- Email: [email protected]
On the convexity of the portfolio choice set
371
Economics Letters 21 (1986) 371-373 North-Holland
ON THE CONVEXITY Thomas
RUSSELL
OF THE PORTFOLIO
CHOICE SET
*
Santa Clara University, Santa Clara, CA 95053, USA
Received Accepted
19 December 1985 27 May 1986
This paper studies the convexity of the set of optimal portfolios in asset space. Recently Dybvig and Ross (1982) have shown by counter example that the optimal set is not always convex. In this letter I provide a necessary condition for convexity and discuss the problem from a geometric point of view.
1. Introduction Most popular discussions of the stock market treat average or market behavior as though it were generated by the calculations of a rational investor. Such anthropomorphosis can only be justified in general if the optimal portfolios of individual investors constitute a convex set in asset space. Otherwise the average holding may lie outside this set, in which case the market would not be optimal for any rational investor. Recently Dybvig and Ross (1982) have shown by counter example that the optimal set is not always convex. In this letter I provide a necessary condition for convexity and discuss the problem from a geometric point of view.
2. The portfolio problem Suppose we are given N assets Xj, i = 1, 2, . . . , N. The return factor on asset i, (1 + pi) = R,, is a random variable defined in general on (- cc, + co) and subject to some joint distribution, say G. A portfolio is a linear combination of assets, say Z, where Z = C(Y~X,, Cai = 1, . . . , (1). Given G, a portfolio is uniquely determined by the specification of (Y= ((~i, (Y*, . . . , 1 - La,), so that a portfolio is simply a point in an n - 1 dimensional space, the unit simplex of R”. Portfolios are valued by an increasing concave utility function U as follows: EU=
co U(W.z) J -cc
dF(z;
cy),
* The author wishes to thank Professor errors are the author’s own.
0165-1765/86/$3.50
Frank
Farris,
Math.
0 1986, Elsevier Science Publishers
Department,
Santa Clara
B.V. (North-Holland)
University
for helpful
discussions.
All
T. Russell / Convexity of the portfolio choice set
372
where w, Z
dF(z;
= initial wealth, = some value of R,, and CI)= the distribution of 2 at (Y.
The portfolio max EU
problem
w.r.t.
is
LY.
3. Portfolio problems in Hilbert space This problem admits the following geometric interpretation. Associate Laguerre sharp transform of its distribution function, which we write as
As Keilson and Sumita (1986) show, this can virtually one-to-one. We therefore have the following associations: (a) Each asset is a point in a Hilbert space. (b) Each portfolio of assets (Yis a point in a Hilbert (c) Each set of portfolios A is a surface in a Hilbert $74
= ~~(4
always
be done,
with each asset
and
X, the
the association
is
space pL,#(cu). space,
LYEA.
(d) Each utility function U is a member of the dual of this Hilbert expected utility make EU a linear functional on flat spaces.
space,
since
the axioms
of
[For details and proofs of the Hilbert space properties of pz, see Keilson and Sumita (1986). For details of the flat space interpretation of expected utility, see Russell (1985a).] Portfolio theory is thus the study of maximised functionals on a surface in Hilbert space. This immediately gives the theorem. Theorem 1. Let A* be defined by (Y E A* - (Y maximises EU for some U with U’ > 0, U” < 0. Then A* convex * p,“( A*) is a convex surface in Hilbert space where p/#( A*) is the set of Laguerre transforms of cy E A*. Proof. Since U is a linear functional on ,g( a), hyperplane. Any set all of whose points are supporting
each maximising point Q.E.D. is convex.
is supported
by
4. Remarks (I ) When moments exist, Laguerre co-efficients are linear transformations [see Keilson and Sumita (1986)]. Thus we have an immediate corollary.
of moment
sequences
a
313
T. Russell / Convexity of the portfolio choice set
Corollary 1. $(A*) a convex surface * pj(A*) moment space of the moment sequences of a E A*.
a convex surface where pj(A*)
is the surface in
This is the easiest way to prove the standard results that, e.g., normal securities or complete Arrow-Debreu securities yield a convex optimal set. (2) The approach taken here is extremely profligate with respect to dimensions. We can associate with the Laguerre coefficients of LY~the Laguerre distance,
This distance imposes a metric g,, on the unit simplex, and thus turns the unit simplex into a Riemannian manifold. Now we can ask the obvious question. When can this manifold be embedded in a flat space in such a way that the resulting surface is convex? This question has a long history in mathematics, going back to Weyl in 1916. It is fully answered in the work of Aleksandrov [see Pogorelov (1973)]. The relevant theorem is: A metric of positive Theorem 2 (Aleksandrov). sphere is realisable by a convex surface. Thus positive
curvature
is the intrinsic
curvature
defined on a manifold
homeomorphic
to a
key to convexity.
5. Conclusion From a geometric point of view, portfolio theory is the study of tensors defined on Riemannian manifolds. To answer Euclidean questions such as the convexity of the portfolio optimal set in asset space, we must embed the Riemannian manifold in some flat space. This point of view is explored further in Russell (1985a,b), and Russel and Schmidt (1985). A lighthearted discussion may be found in Russell (1986).
References Dybvig, P.H. and S.A. Ross, 1982, Portfolio efficient sets, Econometrica 50, no. 6, Nov., 1525-1546. Keilson, J. and U. Sumita, 1986, A general Laguerre transform and a related distance between probability measures, Journal of Mathematical Analysis and Applications 113, 228. Pogorelov, A.V., 1973, Extrinsic geometry of convex surfaces (American Mathematical Society, Providence, RI). Russell, T., 1985a, Economics and psychology: A formal analysis of some common ground, in: Brandstatter, ed., Psychology and economics (Linz). Russell, T., 1985b, A new approach to portfolio theory, Economics Letters 18, no. 1, 59-61. Russell, T., 1986, Dr. Einstein meets a student of financial economics, Mimeo. Russell, T. and C. Schmidt, 1985, Portfolio theory: A differential geometric approach, Mimeo.
Economics Letters 21 (1986) 371-373 North-Holland
ON THE CONVEXITY Thomas
RUSSELL
OF THE PORTFOLIO
CHOICE SET
*
Santa Clara University, Santa Clara, CA 95053, USA
Received Accepted
19 December 1985 27 May 1986
This paper studies the convexity of the set of optimal portfolios in asset space. Recently Dybvig and Ross (1982) have shown by counter example that the optimal set is not always convex. In this letter I provide a necessary condition for convexity and discuss the problem from a geometric point of view.
1. Introduction Most popular discussions of the stock market treat average or market behavior as though it were generated by the calculations of a rational investor. Such anthropomorphosis can only be justified in general if the optimal portfolios of individual investors constitute a convex set in asset space. Otherwise the average holding may lie outside this set, in which case the market would not be optimal for any rational investor. Recently Dybvig and Ross (1982) have shown by counter example that the optimal set is not always convex. In this letter I provide a necessary condition for convexity and discuss the problem from a geometric point of view.
2. The portfolio problem Suppose we are given N assets Xj, i = 1, 2, . . . , N. The return factor on asset i, (1 + pi) = R,, is a random variable defined in general on (- cc, + co) and subject to some joint distribution, say G. A portfolio is a linear combination of assets, say Z, where Z = C(Y~X,, Cai = 1, . . . , (1). Given G, a portfolio is uniquely determined by the specification of (Y= ((~i, (Y*, . . . , 1 - La,), so that a portfolio is simply a point in an n - 1 dimensional space, the unit simplex of R”. Portfolios are valued by an increasing concave utility function U as follows: EU=
co U(W.z) J -cc
dF(z;
cy),
* The author wishes to thank Professor errors are the author’s own.
0165-1765/86/$3.50
Frank
Farris,
Math.
0 1986, Elsevier Science Publishers
Department,
Santa Clara
B.V. (North-Holland)
University
for helpful
discussions.
All
T. Russell / Convexity of the portfolio choice set
372
where w, Z
dF(z;
= initial wealth, = some value of R,, and CI)= the distribution of 2 at (Y.
The portfolio max EU
problem
w.r.t.
is
LY.
3. Portfolio problems in Hilbert space This problem admits the following geometric interpretation. Associate Laguerre sharp transform of its distribution function, which we write as
As Keilson and Sumita (1986) show, this can virtually one-to-one. We therefore have the following associations: (a) Each asset is a point in a Hilbert space. (b) Each portfolio of assets (Yis a point in a Hilbert (c) Each set of portfolios A is a surface in a Hilbert $74
= ~~(4
always
be done,
with each asset
and
X, the
the association
is
space pL,#(cu). space,
LYEA.
(d) Each utility function U is a member of the dual of this Hilbert expected utility make EU a linear functional on flat spaces.
space,
since
the axioms
of
[For details and proofs of the Hilbert space properties of pz, see Keilson and Sumita (1986). For details of the flat space interpretation of expected utility, see Russell (1985a).] Portfolio theory is thus the study of maximised functionals on a surface in Hilbert space. This immediately gives the theorem. Theorem 1. Let A* be defined by (Y E A* - (Y maximises EU for some U with U’ > 0, U” < 0. Then A* convex * p,“( A*) is a convex surface in Hilbert space where p/#( A*) is the set of Laguerre transforms of cy E A*. Proof. Since U is a linear functional on ,g( a), hyperplane. Any set all of whose points are supporting
each maximising point Q.E.D. is convex.
is supported
by
4. Remarks (I ) When moments exist, Laguerre co-efficients are linear transformations [see Keilson and Sumita (1986)]. Thus we have an immediate corollary.
of moment
sequences
a
313
T. Russell / Convexity of the portfolio choice set
Corollary 1. $(A*) a convex surface * pj(A*) moment space of the moment sequences of a E A*.
a convex surface where pj(A*)
is the surface in
This is the easiest way to prove the standard results that, e.g., normal securities or complete Arrow-Debreu securities yield a convex optimal set. (2) The approach taken here is extremely profligate with respect to dimensions. We can associate with the Laguerre coefficients of LY~the Laguerre distance,
This distance imposes a metric g,, on the unit simplex, and thus turns the unit simplex into a Riemannian manifold. Now we can ask the obvious question. When can this manifold be embedded in a flat space in such a way that the resulting surface is convex? This question has a long history in mathematics, going back to Weyl in 1916. It is fully answered in the work of Aleksandrov [see Pogorelov (1973)]. The relevant theorem is: A metric of positive Theorem 2 (Aleksandrov). sphere is realisable by a convex surface. Thus positive
curvature
is the intrinsic
curvature
defined on a manifold
homeomorphic
to a
key to convexity.
5. Conclusion From a geometric point of view, portfolio theory is the study of tensors defined on Riemannian manifolds. To answer Euclidean questions such as the convexity of the portfolio optimal set in asset space, we must embed the Riemannian manifold in some flat space. This point of view is explored further in Russell (1985a,b), and Russel and Schmidt (1985). A lighthearted discussion may be found in Russell (1986).
References Dybvig, P.H. and S.A. Ross, 1982, Portfolio efficient sets, Econometrica 50, no. 6, Nov., 1525-1546. Keilson, J. and U. Sumita, 1986, A general Laguerre transform and a related distance between probability measures, Journal of Mathematical Analysis and Applications 113, 228. Pogorelov, A.V., 1973, Extrinsic geometry of convex surfaces (American Mathematical Society, Providence, RI). Russell, T., 1985a, Economics and psychology: A formal analysis of some common ground, in: Brandstatter, ed., Psychology and economics (Linz). Russell, T., 1985b, A new approach to portfolio theory, Economics Letters 18, no. 1, 59-61. Russell, T., 1986, Dr. Einstein meets a student of financial economics, Mimeo. Russell, T. and C. Schmidt, 1985, Portfolio theory: A differential geometric approach, Mimeo.