Optimal dividends in the dual model
Introduction
The optimal dividends problem goes back to de Finetti (1957). To make the problem tractable, he assumed that the annual gains of a stock company are independent and identically distributed random variables that take on only the values and . How should dividends be paid to the shareholders, if the goal is to maximize the expectation of the discounted dividends before possible ruin of the company?
In Bühlmann (1970), the problem is analyzed in the continuous time model of collective risk theory. In the absence of dividends, the surplus of a company at time is Here, is the initial surplus, and is the constant rate at which the premiums are received. The aggregate claims process is assumed to be a compound Poisson process.
The purpose of this paper is to examine the dual problem, where the surplus or equity of the company (in the absence of dividends) is of the form Here is again the initial surplus, but the constant is now the rate of expenses, assumed to be deterministic and fixed. The process is a compound Poisson process, given by the Poisson parameter and the probability density function , , of the positive gains. In this model, the expected increase of the surplus per unit time is It is assumed to be positive.
Whereas a model of the form (1.1) is appropriate for an insurance company, a model of the form (1.2) seems to be natural for companies that have occasional gains whose amount and frequency can be modelled by the process . For companies such as pharmaceutical or petroleum companies, the jump should be interpreted as the net present value of future income from an invention or discovery. Other examples are commission-based businesses, such as real estate agent offices or brokerage firms that sell mutual funds or insurance products with a front-end load. Postulating that the model might be appropriate for an annuity or pension fund, some authors have derived ruin probability results; see Cramér (1955, Section 5.13), Seal (1969, pp. 116–119), Tákacs (1967, pp. 152–154), and the references cited therein.
Assuming a barrier strategy, we begin by defining a function for the expected value of discounted dividends until ruin. Before displaying general results on the optimal dividend strategy in Section 5, two specific examples of this function are given in Sections 3 Exponential jump distributions, 4 Mixtures of exponential distributions. In Section 6, an alternative approach is introduced and developed for the case where the jump amounts follow a mixture of exponential distributions. With the help of Laplace transforms, Section 7 expands this method for any type of jump distribution. Numerical illustrations are displayed in Section 8. Finally, the method is generalized to every process with independent, stationary, and nonnegative increments.
Section snippets
Barrier strategies
It is assumed that dividends are paid according to a barrier strategy. Such a strategy has a parameter , the level of the barrier. Whenever the surplus exceeds the barrier, the excess is paid out immediately as a dividend. This is illustrated in Fig. 1. Note that dividend amounts are discrete. Remark 2.1 In the case of commission-based gains, this dividend policy is straightforward. When the jump amounts are to be interpreted as the net present value of future income, the excess over the barrier may
Exponential jump distributions
In this and the next section we discuss how can be calculated when has a particular form. In the case where , , the integro-differential equation (2.8) becomes By applying the operator to this equation, we obtain the differential equation From this and condition (2.1) it follows that where and are the solutions of the
Mixtures of exponential distributions
In this section we show how can be calculated when where , , and . The substitution of (4.1) in (2.8) yields By applying the operator to this equation, we obtain a linear homogeneous differential equation (with constant coefficients) of order for the function . Hence, we set where
The optimal dividend barrier
We return to the general case, i.e., we do not make any particular assumption about the form of . From the work of Miyasawa (1962) it follows that the optimal dividend strategy is a barrier strategy. Let denote the optimal value of . For any given value of , is maximized by . This is illustrated in Fig. 2, where . In general, is positive, as can be seen from (2.7). Illustration 5.1 If the jump amount distribution is exponential as in Section 3, there are closed form expressions
Alternative method
The idea is to replace the variable by , the distance between the dividend barrier and the surplus. Let denote the expectation of the discounted dividends until ruin if the barrier strategy with parameter is applied. Thus In particular, and by (2.1).
In this section, we focus our analysis on jump amount distributions of the form (4.1). General jump amount distributions are considered in the next section. Replacing by in (4.4), we
Laplace transforms
In terms of the function , the integro-differential equation (2.8) becomes Originally, is defined for . On the basis of (7.1) the definition can be extended to . Denote the resulting function by the symbol , . Taking Laplace transforms in the integro-differential equation for , we obtain a linear equation for :
Numerical examples
In this section, we apply the method of Section 7 to different types of jump amount distributions. For each, is a rational function, which facilitates the inversion of (7.3). In each case, the mean of is 1 (which can be obtained by an appropriate choice of monetary units) and (which can be obtained by an appropriate choice of the time units). As a consequence, by (1.3).
Table 1 shows the optimal barrier if The variance of this
Processes with nonnegative increments
In this section we assume that the process in (1.2) is a subordinator, i.e., a process with independent, stationary, and nonnegative increments. Such a process is either a compound Poisson process or else a limit of compound Poisson processes; for example, see Dufresne et al. (1991). Prominent examples are the gamma process and the inverse Gaussian process.
We begin by reformulating the results of Section 7 by introducing the jump size frequency function and the function
Acknowledgements
The authors thank the referee for comments. Elias Shiu gratefully acknowledges the support from the Principal Financial Group Foundation.
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Cited by (0)
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Distinguished Visiting Professor of Actuarial Science at The University of Hong Kong.
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Visiting Professor of Actuarial Science at The University of Hong Kong.