Being Locked Up Hurts

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1 Being Locked Up Hurts Frans A. de Roon, Jinqiang Guo, and Jenke R. ter Horst * This version: January 12, 2009 ABSTRACT This paper examines multi-period asset allocation when portolio adjustment is diicult or impossible or some assets due to the existence o lockup periods. Our empirical analysis shows that both unconditional and conditional portolios beneit rom adding hedge unds. More importantly, both unconditional and conditional portolios overestimate their perormance with stocks, bonds and hedge unds when we overlook the eect o a lockup period on perormance. The annualized Sharpe ratio o an unconditional portolio with a three-month hedge und lockup period and monthly rebalancing o stocks and bonds is 1.225, which is signiicantly lower than the annualized Sharpe ratio o the same portolio assuming no lockup period, Investors compensate or the lockup period o hedge unds by making adjustments to their equity holdings. For conditional portolios, the dierence in Sharpe ratios and equity holdings due to a lockup period or hedge unds is also signiicant. Finally, the eect o a lockup period on portolio perormance is less pronounced when investing in unds o unds relative to investing in individual hedge unds, suggesting that unds o unds may help suppress the eect o a lockup period. JEL classiication: G11; G12 Keywords: Multi-period asset allocation; return predictability; hedge unds; lockup period. * First version: August De Roon, Guo and ter Horst are rom Tilburg University (Department o Finance and CentER or Economic and Business Research) and Netspar. Postal Address: Department o Finance, Tilburg University, P.O.Box 90153, 5000 LE, Tilburg, the Netherlands. addresses: F.A.deRoon@uvt.nl (F.A. de Roon), J.Guo@uvt.nl (J.Guo), J.R.terHorst@uvt.nl (J.R. ter Horst). 1

2 I. Introduction An important question or both practitioners and academics in portolio analysis is the multiperiod investment problem. The question is how to rebalance the portolio beore the investment horizon, which is oten complicated by restrictions such as the inability to go short and the act that some positions in illiquid assets cannot be rebalanced easily. In addition, predictability in asset returns has non-trivial rebalancing eects in multiperiod investment problems. There is increasing empirical evidence or predictability in asset returns 1 and or the act that many institutional investors show an increasing allocation to hedge unds, private equity and venture capital (Source: The Russell Investments Survey on Alternative Investing). In this paper, we study the asset allocation problem or an investor when some o the assets such as hedge unds have lockup periods. The analysis extends Brandt and Santa-Clara (2006) where the multi-period investment portolio is solved in a static Markowitz-ramework. We take the perspective o a mean-variance investor who periodically adjusts a portolio that consists o liquid assets and illiquid assets during the investment period. We show that hedge und lockup can be incorporated into the multi-period asset allocation decision by an investor who periodically re-adjusts his portolio. In addition, we ind that lockup periods considered in this paper are empirically highly relevant. Our empirical analysis shows that even with a hedge und lockup, investing in hedge unds can improve portolio outcomes in both the unconditional and conditional context. Moreover, we contribute to the hedge und literature by evaluating investments in hedge unds rom a portolio perspective. The evaluation o hedge und perormance has been studied in several papers, including Agarwal and Naik (2004), Fung, Hsieh, Naik and Ramadorai (2008), Kosowski, Naik, and Teo (2005), Malkiel and Saha (2005). In these studies, the perormance o individual hedge unds or groups o hedge unds is evaluated on the basis o an asset-pricing model. We take the portolio perspective and compute the optimal allocation to dierent asset classes in a portolio. Asset classes such as hedge unds are oten considered attractive investments because o their superior risk-return proile and low correlation with 1 See Campbell (1987), Campbell and Shiller (1988a), (1988b), Cochrane (2007), Fama and French (1988), (1989), Keim and Stambaugh (1986), Hodrick (1992), and Lettau and Ludvigson (2005). 2

3 stocks and bonds. However, investments in hedge unds oten ace more restrictions than investments in stocks and bonds. Most hedge unds impose a lockup period, ranging rom a ew months up to several years, during which investors cannot withdraw their capital. As the lockup periods lengthen, the negative eect o having such illiquid assets grows and may lead to the situation that a portolio o liquid assets and illiquid assets is dominated by a portolio o only liquid assets. As in Brandt and Santa-Clara (2006), we solve a multi-period portolio problem that consists o a set o timing portolios. In a multi-period setting, a timing portolio or a risky asset is a strategy that invests in the risky asset in one period only and in a risk-ree asset in all remaining periods. Thereore, the multi-period asset allocation can be derived by solving the static Markowitz problem on the basis o timing portolios and scaled returns or conditional portolios. We incorporate the constraint o a lockup period or hedge unds to the asset allocation problem. I we assume that the investment horizon is equal to the length o the lockup period, there is no timing portolio or hedge unds, because once an investment in hedge unds is made, the investor has to hold it until the lockup restriction expires. A portolio o stocks, bonds and hedge unds with a lockup will certainly behave dierently than a portolio o the same assets without a lockup, in terms o allocation to dierent assets over time, as well as portolio perormance. Indeed, we ind that the lockup period induces hedge demand or stocks in order to obtain the desired intertemporal equity exposure that cannot be obtained by hedge unds due to the lockup. The paper uses broad market indexes as the proxy or stocks, bonds and hedge unds in the empirical analysis. Our empirical analysis in this paper shows that both unconditional and conditional portolios can be improved upon when adding hedge unds to the stock/bond portolio, but we may overestimate the portolio perormance when we overlook the eect o a lockup on perormance. For instance, the annualized Sharpe ratio or an unconditional portolio with stocks, bonds and hedge unds with a three-month lockup period is 1.225, which is signiicantly higher, both economically and statistically, than the Sharpe ratio o or an unconditional portolio o stocks and bonds only. But when the lockup period is ignored, the investor may believe that the portolio Sharpe ratio with three asset classes is 1.533, which is signiicantly dierent rom the reported at the 1% level. The eect o a lockup period is stronger when the HFRI composite index and the HFRI strategy indexes are considered than 3

4 or the und o unds indices. This suggests that und o und managers may be able to structure their und in such a way that their clients are hurt less by lockup periods. The rest o the paper is organized as ollows. Section II explains the methodology to derive the optimal asset allocation or a mean-variance investor acing lockup periods or some assets. Section III presents the empirical results. Finally, Section IV concludes. II. Asset Allocation with Lockup Periods We consider a conditional asset allocation problem or a risk-averse investor. The portolio consists o liquid assets and illiquid assets. Liquid assets include stocks, bonds, money market instruments, etc., while illiquid assets can be hedge unds, private equity and venture capital investment. The investor can change the allocation to liquid assets every period, but adjusting allocations to illiquid assets is diicult i not impossible. The orm o illiquidity in this paper is restricted to the situation in which a lockup period is imposed or investments in hedge unds. A. Multi-period Asset Allocation with Lockup Constraints We irst illustrate the two-period asset allocation problem with lockup constraints, and generalize the method or longer period setting. There are K 1 liquid risky assets and K 2 illiquid risky assets with a lockup period equal to L. For simplicity, the investment horizon has the same length as the lockup period. Consider the two-period quadratic utility optimization problem or an investor: Et rt γ 2 p p 2 max ( ) t + 2 rt t + 2, (1) where p rt t + 2 is the excess portolio return over two periods and γ is the coeicient o risk aversion. Denote portolio weights on liquid assets and illiquid assets at time t by w, and z t w,, x t respectively. In addition, denote the one-period gross return at time t on the risk-ree asset by R t, and gross returns o illiquid assets by x Rt 1 +. The vector r t + 1 contains one-period excess returns o liquid risky assets. The two-period excess return o the portolio with only liquid assets in Brandt and Santa-Clara (2006) is: ( R + w r )( R + w r ) R R p rt t + 2 = t t t + 1 t + 1 t + 1 t + 2 t t + 1 4

5 Because r t + 1 and t + 2 w ( R ) ( ) ( )( t rt + wt Rt rt + wt rt wt + 1r + 2 ) ( R r ) + w ( R r ) = t t w. (2) t t + 1 t + 1 t + 1 t t + 2 t t + 1 t + 1 t + 2 r are excess returns, the product ( r )( w r ) w is very small at short horizons, so the excess portolio return over the two period is approximately the sum o w ( R r ) and ( R r ) t t +1 t +1 t +1 t t +2 w. z, t t + 1 t + 1 z, t + 1 t t + 2 Brandt and Santa-Clara (2006) interpret w ( R r ) and ( R r ) z, t t + 1 t + 1 portolios. First, ( R r ) w as timing w is the two-period excess return rom investing in risky assets at z, t + 1 t t + 2 time t and then investing in the risk-ree asset. Second, ( R r ) w is the two-period excess return rom investing in the risk-ree asset at time t and then investing in risky assets. When a portolio includes assets with a two-period lockup, the two-period portolio excess return takes the orm o the ollowing: x ( Rt + w z, trt + 1)( Rt w z, t + 1rt + 2 ) Rt Rt wx, trt x ( R r ) + w ( R r ) + w r p rt t + 2 = t + 2. (3) w z, t t + 1 t + 1 z, t + 1 t t + 2 x, t t t + 2 where is the K2 dimensional vector o excess returns o illiquid assets, and or each x rt t + 2 illiquid asset, = or i = 1,2,..., K2. For investment in illiquid assets, x x x ri, t t + 2 Ri, t + 1Ri, t + 2 Rt Rt + 1 one dollar will grow by x x Ri, t 1Ri, t and ater paying back the risk-ree loan, the two-period excess return on illiquid assets is x x Ri, t 1Ri, t + 2 Rt Rt + 1 assets since they are locked up over the two periods. +. There is no timing portolio or illiquid The S dimensional vector o z t is a set o state variables available to investors at time t. The portolio weights are assumed to be linear in state variables. For liquid risky assets, = and wz t 1 β 2z 1, (4) wz, t β1zt =, + t + where the matrix β 1 and β 2 both have a dimension o where β x is a K 2 S matrix. The two-period portolio excess return in (3) then becomes Using some linear algebra, we ind w K 1 S. For illiquid assets, we have x, t = β xzt, (5) = β β. (6) x ( z ) ( R r ) + ( β z ) ( R r ) + ( z ) r p rt t t t + 1 t t + 1 t t + 2 x t t t + 2 5

6 β, (7) ( z ) ( R r ) = vec( β ) R ( z r ) 1 t t + 1 t t + 1 t t + 1 β, (8) ( z ) ( R r ) = vec( β ) R ( z r ) 2 t + 1 t t t t + 1 t + 2 x x ( xzt ) rt t + 2 = vec( β x ) ( zt rt t + 2 ) where vec( β j ) is a vector that stacks the columns o the matrix j β. (9) β, j = 1,2, x, and is the Kronecker product. The investment menu becomes a set o scaled returns or expanded asset return space, ~ r t + 1 = zt rt + 1, ~ r t + 2 = zt + 1 rt + 2 and ~ x x rt t + 2 zt rt t + 2 =. The investor s problem is to choose a set o unconditional weights to maximize the multi-period mean-variance utility: max w ~ E w~ ~ γ r + w~ r~ t t t t t + r~ 2 2 t t w~, (10) where the unconditional portolio weights is w = ( vec( ) vec( β ) vec( β ) ) ~ 1 2 x β, and ~ x r = ( ) ( ) ( ) t t + R ~ t + rt + R ~ t r ~ t + 2 rt t + 2. The portolio weights w ~ that maximize the conditional expected utility at all dates t should also maximize the unconditional expected utility. The optimization still makes use o the static Markowitz approach on the basis o the unconditional moments o scaled returns. The optimal unconditional portolio weights are: ~ 1 [ ~ 1 w Var r ] E[ ~ r ]. (11) γ = t t + 2 t t + 2 The sample analogue o the population moments in the equation (11) leads to a consistent estimate o the unconditional portolio weights w ~. It is a vector o the length ( S K S ) 2 +, K1 2 and we can recover the optimal portolio weights on K 1 risky assets at time t and t + 1, w z, t and w z, t + 1 as ( w~ i w~ i K w~ ( ) ( + ) ( i ( S 1) K ) z 1 + t i,2,, ( w~ ~ ~ ( i+ K S ) w( i+ K + K S ) w( i+ ( S 1) K + K S ) ) z + 1 =, = 1 K1. (12) i w z, t 1 ) w, i = 1,2,, K1. (13) i z, t + 1 = t For illiquid assets, the portolio weights at time t can be derived in the same way as those o liquid risky assets. ( w~ i K S w~ i K K S w~ ( + 2 ) ( ) ( i + ( S 1) K + 2 K S ) z t =, i = 1,2,, K2. (14) i w x, t ) However, the static optimal portolio weights in (11) do not give direct solutions to the portolio weights o illiquid assets at time t + 1. We can normalize the initial portolio value to 6

7 one and the portolio weight o illiquid assets i is the ratio o its value to the portolio value at time t + 1. We can generalize the method above to the L-period asset allocation problem with lockup constraints on certain risky assets. The optimal unconditional portolio weights are 1 [ ~ = Var r ] E[ r ~ ], (15) γ ~ 1 w t t + L t t + L where ~ is a set o timing portolios with scaled returns o liquid assets and L-period r t t + L excess returns o illiquid assets scaled by the inormation set z t. The solution in (15) may produce a negative weight or illiquid assets. In reality, while shorting stocks and bonds is relatively easy, shorting illiquid assets is either too costly or impossible. For instance, investors cannot short hedge unds or transer their stakes in hedge unds to other investors. In this case, investors should add nonnegative constraint on portolio weights o illiquid assets to the analysis. B. Econometric Issues We estimate the set o portolio weights in (15) by sample analogue. In addition, we can test whether state variables are jointly signiicant by a Wald test or F test. The construction o the estimated covariance matrix o w ~ and the test procedure ollow the method by Britten-Jones (1999). Given a time-series sample o asset returns, the estimation o w ~ can be sensitive to the choice o starting date o the sample. Speciically, or a lockup period o L, we have L choices o starting date, and the resulting L sets o the estimated w ~ are all consistent asymptotically. Following Jegadeesh and Titman (1993), and Rouwenhorst (1998), we consider L strategies that contribute equally to a composite portolio. Speciically, at the start o each period, the composite portolio consists o L sub-portolios. Each sub-portolio invests optimally according to one set o estimated w ~ on the basis o an estimation window. For example, suppose the lockup period is two-month and the sample data consists o ten-year monthly asset returns. We can estimate w ~ using two dierent windows: one starting one month later than another in the data. The composite portolio invests one hal according to the irst set o estimated w ~ and one hal according to the second set o estimated w ~. The method is comparable to that in Jegadeesh and Titman (1993), and Rouwenhorst (1998). In those two 7

8 papers, they report the monthly average return o K strategies or K-month holding period in order to evaluate the relative strength portolios. III. Empirical Analysis A. Data For hedge unds, we obtain various hedge und indexes and und o unds indexes rom Hedge Fund Research, Inc. (HFR, Inc.). A und o unds or hedge und o unds is a hedge und that invests with multiple managers o hedge unds or managed accounts. Since a und o unds holds a diversiied portolio o hedge unds, it lowers the risk o investing with an individual hedge und manager and gives access to hedge unds that are closed to new money (Nicholas (2004)). The length o the lockup period depends on the liquidity o the underlying individual hedge unds in the und o unds. Some unds o unds require no lockup periods, but a lockup period o 3 months up to 2 years is not uncommon. An individual U.S. hedge und typically requires a one-year lockup period plus a notice period ranging rom 1 month to 3 months. In contrast, less than 40 percent o unds o unds require a lockup period, and among those unds o unds that do, about two third o them set a lockup period o 6 month or longer (Nicholas (2004)). The HFRI Fund o Funds Composite Index (HFRIFoF) is an equally-weighted index that includes over 800 unds o hedge unds with at least USD 50 Million under management. Monthly returns are net o all ees. HFR, Inc. also provides our equally-weighted sub-indexes according to the classiication o und o unds strategies: Conservative, Diversiied, Market Deensive, and Strategic. A und o unds is classiied as Conservative i it tends to invest in unds with conservative strategies such as Equity Market Neutral, Fixed Income Arbitrage, etc. that exhibit low historical volatilities. A und o unds is Diversiied i it invests with various strategies/managers and exhibits perormance close to that o the HFRIFoF composite index. A Market Deensive und o unds invests in unds with short-biased strategies and exhibits negative correlation with the equity market benchmark. Finally, a Strategic und o unds tends to invest in hedge unds with more opportunistic strategies and exhibits greater volatility relative to the HFRIFoF composite index. For the composite index based on individual hedge unds, we use the HFRI Fund Weighted Composite Index (HFRI), which is an equallyweighted index based on more than 2000 individual hedge unds. Naturally, the HFRI index 8

9 excludes unds o unds to prevent double counting o perormance igures. In addition, HFR, Inc. classiies individual hedge unds into our primary strategies: Equity Hedge, Event-Driven, Macro, and Relative Value. Each primary strategy includes several sub-strategies. HFR, Inc. provides detailed descriptions o primary and sub-strategies in its products and website. From CRSP, we obtain the value-weighted NYSE index as the proxy or stocks, the 1-month Treasury bill as the proxy or the risk-ree asset, and the Fama Bond Portolio (Treasuries) with maturities greater than 10 years as the proxy or bonds. We construct quarterly returns rom monthly index returns o stocks, bonds, and hedge unds. The relatively short sample period or the hedge und data limits the empirical analysis to the sample period rom December 1989 through December Table 1 gives summary statistics o risky asset returns. Over the sample period, the average return and volatility o stocks is 11.4% and 12.6%, respectively. Bonds have an average return o 8.5% and volatility o 7.9%, but the Sharpe ratio o bonds is only slightly lower than that o stocks. The HFRIFoF composite index has a lower average return (9.7%) and volatility (5.5%) compared to stocks, and a Sharpe ratio o 1.033, which is almost twice as large as the Sharpe ratio o stocks or bonds. The HFRIFoF Conservative index has the lowest volatility among all und o unds indexes, consistent with the style classiication. The HFRIFoF Diversiied index shows a similar average return and volatility compared to the composite index. Although average returns and volatilities dier among our HFRIFoF strategy indexes, their Sharpe ratios are not too ar away rom each other. In contrast, the HFRI Relative Value shows a Sharpe ratio that is higher than the other three HFRI strategy indexes and the HFRI composite index, mainly due to its low volatility. Furthermore, the average returns o the HFRI composite index and the HFRI strategy indexes are quite high compared to stocks, bonds and und o unds indexes. The average return o the HFRI composite index is 13.2%, which is 3.5% higher than the average return o the HFRIFoF composite index, while the volatility o the HFRI composite index is about 6.6%, only 1.1% greater than that o the HFRIFoF composite index. The dierence in Sharpe ratios o the two composite indexes is 0.35, so it seems that unds o unds oer lower risk-adjusted returns relative to the aggregate individual hedge unds. The double ees structure o und o unds investments can account or some o the dierence in risk-adjusted returns, but some researchers argue that the greater survivorship bias underlying individual hedge unds may cause the reported under-perormance o unds o unds (See Fung and Hsieh (2000)). 9

10 We obtain the data o state variables rom CRSP. We include two state variables that potentially help predict asset returns: the market dividend-price ratio and the short-term interest rate. For the short-term interest rate, the annualized 1-month Treasury bill is used. The market dividend price ratio is based on the value-weighted NYSE equity index, calculated as the ratio o sum o dividends over past twelve months to the NYSE index level. Many other state variables that potentially help predict asset returns are available, such as smooth earning-price ratio, consumption-wealth ratio, ROE, inlation, and potentially many others. 2 For equity returns, market dividend yields work reasonably well as a predictor, especially at long horizons (see, among others, Campbell and Viceira, (1999), (2002); Cochrane, (2007)). However, Ang and Bekaert (2007) argue that the predictive power o the dividend yield is not robust across sample periods or countries. A univariate dividend yield regression provides weak evidence o predictability when the 1990s bull market period is included. On the other hand, Ang and Bekaert (2007) ind that the short rate is the most robust predictable variable or predicting excess returns at short horizons. Using both the short rate and dividend yield in regressions improves the it, with the short rate dominating the dividend yield. Figure 1 plots the time series o state variables rom December 1989 to December The market dividend price ratio is closely linked to the ups and downs o the U.S. stock market, so the long bull market in 1990s result in a downward trend o the dividend price ratio during this period. The short-term interest rate shows a pattern that is driven by the U.S. business cycle. B. Unconditional Asset Allocation with a Three-Month Hedge Fund Lockup Period Section B.1 reports the portolio weights o the unconditional asset allocation with a threemonth hedge und lockup period. We are interested in the dierence in the allocations to stocks and bonds when hedge unds are added to the portolio, as well as the changes in investment patterns over the three-month investment horizon. We investigate the extent to which the total demand or stocks and bonds in the portolio o stocks, bonds and hedge unds are caused by the speculative demand (Markowitz demand) and the hedge demand due to investments in hedge unds with a three-month lockup period. Section B.2 compares the perormance o 2. Goyal and Welch (2007) and Campbell and Thompson (2007) include a comprehensive list o these variables along with some others as predictors used in predictability studies. 10

11 unconditional portolios with various hedge und indexes. The ocus is to test the dierence in Sharpe ratios o the three-asset portolio with a lockup period and the portolio without a lockup period. In addition, we can test whether adding hedge unds beneits improve the Sharpe ratio o the portolio. B.1. Portolio Weights Table 2 reports the results or the unconditional asset allocations with the three-month hedge und lockup period. The degree o risk aversion o the investor is 10 or all analyses. The estimated parameters, portolio perormance and test statistics are the monthly averages o three-month rolling windows. We can think o this as the result o a strategy that always invests 1/3 o wealth or three months, starting every month, just as in jegadeesh and Titman (1993) and Rouwenhorst (1998) (see Section II.B. Econometric Issues). The t-statistics or the meanvariance portolio weights are based on Britten-Jones (1999). Results or the unconditional asset allocations in Table 2 show that portolio weights vary in a systematic way over the investment horizon. The variation in unconditional portolio weights is caused by the presence o timing portolios. To start out, in the portolio o stocks and bonds only, the allocations to stocks and bonds display distinct patterns over the investment horizon. Over the three months, the allocations to stocks decrease while the allocations to bonds increase. Thus, investors start with a relatively risky portolio and gradually adjust their portolio holdings in order to obtain a less risky portolio by the end o their investment horizon. This is in line with lie-cycle unds where equity exposure decreases over time, due to the autocorrelation in stock returns. Adding hedge unds to the portolio o stocks and bonds reduces the allocation to stocks and increases the allocation to bonds or each month, irrespective o whether or not a hedge und lockup period exists. This relects the act that investing in hedge unds leads to bigger equity exposure relative to bond exposure. Adding hedge unds to the portolio changes the pattern o portolio weights o stocks over the investment horizon, while the pattern o portolio weights o bonds remain monotonically increasing. For example, inclusion o the HFRIFoF to the portolio o stocks and bonds will change pattern o investment in stocks over the three-month rom being monotonically decreasing to being a hump shape, and inclusion o the HFRI will reverse the pattern to be monotonically increasing. 11

12 To urther investigate these changes in the pattern o the portolio weights o stocks, we calculate the Markowitz (or pure speculative) demands and the hedge demands or stocks and bonds in the three-asset portolio with a hedge und lockup period. Investing in a hedge und when there is a lockup period, basically leads to an exogenously given exposure to the hedge und ater the irst period, which may induce a hedge demand or stocks and bonds. The optimal investment in stocks and bonds in the three-asset portolio is the sum o the Markowitz demands and the hedge demand. The Markowitz demand is the optimal portolio weights o stocks and bonds when the investment menu includes stocks and bonds only. The hedge demand arises because the investor wants to hedge the changes in the value o hedge und investment, which is locked up or three months. A negative hedge demand or stocks implies that the overall allocation to stocks will be lower than it would be in the portolio consisting o only stocks and bonds. Table 3 shows the optimal demand or stocks and bonds as the combination o the Markowitz demand 3 and the hedge demand, using either the HFRIFoF or the HFRI composite index as the proxy or hedge unds in the three-asset portolio with lockup restriction. The hedge demand is the product o the optimal demand or hedge unds at time t and the slope coeicients rom the regression o three-month excess returns o hedge unds on a constant and returns o the timing portolios o stocks and bonds: s s s s,1( Rt + 1Rt + 2rt + 1) + bs,2( Rt Rt + 2rt + 2 ) bs,3( Rt Rt + 1rt ) b b b ( R R r ) + b ( R R r ) + b ( R R r ) + ε x rt t b = α + bb 1 t + 1 t + 2 t + 1 b,2 t t + 2 t + 2 b,3 t t + 1 t + 3,, (16) We ind that or each month, the hedge demand is negative or stocks and positive or bonds. Furthermore, the hedge demand or stocks is most negative in the beginning and increases over time, which results in a pattern o the optimal demands dierent rom the Markowitz demands or stocks. For instance, adding the HFRIFoF to the portolio gives rise to a small allocation to stocks relative to the Markowitz demand in the irst month (19% vs. 60%). The Markowitz demand decreases to 58% in the second month, while the total demand increases to 30% due to an increase in the hedge demand. The total demand or stocks decreases to 27%, as the increase in the hedge demand is more than oset by the decrease in the Markowitz demand. For bond investment in the three-asset portolio, the changes in portolio weights are dominated by the 3 The Markowitz demands or stocks and bonds in the three-asset allocation are the optimal allocations to stocks and bonds in the two-asset allocation, i.e. the portolio weights o stocks and bonds in column 2 o Table 2. The dierence between the total demands or stocks and bonds in the three-asset allocation (column 3 o Table 2) and the total demands or stocks and bonds in the two-asset allocation is the hedge demand. 12

13 changes in the Markowitz demands. The hedge demands or bonds are relatively small; the changes over the three-month horizon are not large enough to reverse the patterns o total investment in bonds. The patterns o investment in stocks dier with dierent hedge und indexes used as the proxy. Adding the HFRI to the portolio leads to negative portolio weights or stocks in all three months, and they monotonically increase rom 57% in the irst month to 31% in the last month. Adding the HFRI to the portolio results in the allocation to hedge unds being almost twice as large as the allocation to hedge unds when the proxy or hedge unds is the HFRIFoF. Since the hedge demands or stocks and bonds depend on the allocation to hedge unds, and the covariance between stock/bond returns and hedge und returns, they are larger (in absolute value or magnitude) in the portolio o stocks, bonds and HFRI than those in the portolio o stocks, bonds and HFRIFoF. In act, the hedge demands are so much larger than the Markowitz demands or stocks when the HFRI is included in the portolio that they lead to negative portolio weights o stocks. The patterns o investment in bonds, on the other hand, are similar. Adding either the HFRI or the HFRIFoF would not change the trend o investment in bonds. In all cases, total allocations to bonds increases over the three-month period. The hedge demands or bonds are larger in the portolio o stocks, bonds and HFRI than those in the portolio o stocks, bonds and HFRIFoF, simply because a large portolio weight o the HFRI. However, the hedge demands are small relative to the Markowitz demands, and the variation in the hedge demands is not large enough to make a dierence in the trend o total investment in bonds. For instance, the Markowitz demand or bonds is 55%, 74% and 98% in the irst, second and third month. The corresponding hedge demand or bonds in the portolio o stocks, bonds and HFRIFoF is 11%, 7% and 9% (24%, 13% and 27% in the portolio o stocks, bonds and HFRI). The variation in the Markowitz demands over time is 19% rom month 1 to month 2, and 24% rom month 2 to month 3. In contrast, the variation in the hedge demands is less than 4% (14% in the portolio o stocks, bonds and HFRI). B.2. Portolio Eiciency An investor is interested in knowing the potential beneits rom adding hedge unds to his portolio. Table 4 reports the perormance o unconditional portolios o stocks, bonds and 13

14 hedge unds. The p-values, as they appear in the table, are calculated based on the averaged test statistics over the three overlapping samples. In each case, a dierent hedge und index is used as the proxy. The mean excess return and volatility o the two-asset portolio is 8.1% and 8.9%, respectively. The three-asset portolios with or without a lockup period have noticeably higher mean excess returns and volatilities. Moreover, the Sharpe ratios o the three-asset portolios are much higher than the two-asset portolio. The dierence in mean returns, volatilities and Sharpe ratios o the three-asset portolios is large when dierent hedge und indexes are used. For instance, the portolio o stocks, bonds and HFRIFoF with a lockup has a mean excess return o 14.8% with a volatility o 12.1%, compared to a mean excess return o 23.7% and a volatility o 15.3% or the portolio o stocks, bonds and HFRI. The Sharpe ratio o the irst portolio above is 1.225, lower than the Sharpe ratio o o the second portolio. The test o portolio eiciency ollows Jobson and Korkie (1982) and De Roon and Nijman (2001). Denote the sample Sharpe ratio or the benchmark portolio Sharpe ratio or the portolio o test assets r and benchmark assets Wald statistic o the Sharpe ratio test is: p r by θˆ p, and the sample p r jointly, by θˆ. The 2 2 ˆ θ ˆ θ p 2 ξw T = ~ χ ˆ2 K (17) 1 θ + p where T is the sample size and K is the degrees o reedom. The degrees o reedom are the dierence in the number o parameters between the two asset allocations. From the p-values o the Sharpe ratio test in Table 4, the dierence in Sharpe ratios between the two-asset portolio and each three-asset portolio is statistically signiicant at 1% level, indicating that the twoasset portolio can be signiicantly improved upon by adding hedge unds. An investor who ignores the existence o a hedge und lockup period will get a wrong estimate o portolio perormance. From Table 2 and Table 3, we know that the existence o a three-month lockup period or hedge unds makes a dierence in the allocations to stocks, bonds and hedge unds over the investment horizon. Assuming no hedge und lockup period will produce a portolio o stocks, bonds and hedge unds with a higher mean excess return and volatility, as well as a higher Sharpe ratio, relative to a portolio o stocks, bonds and hedge unds with a lockup period o three months, regardless o the choice o a hedge und proxy. As shown in Table 4, the dierence in Sharpe ratios between the three-asset portolio with hedge und lockup period and the three-asset portolio assuming no hedge und lockup is large and 14

15 statistically signiicant (except or the case when the HFRI Relative Value as the hedge und proxy). For instance, the portolio o stocks, bonds and HFRIFoF with a lockup has the Sharpe ratio o 1.225, but the Sharpe ratio is i the three-month lockup period is ignored. The dierence is statistically signiicant at 1% level. Similarly, or the portolio o stocks, bonds and HFRI, the dierence in Sharpe ratios is (1.549 vs ). Hence, overlooking the existence o a hedge und lockup period may overstate the perormance o a three-asset portolio. C. Conditional Asset Allocation with a Three-Month Hedge Fund Lockup Period This section reports the portolio weights and perormance o various conditional asset allocations. Asset allocations are conditional on a set o state variables, i.e. the market dividend price ratio and the short-term interest rate. We analyze the total demand or stocks and bonds in the conditional portolio o stocks, bonds and hedge unds, as a combination o the speculative demand (Markowitz demand) and the hedge demand due to investments in hedge unds with a three-month lockup period, just like what we did in the previous subsection. We test the dierence in Sharpe ratios o the three-asset portolio with a lockup period and the portolio without a lockup period. Furthermore, we test whether using conditional portolio policy improves the eiciency o the unconditional portolio. C.1. Portolio Decision Conditional on State Variables Table 5 gives the correlation matrix or risky asset returns and lagged state variables. For most hedge und indexes, their correlations to the market dividend price ratio or the short rate are stronger than the correlations o stocks and bonds to each o the two state variables. Notice that the correlation o stock returns to the HFRIFoF composite index returns is high, but lower than the correlation o stock returns to the HFRI composite index returns. This implies that the HFRIFoF is a better diversiier than the HFRI does. On the other hand, a high correlation o stock returns to hedge und returns indicates that hedge unds and unds o unds have large equity exposures. Table 6 reports the results o the conditional asset allocation. State variables are standardized so the intercepts are the average allocations over the sample period. The average allocations to 15

16 stocks and bonds change as the time passes. For the two-asset allocation, the average allocations to stocks are not too dierent among three periods (97%, 93% and 110%), while the average allocation to bonds is 33% in the irst month and increases sharply rom 37% in month 2 to 100% in month 3. This implies that bonds become relatively important in the portolio as the investment horizon approaches. In addition, or all three-asset portolios, the average portolio weights o bonds appear to be increasing over time and the increase is the largest rom month 2 to month 3, a similar pattern to what we ound or the two-asset portolio. The average allocations to stocks in the three-asset allocations with a three-month hedge und lockup period increase over time. Table 7 shows the decomposition o the total demand or stocks and bonds as the combination o the Markowitz demand and the hedge demand. Adding the HFRIFoF to the portolio induces an average hedge demand or stocks in the irst month o -43%. The average hedge demand or stocks in the second or third month is close to zero. The average hedge demand or bonds in each month is negative, ranging rom 38% in month 1 to 7% in month 3. It seems that adding the HFRIFoF to the portolio suppresses the total allocations to stocks and bonds over the three-month horizon. However, the pattern and magnitude o average hedge demands when the HFRI is added to the portolio is quite dierent. The average hedge demands or stocks are large and negative in each month (-171%, -128%, and 110%), dragging down the total demands or stocks in the three-asset portolio. On the other hand, the average hedge demands or bonds are positive in each month (20%, 36%, and 55%). Hence, adding the HFRI to the portolio leads to a sharp reduction in the allocations to stocks and an increase in the allocations to bonds over time. Changes in state variables lead to changes in portolio weights o conditional portolios. The sign o coeicients on state variables in determining portolio weights changes over time and across dierent portolios. For instance, in the two-asset allocation, the change in the market dividend price ratio is positively related to the allocations to stocks in the irst and third month, but not in the second month. The change in the short rate is negatively related to the allocations to stocks in the irst and third month, but is positively related in the second month. Such changes in signs over time exist or other portolios. For three-asset portolios, the signs o coeicients on both state variables in determining bonds weights show consistency over the three periods. Moreover, at a given month, the sign and magnitude o the slope coeicients on state variables is dierent across dierent portolios. Thereore, the investor s response to 16

17 changes in state variables will depend on whether hedge unds are added to the portolio, which hedge und index is used as the proxy, and which month the investment decision is made. C.2. Conditional Allocation vs. Unconditional Allocation A comparison o the conditional asset allocation and the unconditional asset allocation reveals some interesting results. Relative to the unconditional two-asset allocation, the conditional portolio allocates more to stocks in every period. This relects the possibility o portolio adjustment in response to changing market conditions. In addition, the conditional two-asset portolio allocates less to bonds in the irst two months and more in the third month, compared to the unconditional asset allocation to bonds in the same periods. It appears that the ability to adjust portolio weights according to changes in state variables induces the investor to allocate more aggressively to stocks and less to bonds overall. When the conditional portolios include hedge unds with a three-month lockup period, the average allocations to hedge unds become larger compared to the unconditional allocations to hedge unds. While the hedge demands or stocks in the portolio o stocks, bonds and HFRIFoF reduce the total demands or stocks under the unconditional asset allocation in every period, the average hedge demand or stocks is close to zero in the second and third month under the conditional asset allocation. Furthermore, the average hedge demands or bonds become negative or the conditional three-asset portolio, in contrast with the positive hedge demands in the unconditional portolio. Adding the HFRI to the conditional portolio tells a dierent story. Even though the average Markowitz demand or stocks is higher in the conditional portolio, the average hedge demand or stocks in each period is more negative in the conditional portolio (-171%, -128%, and -110%). The net eect is that the conditional portolio has lower average allocations to stocks in the irst month, but higher in the remaining two periods. The average hedge demands or bonds are positive and much larger than the hedge demands or bonds in the unconditional portolio in the second month (36% vs. 13%) and the third month (55% vs. 27%). C.3. Portolio Eiciency 17

18 Table 8 shows the perormance o conditional portolios using dierent hedge und indexes as the proxy or investments in hedge unds. It also reports the results o the Wald test o the null hypothesis that all slope coeicients o the market dividend price ratio and the short-term interest rate are jointly equal to zero. Four questions arise. First o all, is the conditional twoasset portolio mean-variance eicient or does adding hedge unds to the conditional portolio improve the portolio eiciency? Second, is the unconditional portolios mean-variance eicient? Third, is it possible that while the changes in state variables help adjustment in portolio weights o stocks, bonds and hedge unds, the investor does not beneit rom using those state variables in terms o the portolio perormance? Finally, what dierence does a three-month hedge und lockup period make in terms o the portolio perormance? We can use the Sharpe ratio test to determine the portolio eiciency o the conditional twoasset allocation relative to the three-asset allocation, and the unconditional portolios relative to the conditional portolios. Ten dierent hedge und indexes are used as the proxy or hedge unds, including our strategy indexes or HFRI und o unds, and our strategy indexes or HFRI hedge unds. For each case, we have p-values rom our Sharpe ratio tests. For instance, using the HFRIFoF composite index as the proxy, the Sharpe ratio o the conditional threeasset portolio with a lockup period and without a lockup period is and 2.158, respectively. The p-value (0.001) to the right o the Sharpe ratio o the conditional three-asset with a lockup period is based on the Sharpe ratio test o the conditional two-asset allocation as the benchmark portolio (i.e. we test the conditional two-asset portolio vs. the conditional three-asset portolio with a lockup period). The p-value (0.087) next to the Sharpe ratio o the conditional three-asset without a lockup period is based on the Sharpe ratio test o the dierence in Sharpe ratios o two conditional three-asset portolios, i.e. the portolio with a lockup period vs. the portolio without a lockup period. The p-value (0.021) and (0.084) under the Sharpe ratio o the conditional three-asset portolio with a lockup period and the conditional three-asset portolio without a lockup period are based on the Sharpe ratio test o the unconditional portolios vs. conditional portolios, with or without a lockup period, respectively. For all cases, the dierence in Sharpe ratios o the conditional three-asset portolio with a three-month lockup period and the conditional two-asset portolio is signiicant at the 5% signiicance level (the dierence is signiicant at the 1% level or 9 out o 10 cases). Thus, we conclude that the conditional portolio o stocks and bonds only is not mean-variance eicient. 18

19 The investor should add hedge unds to the portolio even though there is a lockup period o three months. The two p-values under the Sharpe ratios o three-asset portolios come rom the Sharpe ratio test o the unconditional asset allocation vs. the conditional asset allocation, with and without a three-month lockup period. For the three-asset portolios with a lockup period and using any und o unds index, the Sharpe ratio o the conditional portolio is signiicantly larger than the unconditional portolio at the 5% level. Adding the HFRI or the HFRI strategy index to the portolio, the dierence in the Sharpe ratios is signiicant at the 5% level in one case (when using HFRI Relative Value as the proxy). When no lockup period is assumed, the dierence in the Sharpe ratios o the conditional three-asset portolio and the unconditional three-asset portolio is signiicantly at the 5% level or 3 out o 10 cases. Interestingly, we can reject the null hypothesis that all slope coeicients o the market dividend price ratio and the short-term interest rate are jointly equal to zero at the 1% signiicance level or all conditional portolios, on the basis o the p-values o the Wald test in Table 8. In other words, changes in state variables lead to signiicant changes in portolio weights o risky assets, and yet the conditional portolios are not necessarily better than the unconditional portolios in terms o portolio perormance. From the Sharpe ratio test o the unconditional portolio vs. the conditional portolio above, or many unconditional three-asset portolios, we cannot reject their eiciency. To answer the last question, we perorm the Sharpe ratio test o the conditional three-asset allocation with a three-month hedge und lockup period against the conditional three-asset allocation assuming no lockup period, in order to assess the eect o a hedge und lockup on portolio eiciency under conditional inormation. When the HFRIFoF and our HFRIFoF strategy indexes are considered as the proxy or hedge unds, the dierence in the Sharpe ratios o the three-asset portolios with a lockup period and the three-asset portolio without a lockup period is not signiicant or 4 cases. The dierence is signiicant at the 5% level only when the HFRIFoF Conservative is used as the hedge und proxy in the three-asset portolio. When the HFRI composite index or the HFRI strategy index is used as the proxy or hedge unds, the dierence is much larger and signiicant at the 5% level regardless o the choice o hedge und proxy. Thereore, having a three-month lockup period implies a signiicant lower Sharpe ratio o the three-asset portolio o stocks, bonds, HFRI (or HFRI strategy). 19

20 As the results o the Sharpe ratio test indicate, a three-month lockup period is less likely to aect the perormance o the three-asset portolios o stocks, bonds and unds o unds. Funds o unds seem to be better able to suppress the eect o lockup periods on conditional portolios perormance than individual hedge unds do. Possible explanations which require urther research include: a und o unds typically has more requent subscriptions and can use new money to pay o redemption requests. Moreover, a und o unds manager can actively manage the lockup periods o the underlying individual hedge unds, such that each und has a dierent lockup expiration date. In this way, a und o unds can still invest in many individual hedge unds with long lockup periods, while imposes a shorter lockup period or und o unds investors. From a conditional portolio investor s perspective, when he decides to add unds o unds to the portolio o stocks and bonds, a three-month hedge und lockup period should not cause great concerns on the basis o portolio outcomes. In contrast, a conditional portolio investor who tries to add individual hedge unds to the portolio should not overlook the eect o a three-month lockup period on the portolio perormance. He would get a wrong impression o the incremental beneits o investing in hedge unds i he ignores the existence o a lockup period. D. Bootstrap Samples with a One-Year Hedge Fund Lockup Period Three-month hedge und lockup period is plausible or many unds o unds, but some unds and unds and individual hedge unds have longer lockup periods. The estimation is problematic with longer lockup periods since the history o hedge und indexes is relatively short. For instance, or the one-year horizon, we have only 18 non-overlapping samples to estimate parameters o interest whose number can be more than 70. Using quarterly returns or ewer state variables will reduce the number o parameters, without decreasing the sample size. We use the bootstrap method to obtain a larger sample size in order to examine the eect o a long lockup period. We ollow the stationary bootstrap method by Politis and Romano (1993) and Sullivan, Timmermann and White (1999) to obtain 5000 bootstrap samples o quarterly data. The smoothing parameter is chosen to be 0.2, so the mean block length is 5. The choice o the smoothing parameter aects the portolio weights and perormance, but the results o the Sharpe ratio rest and the Wald test are not too sensitive to the smoothing parameter. 20

21 Table 9 gives the results o unconditional portolio perormance, using various hedge und indexes as proxy or hedge unds. The signiicantly higher Sharpe ratios or three-asset portolios justiy the inclusion o hedge unds into an investors portolio. Nevertheless, a oneyear lockup period seems to make little impact on the perormance o unconditional three-asset portolios o stocks, bonds and HFRIFoF (or HFRIFoF strategy), as the dierence in Sharpe ratios o the portolios with or without a lockup period is not signiicant. Adding the HFRI or HFRI strategy indexes to the portolio also increases the Sharpe ratio signiicantly. However, having a one-year lockup period causes the dierence in the Sharpe ratios o the three-asset portolios when the HFRI Event-Driven or the HFRI Relative Value is used as the hedge und proxy. Table 10 reports the analysis o the conditional asset allocation. The changes in the market dividend yield aect portolio weights o risky assets in a signiicant way. All p-values are equal to zero, or the Wald test o the null hypothesis that all slope coeicients are jointly equal to zero. In addition, the Sharpe ratios are signiicantly higher under the conditional portolios than those under the unconditional portolios in all cases. Thereore, an investor can beneit rom using conditional inormation in the portolio decision. Relative to the two-asset conditional portolio, adding hedge unds to the portolios improve the portolio payo in terms o Sharpe ratios. However, a conditional portolio investor would overestimate the portolio perormance when he ignores the presence o one-year hedge und lockup period. A one-year lockup period has signiicant impact on the portolio perormance whichever hedge und index is chosen as the proxy. It seems that i the lockup period is long, an investor should be concerned with the eect o a lockup period on the perormance o his portolio, or investments in unds o unds as well as individual hedge unds. IV. Conclusion A lockup period is a realistic eature o investments in hedge unds, private equities and venture capital. This paper considers the impact o hedge und lockup periods on the asset allocation decisions o a mean-variance investor who re-adjusts the portolio periodically. Due to the presence o a hedge und lockup period, the investor can only adjust the allocation o stocks and bonds. The mean-variance ramework in this paper serves to illustrate the eect o hedge und lockup periods on multi-period asset allocation, with the potential to extend to other 21