Journal of Financial Economics 2 (1975) 9S-121. (QNorth-Holland
OPTIMAL,
RULES
FOR
ORDERING
Vijay
Publishing Company
UNCERTAIN
PROSPECTS+
S. BAWA
Bell Laboratories, Holmdel, N.J. 07733, U.S.A.
Received October 1974
In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all
individuals with decreasing absolute risk averse utility functions. The optimal selection rule
minimizes the admissible set of alternatives by discarding, from among a given set of altematives. those that are inferior (for each utility function in the restricted class) to a member of the
given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule
when comparing uncertain prospects with equal means. We also show that in the general case
of unequal means, no known selection rule uses both necessary and sufficient conditions for
dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest
admissible set. The TSD rule is complex and we provide an efficient algorithm lo obtain the
TSD admissible set. For certain restrictive classes of the probability distributions (of returns
on uncertain prospects) which cover most commonly used distributions in Enance and
economics, we obtain the optimal rule and show that it reduces to a simple form. We also study
the relationship of the optimal selection rule lo others previously advocated in the literature.
including the more popular mean-variance rule as well as the semi-variance rule.
1. Introduction
Decision-making
under
uncertainty
may be viewed as choices between
alternative
probability
distributions
of returns, and the individual
chooses
between them in accordance to a consistent set of preferences. Von Neumann
and Morgenstern
(1967) have shown that under reasonable assumptions about
individual preferences, the individual chooses an alternative which maximizes
the expected utility of returns, where the utility function is determined uniquely,
up to a positive linear transformation,
by individual
preferences. However, in
most situations. such a selection is not possible since complete information about
an individual’s
preference set, and hence his utility function, is not available.
Thus, with only partial information
that an individual’s utility function belongs
to a certain restricted class of admissible functions, one is interested in the
optimal selection rules which minimizes the admissible set of alternatives
by
discarding,
from among the given set of alternatives,
those that are inferior
(for each utility function in the restricted class) to a member of the given set.
*We thank Nils Hakansson for stimulating discussions and acknowledge helpful comments
by Nils Hakansson, Karl Borch, Peter Fishbum, and Alex Whitmore on an earlier draft.
96
V.S.Bawa.Oplimalrulesfor or&ring uncertainprospects
The more restrictive the class of utility functions, the smaller will be the
admissible set and thus the more useful will it be in practical situations. However,
more restrictions on the utility functions imply that the admissible set is relevant
for a smaller group of individuals and may involve a severe loss in generality.
Thus, one is interested in determining the admissible set of alternatives for the
most restrictive class of utility functions that is consistent with observed economic
phenomena. Arrow (1971) and Pratt (1964) have pointed out that the observation of certain economic phenomena indicate that individual utility functions
exhibit decreasing absolute risk aversion and to a lesser extent increasing
relative risk aversion. Stiglitz (1970) has raised doubts whether increasing relative
risk aversion is a plausible assumption. Thus, it appears that decreasing absolute
risk aversion is the most restrictive class of utility functions acceptable to most
economists and we are interested in the optimal selection rule for this class of
utility functions.
For the portfolio selection problem, which may be viewed as a canonical
representation for certain types of economic problems involving decisionmaking under uncertainty, Markowitz (1952, 1970) and Tobin (1958, 1965)
proposed for risk averse individuals, a mean-variance selection rule in which,
from among a given set of investment alternatives, the admissible set is obtained
by discarding those investments with a lower mean and a higher variance than
a member of the given set. But even though the mean-variance approach has
spawned a considerable body of literature, including most notably the SharpeLintner-Mossin Capital Asset Pricing model [Sharpe (1964), Lintner (1965),
Mossin (1966)], it has been known for some time [see, for example. Borch
(1969). Feldstein (1969) and Hakansson (1972)] that the approach is of limited
generality since it is the optimal selection rule only if the utility function is
quadratic or the probability distributions of returns are normal. Arrow (1971)
and Hicks (1962) have pointed out that the assumption of quadratic utility is
highly implausible in that it implies increasing absolute risk aversion. Also, the
assumption of normal distribution of return on risky investments is not realistic
as it rules out asymmetry or skewness in the probability distribution of returns.
Cootner (1964) has shown that the returns on financial investments are more
likely to be lognormal than normal. In a recent empirical study, Lintner
(1972) has shown that even returns on portfolios of risky assets are more likely
to be lognormal than normal. Furthermore, progressive taxation and limited
liability of corporations imply that the distribution of net returns is quite likely
to be skewed. Thus a selection rule based on mean and variance alone is indeed
not justifiable on theoretical grounds but is an approximate and computationally
feasible selection rule. Samuelson (1970) and Tsiang (1972) have shown that the
approximation is reasonable only when the ‘riskiness’ of returns is limited.
The use of variance as a measure of risk for non-symmetric distributions has
been questioned by financial theorists [see for example, Hirshleifer (1970,
pp. 278-284). Mao (1970) and Markowitz (1970, pp. 188-201)]. Semivariance,
V.S. Bawa. Optimal rules for ordering uncertain prospccrs
97
rather than variance, has been proposed [Mao (1970) Markowitz (1970)] as a
measure of risk on the grounds that semivariance concentrates on reducing
losses as opposed to variance which considers extreme gains, as well as extreme
losses, as undesirable. Semivariance is less tractable mathematically than
variance and an algorithm for obtaining the mean-semivariance admissible set
has been provided by Hogan and Warren (1972). We note, however, that the
mean-semivariance selection rule, just as the mean-variance rule, is the optima1
selection rule for an increasing absolute risk averse utility function and hence
this approach also appears to be of limited generality.
On recognizing the restrictiveness of the mean-variance rule and its variants,
Quirk and Saposnik (1962) and later Fishbum (1964), Hadar and Russell
(1969, 1971). and Hanoch and Levy (1969) obtained the optimal selection rule
for the entire class of increasing utility functions (i.e., the utility function is an
increasing function of returns). The selection rule, called First Order Stochastic
Dominance (FSD) rule, is that a probability distribution F dominates a probability distribution G if and only if F never lies above and somewhere lies below G;
the FSD admissible set contains distributions that are not dominated by the
FSD rule. Thus, among the class of all distribution functions, the subclass that
can be ordered by the FSD rule will indeed be small. Hence, as is to be theoretically expected, and has been empirically verified by Levy and Hanoch (1970)
Levy and Sarnat (1970) and Porter and Gaumnitz (1972) a large proportion of
the given set of alternatives will still be members of the FSD admissible set;
this restricts the practical applicability of the selection rule.
Perhaps so motivated, and since individual behavior exhibits risk aversion,
Hadar and Russell (1969, 1971), and Hanoch and Levy (1969) considered the
restricted class of increasing utility functions that are risk averse, i.e., utility
functions are increasing and have decreasing marginal utility everywhere, and
obtained the optimal selection rule. This selection rule, called the Second Order
Stochastic Dominance (SSD) rule, is that a probability distribution Fdominates
a probability distribution G if and only if the integral of F never lies above and
somewhere lies below the integral of G; the SSD admissible set contains distributions that are not dominated by the SSD rule. Thus, a larger subclass of distributions will be ordered by the SSD rule and the SSD admissible set will be
smaller than that under the FSD criterion. [This is empirically varified in
Levy and Hanoch (1970), Levy and Samat (1970). and Porter and Gaumnitz
(1972).] Rothschild and Stiglitz (1970, 1971) also obtained SSD rule for the
special case of distributions with equal means and finite range and provided
applications of these to several economic problems.
Whitmore (1970) considered the class of increasing and risk averse utility
functions with the additional restrictions that the third derivative of the utility
functions be positive and for the case when the random variables are deEned
on a finite range obtained the optimal selection rule, called the Third Order
Stochastic Dominance (TSD) rule. The economic grounds for assuming that the
98
VS. &wa,
Opliml n&s for
or&ring
wtccrtain prospects
utility functions have positive third derivative is that it is implied by decreasing
absolute risk aversion; it is more desirable to obtain the optimal selection rule
directly tied to decreasing absolute risk aversion.
The optimal selection rules (FSD and SSD) involve pairwise comparisons
and require complete knowledge of the entire distribution function, as opposed
to the knowledge of only the mean and the variance. With no restrictions on the
probability distribution functions, the selection rule is complex and for practical
applications, one would need an efficient algorithm to obtain the admissible set.
Algorithms to obtain the admissible sets are outlined in Levy and Hanoch
(1970) and Levy and Samat (1970) and an algorithm which is efficient even for
large numbers of alternatives is provided in Porter, Wart and Ferguson (1973).
However, with certain restrictions on the class of distribution functions, the
optimal selection rules reduce to manageable and simple forms; results for
certain special cases are given in Hanoch and Levy (1969) and Hadar and
Russell (1971). As noted earlier, for the case of normal distributions, the SSD
rule reduces to the mean-variance selection rule, but otherwise even for the
restricted class of two parameter distributions, the SSD and mean-variance rules
result in different admissible sets. It is pointed out in Quirk and Saposaik
(1962), Hadar and Russell (1969). Hakansson (1971), and Hanoch and Levy
(1969) that it may be the case that some distribution on the mean-variance
admissible set may be dominated under the SSD (or even the FSD) rule and
hence do not belong on the SSD admissible set. Conversely, a member of the
SSD admissible set may not be on the mean-variance admissible set. Thus,
a major deficiency of the mean-variance rule is that it fails to obtain the SSD
admissible set; it uses neither a necessary nor a sufficient condition for dominance
in eliminating inferior alternatives.
In section 2, we show that for the entire class of distribution functions and for
the class of decreasing absolute risk averse utility functions, the Third Order
Stochastic Dominance (TSD) rule is the optimal selection rule when the distributions have equal means. We also show that for distributions with unequal
means, TSD rule is a sufficient condition for dominance, a large mean is also
a necessary condition for dominance but only a part of the integral condition
underlying TSD rule is necessary for dominance. However, with no restrictions
and the class of distribution functions, there is no selection rule which is both
necessary and sufficient for dominance. In view of these results TSD rule may
be used as a reasonable approximation to the optimal selection rule for the
class of decreasing absolute risk averse utility functions and the entire class of
distribution functions. The TSD rule requires complete knowledge of the entire
distribution function, is complex and involves pairwise comparisons of the
alternatives. In section 3, we provide an efficient algorithm to obtain the TSD
admissible set using the TSD rule. In section 4, we show that Third Order
Stochastic Dominance implies dominance under the mean-lower partial variance
rule: the mean-lower partial variance rule uses a necessary condition for
V.S. Bawa, Optimal rules for or&ring uncertain prospects
99
dominance for the entire class of distribution functions. In view of the complexity
of the TSD rule, this provides a strong rationale for using the mean-lower partial
variance rule (as opposed to the popular mean-variance rule) as a reasonable
approximation to TSD and hence the optimal selection rule for the entire class of
distribution functions. In section 5, we consider certain restricted classes of
distribution functions considered extensively in the literature (and which cover
most distributions of practical interest in economics and finance) and show that
the TSD rule is indeed the optimal selection rule and that it reduces to simple
manageable forms. Some concluding remarks are provided in section 6.
2. Optimal selection rules
We are interested in rules for ordering a pair of uncertain prospects characterized by random variables X and Y with known probability distribution
functions F( *) and G(s), respectively. The random variables X and Y, representing the outcomes of uncertain prospects, may be discrete, continuous or mixed
with range, represented by a closed interval [u, b], u < b, where either one or
both end points may be infinite. The distribution functions F(e) and G( *) are
non-decreasing, continuous on the right with F(u) = G(u) = 0 and F(b) =
G(b) = 1. The analysis is not affected by whether these probabilities are
‘objective’ or ‘subjective’ as long as the distribution functions are completely
specified. The uncertain prospects are thus equivalently characterized by distributions Fand G and we are interested in rules for ordering them.
An individual chooses between F and G in accordance with a consistent set
of preferences satisfying the Von Neumann-Morgenstern
(1967) consistency
properties. Accordingly, F is preferred to G if’ *’
AEu = E+(X)-E&X)
> 0,
(0
(i.e.,
expected utility under F is greater than that under G) where u(x) is determined uniquely, up to a. positive linear transformation, by individual preferences.
Since complete information about individual preference may not be available,
we are interested in ordering rules for certain restricted classes of utility functions.
We define the following sets of utility functions that are commonly used in
economic problems, with R denoting [a, b):
‘The decision-maker is interested in maximizing the expected utility of the end of period
wealth. Hence, Xand Y denote the end of period wealth under the two alternatives being considered. The end of period wealth could equivalently be defined as (W+X)
and (W+ Y),
where Xand Yare now defined as additions (or reductions) to the initial wealth level W. Hadar
and Russell (1971) and Levy and Sarnat (1971) have shown that if W is either a constant or a
random variable distributed independently of X and Y. then the ordering rule is invariant to
the initial wealth position W. Hence, in the ordering rule (1). X and Y may equivalently be
viewed as money payoffs on uncertain ventures that are additions (or reductions) to individual’s
initial wealth position W.
‘It is assumed throughout this paper that the expected utility exists.
100
Dejnition
V.S. Bawo,
Optimal rulesfor
ordering
uncerroin prospects
1:
CJ, = {u(x) 1U(X)is finite for every finite x, u’(x) > 0 Vx E R) ;
Definition 2:
U, = {u(x)~~(x)~U,,
- co < u”(x)~<Otlx~R};
Definition 3:
U, = (u(x)]u(x)~U~,u-(x)
> OVXER};
Definition 4:
U, =
{u(x) 1u(x)
E
U, , r’(x) s
(-u”(x)/u’(x))’ < 0 Vx E R} .
WI is the set of all increasing and continuously differentiable utility functions
u(x) assumed to have finite values for finite values of x. U, is the subset of CJ,
with risk averse utility functions; U, is the subset of U, with utility functions
having positive third derivative, while U, is the subset of II, with decreasing
absolute risk averse utility functions. We are primarily interested in considering
utility functions which vary only in a range bounded from below. This is so if,
for example, an individual cannot be worse off than loosing everything; then
U(X) needs to be defined only for x z 0. Hence, it is assumed throughout (with
no loss in generality) that the interval R( = [a, b)) is a subset of the non-negative
half line (i.e., u 2 0).
The classes (I,, U, and U, of utility functions have been considered extensively
in the literature [Quirk and Sapasnik (1962). Fishbura (1964). Hadar and
Russell (1969, 1971), Hanoch and Levy (1969), Whitmore (1970)], and the
optimal selection rules for these classes (i.e., the FSD, SSD and TSD rules) are
summarized by the following theorems,‘**
Theorem I. For any two distributions
utility functions in II, if and only i/
F(x) s G(x)
Vx E R
and
Theorem 2. For any two distributions
utility functions in U2 if and only if
F and G, F is preferred
to G for all
< for some x.
F and G, F is preferred
to G for all
‘The integrals used throughout are Stieltjcs-Lebcsgues integrals, which are assumed to be
bounded.
.It should be noted that in proof of Theorem 3, it is immaterial whether b is finite or infinite.
Thus our Theorem 3 extends Whitmore’s TSD rule to cover infinite range as well.
V.S. Bawa, Optimal rules for or&ring uncertain prospects
j;F(r)dt
Vx E R
s j: G(r)dr
Theorem 3. For any two distributions
utility functions in U, ifand only if
101
and c for some x.
F and G, F is preferred
to G /or all
cc, 2 c1G
and
j: j: F(r)dr dy < j: j: G(f)dr dy
Vx E R
and < for some x.
We note that the proof of Theorem 3 [in Whitmore (1970)] and Theorems 1 and 2
[in Hadar and Russell (1969, 1971)] is incomplete. We now provide a simple
proof of Theorems l-3.
ProofofTheorems
We note that
1-3.
AEu = E&X)-
E&Y)
= j: u(x) d( F(x) - G(x)).
(2)
Carrying out integration by parts several times with H,(x) = F(x)- G(x),
H,(x) = j: %-,Qdy,
(3)
for n 2 2, we obtain the following equivalent expressions for AEu:
AEu = -j:
u’(x)H,(x)dx,
(4)
AEu = -u’(bW,(b)
+ j: u”(x)Hz(xW,
(5)
AEu = -u’(b)H,(b)
+ u”(b)H,(b) - j.” u”(x)H,(x)dx.
(6)
Using (4)-(6) and noting that H,(b) z pc -pF proves the sufficiency of selection
rules in Theorems l-3. To prove necessity, it needs to be shown that if the
conclusion of Theorem i (i = 1, 2, 3) fails to hold then there exists a utility
function in U, for which the hypothesis of Theorem i is contradicted (i.e., if
for some arbitrary x0, with a < x0 < x,, +b d 6, H,(x) > 0 for x E [x,, x0 +6],
then there is a utility function u, E U, such that AEu, < 0; in addition, for
VS. Bawa, Optimal rules for ordering uncertain prospects
102
Theorem 3, it also needs to be shown that A/J = ~+-c(o >, 0 is also necessary).
The contradiction is provided by means of a utility function defined through the
following function, with E = y6,
a < x < x0,
E,
4(x>=
i
E-Y(X--X0),
x0
<
x < x,+6,
0,
x0
+6
cx.’
To prove Theorem 1, we consider utility function ur(x) defined by u;(x) =
where k > 0; then as u;(x) > 0, ur(x) E WI,’ and we obtain substituting in (4) and noting that the integrals are bounded,
k- f(x),
AEu, = -y
If:+” H,(x)dx+M,.
Thus, by choosing y large enough, one can make AEu, < 0; this completes
proof of Theorem 1.
To prove Theorem 2, we consider utility function uI(x) defined by u’;(x) E
-k + 4’(x), where k > 0. Then as u’;(x) < 0, u;(x) I k, - k(x-a) + b(x) > 0
by appropriate choice of k, , u2(x) E U,, and we again obtain for sufficiently
large Y,
AEu, = M,-Y
,;+,
Hz(x) dx < 0.
To prove Theorem 3, we consider utility function u,(x) defined by u;(x) E
eI - 4’(x), where .sr > 0. We note that uj (x) > 0, u’;(x) = -(k, -Q(x-u))
4(x) c 0 by appropriate choice of k, . Similarly,
u;(x)
3
k, -2
k,-(x-u)
-j;
(x-u)
d(y)dy,
>
where
I
4x -a),
j-10
Ody
=
14x-++~,,l,
a < x < x0,
x0
f
x G x0+6,
‘We note that the differentiability requirements can be satisfied by rounding the edges which
does not change the analysis.
V.S. B4wa, Optimal rulesfor or&ring uncertainprospecrs
and k2 is an arbitrary
u,(x) fz U,. We let
103
constant chosen such that u;(x) > 0 Vx E R; hcna
k, = c,(b-a)(1
+k;),
-;6’
k, = e,(b-a)‘(k;+k;+#+E(x,-o+b)
+k,yb2;
also let
M3 = k;(b-a)2-k;(b-a)H,(b)-Jf
and noting that -H,(b)
= Ap(=j+-po),
AEu, = k,yS2Ap+eIM,-y
<
H,(x)dx,
I:;+” H,(x)dx
vW,W--,l+c,M,,
where for x E [x0, x0 +S], H,(x) 2 M4 > 0.
Thus, if we let E~-+ 0, 6 + 0, y + co such that E = yS + C > 0, then
AEU, < 0 for sufficient small value of 6. This proves that H,(x) < 0 Vx, < 0
for some x is a necessary condition for dominance.
We consider utility function u.,(x) defined by
464 = exp (eSPX),
wherep > 0. Then we note that as uk > 0, u; < 0 and u; > 0, u,&x) E U, and
we obtain, after some algebraic simplifications,
AEu, = -exp(e-pb)[H2(b)+pe-pbH,(b)]
+J:exp (e-b”)A(x)Hs(x)dx,
where ,4(x) z -p2e-P”(1 aemPX) < 0.
AEu,<Ofor
It thus follows that since A(x) +Oasp-,O,ifH,(b)>O,
sufficiently small value of p. This completes proof of Theorem 3.
We are interested in the class U, of decreasing absolute risk averse utility
functions and the optimal selection rule for this class is given by the following:
Theorem 4. For any two distributions F and G with pF = po, F is prefered
(0 G for all utility functions in U4 if and only if
I: JI F(l)dfdy
< J: J: G(t)dldv Vx E R
and
< for somex.
104
VS. Bawa, Optimal rules for or&ring uncertain prospects
Proof of Theorem 4. We obtain from (a), remembering that r(x) E -u’(x)/
u’(x) and H,(b) = 0,
AEu = -u’(ww)+~
u’(x)[“(x)-(4x))2]H3(x)dx.
(7)
For U(X)E Uq, r’(x) < 0, r(x) > 0 and thus H,(x) < 0 Vx and < 0 for some x
is a sufficient condition for dominance. To prove necessity, we consider a utility
function us(x) defined by r;(x) = $‘(x),‘j and note that us(x) E I/,, and we
obtain
AEu, = -E2 j:” &(x)H,(x)dx-y
j:;‘”
u;(x)H,(x)d_X
_ j:;+* 4(x)(&-Y(x-xo))2wx)dx.
We note that since for x E [x,,, x,+6],
VXER, AEu, < Oif
H,(x) Z MI > 0 and since IH3(x)I < M2
YMI j;+* u’(x)& > &2M2j:” u’(x)dx.
(8)
We obtain after some algebraic simplifications,
exp (-j: r(r)df),
x0
u’(x)dx =
remembering
that u’(x) =
e-” _e-exo
e
,
I0
and
j
“,I+”u’(x)dx = e-‘Xo-(N2) Id,_0 e(r/2)(~-*)a dy
>
~e-exo-W/2).
I
hence to prove necessity of the selection rule, it suffices to show that
yM,(Je-txo-W/2))
,
E2~2
e
-m
_,-cxo
.
E
’
that is,
Mle-‘xo-h’/2)
>
M~(~-u__~-IJo).
(9)
We note that with 6 + 0, y + 03 such that E = y6 + 0, inequality (9) will hold
for sufficiently small b. This completes the proof of Theorem 4.
*To insure that r,‘(x) < 0 Vx, we need to define r,‘(x) = d’(x) --e, , where cI > 0. In the
proof, we will take c, arbitrarily close to zero and since it does not effect the analysis, we assume
hereafter for typographical simplicity that c1 = 0.
V.S. Bawa, Optimal rulesfor or&ring wtccrtain prospects
105
For the more general case of distributions with unequal means, our results
on the optimal selection rule are summarized by the following, with H,(x) as
given by (3):
Theorem 5. For any two distributions F and G, F is preferred to G for all
utility functions in U,,
(i) if pr > ~1~and H,(x) < 0 Vx E R, >O for some x,
(ii) only if c(r k ~1~and for some x0 E R, H,(x) Q 0 for (I d x < x0,
<Oforsome x < x0.
Proof of Theorem 5. We note that since U, c U,, it follows that TSD, as
given in (i), is sufficient condition for dominance. To prove (ii), we obtain
from (6), noting that r(x) E -I/(X)/U’(X),
AEu = - u’(b)( H,(b) + r(b)H,(b))
To prove (ii), it needs to be shown that if for some x0 E R, H,(x) Z 0,
a G x < x0 with strict inequality for some x < x0, then there exists a utility
function in u4 such that AEu -C 0. The contradiction is provided by means of
the utility function Q(X) with
U:(x)
=
exp (e-P(x-xO)),
(11)
wherep is a freely chosen positive parameter. We note that u’(x) > 0, Vx E R,
u;(x) = -pe-P’=-%;(x)
< 0,
and
r6(x) = pe-p(“-*o)
> 0,
r:(x) = -p2e-P(X-X0)
< 0.
Thus I+,(X)E U, and we obtain using (10) that for the utility function
by (11).
AEug = - cxp (e -P’b-Xo))[H2(b)+pe-p’b-r~)H)(6)]
+
given
(12)
j:” exp (e -P(“-“0$4(x)H,(x)d_X
+ j:,
where A(x) E r:(x)-r:(x)
exp(e-p’“-‘o’)A(x)H,(x)dx,
< 0. Since exp (e-p(X-*o)) is an increasing
and
106
VS.
Bawa,
Optimal
rules for ordering
uncertain
prospects
unbounded function of p for x < x0, and a decreasing function of p for x > x,, ,
it follows that AEu, can be made negative for a sufficiently large value of p;
similarly, it follows that it is also necessary to have H,(b)(=p,-~1~)
< 0 as
otherwise dEu6 can be made negative for a sufficiently small value of p. This
completes the proof of Theorem 5.
We have shown that the Third Order Stochastic Dominance (TSD) rule that is
optimal for the class U, of utility functions is also the optimal selection rule for
the class U, of decreasing absolute risk averse utility functions when the
distributions have equal means. For the general case of unequal means, TSD rule
is a sufficient condition for dominance, a larger mean is also a necessary condition for dominance, but only a part of the integral condition underlying TSD
are necessary for dominance. Indeed H,(x) Q 0 Vx E R is not necessary for
dominance [see Vickson (forthcoming a, b) for several counter examples], and,
unfortunately, there is no known selection rule which is both necessary and
sufficient for dominance. In view of these results, TSD rule may be used as a
reasonable approximation to the optimal selection rule for the entire class of
distribution functions.
3. Algorithm to obtain the TSD admissible set
The Third Order Stochastic Dominance Rule, involves pairwise comparison
of the given set of alternative probability distributions. It requires complete
knowledge of the entire distribution function, not just means and variances as
needed for the mean-variance selection rule. Thus for a given set of n alternative
distributions, (;) = n(n- I)/2 paired comparisons are needed; in addition, if
each distribution is discrete with M possible values then for each paired comparison, H,(x) has to be computed and checked to be non-positive at 2m values.
Thus, a total of n(n - I)m comparisons would be needed to obtain the admissible
set from a given set of n alternatives. (The same number of comparisons would
be needed for the FSD and SSD rules as well.) Even for moderate values of
n and m, the number of comparisons can be prohibitively large; for example
for n = 10 and m = 10, 900 comparisons would be needed, while for n = 100
and m = 10, 99,ooO comparisons would be needed to obtain the admissible set.
Thus, in order to obtain the admissible set of alternatives in any practical
situation, one needs an algorithm which makes efficient use of all the available
information about properties of the distribution functions and the ordering
rule.
For the case of discrete distributions, Porter, Wart and Ferguson (1973) have
developed an algorithm to obtain the admissible set for the TSD rule,’ which is
‘This algorithm also obtains FSD and SSD admissible sets; Levy and Hanoch (1970) and
Levy and Sarnat (1970) have also provided algorithms to obtain FSD and SSD admissible sets.
Our algorithm is more efficient than these other algorithms as well in obtaining the admissible
sets.
VS.
Bawa, Optimal
rules for ordering
uncertain prospects
107
claimed to be efficient even for large numbers of alternatives.
The efficiency of
this algorithm basically results from the following use of three properties of the
ordering rule :
(1) List alternatives in descending order of mean values. Since larger mean is
a necessary condition
for dominance,
an alternative
cannot dominate
the
alternatives
listed above it. Thus the alternative with the largest mean belongs
to the efficient set.
(2) An alternative F cannot dominate an alternative G if the smallest possible
value of the random variable under F is smaller than that under G; this is so as
in this case F(x)
- G(x)> 0 and hence H,(x) > 0 at the smallest value of x.
hence eliminate from further pairwise
(3) The ordering rule is transitive;
comparisons an alternative dominated by another.
The algorithm
in Porter-Wart-Ferguson
(1973) takes advantage
of these
properties to reduce the number of pairwise comparisons
to a minimum.
In
addition, when comparing an alternative F to an alternative G (with pF 2 po),
the efficiency is increased by computing H,(x) sequentially for increasing values
of x: If H,(x) < 0 for all x, then it is noted that Fdominates
G, but, on the other
hand, as soon as H,(x) > 0 is recorded, further comparisons
are stopped and it
is noted that F does not dominate G.
The algorithm in Porter, Wart and Ferguson (1973) applies only to the case of
discrete distributions.
In addition,
since in comparing
F to G, the algorithm
computes and checks for N,(x) to be non-positive
for all possible values of x
before concluding that Fdominatcs
G, the efficiency of the algorithm is inversely
proportional
to the number of possible values (m) of the random variables. Thus,
for large values of m (which may occur, for example, when approximating
a
continuous
distribution
by a discrete distribution),
the algorithm
in Porter,
Wart and Ferguson (1973) may not be practical.
We now propose an algorithm which is applicable to continuous
as well as
discrete distributions.
The algorithm uses all the clever features used in Porter,
Wart and Ferguson (1973) to reduce the number of pairwise comparisons
to a
minimum, and, in addition, improves the efficiency by making use of available
information
about the distribution
functions. This is done by avoiding, when
comparing
F to G, the computation
of H,(x) for all possible values of x;
instead the algorithm concentrates
on the zero crossings of the function H,(x) E
F(x)-G(x)
and uses the minimum possible number of computation
necessary
to check if F dominates G. For discrete distributions
(with m possible values for
each distribution),
the number of zero crossings is at most 2m but will usually
be much smaller; hence the algorithm will be, in most cases substantially
more
efficient than the one proposed in Porter, Wart and Ferguson (1973). Also the
algorithm can handle continuous
distributions
as well since the number of zero
crossings, except for pathological cases, will be finite for continuous distributions.
We now provide the basic steps of the algorithm used in comparing two un-
108
V.S. Bawa, Optimal rules for or&ring
wucrtain prospect3
certain prospects F and G,s noting that the random variables are defined on
[u, 61 and for the case of continuous distributions the following equivalent
expressions for H,(x) and H,(x) may be used recursively to improve the
c5ciency of computation,
H,(x) = &(9+(x-Wz(Z)+~(x-y)HJ_y)dy,
f < x,
Algorithm to order Fund G (with pp > c(,J
Step I:
If H&z+)
> 0 GO TO STEP 7; otherwise GO TO STEP 2.
Step 2:
(Initialization)
Let N = number of zero crossings from below of H,(x). (Let
X, denote the N zero crossings.)
X*,X2....,
If N = 0 GO TO STEP 6; otherwise let Y, , Y,,. . ., Yn_i denote
the (N- 1) zero crossings from above of H,(x), x E [a, 6).
Let
YN =
last zero crossing from
above of H,(x),
b,
if H,(b) < 0,
otherwise.
Step 3:
Let n = 1. Compute H2( Y,).
If Hz( Yi) < 0 GO TO STEP 4; otherwise compute H,( Y,).
If HJ( Y,) < 0 GO TO STEP 5; otherwise GO TO STEP 7.
Step 4:
Let n = n+l.
If n = N+ 1 GO TO STEP 6; otherwise compute H2( Y,).
If H2( Y.) B 0 GO TO STEP 4; otherwise compute H,( Y,).
If H,( Y,) f 0 GO TO STEP 5; otherwise GO TO STEP 7.
‘The other parts of the algorithm, which essentially reduce the number of pairwix comparisons (between distributions) to a minimum, arc the same as that in Porter, Wart and Ferguson
(1973).
V.S. Bawa, Optimal
rules for ordering
uncertain prospects
109
Step 5:
Letn
= n+l.
Ifn = N+ 1 GO TO STEP 6; otherwise compute
If HJXJ
otherwise
> 0 compute H,(X,):
GO TO STEP 7.
if H,(X,)
H2(X,,).
< 0 GO TO STEP
5;
If H2(Xn) < 0, find Y*, Y,,_, < Y* < X, such that H2( Y*) = 0
and compute H,( Y*).
If H,( Y*) > GO TO
TO STEP 4.
STEP
7; otherwise
let n = n- 1 and
GO
Step 6:
Terminate:
F dominates
G.
Terminate:
F does not dominate
Step 7:
G.
The above algorithm terminates in a finite number of steps and, by noting
that in between successive zero crossings of H,(x), the algorithm checks that the
maximum
of H,(x) is non-positive,
it can be easily verified that when the
algorithm terminates and concludes that F dominates G, it is indeed true that
H,(x) < 0 for all x E [a, 61. This is summarized in the following:
Theorem 6. For discrete or continuous distributions (with finitely many zero
crossings in [a, b]) F and G, the algorithm orders F and G under TSD rule in a
finite number ofsteps.
4. Rationale
for mean-lower partial variance selection rule
The admissible set of alternatives for the class U, of utility functions (as well
as the class U, of decreasing absolute risk averse utility functions when the
distributions
have equal means) is provided by the Third Order Stochastic
Dominance
(TSD) rule. With no restrictions on the class of distribution
functions, the rule involves pairwise comparisons,
is complex and requires complete
knowledge of the entire distribution
function. The efficient algorithm developed
in the last section may be used for practical implementation
of the TSD rule.
This requires that the set of alternatives
being considered is prespecified and
finite. Thus, for problems like capital budgeting under uncertainty,
where the
number of alternatives is finite, the algorithm can be used to obtain the admissible set. For certain other types of problems,
like the portfolio selection
110
V.S. &Iwo, Optimal rulesfor or&ring uncertainprospects
problem, not only the basic alternatives but all linear combinations of these
alternatives are possible alternative choices available to the individual. Thus,
even with a finite number of basic alternatives (e.g., n basic securities in the stock
market) the number of possible alternatives to be considered is infinite.9 The
following theonm provides the result needed to overcome this problem:
Theorem 7.
TSD implies dominance under mean-lower partial variance rule.
Proof.
By definition, the lower partial variance LPVJx)
function F(x) is given as
for distribution
LPV,(x) = j: (JJ- x)~ dF@) ,
and hence,
dLPV(x) E LPV,(x) - LPV,(x)
= j:
(u-x)2dKti).
Integrating by parts, several times, we obtain
LYLPWX)= (Y-x)~H~(Y)
I:--2 j: OI-xW,(y)
= -2 j (v-x)dH2b)
= -2(y-x)H,(y)
I:+2 j; dH,Q
= 2H,(x).
In comparing distributions F and G under the mean-LPV(r), rule F dominates G
if and only if pF-pG 2 0 and dLPV(r) < 0 (where r E R is a prespecified value),
whereas under the TSD rule, F dominates G if and only if pF-pc b 0 and
H,(x) < 0 for Vx E R. Since dLPV(x) = 2H,(x), it follows that dominance
under the TSD rule implies dominance under the mean-semivariance rule. This
completes the proof of Theorem 7.
Theorem 7 provides the rationale for using the mean-lower partial variance
selection rule as an approximation to TSD rule and hence the optimal selection
rule for the class of decreasing absolute risk averse utility functions. Since it uses
a necessary condition for dominance, the mean-LPV(r) admissible set will be
9The literature on stochastic dominance rules make the ad hoc assumption that the number
of alternatives is finite, thus limiting the usefulness of the approach. Theorem 6 overcomes this
and shows the full generality of this approach.
V.S. Bowo, Optimol
rules for ordering
wcertoin
prospects
111
contained in the TSD admissible set; indeed TSD admissible set contains the
inclusion of all the mean-LPV(r) rules. By combining the admissible sets for
the mean-LPV(r) rules for several values of r, one gets sufficiently close to the
TSD admissible set (as the algorithm of last section indicates, only finitely
many r values are needed to obtain the TSD admissible set). Thus, the meanLPV(r) rule for one prespecified value of r can be directly used as an approximation for the TSD rule. Since it can be easily shown that for the class U, (as we!!
as class U4) of utility functions, mean-variance is neither a necessary nor a
sufficient condition for dominance,‘O it appears that on theoretical grounds meanlower partial variance rather than mean-variance should be used as an approximation to the optimalselection rule. Quite remarkably, we have placed no restrictions
on the class of distribution functions, and hence mean-lower partial variance
can be used as an approximation for the entire class of distributions.
We note that the mean-lower partial variance rule, just as the popular meaovariance rule, allows one to consider a!! possible linear combinations of basic
alternatives and can be used to generate the mean-lower partial variance
admissible boundary. [See, for example, Hogan and Warren (1972) for computational feasibility of this approach.] This overcomes the difficulty of infinitely
many alternatives and hence the mean-lower partial variance can be used for the
portfolio selection problem as a suitable approximation to the TSD rule and
hence the optima! selection rule.
It should also be noted that since the lower partial variance function provides
for explicit consideration of asymmetry or skewness of the probability distribution, it is to be preferred to selection rules based on mean, variance and
third moment of the distribution function. Indeed, it can be easily shown that
selection rules based on the first n moments (n > 3) use neither a necessary nor
a sufficient condition for dominance for the class Cl.,of utility functions and
instead the mean-lower partial variance rule should be used. Thus, at least on
theoretical grounds, the approaches recently advocated in Jean (1971) should
be abandoned in favor of the mean-lower partial variance rule.
5. Optimal rule for restricted classes of distribution functions
With no restrictions on the class of distribution functions, the optima!
selection rule is not known and even the TSD rule used as a reasonable approximation is complex and it would be necessary to use the algorithm to obtain
the TSD admissible set. Thus, one might be tempted to consider a certain
restricted family of distribution functions, obtain the optima! selection rule
for that family of distributions and check if the optima! rule reduces to a simple
form involving only certain parameters of the distribution function (e.g., mean
and variance).
“‘We
leave it to the reader to verify this proposition.
112
VS. Bawa, Optimal rules for ordering uncertain prospecfs
We consider a special case of the general problem of decision making under
uncertainty with the number of alternatives considered being finite and prespecified and with the probability distribution for these alternatives belonging
to a restricted class of distributions. rl*l z We consider two classes of distributions
Se and FI which include most distributions that have been extensively coosidered in the literature and are defined as follows:
Definition
A distribution function F(x), x E [a, 61. belongs to class P,,
F(x) = ${(x - IF)/+} for all x E [a, b] and s, > 0.
if
Definition
A distribution function F(x), x E [a, b]. belongs to class 6t if
F(x) = +(( 4(x) - I,)/+} with 4’(x) > 0 for all x E [u, 61 and sr > 0.
4r,, is the class of distributions characterized by a location parameter (I) and
a scale parameter (s), whereas 9, is the class of distributions in which a moootonic transformation (q5(*)) of the random variables belongs to the location and
scale parameter family 9,. Distributions like normal, r-distribution, exponential, uniform and double exponential all belong to the class Fe while the lognormal distribution, which [in view of empirical results of Cootner (1964) and
Lintner (1972)] has a special importance in optimal portfolio selection problem,
belongs to the class 9,.
We are interested in comparing distributions Fand G that belong to class .aC,,.
Thus, by definition
G(x) = t)
and hence
= I,+s,A
;
“The results of this section would thus be useful for problems like capital budgeting under
uncertainty. For portfolio selection problem, the results serve as useful approximations in that
if one is willing to assume that the distribution of returns on the linear combinations of the
basic securities ‘approximately’ belongs to the same restricted class of distributions as the
basic securities, then the optima1 rule obtained herein can be used as a reasonable approximation to the true optimal.
“The results obtained in this section are also useful in deriving the optimal rule for the portfolio selection problem wherein all linear combinations of basic alternatives are also considered.
These results will be presented in a forthcoming paper ‘On Optimal Portfolio Selection Rules’.
V.S. Bawa. Oprinral rules for ordering uncertain prospccfs
113
similarly
EGX = t,+s,A.
of the
where A =jyWCv)dp e en d s u P on the function $ bit is independent
location and scale parameters. We also note that for F, G c 5,,, if we let
x =
max {.v 1 $(x) = 0),
and
2 = min{xj
I(/(x) = 1).
then in comparing
distributions
F and G, the range [u, 61 is quite naturally
defined as the values of the random variables for which either F(x) or G(x)
(or both) are between zero and one, i.e.,
0 = min {IF+s,:,,I,+sG.\-},
and
6 = max {I, + s&
I, + s,.C) ,
[as beyond these values H,(X) E F(x)-G(x)
is identically zero and hence does
not effect the analysis]. It should be noted that a may be -co or finite and
similarly 6(> u) may be finite or co. It appears to be implicitly assumed in the
literature [see for example Hanoch and Levy (1969, Theorem 4)] that [a, b] s
[-co, co] although it need not be necessarily true. We allow for all the possibilities and with x, given (for the case sr # sc) as the solution to (xr -IF)/rF =
the optimal selection rule is given
(x1 - Ml%, i.e., x1 E (I,s, -ICrF)/(rc-rF),
by the following:
Theorem 8. For any two distributions F and G belonging to class 9,,
preferred to G for all utilityfunctions in 11, if and only i/:
(A) For sF = sG,
(B) For sF # sG,
I. for u -z x, c b
z I,+s,A
(9
l,+s,A
(ii)
SF < SG;
and
F is
Y.S. Bawa, Optimd rules for or&ring uncertain prospects
114
II. for x1 2 b
Proof.
For F, G c .9,, ,
H,(x)
Thus, H,(x)
= F(x)-G(x)
2 0 according
as
hence, for s, < sG, H,(x) $ 0 as x g xi , for s, > sG, H,(x) f 0 as x $g x1 ,
andfors,
= sG, H,(x) Z$ OVxasl,
s I,.
Given s, = so, I, > I, satisfies necessary
and sufficient
dominance of Theorem 5. This completes proof of Case A.
conditions
for
Case B: sF # sG
Fora<x,
cb,ifs,>s,,H,(x)2Ofora<x<x,andhenceH,(x)~O
for a < x < xi. Thus F cannot dominate G for s, > sG. However, ifs, < sG,
H,(x) < 0 and hence H,(x) < 0 for u 6 x < x, while for x 2 x,, Hz(x) is an
increasing function of x.
Also if (i) holds, then EFX 2 EGX, i.e., H,(b) z E,X-E,X
< 0. This
implies that Hz(x) < 0 for all x, xi ,< x ,< b; hence H,(x) 6 0 for all x E [a, b].
As (i) and (ii) satisfy the condition
of Theorem 5, this completes the proof of
Case B-I.
For x1 2 b, H,(x) > 0 for all x E [a, b) if sF > sc, whereas H,(x) < 0 for all
x E [u, 6) if sF < sG. Thus sF < sG satisfies the necessary and sufficient condition
for dominance of Theorem 5. This completes proof of Case B-II.
For xi < a, H,(x) > 0 for all x E (a, b] if sF < sc, whereas H,(x) < 0 for all
x E (a, b] if sF > So. Thus sF > So satisfies necessary and sufficient conditions
for dominance of Theorem 5. This completes proof of Case B-III.
Similarly,
if F, G c 9,,
i.e.,
VS. Bawa, Opfinud rules for or&ring uncertain prospecls
fP(-+~F
( 1
for all x E [u, b],
= $ -
F(x)
11s
SF
forallxE[u,b],
and if we let, for the case sp # sG, x2 denote the solution to
+(x2blF
=
SF
i.e.,
x2
=
6
ddx2)-k
SG
,
,
then the optimal selection rule is given by the following:
Theorem 9. For any IWOdistributions F and G rhar belong lo class 9,)
preferred to Gfor all utilityfunctions in U, if and onIy i/:
(A) For SF = SG,
1, > I,.
(B) For SF # SG,
I.
for a < x2 < b
W
ErX 2 EoX,
(ii)
SF < s,;
and
I.I. for x2 z b
sp < s,;
III. for x2 < u
sr > so.
(The proof of Theorem 9 i’ssimilar to that of Theorem 8 and is omitted.)
F is
116
VS. Bawa,
Similarly,
tions, i.e.,
Optimal
if we consider
.%s = {FE 9,I
rules for ordering
a subclass
F2
uncertain
of secontaining
prospects
symmetric
distribu-
F is symmetric},
thensinceF,Gc.%,,A=
0, ErX = l,, EoX = I, and for distributions
with
finite variances, s: = B*VarrX and si = B.Var,X
(where B > 0), Theorem 8
reduces to the following:
Theorem 10. For any two distributions Fand G withjinite variances that belong
to class 9 2 , F is preferred to G for aN utilityfunctions in U4 i/and only lfz
(A) For Var,X
= Var,X,
E,X
> EoX.
(B) For Var,X
I.
fora-zx,
# Var,X,
<b
(0
ErX 2 E,X,
(ii)
Var,X
and
II.
c Var,X;
for x1 2 b
Var,X
< Var,X;
III. for xi G a
Var,X
> Var,X.
It is interesting to note that for the restricted classes 9,, 9, and 9, of two
parameter distributions,
the optimal selection rule naturally reduces to comparison of two parameters
but these parameters
are neither the mean and
variance nor the location (I) and scale (s) parameters
that characterize
the
restricted classes of distributions.
Instead, the selection rule involves comparison
of the mean and the scale parameter. A larger mean is a necessary condirion for
dominance for the entire class of distributions (Theorem 5);’ ’ thus, for a restricted
“The
stochastic dominance rules
may be obtained by placing further
functions) all involve a comparison
universal condition for dominance.
(FSD, SSD and TSD -as well as higher order rules that
restrictions on fourth and higher moments of the utility
of the mean. Thus larger mean may be taken to be a
VS. Bawa, Optimal rules for or&ring
uncertain prospects
117
class of distributions (.%,,, Fl and F,) the mean will obviously be one of the two
parameters being compared. However, as the scale parameter (and not the
variance) is the natural measure of dispersion, it is quite appropriate that it be
the other parameter that is compared in the optimal selection rule. Even for the
class 9z of symmetric distributions and with finite variances, when variance is
indeed proportional to the scale parameter and hence may be used as the second
parameter, the optimal selection rule reduces to the mean-variance rule only if
a < x1 < b. This condition (Case B-I of Theorem 10) does not hold necessarily
for class 4Fz (for example, it always holds for normal distributions but need not
for uniform distributions); hence even for the (unrealistic) case of symmetric
distributions characterized by location and scale parameter that have been used
in the literature to rationalize the mean-variance rule, the mean-variance rule
is not necessarily the optimal selection rule.
For lognormal distributions (that belong to class F1), we note that a < x, < b
obtains and since the scale parameter is the logarithmic variance, Theorem 9
reduces to the following:
Theorem II. For any two lognormal distributions, one is preferred to another
for all utility functions in U, if and only if it has at least as large a mean (in
natural units) and a smaller logarithmic variance.
In view of the empirical study of Lintner (1972) (that the portfolio of lognormal securities is approximately lognormal) this result is of special significance
in portfolio theory. It implies that in order to generate the admissible set of
portfolios, instead of the popular mean-variance rule, the mean-logarithmic
variance rule should be used.
Finally, we consider the class of gamma distributions which is a two-parameter
distribution distinctly different from .%Oand f, . This class is of importance as
in certain situations, the gamma distribution can be used, with appropriate
choice of parameters, to reasonably approximate empirical distributions. The
density function for a gamma distribution is
e-‘“~“~-’
f(n)
*
x 2 0,
and thus the parameters (A, n) (usually called scale and shape parameters
respectively in the statistics literature) characterize the distribution. For this
class of distributions, the optimal selection rule reduces to the following:
Theorem 12. For any two gamma distributions F and G, with (A,, nF) and
(Ai. no) as the respective parameters, F is preferred to G for all utility functions
in U, ifand only if
118
VS. Bawa, Optimal
rules for or&ring
uncertain prospects
with at least one inequality holding strictly.
The proof of this theorem follows simply by noting that n,, 2 no and
E,,X( E n&J z EoX(= n&o) with at least one inequality holding strictly
satisfies necessary and sufficient conditions of Theorem 5.
The different classes of distribution functions considered in this paper have
the common property that any two distributions F, G belonging to the restricted
class cross at most once. Thus we have indeed proved the following:
Theorem 13. For any two distributions F and G that cross at most once F is
preferred to G for all utility functions in U, if and only ifpp 2 po and F initially
lies below G.
We note that in general it will be hard to check if F initially lies below G.
However, for the important classes of distributions considered earlier in this
section, this is easily identifiable by looking at the appropriate parameter that
characterizes the dispersion of the distribution.
6. Conclodlng remarks
We have shown that the Third Order Stochastic Dominance (TSD) rule is the
optimal selection rule for ordering uncertain prospects with equal means which
minimizes the admissible set of alternatives by discarding from among a given
set of alternatives, those that are inferior (for every decreasing absolute risk
averse utility function) to a member of the given set of alternatives. It was also
shown that in ordering uncertain prospects with unequal means, no known
selection rule uses both necessary and sufficient conditions for dominance and
TSD rule is a reasonable approximation to the optimal rule. In view of the
common acceptance among economists of decreasing absolute risk aversion and
almost universal reluctance to place restrictions on higher derivatives of utility
functions, the smallest acceptable admissible set will be given by the TSD rule.
The TSD rule is complex as it involves pairwise comparison of the means
and the lower partial variance function of the probability distributions (which
has to be determined over the entire range of the distribution). We developed an
efficient algorithm which can be used to obtain the admissible set for discrete
as well as continuous probability distributions (of returns on each alternative).
We showed that the two parameter mean-lower partial variance rule (with the
lower partial variance computed at any one point in the range), which uses a
VS. Bawa, Oprimal rules for ordering unccrrain prospeers
119
necessary condition for dominance may be used as a reasonable approximation
to the TSD and hence optimal selection rule for the entire class of distribution
functions. For the class of decreasing absolute risk averse utility functions, and
with no restrictions on the distribution functions, the mean-lower partial
variance rule is thus to be preferred (as an approximation to the optimal
selection rule), at least on theoretical grounds, to the popular mean-variance
rule or the recently proposed mean-variance-skewness preference rules [see, e.g.,
Jean (1971)] which use neither necessary nor sufficient conditions for dominance.
We hope that this will provide a strong impetus for further research in the
development of efficient algorithms for the mean-lower partial variance rule.
We considered certain restricted but important classes of two parameter
distributions that include most distributions of practical interest in economics
and finance and obtained the optima1 rule for decreasing absolute risk averse
utility functions. We also showed that the optimal rule reduces to simple
manageable forms, which involves naturally only two parameters, one of which
is always the mean but the other parameter is the appropriate measure of
dispersion. This second parameter is usually the scale parameter which, except
for the unrealistic case of symmetric distributions, is not the variance. For the
important case of lognormal distributions, the optimal rule reduces to one where
at least as large a mean and smaller logarithmic variance are the necessary and
sufficient conditions for dominance. This result along with the empirical linding
of Lintner (1972) (that linear combination of jointly lognormally distributed
returns is approximately lognormal) allows us to analytically obtain the efficient
set for the portfolio problem and the results will be presented in a later paper.
Our results for the restricted classes of distribution functions considered also
show that the second parameter characterize uncertainty and is the mean
preserving spread parameter of Rothschild and Stiglitz (1970). Hence, for a large
and important class of distribution functions, one can easily obtain unambiguous
economic effects of increasing risk and these results will be presented in a later
paper.
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