A Strategy Which Maximizes the Geometric Mean Return on Portfolio Investments
James H. Vander Weide; David W. Peterson; Steven F. Maier
Management Science, Vol. 23, No. 10. (Jun., 1977), pp. 1117-1123.
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MANAGEMENT SCIENCE
Vol. 23, No. 10, June, 1977
Printed in U.S.A.
A STRATEGY WHICH MAXIMIZES THE GEOMETRIC MEAN RETURN ON PORTFOLIO INVESTMENTS* JAMES H. VANDER WEIDE?, DAVID W. PETERSON? AND STEVEN F. MAIER?
A common formulation of the portfolio selection problem leads to the prescription of a
strategy which maximizes the geometric mean return on investments. In this paper we
examine conditions under which solutions exist for the case where the returns distribution is
discrete. We establish necessary and sufficient conditions for a solution to exist and give a
computationally convenient and exact method for finding a solution in circumstances where
(i) a solution exists and (ii) the number of securities equals or exceeds the number of values in
the returns distribution.
1. Introduction
The geometric mean investment strategy, introduced into the finance and economics literature by Henry LatanC [6] in 1959, has recently received some attention in
scholarly circles after a decade of near neglect. Most of this work has been devoted to
the investigation of various properties of the geometric mean strategy. Among the
properties of optimal geometric-mean portfolios recently discovered are (i) they
maximize the probability of exceeding a given wealth level in a fixed amount of time,
(ii) they minimize the long-run probability of ruin, and (iii) they maximize the
expected growth rate of wealth.' Relatively little attention has been given, on the
other hand, to either the computational problem of finding the optimal geometricmean portfolio2 or the question of the existence of such a p ~ r t f o l i oIn
. ~ this paper, we
analyze both of these latter problems under various assumptions about the investor's
opportunity set and the form of his subjective probability distribution of holding
period returns. We reach two major conclusions. First, optimal geometric-mean
portfolios do not exist for some rather obvious formulations of the investor's decision
p r ~ b l e mSecond,
.~
optimal geometric-mean portfolios are in some circumstances more
easily computed than was previously re~ognized.~
* Accepted by Edwin H. Neave; received June 21, 1976. This paper has been with the authors 1 month,
for 1 revision.
Duke University.
For discussion and derivations of these properties see [2], (51, [6], [lo], and [15].
We know of only three or four papers which purport to discuss the computation of optimal geometric
mean portfolios. The first, by Elton and Gruber [4], assumes that total portfolio returns are lognormally
distributed. But the analytical circumstances under which this would occur seem rather limited, since
lognormal security returns do not imply lognormal portfolio returns. The second, by Ziemba [17], discusses
the problem and proposes nonlinear programming as a solution means. The third, by Zierdba, Parkan and
Brooks-Hill [18], pertains to the problem in which returns are normally distributed. A possible fourth paper
is Ziemba's [16], which discusses general approaches to solving optimization problems with stochastic
objective functions.
Cass and Stiglitz [3] give conditions for a n optimal solution to exist for the case where the returns
distribution is finite and the number of securities exactly equals the number of states in the returns
distribution. Their result is a special case of our Theorem 1. Leland [7] gves a sufficient condition for a
solution to exist, but assumes that there is no direction of recession which offers a nonnegative return for all
states of nature. Bertsekas [l] strengthens Leland's result by extending it to a more general class of utility
functions and shows that for this class it can be made both necessary and sufficient. Though Bertsekas'
approach is applicable to the geometric mean problem, his theorems require some modification to account
for the fact that the logarithmic utility function is defined only for positive values of its argument.
Richard Roll [13] unwittingly formulates a version of the geometric-mean problem for which a solution
is unlikely to exist.
Mossin [lo], for instance, says "The main disadvantage-at least at present---of the growth optimal
policy as compared e.g., with the selection of an E, S efficient policy, is the formidable computational
problem."
1117
'
Copyright 0 1977, The Institute of Management Sciences
1118
J. H. VANDER WEIDE,D. W. PETERSON AND S. F. MAIER
2. The Existence of Optimal Geometric-Mean Portfolios
Consider an investor faced with the repetitive problem of investing his wealth in
some combination of m alternative investment opportunities. Suppose that the investor's objective is to maximize the geometric mean return on his wealth. If xi denotes
the fraction of his wealth the investor invests in opportunity i and r, > 0 denotes the
periodic payout, including return of principal, provided by opportunity i, then we may
identify the task of finding the optimal allocation of wealth for the coming time
period as that of solving the following mathematical programming problem:
Maximize E {log x Tr} subject to x Te = 1,
(I1
and E is the expectation
where x = (x,, . . . , x,)~, r = (r,, . . . , r,JT, e = (1, . . . ,
operator.
As with any mathematical programming problem, there is in connection with (1)
not only the question of finding the best value for x, but also the question of whether
a best value exists. A solution x* is a finite m-vector which imparts maximal value to
the objective function. We discuss the question of whether a solution exists for two
particular cases of the geometric mean problem.
The General Case
The general case of the geometric mean problem is one which imposes no restrictions on the investor's subjective probability distribution of holding period returns.
The existence of a solution to a general geometric mean problem can be assured by
suitably bounding the set of wealth allocations considered feasible. For example, if to
the general problem (1) there are appended the restrictions xi > bi, i = 1, . . . , m,
where the bi-are known, finite constants summing to 1 or less, there results the
problem we call the Constrained General Problem (CGP). The CGP always has a
solution, for it consists of the maximization of a continuous function over a nonempty
compact set.
We refer to the case where the feasible set is unrestricted as the Unconstrained
General Problem (UGP). The UGP has a solution in many circumstances, an
important one being that in which the returns ri are jointly distributed in accordance
with a probability density function which is positive for all r, combinations near zero;
that is P(r) > 0 for all positive r satisfying 0 < rTe < 6, where 6 is a sufficiently small
positive n ~ m b e r One
. ~ example of such a situation is that in which the logarithm of
the return for the ith security is a, PiI E,, where a, and Pi are constants particular
to that security, I is a market index, and E, is an independent normal random variable.
In such circumstances, if one or more of the x, is negative, there is a positive
probability that xTr < 0. Hence any portfolio for which all xi > 0 is at least as
desirable, and one may as well append these constraints. But now a CGP results, and
as noted previously, such problems have solutions.
+
+
The Discrete Case
In the event the r, are discrete random variables which can assume a finite number
n of combinations of values, the Unconstrained General Problem may be stated
Suppose that for some 6 > 0, the probability measure P governing r assigns positive probability to all
open subsets of T = ( r > 0 I r Te < 6 ). Then T is open and nonempty. If some component of x is negative,
then the set S = ( r > 0 I rTx > 0) is open and nonempty, and the intersection S n T is open and
nonempty. Evidently P ( S ) P(S n T ) > 0, so such an x cannot be strictly preferred to any one with
components all of which are nonnegative.
STRATEGY TO MAXIMIZE GEOMETRIC MEAN PORTFOLIO RETURN
11 19
equivalently as the following Unconstrained Discrete Problem (UDP):
n
Maximize
2 [ P, log((x
j= 1
T~
),I
subject to x Te = 1.
(2)
Nature, in this formulation, is capable of assuming any of the states j = 1, . . . , n, and
the probability she will assume the jth is denoted 5.The values of the ri, when Nature
assumes the jth state, are denoted r,. The m x n matrix of r, values is denoted R and
assumed known. One may consider the UDP as an original formulation of the myopic
portfolio selection problem, as did Latank [6],or as an approximation to an Unconstrained General Problem such as (1) permitting other than discrete returns r,. This
latter view is taken in [8].
If there exists a finite and feasible vector x* which imparts a maximal value to the
objective function, then we say x* is a solution to the UDP. The assertion that a
solution exists precludes the possibility that the objective function can be made
arbitrarily large, but does not preclude its being negatively infinite for all feasible
vectors x, as would occur if one or more columns of R were all-zero. To avoid the
latter possibility, we now assume that no column of R contains only zeroes.
The following theorem gives necessary and sufficient conditions that a solution to
the UDP exists.
THEOREM
1. The UDP has a solution if and only if there exists an n x 1 vector y, all
of whose components are positive, such that Ry = e, where e is the m x 1 vector of ones.
PROOF. Suppose there exists y > 0 such that Ry = e, and suppose x satisfies
x T~ 1 0, x Te = 1. Let y,, denote the component of y which is minimal, and note that
= 1/ymin,hence the feasible region
y,, > 0. Then 0 L (x T ~ )5j x T ~ 5e x
of ((x T ~ ) ,. ,. . , (XT ~ ) , )values is bounded. Since the region is also closed, it is
compact, and the objective function, being continuous on this set, assumes a finite
maximum.
If there exists no y > 0 such that Ry = e, then there exists by Tucker's Theorem of
the Alternative [9, p. 341 an x such that x T~ 2 0, x Te = 0.7 Then ax + e l m is feasible
and imparts to the objective function a value which can be made larger than any
preassigned number by taking a sufficiently large. Hence, in such a situation, no
solution to the UDP exists.
The condition Ry = e, y > 0, ensures the objective function assumes a maximal
value for some choice of finite xi's, though the set of xi's imparting maximal value to
the objective need not be bounded. If the condition fails to hold, then the feasible
region is unbounded in such a way that one can make the objective function
arbitrarily large by selling short one or more securities and buying others. A test of
the condition can be implemented by using a linear programming code to find a
feasible solution to Ry = e, y 2 ce, for some small positive c. A geometrical interpretation of the condition Ry = e, y > 0, is that e lies in the interior of the positive cone
spanned by the columns of R .
The condition of Theorem 1 can be restated in equivalent form:
1'. There exists no solution to the UDP if and only if there exists some x
THEOREM
such that x T~ 2 0 and x Te = 0. (We use " > " to mean " 2 but not = ".)
'
Tucker's Theorem of the Alternative, as given in [9, Table 2.4.11 yields the stated result if we set B' = R,
C = 0, D = e T , x = x, and y, = y, where the entities on the left of the equal signs appear in [9], and those
on the right are our own symbols.
1120
J. H. VANDER WEIDE, D. W . PETERSON AND S. F. MAIER
PROOF.^ Tucker's theorem of the alternative [9].
If x satisfies the condition of Theorem It, it has positive components summing to, say,
d > 0, and negative components summing to - d. Denote by x + ( x - ) the vector
obtained by setting to zero the negative (positive) components of x . Then x + / d and
x - / ( - d ) are each nonnegative vectors whose components sum to 1, and may each
be considered a portfolio specification. Furthermore, ( x + x - )=R > 0, SO (X + / d ) = R
> - ( x - / d ) = R . Hence Theorem 1' may be restated as, "An infinite solution to the
UDP exists if and on& if some convex combination of securities dominates some other
(disjoint) convex combination of securities for every state of nature." If an x satisfying
the condition of Theorem 1' is available, one can do arbitrarily well by letting the
portfolio proportions be a x + e l m , and making a a large number.
+
3. The Likelihood of the Existence of a Solution to the Unconstrained
Discrete Problem
Having established exact circumstances under which the UDP has a solution, we
examine next the likelihood with which these circumstances occur. In this regard, it is
helpful to consider a special form of the returns distribution in which one security
dominates all other securities for each state of nature. In addition, we will assume that
for each state of nature all securities besides the dominant one have equal returns. Let
a radial vector be defined as any m-component vector r which has nonnegative
components, all of which are equal, save one which is greater than the rest. Let the n
columns of R be radial vectors selected probabilistically in such a way that the
probability the ith component exceeds the other components of the jth column vectors
is l / m , 1 2 j 2 n, 1 5 i 5 m. For R to be such that there exists a y > 0 satisfying
Ry = e, the columns of R must include at least one radial vector for which the first
component is largest, one radial vector for which the second is largest, and so forth.
This leads at once to the conclusion that if the n columns of R are radial vectors (no
matter how they are chosen) and if n < m , then there is no y > 0 such that Ry = e, and
hence no solution to the UDP exists. If n 2 m and the columns of R are chosen as
described above, then there is some positive probability that a solution to the UDP
exists, and that probability increases with increasing n. For n = m, it happens that the
probability of a solution is m ! / m m ,which equals for m = 2 and decreases rapidly
with increasing m. For n 1 m , the probability of a solution is
+
which for m fixed increases toward 1 as n gets larger.
Using Monte-Carlo methods, we have shown in an unpublished study that even for
more general types of R matrices, the likelihood of the existence of a solution to the
UDP increases as the number n of states of nature increases.
4. Solution of the UDP
A solution to the Unconstrained Discrete Problem (2) can often be obtained with
the aid of an efficient nonlinear programming algorithm. The authors report their
experiences in solving geometric-mean portfolio problems with one particularly
efficient nonlinear programming code in [8]. As we show below, however, if n does
not exceed the number m of securities, it is often possible to obtain optimal alloTheorem 1' may also be obtained as a consequence of Rockafellar's Theorem 27.3 [12, p. 2671 which
asserts the nonexistence of a solution to be equivalent to the existence of a common direction of recession
for the objective function and the feasible set.
STRATEGY TO MAXIMIZE GEOMETRIC MEAN PORTFOLIO RETURN
1121
cations even more efficiently through the solution of a certain set of simultaneous
linear equations. Throughout the discussion of these equations, we assume that (i) a
solution to the UDP exists and (ii) the payout matrix R has rank n.
2. If H is an m X n matrix such that H ~ R= I,,,then x constitutes a
THEOREM
solution to the UDP if and only if
PROOF. (Follows [14, pp. 77-78], with some modifications and extensions).
Necessity. By hypothesis, a solution x* = (x:, . . . , x i ) exists. Since rank(R) = n,
no column of R is 0, and so x* imparts a finite (as opposed to negatively. infinite)
value to the objective function. Hence X * ~ R > 0. Define the Lagrangian function
2 [ Pj log((x R' I),)
n
L(x)
=
- he 'x.
j= 1
It is necessary then that x* satisfy
VL(x*) = Ry* -Ae
= 0,
~ * ~=
e 1,
,
where y * = (y:, . . . , y,*)T,
yJ* = P//(x*~R),,
j
=
1, . . . , n,
and A is a scalar constant.
Premultiplying (4) by x * and
~ noting (5) and (6), we observe that A = 1. By
hypothesis, rank(R) = n so there exists (see Penrose [ll]) an m X n matrix H such
~)~,
that H ~ R= I. From (4), y * = HTe. Using this in (6) yields ( x * ~ R=) ~P ~ / ( H ~ and
necessity has been established.
Sufficiency. Retrace the above steps, bearing in mind that a solution is presumed
to exist, that L(x) is concave, and that a saddle point sufficiency condition is applicable.
This theorem raises the exciting prospect that the UDP often can be solved through
the solution of (3). The latter can be accomplished for fairly large R matrices using a
standard linear programming computer code. Even more straightforward is the
situation in which R is square and of full rank, for then Theorem 2 becomes:
COROLLARY.If rank(R) = n = m, let H T = R
unique solution to (1) is given by x = R -IF, where
-'. Then the x that constitutes the
P/ = P,/(R -le)j, j = 1, . . . , n.
The applicability of Theorem 2 is limited, unfortunately, by two elements of its
hypothesis. The first of these is that the matrix R have rank n. Since R is of dimension
m X n, this restriction requires m > n, or equivalently that the number of securities
exceed or equal the number of states of nature.
The second limiting element in the hypothesis of Theorem 2 is the assumption that
a solution to the UDP exists. As is explained in the previous section, this is unlikely
except in cases where n exceeds m or where the xi are further constrained. With the
imposition of additional constraints (such as nonnegativity), the development of a
theorem analogous to Theorem 2 does not seem possible, and one is apparently forced
to retreat to nonlinear programming codes to obtain numerical solutions.
EXAMPLE.The following example illustrates a situation where the results of the
Corollary to Theorem 2 apply. Suppose that the returns on ten securities in each of
ten states of nature are given by the following return matrix:
J. H. VANDER WEIDE, D. W . PETERSON AND S. F. MAIER
1
2
3
4
5
States of Nature
6
7
8
9
1
0
Security 1
Security 5
Security 10
In addition, suppose that the probability that each state of nature occurs is given by:
Then, since rank(R) = 10 = n = m, the optimal amount to invest in each security is
given by the Corollary to Theorem 2. Using this corollary, we have
xl=-0.857,
x,=14.952,
x , = -0.244,
x4=0.048,
x 5 = -1.284
With this highly unbalanced portfolio we would get an annual return of 66.2%.
5. Conclusion
Though much has been written about the desirable properties of growth-optimal
portfolios, relatively little attention has been given to the problem of computing the
wealth allocation necessar.y to achieve such a portfolio. This latter problem, with
which the present paper is concerned, has two major aspects, one dealing with the
existence of a solution, and the other with the numerical determination of a solution
in those instances where a solution exists.
For a wealth allocation model called the Unconstrained General Problem (UGP)
we show that there are circumstances under which a solution does not exist, and
provide conditions sufficient to ensure that a solution does exist. For this general
problem, in cases where a solution exists, it is suggested that recourse to a nonlinear
programming code may be necessary to obtain a numerical solution.
A special case of the UGP is the Unconstrained Discrete Problem (UDP). For this
problem too it is found that a solution need not exist, and that it probably does not
exist in the case where the number of securities exceeds the number of possible states
of nature. It is, unfortunately, just such a relationship between the numbers of
securities and states of nature which permits a solution, if it exists, to be found readily
as a solution to a set of simultaneous linear equations. In case the number of states of
nature exceeds the number of securities, it is likely that a solution exists. In this case,
though, it seems that a nonlinear programming code must be used to compute the
solution, a task with which our experience is reported elsewhere [8].,
The authors wish to acknowledge Professor Henry Latank's helpful insights, particularly in regard to
the dominance characteristic associated with infinite solutions.
STRATEGY TO MAXIMIZE GEOMETRIC MEAN PORTFOLIO RETURN
1123
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http://www.jstor.org
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A Strategy Which Maximizes the Geometric Mean Return on Portfolio Investments
James H. Vander Weide; David W. Peterson; Steven F. Maier
Management Science, Vol. 23, No. 10. (Jun., 1977), pp. 1117-1123.
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[Footnotes]
1
Capital Growth and the Mean-Variance Approach to Portfolio Selection
Nils H. Hakansson
The Journal of Financial and Quantitative Analysis, Vol. 6, No. 1. (Jan., 1971), pp. 517-557.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197101%296%3A1%3C517%3ACGATMA%3E2.0.CO%3B2-6
2
On the Maximization of the Geometric Mean with Lognormal Return Distribution
Edwin J. Elton; Martin J. Gruber
Management Science, Vol. 21, No. 4, Application Series. (Dec., 1974), pp. 483-488.
Stable URL:
http://links.jstor.org/sici?sici=0025-1909%28197412%2921%3A4%3C483%3AOTMOTG%3E2.0.CO%3B2-K
2
Note on "Optimal Growth Portfolios when Yields are Serially Correlated"
William T. Ziemba
The Journal of Financial and Quantitative Analysis, Vol. 7, No. 4. (Sep., 1972), pp. 1995-2000.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197209%297%3A4%3C1995%3ANO%22GPW%3E2.0.CO%3B2-N
2
Calculation of Investment Portfolios with Risk Free Borrowing and Lending
W. T. Ziemba; C. Parkan; R. Brooks-Hill
Management Science, Vol. 21, No. 2, Application Series. (Oct., 1974), pp. 209-222.
Stable URL:
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NOTE: The reference numbering from the original has been maintained in this citation list.
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2
Solving Nonlinear Programming Problems with Stochastic Objective Functions
William T. Ziemba
The Journal of Financial and Quantitative Analysis, Vol. 7, No. 3. (Jun., 1972), pp. 1809-1827.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197206%297%3A3%3C1809%3ASNPPWS%3E2.0.CO%3B2-9
4
Evidence on the "Growth-Optimum" Model
Richard Roll
The Journal of Finance, Vol. 28, No. 3. (Jun., 1973), pp. 551-566.
Stable URL:
http://links.jstor.org/sici?sici=0022-1082%28197306%2928%3A3%3C551%3AEOT%22M%3E2.0.CO%3B2-H
References
4
On the Maximization of the Geometric Mean with Lognormal Return Distribution
Edwin J. Elton; Martin J. Gruber
Management Science, Vol. 21, No. 4, Application Series. (Dec., 1974), pp. 483-488.
Stable URL:
http://links.jstor.org/sici?sici=0025-1909%28197412%2921%3A4%3C483%3AOTMOTG%3E2.0.CO%3B2-K
5
Capital Growth and the Mean-Variance Approach to Portfolio Selection
Nils H. Hakansson
The Journal of Financial and Quantitative Analysis, Vol. 6, No. 1. (Jan., 1971), pp. 517-557.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197101%296%3A1%3C517%3ACGATMA%3E2.0.CO%3B2-6
8
A Monte Carlo Investigation of Characteristics of Optimal Geometric Mean Portfolios
Steven F. Maier; David W. Peterson; James H. Vander Weide
The Journal of Financial and Quantitative Analysis, Vol. 12, No. 2. (Jun., 1977), pp. 215-233.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197706%2912%3A2%3C215%3AAMCIOC%3E2.0.CO%3B2-V
NOTE: The reference numbering from the original has been maintained in this citation list.
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13
Evidence on the "Growth-Optimum" Model
Richard Roll
The Journal of Finance, Vol. 28, No. 3. (Jun., 1973), pp. 551-566.
Stable URL:
http://links.jstor.org/sici?sici=0022-1082%28197306%2928%3A3%3C551%3AEOT%22M%3E2.0.CO%3B2-H
16
Solving Nonlinear Programming Problems with Stochastic Objective Functions
William T. Ziemba
The Journal of Financial and Quantitative Analysis, Vol. 7, No. 3. (Jun., 1972), pp. 1809-1827.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197206%297%3A3%3C1809%3ASNPPWS%3E2.0.CO%3B2-9
17
Note on "Optimal Growth Portfolios when Yields are Serially Correlated"
William T. Ziemba
The Journal of Financial and Quantitative Analysis, Vol. 7, No. 4. (Sep., 1972), pp. 1995-2000.
Stable URL:
http://links.jstor.org/sici?sici=0022-1090%28197209%297%3A4%3C1995%3ANO%22GPW%3E2.0.CO%3B2-N
18
Calculation of Investment Portfolios with Risk Free Borrowing and Lending
W. T. Ziemba; C. Parkan; R. Brooks-Hill
Management Science, Vol. 21, No. 2, Application Series. (Oct., 1974), pp. 209-222.
Stable URL:
http://links.jstor.org/sici?sici=0025-1909%28197410%2921%3A2%3C209%3ACOIPWR%3E2.0.CO%3B2-3
NOTE: The reference numbering from the original has been maintained in this citation list.