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Portfolio Performance Evaluation with Loss

Aversion∗

Valeri Zakamouline†

This revision: January 4, 2010

Abstract

In this paper we consider a loss averse investor equipped with a specific, but still quite

general, utility function motivated by behavioral finance. We show that under some concrete assumptions about the form of this utility one can derive closed-form solutions for the

investor’s portfolio performance measure. We investigate the effects of loss aversion and

demonstrate its important role in performance measurement. The framework presented in

this paper also provides a sound theoretical foundation for all known performance measures

based on partial moments of distribution.

Key words: utility theory, behavioral finance, portfolio performance evaluation, performance measure, reward-to-risk ratio, loss aversion.

JEL classification: D81, G11.

∗

The author is grateful to Steen Koekebakker for his comments. The usual disclaimer applies.

University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) 38 14

10 39, Valeri.Zakamouline@uia.no

†

1

Electronic copy available at: http://ssrn.com/abstract=1531094

1

Introduction

The literature on portfolio performance evaluation starts with the seminal paper by Sharpe

(1966) who derived a performance measure widely known now as the Sharpe ratio. However,

because it is based on the mean-variance theory, it is valid only for either normally distributed

returns or quadratic preferences. In other words, the Sharpe ratio is a meaningful measure of

portfolio performance when the risk can be adequately measured by standard deviation. When

return distributions are non-normal, the Sharpe ratio can lead to misleading conclusions and

unsatisfactory paradoxes.

As a response, many alternative performance measures accounting for non-normality of

return distributions have been proposed. Most of these alternative performance measures

are reward-to-risk ratios. That is, this type of a performance measure represents a fraction

where a measure of reward is divided by a measure of risk. In a great deal of these alternative

performance ratios the measures of reward and risk (or at least one of them) are based on partial

moments of distribution. The examples of such reward-to-risk ratios include the Sortino ratio,

the Kappa ratio, the Omega ratio, the Upside potential ratio, and the Farinelli-Tibiletti ratio.

Unfortunately, all of these performance measures were presented ad-hoc which left a room

for various critiques. Pedersen and Satchell (2002) were the first to present a well-grounded

justification of the Sortino ratio. In particular, these authors suggested a method which allows

to derive the formula for a performance measure using the shape of the investor’s utility

function. They showed that the Sortino ratio is a performance measure of the investor equipped

with a specific utility function. Therefore, the Sortino ratio is an example of a utility-based

performance measure in the same manner as the Sharpe ratio. Later on Zakamouline and

Koekebakker (2009b) and Zakamouline and Koekebakker (2009a) used a similar idea to that

of Pedersen and Satchell (2002) to derive some new utility-based performance measures.

In this paper we extend the ideas of Pedersen and Satchell (2002) and, specifically, the

framework of Zakamouline and Koekebakker (2009a) in order to provide a sound theoretical

foundation for all known performance measures based on partial moments of distribution and to

derive a generalized version of the performance measure presented in Zakamouline and Koekebakker (2009a). Our main focus is on loss aversion as an important behavioral phenomenon

in the investor’s decision-making. In particular, we consider the investor with a specific, but

2

Electronic copy available at: http://ssrn.com/abstract=1531094

still quite general, utility function that has a reference point. We assume that the investor

regards outcomes below the reference points as losses, whereas outcomes above the reference

point as gains. We suppose that the behavioral utility generally has a kink at the reference

point and different functions below and above the reference point. Consequently, the investor

in our framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in

gains, and aversion to uncertainty in losses. Under some concrete assumptions about the form

of the investor’s utility we derive closed-form solutions for the investor’s portfolio performance

measure.

It turns out that in some cases the investor’s coefficient of loss aversion plays an explicit

role in performance measurement. That is, to compute the value of a performance measure

one needs to take into account the investor’s loss aversion. To demonstrate that loss aversion

plays an important role in performance measurement we conduct an empirical study. In this

study we use a database of hedge fund returns and compute a specific investor’s performance

measure with different levels of loss aversion. We show that investors with different degrees

of loss aversion (generally) rank differently risky portfolios. We also suggest a measure that

evaluates the risk of loss.

The rest of the paper is organized as follows. In Section 2 we first introduce the notion of a

utility-based performance measure and different definitions of loss aversion. Then we present

the generalized form of the investor’s utility and under specific assumptions derive closedform solutions for performance measures. Here we also discuss the effects of loss aversion and

the properties of utility-based performance measures. Finally in this section we consider an

alternative framework that allows to arrive at a formula for the investor’s portfolio performance

measure under more specific, but at the same time weaker assumptions. In Section 3 we perform

the empirical study and demonstrate the role of loss aversion in performance measurement.

Section 4 concludes the paper.

2

2.1

Accounting for Loss Aversion in Performance Measurement

The Notion of Performance Measure

Throughout the paper we denote by x the return on a risky portfolio and by r the risk-free rate

of return. Denote by π(x) the investor’s performance measure of the risky portfolio x, where

3

π(·) is some function. By a performance measure in finance one means a score attached to each

risky portfolio. This score is usually used for the purpose of ranking of risky portfolios. That

is, the higher the performance measure of a portfolio, the higher the rank of this portfolio.

Formally, a performance measure implies the preference ordering among the risky portfolios.

In particular, the investor prefers risky portfolio x1 to risky portfolio x2 if π(x1 ) > π(x2 ). The

goal of any investor who uses a particular performance measure is to select the portfolio for

which this measure is the greatest.

Note that all performance measures are defined over returns only. This property of a

performance measures has two very important consequences:

(a) The ranking of risky portfolios does not depend on the investor’s wealth.

(b) The ranking of risky portfolios does not depend on the investor’s particular risk preferences. All necessary information about the risk preferences of a representative investor

is contained in the expression for a performance measure.

Thus, all investors that agree on some performance measure choose exactly the same risky portfolio (the portfolio with the greatest measure) irrespective of their wealths and some particular

risk preferences, for example, degrees of risk aversion.

Denote by U (W ) the investor’s utility function, where W is the investor’s wealth. The

investor’s objective is to maximize the expected utility of wealth, E[U (W )], where E[.] is the

expectation operator. How to derive the expression for the investor’s performance measure

using the shape of the investor’s utility function? We will use the method that was for the first

time presented by Pedersen and Satchell (2002) (which they call the “maximum principle”) and

later exploited by Zakamouline and Koekebakker (2009b) and Zakamouline and Koekebakker

(2009a). This method is based on finding a solution to the investor’s optimal capital allocation

problem. In particular, assume that the investor’s initial wealth is WI . The investor’s capital

allocation consists in investing a in the risky portfolio and, consequently, WI −a in the risk-free

asset. Thus, the investor’s final wealth is

W (x) = a(x − r) + WI (1 + r).

4

(1)

The investor’s objective is to choose a to maximize the expected utility

E[U ∗ (W (x))] = max E[U (W (x))].

a

(2)

If there exists either an asymptotic or explicit solution for the optimal value of a, then using

this solution in some cases it is possible to show that the investor’s maximum expected utility

can be written as a strictly increasing function (which we denote by f (·)) of a quantity that

can be interpreted as a performance measure

E[U ∗ (W (x))] = f (π(x)).

That is, if π(x1 ) > π(x2 ), then for all investors that have the same utility function (yet

different initial wealths and probably different particular risk preferences) E[U ∗ (W (x1 ))] >

E[U ∗ (W (x2 ))]. Consequently, the goal of maximizing the investor’s expected utility can alternatively be formulated as the maximization of a specific performance measure. Note that

a performance measure is not unique since any positive increasing transformation of a performance measure produces an equivalent performance measure.

2.2

The Notion of Loss Aversion

We suppose that the investor’s utility has a reference point which we denote by W0 . The

investor regards outcomes below the reference point as losses, while outcomes above the reference point as gains. We suppose that the investor’s utility function is everywhere differentiable

except probably at W0 . At W0 the investor’s utility might have a kink. Without loss of generality we also suppose that U (W0 ) = 0. Kahneman and Tversky (1979) introduced the concept

of loss aversion by the following condition

−U (W0 − ∆W ) > U (W0 + ∆W ) for any ∆W > 0.

That is, “losses loom large than gains”. Their definition can be related to a loss aversion

(W0 −∆W )

coefficient of the mean of − U

U (W0 +∆W ) . In Kahneman and Tversky (1979) and later in Wakker

5

and Tversky (1993) the loss aversion is also defined as

U 0 (W0 − ∆W ) > U 0 (W0 + ∆W ) for any ∆W > 0,

where U 0 is the first derivative of U . That is, the investor’s utility function must be steeper in

the domain for losses than in the domain for gains. Their definition can be related to a loss

aversion coefficient of the mean of

U 0 (W0 −∆W )

U 0 (W0 +∆W ) .

A stronger definition was used by Bowman,

Minehart, and Rabin (1999): loss aversion holds if

U 0 (W0 − ∆W1 ) > U 0 (W0 + ∆W2 ) for all ∆W1 > 0, ∆W2 > 0.

That is, the slope of the utility function for losses is everywhere steeper than the slope of the

utility function for gains. For the power utility function of Prospect theory the loss aversion

coefficient is a constant which can be defined in two alternative ways

λ=−

U (W0 − ∆W )

U 0 (W0 − ∆W )

, or λ = 0

for any ∆W > 0.

U (W0 + ∆W )

U (W0 + ∆W )

Yet another measure of loss aversion was proposed by Bernartzi and Thaler (1995) and formalized by K¨obberling and Wakker (2005)

λ=

U 0 (W0 −)

,

U 0 (W0 +)

(3)

where U 0 (W0 −) and U 0 (W0 +) denote the left and right derivatives of U (supposing that they

are positive, finite, but non-zero) at W0 . Observe that if the decision maker does not exhibit

loss aversion, then λ = 1. Loss aversion implies λ > 1. Conversely, loss seeking behavior

implies λ < 1. Since the first-order derivatives of U are positive, the value of λ is also positive,

that is, λ > 0. Finally note that while Kahneman and Tversky (1979) and Wakker and Tversky

(1993) define the loss aversion in a global sense, K¨obberling and Wakker (2005) define the loss

aversion in a local sense, that is, as a peculiarity of the investor’s behavior around the reference

point.

The brief review of the notion of loss aversion presented above reveals that there is no

unique definition of loss aversion. This means, in particular, that a utility function can exhibit

6

loss aversion according to one definition, but not to another. Moreover, the definition of loss

aversion of K¨obberling and Wakker (2005) is based on the existence of positive and non-zero

left and right derivatives. For example, this measure of loss aversion is not defined for the

power utility as in Prospect Theory, since a one-sided derivative of a power function at the

origin is either infinity or zero.

2.3

A Model for the Investor’s Preferences

Consider an investor that has the following generalized form of the piecewise linear plus power

utility function:

1+ (W − W0 ) − γ+ (W − W0 )α ,

if W ≥ W0 ,

α

U (W ) =

³

´

−λ 1− (W0 − W ) + γ− (W0 − W )β , if W < W0 ,

β

(4)

where 1+ and 1− are indicator functions that take values in {0, 1}, γ+ and γ− are real numbers,

and λ > 0, α > 0, and β > 0. This generalized form encompasses the quadratic utility of Tobin

and Markowitz as well as the utility functions used, among others, by Fishburn (1977), Bawa

(1978), Kahneman and Tversky (1979), Holthausen (1981), Jia and Dyer (1996). This utility

function is very flexible and allows to specify different types of the investor’s preferences, in

particular, loss aversion and either risk seeking, risk neutral, or risk aversion attitudes below

and above the reference point.

Since our goal is to solve the investor’s optimal capital allocation problem, we need to use

W (x) given by (1) in the expression for the investor’s utility (4). Let us introduce the following

variable

τ =r−

WI (1 + r) − W0

,

a

supposing that a > 0. Using this variable, the investor’s utility of W (x) becomes

1+ a(x − τ ) − γ+ aα (x − τ )α ,

if x ≥ τ ,

α

U (W (x)) =

³

´

−λ 1− a(τ − x) + γ− aβ (τ − x)β , if x < τ .

β

(5)

It turns out that the investor’s expected utility can be written in terms of lower and upper

partial moments. The definition of a lower partial moment was presented by Fishburn (1977).

7

In particular, a lower partial moment of the distribution of x of order n at level τ is given by

Z

LP Mn (x, τ ) =

τ

−∞

(τ − s)n dFx (s),

where Fx (·) is the cumulative distribution function of x. Similarly, an upper partial moment

of x of order n at level τ is given by

Z

U P Mn (x, τ ) =

∞

τ

(s − τ )n dFx (s).

Therefore, the investor’s expected utility can be written as

µ

¶

γ+ α

γ− β

E[U (W (x))] = 1+ aU P M1 (x, τ )− a U P Mα (x, τ )−λ 1− aLP M1 (x, τ ) +

a LP Mβ (x, τ ) .

α

β

(6)

Observe that τ can be interpreted as the investor’s reference return of the minimum acceptable

return.

Before attacking the optimal capital allocation problem, we need to choose a suitable

reference point, W0 , to which gains and losses are compared. One possible reference point is

the “status quo”, that is, the investor’s initial wealth WI . Unfortunately, with this choice it

is not possible to arrive at a closed-form solution for the optimal capital allocation problem

unless WI = 0. However, according to Kahneman and Tversky (1979) the investor’s initial

wealth does not need to be the reference point. To obtain a closed-form solution we further

assume that the reference point is W0 = WI (1 + r) which is the investor’s initial wealth scaled

up by the risk-free rate.1 With this choice τ = r, that is, the risk-free rate of return serves

as a reference return. Moreover, with this choice the investor’s utility does not depend on the

investor’s initial wealth.

Consider the resulting investor’s optimal capital allocation problem

max

a

1

¡

¢

a 1+ U P M1 (x, r) − 1− λLP M1 (x, r) − aα

µ

¶

γ+

γ−

U P Mα (x, r) + aβ−α λ LP Mβ (x, r) ,

α

β

Barberis, Huang, and Santos (2001) and many others make a similar assumption.

8

subject to a ≥ 0. The first-order condition for the optimality of a gives

¡

¢

¡

¢

1+ U P M1 (x, r) − 1− λLP M1 (x, r) − aα−1 γ+ U P Mα (x, r) + aβ−α λγ− LP Mβ (x, r) = 0. (7)

Observe that it is still not possible to arrive at a closed-form solution for a for all possible

values for α and β. However, as we will se later, under some additional assumptions we can

arrive at a closed-form solutions for a.

2.4

Investor with Piecewise Linear plus Identical Power Utility Function

In this case 1+ = 1 and α = β > 1. The investor with this utility exhibits loss aversion (in

the sense of K¨obberling and Wakker (2005)) if 1− = 1 and λ > 1. Depending on the sign of

γ− , the investor may exhibit either risk aversion (if γ− > 0), risk neutrality (if γ− = 0), or risk

seeking attitude (if γ− < 0) for outcomes below r. Similarly, depending on the sign of γ+ , the

investor may exhibit either risk aversion (if γ+ > 0), risk neutrality (if γ+ = 0), or risk seeking

attitude (if γ+ < 0) for outcomes above r. This utility function reduces to the utility function

of Fishburn (1977) and Bawa (1978) when γ+ = 0, γ− > 0, 1− = 1, and λ = 1. Figure 1

illustrates the form of this utility function.

The first-order condition for the optimality of a reduces in this case to

¡

¢

¡

¢

U P M1 (x, r) − 1− λLP M1 (x, r) − aβ−1 γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r) = 0.

Note that to guarantee the existence of a local maximum (interior solution), the investor’s

expected utility should be concave in a, which means that the following condition should be

satisfied

γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r) > 0.

(8)

In addition, since we require a ≥ 0, the positive solution exists only if the following condition

is satisfied

U P M1 (x, r) − 1− λLP M1 (x, r) > 0.

(9)

Moreover, the solution exits only when β > 1. Otherwise, when 0 < β < 1 and condition (9)

is satisfied, the investor’s expected utility approaches infinity as a → ∞. In other words, in

9

15

γ+=−0.1

10

γ+=0.0

5

γ+=0.1

0

−5

−10

−15

γ−=−0.1

γ−=0.0

−20

γ−=0.1

−25

−30

−10

−5

0

5

10

Figure 1: Some possible forms of the utility function with α = β and 1− = 1. We set λ = α = β = 2

and vary the values of γ− and γ+ . The intersection of the dotted lines shows the location of the reference

point.

this case the investor is willing to borrow an infinite amount to invest in the risky portfolio,

hence, the solution to the optimal capital allocation problem does not exists.

Under the stipulated conditions the solution for the optimal value of a is given by

µ

a=

U P M1 (x, r) − 1− λLP M1 (x, r)

γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r)

¶

1

β−1

.

(10)

Inserting the solution for the optimal a into the expression for the investor’s expected utility

(6) we obtain

β−1

E[U (W (x))] =

β

∗

Ã

U P M1 (x, r) − 1− λLP M1 (x, r)

p

β

γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r)

Note that since we can write E[U ∗ (W (x))] = f (Zγ− γ+ λβ1− ), where f (z) =

!

β

β−1

.

β

β−1 β−1

β z

(11)

is a strictly

increasing function in z, the expression for the performance measure of this investor can be

given by (observe also that U P M1 (x, r) − LP M1 (x, r) = E[x] − r)

E[x] − r − (1− λ − 1)LP M1 (x, r)

Zγ− γ+ λβ1− (x) = p

,

β

γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r)

10

(12)

which we denote as the Z ratio. Seemingly the performance measure Zγ− γ+ λβ1− (x) depends

on five parameters that describe the investor’s preferences. Yet, it turns out that to compute

a performance measure that is equivalent to Zγ− γ+ λβ1− (x), one needs to define maximum four

parameters. Recall that a performance measure is unique up to a strictly increasing positive

transformation. That is, the performance measures π(x) and f (π(x)) produce exactly the same

ranking of risky portfolios if f (·) is a strictly increasing function.

In particular, if γ+ > 0, then an equivalent performance measure is given by

E[x] − r − (1− λ − 1)LP M1 (x, r)

Zθλβ1− (x) = p

,

β

U P Mβ (x, r) + λθLP Mβ (x, r)

where θ =

γ−

γ+

is the relation between the investor’s risk aversions in the domain of losses and

the domain of gains. Similarly, if γ− > 0, then an equivalent performance measure is given by

E[x] − r − (1− λ − 1)LP M1 (x, r)

Zθλβ1− (x) = p

,

β

θU P Mβ (x, r) + λLP Mβ (x, r)

where θ =

γ+

γ− .

If γ+ = 0, then an equivalent performance measure is given by

Zλβ1− (x) =

E[x] − r − (1− λ − 1)LP M1 (x, r)

p

.

β

LP Mβ (x, r)

Observe that the Z ratio generalizes the Sortino2 ratio (when λ = 1, γ+ = 0, β = 2, γ− > 0,

1− = 1), the Kappa3 ratio (when λ = 1, γ+ = 0, γ− > 0, 1− = 1), the Upside potential4 ratio

(when λ = 1, γ+ = 0, β = 2, γ− > 0, 1− = 0), and even the Sharpe ratio of the form √ E[x]−r

E[(x−r)2 ]

(when λ = 1, γ+ = γ− > 0, β = 2, 1− = 1). These ratios are given by

E[x] − r

Sortino ratio = SoR(x) = p

,

LP M2 (x, r)

E[x] − r

,

LP Mβ (x, r)

Kappaβ ratio = Kβ (x) = p

β

U P M1 (x, r)

Upside potential ratio = U P R(x) = p

.

LP M2 (x, r)

2

See Sortino and Price (1994).

See Kaplan and Knowles (2004). This ratio is also known as the Sortino-Satchell ratio.

4

See Sortino, van der Meer, and Plantinga (1999).

3

11

Also seemingly the Z ratio generalizes the Omega5 ratio (when λ = 1, γ+ = 0, β = 1, γ− > 0)

Omega ratio = Ω(x) =

U P M1 (x, r)

E[x] − r

=

+ 1.

LP M1 (x, r)

LP M1 (x, r)

Unfortunately, the Omega ratio requires using β = 1 at which the solution to the investor’s

capital allocation problem does not exists. Yet, the Omega ratio can be considered as a limiting

U P M1 (x,r)

case of, for example, √

ratio when β → 1.

β

LP Mβ (x,r)

Finally it is important to note that when 1− = 1 and the condition (9) is not satisfied, then

the investor’s optimal policy is to avoid the risky asset and invest only in the risk-free asset.6

Yet, when 1− = 0 the condition (9) is always satisfied since U P M1 (x, r) > 0. Somewhat

surprisingly, the investor with this utility and the short sale constraint (a ≥ 0) will find it

always optimal to allocate some amount to the risky portfolio even in the cases where the

risk premium is negative, E[x] < r. The explanation for such a counter-intuitive behavior

lies in the fact that in this case (when 1− = 0) the utility function is globally quasi-concave,

but locally around r it is convex. In other words, the investor with this utility is risk averse

with regard to big risks, but risk seeking with regards to small risks. This might seem not

quite sensible. However, the results of several experimental studies confirm such behavior. For

example, Battalio, Kagel, and Jiranyakul (1990) report that the majority of subjects in their

study accept fair gambles involving small gains and losses, but reject fair gambles involving

relatively large gains and losses.

Note that the utility function considered in this subsection is generally not everywhere

increasing, hence, the Z ratio should be used with caution. This utility is an increasing function

if the investor is risk averse below the reference point and either risk neural or risk seeking

above the reference point. It is worth noting that if the investor’s utility function can decrease

as wealth increases (as, for example, quadratic utility above the satiation point), the investor’s

choice can violate the first-order stochastic dominance principle. Besides, if the investor is

risk-seeking either below or above the reference point, then the solution to the optimal capital

allocation problem may not exist. In case the solution does not exist, the denominator in the

Z ratio becomes a complex number.

5

See Shadwick and Keating (2002). This ratio is also known as the Bernardo-Ledoit gain-loss ratio, see

Bernardo and Ledoit (2000).

6

It can be easily checked that in this case the investor’s expected utility attains maximum at a = 0.

12

3

α=2.0

2

1

α=1.0

α=0.5

0

−1

−2

β=0.5

−3

−4

β=1.0

β=2.0

−5

−6

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 2: Some possible forms of the utility function with λ = 2. The intersection of the dotted lines

shows the location of the reference point.

2.5

Investor with Piecewise Power Utility Function

In this case 1+ = 1− = 0, γ+ = −α, and γ− = β. This is the utility function of Kahneman and

Tversky (1979) and Holthausen (1981). The investor with this utility exhibits loss aversion

(in the sense of Kahneman and Tversky (1979) and Wakker and Tversky (1993)) when λ > 1

(though not for all pairs of parameters α and β). Now it is the values of α and β that define

whether the investor is risk averse, risk neutral, or risk seeking. If α = 1, the investor is risk

neutral above r. If 0 < α < 1, the investor is risk averse above r. In contrast, if α > 1, the

investor is risk seeking above r. Similarly, if β = 1, the investor is risk neutral below r. If

0 < β < 1, the investor is risk seeking below r. Finally, if β > 1, the investor is risk averse

below r. Figure 2 illustrates the form of this utility function.

The first-order condition for the optimality of a reduces in this case to

αU P Mα (x, r) − aβ−α λβLP Mβ (x, r) = 0.

(13)

The existence of the solution to the optimal capital allocation problem requires β > α. Otherwise, since both U P Mα (x, r) > 0 and LP Mβ (x, r) > 0, the investor’s expected utility ap-

13

proaches infinity as a increases to infinity.7 The solution for the optimal a is given by

µ

a=

αU P Mα (x, r)

λβLP Mβ (x, r)

¶

1

β−α

.

Once again, rather surprisingly, the investor with this utility and the short sale constraint

(a ≥ 0) will find it always optimal to allocate some amount to the risky portfolio even in the

cases where the risk premium is negative, E[x] < r. As before, the explanation for such a

counter-intuitive decision-making is that this utility function might be rather tricky. First,

note the investor with this utility function might be risk-seeking either in the domain for losses

or in the domain for gains. Second, when 0 < α < 1 and β > 1, the investor is globally riskaverse, but locally, around r, is risk-seeking. The risk-seeking behavior justifies participation

in small gambles.

Inserting the solution for the optimal a into the expression for the investor’s expected utility

(6) we obtain

! αβ

µ ¶ α "µ ¶ α

µ ¶ β #Ã p

β−α

α

β−α

β−α

β−α

U

P

M

(x,

r)

α

1

α

α

p

E[U ∗ (W (x))] =

−

β

λ

β

β

LP Mβ (x, r)

(14)

Note that since we can write E[U ∗ (W (x))] = f (F Tαβ (x)), where

µ ¶ α "µ ¶ α

µ ¶ β #

αβ

α β−α

1 β−α

α β−α

z β−α

f (z) =

−

λ

β

β

is a strictly increasing function in z (given β > α), the expression for the performance measure

of this investor can be given by the Farinelli-Tibiletti8 ratio

p

α

U P Mα (x, r)

F Tαβ (x) = p

.

β

LP Mβ (x, r)

Observe that the Farinelli-Tibiletti ratio reduces to the Upside potential ratio when α = 1 and

β = 2. Also seemingly the Farinelli-Tibiletti ratio generalizes the Omega ratio when α = β = 1.

Yet, as we have established, when α = β the solution to the optimal capital allocation problem

7

De Giorgi and Hens (2006) who were the first to note that the solution to the optimal capital allocation

problem does not exists if the investor has the utility function as in Prospect Theory with α = β. See also

Sharpe (1998) who showed that the solution to the optimal capital allocation problem does not exists for bilinear

utility which is defined by α = β = 1.

8

See Farinelli, Ferreira, Rossello, Thoeny, and Tibiletti (2008) and references therein.

14

does not exists. Therefore, the Omega ratio can be considered only as a limiting case of, for

example, Farinelli-Tibiletti1β ratio when β → 1.

Since the investor’s utility function is increasing for all values α > 0 and β > 0, the

Farinelli-Tibiletti ratio does not violate the principle of the first-order stochastic dominance.

In particular, Holthausen (1981) provides proofs for the following results: (1) this utility

satisfies first order stochastic dominance for all α > 0 and β > 0; (2) this utility satisfies

second order stochastic dominance for all 0 < α ≤ 1 and β ≥ 1; (3) this utility satisfies third

order stochastic dominance for all 0 < α ≤ 1 and β ≥ 2. Therefore, if one wants to model the

preferences of an investor who is risk-seeking either below or above the reference point, the

Farinelli-Tibiletti ratio should, in most cases, be preferred to the Z ratio.

2.6

Effects of Loss Aversion

The goal of this subsection is to emphasize the effects of loss aversion. First of all, we consider

the loss aversion in the sense of K¨obberling and Wakker (2005). Recall that the loss aversion

coefficient (3) is defined when the left and right derivatives of the investor’s utility function

at the reference point are positive, finite, and non-zero. In this case λ > 0. The loss aversion

coefficient of K¨obberling and Wakker (2005) is a part of the investor’s performance measure

when the investor has piecewise linear plus identical power functions utility with 1− = 1+ = 1.

Under the short sale restriction, this investor finds it optimal to invest in the risky portfolio

when (this follows from condition (9))

E[x] − r > (λ − 1)LP M1 (x, r).

In particular, if the investor is loss neutral, λ = 1, then it is optimal to invest when the risk

premium, E[x] − r, is positive. If the investor is loss averse, λ > 1, then it is optimal to invest

when the risk premium exceeds some particular threshold which depends on the investor’s

coefficient of loss aversion. The more loss averse the investor, the higher must be the risk

premium to induce the investor to undertake the risky investment. Finally, if the investor is

loss seeking, λ < 1, then such an investor finds it optimal to invest even if the risk premium is

negative.

There is a clear link between the loss aversion in the sense of K¨obberling and Wakker

15

(2005) and the notion of “first-order risk aversion” presented by Segal and Spivak (1990).

In particular, the latter authors studied the effects of a kink in the decision maker’s utility

function (when the left- and right-side derivatives exist and non-zero). They showed that,

when there is a kink in a utility function, the decision maker’s attitude towards risk differs

from that when a utility function is everywhere differentiable as in Expected Utility Theory.

These authors studied these effects in the context of insurance, but also mentioned briefly that

a kink in a utility function may cause the avoidance of the risky portfolio in the investor’s

capital allocation.

Second, we consider the case where the loss aversion coefficient of K¨obberling and Wakker

(2005) is not defined (when at least one of the two one-sided derivatives is either zero or

infinity). In this case the investor has the piecewise power utility function. This investor

exhibits loss aversion in the sense of Kahneman and Tversky (1979) and Wakker and Tversky

(1993) (when, for instance, α = β). Yet, irrespective of whether or not the investor exhibits

loss aversion, an investor with this utility finds it always optimal to invest in a risky portfolio

no matter the value of the risk premium. For example, the investor with the piecewise power

utility function as in Prospect Theory always invests some wealth in the risky portfolio no

matter the steepness of the investor’s utility function in the domain for losses. It is worth

noting that for this utility the parameter λ is, to a large extent, equivalent to the risk aversion

coefficient in the mean-variance utility. That is, both coefficients do not influence decision

“invest - do not invest”, but influence the decision of how much to invest.

2.7

Properties of Utility-Based Performance Measures

In the previous subsections we have derived two utility-based performance measures, namely,

the Z ratio and Farinelli-Tibiletti ratio. The goal of this subsection is to review the basic

properties of utility-based portfolio performance measures. First of all, observe that every

performance measure can be interpreted as a reward-to-risk ratio π(x) =

µ(x)

ρ(x) ,

where µ(x) and

ρ(x) are the measures of reward and risk respectively. In the Z ratio the measure of reward is

p

E[x]−r−(1− λ−1)LP M1 (x, r) and the measure of risk is β γ+ U P Mβ (x, r) + λγ− LP Mβ (x, r).

p

p

In the Farinelli-Tibiletti ratio they are α U P Mα (x, r) and β LP Mβ (x, r) respectively.

The investor’s capital allocation between the risky portfolio and the risk-free asset consists

of investing the proportion k in the risky portfolio and the other part, 1 − k in the risk-free

16

asset. We denote the return of the investor’s complete portfolio by y such that

y = k(x − r) + r.

We denote the investment opportunity set with the risky portfolio and the risk-free asset in

some risk-reward space as the capital allocation line.

Property 1. A risk measure satisfies the following properties

ρ(r) = 0,

ρ(x) > 0.

The first equation says that there is no risk associated with investing in the risk-free asset.

The second inequality says that: (1) the risk is a positive real-valued function; (2) the risk

measure of any risky portfolio is strictly greater than zero.

Property 2. A reward measure satisfies the following property

µ(r) = 0.

This property expresses the basic tenet of financial economics: no reward without risk.

Investing in the risk-free asset involves no risk and, therefore, the reward in this case should

be zero.

Property 3. The reward and risk measures exhibit a positive homogeneity property along the

capital allocation line. In particular, for any k > 0

µ(y) = µ(k(x − r) + r) = kµ(x),

ρ(y) = ρ(k(x − r) + r) = kρ(x).

The interpretation of this property is the following. If k = 0, then there is no risk and no

reward according to Properties 1 and 2. When we increase the value of k, the reward and risk

measures should be proportional to k, that is, proportional to the amount invested in the risky

portfolio (in other words, as applied to the risk measures, “twice the risk is twice as risky”).

17

Property 4. The performance measure is the same for any point on the capital allocation line

π(y) = π(k(x − r) + r) = π(x) for any k > 0.

In particular, this property implies that a performance measure cannot be manipulated by

changing the capital allocation.

Property 5. The performance measure is consistent with the first-order stochastic dominance

when the underlying utility function is everywhere increasing.

Property 6. The performance measure is consistent with the second-order stochastic dominance when the underlying utility function is everywhere increasing and concave.

Note, however, that Levy and Levy (2002) and Levy and Levy (2004) developed the

Prospect and Markowitz stochastic dominance (PSD and MSD respectively) theories with

S-shaped and reverse S-shaped utility functions for investors.

Property 7. The performance measure is consistent with the second-order PSD when the

underlying utility function is everywhere increasing, convex below the reference point, and

concave above the reference point.

For example, the Farinelli-Tibiletti ratio with 0 < α < 1 and 0 < β < 1 satisfies the

second-order PSD.

Property 8. The performance measure is consistent with the second-order MSD when the

underlying utility function is everywhere increasing, concave below the reference point, and

convex above the reference point.

For example, the Farinelli-Tibiletti ratio with α > 1 and β > 1 satisfies the second-order

MSD.

2.8

Alternative to the Maximum Principle

Recall that in this section we have employed the method of derivation of a formula for the

investor’s performance measure which is called the “maximum principle”. In short, the implementation of this method is as follows: first we solve the investor’s optimal capital allocation

18

problem and then we show that the investor’s maximum expected utility is as a strictly increasing function of a quantity that can be interpreted as a performance measure. Note that

the maximum principle is based on somewhat strong assumption, namely, that the investor

allocates optimally his wealth. Yet, when the investor’s utility is defined over the risk and

reward measures, and these measures satisfy a few plausible properties, then the maximum

principle might be redundant. That is, we can arrive at a formula for a portfolio performance

measure without the use of the maximum principle. In this case if the investor chooses the

risky portfolio that has the highest performance measure, then this portfolio maximizes the

investor’s utility even if the investor’s capital allocation is not optimal.

In particular, suppose that the investor’s utility function can be represented as

U (x) = V (µ(x), ρ(x)),

where V : R2 → R is continuously differentiable and satisfies

Vµ =

∂V

(µ, ρ) > 0,

∂µ

Vρ =

∂V

(µ, ρ) < 0.

∂ρ

That is, the investor’s utility increases when either the reward increases or the risk decreases.

In addition, we assume that the reward and risk measures exhibit a positive homogeneity

property along the capital allocation line, that is, satisfy Property 3. In this case the capital

allocation line is a straight line in the risk-reward space,

µ(y) =

µ(x)

ρ(y) = π(x)ρ(y).

ρ(x)

Therefore, the investor’s goal is to choose the risky portfolio with the highest slope of the

capital allocation line, and this slope is given by the reward-to-risk ratio π(x) =

µ(x)

ρ(x) .

In

particular, using the risky portfolio with the highest slope of the capital allocation line insures

the highest reward for any arbitrary level of risk. Similarly, the usage of portfolio with the

highest slope of the capital allocation line insures the lowest risk for any arbitrary level of

reward. As an immediate consequence, if the investor uses the risky portfolio with the highest

slope of the capital allocation line, it provides the highest utility for either any arbitrary level

of risk or any arbitrary level of reward.

19

For the sake of illustration, assume the following form of the investor’s utility

U (x) = V (µ(x), ρ(x)) = µα (x) − A ρβ (x),

(15)

where A > 0 is the investor’s coefficient of risk aversion and α > 0, β > 0. This utility satisfies

Vµ > 0 and Vρ < 0. In addition, we assume that the reward and risk measures satisfy Property

3.

Without loss of generality, consider the investor’s choice between two risky portfolios x1

and x2 . Suppose that

π(x1 ) > π(x2 ).

In this case the usage of portfolio x1 in the capital allocation insures the highest utility for any

level of risk. Similarly, the usage of portfolio x1 in the capital allocation insures the highest

utility for any level of reward. More specifically, suppose that the investor wants to attain

some arbitrary level of reward µ∗ . If the investor chooses portfolio x1 , to attain µ∗ he needs

to invest the fraction k1 so that

y1 = k1 (x1 − r) + r,

and

µ∗ = µ(y1 ) = π(x1 )ρ(y1 ).

(16)

On the other hand, if the investor chooses portfolio x2 , to attain µ∗ he needs to invest the

fraction k2 so that

y2 = k2 (x2 − r) + r,

and

µ∗ = µ(y2 ) = π(x2 )ρ(y2 ).

(17)

Now we want to show that in the former case the risk of the investor’s portfolio is less than in

the latter case. To demonstrate this, we solve equation (16) with respect to ρ(y1 )

ρ(y1 ) =

µ∗

.

π(x1 )

20

Similarly, we solve equation (17) with respect to ρ(y2 )

ρ(y2 ) =

µ∗

.

π(x2 )

Then it is easy to see that

ρ(y1 ) < ρ(y2 ),

since π(x1 ) > π(x2 ). Therefore

U (y1 ) = (µ∗ )α − A ρβ (y1 ) > U (y2 ) = (µ∗ )α − A ρβ (y2 ).

Similarly we can show that if the investor wants to limit the risk to some arbitrary level ρ∗

then the usage of portfolio x1 in the capital allocation guarantees higher utility than the usage

of portfolio x2 .

Finally observe that the utility functions studied in Subsections 2.4 and 2.5 can be considered within the framework presented in this subsection. For example, the piecewise power

p

p

utility function is equivalent to (15) when ρ(x) = β LP Mβ (x, r), µ(x) = α U P Mα (x, r),

and A = λ. In contrast, the utility-based performance measures, derived in Zakamouline and

Koekebakker (2009b) using the maximum principle, cannot be considered within the alternative framework presented in this subsection.

3

Empirical Study

The goal of this section is to demonstrate the effects of the loss aversion in the sense of

K¨obberling and Wakker (2005) when the investor exhibits first-order risk aversion in the sense

of Segal and Spivak (1990). In particular, we want to demonstrate that investors with different

degrees of loss aversion (generally) rank differently risky portfolios. Similarly, if an investor is

supposed to construct an optimal portfolio of several risky assets, then the composition of the

optimal risky portfolio generally depends on the investor’s loss aversion. In our demonstration

we employ the following performance measure

Zλ2 (x) =

E[x] − r − (λ − 1)LP M1 (x, r)

p

.

LP M2 (x, r)

21

(18)

Recall that when the investor is neutral to losses, λ = 1, then this measure reduces to the

Sortino ratio which is probably the second most popular performance measure9 after the

Sharpe ratio. Furthermore, when λ → 0, this measure reduces to the Upside potential ratio (or Farinelli-Tibiletti12 ratio). That is, the Upside potential ratio can be considered as a

performance measure of an investor which is extremely loss seeking (at least in the neighborhood of the reference point).

The problem of choosing the best risky portfolio can be formulated either as the maximization of the investor’s expected utility or, alternatively, as the maximization of the investor’s

performance measure. In the latter case the investor’s objective

max

E[x] − r − (λ − 1)LP M1 (x, r)

U P M1 (x, r)

LP M1 (x, r)

p

= max p

− λp

.

LP M2 (x, r)

LP M2 (x, r)

LP M2 (x, r)

(19)

We denote the following ratio as the Lower Partial Moment Ratio (LPRM)

LP M1 (x, r)

LP M R12 (x, r) = p

.

LP M2 (x, r)

Observe that the LPMR represents the expected loss normalized by the corresponding risk

measure (such that this ratio is the same for all portfolios lying along the capital allocation

line). With this notation, the investor’s objective (19) can be interpreted as a double objective:

(1) maximization of F T12 (x, r) and (2) minimization of LP M R12 . Observe that the second

objective is λ times more important than the first one. That is, when the investor’s loss

aversion increases, the minimization of LP M R12 becomes more and more important10 than

the maximization of F T12 (x, r). Consequently, the investors with rather high loss aversion

prefers risky assets with low LP M R12 .

We perform our empirical study using the hedge fund database of the Hennessee Hedge Fund

Indexes (see www.hennesseegroup.com). The Hennessee Hedge Fund Indexes are calculated

from performance data reported to the Hennessee Group by a diversified group of over 1,000

9

Here, in case the investor needs to choose a single risky portfolio from the universe of mutually exclusive

risky portfolios.

10

Note that when the loss aversion coefficient is not quite large, then to minimize LP M R12 onep

needs mainly

to minimize the expected loss (with respect to the benchmark) LP M1 (x, r), because increasing LP M2 (x, r)

is in conflict with the first objective since it decreases

F T12 (x, r). Yet, if the loss aversion is quite large, then to

p

minimize LP M R12 one can try to maximize LP M2 (x, r). Putting it into words, a highly loss averse investor

prefers risky portfolios with large uncertainty in the magnitude of a potential loss given the loss occurs, but

relatively small expected loss.

22

hedge funds. Our sample consists of 24 Hennessee Hedge Fund Indexes which have monthly

data from January 1996 to September 2009. All performance ratios are computed using the

nonparametric estimation method. The rolling 90 day T-bill rate is used as the risk-free rate

of return. Table 1 reports the hedge fund index monthly returns descriptive statistics and the

performance ratios. Table 2 reports the Kendall’s rank correlation coefficients between the

performance ratios.

From Table 1 we can observe how loss aversion influences the ranking of hedge funds. We

would like to draw the readers’ attention to the following most evident and demonstrative

changes in ranking:

• The Technology index is ranked 1st according to the Upside potential ratio. However,

this index has the fourth largest risk of loss as measured by LPMR. A loss neutral investor

downgrade this index by 1 place, whereas a loss averse investor with λ = 1.5 downgrades

this index by 4 places. Obviously, for a more loss averse investor the rank of this index

decreases further.

• The Multiple Arbitrage index is ranked 11th using the Upside potential ratio. Yet, this

index has the lowest risk of loss. The Sortino ratio upgrades this index by 6 places,

whereas the Z ratio upgrades this index by 9 places. If we increase the loss aversion

coefficient a little further, the index will get the highest rank.

• The Merger Arbitrage index is ranked 3rd according to the Upside potential ratio. The

loss risk is the 4rd lowest among all indexes. For either a loss neutral investor or an

investor with relatively low loss aversion the rank of this index becomes the 1st.

• The Arbitrage/Event Driven index has the 3rd lowest risk of loss. A loss averse investor

with λ = 1.5 upgrades this index by 10 places which is the highest upgrade in our sample

of indexes.

• The Asia-Pacific and High Yield indexes have the second and third highest risk of loss

respectively. A loss averse investor with λ = 1.5 downgrades these indexes by 9 and 8

places respectively.

23

24

Mean

0.70

0.74

0.63

0.73

0.72

0.48

0.69

0.55

0.73

0.93

0.39

0.74

0.44

0.99

0.81

0.64

0.37

0.67

0.69

0.80

0.79

0.22

1.25

0.73

Std

2.03

2.38

1.50

2.90

3.51

1.64

2.07

3.92

2.09

3.33

1.64

3.58

2.12

2.59

6.96

2.14

1.15

1.11

1.29

2.39

2.05

5.54

4.14

2.66

Skew

-0.69

-0.22

-2.53

-0.42

0.99

-3.28

-2.13

-1.72

-1.68

-1.49

-1.76

0.09

-1.98

-0.29

-0.56

0.26

-2.48

-1.42

-3.01

-0.34

0.31

1.12

0.98

-1.10

Kurt

5.91

5.12

13.21

6.57

6.15

20.65

10.43

12.89

8.54

12.00

11.18

4.49

10.18

4.89

5.99

4.94

15.59

7.13

16.52

3.42

6.18

9.53

5.96

6.22

FT12

0.727 ( 9)

0.759 ( 7)

0.618 (16)

0.682 (12)

0.755 ( 8)

0.453 (22)

0.600 (17)

0.478 (20)

0.652 (15)

0.652 (14)

0.428 (24)

0.693 (10)

0.468 (21)

0.885 ( 2)

0.554 (18)

0.762 ( 6)

0.440 (23)

0.845 ( 3)

0.686 (11)

0.820 ( 5)

0.843 ( 4)

0.527 (19)

0.957 ( 1)

0.658 (13)

LPMR

0.422 (12)

0.461 ( 7)

0.319 (22)

0.451 ( 8)

0.538 ( 2)

0.315 (23)

0.346 (20)

0.392 (14)

0.362 (17)

0.375 (16)

0.350 (19)

0.496 ( 3)

0.381 (15)

0.436 (10)

0.450 ( 9)

0.490 ( 5)

0.357 (18)

0.328 (21)

0.274 (24)

0.480 ( 6)

0.416 (13)

0.546 ( 1)

0.493 ( 4)

0.423 (11)

Sortino

0.305 ( 7)

0.298 ( 9)

0.299 ( 8)

0.231 (15)

0.218 (16)

0.138 (18)

0.254 (13)

0.086 (21)

0.289 (10)

0.277 (11)

0.078 (23)

0.196 (17)

0.087 (20)

0.448 ( 3)

0.104 (19)

0.272 (12)

0.083 (22)

0.517 ( 1)

0.412 ( 5)

0.340 ( 6)

0.426 ( 4)

-0.019 (24)

0.464 ( 2)

0.235 (14)

Zλ=1.5

0.077 ( 9)

0.055 (12)

0.113 ( 6)

0.005 (15)

-0.042 (17)

-0.016 (16)

0.066 (11)

-0.090 (22)

0.088 ( 7)

0.073 (10)

-0.079 (20)

-0.042 (18)

-0.085 (21)

0.188 ( 3)

-0.099 (23)

0.022 (13)

-0.078 (19)

0.288 ( 1)

0.224 ( 2)

0.082 ( 8)

0.178 ( 4)

-0.239 (24)

0.177 ( 5)

0.019 (14)

Table 1: The table reports the hedge fund index monthly returns descriptive statistics and the performance ratios. The values of means and

standard deviations are given in percents. Numbers in the brackets show the rank of an index using a particular performance ratio. When it

comes to the LPMR, the higher the rank, the higher the risk. For the other three ratios, the higher the rank, the higher the attractiveness

of the index.

Hedge Fund Index

Hennessee Hedge Fund

Long/Short Equity

Arbitrage/Event Driven

Global/Macro

Asia-Pacific

Convertible Arbitrage

Distressed

Emerging Markets

Event Driven

Financial Equities

Fixed Income

Growth

High Yield

International

Latin America

Macro

Market Neutral

Merger Arbitrage

Multiple Arbitrage

Opportunistic

Pipes/Private Financing

Short Biased

Technology

Value

FT12

LPMR

Sortino

Zλ=1.5

FT12

1.00

0.30

0.64

0.48

LPMR

Sortino

Zλ=1.5

1.00

-0.05

-0.22

1.00

0.83

1.00

Table 2: Kendall’s rank correlations between different ratios in the ranking of the hedge fund

indexes.

4

Summary and Conclusions

In this paper we considered the loss averse investor with a specific, but still quite general, utility function that has a reference point. Under some concrete assumptions about the form of

the investor’s utility we derived closed-form solutions for the investor’s portfolio performance

measure. The first contribution of this paper was to provide a sound theoretical foundation

for all known performance measures based on partial moments of distribution. That is, we

established an explicit link between a performance measure based on partial moments of distribution and the shape of the investor’s utility function. Using this link, it becomes possible

to say something concrete about the risk preferences of an investor that uses a specific performance measure. In addition, since the use of several performance measures (for example, the

Kappa and Farinelli-Tibiletti ratios) requires designating the orders of partial moments, this

link facilitates greatly this task. This is because the orders of partial moments are directly

related to the shape of the investor’s utility function.

The second contribution of this paper was to investigate the role of the investor’s loss aversion in portfolio performance evaluation. We found that if the investor exhibits loss aversion

in the sense of Kahneman and Tversky (1979) and Wakker and Tversky (1993), then the investor’s degree of loss aversion plays no role in performance measurement and, thus, does not

influence the decision “invest - do not invest”. This investor’s degree of loss aversion influences

the decision of how much to invest only. In contrast, we found that if the investor exhibits loss

aversion in the sense of K¨obberling and Wakker (2005) (when the investor exhibits first-order

risk aversion in the sense of Segal and Spivak (1990)), then the investor’s degree of loss aversion

plays an important and explicit role in performance measurement. Moreover, this investor’s

degree of loss aversion influences the decision “invest - do not invest”. Namely, for this investor

it is optimal to invest only when the risk premium exceeds some particular threshold which

25

depends on the investor’s coefficient of loss aversion. Otherwise, when the risk premium is

low, this investor avoids the risky portfolio altogether and allocates all wealth in the risk-free

asset. Using the empirical study we demonstrated that investors with different degrees of loss

aversion generally rank differently risky portfolios. We also suggested a measure that evaluates

the risk of loss.

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