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STOCHASTIC DOMINANCE ON OPTIMAL PORTOFOLIO
07/12/2009 00:52
Stochastic dominance on optimal portfolio with one risk−less
and two risky assets
Jean Fernand Nguema
LAMETA UFR Sciences Economiques Montpellier
Abstract
The paper provides restrictions on the investor's utility function which are sufficient for a
dominating shift no decrease in the investment in the respective asset if there are one risk free
asset and two risky assets in the portfolio. The analysis is then confined to portfolio in which
the distributions of assets differ by a first−degree−stochastic dominance shift.
Citation: Nguema, Jean Fernand, (2005) "Stochastic dominance on optimal portfolio with one risk−less and two risky assets."
Economics Bulletin, Vol. 7, No. 7 pp. 1−7
Submitted: June 29, 2005. Accepted: July 25, 2005.
URL: http://www.economicsbulletin.com/2005/volume7/EB−05G10007A.pdf
1
1. Introduction
The question of whether risk-averse investors should or should not choose diversified
portfolio has been analysed quite extensively in the literature; for exemple Fishburn and
Porter (1976), Kira and Ziemba (1980), Meyer and Orminston (1989) and recently Meyer and
Orminston (1994) extend and generalize this literature by considering portfolios containing
more than one risky asset. The first research assumes that the risky returns are independently
distributed and the last ones, focuses on removing this independence restriction, allowing the
risky returns to be stochastically dependent. All of these studies were limited to one decision
variable, so they were not able to analyze the effects of generalized FDS on optimal portfolio
with three assets or more. We extend these earlier works by considering portfolios with one
risk free asset and two risky assets, and we analyze the effects of shift in the sense of firstdegree
stochastic dominance (FDS).
The analysis of comparative static’s for portfo lios with two or more asset is rather
difficult. Following Hart (1975), in order to derive the desired effects of changes in the
investor’s wealth, the utility function must possess a particular type of separation property,
and, as it turn out, not many classes of utility functions satisfy this property. For the case of
portfolio with one risk free asset and two risky assets we were able to determine the effects of
shifts mentioned above.
Our paper provides very simple conditions on the utility function which are necessary
and sufficient for a dominating shift in the distribution of asset i not to cause a decrease in the
investment in that asset. Two implications of independence restriction play an important role
in our analysis. First, independence1 allows one risky return to be altered without changing
the marginal or the conditional cumulative distribution functions (CDF) for the others.
Second, when the return to a risky asset is independent 2 of other risky returns, the marginal
and each of the conditional CDF’s for that return can be changed in exactly the same way.
The paper is organized as follows. In the next section, the notation and assumptions of
one risk free asset and two risky assets portfolio decision model are presented and one main
comparative static result is viewed. Following this, comparative static issues that only arise
with stochastic independence are discussed in section 3. In section 4, for First stochastic
dominance change, we determine the conditions on the decision maker’s preferences that are
necessary and sufficient for the change to cause an increase in the proportion of wealth
invested in the asset whose return is altered. Finally, concluding remarks appear in the end.
2. The model and assumptions
We consider a risk-averse individual endowed with non-random initial wealth who
allocates his wealth normalized to one between one risk free asset (with return x0 ) and two
risky assets with returns y ~ and z~ . The portfolio share of risky assets y ~ and z~ are a1 and
a2 , respectively. The individual’s end of period wealth W
~
is then equal to
( ) ( ) ( ) 0 1 0 2 0
,~,~ 1 ~ ~
~
W ai y z = + x +a y - x +a z - x (1)
Realizations of W
~
depend on realizations of y ~ , z~ and the selected values for a1 and a2 . The
random returns y ~ and z~ are assume to take values in the interval [0, 1].
The joint cumulative distribution function (CDF) for y ~ and z~ is denoted H(y, z). The
conditional and marginal CDF’s for y ~ are denoted F(y z ) and F(y), respectively, and for
1 Hadar and Seo (1990) make this their ceteris paribus assumption.
2 Hadar and Seo (1990) assume this when they require the risky returns to be independent both before and after
one return is altered.
2
z~ there are ( ) y z G and ( ) z G . If y ~ and z~ are independently distributed then F(y z )=F(y)
for all z , G(z y)=G(z ) for all y , and H(y, z )= F(y)×G(z ).
The decision maker is assumed to choose ( ) a1 , a2 to maximize the expected utility (EU) from
the terminal wealth taking random variable y ~ and z~ as given. Formally, ( ) a1 , a2 is selected
to maximize EU, i.e. optimal portfolio solves the following program (P):
ò ò ( ( )) ( )
1
0
1
0
2
,
max , ~, ~ ,
1 2
u W ai y z d H y z
a a
(2)
To simplify notation, the symbol d 2H(y, z ) is used to denoted ( )
dydz
y z
H y z
ú úû
ù
ê êë
é
¶ × ¶
¶ 2 ,
and
u is the von Neumann Morgenstern utility function which is assumed to be three times
continuously differentiable, non decreasing and concave in W , with u'(×)>0 and u''(×)<0 .
Now, assume we have interior solutions, the first order conditions associated to the above
program are:
(~ ) '( ( , ~, ~)) ( , ) 0, 1
0
1
0
2
ò ò y - x0 u W ai y z d H y z = (3)
(~ ) ' ( ( , ~, ~)) ( , ) 0, 1
0
1
0
2
ò ò z - x0 u W ai y z d H y z = (4)
In the case of independence, condition (3) and (4) becomes, respectively
ò ò( - ) ( ( )) ( ) ( )=
1
0
1
0
0
~y x u' W , ~y, ~z dF y z dG z ai ò ò( - ) ( ( )) ( ) ( )
1
0
1
0
0
~y x u' W ,~y , z~ dF y dG z ai (5)
and
ò ò( - ) ( ( )) ( ) ( )=
1
0
1
0
0
z~ x u' W , y~, z~ dF z y dF y ai ò ò( - ) ( ( )) ( ) ( )
1
0
1
0
0
z~ x u' W , y~, z~ dF y dG z ai (6)
Evaluated at a1 = 0 , and a2 =0 (5) and (6) can be written, respectively as
( ( )- )ò ( + + ( - )) ( )
1
0
0 0 2 0
E ~y x u' 1 x a ~z x dG z (7)
( ( )- )ò ( + + ( - )) ( )
1
0
0 0 1 0
E z~ x u' 1 x a ~y x dF y (8)
Which have the sign of the expected excess return ( ( ) ) 0
E x~ - x . It follows that ai is positive if
and only if ( ( ) ) 0
E x~ - x is also positive, that is if x~ offers a positive risk premium.
3. Effects of wealth on optimal portfolio
We investigate the effect on the demand for risky asset when there are changes in
wealth. We design effects of wealth on optimal portfolio by deriving the first order conditions
with respect to the portfolio share of risky asset ai and the agent’s end of period wealth W .
We first have the next result
Proposition 1. Define the function
F( )º ò ( ( ))( - ) ( )
1
0
z u' W ai , y, z y x0 dF y z "z ³ x0 (9)
1) If R'a ³ 0 and F(× ) ³0 , then
3
( *, ~, ~) 0
*
£
dW y z
d
i
i
a
a
, for i =1, 2 (10)
2) If R'a < 0 and F(× ) <0, then
( *, ~, ~) 0
*
>
dW y z
d
i
i
a
a
, for i =1, 2 (11)
where a R' is the derivative of the coefficient of absolute risk aversion
Proof. Assume we have interior solutions, since u is concave, the first order conditions
associate to the above program (P) gives the optimal allocation *
ai satisfying the following
equations:
(~ ) ' ( ( , ~, ~) ( , ) 0, 1
0
1
0
* 2
ò ò y - x0 u W ai y z d H y z = (12)
(~ ) ' ( ( , ~, ~) ( , ) 0, 1
0
1
0
* 2
ò ò z - x0 u W ai y z d H y z = (13)
First, differentiating (11) and (12) with respect to a1 and a2 , respectively, second since u is
concave we have
(~ ) ' '( ( , ~, ~) ( , ) 0, 1
0
1
0
2 * 2
ò ò y - x0 u W ai y z d H y z < (14)
(~ ) ''( ( , ~, ~) ( , ) 0, 1
0
1
0
2 * 2
ò ò z - x0 u W ai y z d H y z < (15)
Defines *
ai as an implicit function of W( i y z )
a *, ~, ~ . We thus obtain
( )
( ) ( ( ) ( )
ò ò ( - ) ( ( ) ( )
ò ò -
= - 1
0
1
0
2 * 2
0
1
0
1
0
* 2
0
*
*
1
~ '' , ~, ~ ,
~ '' , ~, ~ ,
, ~, ~ y x u W y z d H y z
y x u W y z d H y z
dW y z
d
i
i
i a
a
a
a
(16)
and
( )
( ) ( ( ) ( )
ò ò ( - ) ( ( ) ( )
ò ò -
= - 1
0
1
0
2 * 2
0
1
0
1
0
* 2
0
*
*2
~ ' ' , ~, ~ ,
~ '' , ~, ~ ,
, ~, ~ z x u W y z d H y z
z x u W y z d H y z
dW y z
d
i
i
i a
a
a
a
(17)
The denominator is unambiguously negative and clearly the sign of (16) and (17) depends
upon the sign of the numerator. Now, starting from (16)
- ò ò( - ) ( ( ) ( ) = ò ( ) ( )
1
0
1
0
1
0
* 2
0 , ~y x u'' W , ~y, ~z d H y z z dG z ai y (18)
where
( )º - ò ( - ) ( ( ) ( ) º
1
0
*
0
z y~ x u''W , y~, z~ dF y z B y ai (19)
Note that y (z ) is nonnegative if B³ 0
( ) ( ( ) ( ) ( ( )
= - ò - = - ò ( ( ) ( ( )( - ) ( )
1
0
0
*
*
1 *
0
*
0
' , ~, ~ ~ ' , ~, ~
'' , ~, ~ ~ ' ' , ~, ~ u W y z y x dF y z
u W y z
u W y z
B y x u W y z dF y z i
i
i
i a
a
a
a
4
= ò ( ( ) ( ( )( - ) ( )
1
0
0
R W * ,~y, z~ u'W *, ~y, ~z y~ x dF y z a ai ai (20)
= - ò ( ( ) ( ( )( - ) ( )
0
0
0
*, ~, ~ ' *, ~, ~ ~
x
Ra W ai y z u W ai y z x y dF y z (21)
+ ò ( ( ) ( ( )( - ) ( )
1
0
* *
0
, ~, ~ ' , ~, ~ ~
x
Ra W ai y z u W ai y z y x dF y z
" zÎ[x0 , 1] and [ ] " yÎ 0, x0 we have
W( i y z ) x (y x ) (z x ) x (z x ) W( i z )
,~
~ ~ , ~,~ 1 ~ ~ 1
~ *
0
*2
0 0
*2
0
*
0 1
a * = + +a - +a - £ + +a - = a . Hence
R'a > 0 implies that Ra (W( i y z ) Ra (W( i z )
a *, ~, ~ < a* , ~ . Thus
- ò ( ( ) ( ( )( - ) ( ) >
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