Dynamic portfolio strategies of a mean-variance investor are studied using Hilbert space theory which permits a geometrical interpretation to several properties already present in Bajeux and Portait (1993). We characterize the unconstrained dynamic frontier whose slope is compared to the static case, and show hedging strategies to be buy and hold positions in the zero coupon and a portfolio of terminal value inverse to that of the numeraire portfolio. This portfolio is detailed in the case of a market driven by stochastic interest rates. We derive a simplified version of Merton's (1973) separation theorem, involving 2 funds instead of the standard 3 : the zero coupon of maturity equal to the investor's horizon and the instantaneous tangent portfolio. The decomposition thus provides a natural extension of Markowitz (1952) to the dynamic case. Next, we investigate the situation where the investor is required not to default. Following Cox and Huang (1989), his optimal strategy is achieved by setting aside part of his initial capital to buy a put option on the distribution he will manufacture with the balance. The latter distribution also results from an orthogonal projection on the appropriate subspace. We then compare constrained and unconstrained solutions in the case of a market with constant coefficients. Simulations show the 2 distributions to differ substantially with an increasing expectation of returns, which also implies a important loss of efficiency.
NGUYEN, P.D. and PORTAIT, R. (1996). A Risk-return Analysis of Dynamic Portfolio Strategies with a Solvency Constraint. ESSEC Business School.