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OPTIMAL PORTFOLIO STRATEGIES FOR DEFINED- CONTRIBUTION PENSION PLANS: A STOCHASTIC CONTROL APPROACH
BATTOCCHIO, PAOLO
2004-04-22
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Contributor(s)
ZECCHIN, MARCO
•
ZECCHIN, MARCO
Abstract
This work contributes to the analysis of the asset allocation problem for pension funds in a stochastic continuous-time framework. In particular, we focus on the portfolio problem of a fund manager who wants to maximize the expected utility of the fund's terminal wealth, that is to say the wealth accumulated up to the retirement of a representative shareholder. We consider the case of a defined-contribution pension plan. The classical dynamic optimization model, initially proposed by Merton (1969, 1971), assumes a market structure with constant interest rate. We note that the optimal asset-allocation problem for a pension fund involves quite a long period, generally from 20 to 40 years. It follows that the assumption of constant interest rates does not fit with our target. For the same reason, we support the idea that also the inflation risk needs to be considered. According to this assumption, we observe that some of the leading pension funds in U.S. have from 5 to 10 percent of their portfolios allocated just to inflation-indexed instruments (Chicago Mercantile Exchange data). Moreover, the benefits proposed by DC pension plans often require the specification of the stochastic behavior of other variables, such as salaries. Thus, the fund manager must cope not only with financial risks, but also with other risk sources outside the financial market as for example salaries. In this case, we will highlight how the introduction of a stochastic non-financial incarne (in our case contribution) in the optimal portfolio problem causes several computational difficulties. Summing up the above considerations, with respect to the classical Merton's portfolio choice problem, here we include in the model: (i) a stochastic process for the short rate, (ii) the inflation risk, through a stochastic process for the consumer price index, and (iii) the salary risk, through a stochastic process for the contributions. The methodological approach we use to solve the optimal asset-allocation problem of a pension fund is the stochastic optimal control. Alternative approaches (see for instance Deelstra et al., 2003; and Lioui and Poncet, 2001) are based on the so-called "martingale approach" first introduced by Cox and Huang (1989, 1991), where the resulting partial differential equation is often simpler to salve than the Hamilton-Jacobi-Bellman equation coming from the dynamic programming. Nevertheless, in the martingale approach, when a stochastic process for salaries enters the optimization procedure, a submartingale is no more obtained to apply the theory. In the first chapter we present a review of the mathematical tools required for the formal analysis of asset allocation models in continuous-time. Chapter 2 illustrates the use of the stochastic optimal control as optimization engine in the consumption and portfolio choice problems in continuoustime. In Chapter 3 we develop the optimal consumption and investment problem presented by Merton ( 1969, 1971). This model is commonly regarded as the first successful application of stochastic control in economics. Moreover, we present an explicit solution to the control problem for generai hyperbolic absolute risk aversion utility functions. In Chapter 4 we extend the classical Merton's model by allowing interest rates to be stochastic. We illustrate how the introduction of another relevant state variable ( the stochastic short rate) in the control problem, in addition to the wealth, represents a delicate matter, although the methodological approach does not change. Under suitable assumptions on the value function, we derive an exact solution to the control problem by applying the Feynman-Kac Theorem directly to the Hamilton-Jacobi-Bellman equation. Then, we analyse how the short rate dynamics affects the optimal portfolio choice. Actually, the stochastic interest rate introduces a new hedging component in addition to the only speculative component characterizing the optimal portfolio strategy in the Merton's model. Finally, Chapter 5 extends the asset allocation models presented in the previous chapters to the case of a DC pension fund. In order to characterize the accumulation phase, we consider the case of a shareholder who, at each period t E [0, T], contributes a constant proportion of his salary to a personal pension fund. A t the time of retirement T, the accumulated pension fund will be converted into an annuity. Initially, we assume a complete financial market constituted by three assets: a riskless asset, a stock and a bond which can be bought and sold without incurring any transaction costs or restriction on short sales. Then, we take into account two stochastic processes describing the behavior of salaries and the consumer price index. As we have already remarked, the presence of a stochastic process for salaries represents the chief obstacle to a complete solution of the optimal control problem. In fact, if we assume that the salary process is driven by a risk source which does not belong to those defining the financial market, that is a non-hedgeable risk, we obtain that the market is no more complete. In this case, even if we can state the control problem, the corresponding Hamilton-Jacobi-Bellman equation an d the optimal portfolio, we are no t able to apply the Feynman-Kac Theorem and to find the optimal value function in a closed form. Therefore, this prevents us from studying how the coefficients of the salary process affect the optimal portfolio strategies. Here, we propose a model in which the presence of stochastic salaries is consistent with the assumption of complete market. In order to justify this proposition, we link the only non-hedgeable component of the salary process to the consumer price index, whose role in the financial market will be widely investigated. By following this way, we find a closed form solution to the control problem and then we are able to analyze in detail how the risk involved by the stochastic behavior of salary and inflation affects the optimal portfolio composition. We prove that the optimal portfolio is formed by three components: (i) a speculative component proportional to both the portfolio Sharpe ratio and the reciprocal of the Arrow-Pratt risk aversion index, as the one derived in the Merton's model, (ii) a hedging component depending on the state variable parameters as the one derived in Chapter 4, and (iii) a preference-free hedging component depending only on the diffusion terms of both the financial assets and the consumer price index. Furthermore, after working out the expected values characterizing the solution, the optimal portfolio can be simplified to the sum of only two components: one depending on the time horizon, and the other one independent of it. In particular, the optimal portfolio real composition turns out to have an absolutely time independent component. Moreover, the risk aversion parameter determines whether the portfolio is more or less affected by the time-dependent real component. The higher the risk aversion, the more the time-dependent real component affects the optimal portfolio. Accordingly, low values of the risk aversion parameter determine a real portfolio composition that becomes approximately constant through time. Finally, we present a numerical application in order to investigate the dynamic behavior of the optimal portfolio strategy more closely.
Insegnamento
Publisher
Università degli studi di Trieste
Languages
en
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