On a dividend problem with random funding
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Abstract
We consider a modification of the dividend maximization problem from ruin theory. Based on a classical risk process we maximize the difference of expected cumulated discounted dividends and total expected discounted additional funding (subject to some proportional transaction costs). For modelling dividends we use the common approach whereas for the funding opportunity we use the jump times of another independent Poisson process at which we choose an appropriate funding height. In case of exponentially distributed claims we are able to determine an explicit solution to the problem and derive an optimal strategy whose nature heavily depends on the size of the transaction costs. Furthermore, the optimal strategy identifies unfavourable surplus positions prior to ruin at which refunding is highly recommended.
Keywords
Ruin theory Classical risk model Dividends Stochastic control1 Introduction and some first considerations
1.1 Overview
In this article we deal with an extension of the classical dividend maximization problem for an underlying classical (compound Poisson) surplus process. Our proposed extension considers a random funding opportunity which is modelled by the following procedure. The insurer actively searches for investors who are willing to provide additional funding for the insurance portfolio under consideration. If the search is successful, the insurer can choose the height of the funding, increase the surplus and possibly pay out higher dividends in the future. We model the search procedure by means of an intensity \(\beta \ge 0\), such that the insurer finds funding opportunities at the jumptimes of an additional and independent Poisson process. Naturally, new funding is costly and possibly external investors want to participate in future dividends. That is why we weight this additional capital with a factor \(\phi \ge 1\) which plays the role of a proportional transaction cost. The corresponding value function of our problem is the difference of expected cumulated discounted dividends and weighted expected cumulated discounted funding, both up to the time of ruin. In Remark 2 we will discuss the point of view of an external investor and her participation in some detail.
For the case \(\beta =0\) our approach just matches the classical dividend problem, i.e., no additional funding source can be found. Its treatment goes back to Gerber [6] and is analyzed in terms of optimal stochastic control by Azcue & Muler [2, 3] and Schmidli [11]. The opposite extremal case \(\beta \rightarrow \infty \) resembles the situation of a possible capital injection at any point in time, which is discussed in Sect. 4. This problem is by now well known under the keywords maximal dividends and capital injections and was firstly formulated and solved by Kulenko & Schmidli [9] with the subtle difference that the controlled surplus process is not allowed to get ruined and thus resulting in a different value function.
Certainly, the approach of interventions at the jump times of another process is related to the formulation of ruin theoretic problems under random observations. Such a model comprising dividends is introduced by Albrecher et al. [1] and gained some relevance in actuarial research over the last years. We need to emphasize that our present model is continuously monitored, i.e., dividend decisions can be made at any point in time and also the ruin event is immediately observed.
Another framework where dividend maximization problems for firm value determinations play a crucial role is corporate finance. There the underlying process, typically given by a diffusion process, is interpreted as a cash reservoir of a company and the expected value of cumulated dividends reflects the value of this company. The present question is studied in a similar fashion in this financial diffusion framework by Hugonnier et al. [7] in combination with an optimal stopping problem. As mentioned, the problem studied there is based on a continuous sample paths process and also the transaction cost parameter equals one, which results in a common single barrier type optimal strategy, both for dividends and fundings.
Interestingly, the recent paper by Zhang et al. [13] study a compound Poisson risk model with a particular capital injection procedure, which is very similar to the optimal one derived in our contribution. In contrast to our considerations, the focus is put on the determination of discounted penalty functions and dividend decisions are not part of the setup.
The paper is organized as follows. We start with the mathematical formulation of the model and associated stochastic optimization problem. In a next step we establish some basic properties of the value function and study parameter constellations which lead to degenerate optimal strategies. Having understood the crucial dependence on the magnitude of the transaction costs, we can subsequently determine the optimal strategy and corresponding value function. The key in this step is to prove the existence of a solution of a freeboundary value problem comprising two boundaries. We close the paper by some numerical illustrations which focus on the optimal strategy as a function of the transaction cost parameter \(\phi \).
1.2 Model setup
In the subsequent lines we introduce the model of interest and the underlying stochastic protagonists. First of all we set up the stochastic basis of our considered model. We suppose a given probability space \((\varOmega ,\,{\mathbf {F}},\,P)\) which carries the following underlying stochastic processes.
Remark 1
Due to the independence assumptions we have that \(Pa.s.\) the two Poisson processes N and B do not jump at the same time. Since the paths of the dividend process L are leftcontinuous one needs to read \(\varDelta L_t=L_{t+}L_t\).
1.3 Optimization problem and value function
Remark 2
Lemma 1
Let g be a continuously differentiable solution to the HJBequation (3), then g is strictly monotone increasing (\(g'\ge 1\)) and bounded from below by \(\frac{c}{\lambda +\delta }>0\).
Proof
From the equation we directly obtain that \(g'\ge 1\) and if we consider the limit \(x\searrow 0\) we get \(g(0+) \ge \frac{c}{\delta +\lambda }>0\). Since g is monotone increasing and continuous the assertion follows. \(\square \)
Furthermore, we can bound the value function from below similarly as done by Azcue & Muller [2] or by Schmidli [11, p. 80 Lemma 2.37].
Lemma 2
Proof
For the special choice \((L,f)\equiv (L,0)\) we face an admissible dividend strategy for the original dividend maximization problem. The bound follows from the above cited (by now classical) results. \(\square \)
2 Solution of the optimization problem
In the following we assume that the claim size distribution coincides with an exponential distribution with parameter \(\alpha \). We try to identify an optimal strategy and determine an explicit solution to the problem. In case of an arbitrary claim size distribution one can expect a strategy of band type to be optimal. One needs to mention that the presence of the financing control complicates the situation in comparison to other modifications of the dividend problem with exponentially distributed claims in the literature.
We start with identifying parameter sets which lead to somehow degenerate optimal strategies.
2.1 Optimality of keeping the reserve at zero
For a special parameter configuration we obtain that the optimal strategy is to payout the initial reserve immediately and keep on paying dividends such that the current reserve remains zero, which means that the dividend rate is c and the first claim causes ruin. Compare to classical results as presented in [11, p. 93].
Lemma 3
Proof
From the latter result we see that we need to focus on \((\delta +\lambda )^2 < c \alpha \lambda \), which in turn implies that \(c\alpha >\delta +\lambda \), since we assume that all parameters are positive.
2.2 An embedded problem
At the outset of tackling the problem we try as first conjectures some common types of controls such as barrier and simple band strategies. It turned out that they can not be optimal in general. Therefore, in order to get an idea of the shape of the optimal strategy we exploit a numerical approach.
 1.Compute \(V_0\) without additional capital (\(f=0\)), by solvingfor \(0\le x \le b\) and \(g'(x)=1\) for \(x> b\). If the optimal b is not known one can do this for different values of b. Choosing the maximizing b, we obtain an approximation to \(V_0\) with optimal barrier, say \(b_0^*\). Then we can compute the optimal state dependent \(f_0(x)\) by setting$$\begin{aligned} 0=c g'(x)(\delta +\lambda )g(x)+\lambda \int _0^x g(xy)dF_Y (y), \end{aligned}$$$$\begin{aligned} f_0=\underset{f\ge 0}{\text{ argmax }}\{V_0(x+f)V_0(x)\phi f\}. \end{aligned}$$
 2.Compute \(V_1\), where we allow for one financial injection, exactly \(f_0\), at the first jump time of \((B_t)_{t\ge 0}\) and solve for different values of bfor \(0\le x \le b\) and \(g'(x)=1\) for \(x> b\). The usage of the maximizing b results in an approximation of \(V_1\). As next step replace \(V_0\) by \(V_1\) in the first step and go on with computing \(V_2\), where \(f_1\) is obtained according to step 1. Consequently, at the first jump time of \(B_t\) we use \(f_1\), then we switch to the problem with one funding for which the value is given by \(V_1\).$$\begin{aligned} 0=c g'(x)(\delta +\lambda ) g(x) + \lambda \int _0^x g(xy) dF_Y (y) +\beta [V_0(x+f_0)V_0(x)\phi f_0], \end{aligned}$$
This approach reveals a new type of possibly optimal strategy to us, which turns out to be the right conjecture.
2.3 Resulting new strategy
 the dividend strategy is of barrier type at level b,$$\begin{aligned} \varDelta L_{0+}=&(xb)I_{\{x>b\}}\\ dL_t=&c\,I_{\{X_{t}^{L,f}=b\}}dt,\quad t>0, \end{aligned}$$

the financing strategy only applies at reserve levels \(x\in [0,a)\). It is given by \(f(x)=(ax)I_{\{0\le x<a\}}\), with the feature that only below level a we search for a funding source. If one appears, we choose the funding height to such an extent that the surplus jumps up to a and in general not to the barrier level b.
Finally, we have to prove the existence of thresholds \(a^*\) and \(b^*\) such that the smooth fit conditions are fulfilled. In our treatment we are able to derive an interesting condition, which turns out to be equivalent to one of the conditions above.
Remark 3
2.3.1 Extremal behaviour of the optimal strategy
Remark 4
Proposition 1
Let \((\delta +\lambda )^2 < c \alpha \lambda \), such that the optimal barrier in the classical dividend problem \({\tilde{b}}\) is positive. For the optimal levels \(0\le a^* \le b^*\) in the dividend problem with random funding we obtain that \(a^* = 0\) and \(b^*={\tilde{b}}\) if and only if \(\phi \ge {\tilde{V}}'(0;{\tilde{b}})\).
This means that for \(\phi \ge {\tilde{V}}'(0;{\tilde{b}})\) and \((\delta +\lambda )^2 < c\alpha \lambda \) the simple barrier strategy is optimal, and the solution of the classical dividend problem coincides with the solution of the extended problem.
Proof
We assume that \(\phi \ge {\tilde{V}}'(0;{\tilde{b}})\) and have to show that \(a^* = 0\) and \(b^*={\tilde{b}}\) are the optimal thresholds so that \( V(x;0,{\tilde{b}})={\tilde{V}}(x;{\tilde{b}})\) solves the HJBequation, is concave and \({\mathcal {C}}^2\)—the ingredients we later need in the verification theorem.
The above proposition gives us the optimal strategy in the case \(\phi \ge {\tilde{V}}'(0,{\tilde{b}})\). Furthermore, only in that case the usual dividend barrier strategy is optimal. In the next step we consider the lowest bound for the parameter \(\phi \) where a nontrivial strategy appears, namely \(\phi = 1\).
Lemma 4
Let \((\delta +\lambda )^2 < c \alpha \lambda \). For the optimal levels \(a^* \le b^*\) for the dividend problem with random funding we obtain that \(a^* = b^*\) if \(\phi =1\).
In this case we are in the following situation: if an investor occurs we generate external funding to such an extent that we arrive with the surplus process at the dividend barrier, which triggers dividend payments. Hence, there is no gap between the dividend barrier and the funding level. In contrast to the setting by Kulenko & Schmidli [9] where for \(\phi =1\) it is optimal on the one hand to consume the initial capital and all premiums as dividends and on the other hand injections can be chosen to compensate all claims, we get the extremal solution \(a^*=b^*>0\). This is due to the presence of the ruin event in our setting, whereas in [9] injections need to be used to “repair” possible ruin events. The feature \(a^*=b^*\) for \(\phi =1\) shows that “cheap” funding is exploited in full extent to speed up dividend payments.
Proof
Solving the above equations for \(a=b\), we get the function \(V_l(x;a,a)\). It remains to prove the existence of \(a^*=b^*\), resulting in \(V_l(x;a^*,b^*)\), such that the assumptions of the verification theorem are fulfilled. At this point we know that \(V'_l(a;a,a)=1\) and we have to find a such that the smooth fit condition is fulfilled: \(M(a):= V''_l(a;a,a){\mathop {=}\limits ^{!}}0\). Evaluating this function at \(a=0\) yields \(M(0)=\frac{(\delta +\lambda )^2  c\alpha \lambda }{c(\delta +\lambda )},\) which is negative according to our assumptions. Otherwise, we would have a value function of the form \(V(x)=x+\frac{c}{\delta +\lambda }\) as treated in the corresponding lemma above. Furthermore, if \(M(0)=0\) we obtain that \(a^*=b^*=0\). This is also in line with the just mentioned case of a linear value function.
2.3.2 The case of moderate \(\phi \)

\(\phi \ge {\tilde{V}}'(0;{\tilde{b}}) \Rightarrow a^*=0\) and \(b^*={\tilde{b}}\),

\(\phi =1 \Rightarrow a^*=b^*\).
Theorem 1
For \(1<\phi <{\tilde{V}}'(0;{\tilde{b}})\) and \((\delta +\lambda )^2 < c \alpha \lambda \), we have that there exist \(0<a^*<b^*\) such that the smooth fit conditions (12) and (13) are satisfied. The resulting function \(V(x;a^*,b^*)\) from (4) is a twice differentiable and concave solution to the HJBequation (3).
Proof
As a next step we have to make sure that indeed our constructed function solves the HJBequation and is concave.
If \(A_2(a^*,b^*)>0\), then \(V'''_l(x;a^*,b^*)>0\) for all \(0\le x \le a^*\), which implies that \(V''_l(x;a^*,b^*)<0\) for all \(0\le x \le a^*\), since \(V''_l(a^*;a^*,b^*)=V''_u(a^*;a^*,b^*)<0\). Now the concavity together with \(V'_l(a^*;a^*,b^*)=\phi >1\) yields that \(V'_l(x;a^*,b^*)\ge \phi >1\) for all \(0\le x \le a^*\). Additionally, we can deduce that \(V(x;a^*,b^*)>\frac{\phi c}{\delta +\lambda }>0\). This can be shown as already done in Lemma 1, just by using the equation for \(V_l(x;a^*,b^*)\) in \(x=0\) and exploiting that \(\beta (V_l(a^*;a^*,b^*)V_l(0;a^*,b^*)\phi a^*)>0\), due to concavity and \(V'_l(a^*;a^*,b^*)=\phi \).
3 Verification Theorem
Here we state a verification theorem which fits to our constructed function \(V(x;a^*,b^*)\) in (11), (16) and Proposition 1.
Theorem 2
Proof
Since for all parameter constellations our constructed functions are linked to an admissible strategy, are twice differentiable and concave, we have that they dominate the value function. Furthermore, using the band type strategy specified by \((a^*,b^*)\) we have that the corresponding Y from (18) is a martingale. Instead of using dominated convergence in the limitation procedure, one can even use bounded convergence, since \(f^*_s\le a^*\) and observe that \(\lim _{t\rightarrow \infty }\mathbb {E}_x\left[ e^{\delta (t\wedge \tau )}V({\bar{X}}_t^\tau ;a^*,b^*)\right] =0\).
Corollary 1
If \((\delta +\lambda )^2 < c \alpha \lambda \) and \(\phi \ge 1\), the function \(V(x;a^*,b^*)\) is the value function and the corresponding band type strategy is optimal.
4 Extremal case: \(\beta \rightarrow \infty \)
5 Numerical illustration
In this section we present a numerical example which nicely illustrates the dependence of the optimal strategy on the parameter \(\phi \ge 1\). For this purpose we have chosen the parameters as follows. Concerning the reserve process we take \(c = 1.5\) for the premium rate, \(\lambda = 1\) for the intensity of the Poisson process associated with the claim occurrence and \(\alpha = 1.5\) as the parameter of the exponential distribution of the claim size. Furthermore, for the jump process B we take \(\beta = 2\), which corresponds to the expected arrivals of investors per time unit. In terms of the interest rate we choose \(\delta = 0.02\) and in order to illustrate the value function and the smooth fit conditions we fix \(\phi = 1.5\) temporarily.
6 Conclusion and practical aspects
In our contribution we introduce additional funding opportunities at random times into the classical risk model with dividends. In addition to solving the associated optimization problem, we present arguments which link the traditional injection procedure by initial investors to the situation of funding by external investors. Since such external funding problems are heavily discussed in corporate finance, this establishes a further link between company valuations and the optimal dividend problem in insurance. Clearly, the proposed modification leads to an increase of the value of an insurance portfolio (measured by future dividends) over the classical dividend situation. This fact is illustrated in Fig. 3. Hence, the firm value can be raised through the acceptance of capital provided by outside investors. In the case of disproportionate large transaction costs the externally appropriated funding is too costly and therefore the management will not take advantage of this possibility. Thus, the optimal strategy heavily depends on the transaction cost parameter \(\phi \) as exemplified in Fig. 4.
The presence of the funding level \(a^* \ge 0\) stands in contrast to the optimal strategy obtained by Kulenko and Schmidli [9], where the reflection needs to take place at the zero level to prevent ruin. Since in our model the insurance company is still exposed to the risk of possible ruin, the funding level \(a^*\) serves as the lower bound of a  one may call it  stable zone\([a^*,b^*)\), where the optimal strategy tries to keep the surplus process to stay in this area. The region of surplus values larger than \(b^*\) can be seen as comfort zone or zone of preference where the insurance company can afford to pay dividends despite the existing risk of ruin. On the contrary if the current reserve is below \(a^*\) we are in a hazard zone and there is need for action. Hence, the management of the insurance company makes use of every possible funding in order to arrive again at the stable zone. Of course, since our reserve process is bounded from above by \(b^*\), the ruin event is certain, due to that there exists another region, the ruin zone. In this way the obtained optimal strategy leads to a partition of the state space. In the extremal case \(\beta \rightarrow \infty \) this implies that certain claims, those which do not cause ruin but take the surplus to \([0,a^*)\), are immediately partly refunded such that the surplus is pushed back to the stable zone. The determination of the current zone of an insurance portfolio can be a possible guidance for the evaluation of the status quo to help the actuary to stick to the principle of prudence.
Notes
Acknowledgements
Open access funding provided by Graz University of Technology.
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