STOCHASTIC

R. G. VICKSON
(43)-(44), and (45)-(48). In all cases the optimal mutual funds x 1 and x 2
depend strongly on the "exponent" c and on the ratio ajb if this is nonzero.
For quadratic utility with c = 1, x 1 and x 2 are independent of ajb in the
unconstrained problem PO, and are piecewise constant functions of ajb in the
constrained problems Pk (k > 0), varying only according to which segment of
the Markowitz frontier is tangent to a mean-variance indifference curve. When
c # 1 and alb j= 0, x l and x 2 depend on both c and ajb, for unconstrained as
well as constrained problems. For utility functions of the constant absolute
risk-aversion class, «'(£) = ae H , investor preference-orderings and optimal
solutions must be independent of a. Again, this follows directly from Eqs.
(49)-(51).
Throughout this paper the allowed type of investment in an asset has either
been unconstrained or purely nonnegative. A more general case is that which
allows limited borrowing or short sales. It is clear that such a generalization
leads to no substantial changes in the theory. In particular, the conclusions of
Theorems 4 and 5 remain valid in this more general situation. The utility
functions giving local versus global separation in markets with money agree
with those obtained by Hakansson [6] using different methods. (See also
Exercise ME-14.)
REFERENCES
1. VON NEUMANN, J., and MORGENSTERN, O., Theory of Games and Economic Behavior.
Princeton Univ. Press, Princeton, New Jersey, 1953.
2. CASS, D., and STIGLITZ, J. E., "The Structure of Investor Preferences and Asset Returns,
and Separability in Portfolio Allocations: A Contribution to the Pure Theory of Mutual
Funds." Journal of Economic Theory 2 (1970), 122-160.
3. TOBIN, J., "Liquidity Preference as Behavior Toward Risk." Review of Economic
Studies 36 (1958), 65-86.
4. LINTNER, J., "The Valuation of Risk Assets and the Selection of Risky Investments in
Stock Portfolios and Capital Budgets." Review of Economis and Statistics 47 (1965),
13-37, reprinted in this volume.
5. BREEN, W., "Homogeneous Risk Measures and the Construction of Composite Assets."
Journal of Financial and Quantitative Analysis 3 (1968), 405-413.
6. HAKANSSON, N. H., "Risk Disposition and the Separation Property in Portfolio
Selection." Journal of Financial and Quantitative Analysis 4 (1969), 401-416.
7. KUHN, H. W., and TUCKER, A. W., "Nonlinear Programming." Proceedings of the
Symposium on Mathematical Statistics and Probability, 2nd, Berkeley, 1951 (J. Neyman
ed.). Univ. of California Press, Berkeley, 1951.
8. ZANGWILL, W., Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood
Cliffs, New Jersey, 1969.
9. MARKOWITZ, H. M., "Portfolio Selection." Journal of Finance 6 (1952), 77-91.
10. ARROW, K. J., Aspects of the Theory of Risk-Bearing. Yrjo Jahnsson Foundation,
Helsinki, 1965.
170 PART II QUALITATIVE ECONOMIC RESULTS