Phantom aversion and portfolio choices: The
comparative statics
Yoichiro Fujii† , Hideki Iwaki‡ , and Yusuke Osaki§
†,§
‡
Faculty of Economics, Osaka Sangyo University
Faculty of Business Administration, Kyoto Sangyo University
February 14, 2016
1 Introduction
This paper makes compartive statics of the phantom decision model advocated by
Izhakian and Izhakian (2014). In the paper, they says “ This paper poposes a new
model of decision making that aims to capture the multi-dimensional nature of uncertainy, providing a better description of reality.” In the model, a decison maker
(DM) faces three souces of uncertainty: (i) a priori, the event to be realized is
peceived ambigously; (ii) the odds of events are ambiguous; (iii) a priori, the utilities of consequences of events are ambiuous (possibly because the consequences
themselves are vague).
Vague odds of events and vague outcomes, tamed phantoms in the paper, are
framed by phantom numbers. In order to concrete the arguments, we cite a motivating example in the paper. “Consider a lottery, say on a sporting event, that
consists of 10 games. For $2 ticket, a speculator is offered the following bet: if she
guesses the results of all 10 games correctly, she win $200, but may be reduced
to $100, depending on the number of speculators and their betting result. If she
guesses the results of 9 games correctly, the prize is $100 and may be reduced to
$50, otherwise she wins nothing. She estimates her chances to guess the winners
of all 10 games to be 10% with a possible increment up to 30%, if she is “lucky.”
1
She also estimates her chances to guess 9 winners to be 30% but they might drop
down to 20%. The payoff of this decision problem is framed a lottery with phantom
numbers as
P:



X(10 guess) = 198 − P100 7→ 0.1 + P0.2,

X(9 guess) = 98 − P50 7→ 0.3 − P0.1,



X(0–8 guess) = −2 + P0 7→ 0.6 − P0.1.
Here, the payoff of guessing the results of all 10 games is perceived by the phantom number 198 − P100 that has two components. The first component, 198, is
a real term that forms the pivot value. The second component, called the phantom
term and denoted P, is a signed distortion of the pivot value. Since the payoff
subtracting ticket fee $2 of this event may drop from $198 down to $98, the phantom term is minus with the magnitude 100. Similarly, the probability to guess the
winners of all 10 games is perceived by the phantom number 0.1 + P0.2, since it
can jump from 10% up to 30%, the phantom term is positive with magnitude 20%.
The remaing of the lottery is similarly described by the phantom numbers. This
type of representation can be considered as a matter of framing effect.
Given the phantom representation of the lottery, the DM evaluates her expected utility. Unlike classical expected utility theory, the expected utility is also
expressed in phantom terms since the utility function U as well as the probability measure P is also given in term of phantom. Phantom expected utility can
be interpreted as real expected utility with potential distortion from the expectation. To compare different alternatives, the DM ranks her expected utilities using a
phantom-valued function Γ as
(
∑
Γ
)
U (x)P(x) ,
x∈X
where X is a set of consequences.
Based on the above decision making model in the phantom space, we make
comparative statics. First we compare attitudes of the DMs toward phantoms. We
define phantom aversion of the DM in a different manner from that of Izhakian
2
and Izhakian (2014). Their definition of phantom aversion is considered as an extension of “complementary ambiguity aversion”, proposed by Sinscalchi (2009),
where consequences are real and probabilities are ambiguous. That is, they extend the concept of complementary ambiguity aversion to complementary phantom aversion, where both consequences and probabilities are phantom. On the
other hand, our ambiguity aversion is as an extension of risk aversion where both
consequences and probabilities are uniquely given. Since an order of the phantomvalued function Γ preserves that of the utility function u, our definition of phantom
aversion can be characterized by the shape of the utility function, more rigorously
the shape of the phantom term of the utility function. Our definition possesses attractive features comparison to other definitions. Since it is a natural extension of
ordinal definition of risk aversion, ours allows the huge range of comparative static
technique amassed over the years to tackle the questions involving risk under the
expected utility framework.
We aloso investigates the standard static portfolio problem under the model.
As same as the comparative statics of ambiguity aversion for the portfolio problem
(Gollier 2011), the question treated in the paper is parallel to the one of risk aversion. Although since Arrow (1963) and Pratt (1964) it has been well established
that an increase in risk aversion reduces demand for the risky asset in the expected
utility framework, we show that it is not necessarily true that more phantom aversion reduces demand for a phantom asset. For a cleverly–but spurious–chosen set
of phantom consequences and probabilities for the return on the phantom asset,
we show that the introduction of phantom aversion increase the investor’s demand.
This intuition for why such counterexamples may exists can be explained as follows. Compared the ordinary expected utility frameworks under risk, since both
consequences and probabilities are represented by phantom numbers, the expected
utilities under phantom are considered as distortions of the ones under risk. It is
well known from expected utility theory that pessimistic deteriorations in beliefs
do not always reduce the demand for the risky asset.1 As Gollier (2011) pointed
out in the smooth model of ambiguity (KMM 2005), this deterioration in the terms
of trade yields a wealth effect that may in fact raise demand as for Giffen goods in
1
3
consumer theory.
One of the main objective of the paper is to provide conditions under which
more phantom aversion reduces the optimal level of exposure to uncertainty. This
can be done by restricting either the set of utilities functions or the set of possible
phantom probabilities. If the set of phantom probabilities can be ranked according
to their order of first-degree or second-degree stochastic dominance (FSD/SSD),
then some simple sufficient conditions on the utility function result in comparative
static properties similar to those under risk. In the paper, the following questions
are linked:
1. Under the EU model, what are the conditions on the utility function u that
guarantee that any FSD/SSD deterioration in the distribution of excess return
of the risky asset reduces the demand for it? and
2. In the phantom decision model, what are the conditions that guarantee that
any increase in phantom aversion reduces the demand for the phantom asset?
We show the result is given as follows. If the plausible phantom probabilities can
be ranked according to MLR (a special case of FSD), then it is always true that
more phantom aversion reduces the demand for the phantom asset.
2 Phantom Spaces
The space of phantom numbers PH consists of elements of the form z = a + Pb,
where a and b are real numbers called the real term and the phantom term of z,
respectively (resp.). We denote the real term and the phantom term of z by zre and
zph resp., and ẑ = zre + zph . On PH, the addition and multiplication operations
are respectively defined by
(a1 + Pb1 ) ⊕ (a2 + Pb2 ) :=(a1 + a2 ) + P(b1 + b2 ),
(a1 + Pb1 ) ⊗ (a2 + Pb2 ) :=a1 a2 + P((a1 + b1 )(a2 + b2 ) − a1 a2 ).
The unit element is 1 := 1 + P0, and the zero element is 0 := 0 + P0. PH has
zero divisors, i.e., nonzero elements whose product is zero. Zero divisors in PH,
4
denoted Zdiv (PH), are all of the forms z = 0 + Pa or z = a − Pa. The inverse
number z −1 of z is defined by
z
−1
1
:= + P
a
(
1 1
−
ẑ a
)
,
where ẑ = a + b. Thus z ⊗ z −1 = 1 holds. Hence, the division operation ⊘ is
defined by
z1 ⊘ z2 := z1 ⊗ z2−1 .
A phantom number z is said to be strictly positive, written z ≫ 0, if a > 0 and
b > 0, and it is said to be positive, written z≫0, if a ≥ 0 and b ≥ 0. When a > 0
and a + b > 0, then z is said to be pseudo-positive. z is called pseudo-nonnegative
if it is pseudo-positive or 0.
Notation 1. We write z1 z2 for the product z1 ⊗ z2 , z n for z ⊗ · · · ⊗ z with z
repeated n times, and z1 /z2 for the division z1 ⊘ z2 for notational
3 The Setup
Let Ω be a finite nonempty state space, whose elements ω ∈ Ω are states of nature. Let F be a σ-algebra of subsets of Ω. Beliefs are represented by a phantom
probability measure defined as follows.
Definition 1 (Phantom probability measure). Let Λ̄ be a subset of PH defined by
Λ̄ := {z ∈ PH|z = a + Pb, a ∈ [0, 1], 0 ≤ a + b ≤ 1}.
For P : F → Λ̄, we write P = Pre + PPph , where Pre and Pph are real functions
forming the real component and the phantom component of P, respectively (resp.).
P is called phantom probability measure, if it satisfies:
(i) Nonnegativity: 0 ≤ Pre (A) ≤ 1 for every A ∈ F ;
(ii) Normalization: P(Ω) = 1;
(iii) Additivity: P(A∪B) = P(A)+P(B) for any pair of disjoint events A, B ∈
F;
5
(iv) Phantomization: 0 ≤ Pre (A) + Pph (A) ≤ 1 for each A ∈ F.
To avoid nonzero divisors, the target of P is usually taken to be the set Λ =
{λ|λ ∈ Λ̄, λ ∈
/ Zdiv }, called the phantom probability zone, and defines a set of
phantom weights to be ΛX = Λ \ {0, 1}.
The set of consequences X (i.e., alternatives, payoffs, prizes, or outcomes) is a
nonempty set of generic “prizes.” We assume x ∈ X ⊂ PH, that is, a consequence
x ∈ X is a vague outcome consisting of a phantom number. Given a map X : Ω →
X , we write P(X = x) or P(x) for P({ω|ω ∈ Ω, X(ω) = x}) .
The objects of choice are phantom lotteries with finite support, formally defined by
{
L=
}
∑
P : X → Λ #{x ∈ X |P(x) > 0} < ∞,
P(x) = 1 .
x∈X
A constant lottery PC is a lottery that satisfies P(X = x) = 1, for some x ∈ X ,
and write x ∈ L if the notation does not occure confusion.
Let P1 , · · · , Pn be lotteries in L and assume α1 , · · · , αn ∈ Λ sum up to 1. The
compound lottery P̃, written ⊕i αi Pi , is the map
P̃ : X 7→
∑
αi Pi (x).
As to the DM’s preference ≿ over phantom lotteries L, we assume Theorem 1
of Izhakian and Izhakian (2014) holds. That is, for every P, Q ∈ L,
P ⪰ Q ⇐⇒
∑
x∈X
(
⇐⇒ Γ
∑
U (x)P(x) ⪰
∑
U (x)Q(x)
x∈X
)
U (x)P(x)
x∈X
≥Γ
(
∑
)
U (x)Q(x) ,
(1)
x∈X
where U : X → PH is a utility function and Γ : PH → R is a phantom-valued
function.
For the explicit exposition, the utility function U is assumed to be represented
6
as the realization form:2
U (x) = Ure (xre ) + P(Û (x̂) − Ure (xre )),
(2)
where Ure and Û are some real-valued functions.
Example 1 (Izhakian and Izhakian (2014)).
1) Exponential utilities.
e−γx
e−γa
U (x = a + Pb) = −
=−
−P
γ
γ
(
e−γ(a+b) e−γa
−
γ
γ
)
,
γ ∈ R.
,
γ ∈ Q.
2) Power utilities.
x1−γ
a1−γ
U (x = a + Pb) =
=
+P
1−γ
1−γ
(
(a + b)1−γ
a1−γ
−
1−γ
1−γ
)
Remarks 1. In the above examples, both type utilities have the realization forms
and Û = Ure holds.
Hereafter, as to any utility function, we assume the realization form holds.
4 Attitude toward phantoms (ambiguity)
Definition 2. Consider consequenses xi = a + Pbi ∈ X , i = 1, · · · , n, and a
lottery P ∈ L:



P(X = xi ) = pi ∈ ΛX , i = 1, · · · , n;


∑n
p = 1;
 i=1 i


E [X] := ∑n p x = a or ∑n p̂ b = 0.3
P
i=1 i i
i=1 i i
(3)
2
See Definition
1.17 ∑
of Izhakian and Izhakian (2009).
∑
n
Noting n
p̂
=
i
i=1
i=1 pire = 1,
( n
)
( n
)
n
n
n
∑
∑
∑
∑
∑
EP [X] =
pi xi =
pire a + P
p̂i (a + bi ) −
pire a = a + P
p̂i bi .
3
i=1
i=1
i=1
i=1
7
i=1
We say that the DM exhibits
(i) Phantom aversion if X ≾ EP [X];
(ii) Phantom neutrality if X ∼ EP [X];
(iii) Phantom loving if X ≿ EP [X].
Here X denotes the lottery defined by (3) and EP [X] denotes the constant lottery
whose consequence is EP [X].
The above definition of the DM’s attitude toward phantoms is different from
that of Izhakian and Izhakian (2004). For x = a + Pb ∈ X , they say that the
DM exhibits phantom aversion (neutrality, loving) if x̌ ≿ (∼, ≾) 12 x ⊕ 12 x̄, where
x̄ = (a + b) − Pb and x̌ =
xre +x̄re
2
=
a+(a+b)
.
2
Proposition 1. The DM exhibits
(i) Phantom aversion ⇐⇒ Û is concave;
(ii) Phantom neutrality ⇐⇒ Û is linear;
(iii) Phantom loving ⇐⇒ Û is convex.
Proof. We prove the case (i); the other two cases are proved by the same ways.
∑
From (1), we only have to show ni=1 pi U (xi ) ≾ Ure (a) + P(Û (a) − Ure (a)).
From the Jensen’s inequality, if and only if Û is concave, the following holds.
n
∑
pi U (xi ) =
i=1
=
n
∑
i=1
n
∑
{
}
pi Ure (a) + P(Û (a + bi ) − Ure (a))
{
pire Ure (a) + P
i=1
=Ure (a) + P
{
n
∑
p̂i Û (a + bi ) −
i=1
n
∑
n
∑
}
pire Ure (a)
i=1
}
p̂i Û (a + bi ) − Ure (a)
i=1
{ ( n
)
}
∑
⪯Ure (a) + P Û
p̂i (a + bi ) − Ure (a)
i=1
=Ure (a) + P{Û (a) − Ure (a)}.
8
(4)
Definition 3. Consider the preferences ≿A and ≿B of two DMs A and B, resp. We
say that A is at least as phantom averse as B if, for each lottery P ∈ L satisfying
(3),
X ≾B EP [X] =⇒ X ≾A EP [X].
Theorem 1. For DM A’s utility function UA and DM B’s utility function UB , suppose that UAre = UBre holds. Then DM A is at least as phantom averse as DM B
iff there exists a concave and increasing function ϕ : R → R;
{
}
UA (x = a + Pb) =UAre (a) + P ÛA (a + b) − UAre (a)
{
}
=UBre (a) + P ϕ(ÛB (a + b)) − UBre (a)
∀x ∈ X.
(5)
Proof. First, we assume that there exists a concave and increasing ϕ satifying (5),
and that for each lottery P ∈ L satisfying (3), X ≾B EP [X] holds. Then,
n
∑
{
pi UA (a + Pbi ) = UBre (a) + P
n
∑
}
p̂i ϕ(ÛB (a + bi )) − UBre (a)
i=1
i=1
{ (
≾A UBre (a) + P ϕ
{ (
n
∑
i=1
≾A UBre (a) + P ϕ ÛB
)
}
p̂i ÛB (a + bi )
− UBre (a)
( n
∑
))
p̂i (a + bi )
}
− UBre (a)
i=1
{
}
= UBre (a) + P ϕ(ÛB (a)) − UBre (a)
{
}
= UAre (a) + P ÛA (a) − UAre (a) .
This implies X ≾A EP [X]. In the above, the first ≾A holds by the Jensen’s
inequalities since ϕ is concave and the second ≾A also holds by the Jensen’s inequalities since X ≾B EP [X] implies DM B is phantom averse and then ÛB is
concave by Proposition 1.
9
Next, suppose that DM A is at least as phantom averse
( as DM
) B and that there
exists an interval in the image of ÛB such that ϕ = ÛA ÛB−1 is strictly convex.
Consider a lottery X that satifies (3) and suppose that its support is in that interval,
and X ∼B EP [X]. Then
{
UAre (a) + P
n
∑
}
p̂i ÛA (a + bi ) − UAre (a)
i=1
{
= UBre (a) + P
n
∑
}
p̂i ϕ(ÛB (a + bi )) − UBre (a)
i=1
{ (
≻A UBre (a) + P ϕ
n
∑
)
p̂i ÛB (a + bi )
}
− UBre (a)
i=1
{
}
= UBre (a) + P ϕ(ÛB (a)) − UBre (a)
{
}
= UAre (a) + P ÛA (a) − UAre (a) = UA (a).
Here we have used the fact X ∼B EP [X] is equivalent that
ÛB
(a).4
(6)
∑n
i=1 p̂i ÛB (a + bi )
=
(6) implies X ≻A EP [X] and contradicts that A is at least as phantom
averse as B.
Hereafter, for each x ∈ X , we assume Γ : X → R possesses such a functional form that Γ : (xre , xph ) ∈ R2 7→ Γ(xre , xph ), and Γ is differentiable with
respect to (w.r.t.) both xre and xph . We denote the partial derivatives of Γ by
Γre (xre , xph ) =
∂
∂xre Γ(xre , xph )
and Γph (xre , xph ) =
∂
∂xph Γ(xre , xph ).
We abuse
both notations Γ(x) and Γ(xre , xph ) because we think it does not occure confusion.
4
X ∼B EP [X]
{
⇐⇒ UBre (a) + P
n
∑
}
p̂i ÛB (a + bi ) − UBre (a)
i=1
This is equivalent that
∑n
i=1
p̂i ÛB (a + bi ) = ÛB (a).
10
{
}
= UBre (a) + P ÛB (a) − UBre (a) .
5 Applications to the portfolio problem
We consider the static model with two assets. The first asset is the risk-less asset
with a rate of return that is normalized to zero. The second asset is ambiguous and
vague, and its rate of return x̃ has a phantom distribution with phantom realizations. We denote the phantom probability disribution of x̃ as p = {p(s) = P(x̃ =
x(s)) ∈ ΛX : s ∈ Ω} with x(s) ∈ X for each s ∈ Ω.
For a given initial wealth w ∈ R and a portofolio alocation a ∈ R, the welfare
of the DM is measured by V (a) = Γ(EP [U (w+ax̃)]). Since the probability distribution p is decomposed to the real term and the phantom term, it can be represented
as p = pre + P(p̂ − pre ). Hence, if we denote Ere [·] and Ê[·] as the expectation
under the probablity distribution pre and the one under p̂ resp., his/her expected
utility EP [U (w + ax̃)] is decomposed as
(
)
ˆ − Ere [U (w + ax̃re )]
EP [U (w + ax̃)] =Ere [U (w + ax̃re )] + P Ê[Û (w + ax̃)]
∑
=
pre (s)(Ure (w + axre (s))
s∈Ω
+P
(
∑
p̂(s)(Û (w + ax̂(s)) −
s∈Ω
∑
)
pre (s)(Ure (w + axre (s)) .
s∈Ω
Thus, V (a) can be rewritten as
V (a) =Γ(EP [U (w + ax̃)])
(
∑
=Γ
pre (s)(Ure (w + axre (s)),
s∈Ω
∑
p̂(s)(Û (w + ax̂(s)) −
s∈Ω
∑
)
pre (s)(Ure (w + axre (s))
s∈Ω
Hereafter, we assume that V (a) is concave w.r.t. a. One of the sufficient
11
condition the assumption holds is that the Hessian of Γ is negative semi-definite.5
Lemma 1. The demand for the phatom asset is positive(zero, negative) if the equity
premium is positive(zero, negative).
Proof. We only show the case such that the equity premium is positive. The
prooves of the remaining cases are shown quite similarly.
′
V ′ (a)|a=0 =Γre (Ure (w), Û (w) − Ure (w))Ure
(w)Ere [x̃re ]
′
ˆ − Ure
+ Γph (Ure (w), Û (w) − Ure (w))(Û ′ (w)Ê[x̃]
(w)Ere [x̃re ])
′
=Γre (Ure (w), Û (w) − Ure (w))Ure
(w)Ere [x̃re ]
+ Γph (Ure (w), Û (w) − Ure (w))
(
)
′
ˆ − Ere [x̃re ]) + (Û ′ (w) + Ure
× Û ′ (w)(Ê[x̃]
(w))Ere [x̃re ] .
Here, noting the equity premium is given as
(
)
ˆ − Ere [x̃re ] ,
E[x̃] =Ere [x̃re ] + P Ê[x̃]
we obtain the result.
We campare the optimal portfolio when the asset return is phantom with the
one when it is risky and not phantom. We denote the phantom asset return x̃ as
ˆ − x̃re ).
x̃ =x̃re + Px̃ph = x̃re + P(x̃
Let the phantom probability distribution of the phantom asset return be denoted
as (x̃, p) and the probability distribution of the rsiky one as (x̃re , pre ). For each
a ∈ R, we define a function h : R → R by h(x) = U ′ (w + ax)x.
5
(
)
′
′
ˆ x̃]
ˆ − Ere [Ure
V ′′ (a) = Ere [Ure
(w + ax̃re )x̃re ], Ê[Û ′ (w + ax̃)
(w + ax̃re )x̃re ]
(
)(
)
′
Ere [Ure
(w + ax̃re )x̃re ]
Γre2 Γreph
×
′
ˆ x̃]
ˆ − Ere [Ure
Γreph Γph2
Ê[Û ′ (w + ax̃)
(w + ax̃re )x̃re ]
where Γre2 , Γreph , and Γph2 are 2nd order derivatives of Γ, and we abbreviate their arguments for
the notational simplicity.
12
Proposition 2. We assume that the following conditions are satisfied.
(a) h is bounded concave;
(b) Ere [x̃ph |x̃re = x] = 0
∀ x;
(c) p̂ is dominated by pre in the sense of the SSD.
Then the optimal demand for the uncertain (phantom) asset is less than the one for
the risky asset.
Proof. If the conditions (a) and (b) are satisfied, from Theorem 2 of Rothschild
and Stiglitz (1970), we have
Ere [U ′ (w + ax̃re )x̃re ] ≥Ere [U ′ (w + a(x̃re + x̃ph ))(x̃re + x̃ph )]
ˆ x̃].
ˆ
=Ere [U ′ (w + ax̃)
(7)
Furthermore, the condition (c) is satisfied,
ˆ x̃]
ˆ ≥ Ê[U ′ (w + ax̃)
ˆ x̃].
ˆ
Ere [U ′ (w + ax̃)
(8)
Combing the above inequalities, we obtain
ˆ x̃].
ˆ
Ere [U ′ (w + ax̃re )x̃re ] ≥ Ê[U ′ (w + ax̃)
(9)
Hence if we denote are the optimal demand for the non-phantom asset satisfying the FOC; Ere [U ′ (w + are x̃re )x̃re ] = 0 and a∗ the optimal demand for the
phantom asset satisfying the FOC;
V ′ (a) =Γre Ere [U ′ (w + a∗ x̃re )x̃re ]
{
}
ˆ x̃]
ˆ − Ere [U ′ (w + a∗ x̃re )x̃re ] = 0 , 6
+ Γph Ê[U ′ (w + a∗ x̃)
then are ≥ a∗ .
Next we consider conditions in which a change in the phatom distribution of
phantom return increases the demand for the phantom asset. From Propopsitin 1,
6
Here we abbreviate the arguments of both Γre and Γph for the notational simplicity.
13
since a DM is phantom averse is equivalent that his or her utility Û is concave, by
the same arguments of Rothschild and Stiglitz (1970), if ŷ dominates x̂ in the sense
of the MPS(Mean Preserved Spread),
Ê[Û (w + aŷ)] ≤ Ê[Û (w + ax̂)],
for each a > 0. Hence for a given portofolio a, his/her welfare decreases if the
phantom ambiguity increases in the sense of the MPS. However, as Rothchild and
Stiglitz (1971) shows that the demand for the risky asset would not always be
reduced by all risky-averse DMs if its return undergoes an increase in risk in the
sense of the MPS, we can readily show that the demand for the phantom asset
would not always be reduced by all phatom-averse DMs if its return undergoes an
increase of the phantomness in the sense of the MPS.
For a given phatom asset return
ˆ − x̃re ),
x̃ = x̃re + P(x̃
the F.O.C. of the optimal demand for the phantom reterun is given by
(
)
ˆ − Ere [Ure (w + ax̃re )]
V ′ (a; x̃) =Γre Ere [Ure (w + ax̃re )], Ê[Û (w + ax̃)]
′
× Ere [Ure
(w + ax̃re )x̃re ]
(
)
ˆ − Ere [Ure (w + ax̃re )]
+ Γph Ere [Ure (w + ax̃re )], Ê[Û (w + ax̃)]
{
}
′
ˆ x̃]
ˆ − Ere [Ure
× Ê[Û ′ (w + ax̃)
(w + ax̃re )x̃re ] = 0.
(10)
From (10), we can hardly obtain clear sufficient conditions such that a shift of the
phantom return to increase the phantomness in the sense of the MPS leads the DM
to reduce his/her optimal demand for the phantom asset. However, if the function
Γ is assumed to be such an affine function as
Γ(x, y) = αx + βy + γ,
14
with positive constats α and β, and a constant γ, then since (10) reduces to
′
V ′ (a; x̃) =αEre [Ure
(w + ax̃re )x̃re ]
{
}
′
ˆ x̃]
ˆ − Ere [Ure
+ β Ê[Û ′ (w + ax̃)
(w + ax̃re )x̃re ] = 0,
(11)
we obtain a similar proposition to that under risk stated in Rothschild and Stiglitz
(1971) (see also Gollier (2001)).
Let x̃ and ỹ be phantom asset return defined by
ˆ − x̃re )
x̃ =x̃re + P(x̃
ỹ =x̃re + P(ỹˆ − x̃re ).
ˆ
That is, ỹ is different from x̃ in the term of ỹ.
Proposition 3. Assume that Γ is an affine function defined by (). Then a shift of
phantom asset return from x̃ to ỹ decreases the demand for the phantom asset if
ˆ
ˆ Û ′′ (w+aX̃)
1. ỹ is FSD-dominated by x̃, and if relative risk aversion −(w +aX̃)
ˆ
′
Û (w+aX̃)
is less than unity;
ˆ
ˆ Û ′′ (w+aX̃)
2. ỹ is SSD-dominated by x̃, and if relative risk aversion −(w + aX̃)
ˆ
′
is less than unity, increasing and absolute risk aversion
Û (w+aX̃)
ˆ
Û ′′ (w+aX̃)
− ′
is deˆ
Û (w+aX̃)
creasing;
ˆ
ˆ Û ′′′ (w+aX̃)
3. ỹ is SSD-dominated by x̃, and if relative prudence −(w + aX̃)
ˆ is
′′
Û (w+aX̃)
postive and less than 2.
We assume the assumption of Theorem 1 holds and the wealfare of DM A and
DM B are given by
(
(
)
)
ˆ ] − Ere [Ure (w + ax̃re )] ,
VA (a) =Γ Ere [Ure (w + ax̃re )], Ê[ϕ Û (w + ax̃)
(
)
ˆ − Ere [Ure (w + ax̃re )] ,
VB (a) =Γ Ere [Ure (w + ax̃re )], Ê[Û (w + ax̃)]
where ϕ is concave and increasing. That is, DM A is at least as phantom averse as
15
DM B. We define some notations as follows.
aA = arg maxVA (a), aB = arg maxVB (a),
[ (
)]
ˆ
ˆ
âA = arg maxÊ ϕ Û (w + ax̃)
, âB = arg maxÊ[Û (w + ax̃)].
Proposition 4. Suppose that DM A is at least as phantom averse as DM B and that
either âA ≤ aA ≤ âB or âA ≤ aB ≤ âB holds. Then, the optimal investment of
DM A in the phantom asset is less than that of DM B.
Proof. First, we confirm that âA ≤ âB holds. Since ϕ is concave from the assumption, xϕ′ (û(w + ax)) ≤ xϕ′ (û(w)) holds for each x ∈ R and a ≥ 0. It
implies
ˆ Û ′ (w + ax̃)
ˆ x̃]
ˆ ≤ ϕ′ (Û (w))Ê[Û ′ (w + ax̃)
ˆ x̃].
ˆ
Ê[ϕ′ (Û (w + ax̃))
ˆ x̃]
ˆ = 0 implies Ê[ϕ′ (Û (w + ax̃))
ˆ Û ′ (w + ax̃)
ˆ x̃]
ˆ ≤ 0. That
Hence Ê[Û ′ (w + ax̃)
is, âA ≤ âB holds. In the following proof, we only show the case in which âA ≤
aA ≤ âB holds since the other case can be shown quite similarly.
Since âA ≤ aA ≤ âB ,
ˆ Û ′ (w + aA x̃)
ˆ x̃]
ˆ ≤ 0 ≤ Ê[Û ′ (w + aA x̃)
ˆ x̃].
ˆ
Ê[ϕ′ (Û (w + aA x̃))
That is,
′
ˆ Û ′ (w + aA x̃)
ˆ x̃]
ˆ − Ere [Ure
Ê[ϕ′ (Û (w + aA x̃))
(w + aA x̃re )x̃re ]
′
ˆ x̃]
ˆ − Ere [Ure
≤Ê[Û ′ (w + aA x̃)
(w + aA x̃re )x̃re ]
(12)
holds. On the other hand, since ÛA (x) = ϕ(Û (x)) is invariant with respect to
affine transformations, we can assume that
ˆ − Ere [Ure (w + aA x̃re )]
Ê[ϕ(Û (w + aA x̃))]
=Ê[Û (w + aA x̃)] − Ere [Ure (w + aA x̃re )]
holds with out losss of generalities. Hence we can assume following equalities
16
hold.
(
)
ˆ − Ere [Ure (w + aA x̃re )]
Γre := Γre Ere [Ure (w + aA x̃re )], Ê[ϕ(Û (w + aA x̃))]
(
)
ˆ − Ere [Ure (w + aA x̃re )] ,
= Γre Ere [Ure (w + aA x̃re )], Ê[Û (w + aA x̃)]
(
)
ˆ − Ere [Ure (w + aA x̃re )]
Γph := Γph Ere [Ure (w + aA x̃re )], Ê[ϕ(Û (w + aA x̃))]
)
(
ˆ − Ere [Ure (w + aA x̃re )] .
= Γph Ere [Ure (w + aA x̃re )], Ê[Û (w + aA x̃)]
(13)
From (12) and (13), we have
0 =VA′ (aA )
′
=Γre Ere [Ure
(w + aA x̃re )x̃re ]
{
}
′
ˆ Û ′ (w + aA x̃)
ˆ x̃]
ˆ − Ere [Ure
+ Γph Ê[ϕ′ (Û (w + aA x̃))
(w + aA x̃re )x̃re ]
′
≤Γre Ere [Ure
(w + aA x̃re )x̃re ]
{
}
′
ˆ x̃]
ˆ − Ere [Ure
+ Γph Ê[Û ′ (w + aA x̃)
(w + aA x̃re )x̃re ] = VB′ (aA ).
This leads to the desiered result.
17
6 Numerical Examples
Consider two cases where distributions of the phantom asset returns follow (x̃1 , p1 )
and (x̃2 , p2 ), where



(0 − P1,


(x̃1 , p1 ) =
and
(x̃2 , p2 ) =
1/3 − P2/15);
(0 + P0, 1/3 − P2/15);



(0 + P1, 1/3 + P4/15),



(0 − P1,
1/4 − P1/20);




(0 − P1/4, 1/4 − P1/10);


(0 + P3/4,




(0 + P5/4,
(14)
(15)
1/4 + P1/10);
1/4 + P1/20).
(14) and (15) mean
and



(−1, 1/5);


ˆ1 , p̂1 ) = (0,
(x̃
1/5);



(1,
3/5),



(−1,
1/5);




(−1/4, 3/20);
ˆ2 , p̂2 ) =
(x̃


(3/4,
7/20);




(5/4,
3/10).
(16)
(17)
ˆ2 , p̂2 ) is a mean prereserved spread
From (16) and (17), we can readily check that (x̃
ˆ1 , p̂1 ). This means that shift from x̃1 to x̃2 increases the phantomness in the
of (x̃
sense of the MPS.
As to the utility function U , we compare two cases;
Case I
(
)
3
Ure (x) = Û (x) = x ∧ 3 + (x − 3) ,
10
18
x ∈ R.
Case II
Ure (x) = Û (x) = − exp(x),
x ∈ R.
We note that, in the both cases, since Û is concave, the agent is phantom averse.
For each investment amount a into the phantom asset, the welfare function is
given by
V (a) =Ere [Ure (w + ax̃re )] +
}
1{
˜ − Ere [Ure (w + ax̃re )]
Ê[Û (w + ax̂)]
3
Then in each case, the optimal investment a∗ into the phantom asset is given as
follows.
Case I When the phantom asset return follows x̃1 , the optimal investment a∗ is 1.
On the other hand, a∗ = 4/3 when it follows x̃2 .
Case II When the phantom asset return follows x̃1 , the optimal investment a∗ is
ln(3)/2 ≈ 0.55. On the other hand, a∗ ≈ 0.51 when it follows x̃2 .
That is, in Case I, although the agent is phantom averse and shift of phantom
return in the direction of increasing the phatomness, he/she increases the demand
for the phantom asset.
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