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MEAN–VARIANCE VERSUS FULL-SCALE OPTIMIZATION:

BROAD EVIDENCE FOR THE UK

by

BJÖRN HAGSTRÖMER

Aston University

RICHARD G. ANDERSON

Federal Reserve Bank of St Louis

JANE M. BINNER

Aston University

and

THOMAS ELGER

BIRGER NILSSON*

Lund University

Portfolio choice by full-scale optimization applies the empirical return

distribution to a parameterized utility function, and the maximum is

found through numerical optimization. Using a portfolio choice setting

of three UK equity indices we identify several utility functions featuring

loss aversion and prospect theory, under which full-scale optimization is

a substantially better approach than the mean–variance approach. As the

equity indices have return distributions with small deviations from nor-

mality, the findings indicate much broader usefulness of full-scale

optimization than has earlier been shown. The results hold in- and out-

of-sample, and the performance improvements are given in terms of

utility as well as certainty equivalents.

1Introduction

It is well known that asset returns probability distributions in general feature

both skewness and excess kurtosis. Investor preferences of these higher

moments are attracting increasing attention by portfolio choice researchers

and the financial industry, often arriving at more complex utility functions

than traditionally assumed. In full-scale optimization (FSO), originally sug-

gested by Paul A. Samuelson,1empirical return distributions are used in their

entirety, and the choice of utility function is completely flexible. In this paper

* We are grateful for comments and suggestions by Björn Hansson, Mark Kritzman, Paolo

Porchia, Indranarain Ramlall, Jim Steeley, Szymon Wlazlowski and one anonymous

referee, as well as conference presentation attendants at the Financial Management Asso-

ciation European Meeting 2007 (IESE Barcelona), the European Financial Management

Association Annual Meeting 2007 (WU Wien) and Money, Macroeconomics and Finance

(MMF) 2007 (Birmingham University), and seminar attendants at Lund University Eco-

nomics Department. The usual disclaimer applies. Correspondence to the first author,

e-mail hagstrob@aston.ac.uk.

1Samuelson’s article ‘When and Why Mean–Variance Analysis Generically Fails’ was never

published, but the term FSO was picked up by Cremers et al. (2005).

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we assess the performance of FSO using a wide range of utility function

specifications applied to equity indices. We show that when utility functions

include loss aversion or prospect theory, even assets with only small devia-

tions from normality are better selected in an FSO framework than with the

traditional mean–variance (MV) approach.

The strength of FSO is that no analytical solution to the portfolio choice

problem is pursued. This allows the distributional properties of returns to be

left non-parameterized and the utility function to be specified to reflect inves-

tor preferences uncompromised by mathematical convenience. The absence

of simplifying assumptions, yielding theoretical appeal, comes at the cost of

computational burden, however. As the optimization problem is not convex,

a grid search or a global search algorithm has to be used to find the utility-

maximizing portfolio.

Cremers et al. (2005) show in a hedge fund selection problem that the

performance of FSO (in terms of utility) is substantially better than Markow-

itz’s (1952, 1959) MV approach when investor preferences are modeled to

include loss aversion or prospect theory (based on Kahnemann and Tversky,

1979). The results are confirmed in an out-of-sample application by Adler

and Kritzman (2007). Several other portfolio choice papers resembling FSO

but not using the term have appeared lately. Statistical properties of the

estimator have been explored by Gourieroux and Monfort (2005). Scenario-

based approaches (using hypothesized outcomes with probabilities attached

instead of empirical return distributions) have been dealt with by Grinold

(1999) and Sharpe (2007). Higher moments properties of utility-maximizing

portfolios have been investigated by Maringer (2008), who also proposes

heuristic optimization methods to deal with the computational burden of the

optimization problem.

The idea of utility maximization as a methodology for portfolio optimi-

zation problems, based on the utility theory founded by Von Neumann and

Morgenstern (1947), can be traced back at least to Tobin (1958), and also

appearsinseveralassessmentsoftheMVapproach(e.g.LevyandMarkowitz,

1979; Markowitz, 1987). As the latter studies showed that the performance

difference between MV portfolios and utility-maximizing portfolios (using

power utility) was negligible, the less burdensome MV approach became the

model of choice in the financial industry, and the benchmark in academia.2

The relevance of higher moments for investment decisions was pointed

out by Levy (1969) and Samuelson (1970), and in realistic portfolio man-

2In the MV model, risk is defined in terms of the second moment (variances and covariances)

only. This makes the model simple to apply, but it is based on assumptions that are clearly

unrealistic. For the MV solution to be optimal, either the return distribution must feature

spherical symmetry, or investors must be indifferent to higher moments and equally averse

to downside and upside risk (quadratic utility). Financial returns are rarely spherical (first

pointed out by Mandelbrot, 1963), and the quadratic utility is known to be decreasing in

wealth above a certain wealth level.

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agement situations, investors often express preferences that imply more

complex utility functions than power or quadratic utility (Litterman, 2003,

Ch. 2; Meucci, 2005, Ch. 5). It is when implementing such preferences (loss

aversion and prospect theory) that the FSO has been proven superior to

MV (Cremers et al., 2005; Adler and Kritzman, 2007). In these assessments,

however, assets with extremely non-normal returns (hedge funds) have been

used, and the results have only been shown to hold for a few examples of

utility function specifications. Our assessment of FSO features a selection of

equity indices with returns much closer to a normal distribution (in general

still non-normal though) and a much broader spectrum of utility function

specifications. Finding that FSO performance is substantially better than

MV when complex investor preferences (in particular prospect theory)

hold, our results indicate that the earlier studies are robust with respect to

utility function specification, as well as to other asset classes (equity

indices).

This paper is organized as follows. Section 2 discusses the portfolio

choice problem and the characteristics of investors’ preferences. Section 3

presents data, utility functions and methodology used in our assessment of

FSO. The results are presented and analyzed in Section 4, and Section 5

concludes.

2Portfolio Selection and Investor Preferences

2.1 The Portfolio Selection Problem

Single-period portfolio selection models can in general be described as utility

maximization problems such as.

θθ

θ

∈

Ω

Here, R is an (n ¥ T) matrix containing expected returns of n admissible

assets in T different scenarios. q is a vector of length n containing the port-

folio weights for each asset. Utility is a function U of expected portfolio

return, q′R, which q is chosen to maximize, subject to the constraint matrix

W. Typically, W includes a budget constraint such as q′i = 1 (where i is a

vector of ones), but it may also include other constraints, e.g. a short selling

constraint or a loss aversion constraint.

The portfolio return q′R is a vector of length T. Its distribution will

differ depending on the portfolio weights chosen in q. The functional form of

the utility function should mirror the investor’s preferences to determine

which expected portfolio distribution is preferred.

In the MV approach, the utility function is quadratic, which implies that

the mean and variance of the expected returns for each asset in R are all that

determine the utility. In FSO, R contains historical returns of each asset and

θ

* =

′

()

argmax UR

(1)

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the utility for each possible q is evaluated at each point in time available in the

sample. This means that all features of the empirical return distribution are

considered, not only the first two moments.

Setting up the optimization problem in this way typically does not yield

convexity, and analytical solutions cannot always be found. Instead search

techniques must be used, such as a grid search or a stochastic search algo-

rithm. The problem may be thought of as constructing an allocation matrix Q

containing each possible portfolio weight combination q. The allocation

matrix dimension is (n ¥ m), where n is the number of assets considered and

m is the number of possible portfolio allocation combinations, which is a

function of n and a precision parameter p.3Each column of Q represents one

allocation combination vector q (n ¥ 1). To find the optimal q, the utility for

each theta is evaluated for each asset returns vectorRt, which contains returns

on each asset i (i = 1, 2, . . . , n) at time t (t = 1, 2, . . . , T). Each of the T Rt

vectors will have the dimension (n ¥ 1) and elements R

is the price of asset i at time t. The q with the highest average utility over time

will be the optimal allocation combination, qFSO. This is shown formally in

equation (2):

∑

t

1

Θ

A key feature of FSO is that any utility function can be chosen, as the

optimization is carried out numerically. This allows the optimization to

consider more complex investor preferences than traditionally assumed.

PP

i t

,

i t

,

i t

,

=

−1, where Pi,t

θθ

(

θ

θ

FSO=

′

T

)

⎡⎣ ⎤⎦

∈

−

=

argmax TUR

t

T

1

(2)

2.2Investor Preferences

Investor preferences implied by utility functions can be investigated by

expanding the utility function in a Taylor series around the mean (m1) and

taking expectations on both sides, as shown in equation (3). This yields

measures of the investors’ preferences in terms of the distribution’s moments.

Let Ukdenote the kth derivative of the utility function and mjthe jth moment

of the portfolio return; then, set up in the same fashion as in Scott and

Horvath (1980) the expected utility takes the following form:4

()

+

∑

2

The expression shows that the expected utility equals the utility of the

expected returns, plus the impact on utility of deviations from the expected

E U

( ) =

U

UU

i

i

i

i

()+

()

=

∞

μ

μ

μ

μ

!

μ

1

2

1

2

1

3

(3)

3When using a grid search, p is chosen at a level suitable to the problem nature, yielding a finite

dimension m of the allocation matrix. Search algorithms do not always specify p, making m

infinitely large. Then a halting criterion is used to stop the search at suitable precision.

4The term (rp- m1)U′(m1), where rpis the portfolio return, disappears when taking expectations,

as E(q′R - m1) = 0.

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return. The influence of each moment on expected utility is weighted by the

corresponding order derivative of the utility function. Typically, U2(m1) for

variance is negative; U3(m1) for skewness is positive; and U4(m1) for kurtosis

is negative.5

In the MV model the covariances of the assets play an important role.

Implicitly, utility functions with higher moment preferences different from

zero also take co-moments, such as co-skewness and co-kurtosis, into

account.6

In principle there is no limit on the number of moments to consider, but

higher moments than kurtosis (k > 4) have not been considered in the finance

literature, and will not be discussed here. However, according to Scott and

Horvath (1980), most investors have utility functions where moments of odd

order (i.e. k = 1, 3, 5, . . .) have positive signs on the respective derivative and

moments of even order have negative derivatives.

In this paper we consider four families of utility functions. Their general

mathematical forms are presented in Table 1. Here, we define utility in terms

of the portfolio return rather than over wealth. This approach may be inter-

preted as a normalization of initial wealth to one, i.e. W0= 1. A motivation

for defining utility directly in terms of returns can be found in

5As referred to above, the MV approach is based on assuming either quadratic utility or a normal

return distribution. The quadratic utility function is expressed as E(U) = m1- lm2. This

implies U1(m1) = 1, U2(m1) < 0 (usually referred to as the risk aversion parameter, l) and

Uk(m1) = 0 for all k > 2. If normally distributed returns are assumed, all odd moments

(k = 3, 5, . . .) will be zero, and all even moments will be functions of the variance (see the

Appendix to Chapter 1 in Cuthbertson and Nitzsche, 2004).

6Co-skewness is a phenomenon extensively discussed by Harvey et al. (2003).

Table 1

Utility Function Equations

Utility function Utility

Exponential

-exp[-A(1 + rp)]

1

1

−

γ

ln(1 + rp)

ln(1 + rp)

P(rp- x) + ln(1 + x)

−−

()

A zrp

+−

()

B rz

p

Power

1

1

+

()

−

−

rp

γ

for g > 0

for g = 1

for rp3 x

for rp< x, P > 0

for rp2 z

for rp> z

Bilinear

S-shaped

γ1

γ2

Notes: In all the utility functions, rprepresents portfolio return. A repre-

sents the degree of (absolute) risk aversion in the exponential utility

functions. In power utility functions, g is the degree of (relative) risk

aversion. The special case when g = 1 is also called logarithmic utility. In

the bilinear utility specification, x is the critical return level called the

kink. P is the penalty level for returns lower than the kink. In the S-shaped

utility functions, the critical return level (the inflection point) is indicated

by z. A, B, g1 and g2 are parameters determining the curvature of the

S-shaped utility.

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Kahnemann and Tversky (1979). They argue that people focus more on the

return on an investment than on the level of wealth. Figure 1 displays how the

utility varies with returns for certain specifications of the four utility function

families.

The parametric, closed-form utility functions that are most common in

the finance literature are the families of exponential and power utility func-

tions. The former is characterized by constant absolute risk aversion, and the

latter by constant relative risk aversion, meaning that risk aversion varies

with wealth level. In Fig. 1 examples of exponential and power utility func-

tion graphs are given in panels (a) and (b), respectively.

To consider the investor preferences of skewness and kurtosis in particu-

lar, two types of utility functions that have been suggested are the bilinear and

the S-shaped utility function families. Both of these are characterized by a

critical point of investment return, under which returns are given dispropor-

tionally bad utility. Graphical examples of bilinear and S-shaped utility

functions are given in Fig. 1(c) and Fig. 1(d), respectively.

Fig. 1 Utility Function Graphs

Notes: Panel (a) shows how exponential utility varies over returns when the risk aversion

parameter is set to A = 6. In panel (b) the same relationship for power utility is shown, with

risk aversion set to g = 2. The bilinear utility function is shown in panel (c) and has

parameters set to P = 5 and x = 0 per cent (kink). In panel (d) the S-shaped utility function

depicted has parameters A = 1, B = 2, g1= 0.3, g2= 0.7 and z = 0 per cent (inflection point).

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The bilinear functions capture a phenomenon that is central in invest-

ment management today: loss aversion. The objective of limiting losses is

motivated by monetary as well as legal purposes. The issue is traditionally

treated with value-at-risk models, and can also be incorporated in FSO

theory through a constraint on the maximization problem (as shown by

Gourieroux and Monfort, 2005). The bilinear utility functions have a kink

at the critical point and are formed by straight lines of different slope on

each side (i.e. the functions are linear splines), which obviously yields

discontinuity even in the first derivatives. Consisting of straight lines, the

bilinear function does not reflect risk aversion in the sense that marginal

utility is not decreasing in returns (except for the jump at the kink).

This type of function has previously been applied by Cremers et al. (2005)

and Adler and Kritzman (2007). For an example of how power utility can

be combined with bilinear utility to feature risk aversion, see Maringer

(2008).

The S-shaped utility function is motivated by the fact that it has been

shown in behavior studies that an investor prefers a certain gain to an

uncertain gain with higher expected value, but he/she also prefers an uncer-

tain loss to a certain loss with higher expected return (see Kahnemann and

Tversky, 1979). The utility function features an inflection point where these

certainty preferences change. The utility function implies high absolute values

of marginal utility close to the inflection point, but low (absolute) marginal

utility for higher (absolute) returns. The first derivatives are continuous, but

second derivatives are not.

3Empirical Application

We assess the FSO methodology by comparing its performance with port-

folios produced by using the MV methodology. The portfolios are opti-

mized in a setting of three equity indices as admissible assets. The utility

outcome of FSO and MV optima are examined in- and out-of-sample. To

get an economic interpretation of utility differences we also calculate cer-

tainty equivalents. The exercise is repeated for a wide range of utility func-

tion specifications.

3.1FSO Specification

We use a grid search to find the FSO optimum. As the allocation matrix

grows quickly when more assets are added or the allocation precision p is

increased, we use a three-asset setting with p = 0.5 per cent, and we do not

allow for short selling. This yields an allocation matrix of dimension

(3 ¥ 20301), which we evaluate over 96 monthly observations. This rather

limited amount of assets and precision is due to the computational burden of

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the technique.7We analyze the full grid of possible allocations—no search

algorithm is applied.8

The choice of grid precision can be made on the basis of the trade-off

between the marginal utility of increasing p and the additional computational

cost of doing so. Setting the grid precision to 0.1 per cent instead of 0.5 per

cent increases the grid size by a factor of almost 25 (from 20,301 to 501,501).

For most portfolio choice problems, that increased computational cost

cannot be motivated by the increased utility achieved.

An alternative to increasing the precision of the complete grid is to

perform a second grid search around the optimum found in the first search.

We performed a second-step grid search using p = 0.1 per cent around the

p = 0.5 per cent optima, covering all possible allocations within the range of

qi1 1 per cent.9The second grid dimension is in our case 67 2 m 2 331,

depending on whether the first-stage optimum contains allocations close to

the allowed limits 0 and 1 or not. We found that both the computational cost

and the utility improvement of the second grid search are minute. None of the

utility improvements exceeded 1 per cent, and only eight out of 132 utility

functions had improvements exceeding 0.1 per cent. This utility improvement

is what we sacrifice when choosing p = 0.5 per cent rather than p = 0.1 per

cent, a convenience cost. As a grid search never can yield an exact solution,

we also applied simulated annealing on the area around the optimum (for a

description of this technique, see, for example, Goffe et al., 1994). Again, the

convenience cost was very small. We hence concluded that p = 0.5 per cent

was enough for this application.10The computational cost of the second-step

7The number of possible solutions (m) is

C n

(

pp

n

(

1

p

np

pp

+−

)=

+(

)−

−

(

)

!

)

(

)

=

(

)+

(

)∗ (

1

)+

...

(

)∗

∗

∗

1

1 1 1,

11

1

11

1 2

∗ ∗

2!

!

...

n

1 1

(

)

1

pn

)+ −

(

)

−

(

where n is the number of assets and p the precision of the grid. This is the formula for

combinations of discrete numbers with repetition, derived by Leonhard Euler (1707–83).

See, for example, Epp (2003).

8In Cremers et al. (2005) and Adler and Kritzman (2007), a search algorithm is applied to find the

FSO optimum. Such algorithms are necessary when using larger numbers of assets.

Cremers et al. consider 61 assets in their application, and use a precision of 0.1 per cent and

do not allow for short selling. This implies m = 7.23 ¥ 1098, which is the number of vectors

to be evaluated over their 10 annual observations. They do not disclose their search

algorithm. Cremers et al. (2003, 2005), Gourieroux and Monfort (2005) and Adler and

Kritzman (2007) all argue that the computational burden of the FSO technique has become

obsolete with the ample computational power on hand nowadays. To our knowledge,

however, there is no study verifying this.

9This is the range needed to cover all solutions not covered in the previous grid search, as if two

assets change by 0.5 per cent in the same direction, the third has to change by 1 per cent. In

general the range required for the second grid search is p(n - 1), where n is the number of

assets in the problem and p is the precision of the first grid search.

10In portfolio choice problems with large grids resulting from a larger number of assets, the

two-stage technique presented here may be a viable option to decrease computational cost.

Caution is needed, however, as non-convexity may lead to the optimum identified with an

imprecise grid not being in the area of the global optimum.

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grid search, measured in computation time, was also small. Time needed for

performing FSO on each utility function type used in this paper, for first- and

second-step grid searches, respectively, is given in Table 2.

The study is performed in a one-period setting—no rebalancing of the

portfolio is considered.

3.2Data

For the empirical application we use three indices that are published by the

Financial Times, downloaded from Datastream (2007): FTSE100, FTSE250

and FTSE All-World Emerging Market Index (EMI).11The FTSE100

includes the 100 largest firms on the London Stock Exchange and FTSE250

includes mid-sized firms, i.e. the 250 firms following the 100 largest. The EMI

reflects the performance of mid- and large-sized stocks in emerging markets.12

All series are denoted in British pounds (£). We calculate return series for

eight years of monthly observations (January 1999–December 2006), yielding

96 observations. As shown in Fig. 2, the data feature two expansionary

periods and one downward trend.

The data properties are presented in Table 3. All three indices display

positive means over the sample period. The least volatile choice of the three

is the FTSE100, followed by FTSE250 and the FTSE EMI. All of them

feature negative skewness, and excess kurtosis is observed for FTSE100 and

FTSE250. It is shown with a Jarque–Bera test of normality (Jarque and Bera,

1980) that normality can be rejected for the UK indices, but not for the

EMI.13

11The Datastream codes for the indices are FT100GR(PI), FT250GR(PI) and AWALEG£(PI).

12For anexact definition, see http://www.ftse.com/Indices/FTSE_Emerging_Markets/

Downloads/FTSE_Emerging_Market_Indices.pdf.

13The Jarque–Bera test is appropriate for serially uncorrelated data (such as white-noise regres-

sion residuals) but inadequate for temporally dependent data such as certain financial

returns. We do not pursue this further; the interested reader is referred to Bai and Ng

(2005).

Table 2

Computational Cost of FSO

Utility function Time first stepTime second step

Exponential

Power

Bilinear

S-shaped

1.25

1.47

35.78

44.75

0.00

0.01

0.13

0.14

Notes: Time is specified in seconds. Optimization was run over 96 time

periods. The platform used was an Intel Core 2 2.16 GHz processor

with 3 GB RAM. Software used was R v2.6.1, applying the function

system.time.

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In order to mitigate the probability of corner solutions, we scale all

returns to conform to implied returns of an equally weighted portfolio.14This

does not change the shape of the probability distribution, and does not affect

the comparison between FSO and MV.

3.3 Utility Functions

We apply our portfolio selection problem to exponential, power, bilinear and

S-shaped utility functions. The same utility function types have been inves-

tigated before, but only a few cases of each type. We perform the exercise

under several different utility function parameter values, chosen with the

intention to cover all reasonable levels.

The bilinear and S-shaped utility functions are our main interest in this

study. Using these, it has been shown in a hedge fund setting that FSO

14The difference between the average return of all three indices and the return corresponding to

the variance of the equally weighted portfolio is added to each observation. In this case, the

difference added amounts to 0.086 per cent.

FTSE100, FTSE250, FTSE EMI

1999/01/01–2006/12/01?(monthly?data)

200%

100%

FTSE100 FTSE250FTS?EMI

0%

19992000 2001 20022003 200420052006

Fig. 2 Development of the Three Indices over the Time Period Considered

Table 3

Summary Statistics

Mean VarianceSkewness Kurt. J–B stat.p

FTSE100

FTSE250

FTSE EMI

0.0010

0.0096

0.0122

0.0015

0.0025

0.0043

-0.93

-0.83

-0.24

3.91

4.36

2.91

17.07

18.32

0.93

0.00

0.00

0.63

Notes: The table shows the first four moments of the monthly data series. Kurt. is the estimator of Pearson’s

kurtosis using the function kurtosis in R (this is not excess kurtosis). J–B stat. is the Jarque–Bera test statistic

and p is the probability that the series is following a normal distribution according to this test.

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yields portfolio weights that differ substantially from those of MV optimi-

zation (Cremers et al., 2005; Adler and Kritzman, 2007). We seek to test

whether this difference holds in a portfolio selection problem of equity

indices. We also include the traditional investor preferences of exponential

and power utility. Utility maximization using these functions has repeatedly

been shown to differ only marginally from quadratic utility (Levy and

Markowitz, 1979; Markowitz, 1987; Cremers et al., 2005), and we include

them for the purpose of illustration. Gourieroux and Monfort (2005) apply

the FSO model to the exponential and power utility functions, which allows

them to derive the asymptotic properties of the FSO estimator. They estab-

lish in this context that the utility-maximizing estimator yields greater

robustness than the MV counterpart, as no information in the return

distribution is ignored.

The range of utility parameters tested is given in Table 4. For the expo-

nential utility function, the only parameter to vary is the level of risk aver-

sion (A), which we vary between 0.5 and 6. The g parameter in the power

utility function determines the level of risk aversion and how risk aversion

decreases with wealth. As we let it vary between 1 and 5, we include the

special case when the power utility function is logarithmic, which happens

when g is one. The higher g and A are, the higher is the risk aversion. For

the bilinear utility function, we vary the critical point (the kink, x, varied

from -4 per cent to +0.5 per cent) under which returns are given a dispro-

portionate bad utility. We also vary the magnitude, P, of this disproportion

from 1 to 10. In the S-shaped utility function there are five parameters to

vary. We test three levels for the inflection point, z: 0 per cent, -2.5 per cent

and -5 per cent. The parameters g1 and A respectively determine the shape

and magnitude of the downside of the function, whereas g2and B determine

the upside characteristics in the same way. The disproportion between gains

and losses can be determined either by the g parameters or by the A and B

parameters, or both. We perform one set of tests where the gs vary (g23 g1)

and the magnitude parameters are held constant and equal, and one set

of tests where the gs are constant and equal, but where A and B varies

(A 3 B).

Table 4

Utility Function Parameters

Utility function Parameter values

Exponential

Power

Bilinear

S-shaped

0.5 2 A 2 6

1 2 g 2 5

-4% 2 x 2 0.5%; 1 2 P 2 10

-5% 2 z 2 0%; 0.05 2 g12 0.5; 0.95 2 g22 0.5; A = 1.5; B = 1.5

-5% 2 z 2 0%; g1= 0.5; g2= 0.5; 1.5 2 A 2 2.9, 1.5 2 B 2 0.1

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3.4Model Comparison

The methodology for comparing FSO with the MV approach is to a large

extent inspired by that applied by Cremers et al. (2005) and Adler and Kritz-

man (2007), where the performance of the different approaches is measured

in utility.

In order to compare the FSO and MV optima, the resulting total FSO

portfolio return is calculated (R

p FSO

= ′ θ

then the variance-minimizing allocation that yields the same expected port-

folio return, a point on the MV efficiency frontier.15From each solution, a

return distribution over time is calculated (q′Rt). By inserting each of these in

the utility function applied for the FSO method, a measure of the MV

approximation error, eMV, can be calculated as in equation (4):16

θθ

θ

MV

′

()

UR

R

). The MV optimal portfolio, qMV, is

ε

MV

FSO

′

MV

′

=

( )−()

URUR

(4)

For the bilinear and S-shaped utility functions, we also calculate success rates

of the same type as in Cremers et al. (2005). These are the fraction of all

points in time that yield portfolio returns superior to the investor’s specified

critical level (kink and inflection point, respectively).

Measuring the difference in terms of improved utility has the drawback

that utility is hard to interpret in economic terms. Also, different utility

functions yield different magnitudes of utility variation for a set of returns.

An alternative measure, which is more straightforward to interpret, is cer-

tainty equivalents.17The certainty equivalent of a specific risky investment in

terms of return can be defined as the certain return that would render the

investor the same level of utility as the uncertain return of the risky invest-

ment. In other words, the certainty equivalent, rCE, is the solution to the

following equation:

+

( ) =

+

()

CEp

Ur EUr

11

(5)

where E denotes expected value. Provided that the utility function is one-to-

one the solution to this equation exists and is unique:

()

[]−

11

rU EUrp

CE=+

−1

(6)

For the utility functions presented above we are therefore able to compute the

unique return rCE that satisfies equation (5) given the portfolio returns rp

15We use the Markowitz MV model from 1952 as benchmark in this study. In this way, it can be

established whether the FSO model is superior to that model. The rich supply of MV

extensions, however, is yet to be compared with the FSO model.

16The MV solution can be calculated with much higher detail than the FSO portfolio, which is

limitedto0.5percentprecision.Accordingly,aminorsourceofutilitydifferencewillbedue

to this limitation, which in the comparison is to the MV method’s advantage.

17Certainty equivalents have earlier been used for FSO–MV comparisons in a working paper by

Cremers et al. (2003), which however was never published.

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corresponding to the optimal allocation q. Using the same approximation as

in the FSO, the right-hand side of equation (5) can be evaluated. Expected

utility at the optimal weights qFSO is denoted u ¯, and is calculated as in

equation (7):

∑

1

uTURt

t

T

=

′

()

−

=

1

θFSO

(7)

where T is the number of time periods (which differs in- and out-of-sample).

Now, equation (5) is equivalent to the expression.

1+

( ) =

CE

Uru

(8)

which is simple to solve for rCE. The solution for each utility function is

presented in Table 5 (the rightmost column).

When we are evaluating the results of the empirical application in the

next section, we are mainly interested in the difference in certainty equivalents

between the two portfolio selection techniques. Hence, we derive the follow-

ing measure, showing the difference in rCEbetween FSO and MV:

ΔCE CE

r

FSO

CE

r

MV

≡−

(9)

The certainty equivalent difference, DCE, is easy to interpret. It is the certain

return on investment corresponding to the increase in utility (under the utility

function in question).

3.5 Out-of-sample Testing

In order to further examine the robustness of the FSO methodology, we

repeat the procedure described above in an out-of-sample setting. This is

done using essentially the same methodology as in Adler and Kritzman

Table 5

Certainty Equivalent Equations

Utility function Certainty equivalent

Exponential

-exp[-A(1 + rp)]

1

1

−

γ

ln(1 + rp)

ln(1 + rp)

P(rp- x) + ln(1 + x)

−−

()

A zrp

¤ r

A

u

CE= −−

(

)−

1

1 ln

Power

1

1

+

()

−

−

rp

γ

for g > 1 ¤ ru

CE=+−

(

)

[]

−

−

()

111

1 1

γ

γ

for g = 1 ¤ rCE= exp(u ¯) - 1

for rp3 x ¤ rCE= exp(u ¯) - 1

for rp< x ¤a

for rp2 z ¤ r

Bilinear

S-shaped

γ1

z

u

A

CE=−

+( )

−(

)

γ

1

1

γ

+−

()

B rz

p

γ2

for rp> z ¤ rz

u

B

CE=

1

2

Notes:

search algorithm.

aUnder bilinear utility when u ¯ < u(x), no explicit solution exists. This case is handled by a standard

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(2007). For this purpose, the sample is split into two halves. The first half is

used for estimating optimal portfolio allocations, and the performance of the

optimal portfolio retrieved is measured on the second half.

We generate 10,000 samples of cross-sectional monthly returns by

drawing from the second half of our sample (with replacement), which was

not used for the portfolio optimization. This bootstrapping procedure allows

us to study performance of the optimized portfolio out-of-sample.

The average utility difference between FSO and MV portfolios is calcu-

lated. The exercise is repeated using the second half of the sample for esti-

mation and the first half for the diagnostics.

4Results

The portfolio selection problem described was repeated 103 times using dif-

ferent utility function specifications. There were 12 tests with exponential

utility, nine tests with power utility, 31 tests with bilinear utility and 51 tests

with S-shaped utility. All results are presented in the Appendix. Below the test

results are presented and interpreted for each utility function type separately.

The section is concluded with some general observations.

4.1Traditional Utility Functions

The tests performed with exponential utility and differing levels of risk aver-

sion yielded, as expected, portfolios with high weights to risky assets when A

was low and less so as the risk aversion parameter A was set higher (see

Table A1). The portfolios identified were all very close to the MV frontier,

resulting in extremely small utility improvements, if any, when using FSO

instead of MV. As the FSO and MV solutions were close to identical, the

differences in performance out-of-sample were close to non-existent.

Portfolio allocations based on power utility functions were selected

with nine different levels of g, implying different levels of relative risk aver-

sion. As shown in Table A2, the power utility function yielded portfolios

allocating the whole investment to the most risky asset (FTSE EMI) for

g 2 1.5, and then gradually leaned more towards the medium-risky asset

(FTSE250) as g grew. The deviation from the MV frontier was slightly

bigger than in the exponential utility cases, but differences in utility outcome

both in-sample and out-of-sample were still minute. In-sample utility

improvement never exceeded 0.02 per cent, and out-of-sample differences

displayed neither substantial magnitude, nor consistent directions on devia-

tions from zero.

The improvement in terms of certainty equivalents is zero or close to

zero under both exponential and power utility functions. Hence, an investor

following these utility functions is not prepared to pay anything to move from

MV to FSO. These results conform well to those of earlier assessments

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(Cremers et al., 2005; Adler and Kritzman, 2007), showing that portfolio

allocations chosen by the utility-maximizing approaches constitute very small

improvements relative to the MV approach, when the investor’s utility is well

described by the exponential or the power utility functions.

4.2Bilinear Utility Functions

The tests on bilinear utility functions (i.e. linear splines) were performed with

the kink (x) at different levels and various penalties (P) on sub-kink returns.

Each kink value was tested for three different penalty levels. As was shown in

Table 1, the disutility of sub-kink returns is amplified by the factor P. When

P = 1, there is no kink, and the utility function features neither loss aversion

nor risk aversion.

The results retrieved for bilinear utility functions are shown in Table A3.

As expected, the risk level of the optimal portfolios retrieved with FSO

decreased as the kink and penalty parameters increased. When the penalty

parameter was set to 1 all wealth was allocated to the most risky asset (FTSE

EMI), which is due to the lack of risk aversion in this case. For higher penalty

levels, the portfolios were increasingly weighted towards the less risky assets

(FTSE100 and FTSE250). At P = 10, no allocations were made to the most

risky asset. This occurs when the incentive to avoid returns less than those

associated with the kink dominates other investor incentives, such as maxi-

mizing returns or minimizing risk by diversification. Portfolio diversification

was the highest at P = 5.

Whereas many of the portfolios optimized under bilinear utility were

close to the MV frontier, there were cases where the utility improvement was

substantial (up to 4 per cent in-sample and 13 per cent out-of-sample).

Looking at certainty equivalents, there was a positive, fairly consistent, but

small difference in favor of the FSO portfolios. On average, certainty equiva-

lent improvement amounted to 0.02 per cent in-sample and 0.04 per cent

out-of-sample (see bottom of Table A3). This means that usage of FSO

rather than MV when investor preferences are correctly described with a

bilinear utility function yields a utility improvement equivalent to that of a

certain 0.02 per cent annual return. Of the 20,000 draws made for the out-

of-sample test, 14 per cent yielded certainty equivalent improvements higher

than 1 per cent, and 8 per cent yielded less than -1 per cent (in annual terms).

Looking at the distribution of returns out-of-sample, there were on

average (of the 20,000 bootstrapped samples) no differences between FSO

and MV in mean and variance. The solutions under bilinear utility, however,

yielded higher skewness and higher kurtosis than the MV solutions did on

average (as shown in Table A6). Bilinear utility does not punish kurtosis, as

there is no decrease in marginal utility with returns (i.e. no risk aversion)

except for the jump at the kink. Accordingly, it can be seen that the corre-

sponding MV portfolios are more diversified than the FSO portfolios.

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