Journal of Banking & Finance 24 (2000) 1253±1274 www.elsevier.com/locate/econbase

Do constraints improve portfolio performance? Robert R. Grauer a

a,*

, Frederick C. Shen

b,1

Faculty of Business Administration, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6 Canada b Manulife Financial, 200 Bloor Street East, Toronto, Ontario, M4W 1E5 Canada Received 6 May 1998; accepted 28 May 1999

Abstract The discrete-time dynamic investment model, using only historical data in various asset-allocation settings, often produces signi®cant abnormal returns. However, the model does not choose the diversi®ed portfolios that theory suggests it should. Therefore, in this paper, we compare the investment policies and returns of the model with and without constraints on the mix of risky assets. The constraints lead to appreciably more diversi®cation and less realized risk, but only at the cost of less realized return. Visual comparisons of compound returnÐstandard deviation plots and statistical comparisons of JensenÕs alpha suggest that the reduction in return is not worth the reduction in risk. For more risk-averse investors, ex post utility and certainty equivalent returns suggest that it is. The results, however, illustrate the problems associated with using ex post utility to measure performance. Ó 2000 Elsevier Science B.V. All rights reserved. JEL classi®cation: G11 Keywords: Multiperiod power utility asset-allocation

*

Corresponding author. Tel.: +1-604-291-3722; fax: +1-604-291-4920. E-mail addresses: [email protected] (R.R. Grauer), [email protected] (F.C. Shen). 1 Tel.: +1-416-415-7636; fax: +1-416-926-5783. 0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 6 9 - 2

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1. Introduction Grauer and Hakansson (1982, 1985, 1986, 1987) and Grauer et al. (1990) applied discrete-time dynamic portfolio theory 2 in conjunction with the empirical probability assessment approach (EPAA) to examine domestic, global, and industry rotation asset-allocation problems. The results are noteworthy for three reasons. First, the model often generates economically and statistically signi®cant abnormal returns. Second, no attempt is made to correct for estimation error, which is clearly present in the EPAA. Third, the model does not diversify the way that theory suggests it should. 3 Both academics and practitioners have been bothered by the lack of diversi®cation exhibited by portfolio optimizers. Two of the central paradigms of ®nancial economicsÐmodels of asset pricing and the ecient markets hypothesisÐsuggest that investors should hold diversi®ed portfolios. For example, the Sharpe (1964) ± Lintner (1965) mean±variance capital asset pricing model (CAPM) predicts that investors will hold some fraction of the market portfolio. 4 Furthermore, an important implication of the ecient markets hypothesis is that investors not possessing special knowledge would be welladvised to buy and hold diversi®ed portfolios. See, for example, Black (1971) among others. Finally, Black and Litterman (1992) discuss the seemingly ``unreasonable'' and ``unbalanced'' nature of the composition of the portfolios generated by mean±variance optimizers that employ historical data. Together then, the (naive) use of historical data as input in previous applications of the multiperiod model, coupled with the attendant lack of diversi®cation, suggests that it may be prudent to investigate the e€ects of correcting for estimation error. Two methods have been proposed: (1) incorporate corrections for estimation error in the inputs employing either a statistical or a ®nancial model, or (2) constrain the portfolio weights. Grauer and Hakansson (1995) examine the ®rst method in an asset-allocation setting. This paper examines the second. Empirical evidence based on mean±variance portfolio selection, simulation analysis, and out-of-sample portfolio performance suggests that correcting for

2 See Mossin (1968), Hakansson (1971, 1974), Leland (1972), Ross (1974), and Huberman and Ross (1983). 3 With the imposition of non-negativity constraints, mean-variance portfolio optimizers do not diversify widely either. While it is true that investors hold all the assets in positive amounts if the means satisfy the equilibrium relationship of the Sharpe±Lintner capital asset pricing model, small perturbations in these means lead to large changes in portfolio weights. See, for example, Best and Grauer (1991, 1992) who document the extreme sensitivity of the weights to changes in the asset means and provide explicit formulas for how the weights change as a function of changes in the means, the means themselves, and the inverse of the covariance matrix. 4 The arbitrage pricing theory also predicts that investors will diversify widely, holding portfolios whose weights in individual assets are small in absolute value.

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estimation error, particularly in the means, can improve investment performance substantially. See, for example, Jobson et al. (1979), Jobson and Korkie (1980, 1981), and Jorion (1985, 1991). On the other hand, Grauer and Hakansson (1995) compare the investment policies and returns of the discrete-time dynamic investment model for three classes of estimators of the means and ®nd mixed results. In an industry setting, the ®ndings are consistent with those of earlier studies in that the Stein estimators outperform the sample estimator. But the gains are not as great as those reported by others. In a global setting, just the opposite is true: the sample estimator outperforms the Stein estimators. In all cases, the CAPM-based estimator exhibits the worst performance. Alternatively, it has been suggested that one might adjust for estimation risk by constraining portfolio weights. Portfolio managers commonly impose nonnegativity and upper-bound constraints on individual securities to allow for estimation riskÐor to make the portfolio conform to their ideas of what it should look like. More formally, Frost and Savarino (1988) employ simulation to study the e€ects of imposing upper bounds and non-negativity constraints on the portfolio problem. They proceed in four steps. First, they construct a population mean vector and covariance matrix consistent with positive holdings of all the assets in the universe. Second, they generate sample returns based on the population parameters and form mean±variance ecient portfolios from each sample under a set of tightening upper-bound constraints. Borrowing and lending are precluded from the analysis as, for the most part, are short sales constraints. Finally, they measure portfolio performance in terms of certainty equivalent returns, where the certainty equivalent return is calculated using the sample portfolio weights and the population return parameters. Not surprisingly in this contextÐthe population parameters are constructed so that a diversi®ed portfolio is optimalÐthey ®nd that imposing upper bounds both reduces estimation bias and improves performance. The more fundamental question addressed in this paper is whether diversi®ed portfolios perform better out-of-sample where return distributions may not be stationary and markets may not be in equilibrium. Financial theory says they should. On the other hand, the sometimes remarkable performance of the EPAA in various asset-allocation environments together with the estimationerror results in Grauer and Hakansson (1995) suggest they might not. Therefore, in this paper, we add upper and lower bounds on the ``mix of risky assets'' to the usual constraint set employed in the application of dynamic portfolio theory to the asset-allocation problem and compare the returns and investment policies generated under the two sets of constraints. There are two, not necessarily mutually exclusive, reasons for constraining the mix of risky assets. First, it allows us to explore the potential for reducing estimation risk using constraints in an out-of-sample setting. Second, and more importantly, it represents an attempt to marry the strengths of portfolio optimization based on historical data with the theoretically appealing idea that investors should

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``hold the market portfolio''. As noted, with a free hand, optimizers tend to plunge. The constraints temper the propensity of both borrowers and lenders to let their (proportional) risky-asset holdings plunge away from the weights in some benchmark or the ``market'' portfolio. The paper proceeds as follows. Section 2 outlines the basic multiperiod investment model and the method employed to make it operational. Section 3 describes the data. Section 4 records the risk±return trade-o€s and investment policies. Section 5 asks whether there is statistical evidence that the reduction in risk induced by the constraints is worth the reduction in return. Section 6 summarizes the paper. 2. The multiperiod investment model The basic model used is the same as the one employed in Grauer and Hakansson (1986) and the reader is therefore referred to that paper (speci®cally pages 288±291) for details. It is based on the pure reinvestment version of dynamic investment theory. At the beginning of each period t, the investor chooses a portfolio, xt , on the basis of some member, c, of the family of utility functions for returns r given by   X 1 1 c c …1† max E …1 ‡ rt …xt †† ˆ max pts …1 ‡ rt …xt †† ; xt x c c t s subject to xit P 0; xLt P 0; xBt 6 0 X xit ‡ xLt ‡ xBt ˆ 1 ;

all i;

…2† …3†

i

X

mit xit 6 1;

…4†

i

…5† Pr …1 ‡ rt …xt † P 0† ˆ 1; P d is the (ex ante) return on the portfolio where rts …xt † ˆ i xit rits ‡ xLt rLt ‡ xBt rBt in period t if state s occurs, c 6 1 is a parameter that remains ®xed over time, xit is the amount invested in risky asset category i in period t as a fraction of own capital, xLt is the amount lent in period t as a fraction of own capital, xBt is the amount borrowed in period t as a fraction of own capital, xt ˆ …x1t ; . . . ; xnt ; xLt ; xBt †, rit is the anticipated total return (dividend yield plus capital gains or losses) on asset category i in period t, rLt is the return on the d the interest rate on borrowing at the time of the riskfree asset in period t, rBt decision at the beginning of period t, mit is the initial margin requirement for asset category i in period t expressed as a fraction, and pts is the probability of state s at the end of period t, in which case the random return rit will assume the value rits .

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Constraint (2) rules out short sales and ensures that lending (borrowing) is a positive (negative) fraction of capital. Constraint (3) is the budget constraint. Constraint (4) serves to limit borrowing (when desired) to the maximum permissible under the margin requirements that apply to the various asset categories. Constraint (5) rules out any (ex ante) probability of bankruptcy. 5 In this paper, we P also place upper and lower bounds on the mix of risky assets. Let yit ˆ …xit = j xjt † be the proportion of risky assets invested in asset i, yt ˆ …y1t ; . . . ; ynt † be the mix of risky assets, and zit be a target weight for the proportion of risky assets invested in asset i. Finally, let Lit and Uit be lower and upper bounds on the proportion of risky assets invested in asset i. We employ equal and market-value target-weight constraints in the paper. Marketvalue target-weight constraints are more meaningful in the sense that they are consistent with one of the central results found in many asset pricing models, speci®cally, that in equilibrium a representative investor will hold all the risky assets at their market-value weights. These constraints are most easily described in terms of the twelve value-weighted industries employed in the analysis below. The market-value weight of industry i is de®ned as the total market value of industry i divided by the total market value of all twelve industries. In turn, the market value of industry i is the sum of the individual ®rm values (price times number of shares of common stock) in the industry. For example, Table 1 below shows that at the end of December 1995 the proportional market values of Petroleum and Transportation are 6.52% and 1.36%, respectively. We employ two speci®c sets of constraints: market values plus or minus either 5% or 10%. 6 In these cases, we set zit equal to the market value of asset (industry) i, Lit ˆ max…0; ÿc ‡ zit †; and Uit ˆ c ‡ zit ; where c is either 5% or 10%, and the target-weight constraints are , ! X Lit 6 xit xjt 6 Uit all i:

7

…6†

j

In terms of the speci®c example, suppose portfolios were to be chosen at the beginning of the ®rst quarter of 1996 with constraints of market values ‹5%. 5 The solvency constraint (5) is not binding for the power functions, with c < 1, and discrete probability distributions with a ®nite number of outcomes because the marginal utility of zero wealth is in®nite. Nonetheless, it is convenient to explicitly consider (5) so that the non-linear programming algorithm used to solve the investment problems does not attempt to evaluate an infeasible policy as it searches for the optimum. 6 Obviously, any speci®c set of constraints is arbitrary. Therefore, we focus primarily on the constraints targeted on market values ‹5%, but report some results with the looser 10% constraints to show that the results are robust. 7 It is dicult to solve a mathematical programming problem that includes non-linear constraints such as Eq. (6). We thank Michael Best for showing us how to convert Eq. (6) into a system of linear constraints.

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Table 1 Summary description of the US industry groupsa 1925

1933

Percent share of value Petroleum Finance and real estate Consumer durables Basic industries Food and tobacco Construction Capital goods Transportation Utilities Textiles and trade Services Leisure

14.66 1.32 14.35 14.79 9.66 0.40 4.94 20.00 11.41 7.46 0.17 0.85

13.11 2.77 13.74 20.07 12.85 1.47 5.75 8.73 14.09 6.54 0.16 0.72

13.72 2.68 16.82 19.33 5.76 1.68 12.50 3.23 18.05 4.98 0.23 1.04

6.52 21.24 12.85 15.50 9.35 1.31 8.63 1.36 11.44 4.22 4.05 3.52

13.12 6.20 13.88 19.61 8.21 1.58 9.69 4.60 15.04 5.78 0.70 1.61

Total value (US$ billion)

27.29

27.95

521.00

5,446.04

756.90

Percent share of ®rms Petroleum Finance and real estate Consumer durables Basic industries Food and tobacco Construction Capital goods Transportation Utilities Textiles and trade Services Leisure

8.85 3.22 14.69 18.71 12.88 1.01 7.85 15.69 4.83 9.26 0.60 2.41

5.98 5.56 14.39 20.09 11.82 2.71 10.11 10.97 3.70 10.54 0.71 3.42

3.97 6.24 14.75 17.67 8.91 4.13 14.67 5.11 10.70 9.32 0.97 3.57

4.12 36.82 10.79 10.02 2.55 2.55 8.28 1.70 7.64 6.33 4.84 4.37

4.72 10.76 14.62 17.27 8.81 3.04 11.34 6.59 8.41 9.19 1.79 3.45

497

702

1,234

2,355

1,174

Total number of ®rms a

1965

1995

Average

Numbers other than averages re¯ect year-end values. Averages are calculated over 840 months.

Then, the proportion of risky assets invested in Petroleum would have to lie within the range of 1.52% and 11.52% and the proportion invested in Transportation would have to lie in the range of 0% and 6.36%. By way of contrast, equal target-weight constraints provide a simple way of ``forcing diversi®cation'' as well as an opportunity to examine the robustness of the results generated from the more economically meaningful market-value target-weight constraints. To construct equal target-weight constraints with n assets, we set zit ˆ 1=n;

Lit ˆ ÿc ‡ zit ;

and

Uit ˆ c ‡ zit ;

where c is an arbitrary constant set to say 5%. Then, Eq. (6) once again describes the target-weight constraints.

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The inputs to the model are based on the EPAA with quarterly revision. At the beginning of quarter t, the portfolio problem (2)±(5) (or (2)±(6)) for that quarter uses the following inputs: the (observable) riskfree return for quarter t, the (observable) call money rate +1% at the beginning of quarter t, and the (observable) realized returns for the risky asset categories for the previous k quarters. Each joint realization in quarters t ÿ k through t ÿ 1 is given probability 1/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 speci®ed and used, there is no information loss; all moments and correlations are implicitly taken into account. It may be noted that the empirical distribution of the past k periods is optimal if the investor has no information about the form and parameters of the true distribution, but believes that this distribution went into e€ect k periods ago. With these inputs in place, the portfolio weights xt for the various asset categories and the proportion of assets borrowed are calculated by solving system (2)±(5) (or (2)±(6)) via non-linear programming methods. 8 At the end of quarter t, the realized returns on the risky assets are observed, along with the r (which may di€er from the decision borrowing rate realized borrowing rate rBt 9 d rBt ). Then, using the weights selected at the beginning of the quarter, the realized return on the portfolio chosen for quarter t is recorded. The cycle is repeated in all subsequent quarters. 10 All reported returns are gross of transaction costs and taxes and assume that the investor in question had no in¯uence on prices. There are several reasons for this approach. First, as in previous studies, we wish to keep the complications to a minimum. Second, the return series used as inputs and for comparisons also exclude transaction costs (for reinvestment of interest and dividends) and taxes. Third, many investors are tax-exempt and various techniques are available for keeping transaction costs low. Finally, since the proper treatment of these items is non-trivial, they are better left to a later study. 3. Data The data used to estimate the probabilities of next periodÕs returns on risky assets, and to calculate each periodÕs realized returns on risky assets, come from several sources. The returns for the US equal- and value-weighted industry groups are constructed from the returns on individual New York Stock 8

The non-linear programming algorithm employed is described in Best (1975). The realized borrowing rate is calculated as a monthly average. 10 Note that if k ˆ 32 under quarterly revision, then the ®rst quarter for which a portfolio can be selected is b + 32, where b is the ®rst quarter for which data are available. 9

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Exchange ®rms contained in the Center for Research in Security Prices (CRSP) monthly returns data base. The industry data in this paper updates the 1934±86 data in Grauer et al. (1990) through 1995. The ®rms are combined into twelve industry groups on the basis of the ®rst two digits of the ®rmsÕ SIC codes, with equal- and value-weighted industry indices constructed from the same universe of ®rms. Table 1 identi®es the twelve industry groups, the total number of ®rms in the sample and their total market values, together with a breakdown of each industryÕs percentage share of all ®rms and of their total market values at four points in time. In addition, Table 1 shows the average market-value weights for each of the industries. Average market-value weights combined with marketvalue weights at four points in time provide a rough idea of the magnitudes of the period-by-period market-value target-weight constraints employed in the portfolio optimizer. In some cases, the changes over time are quite dramatic. For example, at the end of 1925, ®nance and real estate (transportation) represent 1.32% (20.00%) of the total market value of US$27.29 billion. At the end of 1995, the weights reverse: ®nance and real estate (transportation) represent 21.24% (1.36%) of the total market value of US$5,466 billion. The riskfree asset is assumed to be 90-day US Treasury bills maturing at the end of the quarter; we use the Survey of Current Business and the Wall Street Journal as sources. The borrowing rate is assumed to be the call money rate +1% for decision purposes (but not for rate of return calculations). The apd , is viewed as persisting plicable beginning of period decision rate, rBt throughout the period and thus as riskfree. For 1934±76, the call money rates are obtained from the Survey of Current Business. For later periods, the Wall Street Journal is the source. Finally, margin requirements for stocks are obtained from the Federal Reserve Bulletin. 11 4. Portfolio returns and investment policies Because of space limitations, only a portion of the results can be reported here. However, Figs. 1 and 2 provide a fairly representative sample of our ®ndings. In each comparison, we calculate and include the returns on unlevered and levered equal- or value-weighted benchmark portfolios of the risky assets in the investment universe. E5, for example, is a portfolio with 50% invested in EW, an equal-weighted portfolio of the risky assets, and 50% in riskless lending. Likewise, V15 is a portfolio with 150% invested in VW, a valueweighted portfolio of the risky assets, and 50% in borrowing. 11

There is no practical way to take maintenance margins into account in our programs. In any case, it is evident from the results that they would come into play only for the more risk-tolerant strategies, and even for them only occasionally, and that the net e€ect would be relatively neutral.

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Fig. 1. Geometric means and standard deviations of annual portfolio returns for the value-weighted industry groups, the value-weighted benchmark portfolios, and the power policies with and without target-weight constraints, 1934±1995, borrowing permitted. Target-weight constraints are industry market values ‹5%.

4.1. Compound return±standard deviation pairs Fig. 1 plots the annual geometric means and the standard deviations 12 of the realized returns for two sets of ten power utility strategies, based on cÕs in Eq. (1) ranging from )50 (extremely risk averse) to 1 (risk neutral), under quarterly revision for the 62-year period 1934±95, when the risky portion of the 12

For consistency with the geometric mean, the standard deviation is based on the log of one plus the rate of return. This quantity is very similar to the standard deviation of the rate of return for levels less than 25%.

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Fig. 2. Geometric means and standard deviations of annual portfolio returns for the equalweighted industry groups, the equal-weighted benchmark portfolios, and the power policies with and without target-weight constraints, 1966±95, borrowing permitted. Target-weight constraints are industry market value ‹5%.

investment universe is twelve value-weighted US industry groups. The estimating period is 32 quarters and borrowing is permitted. The ®rst set of strategies (see dark circles) shows the returns generated without target-weight constraints and the second set (see open circles) exhibits those with targetweight constraints: industry market values ‹5%. The ®gure also shows the geometric means and standard deviations of the returns of two sets of benchmark portfolios: the underlying value-weighted industry groups (see dark squares) and the up- and down-levered value-weighted benchmark portfolios (see open triangles).

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The 1934±95 period is characterized by relatively low riskfree rates together with high market returns. Riskfree lending yields a compound rate of return of 4.05%. VW, the passive strategy of buying and holding a valueweighted portfolio of the twelve value-weighted industries, i.e., the CRSP value-weighted portfolio, earns a geometric mean return of 11.57%. Turning to the active strategies, we make two central observations. First, the imposition of target-weight constraints decreases the realized compound returns and the standard deviations of returns of the power investors. (The risk neutral investor is the only exception. In both the full period and the 1966±95 subperiod, this investorÕs compound return increases with target-weight constraints.) When borrowing is precluded, the same basic pattern emerges. Also, with very minor exceptions, the pattern holds as we move progressively from no target-weight constraints to constraints of market-value weights plus or minus 10% ± not reportedÐto constraints of market-value weights ‹5%. Second, although the )50 and )30 powers with target-weight constraints are exceptions, the levered value-weighted benchmark portfolios ``dominate'' the more risk-averse (low) powers. The opposite is true for the less risk-averse (high) powers. Higher riskfree and lower market returns characterize the 1966±95 subperiod. Riskfree lending achieves a 6.97% compound rate of return, while the CRSP value-weighted portfolio earns a geometric mean return of 10.67%. In contrast to the full period, the frontiers (not shown) with and without targetweight constraints dominate the levered value-weighted benchmark portfolios. Turning to the equal-weighted industry groups, we note that with minor exceptions the equal-weighted industry groups and equal-weighted benchmark portfolios realize higher compound returns and standard deviations than their value-weighted counterparts. This re¯ects the well-known small ®rm e€ect, coupled with the strategy of ``selling the winners and buying the losers'' implicit in equal weighting. Interestingly, the realized returns generated from two sets of target-weight constraintsÐmarket-value target weights ‹5% and equal target weights ‹5% (not reported) ± are very close to each other. This is somewhat surprising in light of the theoretic superiority of market-value target weights and the di€erences between the equal (with weights of 8.33% for each industry at each decision point) and market-value target weights (see Table 1). However, as with the value-weighted industries, the most striking results occur in the 1966±95 subperiod. Fig. 2 shows that the ``frontier'' without target-weight constraints clearly ``dominates'' the frontier with target-weight constraints. In addition, although the power strategies with target-weight constraints in turn ``dominate'' the valueweighted benchmark portfolios, the higher powers ``are beaten easily'' by the equal-weighted benchmark portfolios. To provide robustness, we also examined a global universe covering the 1968±95 period. To save space, we simply note that the global results are

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similar to the 1966±95 industry results. The global results, as well as a more complete set of industry results, are available from the authors. 4.2. The investment policies It is not practical to report the full time series of the investment policies. Instead, the broad outlines of the policies without target-weight constraints are compared to the policies with market-value target-weight constraints of ‹5% when the investment universe consists of twelve value-weighted industry groups, borrowing permitted, in the 1934±95 period. The bottom line in this case is that modest di€erences in return space are accompanied by dramatic di€erences in the investment policies. Without target-weight constraints, the most favored industries are: Petroleum, which is chosen from at least 45% of the time by powers )50 through )3, to 16% of the time by power 1, and, Services, which is selected from at least 31% of the time by powers )50 through )3, to 23% of the time by the riskneutral investor. At the other end of the spectrum, Finance and real estate and Transportation are chosen 6% (or less) of the time by all the powers. By way of contrast, with target-weight constraints, Petroleum, Basic, Capital goods, and Utilities are chosen 93.5% of the time by all the powers, while previously outof-favor Transportation and Finance and real estate are selected over 38% and 50% of the time by all the powers. Also, somewhat surprisingly, target-weight constraints cause the power functions to adopt a more conservative approach in their use of ®nancial leverage. With target-weight constraints, the strategies lend (borrow) more (less) often. Without target-weight constraints, powers )1 through 1 often invest over 100% of their capital in an industry group. Not surprisingly, the risk-neutral investor exhibits the extreme form of plunging behavior. In the 21 times he invests in Construction, he places an average of 228.4% of his capital in that category. With target-weight constraints, the risk-neutral investor again places the maximum average percent of capital in an industry. But in this case it is 35.6% of capital in the basic category. Ironically, in the absence of targetweight constraints, this is the only industry in which the risk-neutral (or any other power) investor does not place any capital. Turning to the leverage question, with a few minor exceptions, the average amount lent (borrowed) is larger (smaller) with the imposition of target-weight constraints. (Recall from the previous paragraph that this increase in the average amount lent is accompanied by an increase in the number of times that lending takes place.) Finally, the di€erence in the number of industry groups chosen at a point in time is striking. Without (with) target-weight constraints, the power functions concentrate their holdings in from one to four (eight to eleven) industry groups.

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5. Is the reduction in risk induced by constraints worth the reduction in return? 5.1. The tests Target-weight constraints reduce risk and return. The question is whether the reduction in risk is worth the reduction in return. While the ®gures get at the heart of the matter, unfortunately, they do not give us a sense of how much of the di€erence can be attributed to randomness. In order to shed light on this issue we focus on two paired tests: a paired t-test of the di€erence in Jensen's (1968) alpha and a paired t-test of the di€erence in ex post utility. JensenÕs test embodies both statistical and economic assumptions about the way assets are priced and is not without its critics. 13 In our case, there are at least four reasons for making cautious use of JensenÕs test. First, Roll (1978) pointed out that the results of JensenÕs test are ambiguous because the choice of the benchmark (or market) portfolio a€ects the measures of both systematic risk (beta) and abnormal return (alpha). Second, our empirical work is concerned with sector-rotation strategies. Unfortunately, selecting across industries is neither a pure selectivity strategy, implicit in JensenÕs test, nor a pure market-timing strategy as embodied in Treynor and Mazuy's (1966) or Henriksson and Merton's (1981) tests of market timing. Third, risk may change with time or economic conditions. (Grauer and Hakansson (1999) investigate the performance of the dynamic investment model employing conditional Jensen, Treynor±Mazuy, and Henriksson±Merton measures that, following Ferson and Schadt (1996), make beta a linear function of dividend yields and riskfree interest rates.) Fourth, JensenÕs measure adjusts for systematic risk while expected utility maximizers may care about more than systematic risk. To implement JensenÕs test we run the regression rjt ÿ rLt ˆ aj ‡ bj …rmt ÿ rLt † ‡ ejt ; where rjt is return on portfolio j, rmt the return on the CRSP value-weighted index, and rLt is the return on three-month treasury bills. The intercept aj is the measure of investment performance. Positive (negative) values indicate superior (inferior) performance. The null hypothesis is that there is no superior investment performance and the alternative hypothesis is that there is. Thus, we report the results of one-tailed t-tests. A t-test for paired regression coecients can be used to test for the di€erence in the alphas or betas of portfolios selected with and without target-weight constraints as well any other slope coecients in a more general model. 14 Let vectors and matrices be denoted by boldface type. The general model is 13 14

See, for example, Roll (1978), Dybvig and Ross (1985), Green (1986), and Grauer (1991). We thank Reo Audette for developing the test.

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yj ˆ XBj ‡ ej ;

j ˆ 1; 2;

…7†

where yj is an n-vector of dependent measures, X is an n ´ k matrix of independent measures that includes a column of ones for the intercept, Bj is a kvector of regression coecients, ej is an n-vector of residuals that is distributed normal(0, r2ej I), E…e1 e02 ˆ cov…e1 ; e2 †I, and I is an n ´ n identity matrix. Let ÿ1 bj ˆ …X0 X† X0 yj be the ordinary least squares estimator of Bj . Then, to test the hypothesis that b1i ˆ b2i we employ the t-statistic p …8† tnÿk ˆ …b1i ÿ b2i †= Cii ; where bji is the ith element of the jth vector of regression coecients, ej ˆ yj ) Xbj is the vector of sample residuals, s2e1 ÿe2 ˆ …e1 ÿ e2 †0 …e1 ÿ e2 †= …n ÿ k† is the estimate of the variance of the di€erence in the residuals, ÿ1 C ˆ s2e1 ÿe2 …X0 X† is the sample covariance matrix of the di€erence in b1 and b2 , and Cii is the ith diagonal element of C. The second performance measure is based on an exact knowledge of investorsÕ utility functions rather than on a model of asset pricing. Consequently, we expect it to be more consistent with the ex ante objective functions than the traditional alpha. At each of the 248 quarterly decision points from 1934 to 1995, each investor, employing an estimate of the joint return distribution, maximizes the ex ante expected utility of returns by solving (2)±(5), without target-weight constraints, or (2)±(6), with target-weight constrainfts. Now, suppose each investor is asked to evaluate the ex post utility of the two time series of portfolio returns. De®ne the ex post utility of the time series of (optimal ex ante) portfolio returns as EU ˆ EU ˆ

T X 1 1 c …1 ‡ rt …xt †† ; T c tˆ1 T X 1 tˆ1

T

ln …1 ‡ rt …xt ††;

c 6 1;

c 6ˆ 0; …9†

c ˆ 0;

where each portfolio return 1 ‡ rt …xt † is assigned equal probability of occurrence. To compare the utility of the return series r11 ; . . . ; rn1 with the utility of the return series r12 ; . . . ; rn2 for two di€erent strategies, we calculate the statistic d p ; …10† r…d†= n Pn where d ˆ tˆ1 …u…1 ‡ rt1 † ÿ u…1 ‡ rt2 ††=n, r(d) is the standard deviation of 1 u…1 ‡ rt † ÿ u…1 ‡ rt2 †, and u(.) denotes one in (9). In each   of the  utility2 functions  1 † ˆ E u…1 ‡ r † while the alternative case, the null hypothesis is E u…1 ‡ r t  t    hypothesis is E u…1 ‡ rt1 † > E u…1 ‡ rt2 † . Note that if we set X equal to a tˆ

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vector of ones in (7) or set c ˆ 1 in (9), the paired t-test in (8) or (10) reduces to the well-known paired t-test for the di€erence in two means. While (9) gives the desired ranking of the time series of returns for each of the powers with and without target-weight constraints, the utility numbers are nonintuitive, particularly for the low powers. In addition, the results show that for the more risk-averse powers any unexpectedly large loss su€ered by one policy relative to the other will almost surely result in the latter policy being preferred ex post. Therefore, for reporting purposes, the ex post utility numbers from (9) are transformed into ex post certainty equivalent returns as follows: 1=c

1 ‡ rCE ˆ ‰ cEU Š ;

c 6 1;

c 6ˆ 0;

1 ‡ rCE ˆ exp ‰ EU Š;

c ˆ 0: …11†

At the risk of oversimplifying, we draw the following analogy between the certainty equivalent return and JensenÕs alpha. Alpha reduces the reward (ex post average excess return) to risk (beta) trade-o€ of the mean-variance capital asset pricing model into a single risk-adjusted rate of return. Similarly, we can think of the certainty equivalent return as reducing the reward (ex post average return) to risk (standard deviation) trade-o€ of a speci®c power utility investor into a single risk-adjusted rate of return. 15 In Tables 2 and 3 our comparisons of the di€erences between the returns with and without target-weight constraints focus on the di€erences between these two sets of risk-adjusted returns. 5.2. The results Fig. 1 indicates that with target-weight constraints the reduction in risk may have been worth the reduction in return in the value-weighted industries setting during the 1934±95 period. On the other hand, there is a much stronger impression, drawn from Fig. 2, that the reduction in risk is not worth the reduction in return when target-weight constraints are imposed on the equalweighted industries universe in the 1966±95 period. Table 2 contains the results for JensenÕs test. Columns 1±4 show the arithmetic average excess returns (means) without target-weight constraints (denoted as unrestricted in the table), the alphas without target-weight constraints, the level of signi®cance for the null hypothesis H0: aU ˆ 0 versus the alternative H0: aU > 0, and the betas without target-weight constraints. The corresponding results with target-weight constraints (denoted as restricted) are presented in columns 5±8. Finally, the levels of signi®cance for the hypotheses: H0: lU ˆ lR versus Ha: lU > lR ; H0: aU ˆ aR versus Ha: aU > aR ; and H0: bU ˆ bR ; versus Ha: bU > bR are shown in columns 9±11, 15 The argument is simpli®ed as power utility weighs all the moments of the distribution not just the mean and variance.

0.09 0.14 0.27 0.39 0.72 1.02 1.48 2.03 2.41 2.89

Panel B: Twelve equal-weighted industry groups, 1966±95 )50 0.18 0.09 0.03 0.08 )30 0.30 0.15 0.03 0.12 )15 0.57 0.28 0.03 0.24 )10 0.83 0.41 0.03 0.35 )5 1.47 0.73 0.02 0.61 )3 1.99 0.93 0.04 0.88 )1 3.01 1.33 0.04 1.40 0 4.03 1.93 0.01 1.75 0.5 3.94 1.63 0.05 1.91 1 4.06 1.48 0.10 2.14 0.02 0.04 0.07 0.10 0.18 0.27 0.17 0.39 0.52 0.64

0.06 0.10 0.21 0.27 0.47 0.62 0.71 0.74 0.58 0.39

0.26 0.26 0.26 0.26 0.26 0.24 0.40 0.30 0.26 0.20

0.09 0.09 0.08 0.07 0.04 0.02 0.03 0.03 0.09 0.20

Level of sig. Ha : aR > 0

0.05 0.09 0.17 0.25 0.45 0.62 1.09 1.36 1.57 1.87

0.07 0.11 0.21 0.28 0.46 0.58 0.84 1.07 1.31 1.76

bR

0.01 0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.04 0.13

0.00 0.01 0.01 0.03 0.10 0.13 0.08 0.07 0.06 0.13

Ha : lU > lR

0.02 0.02 0.02 0.02 0.01 0.03 0.02 0.02 0.09 0.21

0.05 0.07 0.14 0.28 0.48 0.59 0.38 0.37 0.38 0.64

Ha : aU > aR

Level of signi®cance

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Ha : bU > bR

Unrestricted is equivalent to no target-weight constraints. Restricted is equivalent to target-weight constraints (industry market values ‹5%). Arithmetic average excess returns are denoted by l. The null hypotheses are H0: aU ˆ 0; H0: aR ˆ 0; H0 : lU ˆ lR ; H0 : aU ˆ aR ; and H0: bU ˆ bR .

a

0.20 0.33 0.65 0.87 1.45 1.86 2.50 3.00 3.36 4.13

aR

lR

bU

Restricted

Level of sig. Ha : aU > 0

lU

aU

Unrestricted

Panel A: Twelve value-weighted industry groups, 1934±95 )50 0.40 0.18 0.04 0.10 )30 0.59 0.25 0.05 0.16 )15 0.93 0.34 0.06 0.28 )10 1.14 0.35 0.10 0.37 )5 1.69 0.49 0.08 0.57 )3 2.15 0.56 0.10 0.75 )1 3.05 0.82 0.07 1.05 0 3.73 0.90 0.09 1.33 0.5 4.37 0.78 0.17 1.69 1 5.13 0.10 0.46 2.37

Power

Table 2 Results for the regression rjt ÿ rLt ˆ aj ‡ bj …rmt ÿ rLt † ‡ ejt . Applied to quarterly returns for twelve value-weighted industry groups, 1934±95, and twelve equal-weighted industry groups, 1966±95. Quarterly portfolio revision, 32-quarter estimating period, leverage permitted.a

1268 R.R. Grauer, F.C. Shen / Journal of Banking & Finance 24 (2000) 1253±1274

lR

Restricted

0.84 1.10 1.86 2.63 4.75 6.52 11.22 13.72 15.47 17.47

1.18 1.65 2.91 3.71 5.69 6.77 9.16 10.82 12.73 16.32

rR

1.62 1.65 1.68 1.69 1.69 1.83 1.77 2.77 3.52 4.59

0.73 0.70 0.53 0.77 1.20 1.82 2.59 3.40 3.96 5.13

CER

0.09 0.16 0.30 0.44 0.75 0.97 1.53 2.00 1.53 1.16

0.19 0.25 0.27 0.27 0.24 0.29 0.55 0.73 1.01 1.00

lU ) lR

0.19 0.36 0.76 1.12 1.67 2.64 2.97 3.55 3.64 4.13

0.75 1.11 1.25 1.45 1.50 2.34 3.09 4.17 6.13 8.71

rU ) rR

Unrestricted ± Restricted

)0.01 )0.01 )0.01 )0.02 0.14 )0.03 0.84 1.51 1.24 1.16

)1.26 )2.30 )1.28 )1.41 )0.45 )0.66 )0.05 0.25 0.58 1.00

CEU ) CER

a Unrestricted is equivalent to no target-weight constraints. Restricted is equivalent to target-weight constraints (industry market values ‹5%). Arithmetic average returns are denoted by l, standard deviations by r, and certainty equivalent returns by CE.

1.79 1.84 1.97 2.09 2.42 2.72 3.18 3.73 4.11 4.59

CEU

Panel B: Twelve equal-weighted industry groups, 1966±95 )50 1.88 1.03 1.61 )30 1.99 1.46 1.64 )15 2.27 2.62 1.67 )10 2.53 3.75 1.67 )5 3.17 6.43 1.83 )3 3.69 9.16 1.80 )1 4.71 14.19 2.61 0 5.73 17.27 4.28 0.5 5.64 19.11 4.76 1 5.76 21.60 5.76

rU 1.20 1.33 1.65 1.87 2.45 2.86 3.50 4.00 4.36 5.13

lU

Unrestricted

Panel A: Twelve value-weighted industry groups, 1934±95 )50 1.40 1.94 )0.52 )30 1.59 2.76 )1.60 )15 1.93 4.16 )0.75 )10 2.14 5.16 )0.65 )5 2.70 7.19 0.75 )3 3.15 9.11 1.16 )1 4.05 12.25 2.54 0 4.73 14.99 3.65 0.5 5.38 18.85 4.54 1 6.13 25.04 6.13

Power

Table 3 Quarterly arithmetic means, standard deviations, and certainty equivalent returns for twelve value-weighted industry groups, 1934±95, and twelve equal-weighted industry groups, 1966±95. Quarterly portfolio revision, 32-quarter estimating period, leverage permitted.a

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respectively. The results for the value-weighted industries in the 1934±95 period are shown in Panel A and for the equal-weighted industries in the 1966±95 period in Panel B. In both periods, the means and betas are uniformly larger without targetweight constraints. In all cases, the betas without target-weights constraints are statistically signi®cantly larger than the betas with target-weight constraints. At the 10% level, so are 7 (9) out of 10 means for the value-weighted (equalweighted) industries. The interrelationship between the alphas is more complex. First, abnormal returns are clearly present: all four sets of alphas are positive. Without target-weight constraints, all but two of the alphas are statistically signi®cantly greater than zero at the 10% level. With target-weight constraints, all but one (none) of the alphas are statistically signi®cantly greater than zero at the 10% level in the value-weighted (equal-weighted) universe. Second, for the most part, the alphas without target-weight constraints are larger than their restricted counterparts. (The exceptions are for the ®ve powers )3 to 1 in the value-weighted universe.) Third, in the value-weighted industries universe, the alphas for the )50 and )30 powers without target-weight constraints are statistically signi®cantly larger than their restricted counterparts at the 10% level. In the equal-weighted industries universe, all the alphas, except for those of the 0.5 and 1 powers, are statistically signi®cantly larger than their restricted counterparts at the 3% level. This result strongly supports the visual impressions drawn from Fig. 2. Table 3 shows quarterly arithmetic average raw returns (means), standard deviations, and certainty equivalent returns. The results without target-weight constraints (unrestricted) are presented in columns 1±3. The corresponding results with target-weight constraints (restricted) are shown in columns 4±6. The ®nal three columns, 7±9, show the di€erence between the unrestricted and restricted means, standard deviations, and certainty equivalent returns. In light of what we have seen, the results may seem surprising. Without target-weight constraints, reward (average return) and risk (standard deviation) are larger, but in 11 of 20 cases the ex post certainty equivalent returns are smaller. In the equal-weighted industry universe, ®ve of the certainty equivalents are smaller, even though the corresponding alphas are statistically signi®cantly larger at the 3% level. In the value-weighted industries universe, four of the certainty equivalent returns are negative. Yet, two of the corresponding alphas (those of the )50 and )30 powers) are statistically signi®cantly larger at the 10% level. The results illustrate the essence of the problems of measuring performance with ex post utility in the presence of estimation risk. Ex ante, investors prefer the policies without target-weight constraints, i.e., they choose the unconstrained policies over the feasible constrained alternatives. However, the optimizations are performed on ex ante joint return distributions, drawn from moving windows of 32 quarters of historical data, that obviously contain estimation error. As a consequence, the realized portfolio returns could be (and,

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in fact, sometimes are) either larger or smaller than the largest or smallest ex ante portfolio returns. As the disutility of a loss outweighs the utility of an equal-size gain for all risk averse investors, either the constrained or unconstrained policies could be preferred ex post depending on the degree of investor risk aversion. The results indicate that the extreme aversion to losses exhibited by the more risk-averse members of the power utility functions will almost certainly result in the policies that lose less being preferred ex post. To illustrate further, in the value-weighted industries universe the results for the )50 power, as well as for those of the powers )30, )15, and )10, are driven primarily by one observation. In the second quarter of 1962, the )50 power investor lost 9.65% (5.32%) without (with) target-weight constraints. While losses this large are rare for the )50 power, measuring 5.7 and 5.5 standard deviations below their averages, the corresponding utilities are much more extreme, plotting 15.7 and 14.2 standard deviations below their averages. Dropping that one observation is enough to change the sign of the certainty equivalent and almost, but not quite, enough to reverse the order of preference. The certainty equivalent without (with) target-weight constraints changes from )0.52 to + 0.84 (0.73 to 0.92). In one sense, it is surprising that the negative certainty equivalent can be traced to a loss of 9.65%. However, in another sense it is not so surprising when we recognize that ex ante the investor, estimating the joint return distribution from a 32-quarter moving window, never envisioned such a loss. If someone that risk averse had foreseen the possibility, he would have altered his investment policy to avoid it completely. 6. Summary Several previous studies employing discrete-time dynamic portfolio theory in conjunction with the empirical probability assessment approach ®nd that the model oftentimes earns economically and statistically signi®cant abnormal returns in a variety of asset-allocation settings. Yet, the model does not diversify the way that asset pricing models and the ecient markets hypothesis indicate it should. The (naive) use of historic return data, coupled with the resulting lack of diversi®cation, suggests that it may be important to correct for estimation error. Two methods have been proposed: (1) correct for errors in the inputs, and (2) constrain the portfolio weights. Grauer and Hakansson (1995) examine the e€ect of correcting for estimation error in the means using statistical and ®nancial models. This paper explores the second. More speci®cally, the paper compares the investment policies and returns of the discrete-time dynamic investment model with, and without, constraints on the ``mix of risky assets''. Two sets of twelve US industries and an eight-country global setting are examined. Target-weight constraints lead to appreciably more diversi®cation and to less use of ®nancial leverage. This is coupled with less

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risk ± measured either as standard deviation of realized return or beta. The cost is less realized return. The key issue is whether the reduction in risk is worth the reduction in return. Visual comparisons of compound return-standard deviation plots and statistical comparisons of JensenÕs alpha suggest that the reduction in return is not worth the reduction in risk, particularly since 1966. On the other hand, for more risk-averse investors ex post utility and certainty equivalent returns suggest that it is. This latter result is somewhat counter intuitive given the compound return ± standard deviation and Jensen results coupled with the fact that ex ante the power utility investors prefer the policies without target-weight constraints. In fact, the results illustrate the problems of measuring performance with ex post utility in the presence of estimation risk. With estimation risk and a large enough sample, some realized returns will be larger and some will be smaller than the largest and smallest ex ante returns. The extreme aversion to these (unexpectedly large) losses exhibited by the more risk-averse members of the power utility functions will almost certainly result in the policies that lose less being preferred ex post. Moreover, the extreme nonlinear transformations involved in calculating expected utilities for these more riskaverse low powers will almost certainly destroy any semblance of normally distributed utility values required in a paired t-test of di€erences in ex post utility. Acknowledgements The research began when Shen was at the University of Waterloo. We gratefully acknowledge ®nancial support from the Social Sciences Research Council of Canada and the Centre for Accounting Research and Education. In addition, we thank Nils Hakansson, John Herzog, Peter Klein and two referees for valuable comments, and Reo Audette, John Janmaat, Maciek Kon, Jaspreet Sahni, and William Ting for most capable assistance.

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