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國立高雄大學統計學研究所
碩士論文
在累積前景理論下的最佳投資策略
The Optimal Solution under Cumulative Prospect Theory
研究生：王立森 撰
指導教授：吳慶堂
中華民國九十六年七月
The Optimal Solution under
Cumulative Prospect Theory
by
Li-Sen Wang
Advisor
Ching-Tang Wu
Institute of Statistics,
National University of Kaohsiung
Kaohsiung, Taiwan 811 R.O.C.
July 2007
Contents
Z`Š
zZ`Š
ii
iii
Chapter 1. Introduction
1
Chapter 2. Theoretical Framework
5
2.1. Preference order and expected utility theory
5
2.2. Violations of the expected utility theory
7
2.3. Prospect theory
9
2.4. Cumulative Prospect theory
10
Chapter 3. The Optimal Solution in Discrete Type
13
3.1. Connection between EUT and CPT
13
3.2. The optimal solution
16
3.3. Example
18
Chapter 4. Conclusion and Future Work
23
Bibliography
25
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ii
The Optimal Solution under
Cumulative Prospect Theory
Advisor: Dr. Ching-Tang Wu
Department of Applied Mathematics
National Chiao Tung University
Student: Li-Sen Wang
Institute of Statistics
National University of Kaohsiung
ABSTRACT
A fundamental problem in the financial mathematics is to find out an optimal strategy
for the investment. And the most used setting to discuss the optimal investment is the
expected utility optimization. But we find that there are some shortages in expected utility theory. In order to overcome these shortages, we use the cumulative prospect theory
to replace the expected utility theory. In this thesis, we make a connection between the
expected utility theory and the cumulative prospect theory. And we consider an optimal
problem for a discrete type under the cumulative prospect theory.
Keywords: cumulative prospect theory, expected utility theory, optimal strategy.
iii
CHAPTER 1
Introduction
A fundamental problem in the financial mathematics is to find out an optimal strategy
for the investment. Such problem has been investigated in a large amount of literatures,
e.g., Lakner (1993), Cvitanic and Karatzas (1992), Boyd [3] and Sundaram [15]. The
most used setting to discuss the optimal investment is the expected utility optimization. Karatzas, Lehoczky, Shreve and Xu (1991) developed a method to get the optimal
investment in a incomplete market under expected utility theory.
Nevertheless, expected utility theory doesn’t characterize the behaviors of the consumers accurately. Kahneman and Tversky (1979) pointed out that the utility functions
of the consumers are not always concave, in other words, people are not always risk-averse.
They found that the consumers’ choices are different from the prediction of expected utility theory. In fact, people are risk-averse when they face gains, but they turn to be
risk-seeking when they face losses.
In order to correct this shortage of expected utility theory, Kahneman and Tversky
(1979) proposed a new theory called the prospect theory. Under expected utility theory,
the utilities of outcomes are weighted by their probabilities. In prospect theory, they use
value function and decision weights to replace the utility function and the probabilities,
respectively. The value function is an S-shaped function. It is concave at positive part
and convex at negative part. The decision weights are the transform of the probabilities
of expected utility theory. The idea of transformation is to assign more weight for the
outcome that has lower probability and assign less weight for the outcome which has
higher probability. So, we can explain the behaviors of the consumers more accurately
with these notations under the prospect theory.
1
2
1. INTRODUCTION
Tversky and Kahneman (1992) proposed a generalized form of prospect theory. This
new form is called the cumulative prospect theory. Cumulative prospect theory was
proposed because the original prospect theory cannot be extended to a continuous case.
There were some examples that prospect theory cannot explain the violations of stochastic
dominance in Tversky and Kahneman (1986). Also, Mohapi (2004) talked about the
weakness of prospect theory. It said that it can not always satisfy stochastic dominance
and not be readily extended to prospects involving a large number of outcomes in prospect
theory.
Under cumulative prospect theory, we transform the entire cumulative probability
distribution rather than the probability of each outcome separately. And in Wakker
and Tversky (1993), they explained why stochastic dominance is still preserved under
cumulative prospect theory. In expected utility theory, stochastic dominance are usually
concerned since stochastic dominance relations offer an efficient way to compare pairs of
strategies. The definitions of several kinds of stochastic dominance can be seen in Hadar
and Russell (1969) and Rothschild and Stiglitz (1970).
First degree stochastic dominance is what we often concern in expected utility theory. We say that a strategy A dominates another strategy B in the sense of first degree
stochastic dominance if all individuals having increasing and continuous utility functions
in wealth either prefer A to B or are indifferent between A and B. However, there is no
first degree stochastic dominance in cumulative prospect theory. Instead of first degree
of dominance, there appears new dominance relations under the cumulative prospect theory. The relations between stochastic dominance and cumulative prospect theory were
discussed in Baucells and Heukamp (2006). That is very helpful for us if the stochastic
dominance can be preserved in cumulative prospect theory.
Although expected utility theory has some shortages, it is still popular in modeling
and there are lots of methods developed in the setting of expected utility theory. We aim
to develop a similar method of expected utility theory to cumulative prospect theory. But
before that, we have to check if this method works or not.
1. INTRODUCTION
3
In this thesis, we will reconsider the fundamental problem in the financial mathematics.
We want to find out an optimal strategy for the investment under cumulative prospect
theory.
In Chapter 2, we introduce the preference order and expected utility theory. And we
state some examples that describe the violations of expected utility theory. Then, we will
introduce prospect theory and the cumulative prospect theory.
In Chapter 3, we make a connection between expected utility theory and cumulative
prospect theory. Here we construct a one-period model under cumulative prospect theory.
Then, we use the method given by Karatzas, Lehoczky, Shreve and Xu (1991) to work
out an optimal solution under this model. In the remainder of this chapter, we give a
simple example to illustrate the differences of the optimal strategies for the investment
between expected utility theory and cumulative prospect theory.
Finally, we list some future works in Chapter 4. Discuss what we can do in the future.
CHAPTER 2
Theoretical Framework
2.1. Preference order and expected utility theory
Let X be a non-empty set. An element X ∈ X may be interpreted as the possible
strategy which a consumer in the market can choose. When we face two strategies X,
Y ∈ X at the same time, there may appear a preference for one over the other. Here,
the preference order, denoted by , is defined. The preference order should satisfy the
following properties:
(1) If X Y , then Y 6 X.
(2) For X, Y, Z ∈ X , if X Y , then either X Z or Z Y or both are true.
We also introduce a weak preference order here. The weak preference order, denoted
by , is defined by
X Y ⇔ Y X.
And a numerical representation of a preference order is a function U : X → R that
satisfies
Y X ⇔ U (Y ) > U (X).
Also, we have
Y X ⇔ U (Y ) ≥ U (X).
Under some restrictions, any preference order admits a numerical representation. If we
have a numerical representation of a preference order, we can compare strategies according
the value on R.
Furthermore, there are two axioms about a preference order.
5
6
2. THEORETICAL FRAMEWORK
(Independent Axiom) For X, Y, Z, if X Y , then ∃Z and α ∈ (0, 1) such that αX +
(1 − α)Z αY + (1 − α)Z.
(Archimedean Axiom) If X Y Z, ∃α, β ∈ (0, 1) such that αX + (1 − α)Z αY βX + (1 − β)Z.
If a preference order satisfies these two axioms, we can prove that there exists a von
Neumann-Morgenstern representation under some restrictions. In Föllmer [7], there are
more detail arguments about the preference order. A numerical representation U of a
preference order on M is called a von Neumann-Morgenstern representation if it is of the
form
Z
U (µ) =
u(x)µ(dx), ∀µ ∈ M,
where M is a subset of X .
A function u : S → R is called a utility function if it is strictly concave, strictly
increasing, and continuous on S, where S is a subset of R. Utility is thought as a numerical
measure of a person’s happiness. Notice that the form of a von Neumann-Morgenstern
representation and we find that a von Neumann-Morgenstern representation will equal
to compute the expectation of utility. So, when we have a von Neumann-Morgenstern
representation of a preference order, we will pick X if the expected utility of X is greater
than the expected utility of Y . In other words, we conclude that
XY
⇔ E[u(X)] > E[u(Y )]
and X Y
⇔ E[u(X)] ≥ E[u(Y )].
This criterion that according the expectation of utility is called expected utility theory,
abbreviated as EUT. Under EUT, we assume that the utility function u is always concave,
i.e., the consumer is risk-averse. Karatzas, Lehoczky, Shreve and Xu (1991) had developed
a method to find the optimal solution to
max E[u(X)]
X∈X
under EUT. Then, we will use a similar argument to complete our objective later.
2.2. VIOLATIONS OF THE EXPECTED UTILITY THEORY
7
2.2. Violations of the expected utility theory
In spite of that EUT is very popular in modeling, there are a few drawbacks in EUT.
For example, it cannot explain the observed behavior accurately. About this, there is a
famous violation of EUT which is called ”Allais paradox”. This paradox is stated as in
the following:
Example 1. Suppose that the utility function for the consumer is u with u(0) = 0.
The consumer will face two situations.
Situation 1. Choose between
A: Get 10000 with probability 33%,
get 9900 with probability
66%,
get 0 with probability
1%;
B: get 9900 with probability 100%.
Situation 2. Choose between
C: Get 10000 with probability 33%,
get 0 with probability
D: get 9900 with probability 34%,
67%;
get 0 with probability
66%.
In situation 1, a rational consumer will prefer B to A. Even he has chance to get a
higher return, he still probably gets nothing. Consequently, he will choose B and he will
get 9900 with certainty. When he is in situation 2, he has similar probabilities to gain
10000 or 9000. But he also has similar probabilities to get nothing. Under this situation,
he is willing to choose C to get the higher return.
In situation 1, we know
E[u(B)] = 1 × u(9900) > 0.33 × u(10000) + 0.66 × u(9900) + 0.01 × u(0) = E[u(A)]. (1)
However, in situation 2, we can get that
E[u(D)] = 0.34 × u(9900) + 0.66 × u(0) < 0.33 × u(10000) + 0.67 × u(0) = E[u(C)]. (2)
8
2. THEORETICAL FRAMEWORK
Due to (2), we have
1 × u(9900) > 0.33 × u(10000) + 0.66 × u(9900)
> 0.34 × u(9900) + 0.66 × u(9900).
This leads to a contradiction. So, we cannot model this problem into expected utility
theory.
Now let us turn to consider another example.
Example 2. Suppose that the utility function for the consumer is u with u(0) = 0.
The consumer will face two situations.
Situation 3. Choose between
E: Lose 10000 with probability 80%,
lose 0 with probability
F : lose 8000 with probability 100%.
20%;
Situation 4. Choose between
G: Get 10000 with probability 80%,
get 0 with probability
H: get 8000 with probability 100%.
20%;
In situation 3, a rational consumer will choose E since there is a chance that he will
lose nothing. However, in situation 4, he would like to choose H that he can get 8000
with certainty. Because in G, there is a chance that he will get nothing. Under EUT, we
always assume that the utility function u is concave, i.e., the consumer is risk-averse. But
if he is risk-averse, he will choose F instead of E in situation 3.
The results of these two examples were from a experiment reported in Kahneman and
Tversky (1979). Their results pointed out that consumers’ choices were different from the
predictions of EUT. They noticed that people are not always risk-averse. If people are
always risk-averse, they will choose F instead of E in situation 3. Precisely, they found
that people are risk-averse and risk-seeking when facing gains and losses, respectively.
That is, the utility function for the consumers should be concave when the wealth is
positive and convex when when the wealth is negative.
2.3. PROSPECT THEORY
9
2.3. Prospect theory
In order to resolve these violations of EUT, Kahneman and Tversky (1979) proposed
the prospect theory. Under prospect theory, we want to characterize the observed behavior more accurately. So, we use value function, v, and decision weight, π, to replace
the utility function and probability measure in EUT, respectively. First, we consider a
prospect Q = (a−m , q−m ; · · · ; a0 , q0 ; · · · ; an , qn ), where ai is a state in Q and qi is the
probability when ai occurs, i.e., qi = P(ai ), −m ≤ i ≤ n. Then, the subject value V
similar to expected utility of EUT is given by the following:
V = V + + V −,
where
V
+
=
n
X
πi+ v(ai )
and
V
−
=
i=0
−1
X
πj− v(aj ),
j=−m
with m, n > 0. We assume that the value function v is concave at x > 0 and convex at
x < 0.
1
0.5
0
−0.5
−1
−1.5
−2
−1
−0.5
0
0.5
1
Figure 1. The shape of the value function v.
10
2. THEORETICAL FRAMEWORK
The decision weight π is a monotonic transformation of the outcome’s probability.
There is an exact definition of π in next section. Now, we reconsider Example 1 under
prospect theory first.
Example 3. Without loss of generality, we may assume that v(0) = 0. Thus, by the
preference in Example 1, we can get the following inequalities:
0.089 × v(0) + 0.334 × v(10000) + 0.577 × v(9900) = VA < VB = v(9900)
0.666 × v(0) + 0.334 × v(10000) = VC > VD = 0.339 × v(9900) + 0.661 × v(0)
Due to the above inequalities, we conclude that
1 > 0.577 + 0.339.
But it doesn’t lead to contradiction since the definition of π does not need to satisfy
π(p) + π(1 − p) = 1. So we can clean up Allais paradox under prospect theory.
There are still some weaknesses in prospect theory. For instance, it only works with
prospects that have finite states since the definition of the decision weight. We have
to know every probability of every state under prospect theory. Mohapi (2004) also
pointed out that prospect theory has some drawbacks. In order to solve these problems,
a generalized form of prospect theory was proposed by Tversky and Kahneman (1992).
2.4. Cumulative Prospect theory
Since the original form is hard to extend, Tversky and Kahneman (1992) proposed the
generalized form of the prospect theory, called the cumulative prospect theory and abbreviated as CPT, to overcome the shortages of the original one. Instead of transforming the
probability separately, the cumulative form transforms the entire cumulative distribution
function. So, we only need to know the cumulative distribution function rather than
knowing every probability of every state under CPT.
2.4. CUMULATIVE PROSPECT THEORY
11
Now, let us give some notations.
Consider a prospect Q = (a−m , q−m ; · · · ; a0 , q0 ; · · · ; an , qn ), where ai is an event in Q
and qi is the probability when ai occurs, i.e., qi = P(ai ), −m ≤ i ≤ n. Without loss of
generality, we assume that a−m < a−m+1 < · · · < a0 < · · · < a( n − 1) < an and a0 = 0.
The value function v is the same as the one in prospect theory. Furthermore, Tversky
and Kahneman (1992) suggested that v(a0 ) = v(0) = 0 and


xα ,
x ≥ 0;
v(x) =
 −λ(−x)β , x < 0.
The estimations of α, β and λ are given by the following:
α̂ = 0.88,
β̂ = 0.88
and
λ̂ = 2.25.
Further, the decision weight π is defined by
πj− = W − (q−m + · · · + qj ) − W − (q−m + · · · + qj−1 ) = W − (F (aj )) − W − (F (aj−1 )) and
πi+ = W + (qi + · · · + qn ) − W + (qi+1 + · · · + qn ) = W + (1 − F (ai−1 )) − W + (1 − F (ai )),
where F (ai ) = q−m + · · · + qi =
i
X
qk , is the cumulative distribution function, the
k=−m
capacity, W , or called the probability weighting function is given by
W + (x) =
xγ
1
(xγ + (1 − x)γ ) γ
and
W − (x) =
xδ
1
(xδ + (1 − x)δ ) δ
.
where γ̂ = 0.61 and δ̂ = 0.69.
The above parameters were estimated from an experiment reported in Tversky and
Kahneman (1992). Note that the parameters α and β denote the elasticities of consumer
in economics here.
We give an example to see the differences of πi and qi .
12
2. THEORETICAL FRAMEWORK
Example 4. Suppose there is a prospect
Q = (a−3 , q−3 ; a−2 , q−2 ; a−1 , q−1 ; a1 , q1 ; a2 , q2 ; a3 , q3 ),
where qi is given by the following. Then we can compute πi according to the definition in
the above.
state
-3
-2
-1
1
2
3
qi
0.083 0.167 0.250 0.250 0.167 0.083
πi
0.152 0.141 0.160 0.130 0.120 0.170
Comparing these values, we find that πi takes more weight than qi when qi is small.
But it takes less weight than qi when qi is large.
Since we transform the cumulative distribution function rather than the probability
separately, we can extend to the continuous form easily. Besides, there is another thing
different from the original one. Under CPT, we can rearrange the outcomes as we want.
Here, we rearrange them from small to large. Similar as in EUT, we make decision
according to the value of V . And notice that we consider the wealth at the terminal time
in EUT. But we consider the change of wealth here.
CHAPTER 3
The Optimal Solution in Discrete Type
3.1. Connection between EUT and CPT
In spite of EUT has many drawbacks, it is still useful and popular in theoretical modeling. Therefore, we want to construct a connection between CPT and EUT. If we can
find it, we may use a similar argument as in EUT to CPT. Next, we give a proposition
that connects EUT and CPT.
Proposition 5. If 1 > γ >
1
, W (F (x)) is a distribution function where F (x) is a
2
distribution function and
W (F (x)) =
[F (x)]γ
1
([F (x)]γ + [1 − F (x)]γ ) γ
.
(3)
Proof. In order to prove that W (F (x)) is a distribution function, we have to prove
the following:
(1) Domain of W (F (x)) = R.
(2) W (F (x)) is increasing and right-continuous.
(3) W (F (−∞)) = 0 and W (F (∞)) = 1.
Since F (x) is a distribution, we can guarantee that its right-continuity and that domain
is equal to R. Further, we also have W (F (−∞)) = 0 and W (F (∞)) = 1. It remains to
prove that W (F (x)) is increasing.
Take natural logarithm on both sides of the equality of (3), we get that
ln W (F (x)) = γ ln(F (x)) −
1
ln([F (x)]γ + [1 − F (x)]γ ).
γ
13
14
3. THE OPTIMAL SOLUTION IN DISCRETE TYPE
Differentiating on both sides with respect to x we obtain
W 0 (F (x))
γF 0 (x) 1 γ[F (x)]γ−1 F 0 (x) − γ[1 − F (x)]γ−1 F 0 (x)
=
−
.
W (F (x))
F (x)
γ
[F (x)]γ + [1 − F (x)]γ
Assume that F 0 exists. So we have
W 0 (F (x)) = W (F (x))
(γ − 1)[F (x)]γ [1 − F (x)]1−γ + (γ − γF (x) + F (x)) 0
F (x).
[1 − F (x)]1−γ F (x)([F (x)]γ + [1 − F (x)]γ )
Since the denominator, W (F (x)) and F 0 (x) are positive, it remains to prove that
F (x)
1 − F (x)
γ
(1 − F (x))(γ − 1) + γ − γF (x) + F (x) > 0
We discuss it in two parts. When F (x) ≤
fact and γ >
1
F (x)
, it implies that
≤ 1. We use this
2
1 − F (x)
1
to get the following inequality.
2
F (x)
1 − F (x)
γ
(1 − F (x))(γ − 1) + γ − γF (x) + F (x)
12
F (x)
1
1 F (x)
≥
(1 − F (x))(− ) + +
1 − F (x)
2
2
2
p
p
1
=
1 + F (x) − F (x) 1 − F (x)
2
1
1
≥
1 + F (x) −
>0
2
2
since x(1 − x) ≤
1
for 0 ≤ x ≤ 1.
4
Now, we turn to consider the other part.
When F (x) ≥
1
, it will imply that
2
F (x)
1
≥ 1. And we use the fact and γ > to get the following inequality.
1 − F (x)
2
3.1. CONNECTION BETWEEN EUT AND CPT
F (x)
1 − F (x)
γ
F (x)
1 − F (x)
γ
F (x)
1 − F (x)
≥
≥
=
15
(1 − F (x))(γ − 1) + γ − γF (x) + F (x)
1
1 F (x)
(1 − F (x))(− ) + +
2
2
2
1
1 F (x)
(1 − F (x))(− ) + +
2
2
2
1
1 F (x) F (x)
+
−
= >0
2
2
2
2
From the above, we conclude that W (F (x)) is increasing. That is, W (F (x)) is a distribution function.
1
Remark 6. When γ = , the result in Proposition 5 still holds.
2
Proof. When γ =
1
, we can use the proof of Proposition 5 to get the following
2
result.
F (x)
1 − F (x)
γ
(1 − F (x))(γ − 1) + γ − γF (x) + F (x)
12
1
1 F (x)
F (x)
(1 − F (x))(− ) + +
1 − F (x)
2
2
2
p
p
1
1 F (x)
F (x) 1 − F (x)(− ) + +
2
2
2
p
p
1
1 + F (x) − F (x) 1 − F (x)
2
1
1
1 + F (x) −
>0
2
2
=
=
=
≥
1
So we can conclude that W (F (x)) is a distribution function when γ = .
2
Notice that W (F (x)) is not always a distribution function when γ <
example is stated below.
1
. A counter
2
16
3. THE OPTIMAL SOLUTION IN DISCRETE TYPE
Example 7. If γ =
1
1
and F (x) = , then
5
10
γ
F (x)
(1 − F (x))(γ − 1) + γ − γF (x) + F (x)
1 − F (x)
51
9
1
4
1
1
1
×
=
× (− ) + −
+
9
10
5
5 50 10
≈ −0.184 < 0.
This means that W (F (x)) is not increasing. Thus, W (F (x)) is not a distribution function
in this situation.
Most studies about CPT followed with the functions and these parameters suggested
by Tversky and Kahneman (1992).
By the above proposition, we know that π is another probability measure. This means
that the structure of CPT is similar as EUT except the utility function is no more concave
on R. So we can use the methods given by Karatzas, Lehoczky, Shreve and Xu (1991) to
find the optimal solution under CPT.
3.2. The optimal solution
We consider a one-period model. If we have a prospect Q = (a−m , q−m ; · · · ; a0 , q0 ; · · · ; an , qn ),
where ai is an event in Q and qi is the probability when ai occurs, i.e., qi = P(ai ), −m ≤
i ≤ n. Price at t = 0 is P0 and price at t = 1 is P1 . P1 and P0 satisfy the following
equation:
P1 = P0 + N, where N ∈ {a−m , · · · , an }.
To keep it easy to compute, we assume that the interest rate is identical to 0. We have the
initial wealth E and we want to invest ρE in the stock market at t = 0. So the outcome
3.2. THE OPTIMAL SOLUTION
17
xi at state i at time 1 is given by the following:
ρE
P1 + (1 − ρ)E
P0
ρE
= ρE +
ai + (1 − ρ)E
P0
ρE
=
ai + E
P0
xi =
Our objective is to choose the optimal ρ̂ which maximizes
V (X) =
n
X
πi+ v(xi )
+
i=0
−1
X
πj− v(xj ).
j=−m
Here we assume that v ∈ C 1 is injective, concave when x > 0 and convex when x < 0.
Further, we suppose that there is no arbitrage in the market. Now, our problem is to
choose ρ that will maximize the subject value V with the constraint q−m x−m +· · ·+qn xn =
E. Because this optimal problem is hard to solve, we turn to consider another problem.
This method is called duality method. We can get what we want from the dual problem.
The solutions of these two problems will be equivalent. So, we consider the dual problem
now.
Define X0 (y) =
P
qi I(yi ), and Y0 = X0−1 , where I is the inverse function of v 0 . Then
the optimal state ξ0x will be given by ξ0x = I(Y0 (x)). Karatzas, Lehoczky, Shreve and
Xu (1991) had guaranteed that ξ0x is the optimal state. And according to the equation
between ξ0x and ρ̂, we can get that
ξ0x = ρ̂
N
E + E.
P0
So the optimal ρ̂ will be given by the following:
ρ̂ =
P0 x
(ξ − E).
NE 0
18
3. THE OPTIMAL SOLUTION IN DISCRETE TYPE
3.3. Example
In this section, we consider a simple example to see the differences of the strategies
for investment between EPT and CPT.
If we have a prospect Q = (a1 , q1 ; a2 , q2 ), where ai is a state in Q and qi is the
probability when ai occurs. Price at t = 0 is P0 and price at t = 1 is P1 . We assume that
a1 < P0 and a2 > 0. P1 and P0 satisfy the following equation:
P1 = P0 + N, where N := {a1 , a2 }.
To keep it easy to compute, we assume that the interest rate is identical to 0. We
have the initial wealth E and we want to invest ρE in the stock market at t = 0. So the
outcome xi at state i at time 1 is given by the following:
ρE
P1 + (1 − ρ)E
P0
ρE
= ρE +
ai + (1 − ρ)E
P0
ρE
ai + E
=
P0
xi =
Here we discuss it in four cases.
Case 1: EUT. The probability is qi and the value function is given by u(x) = xα .
Consider L1 = q1 u(x1 ) + q2 u(x2 ) + θ1 (q1 x1 + q2 x2 − E). Applying Lagrange-Multiplier
method, then
∂L1
= q1 α(x1 )α−1 + θ1 q1 = 0,
∂x1
∂L1
= q2 α(x2 )α−1 + θ1 q2 = 0,
∂x2
∂L1
= q1 x1 + q2 x2 − E = 0.
∂θ1
3.3. EXAMPLE
Thus we can conclude that
(q1 + q2 )
−θ1
α
19
1
α−1
= E.
Since q1 + q2 = 1, we get that
−θ1
α
1
α−1
= E.
So that
1
αE α−1 α−1
θ1 = −αE , x1 = x2 =
= E.
α
From the relation between ρ and xi , we can know that ρ will be equal to 0 in this case.
α−1
This means that the consumer doesn’t want to invest anything under this situation.
Case 2: The value function is the same as in Case 1. But we use the new probability ri
to replace qi . Note that r1 6= q1 , r2 6= q2 , but r1 + r2 = q1 + q2 = 1.
Consider L2 = r1 u(x1 ) + r2 u(x2 ) + θ2 (q1 x1 + q2 x2 − E). Applying Lagrange-Multiplier
method, then
∂L2
= r1 α(x1 )α−1 + θ2 q1 = 0,
∂x1
∂L2
= r2 α(x2 )α−1 + θ2 q2 = 0,
∂x2
∂L2
= q1 x1 + q2 x2 − E = 0.
∂θ2
Thus we can know that
q1
−θ2 q1
r1 α
1
α−1
+ q2
−θ2 q2
r2 α
1
α−1
= E.
Then we can solve it to get
α−1



θ2 = α 

E
q1
−q1
r1
1
α−1
+ q2


1 
α−1

−q2
.
r2
Substitute θ2 into x2 , we know that
1
α−1
q2
E
E
x2 =
= .
1
1
1
r2
q1 α−1
q2 α−1
q1 r2 α−1
q1
+ q2
q1
+ q2
r1
r2
q2 r 1
20
3. THE OPTIMAL SOLUTION IN DISCRETE TYPE
Then, from the equation of ρ and x2 , we can know that


P0
ρ=
a2




q1
q2 r 1
q1 r 2
1
1
1−α
+ q2


− 1 .

If we assume r2 > q2 and r1 < q1 , then we conclude that ρ > 0. It means that the
consumer will invest ρE at t = 0, where ρ is represented as the above. But if we assume
r2 < q2 and r1 > q1 , we will get ρ < 0. However, it is not allowed that the consumer
is able to sell the prospect, i.e., ρ ≥ 0. This means that the consumer will not invest
anything at this situation.
Case 3: The probability is the same as in Case 1. But we use the function v to replace u,
where v is given by the following:
v(x) =


xα ,
x ≥ 0;
 −λ(−x)α , x < 0.
Consider L3 = q1 v(x1 ) + q2 v(x2 ) + θ3 (q1 x1 + q2 x2 − E). Applying Lagrange-Multiplier
method, then
∂L3
= q1 v 0 (x1 ) + θ3 q1 = 0,
∂x1
∂L3
= q2 v 0 (x2 ) + θ3 q2 = 0,
∂x2
∂L3
= q1 x1 + q2 x2 − E = 0.
∂θ3
By using the function I defined by the inverse function of v 0 , then we can get that
v 0 (x1 ) = θ3 ⇒ x1 = I(θ3 ),
v 0 (x2 ) = θ3 ⇒ x2 = I(θ3 ),
q1 x1 + q2 x2 = E.
3.3. EXAMPLE
21
So, we conclude that
x1 =
x2 =
1
1−α
λα
θ3
,
−α
θ3
1
1−α
.
Substituting x1 , x2 into our constraint we know that
θ3 = −α
E
!α−1
.
1
q1 (−λ) 1−α + q2
Then we get that
x2 =
E
1
.
q1 (−λ) 1−α + q2
Then, from the equation of ρ and x2 , we can know that
"
#
1
P0
−1 .
ρ=
1
a2 q1 (−λ) 1−α
+ q2
If we can find λ and α that satisfying
−
1
q2
< (−λ) 1−α < 1,
q1
we can conclude that ρ > 0. That is, we will invest ρ percents of our initial capital.
Case 4: The value function is the same as in Case 3. And we use the decision weight πi
to replace the probability qi .
Consider L4 = π1− v(x1 ) +π2+ v(x2 )+θ4 (q1 x1 +q2 x2 −E). Applying Lagrange-Multiplier
method, then
∂L4
= π1− v 0 (x1 ) + θ4 q1 = 0,
∂x1
∂L4
= π2+ v 0 (x2 ) + θ4 q2 = 0,
∂x2
∂L4
= q1 x1 + q2 x2 − E = 0.
∂θ4
22
3. THE OPTIMAL SOLUTION IN DISCRETE TYPE
By using the function I, then we can get that
−θ4 q1
v (x1 ) =
⇒ x1 = I
π1−
−θ4 q2
v (x2 ) =
⇒ x1 = I
π2+
q1 x1 + q2 x2 = E.
0
0
−θ4 q1
π1−
−θ4 q2
π2+
,
,
So, we conclude that
x1
x2
1
−θ4 q1 α−1
= (−λα)
,
π1−
1
1
−θ4 q2 α−1
1−α
.
= (α)
π2+
1
1−α
Substitute x1 , x2 into our constraint. We can know that


θ4 = α 
E
q1
1
1−α
−
λπ1
q1
+ q2
−π2+
q2
α−1

1 
1−α
.
Then we can get
x2 =
E
.
1
−λq2 π1− 1−α
q1
+ q2
q1 π2+
From the relation between ρ and x2 , we know that


P0
ρ=
a2





1
q1
−λq2 π1−
q1 π2+
1
1−α
+ q2


− 1
.

Similar as in Case 3, we will conclude that ρ > 0 if we can find λ and α that satisfying
1
q2
−λq2 π1− 1−α
− <
< 1.
q1
q1 π2+
Compare the results of the four cases and we find that we want to invest more under
CPT than under EUT. The consumers don’t want to invest under EUT but there ia a
chance that they want to invest under CPT. As we see, CPT will help us to make a better
choice than EUT.
CHAPTER 4
Conclusion and Future Work
In this thesis, we find a way to solve an optimal solution under CPT. And we compare
the results in EUT and CPT. But we consider only the discrete type of models in this
paper. In the future, we will try to consider the continuous type. As we have mentioned
in Chapter 2, it is ready to extend to the continuous case under CPT. In continuous case,
V (x) will be replaced by the following:
Z ∞
Z 0
d d −
v(x)
W (F (x)) dx +
−W + (1 − F (x)) dx.
v(x)
V (x) =
dx
dx
0
−∞
If we replace the subject value with the continuous form, we can find the optimal
strategy for investment. Then we can compare it with the optimal solution of EUT in
order to see the differences between the results of these two theories.
About the continuous form of CPT, Rieger and Wang (2006) has some results. But
there appear some questions at the same time. In Rieger and Wang (2006), the assumption
of Theorem 1 is too strong. We use our notation to restate this theorem in the following.
Theorem 8. Let
Z 0
Z ∞
d d −
W (F (x)) dx +
v(x)
−W + (1 − F (x)) dx,
v(x)
V (x) =
dx
dx
0
−∞
where the value function v is continuous, monotone, convex for x < 0 and concave for
x > 0. Let X be a random variable with E[X] < ∞ and F (x) the distribution function of
X. Assume that there exist constants α, β ≥ 0 such that
v(x)
= v1 ∈ (0, ∞) and
x→∞ xα
lim
|v(x)|
= v2 ∈ (0, ∞).
x→−∞ |x|β
lim
(4)
And the weighting functions W + and W − are continuous, strictly increasing functions
from [0, 1] to [0, 1] such that W + (0) = W − (0) = 0 and W + (1) = W − (1) = 1. Moreover,
23
24
4. CONCLUSION AND FUTURE WORK
assume that W + and W − are continuously differentiable on (0, 1) and that there are
constants δ, γ > 0 such that
W−0 (y)
1 − W+0 (y)
=
w
∈
(0,
∞)
and
lim
= w2 ∈ (0, ∞).
1
y→0 y δ−1
y→1 (1 − y)γ−1
lim
(5)
Then V is finite if α < γ and β < δ.
According the theorem of Arrow (1974), we believe that we don’t need the powerful
assumption such as (4) and (5) to get the result. Since the framework of CPT is similar
to EUT, we should get the result under a similar assumption to the assumption of Arrow
(1974).
Finally, we just discuss a one-period model in this paper. We will extend this model
to two-period, even to multi-period. If we can extend it successfully, we will try to extend
it to a continuous time model in the future.
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