انتخاب سهام بهینه تحت نرخ بهره تصادفی

In an economy where interest rates and stock price changes follow fairly general stochastic processes, we analyze the portfolio problem of an investor endowed with a non-traded cash bond position. He can trade on stocks, the riskless asset and a futures contract written on the bond so as to maximize the expected utility of his terminal wealth. When the investment opportunity set is driven by an arbitrary number of state variables, the optimal portfolio strategy is known to contain a pure, preference free, hedge component, a speculative element and Merton–Breeden hedging terms against the fluctuations of each and every state variable. While the first two components are well identified and easy to work out, the implementation of the last ones is problematic as the investor must identify all the relevant state variables and estimate their distribution characteristics. Using the martingale approach, we show that the optimal strategy can be simplified to include, in addition to the pure hedge and speculative components, only two Merton–Breeden-type hedging elements, however large is the number of state variables. The first one is associated with interest rate risk and the second one with the risk brought about by the co-movements of the spot interest rate and the market prices of risk. The implementation of the optimal strategy is thus much easier, as it involves estimating the characteristics of the yield curve and the market prices of risk only rather than those of numerous (a priori unknown) state variables. Moreover, the investor's horizon is shown explicitly to play a crucial role in the optimal strategy design, in sharp contrast with the traditional decomposition. Finally, the role of interest rate risk in actual portfolio risk management is emphasized.

The overall outburst of interest rate volatility that has plagued recurrently most Western and South-East Asian economies since the late 1970s accelerated the need for and the creation of new speculative and hedging instruments, such as swaps and derivatives. It has also elicited important developments in investment concepts and techniques. This paper examines the issue of optimal portfolio policy in a multi-period model where investors maximizing expected utility of terminal wealth face, in particular, interest rate risk. More precisely, it offers to contribute to the investment and hedging problem in the rather general case where the value of traded and non-traded assets depends, more or less crucially, on the stochastic behavior of interest rates.1 For instance, all financial institutions, most non-financial firms and individual investors do face this situation. In an economy where interest rates and stock price changes follow fairly general stochastic processes, we thus analyze the portfolio problem of an investor endowed with a non-traded cash bond position. He can trade on stocks, the riskless asset and a futures contract written on the bond so as to maximize the expected utility of his terminal wealth. The drift and diffusion parameters of all involved stochastic processes are driven by an arbitrary number of state variables, so that investors face a stochastic investment opportunity set. The traditional route followed in the literature is to use the stochastic dynamic programming technique, leading to the Hamilton–Jacobi–Bellman equation. In such a context, the investor's optimal portfolio strategy is known to contain a pure, preference free, hedge component, a speculative element and Merton–Breeden hedging terms against the random fluctuations of each and every state variable. However, while the first two components are well identified and easy to interpret and work out, the implementation of the last ones is very problematic as the investor must identify first all the relevant state variables and then estimate their distribution characteristics. We follow a different route and use the martingale approach and the methodology developed by Cox and Huang, 1989 and Cox and Huang, 1991. We show that the optimal strategy can be simplified to include, in addition to the pure hedge and speculative components, only two Merton–Breeden-type hedging elements, even though the number of state variables is arbitrarily large. The first one is associated with interest rate risk and the second one with the risk brought about by the co-movements of the spot interest rate and the market prices of risk. The implementation of the optimal strategy is thus much easier, as it involves estimating the characteristics of the yield curve and the market prices of risk only, rather than those of numerous and a priori unknown state variables. Moreover, the investor's horizon will be shown explicitly to play a crucial role in the optimal strategy design, in sharp contrast with the traditional decomposition. Previous research, pioneered by Merton (1971) and Breeden (1979) in a multi-period context, has revealed that investors in general do not make myopic decisions. A portfolio strategy is said to be myopic when each period decision is made as if it were the last one, using no information regarding future investment opportunities. In a complete information economy where the (fully observable) state variables are stochastic and asset returns are correlated with them, myopia results only from logarithmic (Bernoulli) utility.2 Barring such a utility function, optimal portfolios exhibit Merton–Breeden component(s) that are preference-dependent and are used by non-myopic investors as hedges against the unfavorable changes in their investment opportunity set brought about by the economic state variables. Breeden (1984) showed in a continuous time model that if futures contracts existed that were written on the state variables and were of instantaneous maturity, they would be optimally used by investors to hedge against such unfavorable fluctuations. Poncet and Portait (1993), PP thereafter, solve a similar problem by assuming an expected utility maximizer holding a non-tradable position in a long-term bond along with a portfolio of freely traded other financial assets. PP use a two-state-variable model of the yield curve and derive the optimal investment and hedging strategy in a complete market. They actually model the futures contract written on the non-traded long-term bond as a forward. They show that the strategy involving the forward contract has four components, namely the pure hedging and speculative components and the two Merton–Breeden hedging terms associated with the two state variables. The investor achieves these two hedges by using combinations of the forward and a traded discount bond to create two synthetic assets each of which is perfectly correlated with a state variable. The pure hedge component offsets the non-traded position. PP show that this is a one-to-one hedge, i.e. equal in size to the non-traded position. In this paper, we follow a different route in five respects. First, our framework is slightly more general in that the number of state variables is arbitrarily large instead of comprising only the spot interest rate and the long-term bond price. Our model of the term structure, in particular, thus is very general. Second, we use the martingale approach, applied in particular to the yield curve,3 as opposed to stochastic dynamic programming. To the extent that our financial market is complete, this now well-known avenue is appropriate. Third, the derivatives instruments we use for hedging purposes are futures, not forwards. Thus, unlike PP, we do not miss the important effect that interest rate risk has on the (price and) volatility of the futures contract, hence on the investor's optimal strategy, the derivation of which is one of the main objectives of this paper. Fourth, we specialize the investor's utility function to the constant relative risk aversion (CRRA) class, contrasting the isoelastic case and the logarithmic case, in order to derive quasi-explicit solutions in the general framework, draw useful implications and assess the significance of non-myopic (as opposed to myopic) behavior. Fifth, we provide explicit solutions in a particular case that sheds light on the role played by the randomness of the market prices of risk. Our results differ from PPs in three important respects. First, we show that the minimum-variance offsetting component of the futures strategy is not equal in size to the non-traded position and must be continuously rebalanced throughout the investment period. PP found a fixed minimum-variance hedging ratio equal to one because they used forwards instead of futures contracts. By doing so, they miss the effect that (spot) interest rate risk due to the marking-to-market mechanism has on the volatility of the futures position. Even though under complete markets the investor is able to reach a perfect hedge, the hedge ratio is different from one because the volatility of the futures is not equal to that of its underlying bond. Moreover, this hedge ratio also depends on time through the time dimension inherent in the underlying bond price volatility. Continuous rebalancing of the offsetting component thus is called for. The second, more important, main difference with PP lies in the Merton–Breeden components, when they exist, i.e. here in the isoelastic utility case. As mentioned previously, their solution must and does exhibit two such components associated with the two state variables that affect the investment opportunity set, but would have exhibited as many such terms as there are state variables assumed. We also derive two (different from theirs) such components although the number of state variables we postulate is arbitrarily large. One will be shown to be associated with interest rate risk and the other with the risk generated by the co-variations of the spot interest rate and the various market prices of risk. The last difference also concerns the Merton–Breeden terms but has a more methodological flavor. Since in PP's framework there are no traded assets perfectly correlated with their state variables, they solve the hedging problem by manufacturing, in an exogenous manner, synthetic assets that are perfectly correlated with them. In this respect, their contribution is best viewed as devising a method destined to ease the problem of optimally hedging a constrained portfolio in presence of state variables. In our framework, the synthetic assets involved in the Merton–Breeden components of the optimal futures strategy are found endogenously as part of the solution to the investor's problem. It turns out, interestingly, that one of these assets is a discount bond the maturity date of which coincides with the investor's horizon. We then offer novel interpretations of these Merton–Breeden terms. The remainder of the paper is organized as follows. Section 2 details the general economic framework. Section 3 is devoted to deriving the optimal strategy for both the isoelastic and the Bernoulli investors and to interpreting the results. Section 4 provides completely closed-form solutions for the optimal strategies in a special case, which adds further insights to the general results. Section 5 concludes and discusses the prospects of some possible extensions. One tedious technical derivation relative to Section 4 is left to the appendix.

Using the martingale approach, we have investigated the influence of stochastic interest rates on investor's behavior. The optimal strategy of an investor endowed with an interest rate sensitive non-traded cash position and whose utility function exhibits a constant relative risk aversion has been explicitly derived. It is composed of two or four elements, according to whether the investor's behavior is myopic or not. The speculative and pure hedge components are always present. The latter is shown not to be equal in size to the non-traded position and to be time dependent. In contrast to previous studies in which the number of Merton–Breeden hedging terms is equal to that of the state variables, we have shown that the optimal strategy can be simplified to include only two such elements, however large the number of state variables may be. The first one is associated with interest rate risk and the second one with the risk brought about by the co-variations of the spot interest rate and the various market prices of risk. This greatly facilitates the practical implementation of the optimal portfolio strategy. The two Merton–Breeden components, which vanish in the case of a myopic investor, involve a synthetic asset that is found endogenously to be a bond the maturity of which coincides with the investor's horizon. One possible extension of this work would be to consider more general preferences. An obvious candidate would be a general HARA function, of which the isoelastic and logarithmic functions are special cases. This would make the results more intricate but still tractable under the complete market assumption. Another, important, extension would be to examine the effects of incomplete markets. This would occur if the number of sources of risk (Brownian motions) exceeded N in the framework adopted here. This generalization is long-awaited and rather difficult because, although the methodology pioneered by He and Pearson (1991) and Karatzas et al. (1991) is well suited for pure investment decisions, it must be significantly modified when an hedging problem is added.