Comonotonic Approximations for Optimal Portfolio Selection Problems

Abstract

We investigate multiperiod portfolio selection problems in a Black and Scholes type market where a basket of 1 riskfree and "m" risky securities are traded continuously. We look for the optimal allocation of wealth within the class of "constant mix" portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari's dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene et al. (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically. Copyright The Journal of Risk and Insurance.

Convex order approximations in the case of cash flows of mixed signs

“…They generalize portfolio selection problems to the case where a minimal return requirement is imposed. The results that they propose are an extension of the solution of Vanduffel et al (2005) to the more general context of provisioning and saving as described in Dhaene et al (2005). However, the proof of the presented results contains an error.…”

“…We approximate the distribution of the terminal wealth W n by W n = max[V n , 0]. Choosing Λ such that the variance of V n is maximized and hence as close as possible to Var(V n ), results in the optimal conditioning random variable Λ of the form (4), with coefficients β j equal to, see Dhaene et al (2005):…”

“…They allow for liabilities that can be both positive or negative, as opposed to Dhaene et al (2005), where all liabilities have to be of the same sign. They generalize portfolio selection problems to the case where a minimal return requirement is imposed.…”

“…We will illustrate our results with numerical examples. The framework of optimal portfolio selection in which we work is the same as in Dhaene et al (2005) and Van Weert et al (2010) and we refer to those papers for more details, notations and terminology.…”

Minimizing the probability of lifetime ruin under borrowing constraints

“…Also, comonotonicity and its applications to finance and insurance have been extensively studied by Goovaerts, Dhaene, Kaas, Denuit, and their co-workers; for example, see Dhaene et al (2005) and Vanduffel, Dhaene, and Goovaerts (2005). Amongst other applications, they propose comonotonic approximations to tackle multi-period optimal portfolio selection problems under very flexible deterministic saving and consumption patterns.…”