Timing the Market with Stocks, Bonds, and Bills

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1 Preliminary lease do not quote Timing the Market with Stocks, Bonds, and Bills by Robert R. Grauer Faculty of Business Administration Simon Fraser University 8888 University Drive Burnaby, British Columbia, Canada V5A S6 Telehone: (604) FAX: (604) Preliminary version May 2003 Preared for the NFA Meetings in Quebec City Key Words: Dynamic asset allocation, ortfolio choice, return redictability, estimation risk, erformance measurement JEL classification: G * The author thanks the Social Sciences Research Council of Canada for financial suort, Christoher Fong for most caable assistance, and Michael Best for technical suort. Some reliminary results were resented at the VII International Conference on Stochastic Programming at the University of British Columbia.

2 Timing the Market with Stocks, Bonds, and Bills Abstract This aer emloys a discrete-time ortfolio selection model to generate market-timing strategies that combine lending and borrowing (when ermitted) with either: common stocks, or common stocks and long-term government bonds, or common stocks and small stocks, or common stocks, small stocks, and long-term government bonds. In some cases the fraction of wealth committed to equities is constrained to lie between uer and lower bounds. A dividend yield riskfree rate estimator is used to forecast equity means and the yield-to-maturity is used as an estimate of the mean on long-term government bonds. Economically noteworthy results are characterized by statistically significant market-timing ability.

3 I. Introduction Asset allocation strategies take different forms. A ure market-timing strategy combines common stocks and Treasury bills. Variants of this strategy add some combination of government and cororate bonds and/or small stocks to the mix. Industry-rotation strategies, as the name indicates, allocate funds across industries. Lastly, global strategies allocate funds across countries or global industry classes. In earlier work, Grauer and Hakansson (982, 985, 986, and 987) and Grauer, Hakansson, and Shen (990) aly discrete-time dynamic ortfolio theory in conjunction with the emirical robability assessment aroach (EPAA) to examine domestic, global, and industry rotation asset allocation roblems. Grauer and Hakansson (995, 998, and 200) study the effect of allowing for estimation error in the means emloying shrinkage estimators, i.e., James-Stein and Bayes-Stein estimators, Caital Asset Pricing Model (CAPM) estimators, and an inflation-adjustment estimator. Grauer and Shen (2000) exlore the effect of allowing for estimation error by imosing roortional constraints on industry holdings. 2 Finally, Grauer (2002a) investigates the statistical value of emloying historic, Bayes-Stein, CAPM, and dividend yield riskfree rate estimators of asset means and comares it to the economic value of combining the estimators with the discrete-time ortfolio selection model in an industry-rotation setting. The results indicate that the estimators are simultaneously statistical failures and economic success stories. 3 But, some erform areciably better than others. Judged in terms of comound return standard deviation lots and accumulated wealth, the ortfolios generated from the dividend yield riskfree rate estimators erform by far the best and the ortfolios generated from the traditional CAPM estimator erform the worst. With some notable excetions, commonly acceted measures of investment erformance suort these rankings.

4 Given the success of combining the dividend yield riskfree rate estimators with the discrete-time ortfolio selection model in an industry-rotation setting, this aer examines its erformance in four market-timing venues. The four venues combine lending and borrowing (when ermitted) with: common stocks, or common stocks and long-term government bonds, or common stocks and small stocks, or common stocks, small stocks, and long-term government bonds. In some cases, the fraction of wealth committed to equities is constrained to lie between uer and lower bounds. A dividend yield riskfree rate estimator is used to forecast equity means and the yield-to-maturity is used as an estimate of the mean on long-term government bonds. Economically noteworthy results are characterized by statistically significant markettiming ability. Beller, Kling, and Levinson (998), Brennan, Schwartz and Lagnado (997), Connor (997), Ferson and Seigel (997), Klemkosky and Bharati (995), and Solnik (993), among others, studied the economic significance of combining forecasts based on information variables with a ortfolio selection model. With the excetion of Brennan, Schwartz and Lagnado, who emloy a continuous-time ortfolio selection model, the aers examine the ortfolio returns of a mean-variance investor who exhibits an "average" degree of risk aversion and revises his ortfolio monthly. While all the aers reort economically significant returns in U.S. bondstock, U.S. industries, and international settings, none reorts results from before 960. By way of contrast, this aer emloys a discrete-time ortfolio selection model that embodies a broad range of risk-aversion characteristics, quarterly decision horizons, borrowing and lending at different rates, and datasets that san the eriod. Moreover, it emloys 2

5 different ways of measuring the erformance of the ortfolios. The aer roceeds as follows. Section II outlines the basic multieriod investment model and the method emloyed to make it oerational. Section III describes the data. Section IV describes the estimators of the means. Section V evaluates the economic value of the forecasts when they are combined with the discrete-time ortfolio selection model. The economic value is characterized in terms of comound return standard deviation lots and the accumulated wealth of the olicies. Section VI describes four commonly acceted statistical measures emloyed to evaluate investment erformance. Section VII contains the results of the statistical analyses. Section VIII contains a summary and conclusions. 3

6 II. The Discrete-Time Portfolio Selection Model The basic model emloyed here is based on the ure reinvestment version of discrete-time dynamic investment theory. At the beginning of each eriod t, the investor chooses a ortfolio x t on the basis of some member γ of the family of utility functions for returns r given by subject to where ( ) max E ( xt γ γ γ + rt ( xt )) = max π ts ( + rts ( xt )) () xt s γ xit + xlt + xbt =, (2) i ( r ( x )) 0, for all s, (3) + ts t x 0, x 0, x 0, all i, (4) it Lt Bt mx it it, (5) i lower bound x + x uer bound, (6) CSt SSt rts xt = xitrits + xltrlt + xbtr d Bt is the ex ante return on the ortfolio in eriod t if state s i occurs, γ = a arameter that remains fixed over time, x it = the amount invested in risky asset category i in eriod t as a fraction of own caital, and more secifically x CSt and x SSt are the amounts invested in common stocks and x Lt x Bt small stocks, resectively = the amount lent in eriod t as a fraction of own caital, = the amount borrowed in eriod t as a fraction of own caital, x = (,..., x, x, x ), t 4 x t nt Lt Bt 4

7 r it = the anticiated total return (dividend yield lus caital gains or losses) on asset category i in eriod t, r Lt = the return on the riskfree asset in eriod t, d r Bt m it = the interest rate on borrowing at the time of the decision at the beginning of eriod t, = the initial margin requirement for asset category i in eriod t exressed as a fraction, π ts = the robability of state s at the end of eriod t, in which case the random return r it will assume the value r its, and the lower bound in equation (6) takes on the values 0.2 and 0.4 when the corresonding uer bound is 0.8 and 0.6, resectively. Equation (2) is the budget constraint and the constraints embodied in equation (3) rule out any ex ante robability of bankrutcy. 5 Equation (4) rules out short sales and ensures that lending (borrowing) is a ositive (negative) fraction of caital. The utility function in equation (), together with the constraints in equations (2) to (4) (with borrowing set equal to zero), constitute a no-margin roblem. Adding borrowing and the margin constraint (5) serves to limit borrowing (when desired) to the maximum ermissible under the margin requirements that aly to the various asset categories. This makes u a margin roblem. However, many investment managers and ension fund sonsors do not ermit margin urchases nor do they allow any risky asset class to reresent 00 ercent of the ortfolio in their asset-allocation decisions. Rather, they imose uer and lower bounds on equities (or equivalently on fixed income securities). Relacing the margin constraint (5) (and the oortunity to borrow) with equation (6) rovides one way to model of this tye of behavior. It comrises a constrained no-margin roblem. The inuts to the model are based on the "emirical robability assessment aroach" (EPAA) with quarterly revision. At the beginning of quarter t, the ortfolio roblem for that 5

8 quarter uses the following inuts: the (observable) riskfree return for quarter t, the (observable) call money rate +% at the beginning of quarter t, and the (observable) realized returns for the risky asset categories for the revious k quarters. Each joint realization in quarters t k through t is given robability /k of occurring in quarter t. Thus, under the EPAA, estimates are obtained on a moving basis and used in raw form without adjustment of any kind. On the other hand, since the whole joint distribution is secified and used, there is no information loss; all moments and correlations are imlicitly taken into account. It may be noted that the emirical distribution of the ast k eriods is otimal if the investor has no information about the form and arameters of the true distribution, but believes that this distribution went into effect k eriods ago. With these inuts in lace, the ortfolio weights x t for the various asset categories and the roortion of assets borrowed are calculated by solving equation (), together with the relevant set of constraints from equations (2)-(6), via nonlinear rogramming methods. 6 At the end of quarter t, the realized returns on the risky assets are observed, along with the realized borrowing r rate r Bt (which may differ from the decision borrowing rate r d Bt ). 7 Then, using the weights selected at the beginning of the quarter, the realized return on the ortfolio chosen for quarter t is recorded. The cycle is reeated in all subsequent quarters. 8 All reorted returns are gross of transaction costs and taxes and assume that the investor in question had no influence on rices. There are several reasons for this aroach. First, as in revious studies, we wish to kee the comlications to a minimum. Second, the return series used as inuts and for comarisons also exclude transaction costs (for reinvestment of interest and dividends) and taxes. Third, many investors are tax-exemt and various techniques are 6

9 available for keeing transaction costs low. Finally, since the roer treatment of these items is nontrivial, they are better left to a later study. III. The Data The data used to estimate the robabilities of the next eriod's returns on risky assets and to calculate each eriod's realized returns on risky assets come from several sources. Ibbotson Associates' database is the source of the returns for common stocks (Standard and Poor's 500 Index), long-government bonds, and small stocks. The riskfree asset is assumed to be 90-day U.S. Treasury bills maturing at the end of the quarter. The Survey of Current Business and the Wall Street Journal are the sources. The borrowing rate is assumed to be the call money rate +% for decision uroses (but not for rate of return calculations). The alicable beginning of eriod decision rate, r d Bt, is viewed as ersisting throughout the eriod and thus as riskfree. For , the call money rates are obtained from the Survey of Current Business. For later eriods, the Wall Street Journal is the source. Finally, margin requirements for stocks are obtained from the Federal Reserve Bulletin. 9 IV. Correcting the Model for Estimation Error Dividend yields and riskfree interest rates are used to forecast equity means, and the yieldto-maturity of long-term government bonds is used to estimate the quarterly mean return on the bond. The easiest way to exlain these estimates of the means is to begin with the basic EPAA aroach. Under this aroach, means are not used directly but are imlicitly comuted from the realized returns in the estimation eriod. The n-vector of historic (EPAA) means at the beginning of eriod t is denoted as ( r r ) µ,, Ht = t nt ', (7) 7

10 where t it = r i k τ = t k r τ. The EPAA aroach imlicitly estimates the means one at a time, relying exclusively on information contained in each of the time series. To construct the dividend yield riskfree rate estimator, we begin by running the following regression at each time t r + iτ = b0 i + b idyτ + b2irlτ eiτ, for all i and τ, (8) in the t k to t estimation eriod, where the i subscrit denotes either common stocks (the S&P 500 index) or small stocks, dy τ is the dividend yield on the S&P 500 index (whether we are forecasting the means on common stocks or small stocks), and r Lτ is the quarterly riskfree (tbill) rate. Both indeendent variables are "de-meaned." Hence, return on asset i. b 0 i is the historic average rate of To be more secific, the dividend yield series is constructed following the method described in Fama and French (988). The monthly index values P m are created first. Then, the revious twelve months of dividends are summed to create a monthly dividend yield series D / P. m m Following Fama and French, the monthly dividend yield series is lagged one month. The quarterly dividend yields dyτ are the January, Aril, July, and October values of the lagged monthly series, or equivalently, the December, March, June, and Setember values of the original monthly series. The traditional one-eriod ahead forecast of the mean of asset i is rit 0 i + idyt + 2i = bˆ bˆ bˆ r, (9) Lt where b ˆ0i, b ˆ2i, and bˆ 2 i are the estimated coefficients, and dy and r Lt are observable at the beginning of eriod t+. That is, the quarterly variable dy t is lagged one month and there is no need to lag r Lt as it is observable at the beginning of the quarter. However, the one-eriod ahead t 8

11 forecast is extremely variable. Therefore, in the sirit of the Stein estimators we follow Grauer (2002) and "shrink" the forecast of the mean from equation (9) by dividing the difference from the historic mean by two r = bˆ + ( bˆ dy bˆ + r ) / 2. (0) DRit 0i m t 2m Lt The vector of dividend yield riskfree rate (DR) estimators is given by ( r r ) µ,, DRt = DR t DRnt '. () Having calculated the dividend yield riskfree rate and historic means for asset i, we add the difference between the dividend yield riskfree rate and historic means to each actual return on asset i in the estimation eriod. That is, in each estimation eriod, we relace the raw return series with the adjusted return series A ri τ = ri τ + ( rdrit rit ), for all i and τ. Thus, the mean vector of the adjusted series is equal to the dividend yield riskfree rate means of the original series; all other moments are unchanged. By way of contrast, the bond adjustment is extremely simle. The quarterly mean return is set to the annual yield to maturity divided by four. The motivation is as follows. Yields and realized returns on bonds are inversely related. Hence in eriods of rising interest rates, realized bond returns will be low, while in eriods of decreasing interest rates, realized bond returns will be high. (These henomena are illustrated in the increasing interest rate sub-eriod from 966 to 98 shown in Figures 4, and in the decreasing interest rate sub-eriod from 982 to 200 shown in Figure 5.) To claim that the exected return on the bond over the next quarter is one-quarter of the annual yield-to-maturity is undoubtedly an oversimlification because one must hold the bond to maturity (and reinvest any couons at the yield-to-maturity) to realize the yield-tomaturity. It aears to be an imrovement over the average return estimated from historical data, 9

12 however, esecially in eriods of either increasing or decreasing interest rates. Moreover, ractitioners have been using this method to forecast the exected return on bonds since the late 970s. 0

13 V. Comound Return Standard Deviation and Cumulative Wealth Results Figures and 2 show the results for the margin roblem for four different risky-asset investment universes. Figure lots the annual 0 comound return and standard deviation of the realized returns for four sets of ten ower utility strategies, based on γ 's in equation () ranging from -50 (extremely risk averse) to (risk-neutral), for the 68-year eriod from Portfolios are chosen each quarter, emloying a 32-quarter estimation eriod, and dividend yield riskfree rate forecasts of the means for stocks and (annual) bond yields (divided by four) as the estimate of the means of long-term government bonds. Investors are ermitted to invest in T-Bills or to buy on margin, in which case they ay the call money rate lus one ercent on borrowed funds. The assive benchmarks are shown as oen diamonds. They consist of the four asset classes riskfree lending (ninety-day T-Bills) RL, long-term government bonds (GB), common stocks, i.e., the S&P 500 index (SP), and small stocks (SS) lus a weighting of common stocks and T-Bills (SPRL) and a weighting of common stocks and long-term government bonds (SPGB). The first set of active strategies (black circles) shows the returns generated from a ure market-timing strategy, where the "market" is the S&P500 index. The second set (oen circles) dislays the returns when long-term government bonds are added to the risky-asset investment universe. The third set (oen triangles) deicts the returns generated from a universe of common stocks and small stocks. Finally, the fourth set (oen triangles) resents the returns obtained when the investment universe consists of common stocks, small stocks and long-term government bonds. Figure here The full eriod is characterized by relatively low returns for fixed income securities. T-Bills earn a comound annual rate of return of 4.4 ercent and long-term government bonds earn

14 5.4 ercent. Equities do much better with common stocks and small stocks earning.72 and 4.98 ercent, resectively. With a ure market-timing strategy, where the common stocks are the only risky asset, the less risk-tolerant ower utility investors with γ 's of 3 and below realize less comound return and standard deviation than the S&P 500 index, while the more risk-tolerant ower utility investors with γ 's of and above realize more comound return and standard deviation than the S&P 500 index. The growth-otimal strategy, with γ equal to zero, erforms the best earning 5.5 ercent with a standard deviation of 2.93 ercent. By way of contrast, a assive olicy (not shown in the figure) of levering common stock to the maximum level ermitted according to the Federal Reserve initial margin requirements earns 4.25 ercent with a standard deviation of ercent. In sum, according to the "naked-eye" test, the active ower-utility strategies exhibit market-timing ability. When long-term government bonds are added to the investment universe, the ower utility investors earn higher returns with about the same, and often somewhat less, standard deviation than they do in the ure market-timing environment. (The excetion is the risk-neutral investor.) The growth-otimal investor now earns 7.0 ercent with a standard deviation of ercent. When small stocks are substituted for long-term government bonds, the more risk-averse ower utility investors earn slightly less comound return and incur slightly more standard deviation than they do when the risky-asset investment universe consists of common stocks and small stocks. The more risk-tolerant investors earn slightly higher returns accomanied by more risk. The growth-otimal investor earns 7.56 ercent with a standard deviation of 27.4 ercent. Finally, when the investment universe includes all three risky assets common stocks, long-term government bonds and small stocks the ower utility investors earn the highest returns and 2

15 incur slightly more risk than in the other cases. In this case, the growth-otimal investor earns 9.87 ercent with a standard deviation of 27.9 ercent. Even so, rate-of-return lots convey only art of the story. While 5 to 20 ercent returns are not all that remarkable over short time eriods, it is a comletely different story when an investment earns these tyes of returns over a 68-year eriod, esecially when these returns are comared to assive benchmarks returns. Consider, for examle, the wealth accumulated from investing a dollar at the beginning of 934 and holding it until the end of 200. A dollar invested in T-Bills grows to $6, and a dollar invested in long-term government bonds grows to $36. Meanwhile, a dollar invested in common stocks (small stocks) increases to $,876 ($3,239). Finally, an investment of $, where 200 ercent is allocated to common stocks, which is financed by borrowing 00 ercent at the call money rate lus one ercent, grows to $8,587. Comare these benchmark wealth levels to what the growth-otimal olicy accumulates in each of the four risky-asset investment universes. In the common stocks universe, the growthotimal olicy accrues $4,658. In the common stocks and long-term government bonds universe, it accumulates $43,450. In the common stocks and small stocks universe, it accumulates $59,773. But, in the common stocks, long-term government bonds universe, it amasses $225,365! Figure 2 shows the margin results for the four risky-asset investment universes in the 32-year sub-eriod from The sub-eriod is characterized by higher returns for fixed-income securities than in the eriod. T-Bills and government bonds earn 6.65 and 7.75 ercent, resectively. Moreover, The higher return on government bonds is associated with more volatility. Common stocks and small stocks earn lower returns and incur more risk. Nonetheless, the lots for the active strategies are similar to those in the full eriod, which again 3

16 seems to imly that the active ower-utility strategies show signs of market-timing ability. (The Treynor-Mazuy and Henriksson-Merton market-timing tests discussed below rovide statistical confirmation of this result.) Figure 2 here Figure 3 reorts the no-margin results for the four risky-asset investment universes in the sub-eriod. The figure uts to rest any doubts that the forecasting techniques combined with discrete-time ortfolio selection model dislay market-timing ability. In each of the four investment universes, the less risk-tolerant active strategies earn higher returns than the highest return asset category in that universe! This is the only case in the series of discrete-time ortfolio selection aers dating from Grauer and Hakansson (982) in which this result occurs. Figure 3 here Figures 4 and 5 comare the no margin results with the constrained no margin results when the risky-asset investment universe consists of common stocks, small stocks, and long-term government bonds. Figure 4 shows the results for the 6-year sub-eriod from This eriod of raidly rising interest rates is characterized high short-term rates and negative equity remium, i.e., the 6.89 ercent return on T-bills exceeds the 5.95 ercent return on common stocks. Not surrisingly, in a eriod of rising interest rates, the realized return on long-term government bonds is a altry 2.53 ercent. On the other hand, this is the "golden age" of small stocks returns, as they realize 4.08 ercent. In this eriod, it is extremely costly to imose additional constraints on the no margin roblem. Constraining the roortion of the ortfolio held in T-Bills and long-term government bonds to lie between 20 and 80 ercent of the ortfolio lead to lower returns and higher standard deviations for the active ower-utility strategies. 4

17 Tightening the constraints further to kee fixed-income securities between 40 and 60 ercent of the ortfolio lead to still lower realized returns and higher standard deviations. Figure 4 here Figure 5 shows the results for the 2-year sub-eriod. In contrast to the sub-eriod, this sub-eriod is characterized by dramatically falling interest rates. While the realized return on T-Bills is only marginally lower at 6.38 ercent, the return on long-term government bonds reaches a lofty 2.09 ercent. Common stocks earn 5.24 ercent. However, the "small-firm" effect reverses throughout the nineties, as a result small stocks only realize 3.76 ercent for the eriod. The change in the effect of imosing constraints on the no margin roblem is equally striking. Imosing (and tightening) the uer and lower bond constraints has different effects deending on investor risk tolerance. Imosing (and tightening) the constraints increases the return and the standard deviation for the less risk-tolerant investors and decreases the return and the standard deviation for more risk-tolerant investors. Figure 5 here VI. Measures of Investment Performance The figures and cumulative wealth values rovide convincing evidence that dividend yield riskfree rate and yield-to-maturity estimators rovide economic value when combined with the discrete-time ortfolio selection model. However, the figures do not give us a sense of how much of the difference can be attributed to randomness. In order to shed light on this issue we reort the results from a number of commonly acceted statistical measures of erformance. Unfortunately, none is without roblems. First, with the excetion investing in either T-bill and common stocks, the asset-allocation strategies examined here are neither the ure selectivity strategies imlicit in Jensen's (968) test nor the ure market-timing strategies, embodied in 5

18 Treynor and Mazuy's (966) and Henriksson and Merton's (98) tests of market-timing. Second, Roll (978) argues that Jensen's test is ambiguous because the choice of the benchmark (market) ortfolio affects both systematic risk (beta) and abnormal return (alha). 2 Third, exected returns and risk measures may vary with economic conditions. Therefore, this study emloys an eclectic mix of erformance measures that include conditional and unconditional versions of the Jensen, Henriksson-Merton, and Treynor-Mazuy tests as well as Grinblatt and Titman's (993) ortfolio change measure that gauges erformance without reference to a roxy for the market ortfolio. The null hyothesis is that there is no suerior investment erformance and the alternative hyothesis is that there is. Thus, we reort the results of one-tailed tests. All regressions are corrected for heteroskedasticity using White's (980) correction. The unconditional Jensen (968) test is based on the regression R = α + β R + u, (2) t mt t where R t = rt rlt is the excess return on ortfolio over the Treasury bill rate, Rmt = rmt rlt is the excess return on the S&P 500 index, α is the unconditional measure of erformance, and β is the unconditional measure of risk. However, exected returns and betas almost certainly change over time. Therefore, Ferson and Schadt (996) and Ferson and Warther (996), among others, building on the earlier work of Shanken (990), advocate conditional erformance measures. We follow their suggestion that a ortfolio's risk is related to dividend yields and short-term Treasury yields ostulating that β r, (3) = b0 + b dyt + b2 Lt 6

19 where dy t is the S&P 500 index annual dividend yield at the beginning of eriod t and r Lt is the (observable) beginning-of-quarter Treasury bill rate, both measured as deviations from their estimation-eriod means. Substituting equation (3) into equation (2), yields the conditional Jensen test R = α + b R + b [ r R ] + e, (4) t c 0 mt [ dyt Rmt ] + b2 Lt mt t where α c is the conditional measure of erformance, b 0 is the conditional beta, and b and b 2 measure how the conditional beta varies with dividend yields and Treasury bill rates. The unconditional regression secification for the Treynor and Mazuy (966) test is R = α + β R + γ R + u, (5) t mt 2 mt t where α is the measure of selectivity, β is the unconditional beta, and γ is the markettiming coefficient. Substituting for β, the conditional regression secification is R α [ + γ R + e, (6) t = c + b0 Rmt + b dyt Rmt ] + b2 [ rlt Rmt ] 2 mt t where α c, b 0, b, b 2, and γ are defined above. The unconditional Henriksson and Merton (98) test is given by R = α + β R + γ max(0, R ) + u, (7) t d mt mt t where α is the measure of selectivity, β d is the down-market beta, γ is the market-timing coefficient, in this case the difference between the u- and down-market beta, and max(0, R the ayoff on a call otion on the market with exercise rice equal to the riskfree rate of interest. Following Ferson and Schadt (996), the conditional Henriksson-Merton test is mt ) is R + e * * * * * t = αc + bdrmt + b [ dyt Rmt ] + b2 [ rlt Rmt ] + γ Rmt + b [ dyt Rmt ] + b2 [ rlt Rmt ] t, (8) 7

20 where * R mt is the roduct of the excess return on the CRSP value-weighted index and an indicator dummy for ositive values of the difference between the excess return on the index and the conditional mean of the excess return. (The conditional mean is estimated by a linear regression of the excess return of the CRSP value-weighted index on dy t and r Lt.) The most imortant coefficients are b d, the conditional down-market beta, and γ, the market-timing coefficient, which in this case is the difference between the u- and down-market conditional betas. In contrast to most other erformance measures, Grinblatt and Titman's (993) ortfolio change measure emloys ortfolio holdings as well as rates of return and does not require an external benchmark (market) ortfolio. In order to motivate the ortfolio change measure, assume that uninformed investors erceive that the vector of exected returns is constant, while informed investors can redict whether exected returns vary over time. Informed investors can rofit from changing exected returns by increasing (decreasing) their holdings of assets whose exected returns have increased (decreased). The holding of an asset that increases with an increase in its conditional exected rate of return will exhibit a ositive unconditional covariance with the asset's returns. The ortfolio change measure is constructed from an aggregation of these covariances. For evaluation uroses, let PCM t = r it ( xit xi, t j ), i where r it is the quarterly rate of return on asset i time t, x it and xi t j, are the holdings of asset i at time t and time t j, resectively. This exression rovides an estimate of the covariance between returns and weights at a oint in time. Alternatively, it may be viewed as the return on a zero-weight ortfolio. The ortfolio change measure is an average of the PCM t s 8

21 [ rit ( xit xi t j ) T ] PCM =, /, (9) t i where T is the number of time-series observations. The ortfolio change measure test itself is a simle t-test based on the time series of zero-weight ortfolio returns, i.e., ( PCM / σ ( PCM )) T t =, (20) where σ (PCM ) is the standard deviation of the time series of PCM t s. In their emirical analysis of mutual fund erformance, Grinblatt and Titman work with two values of j that reresented one- and four-quarter lags. This aer emloys the same two lags. The ortfolio change measure is articularly aroos in the resent study because the ortfolio weights are chosen according to a re-secified set of rules over the same quarterly time interval as erformance is measured. Thus, we do not have to worry about ossible gaming or window-dressing roblems that face researchers trying to gauge the erformance of mutual funds. VII. Performance Measure Results Tables to 4 resent the results of the investment erformance tests alied to the margin roblem for the four risky-asset investment universes over the , , 966-8, and eriods. Table summarizes the average alhas and betas for the ten owerutility strategies obtained from the conditional and unconditional Jensen tests. There are three main results in Table. First, there is evidence of abnormal conditional and unconditional erformance excet in the sub-eriod. Second, the number of statistically significant alhas increases as the number of risky assets in the investment universe increases. Third, the unconditional test finds more evidence of statistically significant abnormal erformance than the conditional test in the eriod. The reader is cautioned, however, that the Jensen test 9

22 may be an inaroriate measure as it is a measure of micro-forecasting ability (selectivity), while investing in common stocks and T-Bills (or on margin) is clearly a market-timing strategy. Table here Tables 2 and 3 contain the results of the unconditional and conditional Henriksson-Merton and Treynor-Mazuy market-timing tests, resectively. The conditional and unconditional Henriksson-Merton and Treynor-Mazuy tests show strong evidence of market-timing ability in the eriod. At least eight, and more often ten, of the market-timing coefficients are statistically significant at the 5 ercent level in each of the four risky-asset investment universes. In the full eriod, the evidence is somewhat mixed with the unconditional models indicating much more evidence of market-timing ability. Tables 2 and 3 here The results of the ortfolio change measure test are resented in Table 4. The tests encomass the same one-quarter and four-quarter lags emloyed by Grinblatt and Titman (993) in their study of mutual funds. The one-quarter lag results are anomalous. Most of the PCMvalues are negative, and there is not one instance of a ositive statistically significant PCM. However, the four-quarter lag results suort strong statistically significant erformance in all but the sub-eriod. Table 4 here Table 5 shows the results of the investment erformance tests alied to the margin, nomargin and constrained no-margin roblems when the risky-asset investment universe consists of common stocks, small stocks, and long-term government bonds over the , , 966-8, and eriods. The table is more abbreviated than Tables to 4. It shows the average Jensen selectivity coefficients (alhas), the average Henriksson-Merton and 20

23 Treynor-Mazuy market-timing coefficients (gammas), and the average Grinblatt-Titman ortfolio change measures (PCMs) together with the number of times that the coefficients of the ten ower-utility strategies are statistically significant at less than or equal to the 5 ercent level in a one-tailed test in each of the four eriods. The margin results in Panel D, which are abbreviated versions of the S&P500, government bonds, and small stock results reorted in Tables -4, ermit easier comarisons with the no-margin and constrained no-margin results. Table 5 here VIII. Summary This aer examines the erformance sets of ortfolios generated from a discrete-time ortfolio selection model in four market-timing venues. The four venues combine lending and borrowing (when ermitted) with: common stocks, or common stocks and long-term government bonds, or common stocks and small stocks, or common stocks, small stocks, and long-term government bonds. A dividend yield riskfree rate estimator is used to forecast equity means and the yield-to-maturity is used as an estimate of the mean on long-term government bonds. In some cases, the fraction of wealth committed to equities is constrained to lie between uer and lower bounds. Economically noteworthy results are characterized by statistically significant market-timing ability. 2

24 Footnotes Jobson, Korkie, and Ratti (979) and Jobson and Korkie (980, 98) are the first to emloy Stein estimators in the ortfolio selection literature. Jorion (985, 986, 99) examines the effect of correcting for estimation error in with Stein and CAPM estimators. 2 Frost and Savarino (988), emloying simulation and mean-variance analysis with unlimited short sales, reort that imosing uer bounds both reduces estimation bias and imroves erformance. Grauer (2002b), using mean-variance analysis in an out-of-samle industry rotation setting, finds a dramatic imrovement in investment erformance when short-sales constraints are imosed. Moreover, the CAPM estimator of the means, which exhibits the best erformance when short sales are ermitted, exhibits the worst erformance when they are recluded. 3 Dividend yield riskfree rate estimators, and estimators based on other information variables, trace their origin to the return redictability or weak form efficient markets literature. See, for examle, Fama and French (988, 989), Hawawini and Keim (994), and the references in Fama (99). However, the statistical merits of the estimators are not without question. For examle, Goetzmann and Jorion (993) question the long-horizon results. Bossaerts and Hillion (999) confirm the resence of in-samle redictability of a variety of informational variables using monthly data in an international setting. But reort that even the best rediction models have no out-of-samle forecasting ower. Stambaugh (999) oints out that an OLS estimator's finite-samle roerties can deart substantially from the standard regression setting when a rate of return is regressed on a lagged stochastic regressor such as a dividend yield. 4 In this aer, vectors are column vectors and a rime indicates transosition. Vectors and matrices are in bold and scalars are italic. 5 The solvency constraint (3) is not binding for the ower functions, with γ < 0, and discrete robability distributions with a finite number of outcomes because the marginal utility of zero wealth is infinite. Nonetheless, it is convenient to exlicitly consider equation (5) so that the nonlinear rogramming algorithm used to solve the investment roblem does not attemt to evaluate an infeasible olicy as it searches for the otimum. 6 The nonlinear rogramming algorithm emloyed is described in Best (975). 7 The realized borrowing rate is calculated as a monthly average. 8 Note that if k = 32 under quarterly revision, then the first quarter for which a ortfolio can be selected is b+32, where b is the first quarter for which data is available. 9 There is no ractical way to take maintenance margins into account in our rograms. In any case, it is evident from the results that they would come into lay only for the more risk-tolerant strategies, and even for them only occasionally, and that the net effect would be relatively neutral. 0 Annual returns were obtained by comounding the quarterly realized returns. 22

25 For consistency with the comound return (geometric mean), the standard deviation is based on the log of one lus the rate of return. This quantity is very similar to the standard deviation of the rate of return for levels less than 25 ercent. 2 See also Dybvig and Ross (985), Grauer (99), and Green (986). 23

26 Table I Unconditional and Conditional Jensen Alhas for Ten Power Portfolios in Four Investment Universes There are four investment universes in the , , , and eriods. Panel A shows results for an investment universe of S&P500 stocks (SP) only. Panel B shows results for an investment universe of SP and U.S. long-term government bonds (GB). Panel C shows results for an investment universe of SP and U.S. small stocks (SS). Panel D shows results for an investment universe of SP, GB, and SS. Borrowing is ermitted in all universes. Quarterly ortfolio revision with a 32-quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The alhas are measured in ercent er quarter. The -values measure the significance of the coefficients relative to zero. The - values are heteroskedasticity consistent. The alhas and -values are averages calculated over the ten ower ortfolios. Unconditional Conditional Number -tail Number Number -tail Number Category Alha Negative -value 0.05 Beta Alha Negative -value 0.05 Panel A: SP Panel B: SP and GB Panel C: SP and SS Panel D: SP, GB, and SS

27 Table II Unconditional and Conditional Henriksson-Merton Alhas and Timing Coefficients For Ten Power Portfolios in Four Investment Universes There are four investment universes in the , , , and eriods. Panel A shows results for an investment universe of S&P500 stocks (SP) only. Panel B shows results for an investment universe of SP and U.S. long-term government bonds (GB). Panel C shows results for an investment universe of SP and U.S. small stocks (SS). Panel D shows results for an investment universe of SP, GB, and SS. Borrowing is ermitted in all universes. Quarterly ortfolio revision with a 32- quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The alhas are measured in ercent er quarter. Gamma is the timing coefficient. The -values measure the significance of the coefficients relative to zero. The -values are heteroskedasticity consistent. The alhas, gammas, and -values are averages calculated over ten ower ortfolios. Unconditional Conditional Number 2-tail Number -tail Number Number 2-tail Number -tail Number Category Alha Negative -value Gamma Negative -value 0.05 Alha Negative -value Gamma Negative -value 0.05 Panel A: SP Panel B: SP and GB Panel C: SP and SS Panel D: SP, GB, and SS

28 Table III Unconditional and Conditional Treynor-Mazuy Alhas and Timing Coefficients for Ten Power Portfolios in Four Investment Universes There are four investment universes in the , , , and eriods. Panel A shows results for an investment universe of S&P500 stocks (SP) only. Panel B shows results for an investment universe of SP and U.S. long-term government bonds (GB). Panel C shows results for an investment universe of SP and U.S. small stocks (SS). Panel D shows results for an investment universe of SP, GB, and SS. Borrowing is ermitted in all universes. Quarterly ortfolio revision with a 32- quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The alhas are measured in ercent er quarter. Gamma is the timing coefficient. The -values measure the significance of the coefficients relative to zero. The -values are heteroskedasticity consistent. The alhas, gammas, and -values are averages calculated over ten ower ortfolios. Unconditional Conditional Number 2-tail Number -tail Number Number 2-tail Number -tail Number Category Alha Negative -value Gamma Negative -value 0.05 Alha Negative -value Gamma Negative -value 0.05 Panel A: SP Panel B: SP and GB Panel C: SP and SS Panel D: SP, GB, and SS

29 Table IV Grinblatt-Titman Portfolio Change Measures for Ten Power Portfolios in Four Investment Universes There are four investment universes in the , , , and eriods. Panel A shows results for an investment universe of S&P500 stocks (SP) only. Panel B shows results for an investment universe of SP and U.S. long-term government bonds (GB). Panel C shows results for an investment universe of SP and U.S. small stocks (SS). Panel D shows results for an investment universe of SP, GB, and SS. Borrowing is ermitted in all universes. Quarterly ortfolio revision with a 32-quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The Grinblatt-Titman ortfolio change measure (PCM) is based on one quarter and four quarter lags. PCM is measured in ercent er quarter. The -values measure the significance of the PCM relative to zero. The PCM and -values are averages calculated over the ten ower ortfolios. One Quarter Lag Four Quarter Lag Number -tail Number Number -tail Number PCM Negative -value 0.05 PCM Negative -value 0.05 Panel A: SP Panel B: SP and GB Panel C: SP and SS Panel D: SP, GB, and SS

30 Table V Unconditional and Conditional Jensen Alhas for Ten Power Portfolios Subject to Absolute and No Margin Constraints The investment universe consists of S&P500 stocks (SP), U.S. long-term government bonds (GB), and U.S. small stocks (SS) in the , , , and eriods. Portfolios subject to absolute constraints must hold 40 to 60 and 20 to 80 ercent of wealth in Treasury bills and GB combined. The No Margin ortfolio is recluded from borrowing. Quarterly ortfolio revision with a 32-quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The alhas are measured in ercent er quarter. The -values measure the significance of the coefficients relative to zero. The -values are heteroskedasticity consistent. The alhas and -values are averages calculated over the ten ower ortfolios. Unconditional Conditional Number -tail Number Number -tail Number Category Alha Negative -value 0.05 Beta Alha Negative -value 0.05 Panel A: Panel B: Panel C: No Margin

31 Table VI Unconditional and Conditional Henriksson-Merton Alhas and Timing Coefficients For Ten Power Portfolios Subject to Absolute and No Margin Constraints The investment universe consists of S&P500 stocks (SP), U.S. long-term government bonds (GB), and U.S. small stocks (SS) in the , , , and eriods. Portfolios subject to absolute constraints must hold 40 to 60 and 20 to 80 ercent of wealth in Treasury bills and GB combined. The No Margin ortfolio is recluded from borrowing. Quarterly ortfolio revision with a 32-quarter estimation eriod is emloyed. The means emloy a dividend yield riskfree rate adjustment for SP and SS and a bond yield adjustment for GB. The alhas are measured in ercent er quarter. Gamma is the timing coefficient. The -values measure the significance of the coefficients relative to zero. The -values are heteroskedasticity consistent. The alhas, gammas, and -values are averages calculated over ten ower ortfolios. Unconditional Conditional Number 2-tail Number -tail Number Number 2-tail Number -tail Number Category Alha Negative -value Gamma Negative -value 0.05 Alha Negative -value Gamma Negative -value 0.05 Panel A: Panel B: Panel C: No Margin

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