Optimal dividends in the dual model

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Abstract

The optimal dividend problem proposed by de Finetti [de Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2. pp. 433–443] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined or bankrupt. In this paper, it is assumed that the surplus or shareholders’ equity is a Lévy process which is skip-free downwards; such a model might be appropriate for a company that specializes in inventions and discoveries. In this model, the optimal strategy is a barrier strategy. Hence the problem is to determine b, the optimal level of the dividend barrier. A key tool is the method of Laplace transforms. A variety of numerical examples are provided. It is also shown that if the initial surplus is b, the expectation of the discounted dividends until ruin is the present value of a perpetuity with the payment rate being the drift of the surplus process.

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 1 and +1. 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 t is U(t)=u+ctS(t),t0. Here, u0 is the initial surplus, and c is the constant rate at which the premiums are received. The aggregate claims process {S(t)} 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 U(t)=uct+S(t),t0. Here u is again the initial surplus, but the constant c is now the rate of expenses, assumed to be deterministic and fixed. The process {S(t)} is a compound Poisson process, given by the Poisson parameter λ and the probability density function p(y), y>0, of the positive gains. In this model, the expected increase of the surplus per unit time is μ=E[S(1)]c=λ0yp(y)dyc. 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 {S(t)}. 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 {S(t)} 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 b>0, 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 V(u;b) can be calculated when p(y) has a particular form. In the case where p(y)=βeβy, y>0, the integro-differential equation (2.8) becomes cV(u;b)+(λ+δ)V(u;b)λβeβuubV(x;b)eβxdxλβeβ(bu)λV(b;b)eβ(bu)=0. By applying the operator (dduβ) to this equation, we obtain the differential equation cV(u;b)+(λ+δβc)V(u;b)βδV(u;b)=0. From this and condition (2.1) it follows that V(u;b)=κ(eruesu),0ub, where r and s are the solutions of the

Mixtures of exponential distributions

In this section we show how V(u;b) can be calculated when p(y)=i=1nAiβieβiy,y>0, where β1<β2<<βn, Ai>0, and A1++An=1. The substitution of (4.1) in (2.8) yields cV(u;b)+(λ+δ)V(u;b)λi=1nAiβieβiuubV(x;b)eβixdxλi=1nAiβieβi(bu)λV(b;b)i=1nAieβi(bu)=0. By applying the operator(dduβ1)(dduβ2)(dduβn) to this equation, we obtain a linear homogeneous differential equation (with constant coefficients) of order n+1 for the function V(u;b). Hence, we set V(u;b)=k=0nCkerku,0ub, where

The optimal dividend barrier

We return to the general case, i.e., we do not make any particular assumption about the form of p(y). From the work of Miyasawa (1962) it follows that the optimal dividend strategy is a barrier strategy. Let b denote the optimal value of b. For any given value of u, V(u;b) is maximized by b=b. This is illustrated in Fig. 2, where b=0.8816. In general, b 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 u by z=bu, the distance between the dividend barrier and the surplus. Let W(z;b) denote the expectation of the discounted dividends until ruin if the barrier strategy with parameter b is applied. Thus W(z;b)=V(u;b),0zb. In particular, W(0;b)=V(b;b) and W(b;b)=0 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 u by bz in (4.4), we

Laplace transforms

In terms of the function W(z;b), the integro-differential equation (2.8) becomes cW(z;b)+(λ+δ)W(z;b)λ0zW(y;b)p(zy)dyλz[1P(y)]dyλW(0;b)[1P(z)]=0. Originally, W(z;b) is defined for 0zb. On the basis of (7.1) the definition can be extended to 0z. Denote the resulting function by the symbol w(z), z0. Taking Laplace transforms in the integro-differential equation for w(z), we obtain a linear equation for wˆ(ξ): cw(0)cξwˆ(ξ)+(λ+δ)wˆ(ξ)λwˆ(ξ)pˆ(ξ)λξ2[pˆ(ξ)1ξpˆ(0)]+λξw(0)[pˆ(ξ)1]

Numerical examples

In this section, we apply the method of Section 7 to different types of jump amount distributions. For each, pˆ(ξ) is a rational function, which facilitates the inversion of (7.3). In each case, the mean of p(y) is 1 (which can be obtained by an appropriate choice of monetary units) and λ=1 (which can be obtained by an appropriate choice of the time units). As a consequence, μ=1c by (1.3).

Table 1 shows the optimal barrier b if p(y)=(1/3)2e2y+(2/3)0.8e0.8y,y>0. The variance of this

Processes with nonnegative increments

In this section we assume that the process {S(t)} 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 q(x)=λp(x), and the function Q(x)=

Acknowledgements

The authors thank the referee for comments. Elias Shiu gratefully acknowledges the support from the Principal Financial Group Foundation.

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