تجزیه و تحلیل مجانبی و عددی در استراتژی سرمایه گذاری مطلوب برای یک شرکت بیمه

The asymptotic behaviour of the optimal investment strategy for an insurer is analysed for a number of cash flow processes. The insurer’s portfolio consists of a risky stock and a bond and the cash flow is assumed to be either a normal or a compound Poisson process. For a normally distributed cash flow, the asymptotic limits are found in the cases where the stock is very risky or very safe. For a compound Poisson risk process, a composite asymptotic expansion is found for the optimal investment strategy in the case where stock is very risky and the claim size distribution is of an exponential type. In general for a compound Poisson cash flow, the outer asymptotic limit reduces the integro-differential equation describing the optimal stock allocation to an integral equation, which determines the classical survival probability in ruin theory. The leading order optimal asset allocation is derived from this survival probability through a feedback law. Calculation of the optimal asset allocation leads to a difficult numerical problem because of the boundary layer structure of the solution and the tail properties of the claim size distribution. A second order numerical method is successfully developed to calculate the optimal allocation for light and heavy-tailed claim size distributions.

The net cash flow for an insurance portfolio from collected premiums and claims paid on insurance is called the risk process. Ruin theory has traditionally been the study of the probability of ruin of an insurer determined directly from the risk process without allowing for explicit investment of the insurer’s surplus. Generally, an analytical expression for the ruin probability as a function of the current surplus (the ruin function) is only available for claim size distributions of exponential form. Rolski et al. (1999) develop the theory in considerable detail. As analytical results are not available research has concentrated on the behaviour of the ruin function as the current surplus tends to infinity. Lundberg bounds limit the size of the ruin function as long as the claim size distribution is light tailed. For sub-exponential claim size distributions the asymptotic behaviour of the ruin function as the surplus tends to infinity is proportional to the integrated tail distribution of the claims. We consider the optimal investment strategy for an insurer, which has a portfolio of a stock and a bond, where there is a stochastic cash flow arising from the risk process Rt. This problem was first analysed by Browne (1995) who adopted a normally distributed risk process. He found the optimal investment strategy for two different objectives: to minimise the probability of ruin and to maximise the utility of wealth for an exponential utility function. If there is no bond in the portfolio then for both these objectives it is optimal to invest a constant amount of money, independent of the current surplus, in the risky stock. Hipp and Plum (2000) consider a portfolio consisting of just a risky stock, but use a compound Poisson process to model the risk process Rt. They found that the optimal strategy depends critically on the distribution of the claim sizes. In contrast to the normally distributed cash flow, the optimal strategy is to invest a fraction of the current surplus in the risky stock. Analytical results are available if the claim size distribution is exponential, and even here, an explicit optimal investment strategy can be found only for particular parameter choices. Liu and Yang (2004) extend the Hipp & Plum model to include a bond as well as a stock in the portfolio, and calculated the optimal asset allocation numerically for a number of different claim size distributions. A review of the optimal investment problem, as well as other optimal control problems in insurance, can be found in Hipp (2004). There are many refinements to the basic model of ruin in the literature. Of relevance here is the work by Frolova et al. (2002) who consider the effect a fixed investment in a risky asset has on the ruin probability. They use an exponential claim size distribution and a Brownian motion for the risky asset and found the behaviour of the ruin probability for small and large volatility of the asset. Recent research for the optimal investment problem for an insurer has mirrored the development of ruin theory. Hipp and Schmidli (2003) determine the asymptotic behaviour of the ruin probability as the current surplus tends to infinity in the small claims case. Gaier and Grandits (2002) consider a claim size distribution with exponential moments, while Schmidli (2005) has determined the corresponding result for sub-exponentially distributed claim sizes. Here, we study the asymptotic behaviour of optimal investment decision for an insurer using a small dimensionless parameter of the model rather than as the surplus tends to infinity. We use as this small parameter, the amount of risk in the asset defined by equation(1) View the MathML source Turn MathJax on where μ, σ are the drift and coefficient of volatility of the stock and r0 is the risk free rate of interest. All these quantities are taken as constants. The drift μ and the risk free rate r0 are usually quoted in units of percentage per annum. Thus they have dimension per unit time. If the stock is lognormally distributed, the coefficient of volatility σ is also quoted per annum. This represents the observed standard deviation of the log return of the asset after one year, rather than signifying that this quantity has dimension per unit time. The coefficient of volatility has dimension per square root of unit time in order to be consistent with the dimension of the Brownian motion. Consequently, η is a dimensionless quantity. Notice that with this notation the market price of risk is ησ. Asymptotic methods have already been successfully applied to the optimal insurance pricing problem. Emms et al. (2006) use a perturbation expansion to determine the optimal premium in order to maximise the insurer’s expected total wealth. This work represents an extension of those techniques to ordinary differential equations (ODEs) and integro-differential equations arising from the optimal investment problem. Hinch (1991) and Bender and Orszag (1978) develop the general theory of perturbation expansions in a small parameter to give approximate analytical solutions. In Section 2 we identify the limiting behaviour of the optimal strategy as η→0 for a normally distributed risk process. Using this feature of the model allows us to construct approximate optimal strategies when the parameter is not zero, and so describe the qualitative features of the optimal strategy when analytical results are not available. The cash flow is taken as compound Poisson process in Section 3. We obtain approximate optimal strategies for two light-tailed claim size distributions in Section 4. For these distributions the problem reduces to the solution of an ODE and a system of ODEs. It is easy to integrate these equations numerically so the exact solution can be readily compared with the asymptotic results. For some of the analysis, asymptotic results are only available analytically if there is no bond in the portfolio. We examine this case by setting r0=0. Section 5 develops the theory for general claim size distributions and describes three numerical schemes to solve the optimal asset allocation problem when there is no reduction to ODEs. Conclusions are given in Section 6.

There are few analytical solutions for the optimal investment problem for an insurer. Hipp and Plum (2000) found a solution when the risk process is compound Poisson, the interest rate is zero, and the claim size distribution is exponential. Hipp and Plum (2003) found another solution if the income from interest and claims is constant. We have identified the limiting behaviour of the optimal strategy as the asset becomes increasingly risky when the cash flow is normally distributed and when the risk process is a compound Poisson process. If the claim size distribution is exponential we have found a uniformly valid asymptotic expansion for the optimal stock allocation. The optimal strategies for a normal risk process and a risk process with exponentially distributed claims are comparable. For the classical ruin problem there is an analytical expression for the ruin probability if the claims are distributed exponentially. For more general claims distributions the De Vylder approximation uses moment matching in order to derive an explicit expression for the ruin probability (Rolski et al., 1999). It is tempting to use the asymptotic solution for exponential claims and match moments for more general distributions in order to approximate the optimal asset allocation strategy. Given the qualitative differences in the optimal strategy for light and heavy-tailed distributions we can see that this will be a poor approximation over a large range of current surplus values. It is better to calculate the optimal strategy numerically. An understanding of the asymptotic structure of the allocation strategy allows us to interpret the optimal stock allocation and also formulate a good numerical method for its computation. If the stock is risky then there is a thin boundary layer which differentiates the optimal stock allocation for a normal risk process from a compound Poisson risk process. If the current surplus is very small then the latter model says it is optimal to invest very little in the stock. This boundary layer must be resolved, that is, sufficient grid points must be placed across the region of rapid change. Furthermore, as the current surplus becomes large, the tail distribution function decays rapidly, which leads to an unstable numerical problem. If there are few small claims then an intermediate region is present, which means it is optimal to invest in the stock an amount comparable to the mean claim size. The parameters of the model and the claim size distribution change the optimal amount invested in the stock, but the overall strategy can be characterised according to the form of the claim size distribution. A qualitative summary of the optimal stock allocation is given in Fig. 4. For the parameters we have chosen in the paper, the exponential claim size distribution corresponds to 4(a), Erlang distributed claim sizes lead to 4(b), a Pareto distribution yields 4(c), while a Lognormal claim size distribution gives the optimal strategy in 4(d). We have described three numerical methods: the second order method gives greater accuracy for a given surplus step but requires more computational time because it is an implicit method. The numerical results reveal the sensitivity of the numerical approximation to the initial surplus value and the resolution of the finite difference grid. We confirmed the convergence of the schemes using step size reduction and examination of the residual of the integro-differential equation. The application of these techniques to other optimal control problems in insurance (Hipp, 2004) may yield further insight where analytical solutions are not present.