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How inefficient are simple asset-allocation strategies?

Angel-Victor DeMiguel Lorenzo Garlappi Raman Uppal

December 2004

Preliminary

Abstract

In this paper, we wish to evaluate the out-of-sample performance of simple asset-allocationstrategies such as allocating 1/N to each of the N assets available. To do this, we com-pare the out-of-sample performance of these simple allocation rules to several static optimalasset-allocation strategies: the Markowitz (1959) mean-variance portfolio, the Bayes-Steinshrinkage portfolio studied by Jorion (1985, 1986), the data and model approach of Pastor(2000) and Pastor and Stambaugh (2000), and the short-sale-constrained portfolios of Frostand Savarino (1988), Chopra (1993), and Jagannathan and Ma (2003). We also com-pare the simple allocation rules to dynamic models of optimal asset allocation that allowfor changes in the investment opportunity set. In particular, we consider two models ofstochastic investment opportunity sets analyzed in Campbell and Viceira (2002) that al-

low for predictability in expected returns and stochastic interest rates. The out-of-sampleperformance of the static asset-allocation strategies is evaluated for six different data sets,while the out-of-sample performance of the dynamic allocation strategies is evaluated usingthe same data as in the original papers by Campbell and Viceira. We find that the simpleasset-allocation rule of 1/N (with or without rebalancing at each trading date) is not veryinefficient. In fact, it performs quite well out-of-sample: it often has a higher Sharpe ratioand a lower turnover than the policies from the static and the dynamic models of optimalasset allocation.Keywords: Portfolio choice, asset allocation, investment management.JEL Classification: G11.

We wish to thank John Campbell and Luis Viceira for their suggestions and for making available theirdata and computer code. We also gratefully acknowledge comments from Bernard Dumas, Francisco Gomes,Eric Jacquier, Lubos Pastor, Tan Wang and seminar participants at HEC Lausanne, HEC Montreal, LondonBusiness School, University of Texas, University of Vienna, and the International Symposium on AssetAllocation and Pension Management at Copenhagen Business School.

London Business School, 6 Sussex Place Regents Park, London, United Kingdom NW1 4SA; Email:[email protected].

McCombs School of Business, The University of Texas at Austin, Austin TX, 78712; Email:[email protected]. Corresponding author.

London Business School and CEPR; IFA, 6 Sussex Place Regents Park, London, United Kingdom NW14SA; Email: [email protected].


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Contents

1 Introduction 1

2 Methodology 4

3 Simple asset-allocation strategies 7

4 Static asset-allocation strategies 7

4.1 Optimal rules for static asset allocation . . . . . . . . . . . . . . . . . . . . 8

4.1.1 Classical Mean-Variance . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.2 Constrained Mean-Variance . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.3 Bayes-Stein Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.4 Bayesian Data-and-Model Portfolios . . . . . . . . . . . . . . . . . 9

4.1.5 Minimum Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Results for the different data sets considered . . . . . . . . . . . . . . . . . . 104.2.1 US T-bill, 5-year bond and market portfolio . . . . . . . . . . . . . . 11

4.2.2 International equity Indexes . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.3 Ten industry portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.4 US T-bill, US market, SML and HML portfolios . . . . . . . . . . . 15

4.2.5 Size and Book-to-Market sorted portfolios under a single factor model 16

4.2.6 Size and Book-to-Market sorted portfolios under a three factor model 18

4.2.7 Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Conclusion from the analysis of static asset allocation strategies . . . . . . . 21

5 Dynamic asset-allocation strategies 215.1 Stochastic stock returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Conclusions 23

Tables 25

References 48


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List of Tables

1.1 In-sample Sharpe ratios for static asset allocation across US T-bill, 5-year bond andUS market portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Out-of-sample Sharpe ratios for static asset allocation across US T-bill, 5-year bondand US market portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3 Turnover for static allocation strategies across US T-bill, 5-year bond and US marketportfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 In-sample Sharpe ratios for international equity indexes . . . . . . . . . . . . . . . . 28

2.2 Out-of-sample Sharpe ratios for international equity indexes . . . . . . . . . . . . . 29

2.3 Turnover for static allocation strategies across international equity indexes . . . . . 30

3.1 In-sample Sharpe ratios for 10 industry portfolios . . . . . . . . . . . . . . . . . . . 31

3.2 Out-of-sample Sharpe ratios for 10 industry portfolios . . . . . . . . . . . . . . . . . 32

3.3 Turnover for static allocation strategies across 10 industry portfolios . . . . . . . . . 33

4.1 In-sample Sharpe ratios for static allocation across T-bill, market, HML and SMBportfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Out-of-sample Sharpe ratios for static allocation across T-bill, market, HML and SMBportfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Turnover for static allocation strategies across T-bill, market, HML and SMB port-folios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 In-sample Sharpe ratios for static allocation across a T-bill, market, Fama-French sizeand book-to-market, HML and SMB portfolios with a single factor portfolio . . . . 37

5.2 Out-of-sample Sharpe ratios for static allocation across a T-bill, market, Fama-Frenchsize and book-to-market, HML and SMB portfolios with a single factor portfolio . . 38

5.3 Turnover for static allocation strategies across a T-bill, market, Fama-French size andbook-to-market, HML and SMB portfolios with a single factor portfolio . . . . . . . 39

6.1 In-sample Sharpe ratios for static allocation across a T-bill, market, Fama-French sizeand book-to-market, HML and SMB portfolios with three factor portfolios . . . . . 40

6.2 Out-of-sample Sharpe ratios for static allocation across a T-bill, market, Fama-Frenchsize and book-to-market, HML and SMB portfolios with three factor portfolios . . . 41

6.3 Turnover for static allocation strategies across a T-bill, market, Fama-French size andbook-to-market, HML and SMB portfolios with three factor portfolios . . . . . . . . 42

7.1 Sharpe ratios in simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.2 Turnover for static allocation strategies in simulated data . . . . . . . . . . . . . . . 44

8.1 Out-of-sample Sharpe ratios for dynamic asset allocation across T-bill, 5-year bondand market portfolio with stochastic expected returns . . . . . . . . . . . . . . . . . 45

8.2 Turnover for dynamic asset allocation strategies across T-bill, 5-year bond and marketportfolio with stochastic expected returns . . . . . . . . . . . . . . . . . . . . . . . . 46

9.1 Out-of-sample Sharpe ratios for dynamic asset allocation with stochastic interest rates 47


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1 Introduction

In about the 4th century, Rabbi Issac bar Aha proposed the following simple rule for asset

allocation: A third in land, a third in merchandise, a third in cash. After a brief lullin the literature on asset allocation, there have been considerable advances in the last fifty

years.1 Markowitz (1952) developed the optimal rule for allocating wealth across risky

assets in a static setting. Tobin (1958) showed how the optimal portfolio would consist of

only two funds for the case where the investor could hold a riskfree asset in addition to

risky assets, while Sharpe (1964) and Lintner (1965) derived the implications of this two-

fund separation for equilibrium prices. Samuelson (1969) and Merton (1969) showed the

conditions under which these portfolio rules would be optimal even in a multiperiod setting.

Merton (1971) determined the optimal portfolio policies when the investment opportunity

set is stochastic.2 Recent work has considered particular specifications of the stochastic

investment environment (that is, the riskfree rate, expected returns on risky assets, and the

volatilities of the returns on risky assets) and either derived explicit analytic expressions

for the optimal portfolio policies3 or solved these problems numerically.4

Implementing the portfolio policies suggested by the theoretical models described above

requires one to estimate the parameters of the model. In the case of static portfolio models,

these parameters are the expected returns vector and the variance-covariance of returnsmatrix, while for the models of dynamic portfolio choice one needs to estimate the parame-

ters for the processes driving the riskfree interest rate, expected returns on the risky assets,

and the volatilities and correlations of the risky asset returns. Traditionally, this estimation

has been done using methods from classical statistics such as maximum likelihood, ordi-

nary least squares, and generalized methods of moments. But portfolio weights constructed

using point estimates from classical econometric methods lead to extreme portfolio weights

that fluctuate substantially over time.5 One approach adopted in the literature to deal

with these problems has been to use Bayesian shrinkage estimators that incorporate a1For a detailed survey of this literature, see Campbell and Viceira (2002).2The implications for equilibrium asset prices of these optimal portfolio policies in the presence of a

stochastic investment opportunity set are given in Merton (1973).3See, for instance, Brennan and Xia (2000, 2002) Campbell and Viceira (1999, 2001), Campbell, Chan,

and Viceira (2003), Campbell, Cocco, Gomes, and Viceira (2001) Chacko and Viceira (2004), Kim andOmberg (1996), Liu (2001), Skiadas and Schroder (1999), Wachter (2002), and Xia (2001).

4See, for example, Balduzzi and Lynch (1999), Brennan, Schwartz, and Lagnado (1997), Lynch (2001),and Lynch and Balduzzi (2000).

5For a discussion of the problems entailed in implementing mean-variance optimal portfolios, see Hodgesand Brealey (1978), Michaud (1989), Best and Grauer (1991), and Litterman (2003).


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Simple asset-allocation strategies 2

prior.6 A second approach, proposed by Black and Litterman (1990, 1992), combines two

sets of priorsone based on an equilibrium asset pricing model and the other based on the

subjective views of the investor. A third approach imposes portfolio constraints prohibit-

ing shortsales (Frost and Savarino (1988) and Chopra (1993)), which Jagannathan and Ma(2003) show can be interpreted as shrinking the extreme elements of the covariance matrix.

Recently, Michaud (1998) has advocated the use of resampling methods.7

Our objective in this paper is to evaluate the cost of using simple asset-allocation rules

rather than the allocation suggested by models of optimal portfolio selection. We define

simple asset-allocation rules to be those that require no estimation of parameters and no

optimization. So, one simple allocation rule is to just allocate all the wealth in a single

asset (all your eggs in one basket), for instance, the market portfolio. A second simple

allocation rule, and the one we focus on, is the naive 1/N diversification rule where 1/N

is allocated to each of the N assets available for investment. We consider two versions of

this rule: (i) the investor constantly rebalances the portfolio in order to maintain the 1 /N

allocation over time; and, (ii) the investor allocates 1/N at the initial date and then holds

this portfolio until the terminal date (buy-and-hold).

There are several reasons for studying the 1/N simple asset-allocation rule. One, it

does not rely on estimation of moments of asset returns and or optimization and so it is

easy to implement this rule. Two, despite the sophisticated theoretical models developed

in the last fifty years and the advances in methods for estimating the parameters for these

models, investors continue to use such simple allocation rules for allocating their wealth

across assets. For instance, Benartzi and Thaler (2001) and Liang and Weisbenner (2002)

document that investors allocate their wealth across assets using the naive 1/N-rule. There

is evidence also that investors often take the path of least resistance and exhibit inertia

when making investment and rebalancing decisions. For instance, in allocating their wealth

6In the literature, the Bayesian adjustment has been implemented in different ways. Barry (1974),

and Bawa, Brown, and Klein (1979), use either a non-informative diffuse prior or a predictive distributionobtained by integrating over the unknown parameter. In a second implementation, Jobson and Korkie(1980), Jorion (1985, 1986), Frost and Savarino (1986), and Dumas and Jacquillat (1990), use empiricalBayes estimators, which shrinks estimated returns closer to a common value and moves the portfolio weightscloser to the global minimum-variance portfolio. In a third implementation, Pastor (2000), and Pastor andStambaugh (2000) use the equilibrium implications of an asset pricing model to establish a prior; thus, inthe case where one uses the CAPM to establish the prior, the resulting weights move closer to those for avalue-weighted portfolio. The models discussed above use Bayesian estimation methods in the context ofstatic asset allocation. Bayesian methods have also b een used in the context of dynamic asset allocationwhen the investment opportunity set is stochastic by Avramov (2004), Barberis (2000), Cremers (2002),Johannes, Polson, and Stroud (2002), and Kandel and Stambaugh (1996).

7Scherer (2002) describes the resampling approach in detail and discusses some of its limitations, whileHarvey, Liechty, Liechty, and Muller (2003) discuss other limitations and provide an estimate of the lossincurred by an investor who chooses a portfolio based on this approach.


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in pension schemes, employees often accept the default asset allocation decision made by

employers (Madrian and Shea (2000) and Choi, Laibson, Madrian, and Metrick (2001)),

and in contrast to what optimal asset-allocation models would suggest, many employees

never revise these initial allocations (Choi, Laibson, Madrian, and Metrick (2004)). Finally,MacKinlay and Pastor (2000) show that when a risk factor is missing from an asset pricing

model the resulting mispricing shows up in covariance matrix of residuals, and if one exploits

this, then under the assumption that all assets have the same expected return, one gets the

1/N rule.

To evaluate the cost of using the 1/N allocation policy, we compare the out-of-sample

performance of this simple rule to several allocation strategies suggested by various opti-

mizing models using the out-of-sample Sharpe ratio of each asset-allocation strategy. We

also report the turnover (trading volume) required for each portfolio strategy.

We compare the performance of the 1/N rule to the following static models: the

minimum-variance portfolio, the Markowitz (1952) mean-variance optimal portfolio, the

Bayes-Stein shrinkage portfolio studied by Jorion (1985, 1986), the data and model ap-

proach of Pastor (2000), Pastor and Stambaugh (2000) and Wang (2004), and the short-

sale-constrained portfolios of Frost and Savarino (1988), Chopra (1993), and Jagannathan

and Ma (2003). We also compare the performance of the 1/N allocation rule to that of

dynamic models of asset allocation that allow for changes in the investment opportunity

set. Several models of optimal dynamic asset allocation have been studied in the literature.

In order to limit the length of this paper, we restrict attention to two models considered in

the book by Campbell and Viceira (2002) that allow for a stochastic riskless interest rate

and predictability in expected returns. The key reasons for choosing these two models is

that they cover the two important factors of the opportunity set that could be stochastic

(the interest rate and expected stock returns),8 the estimation of the parameters is taken

seriously in these models, the basic underlying framework for both models is very similar,

and these models are ones with which most people are now familiar, and hence, they are

well-suited for serving as benchmarks for the analysis we wish to undertake.

The out-of-sample performance of the static portfolios is evaluated for six different

data sets, which have in addition to the 90-day US T-bill: (i) monthly returns on nine

international equity indexes; (ii) quarterly returns on the US market and the 5-year bond;

8Stochastic volatility has been found to have a much smaller impact on dynamic asset allocation; seeChacko and Viceira (2004).


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(iii) monthly returns on ten industry portfolios; (iv) monthly returns on the US stock-market

portfolio, and the arbitrage portfolios, HML and SMB; (v) monthly returns on the twenty

Fama and French portfolios of firms sorted by size and book to market, in addition to the

the market portfolio and the arbitrage portfolios HML and SMB, under the assumptionthat there is a single factor model generating returns; and (vi) the same data but assuming

a three-factor return generating model. The out-of-sample performance of the dynamic

asset-allocation strategies is evaluated using the same data as in the original papers by

Campbell and Viceira.

Our main finding is that the 1/N allocation rule (with or without rebalancing at each

trading date) is not very inefficient. In fact, it performs quite well out-of-sample: it often

has a higher Sharpe ratio and lower turnover than the policies suggested by both the static

and the dynamic models of optimal asset allocation. The intuition for this result is that

the gain from optimal diversification relative to naive diversification under the 1/N rule

is typically smaller than the loss arising from the error in estimating the inputs to the

optimizing models. That is, the optimizing models do have a higher Sharpe ratio than the

1/N rule in-sample, but the 1/N rule typically has a higher Sharpe ratio out-of-sample.

The exceptions to this are where the gains from optimal rather than naive diversification

are sufficiently large to offset the loss from estimation error.

The rest of the paper is organized as follows. We describe our methodology for com-

paring the performance of different portfolio strategies in Section 2. Simple asset allocation

strategies are defined in Section 3. The description of static models of optimal portfolio

selection, and also the comparison of the strategies from these models to simple strategies is

given in Section 4, while the comparison of simple strategies with dynamic (intertemporal)

allocation strategies is given in Section 5. We conclude in Section 6.

2 Methodology

In this section, we describe the methodology used to compare the performance of simple

asset allocation rules to the strategies from various optimizing models.

The analysis of each optimal strategy consists of the following steps. The first step is to

choose a window over which to estimate the parameters: For monthly data sets, we consid-

ered estimation windows of 60 months and 120 months (corresponding to 5 and 10 years),


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while for quarterly data, we considered estimation windows of 40, 60, 80 and 120 quarters

(corresponding to 10, 15, 20 and 30 years). Because the results are not very different for

different estimation windows, we report the results for only the longest estimation window

considered. We denote the length of this estimation window by M T, where T denotesthe total number of observations.9

The second step is to estimate the parameters required for the particular model being

considered for each estimation window. For instance, in the case of static portfolios, these

are typically the expected returns vector and the variance-covariance matrix of returns.

Third, using these estimated parameters as inputs, we solve the model for the optimal

portfolio weights. Fourth, we measure the return from holding the portfolio with these

weights over the next period, that is, out-of-sample. The fifth step entails repeating this

rolling-window procedure for the next period, by including the data for the new period

and dropping the data for the earliest period. We continue doing this until the end of the

data set is reached. Finally, we calculate the quantities to report.

For each experiment (for the static portfolio strategies), we report three quantities. The

first table reports for each portfolio strategy (simple, optimal and single-asset) the Sharpe

ratio for the time series of in-sample returns. The second table reports the Sharpe ratio

for the time series of out-of sample returns, along with the P-values for the statistical

significance of the difference of this strategys Sharpe ratio from zero and the difference

from the Sharpe ratio for the equal-weight (EW) strategy.10 Finally, the third tables gives

the turnover for the portfolio, as defined below.

In order to define turnover, let wj(t) denote the portfolio weight in asset j chosen at

time t, wj(t) the portfolio weight before rebalancing at t + 1 (but at the prices prevailing

at t + 1), and wj(t + 1) the desired portfolio weight at time t + 1 (after rebalancing). Then,

turnover is defined as the sum of the absolute value of the rebalancing trades across the N

available assets and over the T trading dates, normalized by the total number of tradingdates, T:

Turnover =1

T

Tt=1

Nj=1

wj(t) wj(t + 1)

.

9To ensure that the starting point is the same for different choices of the estimation window, for the caseswhere the estimation window is shorter than 120 periods, we drop the earlier observations. For instance, inthe case of quarterly data where the estimation window is only 40 quarters, we drop the first 80 observationsso that the starting point is the same as that for the case where the estimation window is 120 quarters.

10The measures of statistical significance are calculated using the approach developed in Jobson and Korkie(1981), with the correction for a typo as given in Memmel (2003).


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For example, in the case of the 1/N strategy, wj(t) = wj(t +1) = 1/N but wj(t) is different,

because of the change in asset prices between t and t + 1, which causes a change in the

relative weights in the portfolio.

Note that turnover gives an indication of the trading volume for a particular strategy,

but only an upper bound for the transactions costs that such a strategy would entail, because

in the presence of transactions costs it would no longer be optimal to trade in the same way.

Thus, we focus on the relative turnover across different portfolio strategies rather than on

the absolute value of turnover: in the tables where we report turnover, we give the absolute

turnover for the equal-weighted (EW) strategy, but for all the other strategies we report

their turnover divided by the turnover for the 1/N strategy.

Each table is numbered using two digits: the first digit indicates the number of theexperiment and the list of experiments considered is given below, and the second digit

indicates whether the table is for in-sample Sharpe ratios, the out-of-sample Sharpe ratios,

or turnover.

# Experiment

1 Static allocation across US T-bill, 5-year bond and market portfolio2 Static allocation across nine international equity Indexes3 Static allocation across ten industry portfolios

4 Static allocation across US T-bill, US market, SML and HML portfolios5 Static allocation across size and Book-to-Market sorted portfolios assuming a

single-factor return-generating model6 Static allocation across size and Book-to-Market sorted portfolios assuming a

three-factor return-generating model7 Static allocation using simulated data

8 Dynamic asset-allocation with stochastic stock returns9 Dynamic asset-allocation with stochastic bond returns

In addition to these tables, for each experiment we also plot the path over time of the

portfolio weights for the various portfolio strategies being considered; these plots are not

included in the paper.


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3 Simple asset-allocation strategies

In this section, we provide a definition of the simple allocation strategy that we study in

this paper and also discuss how to interpret it depending on whether one considers the casewith rebalancing or without.

Our definition of a simple asset-allocation rule is one that requires no estimation of

parameters and no optimization. The simple allocation rule that we focus on in this paper

is the naive 1/N diversification rule where 1/N is allocated to each of the N assets

available for investment. So, in our setup where there are a total of N assets available,

of which N 1 are risky and one is riskfree, the simple asset allocation strategy is the

equally-weighted portfolio where the weight in each asset is equal to wj = 1/N.

We consider two versions of the simple asset-allocation strategy defined above one,

with rebalancing each period, and the other and without any rebalancing trades after the

initial date when the portfolio is formed. Thus, for the buy-and-hold strategy, the 1/N

allocation is made at the beginning of the investment horizon and this position is never

rebalanced. In the case with rebalancing, each trading date the asset allocation is revised

so that after rebalancing the weights are such that the amount invested in each of the assets

is again 1/N.

While these two versions of the simple 1/N trading strategies appear very naive, it is

important to realize that do provide the investor with some diversification. Moreover, they

can be interpreted as strategies where the investor has a view about future returns. In

the case without rebalancing, the portfolio over time assigns higher weight to assets that

have done well in the past; so, this can be interpreted as a momentum strategy. On the

other hand, the strategy with rebalancing requires a reallocation of wealth away from past

winners and toward past losers; hence, this can be interpreted as a contrarian strategy.

4 Static asset-allocation strategies

In this section, we describe strategies from a variety of static models of optimal asset-

allocation whose performance we will compare to that of the simple allocation rules discussed

above. Then, we describe the six different data sets that we use and also the simulated data

that we use. For each of these data sets we report the performance of the simple allocation


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rules relative to the various optimal allocation strategies. We conclude this section by

summarizing our results.

4.1 Optimal rules for static asset allocation

The various optimal strategies for asset allocation that we consider are the following.

4.1.1 Classical Mean-Variance

At each point in time the investor chooses a portfolio w that maximizes

w

2

ww, (1)

with a (N 1) 1 vector of expected excess returns (over the risk-free asset) and the

corresponding variance-covariance matrix. The moments and are the sample estimates

of mean and variance-covariance matrix (Maximum Likelihood Estimates).

4.1.2 Constrained Mean-Variance

The investor solves the same problem as above, but with the constraint that short sales

are not allowed, wj 0 for all j = 1, . . . , N 1 and that borrowing is also prohibited,N1i=1 wi 1. The label C is used in the tables to indicate portfolios that are constrained

from shortselling and borrowing.

4.1.3 Bayes-Stein Portfolio

The Empirical Bayes-Stein portfolio is obtained by solving the problem (1) but where instead

of the sample estimates for and the investor uses shrinkage estimators as explained in

Jorion (1986). In particular, the expected returns are calculated according to the following

shrinkage formula

BS = (1 ) + mvp1N1, (2)

where is the sample mean, mvp is the mean return on the minimum-variance portfolio,

=

T +


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=(N + 2)(T 1)

( mvp1N1)S1( mvp1N1)(T N 2), (3)

with S being the unbiased sample covariance. The covariance matrix is computed according

to the formula (see equation (15) in Jorion (1986))

V[r] = (1 +1

T + ) +

T(T + 1 + )

1N11N1

1N111N1

, (4)

and the variance-covariance matrix is obtained by the estimator (see equation (18) in

Jorion (1986))

=T 1

T N 3S.

4.1.4 Bayesian Data-and-Model Portfolios

We now describe the Data-and-Model approach developed inPastor (2000) and Pastor

and Stambaugh (2000), and extended to the case of model uncertainty in Wang (2004).

Under this approach, it is assumed that asset excess returns are generated by

ri = + irm + ui, i = 1, ...N 1,

where rm is the return on the market portfolio and we denote by the variance-covariance

matrix for ui. Wang (2004) shows how to obtain estimators for the expected return and

variance-covariance matrix that account for the belief of a Bayesian investor over the validity

of the asset pricing model (e.g. CAPM). If investors do not use a model to determine

expected return, they estimate the unrestricted regression

ri = + irm + ui, i = 1, ...N 1.

The estimate for i is shown to be

MLEi = i + iM, (5)

where is the MLE estimate and M is the sample mean of rm. In this case, the estimate

for the variance-covariance matrix of returns will be

= 2M + ,

where is the MLE of the residual from the regression.


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If, on the other hand, investors believe in the model dogmatically, they will estimate

by imposing the restriction: = 0. Let and be the restricted MLE estimator in this

case. The estimators of the mean and variance are:

CAPMi = iM,

and

= 2M

+ .

Wang (2004) shows that a Bayesian investor with precision 1/ in the model will use

the following estimate of mean and variance of the assets:

i = iM + (1 )

MLE

i, (6)

where

=1

1 + T1+2

M/2

M

, (7)

and where MLEi is obtained from the unrestricted regression (5) (basically it is the sample

mean). The estimate for the variance is given in equations (15)-(20) of Wang (2004).

4.1.5 Minimum Variance

Even though the minimum-variance portfolio is not an optimal allocation (except in the limit

where the moments of returns on all N assets are equal), and it is also not a simple strategy

according to our definition because it requires the estimation of the variance-covariance of

returns matrix, for completeness we consider also this strategy.

According to the minimum variance portfolio, the weight in the risky assets is given by

w =

1

1N11 1N1

1

1N1. (8)

4.2 Results for the different data sets considered

In this section, we report for several different data sets the performance of the simple

strategies and the strategies from the optimal static models described above.


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4.2.1 US T-bill, 5-year bond and market portfolio

We start with a discussion of the data set with the fewest risky assets. This data consists

of three assets, two of which are risky and one is riskless. The riskless rate is representedby the real Treasury bill rate, constructed as the difference between the yield on a 90-day

T-bill and log inflation. The two risky assets are, respectively, the value-weighted excess

return (including dividends) on the market, which includes all stocks on NYSE, NASDAQ

and AMEX, and the excess return on the 5-year bond. The data are quarterly and span

the horizon 1952:2 until 1999:3. This data is obtained from Campbell, Chan, and Viceira

(2003) and further details about the data construction are available in their paper.

In Table 1.1, we compare the in-sample Sharpe ratios for the simple asset allocation

strategies (equally weighted, with and without rebalancing) and in Table 1.2 we report

the out-of-sample Sharpe ratios. We also give the performance of the minimum-variance

portfolio as well as the four optimal static allocation strategies discussed in the previous

section. The panels of the table are for estimation window lengths of 40 and 120 quarters.

Turnover for the various strategies is given in Table 1.3.

From Table 1.1, we see that in sample, the Sharpe ratio is highest for the mean-variance

portfolio, 0.401, which is what one would expect because this is the optimal portfolio.

The Sharpe ratio for the equal-weights (1/N) portfolio, which ignores information aboutexpected returns and the correlation structure, is 0.376. Note that the difference in the

Sharpe ratio of the mean-variance portfolio and that of the equal-weight portfolio indicates

the potential gains from optimal rather than naive diversification.

Table 1.1 also shows that imposing constraints reduces the in-sample Sharpe ratio of the

mean-variance portfolio to 0.310 because the constraints restrict the investor from holding

the optimal position. Note, however, that as the risk aversion increases and the investor is

less likely to hold extreme portfolio positions, the effect of the constraints declines. Similarly,

because it takes into account only the correlation structure across the risky assets, while

ignoring the expected returns on these assets. the minimum-variance portfolio has a lower

Sharpe ratio of 0.342 than that of the mean-variance portfolio, 0.401. Holding just the

market portfolio, a single asset, has a Sharpe ratio of only 0.310, which is also lower than

that of the mean-variance portfolio. Finally, we note that in-sample the unconstrained and

unconstrained Bayes-Stein strategies have a Sharpe ratio that is lower than that of the

mean-variance strategy but one that is not very different.


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We now examine the portfolio strategies from the various models out of sample. Be-

cause the out-of-sample Sharpe ratio of the equal weight (1/N) strategy is not affected by

estimation error, we see in Table 1.2 that the out-of-sample Sharpe ratio for this strategy is

the same as the in-sample Sharpe ratio, 0.376. And, of all the strategies considered in thistable, this is the highest.

From Table 1.2 we also see that the out-of-sample Sharpe ratios for the mean-variance

strategy are much smaller from their in-sample values. The out-of-sample Sharpe ratio of

the mean-variance portfolio is only 0.236, compared to the in-sample value of 0.401, and this

difference is statistically significant (the P-value of this difference is 0.004) . This difference

between the in-sample Sharpe ratio and the out-of-sample Sharpe ratio is a measure of the

loss from estimation error. The reason why the 1/N portfolio strategy dominates the mean-

variance strategy is because the potential gain from optimal rather than naive diversification

is smaller than the loss from estimation error for this particular data set.

Table 1.2 also shows that constraints improve the out-of-sample performance of the

mean-variance portfolio raising its Sharpe ratio from 0.236 to 0.310 for the case of unit

risk aversion (with a P-value of 0.065); this is because constraints limit the extreme positions

that a mean-variance optimal portfolio would imply. But, as risk aversion increases, the

Sharpe ratio for the constrained policies decreases (along with a big drop in the P-value).

Similarly, the minimum-variance portfolio, which has less extreme positions because it

ignores the poorly estimated expected returns, has a higher Sharpe ratio that is higher than

the mean-variance portfolio, 0.309 compared to 0.236. The single-asset strategy of holding

the market portfolio has an out-of-sample Sharpe ratio of 0.310, which is similar to that

of the minimum-variance portfolio and the constrained mean-variance portfolio but lower

than that from the 1/N strategy (with the P-value for the difference being 0.065). The

Bayes-Stein strategies have Sharpe ratios that are similar to those of the mean-variance

strategies and are dominated by the equal-weight portfolio.

Turnover for the various strategies is given in Table 1.3. Recall that the table reports

the value of turnover for the equal-weight (1/N) strategy, and for all other strategies the

ratio of the turnover for that strategy relative to the turnover for the equal weight strategy.

From this table, we see that the value of turnover for the equal-weight (1/N) strategy is

0.114; the minimum-variance portfolio has a smaller turnover, only 0.343 times as large

as that for the equal weight strategy. But, the turnover for the mean-variance strategy is


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25.266 times as large for the case where RRA = 1. An increase in risk aversion leads to

a decrease in turnover as does the imposition of short-sale constraints. The turnover of

the Bayes-Stein strategy is also much larger than that of the 1/N strategy even with the

imposition of constraints.

4.2.2 International equity Indexes

This data consists of nine international equity indexes whose returns are computed based

on the month-end US-dollar value of the equity index for the period January 1970 to July

2001. The equity indices are for Canada, Japan, France, Germany, Italy, Switzerland,

United Kingdom, United States and the World. Data are from MSCI (Morgan Stanley

Capital International). The risk-free rate is the 90-day T-bill, obtained from CRSP.

Table 2.1 gives the in-sample Sharpe ratios and Table 2.2 reports the out-of sample

Sharpe ratios for different static allocation strategies. The two panels in the tables give the

results for two lengths of the estimation window: 60 months and 120 months. Table 2.3

gives the turnover of the various strategies considered.

We see from Table 2.1 that the in-sample Sharpe ratio is highest for the mean-variance

strategy, 0.213. The equally-weighted portfolio has a Sharpe ratio of only 0.128, with the

difference between 0.128 and 0.213 indicating the potential for gains from naive ratherthan optimal diversification. The Sharpe ratios for the minimum-variance, the constrained

mean-variance, and the Bayes-Stein strategies are lower than that for the mean-variance

but higher than that for the equally-weighted portfolio. Holding the World portfolio (as

defined by MSCI) alone has the lowest Sharpe ratio.

Table 2.2 shows that the out-of-sample performance of the mean-variance portfolio is

very poor: we see that the Sharpe-ratio is negative, 0.003 compared to the in-sample

value of 0.213. The equally-weighted portfolio, which is independent of estimation error,

has a substantially higher Sharpe ratio of 0.128 (and the P-value for the difference in Sharpe

ratios of the equally-weighted strategy and the mean-variance strategy is 0.020). Thus, for

this data set also we see that the loss from estimation error outweighs the potential gains

from optimal rather than naive diversification.

Table 2.2 shows also that the constrained mean-variance strategy, which limits the ex-

treme weights in a particular asset, does better than the unconstrained mean-variance and


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has a Sharpe ratio of 0.075 (and the P-value for the difference in the Sharpe ratio relative

to the equally-weighted portfolio is 0.129).

The performance of the Bayes-Stein portfolio is not very different from that of the mean-

variance portfolio, but the constrained Bayes-Stein strategy does a little better implying

that most of the improvement comes from the imposition of constraints rather than Bayesian

shrinkage.

The single-asset strategy of holding the World portfolio out-performs the strategies from

all the optimizing models, with a Sharpe ratio of 0.124, which is statistically indistinguish-

able from the equally-weighted portfolio (P-value for the difference is 0.432). The highest

Sharpe ratio for this data set is attained by the minimum-variance portfolio, implying that

while the estimates of expected returns are not very reliable, there are still considerable gainsfrom using the estimates of the variance-covariance matrix to diversify. However, even the

Sharpe ratio for this portfolio is indistinguishable from that of the equally-weighted portfolio

(the P-value is 0.209).

Table 2.3 shows that the turnover of the equally-weighted strategy is 0.236, while the

turnover of the minimum-variance portfolio is 6.38 times as large, and that of the mean-

variance portfolio is 3323.42 times as large for the case where risk aversion is 1, and 31.76

times as large when risk aversion is 20. The Bayes-Stein strategies have lower turnover than

the mean-variance strategies. Imposing short-sale constraints also lowers the turnover of

the mean-variance and Bayes-Stein strategies.

4.2.3 Ten industry portfolios

This data consists of ten industry portfolios (Consumer Non Durables, Consumer Durables,

Manufacturing, Energy, High-Tech, Telecommunication, Wholesale and Retail, Health, Util-

ities, and Other). The monthly returns range from July 1926 to December 2003 and were

obtained from Ken Frenchs website. The risk-free rate is the 90-day T-bill, obtained from

CRSP.

Table 3.1 reports the in-sample Sharpe ratio and Table 3.2 reports the out-of sample

Sharpe ratios for different static allocation rules. The two panels of the table are for two

lengths of the estimation window: M = 60 and 120 months. Table 3.3 gives the turnover

for different portfolio strategies.


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Table 3.1 shows that, as one would expect, the in-sample Sharpe ratio is highest for

the mean-variance strategy, 0.195. The Sharpe-ratio of the short-sale constrained port-

folio is only 0.168, and that of the minimum-variance portfolio is still lower, 0.158. The

Sharpe ratio of the Bayes-Stein portfolios (unconstrained and constrained) is similar tothe corresponding mean-variance portfolios. In-sample, the lowest Sharpe ratio is for the

equal-weight portfolio, 0.151.

When we study the out-of-sample Sharpe ratios reported in Table 3.2, we see that

the Sharpe ratio of 0.151 for the equal-weight portfolio is the highest. The out-of-sample

Sharpe ratio of the minimum-variance portfolio is 0.120 (with a P-value of 0.145 for the

difference relative to the Sharpe ratio for the equally-weighted portfolio), and that of the

mean-variance portfolio is still lower, 0.096 (with a P-value of 0.075). Imposing constraints

or using Bayes-Stein shrinkage estimators improves the performance of the mean-variance

portfolio partially.

Table 3.3 shows that the turnover of the equal-weight portfolio is 0.327, while the

turnover of the minimum-variance portfolio is 79.51 times as much and that of the mean-

variance portfolio for an investor with unit risk aversion is 192.16 times as much. Turnover

for the mean-variance portfolio decreases with an increase in risk aversion, with the impo-

sition of short-sale constraints, and with the use of Bayes-Stein shrinkage estimators.

4.2.4 US T-bill, US market, SML and HML portfolios

The next data on which we evaluate the performance of the static allocation strategies is

the one used by Pastor (2000) to illustrate the implementation of the Bayesian Data-and-

Model approach to asset allocation. To accomplish this objective, it is assumed that a

factor model dominates the structure of asset returns. The factor is represented by the

excess return on the market, defined as the value-weighted return on all NYSE, AMEX

and NASDAQ stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson

Associates). The other risky assets considered are the Fama-French portfolios, HML and

SMB. The former is a zero-cost portfolio that is long in high book-to-market stocks and

short in low book-to-market stocks. The latter is a zero-cost portfolio that is long in small

stocks and short in big stocks. We use a series of monthly returns on HML and SMB starting

in July 1927 until December 1996. The data are taken from Kenneth Frenchs website.


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The in-sample and out-of-sample Sharpe ratios for this data are reported in Tables 4.1

and 4.2, respectively. The Sharpe ratios for the portfolios formed using the Pastor (2000)

approach to asset allocation are reported under the heading Data-and-Model, where = 0

represent a prior centered entirely on the data while an = 1 represent a dogmatic belief inthe return generating model. Turnover for different portfolio strategies is given in Table 4.3.

Table 4.1 shows that in-sample the Sharpe ratio of the mean-variance portfolio is the

highest of all strategies, 0.172. In contrast, the Sharpe ratio of the minimum variance

portfolio is only 0.156 and that of the equal-weight portfolio is even lower, 0.134. Note also

that in-sample, of all the strategies using the Data-and-Model approach, the best are for

the case where = 0, implying a prior centered entirely on the data.

Table 4.2 shows that out of sample the Sharpe ratio of the mean-variance portfoliostrategy drops from its in-sample value of 0.172 to 0.139, so that the difference to the

Sharpe ratio from the equally-weighted portfolio strategy is statistically insignificant (P-

value is 0.449). Similarly, there is a drop in the Sharpe ratio of the minimum-variance

portfolio to 0.130. The Sharpe ratio for the portfolios formed using the Data-and-Model

approach when one puts some weight on the model, = 0.5, dominate the Sharpe ratio

of the mean-variance model. But, out-of-sample, the highest Sharpe ratios are for the

Bayes-Stein constrained portfolios, and these are quite close to those for the mean-variance

constrained portfolios. So, in this data set the Sharpe ratio of the 1/N portfolio strategy

does not dominate the Sharpe ratio of the strategies from various optimizing models, though

it is close enough so that the difference is statistically insignificant.

Table 4.3 shows that the turnover for the equal-weighted portfolio is 0.371. Turnover for

the minimum-variance portfolio is 1.26 times as much and for the mean-variance portfolio

is 37.35 times as much when risk aversion is one. As before, turnover decreases with an

increase in risk aversion, with the imposition of short-sale constraints, and with the use

of Bayes-Stein shrinkage estimators. And, for the strategies formed using the Data-and-Model approach, turnover decreases as one puts more weight on the model (that is, as

increases) and less weight on the data.

4.2.5 Size and Book-to-Market sorted portfolios under a single factor model

The next data set is the one used by Wang (2004) to study the data-and-model approach

of Pastor (2000), and Pastor and Stambaugh (2000). The data consist of returns on the


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twenty Fama and French portfolios of firms sorted by size and book to market, in addition

to the Fama-French portfolios HML and SMB, and also the market portfolio for a total

of 23 risky assets.11 Returns are monthly and the data span from July 1926 to December

2002. In our first use of this data, we assume that returns are generated by a single-factormodel, and we assume the market to be this factor; that is, we assume that the CAPM

holds.

Just as for the previous data set, we use the methodology of Pastor (2000) to construct

Bayesian portfolios using the Data-and-Model approach and report the in-sample and

out-of-sample Sharpe ratios in Tables 5.1 and 5.2, respectively. The two panels in the table

correspond to the case where the estimation window has a length of 60 months, and 120

months. Table 5.3 gives the turnover for different portfolio strategies.

Table 5.1 shows that in-sample Sharpe ratio for the mean-variance portfolio strategy

is substantially greater than that for the 1/N strategy, 0.361 compared to 0.148. This

implies that there the potential gains from from optimal rather than naive diversification

are substantial much higher than for any of the previous data sets considered above.

As before, of all the Data-and-Model strategies in-sample, it is optimal to assume that

= 0.0 implying that the prior is centered around only the data with no weight given to

the model.

Table 5.2 shows that even though out-of-sample the Sharpe ratio for the mean-variance

portfolio drops to 0.258 because of estimation error, it is still substantially higher than the

Sharpe ratio for the 1/N strategy, and the difference from the Sharpe ratio for the equally-

weighted portfolio is statistically significant (P-value is 0.010). That is, in this data set the

gains from optimal diversification over naive diversification are sufficiently large that they

are not fully eroded by estimation error. Moreover, the portfolio strategy with the highest

Sharpe ratio is the one using the Data-and-Model approach with a value of = 0.5; that

is, with some weight put on the asset pricing model and some on the data.

Table 5.3 shows that the turnover for the 1/N strategy is 0.590, while all the strategies

from optimizing models have much higher turnover. For instance, the turnover for the

mean-variance portfolio is 587.11 times that of the 1/N strategy, and the turnover for the

strategy using the Data-and-Model approach with = 0.5 is 244.49 times that for the

1/N strategy. And, even when we consider the case where the risk aversion of the investor

11As in Wang (2004), we exclude the five portfolio containing the largest firm since the market, SMB andHML are almost a linear combination of the 25 Fama-French portfolio.


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is 20, turnover for the strategies from optimizing models is substantially greater than that

for the 1/N strategy.

4.2.6 Size and Book-to-Market sorted portfolios under a three factor model

In this specification, we use the same data just described above, but instead of assuming

a single-factor return-generating model, we assume a three-factor model for the return-

generating process. The three factors are the market, HML and SMB portfolios (essentially,

this is an APT setting).

The results for this case are very similar to that for the previous one. The in-sample

quantities are obviously the same for all the portfolio strategies except for those formed

using the Data-and-Model approach. Table 6.1 shows that in-sample Sharpe ratio for

the mean-variance portfolio strategy is substantially greater than that for the 1/N strategy,

0.361 compared to 0.148. As stated above, this implies that there the potential gains from

from optimal rather than naive diversification are substantial much higher than for any of

the previous data sets considered above. As before, of all the Data-and-Model strategies

in-sample, it is optimal to assume that = 0.0 implying that the prior is centered around

only the data with no weight given to the model.

Table 6.2 shows that even though out-of-sample the Sharpe ratio for the mean-variance

portfolio drops to 0.258 for the because of estimation error, it is still substantially higher

than the Sharpe ratio for the 1/N strategy, and as in the earlier case, the difference from the

Sharpe ratio for the equally-weighted portfolio is statistically significant (P-value is 0.010).

That is, in this data set the gains from optimal diversification over naive diversification are

sufficiently large that they are not fully eroded by estimation error. Moreover, the portfolio

strategy with the highest Sharpe ratio is the one using the Data-and-Model approach

with a value of = 0.5; that is, with some weight put on the asset pricing model and some

on the data. This Sharpe ratio is slightly higher than the corresponding one in the previous

section, 0.271 instead of 0.268 , where it was assumed that the asset pricing model is the

CAPM rather than the APT.

The turnover numbers are also the same as they were for the previous section, except for

the strategies using the Data-and-Model approach. Table 6.3 shows that the turnover for

the 1/N strategy is 0.590, while all the strategies from optimizing models have much higher

turnover. For example, the turnover for the mean-variance portfolio is 587.11 times that of


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the 1/N strategy, and the turnover for the strategy using the Data-and-Model approach

with = 0.5 is 223 times that for the 1/N strategy. And, even when we consider the case

where the risk aversion of the investor is 20, turnover for the strategies from optimizing

models is substantially greater than it is for the 1/N strategy.

4.2.7 Simulated Data

We also consider a data set generated using simulations. The objective of this exercise is

to see whether our intuition underlying the results described above is confirmed in a data

set where we know the properties of the data.

Details of the simulation

For the simulations, we assume that the market is composed of N risky assets, K factors

and one risk-free asset. The excess returns over the risk-free rate are generated by the

following model

zt = + BFt + t (9)

where zt is a N 1 vector of excess return on the assets, is a N 1 vector of mispricing

coefficients, B is a N K matrix of factor loadings, Ft is a K 1 vectors of returns on

the factors and t is a N 1 vector of noises. We assume that returns are i.i.d. with the

following moments

E[t] = 0N, t (10)

E[t t ] = =

2

IN, t (11)

E[Ft] = F, t (12)

E[(Ft F) (Ft F)] = F, t (13)

Cov(Ft, t ) = 0KN, t. (14)

The time series of returns on the factors is drawn from a multivariate normal distribution

with mean F and variance-covariance matrix F. The time series of noise is obtained

from a multivariate normal distribution with mean zero and variance-covariance matrix .

Finally we draw the risk-free rate from a normal distribution with mean rf and variance

2rf.

To initialize the simulation we need to choose values for (i) the average risk-free rate rf

and its variance 2rf, (ii) the mispricing , (iii) the factor loading B, (iv) the mean F of


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the factors and their variance-covariance matrix F and (v) the variance of the noise.

These values remain constant across all the simulations.

In our experiment we assume N = 3, K = 1 so that there are 4 risky assets in which

one can invest. We chose the average risk-free rate to be 2% p.a. with an annual variance

of 2rf = 2%. The expected return on the factor portfolio is taken to be 8% p.a. with an

annual standard deviation of 16%. This amount to an annual Sharpe ratio of 0.5 for the

factor. The factor loading B for the three assets are each drawn from a uniform distribution

with support [0.5, 1.5], i.e. B U[0.5, 1.5]. Finally, the variance-covariance matrix of noise

is assumed to be diagonal, = 2

IN with each of the three elements of the vector

drawn from a uniform distribution with support [.10, .20], i.e. the annual volatility of noise

ranges from 10 to 20%.

Given the parameter values described above, we draw 1000 time series of returns for

360 months and for each we compute the out-of-sample Sharpe ratio. In what follows, we

report the average Sharpe ratios across the 1000 simulations.

Results of the simulation

Table 7.1 reports the Sharpe ratios for different portfolio strategies. The row titled Mean-

Variance-True reports the Sharpe ratio of the portfolio strategy where the investor knows

the parameters generating returns; this row corresponds to the tables giving the in-sample

Sharpe ratios in the previous sections. Not surprisingly, this is the strategy that has the

highest Sharpe ratio but even in this case the difference is not statistically significant

(P-value is 0.342). That is, even if one had an infinitely long data series over which to

estimate the parameters so that one knows the true value of the moments, the difference

in the Sharpe ratio of the optimal mean-variance portfolio rule is not statistically different

from that of the 1/N allocation rule.

Moreover, when the investor does not know the true moments and needs to estimate

them, then the strategy with the highest point estimate of the Sharpe ratio is the 1/N

strategy, which has a Sharpe ration of 0.133, compared to the Sharpe ratio of the mean-

variance strategy out-of-sample, 0.086 (the P-value for this difference is 0.054). The Sharpe

ratio of the mean-variance strategy improves with the imposition of constraints, and the

use of Bayes-Stein estimators. And, note that because the data are generated by a factor

model where mispricing is zero (the model is true), of the three Data-and-Model cases

considered, the one that puts the maximum weight on the model ( = 1) does the best.


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From Table 7.2, we see that the turnover for the 1/N strategy is 0.212, while that for

the minimum-variance portfolio is 14.03 times as much and for the mean-variance portfolio

for an investor with unit risk aversion is 181.02 times as much. Turnover decreases with an

increase in risk aversion, the imposition of constraints, and the use of Bayes-Stein estimators.Of the three cases considered for the Data-and-Model approach, the lowest turnover is

for the case where the investor has a dogmatic belief in the market, = 1, and the reason

for this is that the data is being generated from this model.

From the above exercise, we conclude that if one knew the true moments then obviously

the optimal strategy would be the one suggested by the mean-variance model. But, when

the moments have to be estimated, then it may be best to use the 1/N strategy.

4.3 Conclusion from the analysis of static asset allocation strategies

From the above analysis it seems that the simple asset allocation strategy is not very

inefficient: often the simple allocation rules have a higher Sharpe ratio than the strategies

from the models of static portfolio choice, and also the turnover is much lower.

There are two important limitations of the analysis conducted so far: (1) We have as-

sumed that the investor is myopic (i.e. chooses a portfolio without looking at intertemporal

hedging); (2) We have assumed that the moments of asset returns follow very simple pro-

cesses and that the opportunity set of the investor is constant over time. We address both

these limitations in the next section.

5 Dynamic asset-allocation strategies

In this section, we wish to look at dynamic asset allocation models in which a variety of

techniques are used for the estimation of the return processes. Following the classification

of these model in the book by Campbell and Viceira (2002) we consider two alternative

ways of modelling a time-varying opportunity set. In Section 5.1 we study the case where

expected returns on equities are stochastic and in Section 5.2 we consider the case where

interest rates (bond prices) are stochastic. Just as in the previous section, we compare the

optimal allocation strategies with the simple asset allocation strategies.


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5.1 Stochastic stock returns

The setup is the one described in Campbell and Viceira (1999) and Campbell, Chan, and

Viceira (2003). The assets available are the 90-day T-bill, a five-year bond, and the marketportfolio as described in Section 4.2.1. It is assumed that expected returns on the equity

market portfolio are predictable and are described by a VAR model with a suitable number

of state variables. The state variables used to predict returns are the nominal log yield on a

90-day Treasury bill, the dividend-price ratio, and the yield spread between the 5-year zero

coupon bond yield and the T-bill rate.

Investors form portfolios by maximizing Epstein-Zin preferences. As in Campbell, Chan,

and Viceira (2003), we assume that the elasticity of intertemporal substitution is = 1.

Because the opportunity set is stochastic the optimal portfolio will contain both a myopic

and a hedging component.

Table 8.1 gives the out-of-sample Sharpe ratios. We see from this table that out-of-

sample the highest Sharpe ratio is for the 1/N strategy, 0.372.The Sharpe ratio for the

minimum-variance portfolio is 0.301, and the Sharpe ratio for the dynamically optimal

strategy for an investor with unit risk aversion is only 0.204, and it decreases as risk aversion

increases.

The turnover for these strategies is reported in Table 8.2. We see that the turnover for

the 1/N strategy is 0.027, but it is an order of magnitude greater for the dynamic portfolio

strategies, ranging from 310.75 times as much for the case of unit risk aversion to 129.34

times as much for the case where risk aversion is equal to twenty.

5.2 Stochastic interest rates

The setup is the one described in Campbell and Viceira (2001) and Campbell and Viceira

(2002, Chapter 3), where they assume that real interest rates can change over time but

risk premia and variances and covariances of returns are constant. The term structure is

modeled using an AR(1) process with two factors, which allows one to distinguish between

real and nominal bonds.

The assets available for investment are the equity market portfolio, a three-month nom-

inal bond, and a 10-year nominal bond. The two-factor term structure model is estimated

by applying a Kalman filter to data on US nominal interest rates, equities, and inflation.


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We use the same historical data set as in Campbell and Viceira (2001). In particular, we use

nominal zero-coupon yields at maturities three months, one year, three years, and ten years.

For equity, we use the value-weighted return, including dividends, on the NYSE, AMEX,

and NASDAQ markets. For inflation, we use a consumer price index that retrospectivelyincorporates the rental-equivalence methodology. We construct a quarterly dataset. Our

sample begins in 1954:1 and ends in 1996.3.12

Investors form portfolios by maximizing Epstein-Zin preferences. As in Campbell, Chan,

and Viceira (2003), we assume that the elasticity of intertemporal substitution is =

1. Since the opportunity set is stochastic the optimal portfolio policies contain both a

myopic and a hedging component. We consider both unconstrained policies and policies

with borrowing and short-sale constraints.

We then compare the out-of-sample Sharpe ratios from following the unconstrained and

constrained dynamic strategies to the one obtained by using the naive 1/N rule. We report

the out-of-sample Sharpe ratios in Table 9.1. We see from this table that out-of-sample the

1/N strategy has the highest Sharpe ratio, 0.366, while the dynamic optimal strategy has

a Sharpe ratio of 0.284, which declines slightly with an increase in risk aversion.

6 Conclusions

We have compared the performance of 1/N simple allocation rule (without and with rebal-

ancing over time) to the optimal policies from static and dynamic models of asset allocation.

We consider both constrained and unconstrained strategies. The comparison with the static

allocation rules is done for six different data sets. The comparison with the dynamic asset

allocation rules is done using the data set used in the original studies.

Our main finding is that the 1/N allocation rule (with or without rebalancing at each

trading date) is not very inefficient. In fact, it performs quite well out-of-sample: it oftenhas a higher Sharpe ratio and lower turnover than the policies suggested by both the static

and the dynamic models of optimal asset allocation.

Our analysis suggests that if one knew the true moments of asset returns, then it would

be best to use the portfolio weights suggested by an optimizing model. But, when the mo-

12The sample length is slightly different to the one in Campbell and Viceira (2001) because we have datafor long-term real bonds only from 1954:1.


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ments of asset returns are not known and have to be estimated, then the error in estimating

these moments is significant at least for the data sets considered and using the estima-

tion methods that are standard in this literature. Consequently, in many cases, the large

estimation error overwhelms the gains from optimization and so simple allocation strategiesoutperform those from optimizing models of asset allocation.

There are two conclusions that one might wish to draw from the analysis described

in this paper. One, while there has been considerable progress in the design of optimal

portfolios, more energy needs to be devoted to improving the estimation of parameters for

the moments of asset returns or the processes driving these moments. Two, when evaluating

the performance of a particular strategy for optimal asset allocation, the simple 1 /N naive-

diversification rule serves as a useful benchmark.


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Table 1.1: In-sample Sharpe ratios for static asset allocation across US T-bill,5-year bond and US market portfolio

The table reports in-sample Sharpe ratios for static asset allocation across T-bill, 5-yearbond, and the US market portfolio. There are a total of 185 quarterly observations from1952:2 to 1999:3. The riskless rate is represented by the real Treasury bill rate, constructedas the difference between the yield on a 90-day T-bill and log inflation. The two riskyasset are, respectively, the value-weighted excess return (including dividends) on the NYSE,NASDAQ and AMEX market and the excess bond return on the 5-year bond. This dataset is the one used in Campbell, Chan, and Viceira (2003).

Equal Weight 0.376

Min-Variance 0.346RRA()

1 2 3 4 5 10 20

Mean-Variance 0.401 0.401 0.401 0.401 0.401 0.401 0.401Mean-Variance(C) 0.310 0.310 0.314 0.339 0.358 0.398 0.401

Bayes-Stein 0.394 0.394 0.394 0.394 0.394 0.394 0.394Bayes-Stein (C) 0.310 0.325 0.360 0.381 0.393 0.399 0.394

Single-Asset StrategiesMarket 0.310Bond 0.285


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Table 1.2: Out-of-sample Sharpe ratios for static asset allocation across UST-bill, 5-year bond and US market portfolio

The table reports out-of-sample Sharpe ratios for static asset allocation across T-bill, 5-yearbond, and the US market portfolio. There are a total of 185 quarterly observations from1952:2 to 1999:3 and the estimation window is 120 quarters. The riskless rate is representedby the real Treasury bill rate, constructed as the difference between the yield on a 90-dayT-bill and log inflation. The two risky asset are, respectively, the value-weighted excessreturn (including dividends) on the NYSE, NASDAQ and AMEX market and the excessbond return on the 5-year bond. This data set is the one used in Campbell, Chan, andViceira (2003).

Equal Weight 0.376P-value (difference from 0) (0.002)

Min-Variance 0.309P-value (difference from 0) (0.008)

P-value (difference from EW) (0.301)

RRA()1 2 3 4 5 10 20

Mean-Variance 0.236 0.236 0.236 0.236 0.236 0.236 0.236P-value (difference from 0) (0.030) (0.030) (0.030) (0.030) (0.030) (0.030) (0.030)

P-value (difference from EW) (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) (0.004)

Mean-Variance(C) 0.310 0.274 0.260 0.261 0.260 0.260 0.260P-value (difference from 0) (0.007) (0.015) (0.020) (0.019) (0.020) (0.020) (0.020)


Bayes-Stein 0.226 0.226 0.226 0.226 0.226 0.226 0.226P-value (difference from 0) (0.036) (0.036) (0.036) (0.036) (0.036) (0.036) (0.036)

P-value (difference from EW) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014)Bayes-Stein (C) 0.229 0.255 0.250 0.245 0.243 0.243 0.243

P-value (difference from 0) (0.034) (0.022) (0.024) (0.026) (0.027) (0.027) (0.027)


Single-Asset StrategiesMarket 0.310

P-value (difference from 0) (0.007)


Bond 0.285P-value (difference from 0) (0.012)



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Table 1.3: Turnover for static allocation strategies across US T-bill, 5-year bondand US market portfolio

The table reports the turnover for the 1/N strategy (in italics), and for all the otherstrategies their turnover divided by the turnover for the 1/N strategy when allocatingwealth across T-bill, 5-year bond, and the US market portfolio. There are a total of 185quarterly observations from 1952:2 to 1999:3 and the estimation window is 120 quarters.The riskless rate is represented by the real Treasury bill rate, constructed as the differencebetween the yield on a 90-day T-bill and log inflation. The two risky asset are, respectively,the value-weighted excess return (including dividends) on the NYSE, NASDAQ and AMEXmarket and the excess bond return on the 5-year bond. This data set is the one used inCampbell, Chan, and Viceira (2003).

Equal Weight 0.114

Min-Variance 0.343

RRA()1 2 3 4 5 10 20

Mean-Variance 25.266 15.807 12.052 9.821 8.319 4.781 2.615Mean-Variance(C) 0.000 2.909 5.290 4.952 4.341 2.652 1.506



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Table 2.1: In-sample Sharpe ratios for international equity indexes

The table reports in-sample Sharpe ratios for static asset allocation across a T-bill, 8 in-ternational equity indices and the world equity index. There are a total of 379 monthlyobservations over the period January 1970 to July 2001. The nine equity indices are forCanada, Japan, France, Germany, Italy, Switzerland, United Kingdom, United States andthe World. Data are from MSCI (Morgan Stanley Capital International).The risk-free rateis the 90-day T-bill, obtained from CRSP.

Equal Weight 0.128


1 2 3 4 5 10 20Mean-Variance 0.213 0.213 0.213 0.213 0.213 0.213 0.213Mean-Variance(C) 0.163 0.163 0.163 0.163 0.163 0.163 0.163


Single-Asset StrategiesWR 0.124CA 0.053JP 0.062FR 0.098GE 0.084IT 0.091SW 0.100UK 0.120US 0.153


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Table 2.2: Out-of-sample Sharpe ratios for international equity indexes

The table reports out-of-sample Sharpe ratios for static asset allocation across a T-bill, 8international equity indices and the world equity index. There are a total of 379 monthlyobservations over the period January 1970 to July 2001 and the estimation window is 120months. The nine equity indices are for Canada, Japan, France, Germany, Italy, Switzer-land, United Kingdom, United States and the World. Data are from MSCI (Morgan StanleyCapital International).The risk-free rate is the 90-day T-bill, obtained from CRSP.




RRA()1 2 3 4 5 10 20

Mean-Variance -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003P-value (difference from 0) (0.483) (0.483) (0.483) (0.483) (0.483) (0.483) (0.483)






Bayes-Stein (C) 0.077 0.098 0.119 0.123 0.114 0.096 0.096P-value (difference from 0) (0.109) (0.057) (0.028) (0.024) (0.033) (0.062) (0.062)


Single-Asset StrategiesWR 0.124

P-value (difference from 0) (0.023)


CA 0.053P-value (difference from 0) (0.196)


JP 0.062P-value (difference from 0) (0.159)


FR 0.098

P-value (difference from 0) (0.058)P-value (difference from EW) (0.223)

GE 0.084P-value (difference from 0) (0.088)


IT 0.091P-value (difference from 0) (0.071)


SW 0.100P-value (difference from 0) (0.055)


UK 0.120


US 0.153P-value (difference from 0) (0.007)



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Table 2.3: Turnover for static allocation strategies across international equityindexes

The table reports the turnover for the 1/N strategy (in italics), and for all the otherstrategies their turnover divided by the turnover for the 1/N strategy when allocatingwealth across a T-bill, 8 international equity indices and the world equity index. Thereare a total of 379 monthly observations over the period January 1970 to July 2001 and theestimation window is 120 months. The nine equity indices are for Canada, Japan, France,Germany, Italy, Switzerland, United Kingdom, United States and the World. Data arefrom MSCI (Morgan Stanley Capital International).The risk-free rate is the 90-day T-bill,obtained from CRSP.

Equal Weight 0.236





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Table 3.1: In-sample Sharpe ratios for 10 industry p ortfolios

In-sample Sharpe ratios for static asset allocation across 10 industry portfolios (Consumer Non Durables,Consumer Durables, Manufacturing, Energy, High-Tech, Telecommunication, Wholesale and Retail, Health,Utilities and Other). There are a total of 930 monthly observations from July 1926 to December 2003. Thisdata was obtained from Ken Frenchs website.

Equal Weight 0.151



Bayes-Stein 0.187 0.187 0.187 0.187 0.187 0.187 0.187

Bayes-Stein (C) 0.171 0.173 0.171 0.168 0.168 0.168 0.168Bayes-Stein (BH) 0.080 0.080 0.080 0.080 0.080 0.080 0.080Bayes-Stein (BH+C) 0.130 0.124 0.124 0.124 0.124 0.124 0.124

Single-Asset StrategiesNoDur 0.151Durbl 0.122Manuf 0.115Enrgy 0.138HiTec 0.112Telcm 0.108

Shops 0.132Hlth 0.158Utils 0.112Other 0.123


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Table 3.2: Out-of-sample Sharpe ratios for 10 industry p ortfolios

Out-of-sample Sharpe ratios for static asset allocation across 10 industry portfolios (Consumer Non Durables,Consumer Durables, Manufacturing, Energy, High-Tech, Telecommunication, Wholesale and Retail, Health,Utilities and Other). There are a total of 930 monthly observations from July 1926 to December 2003 andthe estimation window is 120 months. This data was obtained from Ken Frenchs website.




RRA()1 2 3 4 5 10 20

Mean-Variance 0.096 0.096 0.096 0.096 0.096 0.096 0.096P-value (difference from 0) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003)








Single-Asset Strategies

NoDur 0.151P-value (difference from 0) (0.000)


Durbl 0.122P-value (difference from 0) (0.000)


Manuf 0.115P-value (difference from 0) (0.001)


Enrgy 0.138P-value (difference from 0) (0.000)


HiTec 0.112P-value (difference from 0) (0.001)


Telcm 0.108P-value (difference from 0) (0.001)


Shops 0.132P-value (difference from 0) (0.000)


Hlth 0.158P-value (difference from 0) (0.000)


Utils 0.112


Other 0.123P-value (difference from 0) (0.000)



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Table 3.3: Turnover for static allocation strategies across 10 industry portfolios

The table reports the turnover for the 1/N strategy (in italics), and for all the otherstrategies their turnover divided by the turnover for the 1/Nstrategy when allocating wealth

across 10 industry portfolios (Consumer Non Durables, Consumer Durables, Manufacturing,Energy, High-Tech, Telecommunication, Wholesale and Retail, Health, Utilities and Other).There are a total of 930 monthly observations from July 1926 to December 2003 and theestimation window is 120 months. This data was obtained from Ken Frenchs website.

Equal Weight 0.327


1 2 3 4 5 10 20Mean-Variance 192.163 141.802 124.510 139.006 58.038 24.508 15.330

Mean-Variance(C) 5.272 4.714 4.313 3.973 3.686 2.768 1.859



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Table 4.1: In-sample Sharpe ratios for static allocation across T-bill, market,HML and SMB portfoliosThe table reports in-sample Sharpe ratios for static asset allocation across a T-bill, the market, defined asthe value-weighted return on all NYSE, AMEX and NASDAQ stocks (from CRSP) minus the one-monthTreasury bill rate (from Ibbotson Associates), and the Fama-French portfolios, HML and SMB. The former

is a zero-cost portfolio that is long in high book-to-market stocks and short in low book-to-market stocks.The latter is a zero-cost portfolio that is long in small stocks and short in big stocks. There are a total of834 observations from July 1927 to December 1996. The data are taken from Kenneth Frenchs website. Inthe following tables an = 0 represent a prior entirely centered on the data while an = 1 represent adogmatic belief in the model.

Equal Weight 0.134




Data-and-Model = 0.00 0.172 0.172 0.172 0.172 0.172 0.172 0.172 = 0.25 0.170 0.170 0.170 0.170 0.170 0.170 0.170 = 0.50 0.164 0.164 0.164 0.164 0.164 0.164 0.164 = 0.75 0.138 0.138 0.138 0.138 0.138 0.138 0.138 = 1.00 0.070 0.070 0.070 0.070 0.070 0.070 0.070

Single-Asset StrategiesSMB 0.066HML 0.156MKT 0.070


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Table 4.2: Out-of-sample Sharpe ratios for static allocation across T-bill, mar-ket, HML and SMB portfoliosThe table reports out-of-sample Sharpe ratios for static asset allocation across a T-bill, the market, definedas the value-weighted return on all NYSE, AMEX and NASDAQ stocks (from CRSP) minus the one-monthTreasury bill rate (from Ibbotson Associates), and the Fama-French portfolios, HML and SMB. The former

is a zero-cost portfolio that is long in high book-to-market stocks and short in low book-to-market stocks.The latter is a zero-cost portfolio that is long in small stocks and short in big stocks. There are a total of834 observations from July 1927 to December 1996 and the estimation window is 120 months. The data aretaken from Kenneth Frenchs website. In the following tables an = 0 represent a prior entirely centeredon the data while an = 1 represent a dogmatic belief in the model.




RRA()

1 2 3 4 5 10 20Mean-Variance 0.139 0.139 0.139 0.139 0.139 0.139 0.139P-value (difference from 0) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)








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