heath windcliff* and phelim p. boyle · 2018-11-07 · the 1/n pension investment puzzle heath...

THE 1/n PENSION INVESTMENT PUZZLEHeath Windcliff* and Phelim P. Boyle†

ABSTRACT

This paper examines the so-called 1/n investment puzzle that has been observed in definedcontribution plans whereby some participants divide their contributions equally among the avail-able asset classes. It has been argued that this is a very naive strategy since it contradicts thefundamental tenets of modern portfolio theory. We use simple arguments to show that thisbehavior is perhaps less naive than it at first appears. It is well known that the optimal portfolioweights in a mean-variance setting are extremely sensitive to estimation errors, especially those inthe expected returns. We show that when we account for estimation error, the 1/n rule has someadvantages in terms of robustness; we demonstrate this with numerical experiments. This rule canprovide a risk-averse investor with protection against very bad outcomes.

1. INTRODUCTION

There is evidence (see Benartzi and Thaler 2001)that many participants in defined contributionplans use simple heuristic diversification rules inallocating their contributions among the availableasset classes. One popular diversification heuristicis often referred to as the 1/n rule. Under this rulethe investor divides his or her holdings equallyamong the available assets. We refer to this portfolioas an equally weighted portfolio. This strategy hasdrawn some criticism since it is not an optimalportfolio, in the sense that, in general, it does not lieon the efficient frontier. In fact, some researchershave suggested that pension plans should offer lessflexibility to avert some of the poor investmentdecisions made by uninformed individuals.

In this paper we demonstrate that the 1/n port-folio is consistent with the Markowitz efficientportfolios, given a limited set of information. Wethen argue that, even with all of the availablehistorical information available to investmentprofessionals, in light of the parameter estimationrisk the performance of the 1/n heuristic is notunreasonable, assuming an appropriate set of in-

vestment choices in the plan. As a result, we feelthat it is difficult to justify limiting the flexibilityof these plans based on the argument that peopleusing the 1/n heuristic are not optimally allocat-ing their pension contributions.

There is an extensive literature on the signifi-cance of estimation risk in the context of theportfolio optimization problem. Over 20 years agoJobson and Korkie (1980, 1981) in a series ofpapers analyzed the problem and proposed someremedies. Since then several contributions to thistopic have been made. In the present paper weuse methods that have been described in thefinance literature. We emphasize that we make noclaims that the techniques used in this paper areoriginal, and almost all the ideas presented herecan be found in the literature. However, we feelthat the paper serves a useful purpose. First, wepresent a short self-contained treatment thatshould be of interest to an actuarial audienceunfamiliar with the finance literature. Second, weprovide an alternative explanation of the 1/n puz-zle in the context of the asset allocation decisionin a defined contribution plan.1 Third, the 1/nrule is sometimes viewed as naive, and we suggestthat this judgment may be somewhat premature.

We begin with a brief overview of the classicalmean-variance portfolio theory. To start with, we* Heath Windcliff currently is employed with Exotic Derivatives, TD

Securities, e-mail: [email protected].† Phelim P. Boyle is Director of the Centre for Advanced Studies inFinance, School of Accountancy, University of Waterloo, 200University Ave. West, Waterloo, Ontario, Canada NZL 3G1, e-mail:[email protected].

1 Benartzi and Thaler (2001) in their influential paper discuss behav-ioral explanations of this puzzle.

32

assume that the true expected return vector andthe variance-covariance matrix are known. Underthese assumptions we provide an example illus-trating that the equally weighted portfolio under-performs the set of optimal portfolios generatedby mean-variance optimization. If the investormakes particularly simple specifications of theexpected returns and covariances, then we showin Section 4 that the equally weighted portfolio isoptimal in a mean-variance sense. This leads usto question how naive this simple specification isfor the input parameters of the mean-varianceanalysis. We find that when we take into accountthe estimation risk when calibrating these param-eters to historical data, this simple specificationof the mean-variance input parameters can beviewed as being quite reasonable.

Before proceeding to the main results, two ad-ditional comments may be helpful. First, there isan obvious risk in using a single set of parametersto build general conclusions on. Hence in Section6 we use a modified example to explore the sen-sitivity of the result to parameter specification.Second, while it is convenient to develop themodel using the analytical structure of the stan-dard mean-variance framework, the basic intu-ition can be obtained from the graphs.

2. MEAN-VARIANCE PORTFOLIO THEORY

The portfolio optimization methods introducedby Markowitz (1952), termed mean-variance op-timization methods, signified the birth of modernquantitative finance. The key insight provided bythis approach is that an investor can use informa-tion describing the relationships between assetsto construct a portfolio with better risk-returncharacteristics than if he or she considers theassets individually. As the name implies, the in-formation that mean-variance portfolio theoryuses to describe the relationships between theavailable assets is the expected returns of theassets, their variances, and correlations. We nowsummarize the basic approach.2

Given a market of N available assets, we definea portfolio to be a vector:

x � �x1, x2, . . . , xN�T subject to eTx � 1, (1)

where xi represents the fraction of wealth held inasset i and where e � (1, 1, . . . , 1)T is the N � 1vector consisting of N 1’s. The constraint eTx � 1represents the budget constraint.

Given a risk-tolerance level,3 � � [0, �), theoptimal portfolio can be found by solving theparametric quadratic program

maxx

�2��Tx � xT�x� (2)

subject to eTx � 1, (3)

where

� � ��1, �2, . . . , �N�T, (4)

and �i is the expected return on asset i, and

� � �ijij�, i, j � 1, . . . , N, (5)

where ij is the correlation between assets i and j,and i is the volatility of asset i. We assume thatthe matrix � is positive definite.

There are three different ways to formulate thisproblem, and they all produce the same solutions.First, we can solve for the portfolio that has thesmallest variance among all feasible portfolioswith the same expected rate of return. Second, wecan solve for the portfolio that has the largest rateof return among all feasible portfolios with thesame variance. Third, we can formulate the prob-lem as in equation (3), where � denotes the tradeoffbetween expect return and variance. We note thatwhen � equals zero, the optimal solution is theportfolio that has the global minimum variance.

In this paper we assume that all of the assetsare risky. First, we consider the case where thereare no restrictions on the portfolio weights. Inother words, short selling is permitted. In this casewe can solve equations (2)–(3) analytically (seePanjer et al. 1998) to find the optimal portfolio

x*�� xmin � ��xrisk, (6)

where

xmin �1

eT� 1e� 1e, (7)

�xrisk � � 1� �eT� 1�

eT� 1e� 1e. (8)

2 See Panjer et al. (1998) for more details.

3 See Panjer et al. (1998) for a definition and interpretation of risktolerance.

33THE 1/n PENSION INVESTMENT PUZZLE

The portfolio xmin represents the minimum-vari-ance portfolio in this market, while �xrisk repre-sents a self-financing portfolio adjustment4 thatoptimally trades off risk versus reward in thismarket. Notice that xmin does not depend on theexpected returns, while �xrisk depends on boththe expected returns and the covariances of theassets.

The expected return on the optimal portfolio isgiven by

�� Tx*��, (9)

and the variance of the return on the optimalportfolio is given by

��2 � �x*��T�x*��, (10)

where the optimal weights are obtained fromequation (6). As the risk tolerance parameter �varies, the points ([�]2, �[�]) trace out the tophalf of a parabola in mean-variance space. Thisprovides a nice geometric interpretation of theoptimal portfolios, and the top half of the parab-ola is known as the efficient frontier. An investorwho cares only about expected return and vari-ance will want to hold a portfolio that is on thisefficient frontier. The precise location will dependon how he or she trades off expected return andvariance, and this will be determined by the in-vestor’s risk tolerance, �.

We can use these explicit solutions to give anintuitive geometric interpretation of the parame-ter �. The solution with � � 0 corresponds to theportfolio with the global minimum variance min

2

given by

min2 �

1eT� 1e

.

The expected return on this portfolio is

�min ��T� 1eeT� 1e

.

With a little algebra we can derive the followingexpression for the expected return on the effi-cient portfolio:

�� min � ��,

where the constant � is given by

� � �T� 1� �eT� 1�

eT� 1e�T� 1e.

As � and �min are constants, higher values of �correspond to higher values of the expected re-turn. We can obtain a similar relation among thevariances:

��2 � �min�2 � �2�.

From the last two equations we obtain

�� min

��2 � �min�2 �1�

.

This means that in expected return-variancespace the slope of the straight line that joins theminimum variance portfolio to the portfolio onthe efficient frontier corresponding to � is equal to1 over �. This corresponds to the investor’s riskaversion, so its reciprocal, �, corresponds to theinvestor’s risk tolerance.

Since we will be analyzing the investment de-cisions of employees in defined contribution pen-sion plans where short sales are not permitted, weimpose the condition

x � 0. (11)

In this case we do not obtain an analytical solu-tion for the optimal portfolio weights. However,we can solve equations (2)–(3) using numericalquadratic programming methods. Details aregiven by Best and Grauer (1990). When the assetweights are restricted to be non-negative, werule out some of the solutions that were feasi-ble for the unrestricted case. Thus the effi-cient portfolio in the case when there is no shortselling lies inside (to the southeast of) the effi-cient portfolio for the case when there is shortselling.5

In practice the decision maker will not knowthe true values of the expected return vector �and the variance-covariance matrix �. Markowitz(1952) suggested that � and � should be esti-mated using forward-looking projections, but usu-ally these parameters are estimated from histori-

4 Self-financing means that the weights sum to zero. It is readilychecked that eT�xrisk � 0.

5 The efficient portfolio with positive weights may sometimes coin-cide with the efficient portfolio when short selling is permitted overa certain segment. Best and Grauer (1992) analyze this.

34 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 8, NUMBER 3

cal data. The historical estimates for � tend to besomewhat robust, but the estimates for � are verynoisy, even in a stationary market, as illustratedby Broadie (1993). This estimation risk has im-portant consequences for the optimal portfolioselection problem. Black and Litterman (1992)suggested using reverse optimization and impliedexpected returns in lieu of the historically esti-mated returns.

3. MEAN-VARIANCE EFFICIENT PORTFOLIOS

FOR A FIVE-ASSET UNIVERSE

We use a five-asset universe to illustrate some ofthese ideas. We assume that an investor wishes toselect an efficient portfolio based on five differentasset classes. This mirrors that of a hypotheticalpension plan participant where the plan sponsorallows participants to allocate their holdingsamong five asset classes. To be specific, we as-sume that the returns on these asset classes cor-respond to portfolios that track the followingindices:

● S&P 500 Large Cap (Large Cap)● S&P Mid Cap 400 (Mid Cap)● Russell 2000 Small Cap (Small Cap)● Morgan Stanley World Equity excluding the

United States (World Index)● Lehman Brothers long-term government bond

index (Long Bond Index).

Holden and Van Derhei’s (2003) descriptive studyof pension plan holdings indicates that these assetchoices are broadly representative of those of-fered to many pension plan participants withintheir 401k plan.

We assume that the annualized expected re-turns, volatilities, and correlations of the five as-sets in the pension plan are given in Table 1.These descriptive statistics were estimated using

15 years of historical data from February 1981through September 1997. We then adjusted theexpected returns of each asset downward by 5%per annum to account for current market condi-tions. Since this study is meant only to be repre-sentative of the conditions that pension plan par-ticipants face, we assume that the marketparameters given in these tables describe the truedistribution of future returns. In other words, theexpected returns, variances, and covariances re-flected in Table 1 are assumed to correspond tothe true population parameters, and thus there isno estimation risk. In Section 5 we will investi-gate the impact of parameter estimation risk onthe mean-variance optimal portfolios.

The efficient frontier for portfolios comprisedof these assets, assuming that short selling is notpermitted, is plotted in Figure 1. This figureshows the optimal tradeoff between reward, mea-sured in units of expected portfolio return, andrisk, measured in units of standard deviations.Note that the lowest point on extreme southwestcorner of the efficient frontier corresponds to thecase when the portfolio is entirely invested in theLong Bond so that the expected return is 5.47%,and the standard deviation is 5.58%. On the otherhand, the highest point on the extreme northeastcorner of the efficient frontier corresponds to thecase when the portfolio is entirely invested in theMid Cap fund so that the expected return is12.88%, and the standard deviation is 15.49%. Inthis figure we also show the risk-reward tradeofffor the equally weighted portfolio. Since the risk-reward tradeoff for the equally weighted portfoliolies below the efficient frontier, it is not optimal inthe sense that preferred portfolios exist that si-multaneously have greater expected return andless risk, for example, the frontier portfolio with� � 0.2 that is labeled in Figure 1.

Table 1Market Data (Assumed True)

Asset Name � � �

Large Cap .1165 .1449 1.0000 0.9180 0.8341 0.4822 0.3873Mid Cap .1288 .1549 0.9180 1.0000 0.9386 0.4562 0.3319Small Cap .0968 .1792 0.8341 0.9386 1.0000 0.4206 0.2002World Index .0921 .1715 0.4822 0.4562 0.4206 1.0000 0.2278Long Bond Index .0547 .0558 0.3873 0.3319 0.2002 0.2278 1.0000

Note: Expected returns, volatilities, and correlations for distribution of returns of assets available in the pension plan described in this section.


4. WHEN IS THE EQUALLY WEIGHTED

PORTFOLIO OPTIMAL?Since there is empirical evidence that some inves-tors use the equally weighted portfolio, it is of inter-est to find a set of simple assumptions that justifythe equally weighted portfolio. The motivation hereis find the assumptions that would lead to investorsselecting the equally weighted portfolio. We showthat the equally weighted portfolio is optimal in avery simple market where the assets are indistin-guishable and uncorrelated.6 Note that this impliesthat all the assets have the same expected returnand the same volatility, and that the pairwise cor-relations are all zero. In this case we can see fromsymmetry that the 1/n portfolio is optimal.

Consider the simple market described abovewhere the asset returns are indistinguishable anduncorrelated. Using a zero subscript to denotethis special market, we can write the expectedreturn vector and variance-covariance matrix as

�0 � �� e, (12)

�0 � � 2I, (13)

where �� and � are the representative levels of theexpected returns and asset volatilities, respec-tively. For any choice of �� , and for any nonzerochoice of � ,7 the optimal portfolios, obtained us-ing equations (7)–(8), are given by

xmin,0 �1N

e, (14)

�xrisk,0 � 0. (15)

In other words, investors who share this particu-larly simple view of the market should selectthe equally weighted portfolio for any risk-tolerance level, �. It is also easy to see that thisportfolio is also optimal when short selling isnot allowed. The portfolio in this case will consistof a single point in mean standard deviationspace.

It is well known that the this simple descriptionof the market is not accurate. Chan, Karceski,and Lakonishok (1999) find that for a sample of500 U.S. stocks the average correlation is 28%.The lowest correlation was minus 37%, and thehighest 92%. Chan et al. also demonstrate thatcorrelation between two stocks is higher whenthey belong to the same industry than when theybelong to different industries. Hence the simpledistributional assumptions that lead to theequally weighted portfolio being optimal are notsupported by the empirical evidence.

We also saw in Section 3 that the equallyweighted portfolio does not lie on the efficientfrontier and hence is suboptimal. However, thisanalysis assumed that the true population param-eters are known. In the following section we ex-amine the impact of parameter estimation on themean-variance optimal portfolios.

5. AN INTRODUCTION TO PARAMETER

ESTIMATION RISK

The efficient frontier computed in Section 3 as-sumed that we knew the both the expected returnvector, �, and the variance-covariance matrix, �.Although Markowitz (1952) in his original papersuggested using a discount dividend model tospecify the expected returns, many practitioners

6 These are not the most general conditions under which we obtainthis result. If the assets all have the same correlation coefficient as wellas identical means and variances, we would still find that the equallyweighted portfolio is optimal.

7 This is consistent with our assumption that all of the assets availablein the pension plan are risky.

Figure 1Risk-Reward Tradeoff

Note: Measured in units of standard deviations of portfolio returns and ex-pected returns, respectively, between portfolios on the efficient frontier and theequally weighted portfolio. Descriptive statistics of the assets available in thepension plan are given in Table 1. Point labeled as Portfolio A will be describedin a later section.


estimate the moments of the future asset returndistributions using historical data. A number ofpapers (Michaud 1998; Broadie 1993; Best andGrauer 1991; Britten-Jones 1999; Bengtsson andHolst 2002) have explored the impact of param-eter estimation risk on mean-variance optimalportfolios.

We denote the mean-variance optimal portfoliobased on the true population parameters � and �by xtrue(�) � xtrue. We drop the � just to simplifythe notation, with the understanding that xtrue isstill a function of �. Suppose that the estimates ofthe expected return and variance-covariance ma-trix based on a sample of M historical observa-tions are denoted by �̂ and �̂. Denote the mean-variance optimal portfolio based on theseestimated parameters by xest(�) � xest. FollowingJobson and Korkie (1981) and Broadie (1993), wecan distinguish three frontiers:

● The true frontier. In this case the expectedreturn and variance are computed as

xtrueT �, xtrue

T �xtrue. (16)

The true frontier is unattainable since in prac-tice we do not know the true parameters andhence cannot realistically compute xtrue.

● The estimated frontier. In this case the ex-pected return and variance are computed as

xestT �̂, xest

T �̂xest. (17)

The estimated frontier is based on the portfolioxest, which is derived using the estimatedweights and hence can be calculated. However,it does not provide a sensible comparison sincein reality the future returns will be drawn fromthe true distribution and not the distributiongiven by the sample estimates, �̂ and �̂.

● The actual frontier. In this case the expectedreturn and variance are computed as

xestT �, xest

T �xest. (18)

The actual frontier depicts the performance ofthe portfolios constructed using the sample es-timates, xest, under the true distribution of re-turns, � and �.

A representative profile of the relationshipsamong the true, estimated, and actual frontiers isillustrated in Figure 2. In this case and in theexample considered by Broadie (1993), the esti-

mated frontier lies above the true frontier,8 butthis need not always be the case. The true frontiermust by construction dominate the actual fron-tier. Although we have depicted a situation wherethe estimated frontier dominates the true fron-tier, this need not always be the case. We notethat the true frontier is the unattainable ideal, theestimated frontier is illusory, and the actual fron-tier is the most realistic one for many comparisonpurposes.

The simplest way to illustrate this is by using anumerical example. Suppose the asset return dis-tributions given in Table 1 follow a joint multi-variate normal distribution. Also suppose that theparameters values given in Table 1 are the truepopulation parameters. Hence the means, volatil-ities, and correlations given in Table 1 are thepopulation values. Of course, an investor who isattempting to construct the mean-variance opti-mal portfolio is not able to observe directly thetrue descriptive statistics of the return distribu-tions. Instead, an investor can observe a sampleof returns generated by the assets. For example,suppose that the investor takes a sample of fiveyears of monthly return data (60 observations)

8 Broadie (1993) found this to be a typical profile based on hisanalysis.

Figure 2Representative Profile of Relationships

among True, Estimated, andActual Frontiers

Note: Points on the frontiers correspond to risk tolerance level of � � 0.2.Estimates of � and � are given in Table 2.


and uses the sample estimates as proxies to de-scribe the future asset return distributions. Rep-resentative sample estimates for a typical numer-ical scenario are provided in Table 2.

Comparing the results provided in Table 2 withthe true parameters from Table 1, we see thatalthough the volatilities and correlations can beestimated with reasonable accuracy, the esti-mates of the expected returns are very poor. Spe-cifically, we note that the sample returns seri-ously overestimate the mean for the Small CapIndex and underestimate the mean for the WorldIndex. To see the effect that this will have on theportfolios generated using mean-variance optimi-zation, in Table 3 we compare the true optimalportfolio and the reportedly optimal portfolioconstructed using the estimated parameters, bothusing a risk-tolerance level of � � 0.2. We findthat the portfolio constructed using the estimatedparameters has overweighted the Small Cap In-dex and underweighted the World Index. Thisdemonstrates the error-maximizing property(discussed by Broadie 1993) of the mean-vari-ance optimization procedure. Notice that assetswhose sample returns have been optimisticallybiased are overweighted in the recommendedportfolio.

We say that a portfolio dominates another if ithas both a higher return and a lower standarddeviation. In Table 3 we see that if the investorknows the true market distributions, then theoptimal portfolio for � � 0.2 dominates theequally weighted portfolio. However, we cannotalways decide whether the portfolio obtainedfrom mean-variance optimization with � � 0.2using the estimated parameters is better than, orworse than, the equally weighted portfolio. Thisportfolio is labeled as Portfolio A in Figures 1 and2. We see that it offers a somewhat higher ex-

pected return, but it has a higher standard devi-ation (15.9%) than the equally weighted portfolio(11.7%).

6. THE NUMERICAL EXAMPLE REVISITED

The data for the numerical example were basedon actual returns. However, it is potentiallydangerous to reach general conclusions fromthe results of a single example. A referee notedthat a special feature of our data set could beviewed as unrealistic and that this feature maydrive some of the results.9 In this section wediscuss this point and make some additionalcalculations to assess the robustness of the per-formance of the equally weighted portfolio tothe input parameters.

The input population parameters for the nu-merical example are given in Table 1. We gener-ated a sample of 60 monthly returns for assetclass from the true population distribution andused the sample to obtain estimates for the ex-pected return vector �̂ and the variance covari-ance vector �̂. Then we computed the optimalportfolio weights, xest, using �̂ and �̂. These esti-mated weights were given in Table 3 for � � 0.2.Note that the amount allocated to the Small Capasset class in this optimization is 60.81%. How-ever, the true expected return on this asset classis 9.68%, which means that it is out of line withthe return profiles of the other major equity in-dices, especially the Large Cap Index and the MidCap Index, both of which have lower standarddeviations and higher expected returns. In addi-tion, all three indices are quite strongly corre-lated, so the Small Cap Index affords only limited

9 We thank the referee for insightful comments on this point.

Table 2Market Data (Estimated)

�̂ �̂ �̂

Large Cap .0991 .1468 1.0000 0.9208 0.8266 0.4017 0.5018Mid Cap .1602 .1620 0.9208 1.0000 0.9413 0.4188 0.4641Small Cap .1830 .1867 0.8266 0.9413 1.0000 0.4386 0.2925World Index .0042 .1747 0.4017 0.4188 0.4386 1.0000 0.1863Long Bond Index .0440 .0489 0.5018 0.4641 0.2925 0.1863 1.0000

Note: Estimates of expected returns, volatilities, and correlations for asset return distributions obtained from one scenario using a sample of 60monthly observations.


diversification benefit. Because of this, the SmallCap Index is not held in the true efficient frontierexpect for a very small percentage holding when �is zero or very close to zero.

However, the sampled data produce a relativelyhigh expected return for the Small Cap Index,18.30%. This high expected return means that theallocation to the Small Cap Index is very high,since xest(3) is over 60%. When we realize that thetrue efficient portfolio hardly ever includes thisasset class, we see the source of the deviationfrom optimality. Our input assumptions concern-ing the expected return on the Small Cap Indexdrive this conclusion. It is therefore desirable tomodify the example to see how the equallyweighted portfolio performs under other, morerealistic, assumptions.

Hence we redo the example using revised inputassumptions by modifying those in Table 1. Spe-cifically we increase the expected return on theSmall Cap Index from 9.68% by 4% to 13.68%. Allthe other input assumptions in Table 1 are un-changed. Note that under this revised assumptionthe Small Cap Index now has the highest ex-pected return as well as the highest volatility. It isconvenient to use graphs to summarize the per-formance of the equally weighted portfolio underthese revised assumptions. In Figure 3 the trueefficient portfolio is shown as the topmost curve.The equally weighted portfolio is marked with aplus sign on the graph. We also generated 100samples of simulated data, where each sampleconsisted of 60 months’ returns. For each samplewe computed the sample expected return vector�̂ and the sample variance covariance vector �̂.Using these inputs we computed the optimal port-folio weights to generate the actual efficient fron-tier. Armed with these weights we use the truepopulation expected returns and covariance ma-

trix to obtain the actual frontier. In this way weobtain 100 actual frontiers. The lower curve inFigure 3 represents the average of these 100 ac-tual frontiers. We note that the equally weightedportfolio lies very close to this average frontier.This result would seem to suggest that the equallyweighted portfolio does not offer much of an ad-vantage in this particular example.

However, there is another feature of the equallyweighted portfolio that may be appealing to someinvestors. A risk-averse investor may be con-cerned about avoiding the really bad outcomes. InFigure 4 we depict the four worst outcomes of the100 frontiers. We see that the equally weightedportfolio now performs well relative to these badoutcomes. In this example the equally weightedportfolio does not dominate the average outcome,but it does perform better than the worst caseoutcomes.

7. PARAMETER ESTIMATION ERROR

In the preceding section we saw that portfoliosgenerated using mean-variance optimization canlie quite far away from the efficient frontier whenthere is parameter estimation error. In this sec-tion we provide a statistical comparison of theperformances of mean-variance optimal portfo-lios with parameter estimation error and theequally weighted portfolio. Assuming that thetrue market parameters are as given in Table 1,10

we generate scenarios of market data from whichwe obtain estimates of the market parameters.Using these estimated parameters we generate

10 Note that we used the original market parameters here rather thanthe revised ones in Section 6. This is reasonable since our main focushere is on the relationship between the length of the historicalwindow and the precision of the estimates.

Table 3Portfolios and Portfolio Performances Under True Market Data

Portfolio

True Market Asset Holdings

� � Large Cap Mid Cap Small Cap World Index Bond Index

xtrue .1070 .1156 .0000 .6796 .0000 .0529 .2675xest .1048 .1590 .0000 .3311 .6081 .0000 .0608Equal weight .0978 .1167 .2000 .2000 .2000 .2000 .2000

Note: Expected returns, volatilities, and holding of the true optimal portfolio for risk tolerance of � � 0.2, the actual portfolio obtained bymean-variance optimization with � � 0.2 using estimated parameters reported in Table 2, and the equally weighted portfolio. Expected returnsand volatilities given in columns 2 and 3 are computed using the true population parameters of asset return distributions specified in Table 1.


mean-variance optimal portfolios and thenstudy their performance under the true marketparameters.

In Figure 5 we investigate the mean-varianceoptimal portfolios computed using estimated pa-rameters for risk tolerances of � � 0 and � � 0.2.In the first panel when � � 0, we see that risk-reward tradeoffs of the estimated portfolios, xest,are all close to the true global minimum-varianceportfolio. This illustrates that the global mini-mum-variance portfolio can be approximatedquite well with the estimated parameters usingfive years of data (60 monthly samples).

For investors who are taking on risk, as in thecase when � � 0.2 shown in panel b, the mean-variance optimal portfolios computed using esti-mated parameters can lie far away from the effi-cient frontier when only five years of data areused. In fact, in panel b we find that there aremany cases where the equal-weight portfoliodominates the reportedly optimal portfolios gen-

erated using estimated parameters. This is be-cause the equally weighted behavioral portfoliodoes not require the specification of the marketparameters using historical data and hence doesnot suffer from parameter sampling risk.

If the market parameters are stationary, we canreduce the parameter estimation risk by samplinga longer history of asset returns. In panel c we use20 years (240 monthly samples) of observed datato estimate the expected returns and covariancesand see that portfolios now lie much closer to theefficient frontier. It is very unlikely that the mar-ket parameters would remain stationary over a20-year window, making this brute force ap-proach inappropriate.

It has been noted by Best and Grauer (1991)and Broadie (1993) that, comparatively speaking,it is more difficult to obtain reliable estimates ofthe expected returns than it is to obtain reliableestimates of the variance-covariance matrix usinghistorical data. We illustrate that it is the mis-

Figure 3Performance of Equal Weight Portfolio Versus Average at

Optimized Portfolios Using Estimated Market Data

Note: Top curve denotes true frontier based on true population parameters. Lower curve represents averageof 100 actual frontiers based on 100 simulated samples. Equally weighted portfolio is denoted by � sign.Values of � and � are the same as those given in Table 1 except that expected return on the Small Cap Indexis 4% higher.


specification of the means that causes the actualperformances of the estimated portfolios to lie offof the efficient frontier in panel d. In this panel weshow that the computed portfolios lie very closeto the efficient frontier if the vector of expectedreturns is assumed to be known, while estimatingthe variance-covariance matrix using only fiveyears of data. This implies that investors shouldtake extra care to specify accurately their views ofthe expected returns on the assets. We will dis-cuss a procedure for improving the estimates ofexpected returns in Section 8.

In summary, we have observed that, at least forthe true market parameters studied in this paper,using a reasonable amount of historical data (fiveyears), the performance of the equally weightedportfolio is comparable with the performance ofthe mean-variance optimal portfolios once we ac-count for the impact of parameter estimationrisk. Simply using a longer window of historicalsamples is not a realistic solution to this problem

because of the nonstationarity of the market pa-rameters. We also demonstrated that the mis-specification of the expected returns has the mostimpact on the performance of the mean-varianceoptimized portfolios.

8. USING SHRINKAGE ESTIMATORS TO

IMPROVE ESTIMATES OF RETURNS

We have seen that the misspecification of theexpected returns sometimes has a dramatic neg-ative impact on the performance of the mean-variance optimized portfolios. In this section weuse a Bayesian shrinkage model suggested by Jo-rion (1986) to quantify how much useful informa-tion is contained in a sample of historical returns.This, in turn, will allow us to quantify the amountof information that is disregarded by investorswho utilize the simple, equally weighted diversi-fication heuristic.

Figure 4Performance of Equal Weight Portfolio Versus Worst of

Optimized Portfolios Using Estimated Market Data

Note: True frontier is topmost curve, and the other four curves represent four worst actual frontiers based on100 simulated samples. Equally weighted portfolio is denoted by � sign. Estimates of � and � are the sameas those given in Table 1 except that expected return on the Small Cap Index is 4% higher.


In Section 4 we defined a market

�0 � �� e, (19)

�0 � � 2I, (20)

in which the assets are statistically indistinguish-able. In this market the optimal risk-rewardtradeoff is obtained with the equally weightedportfolio, a result that is independent of the spec-ified average return �� and average volatility � . Atthe other extreme we have an investor who usesa sample of historical returns to estimate therelevant market parameters, �̂ and �̂. We haveseen that our estimate of the expected returns, �̂,

can be particularly noisy and, due to the error-maximizing property of the mean-variance opti-mization, can result in poor portfolio selection.

For simplicity we assume that the sample vari-ance-covariance matrix, �̂, provides an accurateestimate of the true covariances. We focus onquantifying the benefits of including informationcontained in a sample of historical data whenspecifying the expected return vector. Considerthe Bayes-Stein weighted learning model de-scribed by Jorion (1986):

�̂BS � �0 � ��̂ � �0�, (21)

Figure 5

Note: Actual risk-reward tradeoff under true market parameters, measured in units of standard deviations and expected returns,respectively, of a sample of 100 mean-variance optimal portfolios computed using parameters estimated from a history ofobserved returns.


where

� � 1 �N � 2

N � 2 � M��̂ � �0�T� 1��̂ � �0�

, (22)

and where N is the number of assets and M is thenumber of historical data points in the sample.The parameter, � � [0, 1], specifies the relativeimportance of the prior information, �0, and theinformation contained in the historical sample, �̂.Note that when M 3 �, � 3 1 and all the weightis placed on the sample estimate �̂ from the his-torical data. The estimator in equation (21) isoften called a shrinkage estimator because theestimated expected returns tend to �0 as � 3 0.Although it is possible to consider other priors, inthis paper the prior, �� , is chosen to be the meanof the returns on the minimum-variance portfolio:

�� ̂T� 1eeT� 1e

. (23)

One reason for selecting this prior is that theweights on the minimum variance portfolio de-pend only on the variance covariance matrix �̂and do not depend on the expected returns. Inpractice �̂ is much more stable.

As an aside we note the resemblance betweenthe Bayes-Stein estimator in equation (21) andthe linear credibility approach for estimating in-surance premiums.

In our case we are interested in determiningthe confidence level, �, in a set of historical data.If � is consistently low, then there is little infor-mation contained in the sample mean, and inves-tors who establish equally weighted portfolios are,in fact, doing something that is somewhat reason-able.11 In Table 4 we show the mean and standard

deviation of the weighting parameter, �, for 60and 240 months of historical data. Using fiveyears of data, the Bayes-Stein estimator placesapproximately equal weight on the prior and thesample means, indicating that there is surpris-ingly little useful information about the expectedreturns contained in the sample means of thehistorical data, even if the parameters are station-ary. As more historical data are used, the qualityof the sample mean improves and � increases,but, of course, the benefits will be spurious if themarket parameters are not stationary.

In Figure 6 we plot the risk-reward tradeoff fora sample of portfolios constructed using the sam-ple means and those constructed using the Bayes-Stein estimators. We find that more of the port-folios lie along the efficient frontier when theshrinkage estimators are used.

9. CONCLUSIONS

In this paper we have explored some of the ratio-nale behind the equally weighted portfolio that ispopular with some defined contribution pensionplan participants. If the parameters of the returndistribution are assumed to be known, this simpleheuristic leads to portfolios that lie below theefficient frontier and consequently are subopti-mal. We show that the equally weighted portfoliois optimal in a market where the assets are indis-tinguishable and uncorrelated. This leads us toquestion how much investors lose by implicitlyspecifying the market parameters in this simplemanner.

In practice we do not know the true future assetreturn distributions, and often these are esti-mated using samples of historical data while as-suming that the parameters are stationary. If wetake into account the impact of parameter esti-mation risk, then the computed portfolios usingmean-variance optimization no longer dominatethe equally weighted portfolio. Using a Bayes-Stein learning model we show that, using a suffi-ciently small window of historical data over whichwe might hope that the parameters are station-ary, little weighting is placed on the sample

11 However, as pointed out by Brennan and Torous (1999), they arestill missing out on some of the diversification benefits that can be

obtained by using the correlations between assets, which can beestimated with adequate precision from a reasonable number ofhistorical samples.

Table 4Mean and Standard Deviation of

Weighting Parameter �

Months of Data Used

�

Mean Std Dev.

60 0.4995 0.1611240 0.7028 0.1048

Note: Defined as equation (22) using 5 years and 20 years of historicaldata. Statistics were estimated from a sample of 1,000 scenarios.


means. The prior for the expected returns used inthe Bayes-Stein model is identical in form to thenaive specification implicitly used by investorswho utilize the 1/n heuristic. This indicates that itis surprisingly difficult to improve on this simplediversification rule.

The numerical results given in this paper, andthe conclusions based on them, were derivedfrom a set of market parameters that we feel arerepresentative of the choices offered to many in-vestors within their pension plan. The advantagesof the equally weighted portfolio when there isestimation risk have been noted by other authorsusing different data sets and different time peri-ods. For example, Jobson and Korkie (1980) statethat “naive formation rules such as the equalweight rule can outperform the Markowitz rule.”Michaud (1998) also notes because of that esti-mation risk, “an equally weighted portfolio mayoften be substantially closer to the true MV opti-mality than an optimized portfolio.”

Defined contribution plans are becoming thedominant vehicle for providing pension income.In this connection the portfolio strategies of par-ticipants are a critical factor since the asset allo-cation decision determines the ultimate benefitsavailable under these plans. Here we examinedthe 1/n rule and provided a justification. Ofcourse, the performance of the equally weightedheuristic will depend on the asset choices avail-

able to the plan members. Given that some par-ticipants use the equally weighted heuristic toselect their portfolios, perhaps pension plan spon-sors should take this into account when selectingthe available asset classes.

We emphasize that we have not carried out acomprehensive analysis of the performance of the1/n rule. There are many possible topics of futureresearch. These include

● An investigation of the impact of using differentasset classes on the equally weighted portfolio

● Generating expected returns from a marketequilibrium model such as the Capital AssetPricing Model

● Extensions to the multiperiod case.

These questions are left for future research.

ACKNOWLEDGMENTS

The authors acknowledge support from the Nat-ural Sciences and Engineering Research Councilof Canada. They also appreciate the comments ofthe referees on an earlier version.

REFERENCES

BENARTZI, SHLOMO, AND RICHARD THALER. 2001. “Naive Diversifica-tion Strategies in Defined Contribution Retirement SavingPlans,” American Economic Review 91(1): 79–98.

Figure 6

Note: Actual risk-reward tradeoff under true market parameters, measured in units of standard deviations and expected returns,respectively, of a sample of 100 mean-variance optimal portfolios computed using parameters estimated from a history ofobserved returns. Risk tolerance level was � � 0.2, and 60 months of historical data were available.


BENGTSSON, CHRISTOFFER, AND JAN HOLST. 2002. “On Portfolio Se-lection: Improved Covariance Matrix Estimation for Swed-ish Asset Returns.” Working paper, Lund University.

BEST, MICHAEL, AND ROBERT GRAUER. 1990. “The Efficient Set Math-ematics When the Mean Variance Problem Is Subject toGeneral Linear Constraints,” Journal of Economics andBusiness 42: 105–20.

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BRENNAN, MICHAEL, AND WALTER TOROUS. 1999. “Individual DecisionMaking and Investor Welfare.” Economic Notes by BancaMonte dei Paschi di Siena SpA 28(2): 119–43.

BRITTEN-JONES, MARK. 1999. “The Sampling Error in Estimates ofMean-Variance Efficient Portfolio Weights,” Journal of Fi-nance 54(2): 655–71.

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CHAN, LOUIS K. C., JASON KARCESKI, AND JOSEF LAKONISHOK. 1999.“On Portfolio Optimization: Forecasting Covariances and

Choosing the Risk Model.” Review of Financial Studies12(5): 937–74.

HOLDEN, SARAH, AND JACK VANDERHEI. 2003. “401(k) Plan AssetAllocation, Account Balances, and Loan Activity in 2001,”ICI Perspective 9(2).

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MICHAUD, RICHARD. 1998. Efficient Asset Management: A Practi-cal Guide to Stock Portfolio Optimization and Asset Allo-cation. Cambridge, MA: Harvard Business School Press.

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Discussions on this paper can be submitted untilJanuary 1, 2005. The authors reserve the right toreply to any discussion. Please see the SubmissionGuidelines for Authors on the inside back cover forinstructions on the submission of discussions.


heath windcliff* and phelim p. boyle · 2018-11-07 · the 1/n pension investment puzzle heath...

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