Do constraints improve portfolio performance?
Robert R. Grauer
a,*, Frederick C. Shen
b,1 aFaculty 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. There-fore, 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 postutility and certainty equivalent returns suggest that it is. The results, however, illustrate the problems associated with usingex post utility to measure performance.Ó2000 Elsevier Science B.V. All rights
reserved.
JEL classi®cation:G11
Keywords:Multiperiod power utility asset-allocation
www.elsevier.com/locate/econbase
*Corresponding author. Tel.: +1-604-291-3722; fax: +1-604-291-4920.
E-mail addresses:grauer@sfu.ca (R.R. Grauer), Fred_Shen@manulife.com (F.C. Shen).
1Tel.: +1-416-415-7636; fax: +1-416-926-5783.
0378-4266/00/$ - see front matterÓ2000 Elsevier Science B.V. All rights reserved.
1. Introduction
Grauer and Hakansson (1982, 1985, 1986, 1987) and Grauer et al. (1990) applied discrete-time dynamic portfolio theory2in conjunction with the em-pirical 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 esti-mation 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 di-versi®cation exhibited by portfolio optimizers. Two of the central paradigms of ®nancial economicsÐmodels of asset pricing and the ecient markets hy-pothesisÐsuggest that investors should hold diversi®ed portfolios. For ex-ample, 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 ecient markets hypothesis is that investors not possessing special knowledge would be well-advised 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 eects of correcting for estimation error. Two methods have been proposed: (1) incorporate corrections for esti-mation 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.
4The arbitrage pricing theory also predicts that investors will diversify widely, holding portfolios
estimation error, particularly in the means, can improve investment perfor-mance substantially. See, for example, Jobson et al. (1979), Jobson and Korkie (1980, 1981), and Jorion (1985, 1991). On the other hand, Grauer and Ha-kansson (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 non-negativity 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 eects 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 hold-ings of all the assets in the universe. Second, they generate sample returns based on the population parameters and form mean±variance ecient port-folios 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 pa-rameters. 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 diver-si®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 estimation-error results in Grauer and Hakansson (1995) suggest they might not.
``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 in-vestment model and the method employed to make it operational. Section 3 describes the data. Section 4 records the risk±return trade-os 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 Ha-kansson (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 returnsrgiven by
max
in periodtif statesoccurs,c61 is a parameter that remains ®xed over time,xit
is the amount invested in risky asset categoryiin periodtas a fraction of own capital,xLtis the amount lent in periodtas a fraction of own capital,xBtis the
amount borrowed in period t as a fraction of own capital, xt
x1t;. . .;xnt;xLt;xBt
,rit is the anticipated total return (dividend yield plus
cap-ital gains or losses) on asset category i in period t, rLt is the return on the
riskfree asset in periodt,rd
Btthe interest rate on borrowing at the time of the
decision at the beginning of periodt,mit is the initial margin requirement for
asset categoryiin periodtexpressed as a fraction, andpts is the probability of
statesat the end of periodt, in which case the random returnritwill assume the
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 per-missible under the margin requirements that apply to the various asset cate-gories. Constraint (5) rules out any (ex ante) probability of bankruptcy.5
In this paper, we also place upper and lower bounds on the mix of risky assets. Letyit xit=Pjxjtbe the proportion of risky assets invested in asseti,
yt y1t;. . .;yntbe the mix of risky assets, and zit be a target weight for the proportion of risky assets invested in asseti. Finally, letLit 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. Market-value 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 industryiis de®ned as the total market value of industryidivided by the total market value of all twelve industries. In turn, the market value of industryiis 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 em-ploy two speci®c sets of constraints: market values plus or minus either 5% or 10%.6In these cases, we setzit equal to the market value of asset (industry)i,
Litmax 0;ÿczit; and Uitczit;
wherecis either 5% or 10%, and the target-weight constraints are7
Lit6 xit
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, withc<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.
7It is dicult to solve a mathematical programming problem that includes non-linear
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 Trans-portation 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 withn assets, we set
zit 1=n; Lit ÿczit; and Uitczit;
wherecis an arbitrary constant set to say 5%. Then, Eq. (6) once again de-scribes the target-weight constraints.
Table 1
Summary description of the US industry groupsa
1925 1933 1965 1995 Average
Percent share of value
Petroleum 14.66 13.11 13.72 6.52 13.12
Finance and real estate 1.32 2.77 2.68 21.24 6.20 Consumer durables 14.35 13.74 16.82 12.85 13.88
Basic industries 14.79 20.07 19.33 15.50 19.61
Food and tobacco 9.66 12.85 5.76 9.35 8.21
Construction 0.40 1.47 1.68 1.31 1.58
Capital goods 4.94 5.75 12.50 8.63 9.69
Transportation 20.00 8.73 3.23 1.36 4.60
Utilities 11.41 14.09 18.05 11.44 15.04
Textiles and trade 7.46 6.54 4.98 4.22 5.78
Services 0.17 0.16 0.23 4.05 0.70
Leisure 0.85 0.72 1.04 3.52 1.61
Total value (US$ billion) 27.29 27.95 521.00 5,446.04 756.90
Percent share of ®rms
Petroleum 8.85 5.98 3.97 4.12 4.72
Finance and real estate 3.22 5.56 6.24 36.82 10.76 Consumer durables 14.69 14.39 14.75 10.79 14.62
Basic industries 18.71 20.09 17.67 10.02 17.27
Food and tobacco 12.88 11.82 8.91 2.55 8.81
Construction 1.01 2.71 4.13 2.55 3.04
Capital goods 7.85 10.11 14.67 8.28 11.34
Transportation 15.69 10.97 5.11 1.70 6.59
Utilities 4.83 3.70 10.70 7.64 8.41
Textiles and trade 9.26 10.54 9.32 6.33 9.19
Services 0.60 0.71 0.97 4.84 1.79
Leisure 2.41 3.42 3.57 4.37 3.45
Total number of ®rms 497 702 1,234 2,355 1,174
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 quartert, the (observable) call money rate +1% at the beginning of quartert, and the (observable) realized returns for the risky asset categories for the previous k quarters. Each joint realization in quarterstÿkthrough tÿ1 is given prob-ability 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 pastk periods is optimal if the investor has no information about the form and pa-rameters of the true distribution, but believes that this distribution went into eectkperiods 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.8At the end of quartert, the realized returns on the risky assets are observed, along with the realized borrowing raterr
Bt(which may dier from the decision borrowing rate
rd
Bt).
9
Then, using the weights selected at the beginning of the quarter, the realized return on the portfolio chosen for quartert 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 compli-cations to a minimum. Second, the return series used as inputs and for com-parisons also exclude transaction costs (for reinvestment of interest and dividends) and taxes. Third, many investors are tax-exempt and various tech-niques 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 in-dustry groups are constructed from the returns on individual New York Stock
8
The non-linear programming algorithm employed is described in Best (1975).
9The realized borrowing rate is calculated as a monthly average. 10Note that ifk
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 ®rmsÕSIC 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 in-dustryÕs 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 value weights combined with market-value 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) rep-resent 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 theSurvey of Current Businessand theWall 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 ap-plicable beginning of period decision rate, rd
Bt, is viewed as persisting throughout the period and thus as riskfree. For 1934±76, the call money rates are obtained from theSurvey of Current Business.For later periods, theWall Street Journal is the source. Finally, margin requirements for stocks are obtained from theFederal 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 value-weighted portfolio of the risky assets, and 50% in borrowing.
11There is no practical way to take maintenance margins into account in our programs. In any
4.1. Compound return±standard deviation pairs
Fig. 1 plots the annual geometric means and the standard deviations12 of the realized returns for two sets of ten power utility strategies, based oncÕs 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
12For 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%.
investment universe is twelve value-weighted US industry groups. The esti-mating 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 target-weight 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).
The 1934±95 period is characterized by relatively low riskfree rates to-gether with high market returns. Riskfree lending yields a compound rate of return of 4.05%. VW, the passive strategy of buying and holding a value-weighted portfolio of the twelve value-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 impo-sition 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 subpe-riod. 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 target-weight constraints dominate the levered value-target-weighted benchmark portfolios. Turning to the equal-weighted industry groups, we note that with minor exceptions the equal-weighted industry groups and equal-weighted bench-mark portfolios realize higher compound returns and standard deviations than their value-weighted counterparts. This re¯ects the well-known small ®rm eect, coupled with the strategy of ``selling the winners and buying the losers'' implicit in equal weighting. Interestingly, the realized returns gener-ated 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 dierences 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 value-weighted benchmark portfolios, the higher powers ``are beaten easily'' by the equal-weighted benchmark portfolios.
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 dierences in return space are accompanied by dramatic dierences in the investment policies.
Without target-weight constraints, the most favored industries are: Petro-leum, 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
risk-neutral 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 out-of-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 target-weight 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 ac-companied by an increase in the number of times that lending takes place.)
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 dierence can be attributed to randomness. In order to shed light on this issue we focus on two paired tests: a pairedt-test of the dierence in Jensen's (1968) alpha and a pairedt-test of the dierence inex post utility.
JensenÕs test embodies both statistical and economic assumptions about the way assets are priced and is not without its critics.13In 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 aects the measures of both systematic risk (beta) and abnormal return (alpha). Second, our empirical work is con-cerned with sector-rotation strategies. Unfortunately, selecting across indus-tries 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 Hen-riksson 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
rjtÿrLtajbj rmtÿrLt ejt;
whererjt is return on portfolioj, rmt the return on the CRSP value-weighted
index, andrLtis the return on three-month treasury bills. The interceptajis the
measure of investment performance. Positive (negative) values indicate supe-rior (infesupe-rior) performance. The null hypothesis is that there is no supesupe-rior investment performance and the alternative hypothesis is that there is. Thus, we report the results of one-tailedt-tests.
At-test for paired regression coecients can be used to test for the dierence in the alphas or betas of portfolios selected with and without target-weight constraints as well any other slope coecients in a more general model.14Let vectors and matrices be denoted by boldface type. The general model is
yjXBjej; j1;2; 7
whereyj is an n-vector of dependent measures,Xis an n´k matrix of inde-pendent measures that includes a column of ones for the intercept,Bj is ak -vector of regression coecients,ejis ann-vector of residuals that is distributed normal(0, r2
where bji is the ith element of the jth vector of regression coecients, ejyj)Xbj is the vector of sample residuals, s2e1ÿe2 e1ÿe2
0 e
1ÿe2=
nÿk is the estimate of the variance of the dierence in the residuals,
Cs2
e1ÿe2 X
0Xÿ1
is the sample covariance matrix of the dierence inb1andb2,
andCii is theith diagonal element ofC.
The second performance measure is based on an exact knowledge of in-vestorsÕutility functions rather than on a model of asset pricing. Consequently, we expect it to be more consistent with theex anteobjective 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 theex postutility of the two time series of portfolio returns. De®ne theex postutility of the time series of (op-timalex ante) portfolio returns as
EUX
where each portfolio return 1rt xt is assigned equal probability of occur-rence. To compare the utility of the return seriesr11;. . .;r1nwith the utility of the return seriesr21;. . .;r2n for two dierent strategies, we calculate the statistic
vector of ones in (7) or setc1 in (9), the pairedt-test in (8) or (10) reduces to the well-known pairedt-test for the dierence 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 suered by one policy relative to the other will almost surely result in the latter policy being preferredex post. Therefore, for reporting purposes, theex postutility numbers from (9) are transformed intoex post certainty equivalent returns as follows:
1rCEcEU 1=c
; c61; c60; 1rCE expEU; c0:
11 At the risk of oversimplifying, we draw the following analogy between the certainty equivalent return and JensenÕs alpha. Alpha reduces the reward (ex postaverage excess return) to risk (beta) trade-oof the mean-variance capital asset pricing modelinto a single risk-adjusted rate of return. Similarly, we can think of the certainty equivalent return as reducing the reward (ex postaverage return) to risk (standard deviation) trade-oof a speci®c power utility investor into a single risk-adjusted rate of return.15In Tables 2 and 3 our comparisons of the dierences between the returns with and without target-weight con-straints focus on the dierences between these two sets of risk-adjusted returns.
5.2. The results
Fig. 1 indicates that with target-weight constraints the reduction in riskmay 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 equal-weighted industries universe in the 1966±95 period.
Table 2 contains the results for JensenÕs test. Columns 1±4 show the arith-metic average excess returns (means) without target-weight constraints (de-noted as unrestricted in the table), the alphas without target-weight constraints, the level of signi®cance for the null hypothesis H0:aU 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 hy-potheses: H0:lU lR versus Ha:lU>lR;H0:aUaR versus Ha:aU>
aR; and H0:bU bR; versus Ha:bU >bR are shown in columns 9±11,
15The argument is simpli®ed as power utility weighs all the moments of the distribution not just
Table 2
Results for the regressionrjtÿrLtajbj rmtÿrLt 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
Power Unrestricted Restricted Level of signi®cance
lU aU Level of sig.
Ha:aU> 0
bU lR aR Level of sig.
Ha:aR> 0
bR Ha:lU>lR Ha:aU>aR Ha:bU>bR
Panel A: Twelve value-weighted industry groups, 1934±95
)50 0.40 0.18 0.04 0.10 0.20 0.06 0.09 0.07 0.00 0.05 0.00
)30 0.59 0.25 0.05 0.16 0.33 0.10 0.09 0.11 0.01 0.07 0.00
)15 0.93 0.34 0.06 0.28 0.65 0.21 0.08 0.21 0.01 0.14 0.00
)10 1.14 0.35 0.10 0.37 0.87 0.27 0.07 0.28 0.03 0.28 0.00
)5 1.69 0.49 0.08 0.57 1.45 0.47 0.04 0.46 0.10 0.48 0.00
)3 2.15 0.56 0.10 0.75 1.86 0.62 0.02 0.58 0.13 0.59 0.00
)1 3.05 0.82 0.07 1.05 2.50 0.71 0.03 0.84 0.08 0.38 0.00
0 3.73 0.90 0.09 1.33 3.00 0.74 0.03 1.07 0.07 0.37 0.00
0.5 4.37 0.78 0.17 1.69 3.36 0.58 0.09 1.31 0.06 0.38 0.00
1 5.13 0.10 0.46 2.37 4.13 0.39 0.20 1.76 0.13 0.64 0.00
Panel B: Twelve equal-weighted industry groups, 1966±95
)50 0.18 0.09 0.03 0.08 0.09 0.02 0.26 0.05 0.01 0.02 0.00
)30 0.30 0.15 0.03 0.12 0.14 0.04 0.26 0.09 0.01 0.02 0.00
)15 0.57 0.28 0.03 0.24 0.27 0.07 0.26 0.17 0.01 0.02 0.00
)10 0.83 0.41 0.03 0.35 0.39 0.10 0.26 0.25 0.01 0.02 0.00
)5 1.47 0.73 0.02 0.61 0.72 0.18 0.26 0.45 0.00 0.01 0.00
)3 1.99 0.93 0.04 0.88 1.02 0.27 0.24 0.62 0.01 0.03 0.00
)1 3.01 1.33 0.04 1.40 1.48 0.17 0.40 1.09 0.01 0.02 0.00
0 4.03 1.93 0.01 1.75 2.03 0.39 0.30 1.36 0.01 0.02 0.00
0.5 3.94 1.63 0.05 1.91 2.41 0.52 0.26 1.57 0.04 0.09 0.00
1 4.06 1.48 0.10 2.14 2.89 0.64 0.20 1.87 0.13 0.21 0.02
aUnrestricted is equivalent to no target-weight constraints. Restricted is equivalent to target-weight constraints (industry market values 5%).
Arithmetic average excess returns are denoted byl. The null hypotheses are H0:aU0; H0:aR0;H0:lUlR; H0:aUaR; and H0:bUbR.
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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
Power Unrestricted Restricted Unrestricted ± Restricted
lU rU CEU lR rR CER lU)lR rU)rR CEU)CER
Panel A: Twelve value-weighted industry groups, 1934±95
)50 1.40 1.94 )0.52 1.20 1.18 0.73 0.19 0.75 )1.26
)30 1.59 2.76 )1.60 1.33 1.65 0.70 0.25 1.11 )2.30
)15 1.93 4.16 )0.75 1.65 2.91 0.53 0.27 1.25 )1.28
)10 2.14 5.16 )0.65 1.87 3.71 0.77 0.27 1.45 )1.41
)5 2.70 7.19 0.75 2.45 5.69 1.20 0.24 1.50 )0.45
)3 3.15 9.11 1.16 2.86 6.77 1.82 0.29 2.34 )0.66
)1 4.05 12.25 2.54 3.50 9.16 2.59 0.55 3.09 )0.05
0 4.73 14.99 3.65 4.00 10.82 3.40 0.73 4.17 0.25
0.5 5.38 18.85 4.54 4.36 12.73 3.96 1.01 6.13 0.58
1 6.13 25.04 6.13 5.13 16.32 5.13 1.00 8.71 1.00
Panel B: Twelve equal-weighted industry groups, 1966±95
)50 1.88 1.03 1.61 1.79 0.84 1.62 0.09 0.19 )0.01
)30 1.99 1.46 1.64 1.84 1.10 1.65 0.16 0.36 )0.01
)15 2.27 2.62 1.67 1.97 1.86 1.68 0.30 0.76 )0.01
)10 2.53 3.75 1.67 2.09 2.63 1.69 0.44 1.12 )0.02
)5 3.17 6.43 1.83 2.42 4.75 1.69 0.75 1.67 0.14
)3 3.69 9.16 1.80 2.72 6.52 1.83 0.97 2.64 )0.03
)1 4.71 14.19 2.61 3.18 11.22 1.77 1.53 2.97 0.84
0 5.73 17.27 4.28 3.73 13.72 2.77 2.00 3.55 1.51
0.5 5.64 19.11 4.76 4.11 15.47 3.52 1.53 3.64 1.24
1 5.76 21.60 5.76 4.59 17.47 4.59 1.16 4.13 1.16
aUnrestricted is equivalent to no target-weight constraints. Restricted is equivalent to target-weight constraints (industry market values 5%).
Arithmetic average returns are denoted byl, standard deviations byr, and certainty equivalent returns by CE.
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1253±1274
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 target-weight constraints. In all cases, the betas without target-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 (equal-weighted) industries. The interrelationship between the alphas is more com-plex. First, abnormal returns are clearly present: all four sets of alphas are positive. Without target-weight constraints, all but two of the alphas are sta-tistically 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
con-straints 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.
in fact, sometimes are) either larger or smaller than the largest or smallest ex anteportfolio returns. As the disutility of a loss outweighs the utility of an equal-size gain for all risk averse investors, either the constrained or uncon-strained policies could be preferredex postdepending 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 preferredex 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 thatex antethe investor, estimating the joint return distribution from a 32-quarter moving window, never envi-sioned 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
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 re-turn. 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 thatex antethe power utility investors prefer the policies without target-weight constraints. In fact, the results illustrate the problems of measuring perfor-mance withex postutility 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 pol-icies that lose less being preferred ex post. Moreover, the extreme nonlinear transformations involved in calculating expected utilities for these more risk-averse low powers will almost certainly destroy any semblance of normally distributed utility values required in a paired t-test of dierences 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, Jasp-reet Sahni, and William Ting for most capable assistance.
References
Best, M.J., 1975. A feasible conjugate direction method to solve linearly constrained optimization problems. Journal of Optimization Theory and Applications 16, 25±38.
Best, M.J., Grauer, R.R., 1991. On the sensitivity of mean±variance-ecient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies 4, 315± 342.
Best, M.J., Grauer, R.R., 1992. Positively weighted minimum-variance portfolios and the structure of asset expected returns. Journal of Financial and Quantitative Analysis 27, 513±537. Black, F., 1971. Implications of the random walk hypothesis for portfolio management. Financial
Black, F., Litterman, R., 1992. Global portfolio optimization. Financial Analysts Journal 48, 28± 43.
Dybvig, P.H., Ross, S.A., 1985. The analytics of performance measurement using a security market line. Journal of Finance 40, 401±416.
Ferson, W.E., Schadt, R.W., 1996. Measuring fund strategy and performance in changing economic conditions. Journal of Finance 51, 425±462.
Frost, P.A., Savarino, J.E., 1988. For better performance: Constrain portfolio weights. Journal of Portfolio Management, 29±34.
Grauer, R.R., 1991. Further ambiguity when performance is measured by the security market line. Financial Review 26, 569±585.
Grauer, R.R., Hakansson, N.H., 1982. Higher return, lower risk: Historical returns on long-run, actively managed portfolios of stocks bonds and bills 1936±78. Financial Analysts Journal 38, 39±53.
Grauer, R.R., Hakansson, N.H., 1985. Returns on levered actively, managed long-run portfolios of stocks, bonds and bills, 1934±84. Financial Analysts Journal 41, 24±43.
Grauer, R.R., Hakansson, N.H., 1986. A half-century of returns on levered and unlevered portfolios of stocks, bonds and bills, with and without small stocks. Journal of Business 59, 287±318.
Grauer, R.R., Hakansson, N.H., 1987. Gains from international diversi®cation: 1968±85 returns on portfolios of stocks and bonds. Journal of Finance 42, 721±739.
Grauer, R.R., Hakansson, N.H., 1995. Stein and CAPM estimators of the means in asset-allocation. International Review of Financial Analysis 4, 35±66.
Grauer, R.R., Hakansson, N.H., 1999. Applying the portfolio change and conditional performance measures: The case of industry rotation via the dynamic investment model. Working Paper, Simon Fraser University, Burnaby, BC.
Grauer, R.R., Hakansson, N.H., Shen, F.C., 1990. Industry rotation in the US stock market: 1934± 86 returns on passive, semi-passive, and active strategies. Journal of Banking and Finance 14, 513±535.
Green, R.C., 1986. Benchmark portfolio ineciency and deviations from the security market line. Journal of Finance 41, 295±312.
Hakansson, N.H., 1971. On optimal myopic portfolio policies, with and without serial correlation of yields. Journal of Business 44, 324±334.
Hakansson, N.H., 1974. Convergence to isoelastic utility and policy in multiperiod portfolio choice. Journal of Financial Economics 1, 201±224.
Henriksson, R.D., Merton, R.C., 1981. On market timing and investment performance II. Statistical procedures for evaluating forecasting skills. Journal of Business 54, 513±533. Huberman, G., Ross, S., 1983. Portfolio turnpike theorems, risk aversion and regularly varying
utility functions. Econometrica 51, 1104±1119.
Jensen, M.C., 1968. The performance of mutual funds in the period 1945±64. Journal of Finance 23, 389±416.
Jobson, J.D, Korkie, B., Ratti, V., 1979. Improved estimation for Markowitz portfolios using JamesÐStein type estimators. Proceedings of the American Statistical Association, Business and Economics Statistics Section 41, 279±284.
Jobson, J.D., Korkie, B., 1980. Estimation for Markowitz ecient portfolios. Journal of the American Statistical Association 75, 544±554.
Jobson, J.D., Korkie, B., 1981. Putting Markowitz theory to work. Journal of Portfolio Management 7, 70±74.
Jorion, P., 1985. International portfolio diversi®cation with estimation risk. Journal of Business 58, 259±278.
Leland, H., 1972. On turnpike portfolios. In: Shell, K., Szego, G.P. (Eds.), Mathematical Methods in Investment and Finance. Elsevier Science, Amsterdam, North-Holland.
Lintner, J., 1965. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13±47.
Mossin, J., 1968. Optimal multiperiod portfolio policies. Journal of Business 41, 215±229. Roll, R., 1978. Ambiguity when performance is measured by the securities market line. Journal of
Finance 33, 1051±1069.
Ross, S., 1974. Portfolio turnpike theorems for constant policies. Journal of Financial Economics 1, 171±198.
Sharpe, W.F., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425±442.
