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THE CENTER FOR AGRICULTURAL ECONOMIC RESEARCH
Mean-Gini, Portfolio Theory and the
Pricing of Risky Assets
by
Haim Shalit and Shlomo Yitzhaki
Rehovc:'.1, Israe!, P.O.B. 12
12 .1.11
The working papers in this series are preliminary
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Mean-Gini, Portfolio Theory and the
Haim Shalit and Shlomo Yitzhaki
Rehovot, February 1982
MEAN-GIN:, PORTFOLIO THEORY, AND THE PRICII\,IG .
OF
RISKY . ASSETS
.
by
Haim Shalit:.and ShloMo * Yitzhaki
The Hebrew University of Jerusalem
•
ABSTRACT
' This paper presents the Mean-Gini (MG) approach to analyze risky
prospects and construct optimum portfolios.
The method possesses the
simplicity of the mean-variance model with the efficiency of stochastic
dominance.
Hence, Gini's mean difference is superior to the variance
for evaluating the variability Of a prospect.
The analysis is further
extended with the concentration ratio that permits to classify different securities with respect to their relative riskiness.
The MG
approach is then applied to capital markets and the security valuation
theorem is derived as a general relationship between average return
and risk.
This is further 'extended to include
degree of 'risk aversion.
that can be estimated from capital market data.
This research was supported by a grant from the Government of Israel,
the National Committee for Research and Development, Grant No. 5107.
We ,are grateful to Haim Levy for useful
errors are ours.
comments.
All the remaining
MEAN-GINI, PORTFOLIO ANALYSIS, AND THE PRICING OF RISKY ASSETS
The formal similarity between modelling decision-making under uncertainty
and evaluating and interpreting income inequality has been noted and used
by several economists.
An example is Samuelson
who used the same method
to prove that it pays to diversify risky investment (1967), and that equal
distribution of income among identical Benthamites.maximizes the sum of
social utility (1966).
dominance
Atkinson (1970) showed that ,the rules for stochastic
which were developed as criteria for evaluating risky invest-
ments, can be translated into Lorenz-curve terms in evaluating income
inequality.
The purpose of our paper is to interpret some of the recent results
on income inequality and apply them
o portfolio analysis.
In particular,
we establish that Gini's mean difference, the Lorenz curve, and the
concentration ratio, which are extensively used in the field of income
inequality, can also be used to characterize risky prospectsand construct
optimum portfolios.
The superiority of Gini s mean difference over the variance as an
index of variability has been demonstrated by Yitzhaki (1982).
In parti-
cular, it was shown that using the mean and Gini's mean difference as summary
statistics of the distribution of a risky investment permits the user to
derive necessary conditions for stochastic dominance, enabling him to
discard from the efficient set prospects that are stochastically dominated
by others.
This property means that Gini's mean difference is a better
candidate for evaluating the variability of a distribution.
In the present
and
paper, we argue that Gini's mean difference can replace the variance
ce needed
that the concentration ratio based on it can replace the covarian
in portfolio theory.
.ance
The mean-Gini portfolio selection rule is - superior to the mean-varl
of
selection rule in the Sense that the mean-Gini efficient set consists
be
portfolios that, according to stochastic dominance criteria, cannot
ariance
dominated by any other portfolio. . Furthermore, as with the mean-v
of' risky
model, the mean=Gini ranking can be. used to derive the pricing'
assets in an equilibrium framework.
But it is easier to compute efficient
simpler
portfolios by mean-Gini rules than by mean-variance rules and much
than using stochastic dominance rules.
Moreover, Gini's mean difference can
from
be extended into a family of coefficients of variability, differing
each other by a parameter,
that represents the investor's risk aversion.
It turns out that each member of this family can be used in portfolio
-)
theory for constructing capital-asset pricing models.
In the first section, we define Gini's mean difference and justify
its use.
d
In the second, the properties of a portfolio which is specifie
third section
by its mean and its Gini coefficient are developed, while the
presents some extensions of the Gini
coefficient to portfolio theory, such
as the concentration ratio (Kakwani, 1977, Pyatt
Chen, and Fei, 1980).
The fourth section is devoted to the capital-asset pricing model and
contains a discussion of the properties of the approach, whereas the
last section presents the extended Gini's mean difference (Yitzhaki,
1980) and applies it to portfolio analysis.
Much of the discussion in this paper is based on a reinterpretation
of results on income inequality, in the portfolio context.
Thus, whenever
possible we do not prove the propositions but refer the reader to the
_original papers.
GINI'S MEAN DIFFERENCE
I.
bility of a random variable.
mean difference is an index of the varia
of the absolute difference between every
It is based on the expected value
*
ble. That is, let F(y) and f(y)
pair of realizations of the random varia
distribution and the density function
respectively represent the cumulative
and b < co such that
exist a > 0O
of prospect y and assume that there
difference is defined as follows:1
F(a) = 0, F(b) = 1, then Gini's mean
1
r = 2
(1)
fa
,
a
y - xlf
f(y)dxdy .
e and one finds at least eight
This definition is not easy to handl
mean difference in the literature.
different formulations of Gini t s
with two of them.
our purposes it will be useful to deal
F(y)].:137 a
For
The first is
a
(2)
[1 - F(Y)]
Y
rence in terms of the expected
Equation (2) presents Gini s mean diffe
cumulative distribution function,
value of the distribution, p, and its
F(y).
3
ng with stochastic dominance criteria.
This equation is important when deali
for analysing portfolios is
The other Gini formula which is useful
r =
y[F y
2
a
_
2
f(y)dy
••
that is, Gini's mean difference is twice the covariance of variable
its cumulative distribution.
y
a
For completeness, we state the following pro-
perties of Gini's mean difference:
1.
If
The Gini coefficient is non-negative and bounded from above by
y
is a given constant, F = 0 .
p -
Furthermore, as is shown in Yitzhaki
(1980 ), its maximum value is reached for the distribution
2.
- The - Gini coefficient is sensitive to mean-preserving spreads
Atkinson,
1970).
ay
where
.
Let
where
5.
Let
, where
variables; then
.
Let
y
= a//if
3-
is a constant; then
2
is a constant, then
y 1 and
y . are two independently distributed
+ r
be a normally distributed variable with
p
and
a
2
then
(Nair, 1936)
••
Properties 2 to 5 are similar to those attributed to the standard
deviation.
Hence it is not surprising that the Gini coefficient can be used
to derive the efficient set of uncertain prospects in the same way as is
done using the mean-variance criterion.
This feature is implied-by the
behavior of investors. 'who rank uncertain 'prospects by their mean and the
dispersion of
returns.
Efficient sets
of
uncertain prospects ,,re
••
in the
construeted such that no other feasible prospect will be included
mean for
set unless it has a lower dispersion for a given mean or a higher
given dispersion.
Usually the standard deviation is used as the measure
.
f dispersion; we propose to use instead Gini's, mean difference-
Hence,
obtained by
the 'efficient set that answers the mean-Gini (MG). criterion is
. finding, for each given mean
that prospects xahich are not in the efficient
set have at least larger or equallGini's mean difference.
If combinations of
I
is obtained by
uncertain prospects are allowed to be held, the efficient set
the Gini
constructing, for each given mean, a mix of. prospects that minimize
coefficient of that portfolio.
We advocate the use of the mean-Gini (MG)
t'
method first because, if prospects are normally distributed, the efficien
iance
set of the mean-Gini is identical to the efficient set of the mean-var
(MV) Method (see property 6 above).
Secondly, it is justified by the superiority
s
n
f the mean-Gini over the mean-variance 'approach in ranking uncertai prospect
according to stochastic dominance .(SD) rules as we will now bring forward.
Let y l and y2 be two uncertain prospects.
Proposition
p
1
-
r1 a
p
2
- F
2
y
is a.necessary condition for
Lo first and second stochastic dominance rules
1
The condition
to dominate
Yitzhaki, 1982).
y
2
according
See Appendix
for a proof.
Define SmG the efficient set obtained by the MG method,
the efficient
set obeying the stochastic dominance first and second rules and
obeying proposition 1,
S
is also a subset of
we'assert the following:
S
is a subset o
the set
Scm and
S
MG
.•
•
••••
Mg
'Hence, applyjng.propp,pitionl:to the efficient set constructed by the
method enables. us
obtain an .efficient set which.is a subset of theefficient..
set according to first and second SD rules.
This subset is not liable to 'the
criticism usually advanced against the efficient set constructed by the MV
rule (See 'Rothschild and Stiglitz, 1970;
of proposition 1 can be demonstrated by an example:
uniformly distributed between
between
2
and
4 .
Both
0
and
and
yl
1
y2
to mean-variance and mean-Gini rules,
and
1
2
= 3 r
'
=
2
while
y2
assume that
is
y1
is uniformly distributed
are in the efficient set according
with p 1
=
1
2
r
1
1
6
1
1
But clearly all investors prefer
1
3
The use
Hanoch and Levy, 1969).
y2
over
•
Applying proposition 1 to the set 'obtained by the mean-Gini criterion
enables us to discard '
1
from the efficient set.
The superiority of the mean-Gini approach over the stochastic dominance
(SD) criteria results from its similarity to the MV method.
As far as we know,
there is no method for constructing optimum portfolios by 'SD rules.
Users of
the mean-Gini method may minimize the Gini, coefficient of a linear combination
of prospects subject to a given required mean return.
the user to construct the efficient set
Changing the mean permits
corresponding to the mean-Gini criterion.
This set canbe used for portfolio analysis and capital asset pricing equilibrium
according to MG rules.
Furthermore, by applying proposition 1 to that efficient
set, a subset is obtained which is contained in the efficient set of portfolios
according
o SD rules. .
PORTFOLIO ANALYSIS
portfolio whose performance is summarized by the mean
The properties of
to those of. the regular meat-Standard.
and. the Gini coefficient
These, properties can be illustrated by the familiar
.deviation model.
textbook diagram in which
r,
the Gini coefficient of the return on the
portfolio, is depicted on ,the horizontal axis, while,
the mean of the -
p
one-period return, is on the vertical axis, as in Figure 1.
The performance of two prospects,
in the mean-Gini space.
The return on portfolio
combination .of: the return on
is the share of
is denoted by
and B
A
and
A
wealth invested in
Hence
and
and B
is given by the convex
y
y
A
= aXA -17 (1 - a)X15-Where.
and
XR
are the :one-period..
returns.
As in the ordinary mean-standard deviation model the performance of
depends also on the correlation between
the portfolioY p
on
A
and
B
and
a . •To show this, consider the three special cases as drawn in Figure
First, if prospects
of correlation (p
A
AB) is
and
B
are linearly dependent, the coefficient
equal to unity and, following from the properties of
Gini's mean difference, the line ACB represents all the possibilities of
a portfolio mix composed of A
.
- 0), the curve
pendent (
'
P AB
y
portfolio,
performance.
if
A
and
B
and
B .
ADB
expresses the performance of the
Second, if
A
and
B
are inde-
showing intuitively that diversification improves that
Furthermore, the portfolio returns would be much improved
were negatively correlated as shown by the broken line
for the extreme case of
.
1
AFB
10
•
The effect of the variability of a prospect on the variability of
the portfolio can be presented much as it is in the mean-standard deviaijon.
By equation (3), the Gini coefficient of a portfolio i
model.
y F
(4)
a
=
y
Y
is the cumulative distribution of the portfolio.
F
where
-dF y) = 2cov
p
2
for
E a x.
1=1 i 1
0
i=
is the return on prospect
where
.
Since
1 , we Obtain
•
•
E a.cov[x., F yji
p
1
1
i=1
(6
that is, the risk ofthe portfolio can be decomposed into' a weighted sun)
of the covariance between the variables
x and the cumulative distribution
i
of the portfolio, p.
It
is worth mentioning that the variance of the portfolio can be
written as
(7)
var Y
P
a cov x , y
i=1
-
11
• and the difference in the decomposition of the nondiversifiable risk by
the two methods is that i
(6) the portfolio isrepresented by the cumu-
lative distribution of its. retui-ns,. F', while in (7) it is represented
by its returns
By multiplying and dividing each component of (6) by
where
(8)
where
-
2covk
-F.(
'
F (x') is the cumulative distribution of prospect i, we obtain
i
N
Z. .a.R.r.
-
iii
R. = cov[xi, Fp(yp)J/cov[xi
F (x)I which represents the ratio
non-diversifiable risk tothe variability of prospect
7
r. =
)]
-
12
-
THE CLASSIFICATION OF PROSPECTS BY RELATIVE RISK
III.
, F (y )]
One can classify a security by its risk by using cov[xi
p p
as an
whereas the diversiindex of the undiversified risk carried by a security i
by including
fied risk is the share of total risk that can be reduced
in the portfolio
security i
-
However, this classification is silent
yp
about the possibility that security
I
can be riskier than security
in one situation and less risky in another.
n curves
ne way to improve the classification is' o'use concentratio
on
us to determine the security's risk according to the return
that enable
the portfolio.5
' prospects
prospect
•I
ine, for any two
This classification enables us to determ
within portfolio
A and B
A
p , whether all investors agree that
•
is riskier than
B
with regard to portfolio
relative riskiness of
may disagree among them about the
p
A.
or that they
Assume two
ed rates of return;
securities with identical positive expect
expectation of rate of return
( 9)
g.(y ) as the conditional
J P
„ given the portfolio y ; that is,
x
Now define the function
E (x ) = E (x ) = p
J
y)
= E
g(y)
j jp
JP
We assume that
>
y
P
exists.
If E[g.(y )]
J P
security
j
, g.(y). > 0
J P -
and that the first derivative of
g( )
tration curve of
= P. , one can define the concen
as the relationship between
13
-
Y)f(Y)dY
(10)
f (y)dy .
=
F y
P P
P
Concentration curves are plotted in Figure 2 below.
seen that
if x. =
It can easily be
y, equation (10) represents the Lorenz curve of port-
.)
folioyp (curve B in the figure).
The relative riskiness of the securities
in a portfolio can be compared according to the following proposition:
Proposition 2:7
The concentration curve for the function
above (below) the concentration curve for the function
less (greater) than
f r all, y
,) where!:n
g (y ) will b
P
g (y ) ifn.(Y p) is
i p
is the elasticity of
with reapect to
Two corollaries follpw'from proposition
corollaryl:neconcentrationcurveforthefunction-W
gJ
(below) the egalitarian line (45
line) if
(y)
will be above
is less.(greater) than
zero, for all .y .
That is, in Figure 2, the 45degree line represents the risk free asset.
Stocks which are always negatively correlated with the portfolio have a
0
concentration curve which is above the 45 -degree line.
14
•••
The second corollary permits comparison between the securities and, the
portfolio they make up.
Corollary 2:
The concentration curve for the function
(below) the Lorenz curve for the distribution
(greater) than unity for all y
F (y ) if
P P
.(y)
lies above
1-1.(y ) is' less .3 P
> 0'
Corollary 2 permits us to distinguish between two kinds of securities
in a
, portfolio
First, we observe aggressive securities, with concent-
ration curves below the Lorenz curve of the portfolio, whose high.degree of
responsiveness to the
.portfolio
leads to considerable instability.
Second, we have defensive securities, with concentration curves above the
Lorenz curve of the portfolio which reduce instability because they are less
responsive..
These results are summarized
Figure .2.
Let OAB be the Lorenz. curve of the portfolio.
The aggressivestock will b
represented by OCA while the • defensive stock is represented by the concentration
curve ODA.
0
The 45 line portrays the risk-free asset (if it exists) while
OFA represents a stock which is negatively correlated with the portfolio.
By construction and definition, the relative Gini coefficient is 1-2 (area
under
j
is
the
Lorenz curve) and the relative concentration ratio for security
area under the concentration curve for security
Hence,
-
16
whenever concentration curves do not intersect:, the classification o
securities by the Gini coefficient and the concentration ratio truly
represents their relative riskiness.
That is, if cov[
F (y )] >
P P
cov[y,F (y )1, . security i is said to be aggressive, and it .1S
P P
P
said to be defensive if the inequality is reversed. However, when
concentration curves intersect, the relative riskiness of security
depends on the investor aversion towards risk.
Thus, different investors
may disagree about the relative riskiness of securities.
1.7
IV.
THE PRICING OF RISKY ASSETS
In this sectionwe develop the security valuation theorem for investors
'holding mean-Gini efficient portfoiios.
The
CAPM
market-equilibrium
relationship has been formulated for MV efficient portfolios by Treynor
•(1961), Sharpe (1960., Lintner (1965), and Mossin (1966).
The theorem
states that for any security, the higher its nondiversifiable risk,
the higher will be its expected return.
Nondiversifiable (or systematic)
risk is that part of the security's total risk that cannot be reduced
by diversifying a portfolio without reducing its expected rate of
return.
The theorem is stated in the context of competitive financial
markets without taxes and without restrictions on short selling and
borrowing.
In these markets investors trade risky assets whose
quantities are known and fixed, to build efficient portfolios that
answer their preferences.
By doing so, they actin.the securities
••
market, building forces that influence and determine the value of
these securities.
Assuming that investors build their portfolios
according to a ,MV utility, ,the familiar
CAPM
expected return and risk is expressed as
[cov(x., y )]/0
m
relationship between
18! -
where
= E(x
p
is the expected rate of return on security
=is the rate of,return on a risk-free security.
= E(y
a
2
is the expected rate of return on the market portfolio
= variance of the rate o
return on_theimarket portfolio.
Equation (11) was derived under several assumptions of which the most important
are (a) single-period analysis, (b) the existence of a risk-free asset, and
) perfect competition in the securities market.
Since then, most efforts in modern financial theory have been directed
o adapting the model, to different. contexts and testing it empirically.
Much less attention has been devoted to whether its theoretical foundations were sound, with the notable exception of stochastic dominance theory
(SD).
Unfortunately, the SD approach was found to be computationally
cumbersome and cannot provide answers to investors' questions on security
valuation in terms,of portfolio diversification.
The valuation theorem proposed here retains the main assumptions of the
classical CAPM.
However, instead of, holding MV efficient portfolios,
investors construct market portfolios which are SD efficient
by the mean-Gini approach.
This is done
-
19
by choosing a securities
Each investor determines his optimum portfolio
of the portfolio given its
mix that minimizes the Gini's mean difference
' expected rate of return.
j's portfolio is given by
The rate of return of investor
riskless rate,
(12)
Investors are permitted to borrow and lend at the
Eai <1
ix
• • •-1-.
.
Y. =a
11
•
where
x,
is the rate of return on security
. is the share of investor
1,
wealth invested in security
investors' problem is to minimize
.
+ (1
.
a3
11
i=1
=
, ,
007
i.
Hence
the
subject to
N
(13) •,E
Iaj
i=1
lio can be written as
Recall that the Gini coefficient of the portfo
= 2cov[y.
J
F(y)] =
2 E a cov x , F.(37.)]
J
i=
simply:
Thus the necessary conditions for a minimum are
N
:(15) 2
COV
and equation
investor
a
k=1 k
cov[xk,Fj(y.j)]
1
is the Lagrange multiplier ,associated with
-
As we saw earlier,
between security
20
-
cov[x., F (y )]
j
1
is the concentration ratio8
It represents.
and .the.portfolio'of individual
i
the degree by which the risk of security
i
cannot be 'diversified by
including it in j's portfolio.
By property 3 of the Gini coefficient, we know that
homogeneous of degree one in
a4
.
is
Therefore
r.
N
E a3.
1
i=1
*
jt
(16) • r..
J
aa.
i
or
N ,
3
F(y)]
E a.
cov[x'
i=1
(16a)
implying
N
N
E sot
2 E
i=1 k=1
.
a3
k
.coV
xk, F.
kt.
by equation (14) that the double sum on the right-hand-side
vanishes.
By multiplying each of the conditions i
summing over all the securities
(17)
= A.E
r.
j
a.
N
.
(15) by its share et4
and
we obtain for every investor
or
(18)
F.
3 1=1
1
1
and we have obtained the relation between risk (expressed by the Gini)
f the portfolio and its expected return as
(19)
r. = A. E( .) - rf
3
3
3
-
21
Equation (19) represents the highest feasible straight line
• in the (P,
r)
ions of
space given an efficiency frontier constructed by portfolio combinat
that space
risky assets. ,Since the efficiency frontier is concave within
a minimum are
due to properties - 3 and 5, the second-order conditions for
satisfied.
In that context 1/A, is .the investor subjective price of risk
portfolio to its
since it relates the expected rate of return of the chosen
•
mix along that line that maximizes
o
portfoli
a
chose
will
investor
The
risk.
•-•
his utility.
have
Now assume a market of similar investors who are risk averse
of.
identical investment opportunities, and minimize the Gini coefficients
their portfolios subject to their expected rates of return.
In that case,
equation (19) will be identical for all the investors in that market.
For a given risk-free rate of return, the unit price of risk will be
equal and determined by the slope of the market line in the
, r)space
(see Figure.3)
For an investor which does not borrow nor lend, all his wealth will be
invested in risky secdrities whose portfolio is the market portfolio
in Figure 3.
where
p
coefficient.
investor
pictured
Thusfbr all investors, the price of risk will be
the expected return on the market portfolio and
r
is its Gini
In that case, the Gini coefficient of the portfolio held by
is equal to
'since the optimum ranking,
remains unchanged whether or not •
F (yi)
investor
risk..-free'security is added and thus
for all investors. 'Thus, from (15)
is equal to F(y)
m m
)]
F
3 cov[x
N
' m m
k
E a3
(21)
Da:
k
k=1
and the equilibrium condition for every security
and thvestor
becomes
- market portfolio
23
(22)
p.A
r
• 2cov x,, F
m
1
M
y
M
/ F
This is essentially the CAPM valuation relationship for a market o
investors using the mean-Gini approach.. •To understand the dependence between
systematic risk and expected return, let us rewrite
R F
im i
2cov[x , F(y)]
m m
as
where
- F(y )]
, coy
m m
R. =
cov[x. F. x.)]
im
1.
a sort of rank correlation coefficient and
Thus
(23)
,
Therefore, security
(24)
beta is simply
= R. (r./r )
im 1 m
As is well known,
of return of security
represents the degree of responsiveness of the rate
to changes in the market, and
Rim is the
proportion of total risk .(expressed by the Gini coefficients of security
I,r
that cannot be/eliminated by the market without reducing the expected rate o
return.
24
At this point, it is important to draw the analogy between the
in (24) and the
f3.
derived from the MV-CAPM.
normally distributed securities,
and
rn
. =
/ill
I.
It can'be shown that for
r'=-
and
aff.
where
are the standard deviations of i and
Therefore, R
does converge to. the Pearson correlation coefficient
im
between security
i
and the market
This assertion is intuitively deduce
since in the case of normally distributed prospects
coincide for the same set of observations.
the MV and MG 'betas'
However this is not always true
for prospects that are not normally distributed.
In general, MV and MG betas
will be different; with the MG betas corresponding to SD efficient securities
markets.
Furthermore when investors use the Tean-extended-Gini (MEG) approach
to evaluate risk and construct efficient portfolios, the computation and
",.
estimation of betas will depend on their attitude towards risk, as will now
be shown, when we" extend the analysis with the MEG.
—
25
THE EXTENDED GINI COEFFICIENT
In this section we develop the extended Gini coefficient and apply it to
portfolio analysis and the CAPM.
Gini'a mean difference may be extended
into a family of coefficients of variability differing from each other in
the decision-maker's degree of risk aversion, which is reflected by the
Rewrite equation (2) as
parameter
JU
F(y)]
(25.)
where 1< v <
is a parameter chosen by the user of the method.
We define
(25) as the extended Gini coefficient whose properties are (Yitzhaki, 1980)
The extended Gini coefficient is non-negative and bounded from above
1.
v .
for all
That i
0 S r(v) < 1.1 - a
It is anon-decreasing function
The proof of this property is immediate if we remember that
1
2.
F(x) >'O..for ,
For
simply
x < b .
, 4
r(v)
, x)]
X'c)b
min(xi)
Thus,
integers, the extended Gini coefficient is
E[min (x.)]
.v.
i=1,2
1.1 (x.) < yl =
i=1,2..v
rb
1 [1-F y)]
•o)
y for
since
6b
1
Y
-F
7)1 ,
integer
The extended Gini coefficient is sensitive to mean-preserving,'spreads.
.
If
=
hen
26 - -
• Proof:
Since
a
F
for a'> 0
we get
ab
j
{[
a
] - [1
Let
Y 2 = Y1
Let
y1, y
for
F ( )]v)
=.T"1(v)
then
be two prospects; then[p-r
, 2, 3, •
1
]-
is the necessary condition for first and second degree
stochastic dominance (Yitzhaki 1982).
•
7.
Let
yl, y2 be two prospects with cumulative distributions that intersect
most once.
Then [111 - r i( )]
is a sufficient condition for
(Yitzhaki
y1
- [P2 - r2(v)]
v = 1, 2,:3
>,0 for all
to stochastically dominate
1982).
The interpretation of
can best be seen by looking at
v
Its value for different values of
p
'r(v).
are
v= 1
11(v) =
v= 2
V
and it is a non-increasing function of
v . Thus one can view F(v) as
the risk premium that should be substracted from the expected value of the
distribution.
while
v
00
The case
V
= 1 represents the risk-neutral investor
represents the investor who is interested in the minimum
value of the distribution, and maximizes this minimum.9
27
portfolio can be decomposed -into •
The extended Gini coefficient
an equation similar to (6). "That i
(26)
r v) = -v
Proof:
E a cov x ,
1=1
Let
y = E ,a x
i=1
P
1
F
v
and
v =
)]
F (y)
u
}
hen
r' (v) "=P
Defining
•
dy .
as
1 - F(y)]
F(y)
(y)
•
U= y
and applying integration by parts, we get
)]vy
- F (y)]v- yf (y)dy
P
P
[1- F(y
- 28
t
y)
F (y)
y dy =-[
thus
1[1 - F
30]
1
dy
Yfp Y
yfp(y)dy
F(y)
F
= -VCOV
1_ 1
v
F (y)
v-1
Therefore
(27) r (v) =
1 - F
cov x.,
i
-1}
since y• =
1
Q.E.D.
prospect, is its covariance with
That is, •the nondiversified risk
the portfolio rank
the power
.
The higher
the greater the
weight given to the performance of the prospect when the yield of the
portfolio is low.
Note that if
v = 2 , then we have the regular Gini
coefficient.
We derive the CAPM, using the extended Gini of degree v.
, the extended-Gini of his portfolio is given by
F (v) = -v E a4covf[
1
i=1
F.(y.)]Vl, x.}
1
J 3
For investor
-
29
that
chooses
Investor
F.( ) subject
minimize
to the expected rate of return on the portfolio given by (12).
v . reflects the degree of risk aversion.
As •shown,
Hence, it is .possible to model
a securities market that will exhibit different market portfolios because
Of different degrees of risk aversion.
For the present, we require that .
investors are similar and that they display identical
all j
.
Therefore
the necessary conditions for a minimum are
-vcovf 1 -
J
J
(28)
the market portfolio's extended Gini coefficient o
where
r() is
degree
v , and the market-equilibrium relation becomes
v-1
(29)
If. the !rank ,correlation .ratio
degree
and the _market is
covU1
(v) =
•••
cov{0.
m
m W-
-1
, x.}
x}
v
between security.
30
the market—equilibrium valuation for every
and any
is given by .
r(v)
- (30)
(v)
t (11)R.
f im
r m(v)
Thus,
(31)
= R.
im
Ti
. rm(v)
Thus, even if investors have the same attitude towards risk as expressed
by
v, different efficient market portfolios and different systematic risks
for each security will be obtained.
estimating betas.
This feature must be borne in mind when
It must be added that if securities are normally distributed,
the MEG 6's will b .identical to the MV betas, independently of
But our
concern for the existence for different efficient market portfolios is
-principally directed towards distributions other than normal such as the
log normal and the uniform distribution.
For these 'distributions, the MV
approach is not consistent with expected utility maximization and stochastic
dominance, whereas the mean—Gini i
10
VT. CONCLUSION
We have presented a new approach to analyze risky prospects and construct
optimal portfolios.
This method owns the simplicity'of a two-parameter
model with the efficiency
f Stochastic dominance.
To that extent,
we claim that Gini's mean difference is superior to the variance for
evaluating the variability of a security.
Furthermore, the concentration
ratio based on the Gini coefficient permits us to classify different
securities with respect to their relative riskiness.
Finally we have
applied the MG approach to capital markets and the security valuation
theorem was derived on a general relationship between average return
and risk.
By extending
we have explicitly
the analysis with the mean-extended-Gini method,
introduced the degree of risk aversion as a parameter
that can determine the specific composition of market portfolios.
The main implications of the model reside as whether investors,
in general, behave more in a MG or MEG framework rather than follow the
MV approach.
These implications can be empirically tested by estimating
the performance of CAPM for different degrees of
these with the results obtained from
MV-CAPM.
models were tested using regression techniques.
v
and comparing
Two-parameter portfolio
This was notably
exemplified by Fama and MacBeth (1973) who supported the hypothesis
that risk-averse investors hold efficient portfolio in terms of mean
and standard -deviation of the returns.
Recently, certains doubts were
-
32
raised as to whether the ,different regression procedures were
to test the CAPM (see Ross, 1980)
By proposing the mean7Gini and .
mean-extended-Gini models as means of evaluatingcapital market data,
we add a new dimension to modern, finance theory, suggesting that we
should return to the drawing •board.
.•
•
33
Appendix
.
(Yitzhaki, 1982).
Proposition
The condition p i -
a necessary condition for y
is
to dominate y
first and second stochastic rules for, v
Proof:
according to
0.
For y l. o dominate y2 according to F§D and SSD rules it is
necessary that
for all y
(A.1)
and
ry
for all y.
t dt
t)dt 5_
implies
The condition
(A.3)
If
af
-F
v
1-F 37) dy
2
dy
for V > 1.
v = 1, the proposition is proved directly since (A.3) holds whenever
(A.1) and (A.2) exist.
For v > 1
v
z
strictly convex for positive
'(A4
[1-F
1
O]v >
we know that since the function
-F(y)]
+
z.v
v[i-F2
v-1
(z-z ).
V z
y)]
y)
is
Thus
(y)]
and
fb
5),
y > v [ 1 -F
a
jb
1[1-F (y)]v-
(A.5)
1-F
a
]dt
since
a
0
for ally by
A.2)
afy
and
[1-F
v-1
is non negative and non increasing in y,
y)]dy
34
3.
7)]V71.
(A.6)
(y) - F (y)]dy
2
a
0
for v > 1.
by the following lemma.
Thus from (A.5) we have
. (A.7) .1
whenever
Lemma:
Proof:
dy
y)
stochastically dominates
Let h(z) be a non-negative and non-increasing function of z, and
let . g(z) be a :function with the property that ixg(z)dz
a).
Then I
z)-g(z)dz
0 for all x.
a
0
Assume
at
g(x) changes signs
X) .
' 2'
.
n
n
times between
Then g(x)r
< x <:x , and so on.
).11(z)dz
h
jg
a
we have
z)-h z dz
0
and
b
0 -for a <
< x and g(x) < 0 for
x
2 .
Thus f g(z)-h(z d ?. h(x )1x2g(z)dz ?.. 0.
1 j
a
'
ince this argumOnt can be repeated for
aT
a
for all x.
for all x.
)dz
. when i =
0 ,
- 35
FOOTNOTES
For simplicity of presentation, continuous and bounded random variables
wiT1 be used throughout the paper, keeping in mind that most of the
results can be applied to discontinuous and unbounded distributions.
The index
I
will be omitted wheneverit is not necessary to disting-
uish between two variables.
mean difference.
Note that we use half of Gini's original
Note too that we here use the absolute forms of
Gini's mean difference and the concentration ratio; that is, we do
not follow the more usual practice of dividing by the mean (see Kendall
and Stuart, 1977).,
2For derivation of equation (2) for continuous, discrete and unbounded
distribution see Dorfman (1979).
3For-the definitions of the stochastic dominance rules see Hanoch and .
Levy (1969) and Rothschild and Stiglitz (1970).
'For derivation of formula (3) see Kendall and Stuart (1977) for the
continuous case and Pyatt, Chen, and Fei (1980), in the case of discrete
distributions.
5For a discussion of the properties of.the concentration curve see
Kakwani (1977) and Pyatt et al
6The restrictions
g.( )
(1980)
0 and yp > 0 should be interpreted as
shift of the origin.
7This proposition and the two corollaries were proved by Kakwani (1977).
8Note that to obtain the concentration ratio .as defined by Pyatt'et al;
(1980), covix., F(yi)]
9
should be divided by
Ei(xj)
Values of 0 < v < 1 represent the case of the risk lover, the extreme
case
v = 0 representing the investor interested in the maximum value of
the distribution, b (the max-max investor)
In this paper, we restrict
ourselves to risk-averse investors.
10See Yitzhaki (1982) for a consistency analysis of the mean-Gini
approach for different distributions.
3
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