Flow of Dividends under a Constant Force of Interest

This study addresses the issue of maximization of dividends of an insurer whose portfolio is exposed to insurance risk. The insurance risk arise s from the classical surplus process commonly known as the Cram ér-Lundberg model in the insurance literature. To en hance his financial base, the insurer invests in a risk free asset whose price dy namics are governed by a constant force of interest . We derive a linear Volterra integral equation of th e second kind and apply an order four Block-byblock method of Paulsen et al. in conjunction with the Simpson rule to solve the Volterra integral equations for each chosen barrier thus generating c orresponding dividend value functions. We have obtained the optimal barrier that maximizes the div idends. In the absence of the financial world, the analytical solution has been used to assess the acc uracy of our results.


INTRODUCTION
The model we consider in this study derives its name from the path breaking work of Lundberg [2] and Cramér [3][4][5] . In this model, at time t, the surplus t Y of an insurance company evolves according to: where 0 0 ≥ = Y y is the initial reserve, p > 0 is the is a Poisson process with intensity λ , modelling the number of claims (jumps) in (0, t] and { } N i i S ∈ is an independent and identically distributed sequence of positive random variables (with distribution F) independent of N, modelling the jump sizes. The distribution F has finite expectation µ and finite variance 2 σ .
The surplus process (1) with the risk averse condition that ( ) Hence the need to modify the surplus process to take care of this. One way is to predetermine a surplus level b in such a way that when the surplus level hits b, premium income is paid to shareholders as dividends. If the surplus is b and a claim occurs, the claim is paid and no dividends are paid until the surplus hits b again. Eventually and with certainty, ruin will occur at some stage in the future. When ruin occurs, the process stops. The question at hand is 'what b ensures that the insurer pays maximum dividends to shareholders before ruin/bankruptcy?' We concentrate on barrier strategies in this study although they may not be optimal when compounded by a diffusion in a vibrant financial market in which case other strategies e.g. b and strategies take an upper hand. Under the barrier strategy, we study the expected discounted dividend payments prior to ruin with δ as the discount factor when the model (1) is compounded by a constant force of interest. According to Miller Modigliani theory, this approach can be used as a valuation tool for companies, since the value of a company is exactly expected present value of future dividends [6] . Their theory was developed under the assumption of perfect capital markets, rational behavior and perfect certainty.
De Finetti [7] underscored the importance of dividend payments in the economic considerations and management of insurance companies. His work culminated into a barrier strategy as being the optimal strategy. Dividend barrier models have a long history of risk theory [8][9][10] . The later work on optimal dividends was extensively discussed by Borch [11,12] . Gerber [13,14] studied dividend payouts in (1) with the safety of the company in mind. Most literature is available in Gerber [15] and the references therein. In many later papers, the problem is formulated and solved in the framework of the Brownian motion model, see e.g. Jeanblanc-Picqué and Shiryaev [16] , Radner and Shepp [17] , Paulsen and Gjessing [18] , Højgaard and Taksar [19] , Taksar and Zhou [20] , Asmussen and Taksar [21] , Asmussen et al. [22] , Taksar [23] , Chouilli et al. [24] and Paulsen [25] . None of the papers in the literature take advantage of the vast knowledge of Volterra integral equations as we do with this work. More to integral equations, we use numerical methods in this study. Before moving a step, we define a dividend and a barrier strategy.

Definition 1:
Dividends are taxable payments declared by the insurer's board of directors and given to shareholders out of the company's current or retained earnings.

Definition 2:
A barrier strategy b pays out any surplus above b as dividends immediately so that the surplus is brought back to b. Below b, nothing is paid out.

EXACT SOLUTION TO (1)
To give a mathematical formulation of the models in this study, we take it that all processes and random variables are defined on the stochastic basis Using Itô's formula, the infinitesimal generator for Y is given by the following integral-differential operator: Ag y pg y g y x g y dF x .
Now introduce a dividend barrier strategy b so that in the infinitesimal time interval of length dt, the insurer pays the non-negative amount b t dD in dividends. With such a strategy, (1) becomes: denote the expected present value of dividend payments prior to ruin, that is, our dividend value function. Here, 0 > δ is the discount factor and time of ruin, In the said time interval [t, t + dt], a claim occurs with probability dt P λ . Hence, there will be no claim with probability dt λ − 1 . By law of total probability: and simplifying gives: Specifically, let the jumps be ( ) Exp α , so that the expectation 1 µ = α . Then: Put x for y -s into (6), partially differentiate and eliminate integral term to get: The solution to the specific case (7) which is also a solution for the general case (5), up to the barrier, is of the form [25] : where, in the case of (7),  (7). Above the barrier, the solution becomes where A is given by (2). We observe that (5) takes this form.
Remark 1: For a given initial y, let * b denote the optimal value of b, thmaximizese value that maximises the expected present value of dividends. We see that this b is the value that maximizes (8) thus we numerically compute the same dividend value functions using the block-by-block method developed in Paulsen et al. [1] and compare the results. These results and ideas herein will be handy in numerically computing value functions in the Cram´er-Lundberg model compounded by a constant force of interest.

NUMERICAL SOLUTION TO (1)
Here, our risk process is still (1) whereby Y is the same process as the one described earlier 2 and The dividend process b D is nondecreasing and continuous and F (0) = 0. The total expected present value of dividends with discount factor δ and y b Y = 0 is given by (3) where b y T is ruin time, equal to infinity if ruin never occurs. Note that when a.s. We use Theorem 1 below to find solutions to (3). Our strategy is to find the associated Volterra integral equation so that we use the methods in Paulsen et al. [1] .
Theorem 1: Assume model (9) and let A be given by (2). Let together with the conditions: is given by (3).

Remark 2:
In addition to Remark 1, * b has geometric characterization. We have the expected present value of dividends: is continuous and has a continuous first derivative since This geometric characterization of the optimal value is called the high contact condition in the finance literature and equivalently, the smooth pasting condition in the optimal stopping literature. For details, Dixit and Pindyck [26] .
Theorem 1 together with (2) gives the following version of (5): Integrating (10) is a linear Volterra integral equation of the second kind where the kernel is given by and the forcing function, For any arbitrary starting g (0), g (y) will be the ( ) 4 h O [1] numerical solution from the block-by-block method. Then the solution to (11) is given by (8). To find , where h is the grid size.
We have solved (11) for several values of b. At each run using a FORTRAN program that gives ( ) 4 h O solution to Volterra equations of the second kind (confer Paulsen et al. [1] ), we have computed The results indicate that for any two barriers Eventually, some interval [ ] for the first time. This interval contains the optimal * b which gives the optimal value function ( ) implying that the * b lies in the interval [10,12].
To locate the actual * b , we use our numerical solution of ( ) ( ) to calculate all the derivatives: (12) and then locate the minimizing one. Then value is then used to compute the numerical value function which is given by:   Table 2 shows the absolute percentage relative error: Table 2 that the results are extremely good. This then gives us confidence to apply the numerical method to the model later.

MODEL (1) COMPOUNDED BY A CONSTANT FORCE OF INTEREST
The Cram´er-Lundberg model for the reserve t Y of an insurance company at time t with a dividend process D b is given by equation (9). A natural generalization of this model is to allow reserves to earn interest. If there is a positive constant force of interest r, the reserves now become Presence of a positive constant force of interest r implies modifying (2) to a new infinitesimal generator for Y now given by the following integro-differential operator: Ag y ry p g y g y x g y dF x Then Theorem 1 together with (13) gives the integral-differential equation of Volterra type (for b y ≤ ≤ 0 ):  Theorem 2: The integral-differential equation (14) can be represented as the linear Volterra integral equation of the second kind: where the kernel: p ry Replacing z with y and v with x in (15) completes the proof.
We know from Theorem 1 that when . Hence, our value function takes the form: We carry with us the analysis, results and discussion from the previous section onto (16). The values of ( )   Figure 1 is a graph of dividend value functions for some select barriers. The optimal dividend value function is the one corresponding to * b .
When we expect heavy claims, then adequate probability should be assigned to the higher jumps by the distribution function F. Some of the distributions in this class are Lognormal, Pareto and Mixture of exponentials. These distributions are said to have heavy tails. We now report the results when the distribution for S (generic for the S i ) is Pareto. The density function is given by: If we set E(S) = 2, then α = 1.5. To locate actual b * , we again calculate all the derivatives using (12minimizing locate the minimising one. The results indicate that i * = 1279 thus giving b * = 12.79. Figure 2 is a graph of dividend value functions for some select barriers. The optimal dividend value function is the one corresponding to b * .

CONCLUSION
We numerically obtained maximal dividend function in the Cram´er-Lundberg model compounded by a constant force of interest using the block-by-block method and a barrier strategy. The results in Table 2 indicate its quality. An extension of this problem is to allow a diffusion in (1) thus: With another diffusion, the return on investment process will no longer be rt t R = but: For a detailed description of such a model, see e.g. Paulsen et al. [1] . However, when there is a diffusion, band strategies might take an upper hand over barrier strategies and the mathematics changes accordingly.