibbotson SBBI and Torchio-Surana Long-Term Returns in Excess of Estimated CAPM Returns for Sub-Groups of the Tenth Size Decile Size Group Beta Actual Arithmetic Mean Return Actual Return in Excess of CAPM Return in Excess of Size Premium (Return in Excess of CAPM Return) [1] [2] [3] [4] = [2] - [3] [5] [6] [7] = [5] - [6] [8] = [5] - 5.17% [9] = [2] * (11.88%-5.17%) [10] = [8] - [9] [11] [12] = [10] - [11] TS SBBI TS - SBBI TS SBBI TS - SBBI TS TS TS SBBI TS - SBBI 10 w 1.40 1.39 0.01 18.67% 18.52% 0.15% 13.50% 9.36% 4.14% 3.99% 0.15% 10 x 1.44 1.45-0.01 21.42% 19.88% 1.54% 16.25% 9.69% 6.55% 4.96% 1.59% 10 y 1.41 1.40 0.01 23.61% 23.72% -0.11% 18.44% 9.43% 9.01% 9.15% -0.14% 10 z 1.34 1.34 0.00 28.37% 26.25% 2.12% 23.20% 9.00% 14.20% 12.06% 2.14% Notes: Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). V.a Missing Volume Data Because the monthly stock data contains missing volume data for some months, the number of observations used to compute the liquidity premiums is less than the number of observations used to compute the size premiums. Table 8 compares the number of observations used to compute the liquidity premiums and the number of observations used to compute the size premiums. The table also compares the mean annual returns and the standard deviation of annual returns for the two data samples. 24

Table 8 Number of Observations, Mean Annual Returns, and Standard Deviation of Annual Returns Decile Total Number of Observations Arithmetic Mean of Annual Returns Standard Deviation of Annual Returns Size-based Partial Size-based Partial Size-based Partial 1 125,566 125,027 10.87% 10.88% 19.47% 19.46% 2 129,233 128,360 13.02% 13.06% 22.31% 22.33% 3 137,118 134,914 13.44% 13.59% 23.72% 23.80% 4 144,318 140,964 14.04% 14.14% 26.13% 26.17% 5 156,495 150,852 14.66% 14.97% 26.83% 26.98% 6 180,032 170,476 14.93% 15.20% 27.45% 27.64% 7 208,066 194,967 15.24% 15.48% 29.84% 30.13% 8 253,749 236,337 16.20% 16.45% 34.14% 34.37% 9 372,783 344,372 17.01% 17.31% 36.48% 36.60% 10 1,360,963 1,168,794 21.49% 21.67% 46.18% 46.27% 10 w 131,871 121,312 18.67% 18.88% 42.67% 42.84% 10 x 171,610 157,854 21.42% 21.71% 46.36% 46.48% 10 y 263,696 240,817 23.61% 23.83% 54.63% 54.71% 10 z 793,786 648,811 28.37% 28.54% 53.63% 53.72% 10w H 43,843 15.41% 46.65% 10w L 77,469 22.05% 50.05% 10x H 50,427 20.57% 54.13% 10x L 107,427 24.63% 46.46% 10y H 65,780 20.34% 55.68% 10y L 175,037 26.79% 59.96% 10z H 131,697 21.07% 53.45% 10z L 517,114 30.96% 59.49% Notes: Liquidity categories are defined independently of the size groups. Since several observations have stock price return data but do not have volume data, the liquidity analysis uses fewer observations than the purely size-based analysis. Partial denotes that some observations are lost during the liquidity categorization. Using the sub-set of monthly return data used to compute liquidity premiums, we recompute size premiums for each size-decile and for the sub-groups of the tenth decile and 25

compare the resulting premiums to the premiums previously computed. As can be seen in Tables 9 and 10, the loss of data has de minimis effects on the results of the computed size premiums. Table 9 Impact of Loss of Data on Size-Decile Portfolios Actual Return CAPM Return Decile Beta Actual Arithmetic Mean Return in Excess of in Excess of Size Premium (Return in Excess of CAPM Return) [1] [2] [3] [4] = [2] - [3] [5] [6] [7] = [5] - [6] [8] = [5] - 5.17% [9] = [2] * (11.88%- [10] = [8] - [9] [11] [12] = [10] - [11] Partial Size-based Partial - Partial - Partial - Partial Size-based Partial Partial Partial Size-based Size-based Size-based Size-based 1 0.92 0.92 0.00 10.88% 10.87% 0.00% 5.71% 6.14% -0.44% -0.44% 0.00% 2 1.03 1.03 0.00 13.06% 13.02% 0.04% 7.89% 6.89% 1.00% 0.96% 0.04% 3 1.10 1.10 0.00 13.59% 13.44% 0.15% 8.42% 7.39% 1.03% 0.88% 0.15% 4 1.13 1.13 0.00 14.14% 14.04% 0.10% 8.97% 7.58% 1.39% 1.30% 0.09% 5 1.17 1.17 0.00 14.97% 14.66% 0.31% 9.80% 7.83% 1.97% 1.67% 0.30% 6 1.19 1.19 0.00 15.20% 14.93% 0.27% 10.03% 7.99% 2.04% 1.79% 0.25% 7 1.24 1.23 0.01 15.48% 15.24% 0.24% 10.31% 8.30% 2.02% 1.82% 0.19% 8 1.30 1.30 0.01 16.45% 16.20% 0.25% 11.28% 8.74% 2.55% 2.33% 0.22% 9 1.35 1.34 0.01 17.31% 17.01% 0.30% 12.14% 9.04% 3.10% 2.85% 0.25% 10 1.41 1.40 0.01 21.67% 21.49% 0.18% 16.50% 9.45% 7.05% 6.93% 0.12% Notes: See the summary statistics for the data used in the liquidity analysis and that used in the size-based analysis. Partial denotes that some observations are lost during the liquidity categorization. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). 26

Size Group Beta [1] [2] [3] [4] = [2] - [3] Partial Size-based Table 10 Impact of Loss of Data on Sub-Groups of the Tenth Decile Partial - Size-based Actual Arithmetic Mean Return [5] [6] [7] = [5] - [6] Partial Size-based Partial - Size-based Actual Return in Excess of [8] = [5] - 5.17% CAPM Return in Excess of [9] = [2] * (11.88%-5.17%) Size Premium (Return in Excess of CAPM) [10] = [8] - [9] [11] [12] = [10] - [11] Partial Partial Partial Size-based Partial - Size-based 10 w 1.40 1.40 0.01 18.88% 18.67% 0.21% 13.71% 9.41% 4.29% 4.14% 0.16% 10 x 1.45 1.44 0.01 21.71% 21.42% 0.30% 16.54% 9.74% 6.81% 6.55% 0.25% 10 y 1.41 1.41 0.01 23.83% 23.61% 0.21% 18.66% 9.47% 9.18% 9.01% 0.17% 10 z 1.35 1.34 0.01 28.54% 28.37% 0.16% 23.37% 9.07% 14.30% 14.20% 0.10% Notes: See the summary statistics for the data used in the liquidity analysis and that used in the size-based analysis. Partial denotes that some observations are lost during the liquidity categorization. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). VI. Liquidity versus Size Effect As mentioned, several studies have shown that less liquid stocks earn higher returns than more liquid stocks. Since liquidity risk is priced in stock returns, we examine whether and to what extent stock liquidity accounts for the commonly used size premiums. We use the methodology described above to separately estimate premiums (the difference between the actual returns in excess of the riskless rate and the estimated returns based on CAPM also in excess of the riskless rate) for low and high liquidity portfolios for each of the ten size deciles and also for the sub-groups of the tenth size decile. Results for the size decile portfolios are presented in Table 11. 27

Table 11 Long-Term Returns in Excess of Estimated CAPM Returns for Size Decile Portfolios Split into High and Low Liquidity Categories Group [1] Beta Actual Arithmetic Mean Return Actual Return in Excess of [2] [3] [4] = [3] - 5.17% CAPM Return in Excess of [5] = β * (11.88% - 5.17%) Premium (Return in Excess of CAPM Return) [6] = [4] - [5] 1 H 1.12 11.32% 6.15% 7.50% -1.35% L 0.81 10.71% 5.54% 5.41% 0.13% 2 H 1.18 12.92% 7.75% 7.90% -0.16% L 0.87 13.28% 8.11% 5.87% 2.25% 3 H 1.25 13.49% 8.32% 8.37% -0.05% L 0.93 14.28% 9.11% 6.22% 2.88% 4 H 1.31 14.04% 8.87% 8.80% 0.07% L 0.90 14.49% 9.32% 6.07% 3.25% 5 H 1.33 14.66% 9.49% 8.92% 0.57% L 0.97 15.68% 10.51% 6.50% 4.01% 6 H 1.37 14.04% 8.87% 9.21% -0.33% L 0.98 16.62% 11.45% 6.55% 4.90% 7 H 1.41 14.68% 9.51% 9.45% 0.06% L 1.06 16.61% 11.44% 7.10% 4.34% 8 H 1.46 15.19% 10.02% 9.83% 0.19% L 1.13 18.18% 13.01% 7.61% 5.40% 9 H 1.51 17.27% 12.10% 10.10% 1.99% L 1.20 18.49% 13.32% 8.08% 5.25% 10 H 1.62 18.52% 13.35% 10.90% 2.46% L 1.31 25.11% 19.94% 8.76% 11.18% Notes: Liquidity categories are defined independently of the size groups. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). Clearly, higher liquidity stocks have significantly smaller size premiums than their less liquid counterparts within each size decile. In fact, there is virtually no size premium for the higher liquidity groups of the first eight size deciles. For the two smallest size deciles, the ninth 28

and the tenth deciles, the premiums for the high liquidity group are only 2% and 2.5%, respectively. Results for each of the sub-groups of the tenth decile (groups 10w, 10x, 10y, and 10z) are shown in Table 12. As before, for each of the sub-groups of the tenth decile, premiums for the portfolios of high liquidity stocks are considerably smaller than the premiums for their lower liquidity counterparts. Indeed, the premium is not greater than 5% in any of the sub-groups high liquidity portfolios of stocks. Table 12 Long-Term Returns in Excess of Estimated CAPM Returns for Sub-Groups of the Tenth Size Decile Split into High and Low Liquidity Categories Group [1] Beta Actual Arithmetic Mean Return Actual Return in Excess of [2] [3] [4] = [3] - 5.17% CAPM Return in Excess of [5] = β * (11.88% - 5.17%) Premium (Return in Excess of CAPM Return) [6] = [4] - [5] 10 w H 1.58 15.41% 10.24% 10.61% -0.37% L 1.31 22.05% 16.88% 8.80% 8.08% 10 x H 1.61 20.57% 15.40% 10.84% 4.57% L 1.35 24.63% 19.46% 9.06% 10.40% 10 y H 1.76 20.34% 15.17% 11.83% 3.34% L 1.31 26.79% 21.62% 8.77% 12.85% 10 z H 1.84 21.07% 15.90% 12.33% 3.57% L 1.23 30.96% 25.79% 8.24% 17.55% Notes: Liquidity categories are defined independently of the size groups. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). One observation from Tables 11 and 12 is that within each size group the estimated beta is greater for the high liquidity stocks than for the low liquidity stocks. Since higher liquidity 29

stocks by definition trade more frequently than less liquid stocks, their prices more quickly reflect the movements of the broader market. Thus, the high liquidity stocks are less prone to the under-estimation of beta that low liquidity stocks suffer because of a lagged response to the market movements. Hence, the betas and therefore the estimated CAPM returns are higher for more liquid stocks than less liquid stocks within the same size portfolio of stocks. Beta estimates for low liquidity stocks can be improved by accounting for their lagged response to market movements. We discuss one such methodology in the Appendix. The second observation from Tables 11 and 12 is that within each size group, the average actual returns for high liquidity stocks are greater than that for low liquidity stocks. This is consistent with the findings in the general liquidity literature that investors require compensation for holding low liquidity stocks, even within the same size group. The higher actual returns combined with the lower CAPM returns explain the significantly higher premiums for the less liquid stock relative to their more liquid counterparts within the same size group. VI.a Comparison of Size Premiums of Higher Liquidity Stocks to Ibbotson SBBI Size Premiums We next compare the size premiums of higher liquidity stocks to the frequently used size premiums published in Ibbotson SBBI. Table 13 shows that for all size portfolios, the size premiums for higher liquidity stocks are substantially smaller than the size premiums published in Ibbotson SBBI. 30

Table 13 Comparison of Size Premiums of Higher Liquidity Stocks to Ibbotson SBBI Size Premiums Decile Size Premium Ibbotson SBBI High Liquidity 1-0.38% -1.35% 2 0.81% -0.16% 3 1.01% -0.05% 4 1.20% 0.07% 5 1.81% 0.57% 6 1.82% -0.33% 7 1.88% 0.06% 8 2.65% 0.19% 9 2.94% 1.99% 10 6.36% 2.46% 10 w 3.99% -0.37% 10 x 4.96% 4.57% 10 y 9.15% 3.34% 10 z 12.06% 3.57% Notes: [1] Ibbotson SBBI size premiums are from the 2011 Yearbook. [2] See Tables 11 and 12 for the computation of size premiums for the high liquidity stocks. In summary, our research shows that, within each size portfolio, higher liquidity stocks have substantially smaller size premiums than the size premiums computed for all (higher and lower liquidity) stocks. This finding holds across all size portfolios. The lowest size portfolios exhibit the largest difference in size premiums between higher liquidity stocks and all stocks. The main inference from our findings is that the commonly used size premiums from Ibbotson SBBI are overestimates of the true size premiums for higher liquidity stocks. This arguably has significant implications in valuations for which the purpose is to compute fair 31

value, which requires no reduction to value for lack of liquidity. The implicit illiquidity discount from using the commercial size premiums can be substantial. For example, for sub-group 10z the difference between Ibbotson SBBI size premium of 12.06% and the 3.57% size premium for higher liquidity stocks in sub-group 10z is 8.5%. This 8.5% difference results in an implicit illiquidity discount of one-third for the average stock in sub-decile 10z (assuming a discount rate of 17.5%). Hence, the effect is non-trivial. VII. Conclusion In many circumstances, it is necessary to compute a stock s fair value, which by definition eliminates any reduction to value because of a lack of marketability or liquidity. The method of computing fair value most frequently used by practitioners is the DCF analysis. A critical parameter of a DCF analysis is the computation of the cost of equity. Over the last decade, many practitioners have included in the computation of the cost of equity, a premium based on the finding that historic returns for firms with lower market capitalization are systematically greater than the returns implied by the standard CAPM. This difference between the observed returns for small-sized firms and the computed CAPM returns is called a size premium. Ibbotson SBBI is the most common source of size premiums used by practitioners. In theory, the size premium compensates investors for the systematic risk of holding small capitalization companies. Researchers have recognized that the measurement of size premiums can also include the effects from the lack of liquidity that disproportionately affects stocks of smaller-sized companies. The lack of liquidity for small-sized firms causes transactions costs of trading a share of stock to be greater, which in turn results in a premium to properly compensate investors for holding these stocks relative to more liquid stocks. 32

Our research builds on the general research on liquidity premiums. We stratify the stocks used by Ibbotson SBBI to compute size premiums by a measure of liquidity used by Ibbotson et al. (2013). For each size grouping used by Ibbotson SBBI, we divide the size group into high and low liquidity groups. Our findings show that the frequently used measure of size premiums includes a substantial fraction that is explained by illiquid trading. For example, the size premium in the smallest size grouping by Ibbotson SBBI (sub-group 10z) is 12.06%. When we stratify the stocks by liquidity, the size premium for the high liquidity stocks in sub-group 10z is reduced to 3.57%. Thus, the majority of the measure of commonly used size premiums is attributable to the lack of liquidity. This finding has implications for computing fair value which is meant to abstract from reductions due to illiquidity. Specifically, valuations of small capitalization stocks that reflect the Ibbotson SBBI size premium will cause the fair value to be reduced because of the effect of illiquidity. The smaller the size, the greater is the reduction due to illiquidity from using the Ibbotson SBBI size premiums. Is the value obtained from such a DCF analysis that uses the standard size premium really the fair value of the stock if that value reflects an implicit discount for illiquidity? Because fair value is a legal concept, the answer to this question is left to the courts and legal scholars. This paper, however, provides an economic context to assist in answering this question by quantifying the liquidity premium reflected in the size-decile premiums that are currently used by many practitioners. 33

APPENDIX Effect of the Sum Beta Methodology One method suggested to provide a better estimate of beta, one that reduces the underestimation problem in less liquid stocks, is by accounting for the lagged response of small-sized companies to market movements by including in the regression a lagged market return in addition to the current market return. We use the method suggested by Ibbotson, Kaplan, and Peterson (1997) and calculate a current and a lagged beta coefficient and then sum the two coefficients to arrive at the beta estimate (called the sum beta). Tables 14 and 15 present the premiums for the high and low liquidity groups for the ten size-decile portfolios and for the subgroups of the tenth decile, respectively, using the sum beta methodology. As expected, by better capturing the response to market movements, the sum beta methodology lowers the estimated premiums. 34

Table 14 Long-Term Returns in Excess of Estimated Returns using the Sum Beta Methodology for Size Decile Portfolios Split into High and Low Liquidity Categories Group [1] Sum Beta Actual Arithmetic Mean Return Actual Return in Excess of [2] [3] [4] = [3] - 5.17% Estimated Return in Excess of [5] = β * (11.88% - 5.17%) (Return in Excess of Estimated Return) [6] = [4] - [5] 1 H 1.12 11.32% 6.15% 7.54% -1.39% L 0.78 10.71% 5.54% 5.25% 0.29% 2 H 1.19 12.92% 7.75% 8.00% -0.25% L 0.91 13.28% 8.11% 6.08% 2.03% 3 H 1.26 13.49% 8.32% 8.44% -0.11% L 0.99 14.28% 9.11% 6.64% 2.46% 4 H 1.35 14.04% 8.87% 9.06% -0.20% L 1.01 14.49% 9.32% 6.75% 2.56% 5 H 1.37 14.66% 9.49% 9.17% 0.32% L 1.09 15.68% 10.51% 7.33% 3.18% 6 H 1.45 14.04% 8.87% 9.73% -0.85% L 1.13 16.62% 11.45% 7.60% 3.85% 7 H 1.48 14.68% 9.51% 9.96% -0.45% L 1.25 16.61% 11.44% 8.39% 3.05% 8 H 1.63 15.19% 10.02% 10.94% -0.92% L 1.39 18.18% 13.01% 9.33% 3.68% 9 H 1.67 17.27% 12.10% 11.22% 0.88% L 1.46 18.49% 13.32% 9.82% 3.50% 10 H 1.82 18.52% 13.35% 12.20% 1.15% L 1.67 25.11% 19.94% 11.23% 8.71% Notes: Sum betas are estimated based on the methodology proposed by Ibbotson, Kaplan, and Peterson (1997). Liquidity categories are defined independently of the size groups. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). 35

Table 15 Long-Term Returns in Excess of Estimated Returns using the Sum Beta Methodology for Sub-Groups of the Tenth Size Decile Split into High and Low Liquidity Categories Group [1] Sum Beta Actual Arithmetic Mean Return Actual Return in Excess of [2] [3] [4] = [3] - 5.17% Estimated Return in Excess of [5] = β * (11.88% - 5.17%) Premium (Return in Excess of Estimated Return) [6] = [4] - [5] 10 w H 1.70 15.41% 10.24% 11.39% -1.15% L 1.61 22.05% 16.88% 10.80% 6.08% 10 x H 1.91 20.57% 15.40% 12.79% 2.61% L 1.75 24.63% 19.46% 11.72% 7.74% 10 y H 1.94 20.34% 15.17% 13.04% 2.13% L 1.68 26.79% 21.62% 11.24% 10.38% 10 z H 2.34 21.07% 15.90% 15.73% 0.17% L 1.61 30.96% 25.79% 10.83% 14.95% Notes: Sum betas are estimated based on the methodology proposed by Ibbotson, Kaplan, and Peterson (1997). Liquidity categories are defined independently of the size groups. Historical riskless rate is measured by the arithmetic mean income return component of the 20-year government bond (5.17%). Long horizon equity risk premium is estimated by the arithmetic mean total return of the S&P 500 (11.88%) minus the arithmetic mean income return component of the 20-year government bond (5.17%). The results show that by using sum betas, the beta estimates increase as compared to the beta estimates from the single beta models used in Tables 11 and 12. Interestingly, the higher betas that obtain from using the sum beta approach almost eliminate any size premium for size deciles 1-10 for the higher liquidity stocks. Within decile 10, the results are somewhat mixed with the size premium for higher liquidity stocks virtually zero for sub-groups 10w and 10z, but positive 2.61% to 2.13% for sub-groups 10x and 10y, respectively. 36

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