Economic Valuation of Liquidity Timing

Transcription

1 Economic Valuation of Liquidity Timing Dennis Karstanje a,b,, Elvira Sojli a,c, Wing Wah Tham a, Michel van der Wel a,b,d a Erasmus University Rotterdam b Tinbergen Institute c Duisenberg School of Finance d CREATES May 11, 2012 Preliminary Draft Please do not distribute without authors permission Abstract This paper examines whether liquidity timing strategies yield any economic benefits, and which liquidity proxy should be used to forecast excess returns. We evaluate the impact of predictable changes in conditional returns on the performance of short-horizon dynamic asset allocation strategies, to assess the economic value of forecasts of different liquidity measures. We find that liquidity timing leads to tangible economic gains and the Lesmond, Ogden, and Trzcinka (1999) Zeros measure to the best market timing returns. JEL classification: G11; G12; G17. Keywords: Liquidity; forecasting; expected returns; economic valuation. Sojli gratefully acknowledges the financial support of the European Commission Grant PIEF-GA Tham gratefully acknowledges the financial support of the European Commission Grant PIEF-GA Michel van der Wel is grateful to Netherlands Organisation for Scientific Research (NWO) for a Veni grant; and acknowledges support from CREATES, funded by the Danish National Research Foundation. Corresponding author. Address: Econometric Institute, H9-01, Erasmus University Rotterdam, PO Box 1738, Rotterdam, 3000DR, The Netherlands. Tel.: karstanje@ese.eur.nl. Other author s addresses: esojli@rsm.nl (Sojli), tham@ese.eur.nl (Tham), vanderwel@ese.eur.nl (van der Wel).

2 1 Introduction There is ample evidence that liquidity can be timed, i.e. liquidity is predictable and liquidity predicts expected returns over time. Amihud (2002) and Jones (2002) show that when market liquidity is expected to be low expected returns are higher, in the U.S. market. Baker and Stein (2004) theoretically model and empirically find that liquidity variables have large forecasting power in economic terms. Furthermore, Bekaert, Harvey, and Lundblad (2007) find that also in emerging markets liquidity significantly predicts future returns. If liquidity is able to predict expected returns, liquidity information can be used to time the market. Cao, Chen, Liang, and Lo (2011) provide evidence that many hedge fund managers behave like liquidity timers, adjusting the market exposure of their portfolios based on the equity-market liquidity. They show that the top timing funds outperform the bottom timing funds by 5% per year on a risk-adjusted basis. There are many reasons why liquidity may predict returns in the timeseries. Amihud and Mendelson (1986) and Vayanos (1998) argue that investors anticipate future transaction costs and discount assets with higher transaction costs more. This explanation is reasonable for individual stocks but is less straightforward for the market as a whole. Baker and Stein (2004) develop a model in which market liquidity is seen as a sentiment indicator. They relate increased liquidity to a behavioral aspect, namely irrational investors who underreact to information in order flow. Under the assumption that these investors are restricted by short-sales constraints, they will only be in the market when their sentiment is positive, i.e. when they overvalue the market relative to rational investors. Hence when the market is more liquid, it is overvalued and expected returns are lower. Liquidity is an elusive concept because it is not directly observable. Moreover, liquidity has multiple aspects that cannot be captured in a single measure and the number of available proxies is large. Examples of articles that develop low-frequency spread proxies are Roll (1984), Lesmond, Ogden, and Trzcinka (1999), Hasbrouck (2009), Holden (2009), and Goyenko, Holden, and Trzcinka (2009). Measures of price impact are used in Amihud, 1

3 Mendelson, and Lauterbach (1997), Berkman and Eleswarapu (1998), Amihud (2002), and Pástor and Stambaugh (2003). Turnover is also often used as a proxy for liquidity (e.g. Baker and Stein (2004)). Given the empirical evidence that liquidity predicts expected returns and since the liquidity of a financial asset is unobservable, we are interested in which proxy a liquidity timer should use. In this paper, we examine which proxy a liquidity timer should use, by measuring the economic value of liquidity forecasts from different liquidity proxies for investors, who engage in short-horizon asset allocation strategies. We use economic valuation of liquidity proxies because we want to assess which measure captures returns the best, or following the explanation of Baker and Stein (2004) which is the best sentiment indicator. Furthermore, a timing exercise is most relevant from an investor s point of view, thus using economic evaluation as the performance metric is natural. Last, by comparing the economic value of different measures, the performance of all proxies is expressed in the same units, irrespective of the aspect of liquidity they measure. To gauge the economic value of liquidity timing, we first construct four low-frequency liquidity measures suggested by the current literature: illiquidity ratio (ILR) (Amihud, 2002), Roll (Roll, 1984), Effective Tick (Holden, 2009; Goyenko et al., 2009), and Zeros (Lesmond et al., 1999). Using these liquidity measures, we form conditional expectations about stock returns for the next period. Building on previous research by West, Edison, and Cho (1993), Fleming, Kirby, and Ostdiek (2001), Della-Corte, Sarno, and Thornton (2008), Della-Corte, Sarno, and Tsiakas (2009), and Thornton and Valente (forthcoming), we employ mean-variance analysis as a standard measure of portfolio performance and apply quadratic utility to examine whether there are any economic gains for an investor who uses conditional excess return forecasts based on liquidity measure A relative to conditioning on liquidity measure B. Economic gains are evaluated mainly using two measures: the Sharpe ratio and the performance fee. The Sharpe ratio is the most common measure of performance evaluation employed in financial markets to assess the success or failure of active asset managers; it is calculated as the ratio of the average 2

4 realized portfolio excess returns to their variability. The performance fee measures how much a risk-averse investor is willing to pay for switching from one strategy to another. In addition, we calculate the break-even transaction cost, which is the transaction cost that would remove any economic gain from a dynamic asset allocation strategy. Furthermore we implement some well known improvements of mean-variance portfolios, as suggested by Jagannathan and Ma (2003). Based on NYSE-listed stocks for the period , we find evidence of economic value in liquidity timing. The Zeros measure outperforms the other measures: ILR, Roll and Effective Tick. The Zeros measure achieves a Sharpe ratio of 0.51, followed by the ILR with a SR of A risk-averse investor with quadratic utility would pay an annual fee of almost 200 basis points to switch from the other liquidity proxies to condition on the ILR or Zeros liquidity measure. We show that these results are not driven by alternative return predictors such as the dividend yield or book-to-market ratio. Furthermore, the ranking of the liquidity measures based on economic value is robust to different specifications, e.g. imposing weight restrictions, and parameter settings. A related paper is Goyenko, Holden, and Trzcinka (2009) that addresses the crucial issue of how well do low frequency liquidity measures approximate true transaction costs for market participants. The authors find that low frequency measures provide good approximations to high frequency transaction costs. Effective Tick is the best low frequency measure for effective and realized spread, and ILR is the best measures for price impact. Although the analysis of Goyenko et al. (2009) is also a relative comparison, we investigate which liquidity measure is the best sentiment indicator. Our results show that the Zeros measure is most relevant for liquidity timing. On the other hand, Effective Tick shows no economic value, despite its ability to approximate high frequency transaction costs well. This implies that the best proxy for transaction costs is not necessarily the proxy that an investor should use for liquidity timing. Portfolio allocation using a mean-variance framework to investigate the economic value of a timing strategy is not new. West et al. (1993) use the mean-variance setting and quadratic utility to rank models based on utility gains. Fleming et al. (2001) generalize 3

5 the approach of West et al. (1993) and calculate switching fees to investigate volatility timing in equity markets. Della-Corte et al. (2008) apply the approach to the interest rate market, while Della-Corte et al. (2009) use the framework for exchange rate data. Most recently, Thornton and Valente (forthcoming) investigate the economic value of long-term forward interest rate information to predict bond returns. We differ from these articles because we use the framework to investigate liquidity timing in equity markets. The paper proceeds as follows. In the next section we present the methodology: how we form conditional expected excess returns, construct dynamic trading strategies, and evaluate them. Section 3 presents the data, the liquidity measures we construct, and the descriptive characteristics. The results are presented in Section 4, followed by a discussion of the robustness of the results in Section 5. Section 6 concludes the paper. 2 Methodology Our methodology can be separated into three steps. First, we form conditional expectations of returns based on different liquidity measures. Second, we construct dynamically rebalanced mean-variance portfolios based on these return predictions. Third, we evaluate the performance of these strategies. 2.1 Forecasting Liquidity and Expected Returns We model liquidity in order to get an expression for expected liquidity in the next period. The high persistence in liquidity series suggests that autoregressive models are most appropriate. Based on the Akaike Information Criterion and the Bayesian Information Criterion, we select an AR(2) model to capture all autocorrelation in liquidity: LIQ k,t = φ 0,t + φ 1,t LIQ k,t 1 + φ 2,t LIQ k,t 2 + η k,t, (1) where LIQ k,t is the liquidity of asset k at time t. Iterating forward Eq. 1, liquidity predictions for the next period are given by E t [LIQ k,t+1 ] = φ 0,t +φ 1,t LIQ k,t +φ 2,t LIQ k,t 1. 4

6 Integrating expected liquidity in a model for conditional expected excess returns that is solely driven by liquidity, gives: E t [r k,t+1 r f,t ] = δ 0,t + δ 1,t E t [LIQ k,t+1 ] + ε k,t = δ 0,t + δ 1,t (φ 0,t + φ 1,t LIQ k,t + φ 2,t LIQ k,t 1 ) + ε k,t = β 0,t + β 1,t LIQ k,t + β 2,t LIQ k,t 1 + ε k,t, (2) where β 0,t = δ 0,t + δ 1,t φ 0,t, β 1,t = δ 1,t φ 1,t, and β 2,t = δ 1,t φ 2,t. We are interested in return predictions generated by Eq. 2, hence we only need estimates for the β-parameters and we do not estimate Eq. 1. The coefficients β 0,t, β 1,t and β 2,t are allowed to vary over time and are estimated using a rolling window of length L. To minimize the effect of the choice of L on the results, Pesaran and Pick (2011) suggest to average predictions generated using different rolling window lengths. We estimate the parameters in Eq. 2 both using a particular choice for the window length (L = 120 monthly observations) and following Pesaran and Pick (2011) using different window lengths (L = 60, 120, 240 monthly observations). In the case of the Pesaran and Pick (2011) approach, we take the ( average of the three different predictions for next month s excess return r k,t+1 r f,t ), whereby these predictions are based on the three sets of estimated parameters. The first return prediction is made for January 1967, in order for the longest moving window of 20 years to be estimated. For the 10 year moving window, we estimate the regression in Eq. 2 using data from January 1957 to December Using the estimated coefficients we make a forecast for the next month, January Then we shift our window one period ahead. Thus, the second estimation window runs from February 1957 to January Again we estimate the coefficients and make a prediction for February This procedure is repeated for all months t = Jan 1967, Feb 1967,..., Dec 2008 and all assets k = 1, 2,..., K, for each liquidity measure. For the 5 year moving window, the first window is January 1962 to December 1966 and for the 20 years, the first window is January 1947 to December

7 2.2 Asset Allocation We use dynamic trading strategies to asses the economic value of liquidity timing. We employ mean-variance analysis as a standard measure of portfolio performance to calculate Sharpe ratios. Assuming quadratic utility, we also measure how much a risk-averse investor is willing to pay for switching from one liquidity measure to another. An investor invests every month in the K risky assets and one riskless U.S. Treasury bill. She chooses the weights to invest in each risky asset by constructing a dynamically re-balanced portfolio that maximizes the conditional expected return subject to a target conditional volatility. Her optimization problem is given by max w t { rs,t+1 t = w tr k,t+1 t + (1 w t1)r f,t } s.t. (σ s) 2 = w tσ t+1 t w t, (3) where r s,t+1 t is the conditional expected return of strategy s, w t is the vector of strategy weights of the risky assets, r k,t+1 t is the vector of conditional risky asset return predictions, σs is the target level of risk for the strategy, and Σ t+1 t is the variance-covariance matrix of the risky assets that is estimated recursively as the investor updates return predictions and dynamically balances her portfolio every month. The solution to this maximization problem yields the risky asset investment weights: w t = σ s Qt Σ 1 t+1 t ( rk,t+1 t 1r f,t ), (4) where r k,t+1 t 1r f,t is the conditional excess return and Q t = ( r k,t+1 t 1r f,t ) Σ 1 t+1 t ( rk,t+1 t 1r f,t ). The weight invested in the risk free asset is 1 w t1. The covariance matrix is estimated by the sample covariance matrix over a 10 year rolling window, thus, the covariance matrix is time-varying. 6

8 2.3 Evaluation We evaluate the dynamic strategies based on Sharpe ratio (SR), performance fee, and transaction costs. In contrast to other work, we compare all different liquidity measures among each other, irrespective of the aspect of liquidity they are measuring. The advantage is that we get one ranking for all investigated liquidity measures based on economic value and not multiple rankings per liquidity aspect (e.g. bid-ask spread or price impact). Sharpe Ratio The first economic criterion we employ is the Sharpe ratio (SR), or return-to-variability ratio, which measures the risk-adjusted returns from a portfolio or investment strategy and is widely used by investment banks and asset management companies to evaluate investment and trading performance. The ex-post SR is defined as: SR = r s r f σ s, (5) where r s r f is the average (annualized) excess strategy return over the risk free rate, and σ s is the (annualized) standard deviation of the investment returns. This measure is commonly used to evaluate performance in the context of meanvariance analysis. However, Marquering and Verbeek (2004) and Han (2006) show that the SR can underestimate the performance of dynamically managed portfolios. This is because the SR is calculated using the average standard deviation of the realized returns, which overestimates the conditional risk (standard deviation) faced by an investor at each point in time. Thus, we use the performance fee as an additional economic criterion to quantify the economic gains from using the liquidity models considered. Performance Fees Under Quadratic Utility We calculate the maximum performance fee a risk-averse investor is willing to pay to switch from the strategy based on liquidity measure A to an alternative strategy that is based on liquidity measure B. The specific measure adopted is based on mean-variance 7

9 analysis with quadratic utility (West et al., 1993; Fleming et al., 2001; Della-Corte et al., 2008). Under quadratic utility, at the end of period t + 1 the investor s utility of wealth can be represented as: U (W t+1 ) = W t+1 ϱ 2 W 2 t+1 = W t (1 + r s,t+1 ) ϱ 2 W 2 t (1 + r s,t+1 ) 2, (6) where W t+1 is the investor s wealth at t + 1; r s,t+1 is the gross strategy return; and ϱ determines her risk preference. To quantify the economic value of each model the degree of relative risk aversion (RRA) of the investor is set to δ = ϱw t 1 ϱw t, and the same amount of wealth is invested every day. Under these conditions, West et al. (1993) show that the average realized utility (U) can be used to consistently estimate the expected utility generated from a given level of initial wealth. The average utility for an investor with initial wealth W 0 = 1 is: U = 1 T T 1 t=0 ( 1 + r s,t+1 ) δ 2 (1 + δ) (1 + r s,t+1) 2. (7) At any point in time, one set of estimates of the conditional returns is better than a second set if investment decisions based on the first set leads to higher average realized utility, U. Alternatively, the optimal model requires less wealth to yield a given level of U than a suboptimal model. Following Fleming et al. (2001), we measure the economic value of liquidity by equating the average utilities for selected pairs of portfolios. Suppose, for example, that holding a portfolio constructed using the optimal weights based on liquidity measure A yields the same average utility as holding the optimal portfolio implied by the liquidity measure B, that is subject to daily expenses Φ, expressed as a fraction of wealth invested in the portfolio. Since the investor would be indifferent between these two strategies, we interpret Φ as the maximum performance fee she will pay to switch from strategy A to strategy B. In other words, this utility-based criterion measures how much a mean-variance investor is willing to pay for conditioning on a particular liquidity measure for the purpose of forecasting stock returns. The performance fee will depend on the investor s degree of risk aversion. To estimate the fee, we find the value of Φ that 8

10 satisfies: T 1 t=0 { 1 + r A s,t+1 } δ ( ) 1 + r A 2 2 (1 + δ) s,t+1 = T 1 t=0 { (1 + r B s,t+1 Φ ) δ ( 1 + r B 2 (1 + δ) s,t+1 Φ ) } 2, (8) where rs,t+1 A is the strategy return obtained using forecasts based on the liquidity measure A, Φ is the maximum performance fee an investor wants to pay to switch from strategy A to strategy B, and δ is the degree of relative risk aversion (RRA) of the investor. The switching fee Φ depends on the value of RRA δ, therefore we show in the robustness section that our results are quantitatively similar for different values of δ. Transaction Costs In dynamic investment strategies, where the individual rebalances the portfolio every month, transaction costs can play a significant role in determining returns and comparative utility gains. However, traders charge transaction costs according to counter-party types and trade size. Thus, instead of assuming a fixed cost, we compute the break-even transaction cost τ, which is the minimum monthly proportional cost that cancels the utility advantage of a given strategy. A similar measure of break-even transaction costs has been previously used by Han (2006), Marquering and Verbeek (2004), and Della-Corte et al. (2009). We assume that transaction costs at time t equal a fixed proportion τ of the amount traded in asset k: τ K k=1 ( ) 1 + rk,t + r f,t 1 A k,t w k,t w k,t 1, (9) 1 + r s,t where k = 1,..., K refers to the risky assets and A k,t is a scaling factor that expresses the break-even transaction costs τ in terms of asset K. The scaling factor takes the difference in trading costs between the different assets into account. Previous articles assume that the transaction costs of all assets in their analysis is the same, i.e. A k,t = 1. We cannot make that assumption because we focus on the liquidity differences between assets, which implies that the transaction costs of the different assets are unlikely to be the same. In our results we report both τ 1 based on A k,t = 1 and τ A based on a time and portfolio 9

11 dependent A k,t. To quantify the transaction costs differences we use the ratio of Effective Tick estimates for the scaling factor A k,t = Eff.T ick k,t Eff.T ick K,t, such that we express all transaction costs in terms of asset K. The choice for Effective Tick is based on the finding of Goyenko et al. (2009) that this measure is the best at approximating the high-frequency effective spread, which is exactly what we want the scaling factor to represent. 3 Data We use data of common stocks listed on the New York Stock Exchange (NYSE) from All data is obtained from the Center for Research in Security Prices (CRSP). We collect daily data and use this data to construct the monthly variables. Using daily data, instead of high frequency data, enables us to investigate a longer sample period. We only include stocks that have sharecode 10 or 11 and do not change ticker symbol, CUSIP or primary exchange over the sample period, as is standard in the literature (see e.g. Chordia et al. (2000), Hasbrouck (2009), and Goyenko et al. (2009)). Days with unusually low volume due to holidays are removed. Our final sample includes 4348 stocks with 16,083,228 stock/day observations. 3.1 Returns The dependent variable in all our regressions is monthly excess returns. All monthly stock returns are adjusted for delisting bias following Shumway (1997). 1 Excess returns ri,t e are calculated above the 1-month Treasure bill rate from Ibbotson Associates as provided on Kenneth French s website. 2 1 For all delistings we use the delisting returns available in CRSP. Only if this return is not available we follow Shumway (1997) and use a return of 30% if the delisting code is 500, 520, , 580, or

12 3.2 Liquidity Variables The liquidity variables are the explanatory variables in the regressions. We consider a variety of monthly liquidity measures which together capture all aspects of liquidity. Only measures that can be constructed over the entire sample period are used, which implies working with liquidity measures that are constructed from daily prices, volumes, or returns. 3 We construct four measures: Roll, Effective Tick, Zeros and Illiquidity Ratio (ILR). The first three variables proxy for the bid-ask spread and the fourth variable is a price impact proxy. All liquidity variables measure illiquidity, i.e. higher estimates correspond to lower liquidity. Roll Roll (1984) shows that trading costs lead to a negative serial correlation in subsequent price changes. In other words, the effective bid-ask spread is inversely related to the covariance between subsequent price changes. The Roll measure is calculated as: 2 Cov ( p i,t,d ; p i,t,d 1 ) if Cov ( p i,t,d ; p i,t,d 1 ) < 0 Roll i,t = 0 if Cov ( p i,t,d ; p i,t,d 1 ) 0 (10) where p i,t,d is the price change for stock i in month t on day d with d = 1, 2,..., D i,t and D i,t the total number of trading days of stock i in month t. Effective Tick Holden (2009) and Goyenko et al. (2009) jointly develop a liquidity measure based on price clustering, which builds on the findings of Harris (1991) and Chrisie and Schultz (1994). If one assumes that the spread size is the only cause of price clustering, observable price clusters can be used to infer the spread. If prices are exclusively quoted on even eight increments ( $ 1 4, $ 1 2, $ 3 4, $1) the spread must be $ 1 4 or larger. However when prices are also 3 Some liquidity measures that we leave out are: the measures developed in Chordia, Huh, and Subrahmanyam (2009) because they require analyst data and the Sadka (2006) measure based on highfrequency data. 11

13 quoted on odd eight increments ( $ 1, $ 3, $ 5, $ ) the spread must be $ 1. If the minimum 8 tick size is $ 1 8, there are J = 4 possible spreads: s 1 = $ 1 8 ; s 2 = $ 1 4 ; s 3 = $ 1 2 ; s 4 = $1. The observed fraction F j of odd $ 1 8, $ 1 4, $ 1 2, $1 prices can be used to estimate the probability γ j of a certain spread s j. The unconstrained probability U i,t,j of the j th spread s j for stock i in month t is: 2F i,t,j j = 1 U i,t,j = 2F i,t,j F i,t,j 1 j = 2, 3,..., J t 1 F i,t,j F i,t,j 1 j = J t, (11) where F i,t,j is the observed fraction of trades on prices corresponding to the j th spread for stock i in month t: F i,t,j = N i,t,j Jt j=1 N i,t,j for j = 1, 2,..., J t. with N i,t,j the number of positive volume days in month t that correspond to the j th spread. The unconstrained probabilities U i,t,j can be below zero or above one, so we add restrictions to make sure the γ i,t,j s are real probabilities: min[max(u i,t,j, 0), 1] j = 1 γ i,t,j = min[max(u i,t,j, 0), 1 j 1 m=1 γ i,t,m] j = 2, 3,..., J t. (12) The effective tick measure is the expected spread scaled by the average price over that month. Eff. Tick i,t = Jt j=1 γ i,t,js i,t,j p i,t. (13) Zeros Lesmond et al. (1999) develop a liquidity measure based on the proportion of days with zero returns. In a day with zero return, the value of trading on information does not exceed transaction costs for an investor on that day. A less liquid asset with high transaction costs is less often traded than a more liquid asset, and the less liquid asset has a higher proportion of days with zero returns. Zeros is measured as: 12

14 Zeros i,t = Di,t d=1 I {r i,t,d =0} D i,t, (14) where I {ri,t,d =0} is an indicator function that takes the value 1 if the return of stock i on day d in month t is zero. Amihud The measure developed in Amihud (2002) proxies for the price impact of a trade, in contrast to the other measures which proxy the bid-ask spread. Price impact refers to the positive relation between transaction volume and price change. The measure is defined as the ratio between the absolute daily return over dollar volume, averaged over the month: ILR i,t = 1 D i,t D i,t d=1 r i,t,d V i,t,d, (15) where V i,t,d is the dollar volume of traded stocks i (in millions) on day d and D i,t is the number of trading days in month t. To be able to compare the ILR over time, we correct it for inflation and the increased size of financial markets. When adjusting the series we cannot use future information that is not available to a real-time investor. We follow Acharya and Pedersen (2005) and Pástor and Stambaugh (2003) and scale the liquidity measure by a ratio of market capitalizations: ILR adj i,t = ILR i,t M m,t 1 M m,1, (16) where ILR i,t is the illiquidity ratio in month t of stock i and M m,t 1 is the market capitalization in month t 1 and M m,1 is the market capitalization in January In the remainder of this paper we drop the superscript and refer to the adjusted Amihud illiquidity ratio with ILR. 13

15 3.3 Size Portfolios In the regressions we use liquidity and excess return series of size portfolios instead of individual stocks. The aggregation of individual stocks into portfolios is necessary to reduce the number of assets in the asset allocation. It is also a good way to deal with individual stocks that enter and leave the sample, due to delistings and IPOs. Before aggregating the individual stocks into portfolios, we filter the individual observations based on the level of the stock price, the number of daily observations within the month, and the availability of size, liquidity and return information. 4 Stock i is included in a portfolio in month t if it satisfies the following criteria: (1) The preceding month-end stock price is between $5 and $1000 ( 5 < p i,t 1,Di,t 1 < 1000 ), where p i,t 1,Di,t 1 is the stock price of stock i on day D i,t 1 in month t 1. This rules out returns that are greatly affected by the minimum tick size. (2) The preceding month-end market capitalization information (M i,t 1 ) is available, which we need for sorting. (3) LIQ i,t 1 is available and is computed using at least 15 daily observations to ensure the quality of the measure. (4) After excluding individual monthly observations that do not satisfy conditions (1) to (3), we winsorize each month across all remaining stocks to the top and bottom 1% of the liquidity variables to avoid outliers. After filtering, we sort the stocks based on previous end-of-month market capitalization in K = 10 size portfolios. The portfolio liquidity and return series are simply the crosssectional averages of the included individual stocks. Directly sorting on the liquidity measure of interest is not possible because we analyse various liquidity measures, which would lead to different rankings and different portfolio components for each measure. If we construct different portfolios for each individual liquidity measure, we will not be able 4 The filtering criteria are in line with Amihud (2002), Pástor and Stambaugh (2003), Acharya and Pedersen (2005), and Ben-Rephael et al. (2010). 14

16 to disentangle whether performance differences are due to the different composition of the risky assets or to the better predictive quality of the liquidity measure. Furthermore, Amihud (2002) finds that the effect between liquidity and expected returns is stronger for small firms than for large firms. This set-up allows for different effects of liquidity on expected returns for various size portfolios. 3.4 Preliminary Statistics Table 1 shows the descriptive statistics for the market portfolio excess returns and the three size portfolios, over the whole sample and three subsamples of each 20 years. In line with Fama and French (1992), we find that average returns for small firms are higher than the average returns for large firms, but so is their volatility. These returns vary considerably over time. The highest returns and lowest volatility are generated in the postwar period ( ), with a market return of 12% and a volatility of 14%. The returns for the and periods are much lower and exhibit higher volatility. This is mainly because there have been several crises between , amid periods of high asset appreciations. [insert Table 1 here] Figure 1 shows a time series of the liquidity measures. The Roll liquidity measure hovers around We find relatively illiquid periods around 1975 (oil crisis), around 2000 (burst of internet bubble), and in (global financial crisis). The Effective Tick liquidity measure has a similar level until 1997 and also shows an illiquid period around In contrast to Roll, the Effective Tick measure clearly shows the two minimum tick size changes in this NYSE data sample, the first in 1997 and the second in These changes in market conditions stress again that it is important to allow the relation between liquidity and conditional expected returns to vary over time by using rolling estimation windows. The Zeros measure shows similar patterns as the Effective Tick measure, again showing the tick size changes. Our last measure, the ILR, shows periods of illiquidity around 1970, around 1975, in the beginning of the 90 s, around 2000, and in

17 [insert Figure 1 here] Table 2 and 3 show the preliminary statistics for the portfolio liquidity series. If liquidity has valuable information for returns and returns of small firms exhibit higher volatility compared to large firms, we would expect that the liquidity of small firms is also more volatile. Table 2 confirms our expectation and shows that for all four liquidity measures the portfolio with small firms is the least liquid and also shows the most variability over time. Furthermore, the spread estimates of Roll and Effective Tick have the same magnitude, while the Zeros measure has higher estimates. The liquidity statistics are split in the same three subsamples as the excess returns. The ILR measure shows that the price impact is the lowest in the post-war period ( ) with a value of and a volatility of In the subsequent periods the price impact is higher (3.258 and respectively) and more volatile (1.752 and respectively). This is mainly because there have been several crises between 1968 and 2008, during which liquidity was low. Of the other three liquidity measures Effective Tick, Roll, and Zeros, only Roll shows a similar but less strong pattern. The averages of Effective Tick and Zeros are the lowest in the most recent period ( ) but at the same time also the most volatile. This indicates that liquidity has increased over time but during the crises in the period there have been shocks to liquidity. [insert Table 2 and 3 here] The preliminary statistics provide some initial evidence of a possible link between liquidity and returns. First, for portfolios where returns are volatile and high, illiquidity is also volatile and high. Second, the volatility of market liquidity seems to coincide with volatile returns. Last, the volatility of returns shows that timing the market and investing at the right time can be worthwhile. 16

18 4 Results Table 4 shows the performance of the liquidity timing strategies. In Panel A the strategies use only return predictions based on a 10 year rolling window. Each row corresponds to the characteristics of a strategy that conditions on a particular liquidity measure. The first column shows the Sharpe ratio (SR), columns (2) - (4) show the relative performance expressed in switching fees and columns (5) - (8) show additional performance statistics of the individual strategies. The SR of the Zeros strategy is 0.38 and is higher than the SR of ILR (0.13), Roll (-0.04), and Effective Tick (-0.07). The columns (2) - (4) of the table show the switching fee in annual basis points. We show only the below diagonal elements because the fees are symmetric. To switch from the ILR strategy to a strategy that conditions on Zeros, a risk-averse investor would pay basis points per year. The switching fees in this table are based on a relative risk aversion coefficient δ of 5. The Zeros strategy performs best as indicated by the positive switching fees in that row. The Effective Tick strategy has the worst performance because a risk-averse investor does not want to pay a positive fee to switch from any other strategy to the Effective Tick strategy. The break-even transaction costs are not computed for the Effective Tick and Roll strategy because their excess returns are negative. The Zeros break-even transaction costs (τ 1 ) are 4.2 basis points, if we assume that transaction costs are the same for all risky assets. When we incorporate the cost differences and express the break-even transaction costs in terms of the most liquid asset, we find τ A = 2.0 basis points. The final two columns show the excess returns and their volatilities. The excess return of Zeros is the highest, 4.57%. The returns of the Effective Tick strategy are the least volatile, 11.41%. The ex-post volatilities in column (8) are above the ex-ante target volatility of σs = 10%. Panel B shows the strategy characteristics when the return predictions are based on the approach of Pesaran and Pick (2011) where we use the average return prediction based on three different rolling windows (5, 10, and 20 years). The Zeros strategy has both the highest excess return (6.00%) and SR (0.51). The switching fees in the row of the Zeros strategy are all positive, hence a risk-averse investor wants to pay a positive fee 17

19 to switch to condition on the Zeros measure. Compared to Panel A, the excess returns in Panel B are higher for three of the four measures and the return volatilities remain similar. Averaging the return predictions of different rolling windows seems to deliver more performance, which could be related to more accurate return forecasts (in line with Pesaran and Pick (2011)). The increase in performance is especially pronounced for the best performing strategies, ILR and Zeros. [insert Table 4 here] Figure 2 shows the cumulative returns of all four strategies based on the 10 year rolling window predictions. The solid lines correspond to the best performing strategies based on the ILR and the Zeros measure. The dark solid line of the Zeros strategy is steadily increasing over the entire sample period, hence the outperformance of this strategy is not caused by a particular period. The biggest downturn is around 1973 but this is offset by an increase around 1970 and The ILR strategy performs quite well until 1995 but thereafter the cumulative returns do not achieve a higher level, this finding is in line with Ben-Rephael et al. (2010). The Roll strategy shows good performance until 1980, however from there onwards the performance is negative. Finally, the Effective Tick measure loses money until Then it sharply increases until the beginning of 20th century, and in the final years of our sample the performance is flat. [insert Figure 2 here] 4.1 Control variables This section deals with the possibility that an omitted variable, which is correlated with our liquidity measures, is driving our results. We show that including control variables lowers the performance of the best liquidity strategy. Hence including more information leads to worse return predictions, which implies that the control variables only add noise. An additional analysis on the individual control variables confirms this. When we condition only on one individual control variable, we do not find a strategy that gets close to our best liquidity strategy in terms of SR. 18

20 Welch and Goyal (2008) examine the predictive ability of many of the return predictors suggested in the existing literature. Examples are the dividend yield, book-to-market ratio, earning price ratio, and interest rate variables. They show that almost none of these variables has out-of-sample predictive power. To correct for the possibility that another variable is driving our results and to show that our liquidity variables contain more information for returns than other predictors, we correct for alternative return predictors. The control variables we include are: dividend yield, earnings price ratio, dividend payout ratio, stock variance, book-to-market ratio, term-spread, default yield spread and inflation. All variables are the same as in Welch and Goyal (2008), that is they are constructed for the market and not for the individual size portfolios. 5 A variable that is available but not included is the net equity expansion variable. The net equity expansion variable is excluded because Baker and Stein (2004) argue that it is (inversely) related to liquidity, hence we do not want to include two liquidity variables at once. When we add the control variables to the model in (2) for conditional expected excess returns, we get: N E t [r k,t+1 r f,t ] = β 0,t + β 1,t LIQ k,t + β 2,t LIQ k,t 1 + γ n,t f n,t + ν k,t, (17) where LIQ k,t is the liquidity of asset k at time t and f n,t are the n = 1, 2,..., N (= 8) control variables. We estimate the β- and γ-parameters in the same way as in our main analysis. Table 5 shows the results when the control variables are included. The set-up of the table is identical to Table 4. All strategies make predictions according to the expression in (17), so using both liquidity information and the control variables. Since more information is included, we would expect that the strategies perform better. A comparison of Table 4 and 5 indicates that only in three cases the SR has increased by including control variables. The most pronounced increases are found for the ILR strategy in Panel A (from 0.13 to 0.33), and for the Effective Tick strategy in Panel B (from to 0.10). In all other 5 We kindly thank Amit Goyal for providing the data on his website agoyal/. 19 n=1

21 cases the performance remains similar or declines. These findings suggest that the control variables only add more noise and do not lead to better return predictions, which leads to the impression that our results are not driven by an ommited variable. [insert Table 5 here] In an additional analysis we investigate the control variables individually, to make sure that not one of them has superior predictive ability compared to the model with only a liquidity term. To be more precise, in this set-up we assume that the expected excess returns are only driven by a constant and one of the control variables. Hence in total we have 8 different strategies of which the results are presented in Table 6. The highest SR in Panel A is 0.21, which is lower than the highest SR (0.38) of a strategy conditioning only on liquidity. In Panel B the best control variable also underperforms the best liquidity strategy, the SR is 0.28 versus The last rows of both Panel A and B show an interesting finding. Here we show the performance of the net equity expansion variable (ntis) that was excluded in the previous analysis because Baker and Stein (2004) argue that it is related to liquidity. In contrast to the 8 other controls, this variables performs really well with a SR of 0.42 in both panels. This supports our idea that liquidity has predictive ability for returns. [insert Table 6 here] To conclude, the addition of alternative return predictors shows that the results are not driven by an omitted variable bias. Furthermore, the control variables are individually not able to deliver the same performance as the liquidity strategies. 4.2 Benchmarks In this section we show if the liquidity timing strategies are related to other strategies, which we refer to as benchmarks. The benchmarks either predict returns by conditioning on other information than liquidity information, or they make no use of return predictions at all. The first group of strategies is estimated in the same way as the liquidity models. 20

22 These strategies condition on the historical average return (Prevailing mean), a constant and a lagged return term (Lagged return), or on a constant and 8 other return predictors (Control variables). The second group does not use time varying return predictions. The strategies belonging to this group are an equally weighted, a volatility timing, and a minimum variance strategy. Some of the benchmarks that we use are related to existing literature. The prevailing mean model is used in Welch and Goyal (2008) and Campbell and Thompson (2008). A simple extension to this prevailing mean model is the addition of a lagged return term. If monthly returns are correlated over time this term would improve the quality of return predictions. The third strategy that makes return predictions each month, conditions on the 8 control variables. Table 7 shows the results of the aforementioned strategies in Panel A and B. As in Table 4, the difference between Panel A and B is the length of the estimation window. The three strategies in Panel C, do not make explicit return predictions. The equally weighted strategy simply attaches the same weight to all available stocks. DeMiguel et al. (2009) show that sample-based mean-variance models have difficulties outperforming such a naive 1/N portfolio. The second is a volatility timing strategy, similar as in Fleming et al. (2001). The variation in weights of this strategy is fully determined by changes in the conditional covariance matrix. The estimate of this matrix is the same as for the liquidity timing strategies, which is simply the sample covariance matrix estimated over the past 10 years of data. To compute the asset weights for this strategy, we still need an estimate of future returns. To eliminate possible predictive power from these predictions, we set them at their unconditional mean. Note that this introduces a look-ahead bias, although this is only in favor of this benchmark. The last is a minimum variance strategy. As the name suggests its objective is to find the portfolio with the lowest possible variance and therefore returns are not part of its optimalization problem. [insert Table 7 here] Table 7 shows that the SR of the benchmarks is lower than the SR of the best per- 21

23 forming strategies in Tables 4 and 5. The prevailing mean model has a low SR, which is in contrast to the findings of Welch and Goyal (2008). This difference can possibly be explained by the differences in data. We use both a shorter data sample and we predict the returns of size portfolios, not of the market as a whole. Furthermore, we make use of a rolling window approach whereas they use an expanding window. The addition of a lagged return term leads to an higher SR in both Panel A and B. Especially when we combine the predictions of three rolling window (Panel B), the SR increases from 0.06 to The third model that is included in both Panel A and B is conditioning on the control variables. In Panel A, the SR is 0.30, which is slightly lower than the SR of 0.33 in Table 5. In Panel B the SR is 0.27, versus an SR of 0.32 when we also include liquidity information. This difference seems small due to the noise introduced by some of the control variables, as shown in the previous section. Panel C in Table 7 shows the performance of the three strategies that do not make return predictions. The good performance of the equally weighted strategy is in line with the findings in DeMiguel et al. (2009). Its SR is equal to 0.28 and its turnover is lower than that of the other strategies, as reflected by the high break-even transaction costs. The results of the volatility timing strategy are worse than the liquidity timing strategies, hence in the main analysis we are really improving on the quality of the return predictions and are not timing volatility. The last row indicates that not predicting returns at all but only minimizing volatility also gives lower performance. The best performing benchmark strategies achieve an SR of 0.30 (Control variables Panel A), 0.27 (Control variables Panel B), and 0.28 (Equally weighted strategy). This is sligthly lower than the best performing liquidity strategies with control variables that have SRs of 0.33 (Panel A) and 0.32 (Panel B), see Table 5. however, it is a lot lower than the best performing liquidity strategies without control variables that have SRs of 0.38 (Panel A in Table 4) and 0.51 (Panel B in Table 4). 22

24 5 Robustness We implement and discuss alternative set-ups to assess the robustness of the results found in the previous section. The robustness analyses covers the weights of the asset allocation and the sensitivity to the parameter values. Weight restrictions Jagannathan and Ma (2003) show that restricting the weights in the asset allocation reduces the estimation error and hence should improve the return predictions and increase the performance of the optimal portfolios. They show that imposing weight restrictions has similar effects as shrinking the return predictions. In that sense, putting more tight restrictions on the investment weights would lead to a more similar performance of the different strategies. Besides the possible performance improvements, weight restrictions are also practical for investors. In this section we restrict the weights in the asset allocation to be between -1 and 2 and see if our ranking is robust to this change. We allow for short-selling in our strategy, but we restrict how much the investor can short-sell. This feature reflects the ability to short-sell of many investors. In this way we prevent the strategies from taking extreme positions in the risky assets. Similar restrictions are used by Thornton and Valente (forthcoming). [insert Table 8 here] Table 8 shows the performance of the dynamic strategies when the weights are restricted to be between -1 and 2. The set-up of this table is comparable to the main results in Table 4, where the weights are unrestricted. As weight restrictions should show similar effects as shrinkage of the predictions, we expect the performance of the strategies to be more similar. The results are not in line with this expectation. In Panel A, where return predictions are based on a 10 year rolling window, the Zeros strategy achieves the highest return (3.84%) and SR (0.33). Compared to the unrestricted case, the SR is slightly lower, 0.33 compared to 0.38 respectively. However, the worst performing strategies, Roll and Effective Tick, still have negative SR s that are even lower than in the 23

25 unrestricted case. Therefore the performance of the strategies has not become more similar by the introduction of weight restrictions. In Panel B, we see similar results. The SR of the Zeros strategy deteriorates, while the SR of the other strategies remains the same. The main point of this analysis is that the ranking of the restricted strategies is the same as in the unrestricted case. Sensitivity Analysis All of our previous results are based on a particular target conditional volatility σs. Furthermore the switching fees depend on the value of Relative Risk Aversion (RRA), which we denote by the parameter δ. We show in this section that the results are robust to changes in these parameters and that our findings apply to investors in general, irrespective of their specific target portfolio volatility or RRA coefficient. [insert Table 9 here] Table 9 shows the results when we change the target conditional volatility from 10% to either 15% or 20%. In all cases the SR s slightly decrease with a 0.02 to 0.06 drop, still the ranking of the strategies remains the same. The switching fees between the strategies are now more extreme because the strategies are more volatile, which a risk-averse investor dislikes. [insert Table 10 here] The sensitivity results of the RRA parameter δ are presented in Table 10. In Panel A the RRA is set to δ = 1 and in Panel B we set it to δ = 10. Similar RRA values are used in Fleming et al. (2001) and Della-Corte et al. (2009). The target conditional volatility is in both panels set to the same value as in the main analysis, σs = 10%. In Panel A we see that an investor is willing to pay a higher switching fee to switch to the high return Zeros strategy, which is also most volatile. For example in the last column of the Zeros row, the switching fee from the Roll strategy to the Zeros strategy increased to basis points per year. This is as expected because a lower parameter 24