سیاست های سهام مطلوب با استقراض و محدودیت های حراج کوتاه

We characterize optimal intertemporal portfolio policies for investors with CRRA utility facing either a borrowing constraint, or shortsale restrictions, or both. The optimal constrained portfolios are identified as optimal unconstrained portfolios for a higher riskless rate, or for a subset of the risky assets, or for a combination of the two settings. Our characterization is based on duality results in Cvitanić and Karatzas (1992, Annals of Applied Probability 2, 767–818) for optimal portfolio investment when portfolio values are more generally constrained to a closed, convex, nonempty subset.

This paper characterizes optimal intertemporal portfolio policies for CRRA-utility investors facing either a borrowing limit on the total wealth invested in the risky assets, or shortsale restrictions on all risky assets, or both. The characterization is based on the first-order conditions to a minimization problem identified by Cvitanić and Karatzas (1992) as underlying the dual formulation of the optimal portfolio investment problem when portfolio values are more generally constrained to a closed, convex, nonempty subset of (when there are n risky assets). In each setting, the optimal constrained portfolio is identified as an optimal ‘unconstrained’ portfolio. Specifically, with borrowing constraints only, CRRA-utility investors act as if unconstrained but facing a higher interest rate. With shortsale constraints only, these investors act as if unconstrained when investing only in a subset of the risky assets. With borrowing and shortsale constraints, both effects obtain. Specifically, the optimal portfolio is equivalent to the optimal borrowing-constrained-only portfolio for a subset of the risky assets, and thus to the optimal unconstrained investment in these assets at a higher interest rate. Results closely related to a number of those derived here in a dynamic setting have previously been identified as holding in a one-period, mean-variance or Markowitz framework. Black (1972) establishes that an investor who cannot borrow at all chooses a tangency portfolio corresponding to a higher interest rate. Brennan (1971) considers the setting in which the investor can borrow without limit, but faces a borrowing rate which is greater than the lending rate. The optimal portfolio is again equivalent to a tangency portfolio, in this case corresponding to one of three possible ‘risk-free’ rates.1 Separately, Lintner (1965) identifies the optimal shortsale-constrained Markowitz portfolio as the optimal unconstrained portfolio for a subset of the risky securities. The fact that we obtain very similar results for CRRA-utility investors in the dynamic setting is not entirely unexpected given that it is well-known that, for the model we consider, these investors’ optimal unconstrained portfolios are instantaneously mean-variance efficient.2 Nonetheless, to date, some of these results, particularly those concerning shortsale constraints, have been missing from the continuous-time literature. Grossman and Vila (1992), using a stochastic dynamic programming approach, study the optimal intertemporal portfolio policies of a borrowing-constrained power-utility investor in the standard Merton (constant-coefficient) setting. Rather than restricting investment in the risky assets to be less than some constant proportion of wealth, as we do here, Grossman and Vila consider the effects of a borrowing limit which is affine in wealth.3 Because their model features only one risky asset, it does not identify how a borrowing restriction affects relative investment in different risky assets. Fleming and Zariphopoulou (1991) arrive at results analogous to those in Brennan (1971) for a power-utility investor facing differing borrowing and lending rates. Their model also features one risky asset. Xu and Shreve 1992a and Xu and Shreve 1992b use a martingale approach and duality methods to characterize solutions subject only to shortsale constraints in a model with multiple risky assets. Although Xu and Shreve establish that, in the constant-coefficient setting, a ‘mutual fund’ theorem still holds,4 they do not explicitly identify this mutual fund. Optimal portfolio policies when portfolio values are more generally constrained to a closed, convex, nonempty subset, K, of , are studied by Cvitanić and Karatzas (1992). These results form the departure point for the analysis in this paper. In their paper, Cvitanić and Karatzas treat an example, with borrowing and shortsale constraints, for a log-utility investor with two risky assets having uncorrelated returns of equal volatility. Their approach, which rests on exhaustive enumeration of optimal portfolios given various relationships between different parameters of the model, is not ideally suited to a setting with a much larger number of risky assets, whose returns may be correlated. In the following section we describe the economy and formulate the unconstrained investor's problem. Section 2 reviews key results for the constrained problem from Cvitanić and Karatzas (1992). Building on these results, Section 3 examines the optimal policies of the CRRA-utility investor when borrowing and shortsale constraints are imposed, first separately and then concurrently. In the borrowing-constrained-only setting, the investor's optimal portfolio is identified explicitly. Whenever shortsale constraints are present and binding, it is not known a priori which assets are held in positive amounts in the optimal portfolio. In such instances, further characterization of the optimal constrained portfolios is provided, leading to an algorithm for their calculation. Section 4 concludes.

The goal of the above analysis has been to determine the investor 's optimal portfolio policy under various constraints by identifying the vector v H Summarizing our results, we have shown that v H is characterized as follows: Thus, in the presence of both borrowing and shortsale constraints,v H can be broken down into a borrowing-constrained component, /)1, and a shortsale- constrained component, u H.ThisdecompositionallowsustolinkPropositions 2 and 3. While having very similar interpretations, the conditions underlying these two propositions (v H i 5 0and v H i'/,respectively, in each case for any asset i for which the shortsale constraint is binding) appear somewhat di !erent in nature. However, from (iii) above we see that the condition in Proposition 3 can be restated in terms of the shortsale-constrained component of v H as u H i 0. Thus the condition being interpreted is essentially the same in both propositions.