CEFAGE-UE Working Paper
2014/06
Portfolio Selection Optimization under Cumulative Prospect
Theory – a parameter sensibility analysis
Luis Alberto Godinho Coelho
CEFAGE-UE and Management Department, Évora University
CEFAGE-UE, Universidade de Évora, Palácio do Vimioso, Lg. Marquês de Marialva, 8, 7000-809 Évora, Portugal
Telf: +351 266 706 581 - E-mail: [email protected] - Web: www.cefage.uevora.pt
Portfolio Selection Optimization under Cumulative Prospect
Theory – a parameter sensibility analysis
Luís Alberto Godinho Coelho
CEFEGE-UE and Management Department, Évora University
Email: [email protected]
ABSTRACT
The Cumulative Prospect Theory (CPT) is one of the most popular theories for evaluating the behavior
of decision makers in the context of risk and uncertainty. This theory emerged as a generalization of
the Expected Utility Theory (EUT) and being a relatively recent theory, its application has been
somewhat reduced, especially when linked to optimization models. This paper intends to analyze the
behavior of CPT, with a power value function and a two-parameter probability weighting function, as an
objective function of a portfolio selection model. The parameterization of the objective function
parameters allows us to analyze the composition of portfolios such as loss aversion, risk aversion in gains
and risk preference in the range of losses. The results suggest that loss aversion and risk aversion in gains
lead to the choice of portfolios with lower profitability and variability and that the risk preference in
losses leads to the choice of portfolios with higher profitability and variability. The results are also
compared with those obtained with EUT, and allow us to conclude that CPT leads to more diversified
solutions which are therefore more easily adjusted to the investors’ behavioral profile.
Key words: Cumulative Prospect Theory, Portfolio Selection, Loss Aversion, Risk
Aversion, Expected Utility
JEL classification: C61, D81, G11
1. Introduction
Expected Utility Theory (EUT) has been the most important theory used in economics to
model choice under risk and uncertainty. Since its creation (von Neumann and
Morgenstern, 1944) this theory has been subject to vast theoretical and empirical
development that brought some violations of axioms and motivated the appearance of
alternative modeling approaches. These new theories, called non-expected utility theories,
developed new approaches to respond to the continuous violations of EUT and to improve
the decision making process.
The most important theory developed to describe the decision making process is the
Cumulative Prospect Theory (CPT) (Kanheman and Tversky, 1979; Tversky and
Kanheman, 1992). This descriptive theory was the first to incorporate irrational behavior in
an empirical realistic manner, while at the same time being systematic and tractable
(Wakker, 2010). This theory takes into account the fact that decision makers evaluate
decisions in terms of gains and losses with respect to a reference point and that they do not
weigh probabilities linearly. The CPT has two key elements: i) a concave function for gains
and a convex function for losses and a steeper function for losses than for gains; ii) a nonlinear transformation of the scale of probabilities that overweigh the low probabilities and
underweigh the moderate and high probabilities. This theory transforms probabilities in
decision weights through a cumulative probability function, which is applied to gains and
losses separately. CPT has gained importance in recent years and papers published with
empirical applications of this theory have greatly increased.
The financial area, specifically portfolio selection (Markowitz, 1952) offers a very broad
field for the application of CPT. Portfolio selection was developed based on the meanvariance model (MVM) aiming to find the optimal frontier between risk and return. Since
then, many developments have followed, and recently several authors have adopted CPT to
the portfolio selection model and compared the predictive power of CPT with EUT and
MVM. Levy and Levy (2004) compared CPT with MVM, and concluded that the efficient
sets almost coincide, suggesting that the MVM model can be used to construct a CPT
efficient set. Gurevich, et al (2009) performed an empirical study which estimated the value
function and probability weighting function using US stock option data. Comparison of the
results found in that study with laboratorial results allowed them to conclude that the
estimated functions are close to linearity and loss aversion is less pronounced. At the same
time, Kliger and Levy (2009) compared the three most popular theories of choice under risk
2
(EUT, CPT and Rand Dependent Expected Utility (Quiggin, 1982)) based on the call
options of the SP500 index. Non-linear probability weighting, loss aversion and
diminishing marginal sensitivity are clearly manifested by the data. De Giorgi and Hens
(2006) suggest that an exponential function should be used to implement CPT in portfolio
selection. Bernard and Ghossoub (2010) and Pirvu and Schulze (2012) find the optimal
portfolio for a static model of one period for a decision maker behaving according to CPT.
All those studies are based on estimation of the parameters of the value function and
probability weighting function and on comparison of the results obtained with the results of
laboratory experiments. None of them use a classical optimization model of portfolio
selection with decision maker preferences described by CPT. Applications of CPT in
optimization models have been limited, with the exception of Coelho, et al (2012), who by
including CPT in an objective function of an optimization model, predicts the behavior of
Portuguese farmers regarding the Common Agricultural Policy. Studying the behavior of
CPT parameters in a discrete optimization model of portfolio selection is the goal of this
study.
The remainder of the paper is organized as follows. Section 2 presents the theoretical
foundations of CPT and the optimization model used in this study. The third section shows
the dataset used in the optimization procedure. Section 4 starts with analysis of the results
obtained and the sensibility analysis of the CPT parameters, and ends with the comparison
with EUT. The paper ends with the conclusions.
2. Theoretical foundations
First, the theoretical foundations of CPT are presented. The second part develops the
optimization model used to find the best portfolio and test CPT.
2.1. Cumulative Prospect Theory
Let S denote the finite set of states of nature and X denote the set of possible results. It is
assumed that X includes a neutral result (0) and that the remaining elements of X are gains
or losses. The lottery y is a function from S to X, which allocates to each state of nature s∈S
a consequence y(s) = x, with x∈X. The lottery y is represented by a sequence of pairs (xi,
Ai), which originates xi if the event Ai happens (Ai is a subset of S). The positive part of y,
3
represented by y+, is obtained by y+(s) = y(s) if y(s) > 0, and y+(s) = 0 if y(s) ≤ 0. The
negative part of y, represented by y -is defined in a similar way (Tversky and Kahneman,
1992).
The objective function is defined by the value of the lottery (V(y)) and is equal to the
sum of its positive and negative components, with –m ≤i≤n:
!
! ! =! !
!
+! !
!
!
!
=
ℎ ! ! ! !!
ℎ ! ! !! +
!!!!
!!!
Where hi are the decision weights and v is the value function.
The value function has the following characteristics: (i) it is defined for changes with
respect to the reference point; (ii) it is concave for gains and convex for losses; (iii) it has a
steeper slope for losses than for gains.
To represent the value function, the classical power function is used, as in the original
paper by Tversky and Kahneman (1992):
! !! =
!!! !" 0 ≤ ! ≤ !
−! −!! ! !" − ! ≤ ! < 0
where λ, α and β are value function parameters.
The parameterλ, with λ≥1, is associated with the degree of loss aversion. The parameter
α, with 0<α≤1, represents risk aversion in gains and the parameterβ, with 0<β≤1,
represents risk preference in losses. This function constitutes the necessary and sufficient
conditions for CPT and, according to Stott (2006), the available tests seem to favor this
function over logarithmic and exponential functions. This function is presented in the
following figure (with α=β=0.5 and λ=3):
4
Figure 1 –Value Function
v
x
The decision weights hi– and hi+are defined in a cumulative way through the following
expressions:
!
ℎ!!
=!
!
!! andℎ!!
=!
!
!
!! − !
!
!
!!!
!
!
ℎ!!
=
! ! !!! andℎ!!
!! for 0 ≤ ! ≤ ! − 1
= !!
!!!
!! − ! !
!!
!! for 1 − ! ≤ ! ≤ 0
!!
where! ! and ! ! are the probability weighting function that has the ability to capture the
agent’s probability distortions in psychological terms. Because the differences between the
parameters of positive and negative functions are very small (Tversky and Kahneman
(1992), Gonzalez and Wu (1999), Abdellaoui (2000)), this paper does not distinguish
between the positive and negative function. The function f and is strictly increasing in the
interval [0, 1], with f (0) = 0 and f(1) = 1. This paper uses the following two parameter
function, conceived by Goldstein and Einhorn (1987) and analyzed by Gonzalez and Wu
(1999):
!! ! = !! ! =
!! !
!! ! + 1 − !
!
where γ and δ are probability weighting function parameters. The function, with γ =0.44
and δ= 0.77, is presented in the following figure (with γ=0.44 and δ=0.77):
5
Figure 2– Probability Weighting Function
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
According to Gonzalez and Wu (1999), this function is able to capture two types of
behavior: (i) diminishing sensitivity; (ii) attractiveness. According to the diminishing
sensitivity concept, an increase in p has a greater effect on f if p is closer to its extreme
points than if p has an intermediate value. This behavior is captured by parameter γ. The
probability weighting function can be below or above the identity line and it can cross the
identity line at any point. The higher the function, the greater the attractiveness of the
prospect for the decision maker. This behavior is reflected in parameter δ.
2.2. Portfolio selection model
This section presents the model. This model is based on the Mean-Variance optimization
model (MVM), which provides a mechanism for selecting efficient portfolios based on
profitability and risk. Usually, based on efficient portfolios found in MVM, the decision
maker chooses the optimal portfolio, according to his preferences. The model with CPT
does not work this way. This model incorporates the decision maker’s preference in the
objective function, so does not generate an efficient frontier, but an optimal portfolio. Levy
and Levy (2004) suggest the use of mean-variance efficient solutions to generate efficient
solutions of CPT.
The objective function of this model is Cumulative Prospect Theory. This function is
subject to four types of constraints: the first constraint requires that the total investment
equals 100 percent, the second type of constraint calculates profitability and will be as
many as the states of nature considered, and the third and fourth constraints calculate the
expected profitability and variance of the portfolio. The simplified model is:
6
!
!
!
max ! ! =
ℎ ! ! ! !!
ℎ ! ! !! +
!!!!
!!!
!. !.
!
!! = 100
!!!
!
!! !! − !! = 0 !"#ℎ ! = ! + !
!!!
!
!! !! − ! = 0
!!!
!
!! ! !! − Z = 0
!!!
!! ≥ 0 and!! free
where aj are the risk prospects, rj the profitability of the prospects, xi the profitability for
each state of nature, µj the expected profitability, σj the standard deviation, P the portfolio’s
expected profitability and Z the portfolio variance. The third and fourth constraints do not
affect the solution of the model, but only compute the expected profitability and variance of
the optimal portfolio.
3. Data
To test the model ten prospects were created, intended to reflect several levels of risk.
The first prospect (a1) is risk free and the last (a10) is a high risk prospect. These ten
prospects were grouped into four broad categories reflecting in a generic way the degree of
risk: risk-free, low risk, moderate risk and high risk. The following table presents the
features of the ten options:
Table 1 - Prospects
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
Expected profitability (µ)
0,040
0,040
0,045
0,045
0,050
0,050
0,050
0,075
0,075
0,075
Standard deviation (σ)
0,000
0,005
0,010
0,025
0,025
0,050
0,075
0,075
0,100
0,125
Risk Classification
Risk-free
Low risk
7
Moderate risk
High risk
These ten risk prospects are sufficient in number and risk characteristics to conclude on
the suitability of CPT for portfolio selection in an optimization context. The highest
profitability is associated with high risk and the lowest profitability is associated with riskfree option. Based on the number of states of nature, two models were defined: one model
with five states of nature and the other with nine states of nature. With five states of nature,
the profitability of each state was defined as: ! − 2!, ! − !, !, ! + !, ! + 2!. With nine
states of nature the profitability is: ! − 2!, ! − 1.5!, ! − !, ! − 0.5!, !, ! + 0.5!, ! + !,
! + 1.5!, ! + 2!. That is, these values identify intermediate, very good and very bad states
of nature.
In addition to the profitability and risk of each state of nature, another important input of
the model is the probabilities associated with the states of nature. To illustrate this question,
the paper defines three types of probability distribution that can portray extreme situations
the decision maker might be confronted with: a symmetric distribution, a negative skew
distribution and a positive skew distribution. The probabilities affected to the model with
five states of nature and the model with nine states of nature are presented in the following
tables:
Table 2 – Probabilities for 5 states of nature
SN 1
SN 2
SN 3
SN 4
SN 5
Symmetric
0.05
0.15
0.60
0.15
0.05
Negative skew
0.05
0.05
0.15
0.15
0.60
Positive skew
0.60
0.15
0.15
0.05
0.05
Table 3 – Probabilities for 9 states of nature
SN 1
SN 2
SN 3
SN 4
SN 5
SN6
SN7
SN8
SN9
Symmetric
0.025
0.075
0.100
0.150
0.300
0.150
0.100
0.075
0.025
Negative skew
0.290
0.175
0.150
0.125
0.100
0.075
0.050
0.025
0.010
Positive skew
0.010
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.290
The number of states of nature, profitability and the type of probability distribution
define the context of the decision. To evaluate CPT in the context of portfolio selection, it is
necessary to define values for the parameters of the value function and probability
weighting function. The value function parameters define the degree of risk aversion in
gains, the degree of risk preference in losses and the degree of loss aversion. The
probability weighting function parameters define the degree of probability weighting. The
following table presents the values tested for the CPT parameters.
8
Table 4. Values of the CPT parameters
Parameters
Values
λ
1.0
No loss aversion
3.0
High loss aversion
1.0
Risk neutral
0.5
Risk aversion (risk lover)
0.1
High risk aversion (high risk lover)
1.0
Linear Probability Weighting
0.44
Non-Linear Probability Weighting
1.0
Linear Probability Weighting
0.77
Non-Linear Probability Weighting
α, β
γ
δ
Meaning
(.) negative part of the value function
The loss aversion parameter (λ) has two values: 1 to represent the absence of this
behavior and 3 to represent a high degree of loss aversion (Tversky and Kahneman (1992)
find the value 2.25). For the probability weighting function parameters, two situations were
selected: one with linear probability weighting and other with non-linear probability
weighting by the median values obtained by Gonzalez and Wu (1999).
4. Model results analysis
4.1. Parameters of Cumulative Prospect Theory
To answer the objectives proposed in this paper, the optimization model of portfolio
selection with Cumulative Prospect Theory was solved by varying each of the parameters of
CPT, number of states of nature and types of probability distribution. The average results
for 5 and 9 states of nature, type of probability distribution and probability weighting
function parameters are presented in the following table:
Table 5 – Analysis by number of states of nature
Risk-free
(*)
Type of probability distribution:
Symmetric
SN=5
12
SN=9
12
Positive skew SN=5
5
SN=9
3
Negative skew SN=5
54
SN=9
51
Probability weighting function parameters
23
SN= 5:
γ=δ=1.0
24
γ=0.44; δ=0.77
21
SN=9:
γ=δ=1.0
23
γ=0.44; δ=0.77
Low risk
(*)
Moderate risk
(*)
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
32
33
25
23
15
17
16
21
9
17
8
14
40
34
61
57
23
18
6.07
6.04
6.57
6.68
5.12
5.14
0.71
0.64
1.01
1.98
0.40
0.35
20
28
22
26
15
7
21
15
42
41
36
36
6.06
5.79
6.07
5.86
0.73
0.68
0.68
0.64
(*) Values as a percentage of the portfolio
9
The portfolios chosen, defined according to probability distributions, are not very
different when working with 5 and 9 states of nature. Put another way, despite the great
difference between the probabilities presented for 5 and 9 states of nature, the difference
between the portfolios is negligible in risk-free/less risk prospects but there is investment
transference from prospects with high risk to prospects with moderate risk. Analyzing the
impact of probability distortion by varying the probability weighting function parameters,
as in the previous analysis, the difference between portfolios is found to be minimal, but
there is a small change between investments in prospects with low risk and moderate risk.
With respect to loss aversion (parameter λ), the summary of results is presented in Table
6, which presents the average results for each type of probability distribution, the global
average results and the average results for the value function parameters for symmetric
distribution.
Table 6–Loss Aversion (parameterλ)
Risk-free
Low risk
(*)
(*)
Type of probability distribution:
9
31
Symmetric:
λ=1
15
35
λ=3
4
18
Positive skew: λ=1
4
29
λ=3
47
14
Negative skew: λ=1
58
17
λ=3
Risk parameters with symmetric probability distribution
0
0
α=1.0; β=1.0:
λ=1
19
14
λ=3
0
0
α=1.0; β=0.5:
λ=1
0
0
λ=3
0
0
α=1.0; β=0.1:
λ=1
0
0
λ=3
0
77
α=0.5; β=1.0:
λ=1
0
87
λ=3
3
14
α=0.5; β=0.5:
λ=1
30
53
λ=3
0
0
α=0.5; β=0.1:
λ=1
0
0
λ=3
12
79
α=0.1; β=1.0:
λ=1
13
83
λ=3
26
64
α=0.1; β=0.5:
λ=1
44
39
λ=3
36
43
α=0.1; β=0.1:
λ=1
32
35
λ=3
20
21
Global Mean:
λ=1
26
27
λ=3
Moderate risk
(*)
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
19
19
13
13
13
10
41
31
65
54
26
15
6.27
5.87
6.79
6.47
5.33
4.94
0.76
0.59
1.08
0.91
0.47
0.28
26
66
0
0
0
0
23
13
83
17
1
25
9
4
10
15
21
27
15
14
74
1
100
100
100
100
0
0
0
0
99
75
0
0
0
2
0
6
44
33
7.50
6.44
7.50
7.50
7.50
7.50
5.18
4.90
6.98
4.82
7.50
7.49
4.69
4.58
4.57
4.62
5.00
4.95
6.13
5.76
1.3
0.39
1.56
1.56
1.56
1.56
0.14
0.08
0.47
0.10
1.56
1.31
0.06
0.03
0.05
0.10
0.13
0.19
0.77
0.59
(*) Values as a percentage of the portfolio
It is found that when λ changes from 1 to 3, portfolios with lower associated risk are
selected (there is clearly an investment transfer from prospects with high/moderate risk to
10
prospects with risk-free/lower risk), because the prospects with higher profitability have
also the highest level of loss. The only exception is the situation of high-risk aversion (α =
0.1), where this effect eliminates the loss aversion effect, i.e., when risk aversion is high
loss aversion has little importance. Another interesting fact is when the parameter β (risk
preference in losses) is smaller than α, it leads to the choice of portfolios with higher risk,
despite loss aversion tending towards more risk-free/low risk portfolios.
The analysis of the risk aversion parameter (α) is shown below:
Table 7 – Risk aversion in gains (parameterα)
Risk-free
Low risk
(*)
(*)
Type of probability distribution:
3
3
Symmetric:
α=1.0
9
44
α=0.5
29
54
α=0.1
0
0
Positive skew: α=1.0
2
20
α=0.5
10
52
α=0.1
39
2
Negative skew: α=1.0
52
18
α=0.5
66
28
α=0.1
Risk parameters with symmetric probability distribution
10
7
β=1.0: α=1.0
0
82
α=0.5
13
81
α=0.1
0
0
β=0.5:
α=1.0
27
49
α=0.5
40
43
α=0.1
0
0
β=0.1:
α=1.0
0
0
α=0.5
34
39
α=0.1
14
1
Global Mean
α=1.0
22
27
α=0.5
35
44
α=0.1
(*) Values as percentage of the portfolio
Moderate risk
(*)
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
15
18
15
0
19
19
17
12
6
79
29
2
100
59
19
42
18
0
7.32
5.87
4.75
7.50
6.85
5.53
6.07
5.04
4.28
1.32
0.56
0.10
1.56
1.03
0.39
0.75
0.34
0.04
45
18
6
0
24
15
0
13
24
11
16
14
38
0
0
100
0
2
100
87
3
74
35
7
6.97
5.04
4.64
7.50
5.08
4.63
7.50
7.49
4.98
6.97
5.92
4.85
0.84
0.11
0.05
1.56
0.14
0.10
1.56
1.43
0.16
1.21
0.64
0.18
As risk aversion increases (α decreases), lower risk portfolios are selected, regardless of
the type of probability distribution. It is also seems clear (Tables 7 and 8) that as the risk
preference for loss increases (β decreases), more profitable portfolios are selected but with a
higher degree of risk.
11
Table 8 – Risk preference in losses (parameterβ)
Risk-free
(*)
Type of probability distribution:
8
Symmetric:
β=1.0
21
β=0.5
11
β=0.1
0
Positive skew: β=1.0
10
β=0.5
2
β=0.1
100
Negative skew: β=1.0
37
β=0.5
20
β=0.1
36
Global Mean
β=1.0
23
β=0.5
11
β=0.1
Low risk
(*)
Moderate risk
(*)
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
57
33
13
43
23
5
0
31
16
34
29
11
22
12
16
23
7
9
0
22
12
15
14
13
13
34
60
34
60
84
0
10
52
15
34
65
5.50
5.73
6.65
6.21
6.38
7.28
4.00
5.19
6.20
5.24
5.77
6.71
0.32
0.59
1.01
0.66
0.96
1.36
0.00
0.27
0.86
0.33
0.61
1.08
(*) Values as percentage of the portfolio
That is, CPT seems to have two antagonistic effects, in which the loss aversion and risk
aversion in gains leads to the choice of portfolios with lower risk and lower profitability,
while on the other hand, since the risk preference in losses tends to avoid losses, it leads to
riskier but also more profitable portfolios.
For the Probability Weighting Function two alternatives were defined: linear probability
weighting (γ = δ = 1) or non-linear probability weighting (overweight of low probabilities
and underweight of high probabilities). According to Gonzalez and Wu (1999), who worked
this function, average values for γ and δ were 0.44 and 0.77, respectively.
12
Table 9 – Probability Weighting Function (parameters γ and δ)
Risk-free
(*)
Type of probability distribution:
9
Symmetric: γ=δ=1.0
15
γ=0.44; δ=0.77
2
Positive skew: γ=δ=1.0
5
γ=0.44; δ=0.77
55
Negative skew:γ=δ=1.0
50
γ=0.44; δ=0.77
Loss aversion parameter
19
λ=1:
γ=δ=1.0
21
γ=0.44; δ=0.77
25
λ=3:
γ=δ=1.0
26
γ=0.44; δ=0.77
Risk aversion in gains parameter
15
α=1.0:
γ=δ=1.0
13
γ=0.44; δ=0.77
21
α=0.5:
γ=δ=1.0
19
γ=0.44; δ=0.77
30
α=0.1:
γ=δ=1.0
38
γ=0.44; δ=0.77
Risk preference in losses parameter
33
β=1.0: γ=δ=1.0
38
γ=0.44; δ=0.77
22
β=0.5:γ=δ=1.0
21
γ=0.44; δ=0.77
11
β=0.1: γ=δ=1.0
11
γ=0.44; δ=0.77
22
Global Mean: γ=δ=1.0
24
γ=0.44; δ=0.77
(*) Values as percentage of the portfolio
Low risk
(*)
Moderate risk
(*)
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
29
36
16
32
18
13
25
12
17
10
12
11
37
37
65
53
15
26
6.31
5.83
6.91
6.34
4.96
5.30
0.71
0.64
1.11
0.89
0.29
0.46
18
24
24
30
19
10
17
12
44
45
34
32
6.25
6.01
5.87
5.64
0.78
0.76
0.63
0.56
2
2
18
33
44
47
11
10
27
12
16
11
72
75
34
36
10
4
6.94
6.99
6.19
5.84
5.06
4.64
1.19
1.23
0.68
0.64
0.24
0.11
27
39
25
31
11
12
21
27
23
8
18
14
13
10
18
11
17
15
35
34
65
67
39
38
5.53
4.98
5.92
5.79
6.74
6.69
6.06
5.82
0.39
0.27
0.64
0.61
1.08
1.10
0.70
0.66
When non-linear probability weighting is introduced in the model, the average
profitability decreases, except the probability distribution with negative skew, in which the
probability of the more favorable state of nature is extremely low and then is overweight,
leading to selection of portfolios with a higher degree of risk. Another conclusion is that
when non-linear probability weighting is introduced, in general, portfolio composition does
not change much (the biggest change is switching between prospects with low and
moderate risk), unless risk aversion is high.
4.2. Comparison with Expected Utility Theory
Since Expected Utility Theory (EUT) has been considered, the decision theory in the risk
context par excellence, comparison of the results obtained with this model with those
obtained with Cumulative Prospect Theory is in the best interests of this work. The model is
the same in terms of constraints, but for the objective function the following power function
is used:
13
!
max ! ! =
!! 17.5 + !!
!
!!!!
where pi are the states of nature probabilities, xi the profitability for each state of nature and
α is the risk aversion parameter.
To be able to make this comparison, the model with EUT was solved for 5 and 9 states
of nature, for the three types of probability distribution and α = 1.0, 0.5 and 0.1. The results
obtained are presented in the following table:
Table 10 – Comparison between EUT and CPT
Risk-free
Low risk
Moderate risk
(*)
(*)
(*)
Type of probability distribution:
Symmetric:
SN=5
0
0
67
SN=9
0
0
55
Positive skew SN=5
0
0
0
SN=9
0
0
0
Negative skew SN=5
100
0
0
SN=9
86
14
0
Degree of risk aversion with symmetric distribution and EUT
0
0
0
α=1.0
0
0
100
α=0.5
0
0
83
α=0.1
Degree of risk aversion with symmetric distribution and CPT (**)
0
0
2
α=1.0
0
56
44
α=0.5
0
83
17
α=0.1
High risk
(*)
Portfolio
Profitability
Portfolio
Variance
33
45
100
100
0
0
7.50
7.46
7.50
7.50
4.00
4.07
0.90
0.94
1.56
1.56
0.00
0.00
100
0
17
7.50
7.50
7.44
1.56
1.12
0.63
98
0
0
7.50
5.81
5.01
1.54
0.25
0.10
(*) Values as percentage of the portfolio; (**) With λ = β = γ = δ=1
As the EUT function has only one parameter (α), it tends to choose extreme portfolios,
for probability distributions with negative skew choosing portfolios with low risk and for
probability distributions with positive skew or symmetric ones choosing high risk
portfolios. When varying the parameter of risk aversion, compared to CPT, portfolios with
greater profitability and risk are chosen.
5. Conclusions
This paper intends to study the behavior of Cumulative Prospect Theory parameters in
an optimization model of portfolio selection. The study develops a model with ten prospects
with different degrees of profitability and risk. The model was solved for two possibilities
of states of nature, three types of probability distribution and several values of CPT
parameters. The sensibility analysis of the value function parameters, which represent the
concepts of loss aversion, risk aversion in gains and risk preference in losses, and
14
probability weighting function, leads to the conclusion that they interfere in a different way
in the model.
Increased loss aversion and risk aversion in gains leads to the choice of low profitability
and risk-free/low risk portfolios. On the other hand, increased risk preference in losses leads
to the choice of high profitability and high/moderate risk portfolios. This antagonistic effect
allows us to conclude that when the risk preference in losses is higher than the risk aversion
in gains (β<α), the model tends to choose high-risk portfolio solutions. In contrast, when
the risk aversion in gains is high the effect of loss aversion is very small. The average
values chosen for the probability weighting function parameters do not lead to different
portfolios, except when the probabilities are very low (tending to be over weighted). The
comparison of CPT and EUT lets us conclude that EUT tends towards less diversification
and high-risk portfolios.
This paper has the disadvantage of not using real data. This choice was deliberate, since
it was intended to study the behavior of CPT parameters and real data, not being so well
behaved, would not allow such a precise analysis. This study proves that CPT responds
quite well as an objective function of portfolio selection optimization models. It gives the
researcher the opportunity to model different characteristics of decision makers. Analysis of
the behavior of other value and probability weighting functions, despite the work already
done, is an interesting field of research.
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