Estimation Risk and Portfolio Selection in the Lower Partial Moment

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Estimation Risk and Portfolio Selection in the LowerPartial Moment

Mattias Persson

Department of Economics

Lund University

P.O. Box 7082

220 07 LundSweden

Email: [email protected]

June 8, 2000

Abstract

Portfolio selection models generally assume that the investor knows the param-

eters of the probability distribution of security returns. In practise the investor

must, however, employ estimates of the necessary parameters. In this paper we

investigate the effect of estimation risk on the efficient frontier in the lower partial

moment framework. The results of the average difference between the actual and

estimated portfolios show that the estimated portfolios are biased predictors of the

actual portfolios. However, the estimates of the optimal portfolios can be improved.

If our concern is the uncertainty in the optimal portfolio weights, then a bootstrap

approach should be used to improve the optimizations. On the other hand, if our

concern is related to the risk and portfolio mean returns of the optimized portfolios,

then a James-Stein approach should be used .

I thank seminar participants at Lund University, at the Nordic Econometric meeting , Uppsala, 1999,

and at the 28th Annual Conference of Economists, Melbourne, 1999, for valuable comments.

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the MLPM efficient portfolios, although the poor performance of mean-variance optimized

portfolios that use sample estimates as inputs has often been noted. For example, Best

and Grauer (1992) analyzed the effects of changes in assets means, and found that such

changes had large effects on the composition of the mean-variance efficient set. Chopraand Ziemba (1993), studied the relative importance of errors in mean returns, variances,

and covariances. Jobson (1991), investigated the effect of errors in the input parame-

ters, and Broadie (1993), showed in a simulation study, that the estimation errors in the

mean-variance efficient frontier can be surprisingly large, and that estimates of portfolio

performance are optimistically biased predictors of actual portfolio performance. Michaud

(1989), discussed the implications of estimation error for portfolio managers, and Jobson

(1991) investigated the effect of input parameters, but without restriction on short-selling.

Simaan (1997), compared the estimation risk in the mean-absolute deviation model andthe mean-variance model, and showed that ignoring the covariance matrix resulted in

greater estimation risk.

The purpose of this paper is to investigate the effect of estimation errors in the MLPM-

model. We particularly address the following question; How far away is the optimized

portfolio, based on sample data, from the true efficient portfolio? The issue is analyzed

using a simulation approach similar to the one employed by Broadie (1993) and Jobson

and Korkie (1980).1 A simulation approach is employed because it can directly show the

effect of estimation error on the results of the efficient portfolios in the MLPM-model. Inorder to analyze how large the estimation errors are in the MLPM-model, we compare

the results to the estimation errors in the mean-variance model. Finally, three different

approaches to improve the portfolio optimization are employed. Two of the methods have

previously been used in the mean-variance model and belong to the class of shrinkage

estimators which improve the asset means. The third approach used to improve the

performance is a bootstrap approach, which in contrast to the other two methods are

employed to improve not only the assets mean returns but also the estimates of the risk

measure and the optimized portfolio weights.

The remainder of this chapter is organized as follows: section 2 gives a brief review

of portfolio selection in the lower partial moment, in section 3 the methodology of the

simulation study is presented, the results are presented in section 4 and the summary and

1 This approach is also proposed by Michaud (1998) as a method to improve the estimates of mean-

variance efficient portfolios.

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concluding remarks are provided in section 5.

2 Portfolio Selection in the Lower Partial Moment

Bawa (1975, 1978), Bawa and Lindenberg (1977) and Fishburn (1977) have shown that the

lower partial moment provides a more general alternative to the traditional mean-variance

model. Consider an investor who is averse to downside risk with target rate of return, .

Let X denote the investors portfolio allocation across k assets with X0 = (X1, X2,...,Xk)

and let R represent the vector of security returns R0 = (R1, R2,...,Rk) . Let F and FX

denote the joint distribution of security returns and the probability distribution of returns

of the portfolio respectively. The nth order lower partial moment of the distribution of

returns under the allocation X about , LPMn(; X) is then

LP Mn(; X) =

Z

(RX)ndFX(RX) =Z

(X0R)ndF(R). (1)

The investors optimization problem is then to minimize (1) subject to

k

Xi=1

XiE(Ri) = (2)

kXi=1

Xi = 1 orkXi=1

Xi = 1, Xi > 0

Bawa (1976) has shown that the LPMn(; X) is a convex function ofX. Furthermore,

the optimal value of LPMn(; X), as a function of , is increasing and convex for all

greater than the mean of the portfolio that yields the minimum lower partial moment over

the feasible set. This is the same property exhibited by the admissible boundary in the

mean-variance model. The target rate of return, , can be fixed, variable but deterministic

and stochastic. For example, when the downside risk is measured against some benchmark

portfolio we have a stochastic target rate of return.2 The MLPM-model was proposed as

2 In terms of equilibrium valuation, a target rate of return equal to the riskless rate of return and

normality is both required for the MLPM-CAPM to be reduced to the standard CAPM, see Bawa and

Lindenberg (1977).

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an approximation for arbitrary distributions by Bawa (1975) and Bawa and Lindenberg

(1977), and is a way of reducing the dimensionality of stochastic dominance to a two-

parameter framework. In fact, the admissible set obtained in the MLPM-model for a

fixed is a subset of the admissible set under second order stochastic dominance whenn is equal to zeros or one, and third order stochastic dominance when n is equal to two.

The analysis in the MLPM-model holds exactly if the distribution of stock returns belong

to the two-parameter location scale family, which includes normal distributions, Student

t-distributions with the same degree of freedom, and stable distributions with the same

characteristic exponent (between one and two) and skewness parameter (not necessarily

zero).

The MLPM-model is justified for a general set of utility functions. The order, n, of the

LPMn measure, determines the type of utility functions consistent with that risk measure.LPM

0

3 is consistent with all utility functions that prefer more to less (u0 > 0), and LPM1

with all risk averse utility functions (u0 > 0 and u00 < 0), while LPM2

is valid for all

risk averse functions displaying skewness preference (u0 > 0, u00 < 0 and u000 > 0). That

is, LPM1

is consistent with the familiar HARA class of utility functions, while LPM2

is

consistent with the DARA class of utility functions.

Many popular notations of risk are special cases of the MLPM-model. For example,

with n = 0 and the target rate of return equal to zero, LPM0

is the probability of a loss.4

For n = 2 and a target rate of return equal to the mean of the distribution, LPM 2 becomesthe traditional semivariance measure. Moreover, under the normal distribution LPM

2is

proportional to variance, and would result in the same ordering of the risky assets as the

mean-variance model.

Therefore, the power and flexibility of the downside-risk framework stem from the

joint set of assumptions regarding investors preferences and asset return distributions.

Unlike other frameworks, which place restrictions on preferences or on distributions, the

lower partial moment approach uses a combined set of reasonable and less restrictive

assumptions. The downside risk approach in the lower partial moment is not only more

attractive in terms of its consistency with the way investors actually perceive risk5, but

3 The LPM0

efficient set is not convex, and can therefore not be used for the general purpose of portfolio

selection.

4 The probability of loss, has, among others, been studied by Kataoka (1963) as a measure of risk.

5 See for example Fishburn (1977) and Mao (1970).

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it is also valid under a broader range of conditions.

3 Methodology

A simulation approach is employed for the analysis of estimation risk in the MLPM-

model because it can directly show the effect and magnitude of the estimation error on

the portfolios. In addition, a simulation approach allows us to concentrate on the effect of

estimation risk on the outcome of the MLPM analysis. An analytical approach would be

difficult since the lower partial moments for a given portfolio depend on the actual location

of the discrete observations, and the discrete observations directly affect the estimated

portfolios which in turn implies that a closed form solution does not exist. Furthermore,

investors are probably not directly interested in the magnitude of the estimation errors inthe parameters, but rather in their effect on the outcome of the portfolios. Since the sim-

ulations and optimizations are computer intensive we restrict our analysis to incorporate

only LPM of the second order, which is also the most commonly used measure.

3.1 Population parameters

We simulate returns from a multivariate normal distribution, and this is done for two

reasons. First, under multivariate normality both the mean-variance and MLPM-modelproduce identical orderings of the portfolios, and thus the population MLPM e fficient

frontier can be calculated. Second, the results from the MLPM-model can, under mul-

tivariate normality, be compared to the results from the mean-variance model in order

to analyze whether estimation errors are larger in the MLPM-model than in the mean-

variance model.

To asses realistic population parameters for the assets we draw N, N = 5 and 25,

assets from the monthly return series from the Center for Research in Security Prices

(CRSP) database on the New York Stock Exchange between January1

989 to December1995 without replacement. The returns on the N assets are then used to compute a

population mean vector, N, and a covariance matrix,P

N . The target rate of return

used is, = M, where M is the estimated mean return on the value weighted index in

the CRSP database during the same period as above. M is fixed, and was estimated to

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1.02 percent on a monthly basis.6 For each N we end up with a N,and aP

N, and the

population portfolio weights XM.

The next step is to calculate the population efficient set in the MLPM-model, and to

obtain the population portfolio weights, mean returns and lower partial moments. Thepopulation weights were obtained through optimization of equation (1), where equation

(1) was solved for a portfolio with normally distributed returns. Next, the optimal weights

obtained through optimization of equation (1) were controlled against the weights from

the mean-variance model since the MLPM model should at least produce a subset of

the mean-variance efficient set. It is true that the mean-variance model and MLPM-

model produce identical rankings of the portfolios, but the efficient sets do not have to

be identical. The population efficient sets are obviously identical when the target rate of

return is set equal to the mean return of the assets, since the LPM2 is equal to one-halfof the variance. Remember that the efficient set in the MLPM-model for a fixed and

when n = 2 is a subset of third order stochastic dominance, and it is well known that

the portfolio that yields the minimum variance (MVP) on the mean-variance efficient set

belongs to second order stochastic dominance when returns are normally distributed. This

implies that the portfolio that yields the minimum lower partial moment on the e fficient

set must be equal to the MVP or be a mean-variance efficient portfolio with higher mean

return than MVP. However, the portfolio that yields the minimum lower partial moment

must still be a mean-variance efficient portfolio, to be more specific, all efficient portfoliosin the MLPM-model belong to the MV efficient set under normally distributed returns.

3.2 Details of the simulation and optimization procedure

The simulation and optimizations are carried out in two steps. First, we simulate T

returns, R, from the multivariate normal distribution N(N,P

N). This simulation is

carried out through a linear transformation and the problem of simulating multivariate

normally distributed returns is reduced to a simulation of independent and identicallydistributed standard normal random variables.7 Second, the T observations on the N

6 The analysis has also been conducted with a target rate of return equal to zero, but the results are

similar to the results under = M. The results are available upon request.

7 The Randn function in Matlab was used to simulate standard normal random variables, and the seed

of the random generator was set equal to the sum of the current time.

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assets are used in the following optimization to obtain the estimated MLPM efficient

portfolios:

Min{x}

1T

TXi=1

min(RX0 , 0)2 bX0 (3)subject to X0I = 1 (4)

and X 0. (5)

Where R is the return matrix, b is the estimated mean return vector of the assets, andI is a vector of ones. is a trade-off parameter between risk and return, and the target

rate of return, , is set equal to M. The critical-line algorithm of Markowitz (1992) and

Markowitz et al. (1993) was used to solve the optimization problem. This algorithmsolves the optimization problem efficiently and yields a set of corner portfolios that can

be used to compute the efficient portfolio for a specific . Since the portfolio weights

for a portfolio between two corner portfolios are linear combinations of the two corner

portfolios, the algorithm traces out the whole efficient set. The critical-line algorithm also

has the advantage that we know that we will always end up with the sample efficient set, in

contrast to the optimization setup where a restriction is put on the expected return on the

portfolio. Hence, for simulation purposes of efficient portfolios this is the best optimization

setup.8

Furthermore, let cXl denote the vector of portfolio weights that solves (3) for aspecific , = l, and let b() and dLP M2() denote the estimated, mean return and lowerpartial moment respectively for the efficient portfolio at = l. The two steps, described

above, are repeated 10 000 times for the target rate of return, and for T equal to 36, 60,

120 and 240 observations. In order to analyze how severe the estimation errors are we

also estimate mean-variance efficient portfolios which will be used for comparisons. In

this case we have the following optimization problem:

Min{x}

XbCX0 bX0 (6)subject to X0I = 1 (7)

and X 0. (8)

8 If a restriction is put on the expected return on the portfolio and a simulation or resampling approach

is used, we cannot be sure that the resulting portfolio belongs to the e fficient set.

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Where bC is the estimated variance-covariance matrix and as above, the critical-linealgorithm of Markowitz (1987) is used to solve the optimization problem.

3.3 Portfolio optimization with improved estimates

The estimation risk literature suggest, that we should be able to improve the portfolio

optimization by using improved estimates of the assets mean returns at least in the

mean-variance model. In the mean-variance framework several studies have found that

the portfolio optimization is extremely sensitive to the estimates of the mean returns, and

that small changes in the mean returns can have large effects on the optimized portfolios,

e.g. Chopra and Ziemba (1993), Best and Grauer (1992), and Michaud (1989). The

present study compares the portfolio optimization and the estimation errors for three

different approaches, two shrinkage estimators of the means and a bootstrap approach.

Shrinkage estimators have been studied by Jorion (1985, 1991), Dumas and Jacquillat,

Grauer and Hakansson (1998), and Michaud (1998).

Steins (1955) suggestion that the efficiency, in a mean squared error sense, of the

estimate should be improved by pooling the information across series leads to a number

of so-called shrinkage estimators that shrink the historical means to some grand mean.

The first estimator used to improve the means is the classical James-Stein (JS) estimator,

which takes the form9

bJS = (1 w)b + wrGI (9)where I is a vector of ones and as above b is the sample estimate, rG = 1NPbi is thegrand mean,

wt = min h1, (n 2)/(T(b rGI)0

bC1(b rGI))i

is the shrinking factor, and bC is the sample variance-covariance matrix calculated with Tobservations. In this case the simple sample means are shrunk toward the arithmetic

average of the sample means. As noted by Jorion (1985), this shrinking is difficult to

9 For a discussion of James-Stein estimators, see Efron and Morris (1973, 1977).

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reconcile with the generally accepted trade-off between risk and expected return, unless,

of course, all assets fall within the same risk class.

The second estimator used to improve the assets means is the Bayes-Stein (BS) esti-

mator of Jorion (1985, 1991). This estimator takes the form

bBS = (1w)b + wrGBSI (10)where

rGBS = I0bC1b/(I0bC1I),

w = /( + T)

and the shrinking factor is

= (N + 2)/h

(b rGBSI)0bC1(b rGBSI)i .In this case, the grand mean is the mean of the MVP portfolio in the sample. It should be

noted, that the shrinkage estimators are biased and nonlinear, since the shrinkage factor

is itself a function of the data.

The third method used to improve the portfolio optimizations is a bootstrap approach.The bootstrap method, introduced by Efron (1979), is a computer-intensive method for

estimating the distribution of an estimator or statistic by resampling the data at hand. A

bootstrap approach to portfolio selection in the mean-variance framework has previously

been used by Liang et al. (1996) and Hansson and Persson (2000). Let Rp be the simulated

return matrix for the N assets of length T from simulation trail p. We resample the Rp

matrix T times with replacement so that we end up with a bootstrapped return matrix,

R, of length T. In the resampling, we resample cross-sectionally so that the correlation

structure among the assets is not destroyed. From each resampled R we optimize the

portfolios according to (3) and the estimated efficient portfolios for R are obtained.

Then the portfolios of interest are obtained from the corner portfolios that solve (3). This

procedure is repeated B times for each R, and in the end we have a set of bootstrapped

observations for each estimated efficient portfolio. The average portfolio weights, portfolio

returns and LP M2 s of the B resampled estimated efficient portfolios corresponding to

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the portfolio at are then taken as the solution to the portfolio optimization for the

return matrix Rp. That is, the solution for the portfolio corresponding to a specific for

the return matrix Rp in simulation trail p is given by

bxRp() = 1BBXb=1

bxb(), bRp() = 1BBXb=1

bb() and dLP MRp() = 1BBXb=1

dLP Mb()In order to investigate how the efficiency of the bootstrap approach depend on B,

B was set in the interval 50 to 300. The simulations are only repeated with improved

estimates for N = 5 but for all values of T, i.e. T = 36, 60, 120 and 240. When JS and

BS are used to improve the estimates, the number of simulations, P, is 10 000, while the

number of simulations is reduced to1

000 in the bootstrap approach.

3.4 Quantitative measures of estimation errors

The effect of using estimated parameters instead of the true parameters when computing

the MLPM efficient set can be studied through the true-, actual- and estimated-efficient

set. This approach was proposed by Broadie (1993).10 The true efficient set is the

population efficient set which is unknown to the investors. The estimated efficient frontier

is the efficient frontier based on the estimated parameters and is the sample efficient

frontier. The actual efficient set is defined as follows: Each point on the estimated efficient

frontier corresponds to a portfolio of the N assets, using the portfolio weights from the

estimated parameters, i.e. the sample efficient portfolio, and then applying them to the

true parameters. That is, the expected returns and variance-covariance matrix in (1)

yields the actual efficient frontier. The characteristics of the actual portfolio are what

the investor with the estimated portfolio weights actually holds. In short, the estimated

frontier is what appears to be the case based on the data and the estimated parameters,

but the actual frontier is what really occurs based on the true parameters.

The estimation errors that we focus on are the errors between the actual efficient

frontier and the true efficient frontier. The estimation errors are measured through the

following root-mean-squared errors (RMSE) measures:

10 Other measures used are the cash equivalent loss in Chopra and Ziemba (1993) and the utility loss

used in Jorion (1985) and Simaan (1997).

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f() =

vuut

1

P

PXp=1

()

ep()

2

(11)

fLPM() =

vuut 1P

PXp=1

LP M()

1

2 gLP Mp()122 . (12)Where, p() and LP M() denote the target point on the true efficient set for the portfolio

at . ep() and gLP Mp() 12 represent the mean return and LPM of the actual portfoliocorresponding to that specific for simulation trail p. P is equal to the number of

simulations, 10 000.

Investors observe the estimated frontier but portfolio performance is given by the

actual frontier. In order to answer the question How far is the estimated portfolio from

the actual portfolio?, a second measure of estimation errors is used. Let bp() anddLP Mp()12 denote the estimated parameters corresponding to the portfolio on theestimated efficient frontier for simulation trail p. Then the average difference of the mean

returns of the portfolios between the actual and estimated portfolios is given by g(),

and the average difference between the standardized LPM1

2

2is given by gLPM().

g() =1P

PXp=1

bp() ep() (13)gLPM() =

1

P

PXp=1

dLP Mp() 12 gLP Mp() 12 (14)

4 Results

The effect of using estimated parameters instead of true parameters on the MLPM efficient

frontier is graphed in Figure 1. The estimated efficient frontier is located to the left of boththe true and actual efficient frontiers, which implies that the estimated efficient frontier

underestimates the risk of the portfolios. Recall that the actual efficient portfolio frontier

is how the estimated portfolios really look like and it is also these portfolios that the

estimated portfolios are evaluated against. It can, however, be seen in Figure 1 that we

generally end up with inefficient portfolios. What seems most striking about Figure 1, is

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the large difference between the estimated frontier and the actual frontier, the estimated

frontier overestimates the expected return for the portfolios and underestimates the risk

of the portfolios. This seems to indicate that the estimated portfolios are optimistically

biased predictors of portfolio performance. However, the difference between the true andactual portfolios are not constant over the frontiers but rather the di fference seems to

increase with the portfolio mean return.

2 3 4 5 6 7 8 9 10 11

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Standardized LPM

M

ean

Portfolio

Return True Efficient Frontier

Estimated Frontier

Actual Frontier

Figure 1: Mean-Lower Partial Moment frontiers using 36 observations.

In Table 1, the estimation errors between the actual and the true portfolios are pre-

sented. In Panel A it can be seen that the RMSE are surprisingly large when there are five

assets in the frontiers. The RMSE for both LPM and the mean returns on the portfolios

fall with and the number of observations used to estimated the frontier. Our estimate

of the portfolio that yields the minimum lower partial moment is much better than for

portfolios with a higher mean portfolio return. The relative efficiency between portfolios

at = 2000 and = 0 is as high as 43 when 36 observations are used to estimate the risk

of the portfolios, and the relative efficiency is almost 74 when 240 observations are used.11

11 The relative efficiency is the ratio between the two mean-squared errors. The higher the ratio, the

higher is the relative efficiency.

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The results for the MV-model are presented in Table 2. The RMSE decrease, as in the

MLPM-model, with the number of observations and with . Furthermore, the errors for

high return portfolios in the 25 asset case have, just as in the MLPM-model, decreased. If

the sizes offLPM and f are compared to the corresponding f and f in the MV-model, weobserve that the errors in the MLPM-model are larger than in the MV-model, especially

for low portfolios. This is especially true for the difference in efficiency between fLPM

and f

. In fact, the difference in efficiency between the two models increases with the

number of observations used in the optimization.

Table 2.

Root Mean-Squared Errors between Actual and True Portfolios

in the Mean-Variance Model

Panel A: N = 5T = 36 T = 60 T = 120 T = 240

f

f f f f f f f2000 6.94 0.40 6.89 0.38 6.72 0.35 6.49 0.31100 4.90 0.25 4.50 0.23 3.85 0.19 2.99 0.1550 3.67 0.21 3.04 0.19 2.16 0.15 1.45 0.1110 1.02 0.10 0.71 0.08 0.41 0.06 0.23 0.045 0.55 0.07 0.35 0.05 0.19 0.04 0.10 0.031 0.25 0.05 0.15 0.04 0.08 0.03 0.04 0.02

0.1 0.23 0.04 0.14 0.04 0.07 0.03 0.04 0.020 0.23 0.04 0.14 0.04 0.07 0.03 0.04 0.02

Panel B: N = 25T = 36 T = 60 T = 120 T = 240

f

f f f f f f f2000 6.50 1.00 6.12 0.90 5.47 0.74 4.37 0.58100 3.89 0.81 2.94 0.67 1.83 0.49 1.23 0.3450 3.00 0.64 2.26 0.52 1.58 0.36 1.18 0.2410 1.64 0.31 1.24 0.25 0.82 0.19 0.56 0.155 1.39 0.21 0.97 0.18 0.55 0.14 0.30 0.101 0.53 0.12 0.31 0.09 0.15 0.06 0.08 0.05

0.1 0.39 0.09 0.23 0.07 0.12 0.05 0.06 0.040 0.38 0.09 0.23 0.07 0.11 0.05 0.06 0.04

All values in percent, except

The difference in efficiency between the two models continuous to hold independently

of the number of assets used. If we look at an increase in RMSE when we go from five

assets to 25, the relative increase in fLPM is actually smaller than the increase in f, while

the increase is higher in f for the MLPM-model.

The fact that the errors are larger for high return portfolios in both models is a dark

side of portfolio optimization that cannot be ignored. In its search for assets with high

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mean returns the portfolio optimization favors assets with overestimated mean returns

and underestimated risk. This error-maximization property of the portfolio optimization

also pertains to errors in our risk estimates (Michaud (1989)). However, the results in

Table 1 and 2, actually indicate that the severity of the error-maximization problemdiminishes as the number of assets in the asset universe is increased.

The result that the errors are larger in the MLPM-model than in the MV-model should

not come as a surprise since the LPM risk measure is a partial domain risk and hence only

uses a truncated part of the available data. This in turn translates into larger estimation

errors in the MLPM-model.

In Table 3, the results from the optimizations with James-Stein improved estimates

of the mean returns are presented. It can clearly be seen that the JS improved estimates

make the errors between the actual and true portfolios even larger as compared to thesituation in Table 1 in which the sample estimates were used. In addition, the errors

between the portfolios increase with the number of observations used in the portfolio

optimization. For example, the errors are higher for the minimum lower partial moment

portfolio, i.e. = 0, when 240 observations are used. The relative efficiency between the

simple sample estimates is almost 20 for fLPM and 4 for f. The JS approach is in the

MLPM-model outperformed by the classical approach of using simple sample estimates,

and this approach has been shown to improve the performance of the MV-model, e.g.

Jorion (1985) and Dumas and Jacquillat (1990).

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Table 3.

Root Mean-Squared Errors between Actual and True Portfolios

in the MLPM-model with James-Stein Estimator

= M T = 36 T = 60 T = 120 T = 240

fLPM f fLPM f fLPM f fLPM f2000 5.59 0.41 5.55 0.39 5.32 0.36 4.90 0.31100 3.18 0.24 3.10 0.23 3.16 0.20 3.23 0.1850 2.96 0.21 2.77 0.20 2.73 0.18 2.75 0.1610 1.79 0.14 1.60 0.13 1.91 0.13 2.30 0.135 1.39 0.12 1.36 0.11 1.84 0.12 2.28 0.131 1.07 0.10 1.23 0.10 1.81 0.12 2.28 0.14

0.1 1.02 0.10 1.21 0.10 1.80 0.12 2.28 0.140 1.02 0.10 1.21 0.10 1.80 0.12 2.28 0.14

All values in percent, except and N=5

Turning to Table 4, the results of the Bayes-Stein improved portfolios are presented.

When the BS estimator is used to improve the assets mean returns, there is some im-

provement in the RMSE for both fLPM and f. The improvement is particularly eminent

for the portfolios in the interval between 100 and 5, where the efficiency has increased for

fLPM. For the low -portfolios the fLPM is somewhat larger with the BS improved mean

returns when compared to the simple sample estimates in Table 1. In fact, it is somewhat

surprising that the BS approach does not improve the estimates of the actual portfolios

more, since the simulation is based on normally distributed returns and we know that the

minimum variance portfolio is close to the MLPM efficient set.

Table 4.

Root Mean-Squared Errors Between Actual and True Portfolios

with Bayes-Stein Estimator

= M T = 36 T = 60 T = 120 T = 240

fLPM f fLPM f fLPM f fLPM f2000 4.77 0.40 4.76 0.38 4.69 0.34 4.55 0.31

100 3.27 0.25 3.01 0.23 2.68 0.19 2.34 0.1650 2.86 0.21 2.48 0.19 1.97 0.16 1.51 0.1210 1.52 0.14 1.14 0.12 0.79 0.09 0.63 0.075 1.15 0.12 0.87 0.10 0.65 0.08 0.56 0.071 0.85 0.10 0.67 0.08 0.58 0.08 0.54 0.07

0.1 0.80 0.09 0.66 0.08 0.57 0.08 0.53 0.080 0.78 0.09 0.65 0.08 0.57 0.08 0.53 0.08

All values in percent, except , and N=5.

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The results from the bootstrap approach are presented in Table 5. 300 bootstrap

resamples were used to compute the portfolio characteristics. With this approach all

estimates, except for the portfolio when = 2000, were improved as compared to the

results in Table 1, where the sample estimates were used in the optimization. The RMSEfor both fLPM and f have decreased, which implies that the bootstrap approach is the

most efficient method for improving the estimates as compared to the classical, JS and BS

methods. Furthermore, the relative efficiency between the simple sample estimates and

the bootstrap approach increases with the number of observations used in the portfolio

optimization. The RMSE are larger than those in Table 5 when less than 300 bootstrap

samples were used to estimate the portfolios, but the RMSE were still smaller than the

RMSE when the simple sample estimates were used. Since 300 bootstrap samples are

the maximum of resamples used in the present study it is possible that the RMSE coulddecrease further if the number of bootstrap resamples are increased beyond 300.12

Table 5.

Root Mean-Squared Errors Between Actual and True Portfolios

with a Bootstrap Approach

= M T = 36 T = 60 T = 120 T = 240

fLPM f fLPM f fLPM f fLPM f

2000 5.50 0.34 5.52 0.33 5.20 0.30 5.04 0.27100 2.73 0.18 2.55 0.17 2.82 0.16 2.68 0.1450 2.80 0.17 2.52 0.16 2.64 0.15 2.28 0.1310 2.04 0.13 1.51 0.11 1.17 0.10 0.78 0.085 1.52 0.11 1.03 0.10 0.74 0.08 0.53 0.071 0.79 0.08 0.54 0.07 0.41 0.06 0.35 0.06

0.1 0.56 0.07 0.42 0.06 0.35 0.06 0.32 0.050 0.52 0.07 0.41 0.06 0.34 0.06 0.31 0.05

All values in percent, except . N=5 and 300 bootstrap samples usedto compute the bootstrap estimates.

Lets turn to the second question raised above: How far is the estimated portfoliofrom the actual portfolio? In Table 6, the results for the portfolio optimizations based

on the simple sample estimates are presented. Again the results illustrate the error-

maximization properties of portfolio optimization, i.e. the actual portfolios mean returns

12 The results for the bootstrap approach when 50, 100 and 200 bootstrap resamples were used are

available upon request.

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are overestimated and the risk underestimated. The bias is surprisingly large in both

parameters, and the number of observations used in the portfolio optimization have a

large impact on the bias. For example, when 36 observations are used g is three times

as large as the corresponding g when 240 observations are used to estimate the portfoliolocated at = 2000. The results in Table 6 are important because they indicate the large

degree to which estimated frontiers are optimistically biased predictors of actual portfolio

performance.

The James-Stein estimator actually performs better for estimating the characteristics

of the actual portfolios than for estimating the characteristics of the true portfolios. The

results for when the JS-estimator is employed to improve the asset mean returns are

presented in Table 7. The bias in the portfolios mean returns has decreased, but the bias

in the risk measure is more or less unaffected by the JS-estimator.

Table 6.

Average Difference Between Estimated and Actual Portfolios

in the MLPM-model

= M T = 36 T = 60 T = 120 T = 240

gLPM g gLPM g gLPM g gLPM g2000 -1.01 1.54 -0.73 1.17 -0.48 0.80 -0.32 0.54100 -1.04 1.54 -0.75 1.18 -0.49 0.80 -0.33 0.5450 -1.04 1.51 -0.75 1.14 -0.49 0.75 -0.31 0.48

10 -0.93 1.19 -0.62 0.82 -0.35 0.48 -0.20 0.275 -0.84 0.99 -0.54 0.66 -0.30 0.37 -0.16 0.201 -0.68 0.64 -0.42 0.41 -0.22 0.22 -0.12 0.13

0.1 -0.60 0.49 -0.37 0.32 -0.20 0.18 -0.11 0.110 -0.59 0.47 -0.36 0.31 -0.20 0.18 -0.11 0.11

Table 7.


with James-Stein Estimator

= M T = 36 T = 60 T = 120 T = 240

gLPM g gLPM g gLPM g gLPM g2000 -0.93 0.65 -0.66 0.45 -0.43 0.27 -0.29 0.16

100 -0.93 0.64 -0.65 0.46 -0.41 0.29 -0.27 0.1650 -0.91 0.63 -0.62 0.44 -0.38 0.26 -0.23 0.1510 -0.78 0.49 -0.49 0.31 -0.26 0.15 -0.15 0.075 -0.71 0.41 -0.44 0.25 -0.22 0.11 -0.13 0.041 -0.60 0.28 -0.36 0.16 -0.18 0.07 -0.10 0.017

0.1 -0.56 0.23 -0.33 0.13 -0.17 0.05 -0.10 0.010 -0.56 0.22 -0.33 0.12 -0.17 0.05 -0.10 0.01

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In Table 8, the results for when the Bayes-Stein estimator was used to improve

the mean returns of the assets are presented. The results are comparable, in size, with

the results when the JS-estimator was used to improve the asset means. However, the

bias in the mean returns of the portfolios has decreased although the bias in the risk ofthe portfolios has not improved.

The results for the bootstrap approach are presented in Table 9, and they are somewhat

mixed. The bootstrap approach results in a lower bias for the risk measure for high

portfolios when 36 or 60 observations are used in the optimization, but in all other cases

the bias is larger than when the sample estimates are used.

Table 8.


with Bayes-Stein Estimator

= M T = 36 T = 60 T = 120 T = 240

gLPM g gLPM g gLPM g gLPM g2000 -1.01 0.57 -0.74 0.40 -0.50 0.23 -0.33 0.12100 -1.03 0.58 -0.73 0.42 -0.47 0.25 -0.30 0.1550 -1.00 0.56 -0.69 0.39 -0.42 0.23 -0.25 0.1310 -0.83 0.42 -0.53 0.26 -0.29 0.13 -0.16 0.065 -0.76 0.36 -0.47 0.21 -0.25 0.10 -0.14 0.041 -0.66 0.26 -0.40 0.15 -0.21 0.07 -0.12 0.03

0.1 -0.62 0.23 -0.37 0.13 -0.20 0.06 -0.11 0.02

0-0.61 0.22 -0.37 0.13 -0.20 0.06 -0.11 0.02

Table 9.


with a Bootstrap Approach

= M T = 36 T = 60 T = 120 T = 240

gLPM g gLPM g gLPM g gLPM g2000 0,082 2,23 0,49 1,68 0,61 1,22 0,82 0,82100 -0,13 2,24 0,24 1,68 0,35 1,22 0,49 0,8350 -0,29 2,22 0,04 1,66 0,13 1,19 0,22 0,7910 -0,86 1,95 -0,60 1,37 -0,46 0,88 -0,39 0,53

5 -1,03 1,72 -0,78 1,17 -0,58 0,72 -0,47 0,431 -1,09 1,22 -0,85 0,80 -0,61 0,48 -0,47 0,31

0.1 -1,01 0,94 -0,80 0,63 -0,57 0,41 -0,45 0,270 -0,98 0,90 -0,79 0,61 -0,56 0,40 -0,45 0,27

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5 Summary and Concluding Remarks

The present study has analyzed the effect of estimation risk on the portfolio selection

problem in the lower partial moment framework. A simulation study was used to capture

the effects of estimation errors and the important distinction between the true, estimated

and actual efficient set was made. In short, the estimated frontier is what appears to be

the case based on the data and the estimated parameters, but the actual frontier is what

really occurs based on the true parameters. The true efficient set is based on the unknown

parameters and is, as the actual frontier, unknown to the investor. In addition, three

different methods for improving the portfolio optimizations have been employed in this

chapter and they were compared to the classical approach in which the sample estimates

are used as inputs into the portfolio optimizations. The three different methods were two

different shrinkage estimators; James-Stein and Bayes-Stein, and the third method was a

bootstrap approach.

The results showed that the errors in the optimized portfolios, that is the di fference

between the actual and true portfolios, can be surprisingly large in the mean-lower partial

moment framework. This is especially true for portfolios at low levels of risk tolerance and

for portfolios with high portfolio mean returns. That is, the portfolio optimizations suf-

fered from error-maximization which implies that the optimization setup favors assets with

overestimated mean returns and underestimated risks. Thus, the results in the presentstudy are well in line with the well known result from the mean-variance model, that our

estimates of portfolios close to the minimum risk portfolio are better than portfolios with

higher portfolio mean returns.

The bootstrap approach performed best of the three methods used to improve the

portfolio optimizations, and the James-Stein approach actually performed worse than the

classical approach of using sample estimates as inputs into the portfolio optimizations.

The results of the average difference between the actual and estimated portfolios show

that the estimated portfolios are biased predictors of the actual portfolios in that theyunderestimate the risk in the portfolios and overestimate the portfolio mean returns. In

this case the James-Stein approach produced the smallest bias, and the performance of

the bootstrap approach was comparable with the results from using sample estimates.

Hence, the estimated optimal portfolios in the lower partial moment suffer to a larger

extent from estimation errors, but the estimates can be improved. If our concern is the

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uncertainty in the optimal portfolio weights, then a bootstrap approach should be used

since this approach produce the lowest root-mean squared errors between the actual and

true efficient portfolio. On the other hand, if our concern is related to the risk and portfolio

mean returns of the optimized portfolios a James-Stein approach should be used.

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6 References

Barry, C. B. 1974. Effects of Uncertain Means, Variances and Covariances, Journal of

Finance, Vol 29, 515-524

Bawa, V. S. 1975. Optimal Rules for Ordering Uncertain Prospects, Journal of Finan-

cial Economics, 2, 95-121.

Bawa, V. S. 1976. Admissible Portfolios for All Individuals, Journal of Finance, 31 (4),

1169-1183.

Bawa, V. S. and R. W. Klein. 1979. The Effect of Estimation Risk on Optimal Portfolio

Choice, Estimation Risk and Optimal Portfolio Choice, Amsterdam, North-Holland.

Bawa, V. S. and E. B. Lindenberg. 1977Capital Market Equilibrium in a Mean-Lower

Partial Moment Framework, Journal of Financial Economics, 5, 189-200.

Best, M. J. and R. P. Grauer. 1992. The Analytics of Sensitivity Analysis for Mean-

Variance Portfolio Problems, International Review of Financial Analysis, 1(1), 17-

37.

Broadie, M. 1993. Computing Efficient Frontiers Using Estimated Frontiers, Annals of

Operations Research: Financial Engineering, 45 (Special Issue), 21

-58.

Chopra, V. K. and W. T. Ziemba. 1993. The Effects of Errors in the Means, Variances

and Covariances, Journal of Portfolio Management, 4, 6-11.

Efron, B. 1979. Bootstrap Methods: Another Look at the Jackknife, Annals of Statis-

tics, 7: 1-26.

Eftekhari, B. 1998. Lower Partial Moment Hedge Ratios, Applied Financial Economics,

8, 645-652.

Fishburn, C. P. 1977. Mean-Risk Analysis with Risk Associated with Below-Target

Returns, American, Economic Review, 67(5), 116-126.

Grauer, R. and N. Hakansson. 1998. Applying the Grinblatt-Titman and the Condi-

tional (Ferson-Schadt) Performance Measures: The Case of Industry Rotation Via

the Dynamic Investment Model Manuscript.

23


24/25

Grootveld, H. and W. Hallerbach. 1999. Variance vs Downside Risk: Is there really

that much difference?, European Journal of Operational Research, 114: 304-319.

Hansson, B. and M. Persson. 2000. Time Diversification and Estimation RiskFinancial

Analysts Journal, (May/June).

Harlow, V. W. 1991. Asset Allocation in a Downside-Risk Framework, Financial Ana-

lysts Journal, Sept/Oct: 28-40.

Harlow, W.V. and Rao. R.K. 1989. Asset Pricing in a Generalized Mean-Lower Partial

Moment Framework: Theory and Evidence, Journal of Financial and Quantitative

Analysis, 24: 285-311.

Jobson, J.D. 1991. Confidence Regions for the Mean-Variance Efficient Set: An alterna-tive Approach to Estimation Risk, Review of Quantitative Finance and Accounting,

1: 235-256.

Jobson, J. D. and B. Korkie. 1980. Estimation for Markowitz Efficient Portfolios,

Journal of American Statistical Association, 75: 544-554.

Jorion, P. 1985. International Portfolio Diversification with Estimation Risk, Journal

of Business, 58: 259-278.

Jorion, P. 1991. Bayesian and CAPM Estimators of the Means: Implications for Port-

folio Selection, Journal of Banking and Finance, 15: 717-727.

Kataoka, S. 1963. A Stochastic Programming Model, Econometrica, 31: 181-196.

Liang, Y., F.C.N. Myer, and J.R. Webb. 1996. The Bootstrap Efficient Frontier for

Mixed-Asset Portfolios Real Estate Economics, 24, no. 2 (Summer):247-256.

Lien, D. and Y. K. Tse. 2000. Hedging Downside Risk with Futures Contracts, Applied

Financial Economics, 10: 163-170.

Lewellen, J. and J. Shanken. 1998. Market Efficiency, Rational Expectations, and

Estimation Risk. paper presented at the annual meeting of American Finance As-

sociation, New York (1999).

24


25/25

Markowitz, H. 1987. Mean-Variance Analysis in Portfolio Choice and Capital Markets.

Oxford, Basil Blackwell.

Markowitz, H. 1992. Portfolio Selection - Efficient Diversification of Investments. Ox-

ford, Basil Blackwell (First published 1959)

Markowitz, H., P. Todd, G. Xu and Y. Yamane. 1993. Computing the Mean-Semivariance

Efficient Set by the Critical Line Algorithm Annals of Operations Research: Finan-

cial Engineering, 45 (Special Issue), 307-317.

Michaud, R. O. 1989. The Markowitz Optimization Enigma: Is Optimized Optimal?

Financial Analysts Journal, (January/February): 31-42.

Michaud, R.O. 1998. Efficient Asset Management - A Practical Guide to Stock PortfolioOptimization and Asset Allocation. Boston, Harvard Business School Press.

Roy, A. 1952. Safety-First and the Holding of Assets Econometrica, 20, 431-449.

Rudolf, M., H. Wolter. and H. Zimmerman. 1999. A Linear Model for Tracking Error

Minimization, Journal of Banking & Finance, 23, 85-103.

Sortino, F. and R. van der Meer. 1991. Downside Risk, Journal of Portfolio Manage-

ment, 2, 27-31

.

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