Value Premium and Liquidity Risk

Value Premium and Liquidity Risk Wenjin Kang, Nan Liyz June 24, 2010

Abstract In this paper, we investigate the risk factors that drive value premium. We found theoretical link as well as the empirical evidence between the asset liquidity and book-to-market ratio. We further investigate the relationship between asset liquidity risk and stock liquidity risk using value premium as the proxy for the asset liquidity risk premium. We found that value …rms have lower liquid-asset ratio as compared with growth …rms. Value stocks are signi…cantly less liquid than growth stocks both statistically and economically, and liquid stocks are more likely to be growth stocks. The liquidity di¤erence between value and growth stocks are much more prominent when the aggregate market is less liquid, when the market valuation is down, and when the economy is in recession.The value premium can explain around 30% of the liquidity premium in the US stock market and vice versa. We argue that the liquidity risk and risk factor that drives the value premium are important fundamental risks correlated with each other and help to explain the cross-section di¤erence in returns. Keywords: Value Premium; Liquidity Risk; Asset Liquidity; Adjustment Cost JEL classi…cation:

Department of Finance, NUS Business School, National University of Singapore, Singapore 117592 E-mail address: [email protected] y Department of Finance, NUS Business School, National University of Singapore, Singapore 117592 E-mail address: [email protected] z Prelimnary, please do not circulate without permission from authors. We would like to thank Amihud Yakov, Steven Davis, Philip Dybvig, Takashi Yamada for their helpful comments and suggestions. We also thank Cheng Si for her excellent research support. We gratefully acknowledge …nancial support from the Ministry of Education of Singapore under Academic Research Fund R-315-000-071-112. All the errors are on our own.

Value Premium and Liquidity Risk

1.

2

Introduction

Value premium has attracted both academic and professional attention for many years. While people agree on the existence of value premium, they debate over the interpretation of why the stock with higher book-to-market ratio earns higher returns. Lakonishok, Shleifer, and Vishny (1994) use post 1963 data to show that value strategies earns value premium because value strategies exploit the suboptimal behavior of the typical investor, not because value stocks are fundamental riskier than growth stocks. Using data from 1963-2005 in the US stock market, we found similar result as Lakonishok, Shleifer, and Vishny (1994). However, using data from 1931-1962, we found that value stocks under perform glamour stocks in down market, which support the argument that value stocks are fundamentally riskier than glamour stocks. Fama and French (1992), Zhang (2005) and Yogo (2006) think it is the fundamental risk that drives the di¤erence between returns of high book-to-market and low book-to-market portfolios, but they identify di¤erent types of the fundamental risk. Daniel and Titman (2006) argue that value stocks contain more intangible risk so earn higher return but they did not identify what is the source of this intangible risk. To understand the underlying risk that drives the value premium, it is important to identify the factors that drive the di¤erence between market value and book value of the capital. If the market is e¢ cient, this di¤erence could be due to the missing book value of intangible capital or adjustment cost of investment. Abel and Eberly (1994) summarize adjustment costs of investments in three categories, and one of which is a wedge between the purchase prices and sale prices of capital. The larger is this wedge, the more costly it is to resell the installed assets, that is the more illiquid is the asset. In a simpli…ed model of investment under certainty of Abel and Eberly (1994), we explore the theoretical relationship between liquidity of the asset and the di¤erence between market value and book value of the …rm. We found that …rms with less liquid assets tend to be the value …rms with high book to market ratio. Our results are consistent with that of Cooper (2006), who found that if real investment is largely irreversible, the book value of a distressed …rm is high relative to its market value. If illiquidity of the real asset is one the important factors that drive the di¤erence between market value and book value of the …rm, then liquidity risk of real asset should be one of the underlying risk factors that drive the value premium. Firms will less liquid assets su¤er more when the market liquidity is low, hence investors require higher risk premium to hold the stocks of such …rms. The liquidity risk of real asset discussed here is di¤erent from the liquidity risk of stocks, however, are they related to each other? Pastor and Stambaugh (2003) and others have identi…ed liquidity risk of stocks as one risk factors in addition to market, size and book-to-market to account for the crosssection variation in stock returns, but no one has studied the relationship between the liquidity risk of stocks and that of the real asset. Cochrane (2005) argues that the separation between macroeconomics-based asset pricing and market microstructure-based asset pricing has begun to erode. While many studies focus on the spill-over e¤ect of volume, trading, liquidity and market structure on the level of prices, we took the other direction to study linkage between liquidity risk of real asset and stocks. In this paper, we indirectly investigate the relationship between the liquidity of real asset and that of the stocks, through the relationship between value premium and liquidity premium of stocks. It is hard to measure the risk premium of real asset illiquidity, but


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given the theoretical link between value premium and liquidity of asset, using value premium as the proxy of the risk premium of real asset illiquidity may shed some lights on the relationship between the liquidity of real asset and that of the stocks. We found some empirical evidence that liquidity risk of stocks and that of real asset are indeed closely linked with each other. However, they are not perfectly correlated. The correlation between value-growth liquidity di¤erence and market liquidity is strongly positive, and this is mainly caused by the periods of low-liquidity aggregate market. It suggests that when there is an illiquidity shock, the value stocks are the one most a¤ected. In addition, we construct the illiquidity index of value …rms and growth …rms using measures of Amihud (2002). We found that value stock is signi…cantly less liquid than growth stocks both statistically and economically. We also found that the liquidity di¤erence between value and growth stocks are much more prominent when the aggregate market is less liquid, when the market valuation is down, and when the economy is in recession. And the di¤erence of liquidity di¤erence is again statistically signi…cant. Section 2 derives the relationship between liquidity of asset and Tobin’s Q of …rms in a simple model of investment under uncertainty. Section 3 presents empirical evidence on the links between liquidity of real asset, value premium and stock liquidity. Section 4 contains a summary and concluding remarks. 2.

A Simple Model of Investment Under Uncertainty

We start with an extended version the production-based asset pricing model in Cochrane (1991), which is similar to that of Hansen, Heaton, and Li (2005) The …rm chooses investment and labor inputs to maximize the present value of the …rms. " +1 # X V0 (K0 ; z0 ) = max E0 S0;t [f (Kt ; nt ; zt ) wt nt pt it ] it ;Lt

s:t: kj;t+1 = (1 zt+1 = (1

t=0

)kj;t + ij;t cj (ij;t ; kj;t ) ) z + zt + "t+1

where S0;t+1 is the stochastic discount factor f (Kt; Lt ; zt ) = zt k1;t1 k2;t2 n1t Kt =

k1;t k2;t

1

2

;

de…ne the adjustment cost for type j capital, for j = 1; 2; as following 8 + bj ij;t 2 > > ( ) kj;t if ij;t 0 < 2 kj;t cj (ijt ; kjt ) = b i > > : j ( j;t )2 kj;t if ij;t < 0 2 kj;t

Notice that cj is linear homogenous in i and k; and we can rewrite it as cj (ijt ; kjt ) = gj (

ij;t )kj;t kj;t

(1)


4

where

8 + b > < j x2 if x 0 2 gj (x) = b > : j x2 if x < 0 2 which is continuous, strictly convex and continuously di¤erentiable with respect to x everywhere except at x = 0: The marginal cost of increase or decrease one unit of uninstalled capital is given by 8 ijt ijt > b+ if 0 < j @cj (ijt ; kjt ) kj;t kj;t 0 ijt = gj ( ) = i i > @ijt kj;t : bj jt if jt < 0 kj;t kj;t and marginal cost of increase installed capital is given by 8 b+ > j > < @cj (ijt ; kjt ) ijt i i 2 jt j;t = gj ( ) gj0 ( ) = b @kjt kj;t kj;t kj;t > j > : 2

:

ijt kj;t ijt kj;t

2

ijt 0 kj;t ijt pj ; while the sale price of one unit of uninstalled capital uninstalled capital is pj +b+ j kj;t b+ bj ij;t j is pj + bj < pj : Hence pj is in fact the average of bid and ask price and ; that kj;t 2 is the depth. While people using depth to construct illiquidity index in the stock liquidity literature, we …nd it also make sense to use it to measure the illiquidity of investment goods. Larger b+ j means that the supply of investment good is more inelastic and the …rm has to pay more when it need to increase investment rapidly. While larger bj means the demand for investment good is more inelastic or the market is more illiquid, hence it is more di¢ cult to sell and the depth is bigger. We assume that liquid asset has larger b+ but lower b : This is to capture the idea that illiquid asset are those asset that are more …rm speci…c, the adjustment cost to increase is smaller than that of liquid asset, while the discount to sell is larger1 . Without loss of generality, suppose type1 asset is more illiquid than type 2 asset, then + b+ 1 < b2 ; b1 > b2

The Lagrangian is E0

+1 X t=0

1

S0;t ff (Kt; nt ; zt )

wt nt

p t it +

t

((1

)Kt + It

C(Kt ; It )

Kt+1 )g

Think of investing in a production line of automobile. It takes more to build a production line that could be used to produce cars as well as vans, and the production line would worth more when the …rm need to sell. That is, the liquid asset embeds a growth option, hence it is more expensive to purchase.


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The …rst order conditions of the maximization problems are [nt ] : wt = (1 [ij;t ] : pj;t =

2 )zt k1;t k2;t nt 2

1

1

1

2

@cj (ij;t ; kj;t ) ij;t = j;t 1 bj;t @ij;t kj;t @f (Kt+1 ; nt+1 ; zt+1 ) + j;t+1 [(1 ) @kj;t+1

1

j;t

[kj;t+1 ] : Et S0;t+1

@cj (ij;t+1 ; kj;t+1 ) ] @kj;t+1

= S0;t

Note that j;t is the shadow price of the installed capital, hence the …rst order condition with respect to to investment link the investment to capital to Tobin’s q j;t

qj;t =

where bj;t = b+ fij;t j

pj;t = b+ j 0 pj;t = bj

ij;t = kj;t > > 1 > > :

1

bj;t

ij;t kj;t

fij;t < 0g, which implies that

0g + bj 8 1 > > > >
0

pj;t or qj;t > 1

if

j;t

> pj;t or qj;t = 1

if

j;t

< pj;t or qj;t < 1

+ When ijt is positive, that is, when the …rm is purchase capital good, b+ 1 < b2 ; it is more expensive to get liquid asset, q1 < q2 : When ijt is negative, it is more di¢ cult to sell uninstalled capital for illiquid asset, b1 > b2 ; while again implies that implied q1 < q2 . In summary, Tobin’s q is lower for illiquid asset. Aggregate Tobin’s q for the …rm is then given by weighted average of q1 and q2

qt =

q1t + (1

= 1

b1;t

1+

)q2t

i1;t k1;t

(1

1

i2;t k2;t i2;t b2;t k2;t b2;t

i1;t k1;t i1;t b1;t )(1 k1;t

b1;t

=

1

+

b2;t

b1;t

i1;t k1;t

i2;t ) k2;t

+ where = k1t =(k1t + k2;t ): When ijt > 0; b+ ; when 1 < b2 ; hence qt is decreasing in ijt < 0; b1 < b2 ; qt is again decreasing in ; that is the more illiquid capital the lower is Tobin’s q 2 . From this adjustment cost model, we found that …rms with more illiquid asset measured by the relative size of illiquid asset and/or subject to higher adjustment 2 In fact, we compare Tobin’s q of two …rms with di¤erent equilibrium composition of liquid assets and illiquid assets, assuming the asset compositions of …rms are in equilibrium. In further study, we need to solve for the equilibium investment-capital ratio of the …rms as functions of fundamentals. In a model with adjustment cost and irreversiblility, Cooper (2006) found that "if real investment is largely irreversible, the book value of assets of a distressed …rm is high relative to its market value because it has idle physical capital."

j;t


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cost of the asset, have lower q or higher book-to-market ratio, that is, they are value …rms. The …rst order condition with respect to capital implies the Euler Equation 2 3 @f (Kt+1 ; nt+1 ; zt+1 ) @cj (ij;t+1 ; kj;t+1 ) @cj (ij;t ; kj;t ) + (1 ) 6 S0t+1 7 @kj;t+1 @kj;t+1 @ij;t 6 7 Et 4 5 @cj (ij;t ; kj;t ) S0t @kj;t = Et [St;t+1 Rj;t+1 ] = 1 where the return to the investment in the capital good is

Rj;t+1 =

=

@f (Kt+1 ; nt+1 ; zt+1 ) @kj;t+1

@cj (ij;t+1 ; kj;t+1 ) @cj (ij;t ; kj;t ) + (1 @kj;t+1 @ij;t @cj (ij;t ; kj;t ) @kj;t

@f (Kt+1 ; nt+1 ; zt+1 ) bj;t+1 + @kj;t+1 2

ij;t+1 kj;t+1 pjt + bj;t

)

2

+ (1

) pj;t+1 + bj;t+1

ij;t+1 kj;t+1

ij;t kj;t

which says to increase one unit of investment good, the cost in market value is pjt +bj;t and the bene…t is increase in marginal product

ij;t kj;t

@f (Kt+1 ; nt+1 ; zt+1 ) and save on the @kj;t+1

market value of next period investment good bj;t+1 2

ij;t+1 kj;t+1

2

+ (1

) pj;t+1 + bj;t+1

ij;t+1 kj;t+1

In the following section, we …rst study the empirical link between the asset liquidity and book-to-market ratio, then the relationship between liquidity of real asset and that of stocks. We test whether …rms with higher book-to-market ratio are those …rms with more illiquid asset and su¤ers larger liquidity risk in the stock market. 3.

Empirical Evidences

In this section, we present empirical evidence on the relationship between asset liquidity, value premium and stock liquidity. 3.1. Value Premium and Asset Liquidity Table 1 summarizes the statistics of book-to-market (B/M) portfolio in whole sample period from 1931 to 2005 as well as two sub-sample periods, pre and post 1963 periods.. Panel A report the average value-weighted monthly return of 10 B/M portfolios, market portfolio and risk free rate3 . The value premium4 remains stable around 0.67% per month 3

We use return on 30 day Treasury Bills as risk free rate. Value premium is de…ned as the return of value stocks with the highest decile B/M ratio minus the return of growth stocks with the lowest decile B/M ratio. We follow Davis, Fama, and French (2000) to construct the B/M portfolio. We use the data from 1931-2005. 4


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in the two sub-sample periods, while the equity premium which is the di¤erence between market portfolio and risk free rate is halved from 0.92% to 0.48%. However, this is mainly due to the increase in the risk free rate instead of the decline in the market portfolio return. In addition, market volatility on the average declines from pre-1963 period to post 1963 period. Davis, Fama, and French (2000) found that value premium is robust from 1926 to 2000 in the US stock market and returns to book-to-market portfolios are always positively correlated with market return, but they did not studied the relationship between value premium and equity premium. Chen, Petkova, and Zhang (2006) found that expected value premium is countercyclical from 1941 to 2002, as measured by the correlation with real consumption and real investment. In addition, he found mixed evidence suggest that value premium has declined over time. We found that the value premium did not decline over time and is positively correlated with market premium before 1960s and negatively correlated with market return afterwards as shown in Figure 1. In addition, Figure 2 shows that value premium is counter cyclical before 1960 and exhibit very weak cyclically afterward. Does this evidence implies that value stocks provide a better hedge for market risk and should require high return compensation? This may explain the phenomena that equity premium has declined over time but value premium only has mixed evidence. What explains the change in the correlation between value premium and equity premium? Why value premium exhibit di¤erent cyclical behavior over time? To address these questions, we …rst look at the behavior of asset liquidity and book-to-market ratio over time. From Panel B in Table 1, it is clear that not only the di¤erence in B/M ratios between value stocks and growth stocks but also the market average B/M ratio declined signi…cantly from pre-1963 period to the post-1963 period. It is interesting to notice in Panel D and E that the market average asset liquidity either measured as the Asset Liquidity Ratio5 or Cash Ratio increased from pre-1963 period to the post-1963 period. From Panel D and E, we also …nd value …rms hold less liquid asset than growth …rm, and the di¤erence in asset liquidity declines from pre-1963 period to post-1963 period. This is consistent with our result in the section 2, that is, when the …rms in the market on average hold more liquid assets, the market book-to-market ratio decreases; value …rms who have high book-to-market ratio or low Tobin’s q hold less liquid asset than growth …rms. In Figure 3, we plot the time series of value-weighted asset liquidity and book-to-market portfolio from 1952 to 2005. We …rst notice that the negative correlation between asset liquidity and book-to-market ratio, then not only the level but also the volatility of asset liquidity and book-to-market changes a lot from pre and post 1963 period. It seems that the change in the asset liquidity may help to explain the time variation in the correlation between value premium and equity premium. In the post 1963 period, the market average liquid asset increases, hence investors require less equity premium to compensate for the risk associated with illiquid asset; on the other hand, it is mainly the growth …rms who increased their holdings of liquid asset, and value …rm hold about the same level of liquid asset, which leads to the increase in the spread between the asset liquidity of growth …rms and value …rms. The investors requires more premium to hold value stocks, which have relatively larger exposure to the risk of asset illiquidity. Hence overall, we 5

Asset Liquidity Ratio is de…ned as total current assets minus total inventories minus total current liabilities divided by total assets in last year. Cash Ratio is de…ned as cash and short minus term investments divided by total assets in last year. We take annual data from COMPUSTAT which is available from 1952.


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observe the market equity premium decreases, value premium remains about the same, and the correlation between market equity premium and value premium decreases in the post 1963 period as compared with the pre 1963 period. Next, we investigate whether the asset liquidity is related to the stock liquidity. In Panel C, we report the turnover rate of B/M portfolios and the evidence is mixed. In pre-1962 period, the turnover rate of value stocks is higher than that of growth stock, while in the post 1963 period, the overall turnover rate increased signi…cantly and growth stocks has slightly higher turnover rate than value stocks. In fact, the average turnover rate exhibit a "smile" shape instead of monotonically change from growth stocks to value stocks. However, using turnover rate as measure of stock liquidity is problematic. In the next subsection, we use liquidity index from Amihud (2002) to measure the stock liquidity, which proxies for the e¤ect on returns of a given trading volume. Intuitively, it seems that asset liquidity and Amihud (2002)’s stock liquidity measure are totally di¤erent, as asset liquidity measures how liquid is the real asset or how easy to convert the real asset to the cash. However, we argue that when stocks are less liquid or the market liquidity is low, investors anticipate the increase in illiquidity risk of the real asset, that is, it would be harder for …rms to convert real asset to cash. Hence investors would require higher risk premium compensation to hold value stocks. We test whether …rms with higher book-to-market ratio are those …rms with more illiquid asset and su¤ers larger liquidity risk in the stock market. 3.2. Value Premium and Stock Liquidity In this subsection, we investigate the relationship between liquidity risk and value premium as suggested in the model of investment under uncertainty in Section 2. In Table 2, we present the returns and other characteristics of …ve portfolios sorted on Amihud (2002) liquidity measure, with portfolio 5 (High)as the least liquid portfolio (most illiquid) and portfolio 1(Low) as the most liquid portfolio. Panel A shows that the B/M ratio increases monotonically with the illiquidity of the portfolios, and the di¤erence in the B/M ratio between low liquid portfolio and high portfolio is signi…cant at 5% level. The di¤erence between return on low liquid portfolio and high liquid portfolio, that is, the liquid premium is 0.68% per month. Panel B shows that after controlling for the di¤erence in B/M, the liquidity premium drops from 0.68% to 0.45% and the di¤erence in B/M explains about 35% of the monthly liquidity premium, which is signi…cant at 5% level statistically and 0.24% of the monthly liquidity premium is also economically important. Panel C presents the results based on the B/M-then-Liquidity two-way sort, which show that the liquidity premium increases with the B/M and for growth stock (low B/M), liquidity premium is not signi…cantly from zero. Table 3 presents the returns and other characteristics of …ve portfolios sorted on book-to-market ratio, with portfolio 5 as the value portfolio (high B/M) and portfolio 1 as the growth portfolio (low B/M ). Panel A shows that the illiquidity of stocks increases monotonically with the B/M ratio, and the di¤erence in the illiquidity between value and growth portfolio is signi…cant at 5% level. Panel B shows that after control for the di¤erence in liquidity, the value premium, that is, the return di¤erence between value stocks and growth stocks drops from 0.75% to 0.50% and the di¤erence in liquidity explains about 33% of the monthly value premium, which is signi…cant at 5% level statistically and 0.25% of the monthly value premium is also economically important. Panel C presents the results based on the Liquidity-then-B/M


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two-way sort, which shows that the value premium increases with the illiquidity. Next, we look at the liquidity di¤erence between value and growth stocks conditional on market liquidity and economic status. Table 4 report Amihud liquidity measure6 of B/M portfolios and market portfolios in di¤erent market conditions7 . In Panel A, we compute the conditional average Amihud liquidity of B/M portfolios when market liquidity is in low 30 percentile, medium 40 percentile and high 30 percentile. Growth stocks are more liquid than value stocks in all of the conditions, and the di¤erence between the liquidity of growth stocks and value stocks is signi…cantly larger when market liquidity is low. In Panel B, we compute the average liquidity of B/M portfolio conditional on the NBER business cycle recession and expansion periods. Again the value stocks are more illiquid than growth stocks in recession as well as in economic expansion, and the di¤erence between the illiquidity of value stocks and growth stocks is slightly larger in recession. In Panel C and Panel D, we found that value stock are more illiquid than growth stocks and the liquidity di¤erence are much more prominent in down market and when the market is more volatile. In all of the four panels, we found that the di¤erence between the liquidity in value stocks and growth stocks strongly correlates with market liquidity, and this correlation is more prominent during the periods of low-liquidity aggregate market, economic recession, down market and high market volatility, which suggests value stocks are more a¤ected by the illiquidity shock than growth stocks. These results are consistent with our hypothesis posted earlier, that value stocks have larger exposure to the asset liquidity risk and asset liquidity risk is closely related to the market liquidity, hence value stocks have larger exposure to the stock liquidity risk as well. Given the close link between value premium and liquidity risk, it is natural to ask whether liquidity is a risk factor that is priced in the market and what is the relationship between liquidity risk factor and HML risk factor in the Fama and French (1993) threefactor model. 3.3. Value Premium and Liquidity Premium In this subsection, we study the relationship between value premium and liquidity premium. In Table 2, we report the statistics of liquidity portfolio formed based on Amihud Liquidity measure from 1931 to 2005, and in two sub-sample periods (pre and post 1963). IML (Illiquid minus Liquid) is the liquidity premium, which is the di¤erence between returns of the most illiquid portfolio (portfolio 10) and the least illiquid portfolio (portfolio 1). The sample mean of liquidity premium is stable over time, only drops a little in post 1963 period. The B/M ratio is monotonically increasing with the illiquidity of the portfolio. Similar as in Table 1, B/M ratio and market volatility decline over time. However, the stock illiquidity on the average increases instead of decreases over time and illiquid stocks become more illiquid in post 1963 period. This pattern is opposite to the pattern of asset liquidity we saw in Table 1, which implies that although stock liquidity and asset liquidity are linked, but they are not the same. In Table 6 we report the correlation between market excess return, SMB, HML and IML, where SMB is de…ned as average return on the smallest 10 percentile stocks minus 6

Note that the larger is Amihud liquidity measure the more illiquid or less liquid is the stock. Numbers in bold are signi…cant at 1% level, numbers in italic are signi…cant at 5% level and numbers marked with * are signi…cant at 10% level. 7


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the average return on the biggest 10 percentile stocks, HML and IML are value premium and liquidity premium de…ned earlier8 . From Table 6, we …nd several interesting pattern: 1. IML is much less correlated with market excess return than HML SMB. If IML is a priced risk factor in the market, the low correlation with market return implies that a factor model with IML included should do a better job than HML or SMB in explaining the cross-section di¤erence in returns. We will revisit this point in the next table. 2. Correlation with market declines over time, and this is true for all of the three factors, HML, SMB and IML. In particular, for HML and IML, the correlation even drops from positive or zero in pre-1963 period to negative in post-1963. 3. Correlation between HML and IML and correlation between HML and SMB signi…cantly declines over time, while the correlation between IML and SMB remain stable over time. Given the above observation, we are compelled to analyze the relationship between value premium and liquidity premium through regression analysis. In Table 7, we present the results of time series regression in three-factor and two-factors model. In Panel A, we regress IML on market, HML, SMB, HML on market, IML and SMB, and SMB on market, HML an IML. IML has signi…cant positive alpha, about 2/3 of the liquidity premium cannot be explained by the three factor model. If we replace SMB by IML in the three factor model, and regress SMB on market , HML and IML, SMB has a signi…cantly negative alpha. This result is consistent with the …ndings in Amihud (2002). If we replace HML by IML in the three factor model, and regress on market, SMB and IML, HML does not have a signi…cant alpha. In Panel B, we present the results of regression in two-factor models, with market excess return as one factor and HML or SMB or IML as another factor. In this two factor models, only IML has signi…cantly positive alpha either regressed on market and HML or on market and SMB, while SMB and HML have either insigni…cant or signi…cant but negative alphas when regressed on the market and other factors. These results seem to imply that IML is an important risk factor that are priced in the market and when it is added to the three factor model, it can price SMB and HML. Furthermore, we use liquidity portfolio and 25 BM and size portfolio as testing portfolios and regress the returns of these test portfolios on three factor models with di¤erent factors. In Figure 4, top panel plots the …tted value of expected return again expected return of testing portfolios regressed on market, HML and SMB; middle panel plots the results of regressing testing portfolios on market, SMB and IML; bottom panel plots the results of regressing testing portfolios on market, HML and IML. The Fama French three-factor model can not explain the return di¤erence among liquidity portfolio, while the three factor models with IML did better job in pricing liquidity portfolio but did worse in explaining the return di¤erence among 25 BM size portfolios. Hence although liquidity premium is an important risk factor, it can not replace HML or SMB. If we add IML to the Fama French three-factor model, the model perform better in accounting for the return di¤erence among liquidity portfolios as well as 25 BM size portfolios as shown in the bottom Panel of Figure 5, which plots the …tted value of expected return again expected return of testing portfolios regressed on market, IML, HML and SMB. The top 8

We also checked the result using the Fama French Factor SMB and HML and the results still hold.


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panel plots the …tted value of expected return again expected return of testing portfolios regressed on market, and IML only. In summary, we …nd important empirical evidence that liquidity premium and value premium are closely linked with each other. Liquidity premium are not fully priced by Fama French three factor model, however, we think more evidences are needed to argue for adding IML as an additional risk factor. In addition, as we have noticed in Table 6, the correlation among these factors changes signi…cantly over time. Are the results presented in Table 7 robust in sample periods? In Table 8 and Table 9 we present the regression results using pre-1963 and post 1963 data, respectively. Most of the results are robust in the sub-sample periods. However, R2 in the regressions with HML as the regressor drops signi…cantly in the pre-1963 sub-sample period. 4.

Conclusion

In this paper, we investigate the driving force of the value premium. We found that both asset liquidity and stock liquidity are closely related to the value premium. In an investment-based asset pricing model, we have shown that value …rms have less liquid asset on their balance sheet. Hence investors required compensation for the risk exposure to the asset illiquidity to hold value stocks. Furthermore, when the market is more volatile, less liquid, or in economic recession, investors require larger compensation. In our empirical analysis, we do …nd that value stocks are less liquid than growth stocks and such di¤erence amplify in down market, economic recession or volatile market, and liquid stocks are more likely to be growth stocks. We also found value stocks exhibit lower market liquidity, and liquidity premium and value premium are correlated. These evidence support our argument that risk of asset illiquidity is one of driving force of value premium. Why do stocks of value …rms exhibit lower market liquidity? One possible explanation is that it is di¢ cult for the value …rms with less liquid asset on the balance sheet to raise cash meeting outstanding claims, since the less liquid asset can only be sold at a deep discount as the model assumes. The investors realize that value …rms are more likely to experience …nancial distress and hence will be less willing to trade it, (or the market maker is less willing to own inventory position on such value stocks), and hence the market liquidity for value stocks also becomes lower than the growth stocks. The model we presented in this paper is a static model, which allows us to compare the equilibrium book-to-market value of …rms with di¤erent composition of liquid asset and illiquid asset on their balance sheets. However, the current can not answer the question why …rms choose di¤erent composition of assets. We need to solve for the equilibrium investment-to-capital ratio as functions of fundamentals of …rms to address this question.


12

References Abel, A. B. and J. C. Eberly (1994). A uni…ed model of investment under uncertainty. American Economic Review 84, 1369–1384. Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and time-series e¤ects. Journal of Financial Markets 5, 31–56. Chen, L., R. Petkova, and L. Zhang (2006). The expected value premium. Working Paper. Cochrane, J. (1991). Production-based asset pricing and the link between stock returns and economic ‡uctuations. Journal of Finance 46, 209–237. Cochrane, J. (2005). Liquidity, trading and asset price. NBER Reporter Winter, 1–12. Cooper, I. (2006). Asset pricing implications of nonconvex adjustment costs and irreversibility of investment. Journal of Finance LXI, 139–170. Daniel, K. and S. Titman (2006). Market reactions to tangible and intangible information. Journal of Finance 61 (4), 1605–1643. Davis, J. L., E. F. Fama, and K. R. French (2000). Characteristics, covariances and average returns: 1929-1997. Journal of Finance 55 (1), 389–406. Fama, E. and K. French (1992). The cross-section of expected stock returns. Journal of Finance 47, 427–465. Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3–56. Hansen, L. P., J. Heaton, and N. Li (2005). Intangible risk? In C. Corrado, J. Haltiwanger, and D. Sichel (Eds.), Measuring Capital in the New Economy. The University of Chicago Press. Lakonishok, J., A. Shleifer, and R. Vishny (1994). Contrarian investment, extrapolation, and risk. Journal of Finance 50, 185–224. Pastor, L. and R. F. Stambaugh (2003). Liquidity risk and expected stock returns. Journal of Political Economy 111, 642–685. Yogo, M. (2006). A consumption-based explanation of expected stock returns. Journal of Finance 61 (2), 539–580. Zhang, L. (2005). The value premium. Journal of Finance 60 (1), 67–103.


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1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

1940

1950

1960

1970

1980

1990

2000

Figure 1: Daily Correlation of Value Premium and Market Returns


14

market returns 1

40 30

0.8

20 0.6

10

0.4

0 -10

0.2 0

-20 1930

1940

1950

1960

1970

1980

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-30

HML 1

80 HML value-growth

0.8

60 40

0.6 20 0.4 0 0.2 0

-20

1930

1940

1950

1960

1970

1980

1990

2000

-40

Figure 2: Market Return and Value Premium with NBER Business Cycles


15

Figure 3: Value-Weighted Asset Liquidty and Book-to-Market Ratio of market portfolio, annual data from 1952-2005


Fitted E(R)

Liquidity Portf olio

25 BM Size

0.02

0.02

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0

Fitted E(R)

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0 0

Fitted E(R)

16

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0

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0

0 0

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Figure 4: Regression of Liquidity Portfolios and 25 BM-Size Portfolio in Three-Factor Model


Fitted E(R)

Liquidity Portfolio

Fitted E(R)

25 BM Size

0.02

0.02

0.015

0.015

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0

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17

0.02

0

0

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0.015

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0

0.005

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0.015

0.02

Figure 5: Regression of Liquidity Portfolios and 25 BM-Size Portfolio in Two Factor Model and Four Factor Model with IML

0.25 0.33 0.20

4.09 1.06 6.35

0.38 0.15 0.44

0.52 0.26 0.52

1931-2005 1931-1962 1963-2005

1931-2005 1931-1962 1963-2005

1952-2005 1952-1962 1963-2005

1952-2005 1952-1962 1963-2005

0.54 0.23 0.54

0.45 0.13 0.53

3.82 1.43 5.59

0.45 0.56 0.37

Sample 1(Growth) 2 Periods 1931-2005 0.86 1.00 std. dev. 5.71 5.47 1931-1962 0.93 1.00 std. dev. 6.37 6.27 1963-2005 0.80 1.00 std. dev. 5.16 4.80

Table 1: Statistics of B/M portfolios 3 4 5 6 7 8 9 10(Value) Value - Growth Market Rf Panel A: Average Value-Weighted Monthly Returns (%) 0.96 0.98 1.07 1.12 1.20 1.32 1.51 1.53 0.67 0.99 0.31 5.31 6.12 5.77 6.28 6.82 6.89 8.16 9.53 6.81 5.41 0.26 0.91 0.95 1.15 1.16 1.16 1.43 1.64 1.61 0.67 1.02 0.09 6.04 7.65 7.22 8.17 9.16 9.29 11.26 13.19 9.03 6.56 0.08 1.00 1.00 1.01 1.08 1.23 1.24 1.41 1.47 0.67 0.96 0.47 4.69 4.67 4.40 4.37 4.33 4.31 4.69 5.38 4.50 4.38 0.23 Panel B: The Book-to-Market Ratio 0.61 0.75 0.91 1.09 1.31 1.65 2.29 4.74 4.49 0.77 0.75 0.93 1.15 1.41 1.76 2.33 3.47 8.12 7.80 0.98 0.50 0.61 0.73 0.85 0.98 1.15 1.42 2.22 2.02 0.61 Panel C: Turnover Rate (%) 3.58 3.63 3.48 3.75 3.77 3.99 4.33 5.00 0.90 3.81 1.28 1.46 1.57 2.04 2.14 2.39 2.90 3.64 2.58 1.49 5.29 5.24 4.89 5.03 4.99 5.18 5.40 6.01 -0.34 5.53 Panel D: Asset Liquidly 0.31 0.18 0.17 0.13 0.17 0.18 0.09 0.10 -0.29 0.22 0.13 0.11 0.10 0.10 0.11 0.11 0.09 0.08 -0.07 0.11 0.36 0.20 0.19 0.14 0.19 0.20 0.09 0.10 -0.34 0.25 Panel E: Cash Ratio 0.34 0.22 0.20 0.16 0.18 0.18 0.12 0.14 -0.38 0.26 0.20 0.17 0.18 0.17 0.17 0.15 0.14 0.11 -0.15 0.18 0.34 0.22 0.20 0.16 0.18 0.18 0.12 0.14 -0.39 0.26

Value Premium and Liquidity Risk 18


Table 2: Liquidity Sorted Portfolio Panel A: Liquidity-Sorted Portfolios 1(Low) 2 3 4 5(High) Low - High Return (%) 1.00 1.13 1.24 1.33 1.68 0.68 *** B/M ratio 0.90 1.04 1.17 1.36 1.89 0.98 *** Amihud 0.04 0.14 0.33 0.68 1.50 1.46 *** Panel B: B/M-Controlled Liquidity-Sorted Portfolios 1 2 3 4 5 Low - High Return(%) 1.09 1.21 1.21 1.35 1.54 0.45 *** Controlled Liquidity premium 0.24 *** B/M ratio 1.23 1.24 1.25 1.26 1.39 0.15 *** Amihud 0.05 0.17 0.37 0.69 1.40 1.35 *** Panel C: B/M-then-Liquidity Two-Way-Sorted 5x5 Portfolios Return(%) 1 2 3 4 5 Low - High 1 (Growth) 0.85 0.87 0.84 0.94 1.05 0.20 2 0.98 1.05 1.11 1.24 1.36 0.38 ** 3 1.04 1.25 1.31 1.34 1.49 0.45 *** 4 1.21 1.25 1.32 1.46 1.72 0.51 *** 5 (Value) 1.37 1.60 1.45 1.78 2.08 0.71 *** 0.51 ***

Table 3: B/M sorted Portfolio Panel A: B/M-Sorted Portfolios 1 (Growth) 2 3 4 5 (Value) Value - Growth Return(%) 0.91 1.15 1.29 1.39 1.65 0.75 *** B/M ratio 0.35 0.65 0.94 1.35 3.07 2.72 *** Amihud 0.30 0.42 0.51 0.62 0.83 0.53 *** Panel B: Liquidity-Controlled B/M-sorted Portfolios 1 (Growth) 2 3 4 5 (Value) Value - Growth Return(%) 1.00 1.19 1.32 1.37 1.50 0.50 *** Controlled Value premium 0.25 *** B/M ratio 0.40 0.69 0.96 1.34 2.97 2.57 *** Amihud 0.51 0.52 0.53 0.54 0.57 0.06 *** Panel C: Liquidity-then-B/M Two-Way Sorted 5x5 Portfolios Return(%) 1 (Growth) 2 3 4 5 (Value) Value - Growth 1 (High Liquidity) 0.82 0.92 0.97 1.10 1.19 0.37 ** 2 0.88 1.03 1.16 1.25 1.35 0.47 *** 3 0.89 1.20 1.32 1.28 1.52 0.63 *** 4 1.08 1.24 1.44 1.41 1.48 0.40 *** 5 (Low Liquidity) 1.35 1.56 1.69 1.83 1.99 0.63 *** 0.27 *

19

Table 4: Amihud Liquidity Measure for the B/M Portfolios B/M Ranking 1 2 3 4 5 6 7 8 9 Mean 0.698 0.791 0.996 1.163 1.367 1.500 1.755 2.001 2.578 Panel A: Conditional on the State of Amihud Market Liquidity Low 1.168 1.402 1.839 2.083 2.535 2.810 3.241 3.773 5.055 Medium 0.698 0.725 0.876 1.077 1.186 1.291 1.553 1.698 2.080 High 0.219 0.261 0.301 0.343 0.422 0.451 0.518 0.607 0.734 Low - High 0.949 1.141 1.538 1.740 2.113 2.359 2.723 3.166 4.321 Panel B: Conditional on NBER Business Cycles Recession 0.642 0.887 1.127 1.266 1.471 1.668 2.008 2.257 2.876 Expansion 0.711 0.769 0.965 1.139 1.342 1.461 1.696 1.941 2.509 Reces - Expan 0.070 -0.118* -0.162* -0.128 -0.128 -0.207* -0.312 -0.315* -0.367* Panel C: Conditional on the State of Market Returns Down 0.773 0.928 1.172 1.355 1.604 1.770 2.049 2.310 3.039 Up 0.652 0.707 0.888 1.045 1.220 1.334 1.575 1.810 2.295 Down - Up 0.121 0.221 0.284 0.310 0.384 0.436 0.474 0.500 0.743 Panel D: Conditional on the State of Market Volatility High 0.761 0.957 1.210 1.402 1.704 2.019 2.291 2.622 3.418 Medium 0.658 0.758 0.957 1.094 1.300 1.357 1.642 1.807 2.329 Low 0.688 0.670 0.832 1.013 1.116 1.168 1.366 1.632 2.064 High- Low 0.074 0.287 0.378 0.389 0.588 0.851 0.926 0.990 1.354

3.057 1.451 0.460 2.533 1.670 1.567 -0.103 1.847 1.427 0.420 1.974 1.482 1.335 0.639

5.573 2.152 0.668 4.905

6.742 2.850 0.887 5.855

3.572 2.931 3.412 2.701 -0.160 -0.230 4.040 3.267 3.075 2.423 0.966 0.845 4.417 3.174 2.819 1.598

3.655 2.516 2.131 1.525

10-1 Market 2.744 1.587

10 3.443


0.927 5.845 0.858 0.020 0.878 7.215 1.144 0.023 0.964 4.574 0.646 0.018

Mean Return (%) Std. Dev. of Return(%) BE/ME Amihud Liquidity



1(Liquid)

Table 5: Statistics of Liquidity Portfolio 2 3 4 5 6 7 Panel A: 1931-12005 1.071 1.097 1.168 1.236 1.251 1.309 6.315 6.352 6.825 6.784 6.867 6.826 0.948 1.018 1.060 1.157 1.188 1.302 0.055 0.104 0.177 0.265 0.386 0.559 Panel B: 1931-1962 1.042 1.043 1.190 1.241 1.330 1.355 7.906 7.636 8.419 8.285 8.337 8.220 1.214 1.370 1.420 1.604 1.617 1.785 0.060 0.107 0.165 0.230 0.304 0.407 Panel C: 1963-12005 1.093 1.137 1.152 1.232 1.192 1.274 4.809 5.202 5.347 5.412 5.533 5.576 0.749 0.757 0.793 0.823 0.868 0.942 0.051 0.103 0.186 0.291 0.447 0.672 1.332 5.649 0.995 0.985

1.387 8.636 2.009 0.554

1.356 7.075 1.428 0.801

8

1.440 5.502 1.098 1.461

1.610 8.882 2.190 0.743

1.512 7.138 1.564 1.155

1.850 5.445 1.346 2.319

1.863 9.620 3.379 1.197

1.856 7.511 2.213 1.840

9 10(Illiquid)

0.886 4.042 0.700 2.301

0.985 5.439 2.235 1.173

0.928 4.687 1.355 1.820

IML



22

Table 6: Correlation Between Risk Factors Market-Rf SMB HML IML Panel A: 1931-2005 Rm-Rf 1.000 SMB 0.383 1.000 HML 0.374 0.706 1.000 IML 0.076 0.770 0.507 1.000 Panel B: 1931-1962 Rm-Rf 1.000 SMB 0.491 1.000 HML 0.597 0.832 1.000 IML 0.198 0.769 0.595 1.000 Panel C: 1963-2005 Rm-Rf 1.000 SMB 0.149 1.000 HML -0.105 0.325 1.000 IML -0.106 0.819 0.358 1.000

IML t-stat HML t-stat SMB t-stat IML t-stat IML t-stat HML t-stat HML t-stat SMB t-stat SMB t-stat

Table 7: Full Sample Regression 1931-2005 Constant Rm-Rf HML SMB IML Panel A: Three-Factor Model 0.69 -0.220 -0.014 0.526 8.291 -7.277 -0.386 10.226 0.13 0.143 0.593 -0.041 0.763 2.940 7.274 -0.392 -0.61 0.313 0.377 0.985 -5.429 8.340 4.668 21.962 Panel B: Two-Factor Model 0.76 -0.114 0.383 6.308 -2.901 6.526 0.68 -0.222 0.518 8.291 -6.862 9.434 0.11 0.152 0.572 0.673 3.085 13.693 -0.29 0.424 0.700 -1.707 4.899 9.999 0.14 0.201 0.754 0.841 4.260 8.612 -0.72 0.473 1.249 -6.078 5.870 17.337

R2 65% 51% 77%

27% 65% 51% 37% 51% 70%


Table 8: Pre-1963 Sample Period Regression 1931-1962 Constant Rm-Rf HML SMB IML R2 Panel A: Three-Factor Model IML 0.68 -0.204 0.058 0.660 73% t-stat 7.178 -6.971 1.906 16.358 HML 0.39 -0.113 0.156 0.227 14% t-stat 1.917 -1.954 1.439 1.986 SMB -0.68 0.279 0.062 1.038 73% t-stat -5.957 8.688 1.616 20.629 Panel B: Two-Factor Model IML 0.73 -0.063 0.313 13% t-stat 4.364 -1.276 6.989 IML 0.71 -0.213 0.678 72% t-stat 7.702 -7.575 18.079 HML 0.55 -0.162 0.310 13% t-stat 3.025 -3.033 5.079 HML 0.29 -0.070 0.393 13% t-stat 1.548 -1.301 6.710 SMB 0.08 0.214 0.387 14% t-stat 0.374 4.192 6.833 SMB -0.66 0.275 1.063 73% t-stat -5.810 8.466 25.351 Table 9: Post-1963 Sample Period Regression 1963-2005 Constant Rm-Rf HML SMB IML R2 Panel A: Three-Factor Model IML 0.61 -0.197 -0.001 0.461 63% t-stat 4.243 -4.306 -0.020 5.142 HML -0.37 0.343 0.612 -0.003 74% t-stat -1.520 5.924 8.770 -0.020 SMB -0.19 0.162 0.585 0.866 81% t-stat -0.865 2.734 5.726 12.742 Panel B: Two-Factor Model IML 0.87 -0.203 0.446 39% t-stat 4.618 -3.625 5.415 IML 0.61 -0.197 0.460 63% t-stat 4.384 -3.986 6.293 HML -0.37 0.344 0.611 74% t-stat -1.615 5.990 15.841 HML -0.76 0.689 0.823 59% t-stat -2.790 6.472 7.166 SMB 0.56 -0.014 0.972 69% t-stat 2.053 -0.211 9.644 SMB -0.63 0.565 1.347 71% t-stat -2.772 5.001 11.754

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