استراتژی های سرمایه گذاری بهینه در حضور حداقل تضمین

In a continuous-time framework, we consider the problem of a Defined Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution flow into the financial market, in order to maximize the expected utility function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee.We assume that the stochastic interest rates follow the affine dynamics, including the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985) 385] and the Vasiˇcek model. The optimal investment strategies are obtained by assuming the completeness of financial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for different examples of guarantees and contributions.

Since Merton (1971), the optimal consumption-investment modelization is a central topic in mathematical finance. An important aspect is the introduction of a stochastic term structure for the interest rates. In the existing literature, a first stream of papers (e.g. Karatzas et al., 1987; Karatzas, 1989; El Karoui and Jeanblanc-Picqué, 1998) do not specify the stochastic process which leads to very general results. However, the general feature of the interest rates does not permit to test the results of those papers by comparing them with reality, since their solutions are not explicit.Afterwards, some authors focused on the possibility to obtain closed form solutions in order to test the behaviour of the optimal portfolio. For this reason, Bajeux-Besnainou et al. (1998, 1999) and Lioui and Poncet (2001) choose a Vasiˇcek specification of the term structure. In Deelstra et al. (2000), we investigated the case where interest rates follow the Cox–Ingersoll–Ross (CIR) dynamics. Assuming completeness of the markets and power utility function, we obtained by the use of the Cox and Huang (1989) methodology closed-form solutions for a utility maximization problem of terminal wealth, without considering contributions or a positive guarantee. In this paper, we are interested in obtaining explicit optimal strategies for a defined contribution pension fund in the presence of a minimum guarantee in a continuous-time framework. Since this is a long-term investment problem of typically 30 or 40 years, it is crucial to allow for a stochastic term structure for the interest rates. In the Vasiˇcek framework, Jensen and Sørensen (2000) measure the effect of a minimum interest rate guarantee constraint through the wealth equivalent in case of no constraints and show numerically that guarantees may induce a significant utility loss for relatively risk tolerant investors. Boulier et al. (2001) also study a problem closely related to ours, namely the optimal management of a defined contribution plan in the presence of a minimum guarantee. Nevertheless, in their framework, the contribution flow is a deterministic process, and the guarantee has a very specific form, which is an annuity paid out from the date of retirement until the date of death, where both annuity and date of death are supposed to be deterministic. Moreover, they choose the Vasiˇcek specification of the term structure in the spirit of Bajeux-Besnainou et al. (1998, 1999). We investigate the case in which interest rates follow the affine dynamics of Duffie and Kan (1996) in its one-dimensional version, which includes as special cases the CIR (Cox et al., 1985) model and the Vasiˇcek (1977) model. Moreover, the problem of the fund manager is to invest the initial wealth and the stochastic contribution flow into the financial market, in order to maximize the expected utility function of the terminal wealth, which should exceed the minimum guarantee, a general random variable. The paper is organized as follows: in Section 2, we define the market structure and introduce the optimization problem under consideration. We also show how this problem is related to the pension fund management. In Section 3, we transform the initial problem into an equivalent one, which we solve explicitely in the power utility case (Section 4). In Section 5 we come back to the solution of the initial problem and we find it explicitely by specifying the form of the contribution process in some interesting cases. Section 6 is devoted to numerical applications and Section 7 concludes the paper.

We considered a model for a Defined Contribution Pension Fund: we studied the problem of a fund manager to invest some given initial wealth and a contribution flow into the financial market in such a way that the expected utility of the terminal wealth is maximized and a minimum guarantee is satisfied at the final date. We investigated the case of affine interest rates and we supposed the markets to be complete. From the point of view of pension fund modelling, it might seem indispensable and trivial to include a guarantee and a contribution flow, but from the mathematical point of view, the problem turns out to be a no standard investment problem. By introducing an auxiliary process, called surplus process, we reduced to a purely investment-problem. This problem has been explicitly solved under the assumption that the utility function of the fund manager belongs to the CRRA family. Finally we came back to the solution of the initial problem by specifying the contribution process and the guarantee. A numerical analysis is included. There are several directions for future research. First, it would be interesting to extend our approach to the case of incomplete markets, since the contribution process is not necessarily generated by the market. We further notice that the constraint on the surplus does not imply that the wealth at date t,Wt , is positive almost surely. In fact, the closed-form solution can be negative with a strictly positive probability. It would be interesting to study also the constrained case (W(t) ≥ 0, ∀t ≥ 0). In a consumption-investment framework such problem has been solved by El Karoui and Jeanblanc-Picqué (1998). In a pension fund context, however, the fund manager could be more interested in constraints on the surplus process or he could be interested in ways to diversify the risk that the global wealth becomes negative over the fund population. It would therefore be interesting to find some actuarial hypotheses on the fund population under which this implication is valid for the global fund wealth.