Optimal capital injections and dividends with tax in a risk model in discrete time

We consider a risk model in discrete time with dividends and capital injections. The goal is to maximise the value of a dividend strategy. We show that the optimal strategy is of barrier type. That is, all capital above a certain threshold is paid as dividend. A second problem adds tax to the dividends but an injection leads to an exemption from tax. We show that the value function fulfils a Bellman equation. As a special case, we consider the case of premia of size one. In this case we show that the optimal strategy is a two barrier strategy. That is, there is a barrier if a next dividend of size one can be paid without tax and a barrier if the next dividend of size one will be taxed. In both models, we illustrate the findings by de Finetti’s example.


Introduction
We consider the surplus process of an insurer. A classical measure for the risk are the ruin probabilities, see for example [2,10]. Even though ruin probabilities have to be seen as a technical measure, they were criticised since neither the time to ruin nor the deficit at ruin is taken into consideration. De Finetti [4], see also [6], proposed a more economical tool: the expected value of discounted dividends (until ruin). We will consider this, as the further measures mentioned, as a stability criterion. Dividends have, like ruin probabilities, the disadvantage that the deficit at ruin is not taken into account. This was the reason that Kulenko and Schmidli [5], motivated by a remark in [3], introduced capital injections. The insurer is not allowed to get ruined and has to make a capital injection whenever the surplus becomes negative. The measure is then the discounted value of dividends minus the penalised capital injections. The discounting rate has to be seen as a preference parameter. That is, dividends today are preferred to dividends tomorrow, and capital injections tomorrow are preferred to capital injections today. Since the capital injections are mandatory, the deficit at ruin matters. In a Markovian environment, the value function turns out to be concave, see [5]. This implies that the optimal strategy is of barrier type. That means, all capital above a barrier is paid as dividends, and no dividend is paid below the barrier.
Since capital injections can only be avoided by the price of no dividend payments, Schmidli [7][8][9] introduced tax payments for dividends. Capital injections lead to an exemption from tax. That is, the amount injected can later be paid as dividends without paying tax. For example in Switzerland a capital injection can be accounted as capital reserve and later be paid back as dividend without taxes. Alternatively, dividends without tax are paid at times when the surplus without dividends is at a maximum. Therefore, the tax can also be interpreted as a tax for the company and so less surplus is present to pay as dividends, see also [1]. In the continuous time models mentioned above, the optimal strategy turns out to be very simple. When an instantaneous dividend can be paid without tax, a barrier strategy with the same barrier as in the case without tax is applied. When taxes have to be paid, also a barrier strategy is applied. Unless both barriers are at zero, the second barrier is (as expected) higher than in the case without tax.
We now work on a probability space (Ω, F, IP) . Let {Y i } be iid modelling the profit of an insurance portfolio in period i. Then the post-injection-pre-dividend surplus at time n is where x is the initial capital, U n−1 the dividend at time n − 1 and L n the capital injection at time n. In order that the problem is well defined and not trivial, we assume The information available at time n is given by the natural filtration of the {Y k } , IF = (F n ) n∈IN . The processes {U n } and {L n } have to be adapted to IF.
For the problems below we only have to choose the dividend strategy {U k } because making an injection before it is necessary is not optimal. Since there is no advantage to pay dividends and make a capital injection at the same time, we only allow dividends such that X n − U n ≥ 0 . The capital injection process {L t } will then be determined as the minimal process that keeps the surplus positive. That is, {L t } is given through where a − ∶= max{−a, 0} and a + ∶= max{a, 0} denote the negative part and positive part of a, respectively. We aim to determine the optimal strategy and the corresponding value function. We will start with the model without tax and then deal with the model with tax. In both cases, we prove Bellman's equation and will show that the optimal strategy is of barrier type. Further, as an example we look at a model considered by de Finetti [4].

The model without tax
The results in this section follow similarly as in the model without injections, see [6, Section 1.2].

The value function
We denote the value of a dividend strategy {U t } by where > 1 is a penalty factor (higher administration costs for injections than for dividends) and ∈ (0, 1) is the preference factor. Choosing = 1 would lead to X n − U n ≡ 0 under the optimal strategy. We are looking for the value function and for the optimal strategy {U * t } , for which In particular, this implies that it cannot be optimal to inject capital before it is necessary since later injections are discounted stronger. We start by proving some useful properties.
Lemma 1 V(x) has the following properties:

Proof
(i) For the upper bound we consider the pseudo-strategy U ′ with U � 0 = x and U � n = Y + n for n ≥ 1 and L � n = 0 . Because X ′ n ≤ 0 , we have W ′ n ≥ W n for any strategy. From (3) we get Changing the injection strategy to L U � n = Y − n we get an admissible strategy. The value is which yields the lower bound.
(ii) For a strategy U with initial capital y we define a strategy U ′ with initial capital x ≥ y through U � 0 = U 0 + x − y and U � n = U n for n ≥ 1 , and L U n = L U � n for all n. Then X n = X � n for all n ≥ 1 . This causes V(x) ≥ V U � (x) = V U (y) + x − y which leads the assertion by taking the supremum. Concavity follows as in [5].

◻
In the following we define V(x) = V(0) + x for x < 0 , and we let Y be a random variable with the same distribution as Y k . The following is proved in [6, Lemma 1.1].

◻
Note that u can only take values in a compact interval. Since V(x) is concave, the function u ↦ IE[V(x − u + Y)] is concave, too, and therefore continuous.
Thus there is an u(x) where the supremum is attained. If more than one value u exists at which the supremum is attained, then by concavity the supremum is also attained at all values between. We therefore can choose u(x) as the maximal value at which the supremum is attained.

The optimal dividend strategy
We can now characterise the optimal strategy.

Theorem 1
The function u(x) has the following properties: , for x large enough a dividend should be paid immediately.
, the optimal strategy is of barrier type.
We postpone the proof to Appendix 2.
A numerical solution can be obtained recursively as described in [ We next show that it suffices to consider integer initial capital only. The proof can be found in Appendix 1.1.

Lemma 2 We can choose
Moreover, under the optimal strategy the process visits integer points only after the first injection or the first dividend.

All surplus as dividend
A possible strategy is to pay all positive capital as dividend and make an injection whenever the surplus is negative. This is optimal if the value function is . We now look for conditions that this is the case. That is, the optimal strategy yields X * n = 0 . We need to show that f(x) solves (5). For x = x − u , this is equivalent tõ

This is equivalent to
The left hand side is convex in x . For x = 0 we obtain equality by the definition of c. The inequality is therefore fulfilled, if it is fulfilled for x = 1 . That is, . We therefore get the condition E.g., the condition is fulfilled if ≤ 1 . Note that, in contrast to [5], the condition depends on the distribution of Y.
This has the solution We want to maximise V(0) = C 1 + C 2 , or equivalently to maximise The first order condition is or equivalently The solution has to be found numerically.

The value function
We now introduce tax on dividends. Let 1 − ∈ (0, 1) denote the tax rate. As in [7][8][9], a capital injection leads to an exemption for tax up to the injected amount. By Z n we denote the amount we could immediately pay as dividend without having to pay tax. That is, letting Z 0 = z be the initial exemption, Z n+1 ∶= (Z n − U n ) + + L n+1 . The value of a dividend strategy U is then defined as It cannot be optimal to inject more capital than needed. We therefore omit the letter L in V U (x) because the injection strategy is determined by the dividend strategy U. The value function is defined as where the supremum is taken over all adapted dividend strategies U. Because of the tax we can allow here the value = 1.
We will need below the value of the strategy if we pay all surplus as dividend. Call this reference value (x, z) . We will calculate (x, z) later.
The value function has the following properties. The proof is found in the Appendix 1.2 A verification theorem is proved in Appendix 3. z) . On the other hand, V(x, z) fulfils (10).

The optimal strategy
As in Sect. 2 it follows that an optimal strategy exist. We denote by u(x, z) the optimal dividend and we choose the largest value for u if the maximiser is not unique. The optimal strategy has the following properties.

Theorem 3
(i) The number x 1 ∶= sup{x ∶ u(x, z) = 0 for some z ≥ 0} is finite; i.e., for x large enough a dividend should be paid.
Proof The first assertion follows similarly as in the proof of Theorem 1. The second assertion follows from the concavity of V(x, z) in the first argument. ◻ The optimal strategy is thus a barrier strategy, where the barrier may depend on Z t .
Similarly as in Lemma 2 it follows that we can restrict to integer values. We thus assume in the following that x, z, u ∈ IN.

Premia of size one
We now make the additional assumption IP[Y ≥ 2] = 0 and p 1 > 0 . That is a risk model in discrete time where the monetary unit is the premium income of a period.
The solution can then be calculated recursively for n ∈ IN from (10).
. We choose the larger of the two values. If p 0 > 0 , then either , whichever is larger.

Preliminaries
Let > 1 be the solution to the equation where f 0 (0) is the value at zero of the model without tax. From we see that the solution exists and is unique. Note that for x ≤ x 0 + 1 , where x 0 was defined in Lemma 2, f 0 (x) is the value of the model without tax.
We prove the result in Appendix 1.3. The result shows that in the model without tax we have to choose the initial value such that the minimal increment is one. In particular, f 0 (x) is strictly increasing with Let We let x 1 be the argument for which the minimum is attained. Note that This implies that x 0 ≤ x 1 . We need to verify the existence of x 1 .

Lemma 5
The constants C and x 1 exist.
For the proof see Appendix 1.4.

The solution to the problem
We can now solve the problem.

Theorem 4 If IP[Y ≥ 2] = 0 and p 1 > 0 then
The constant x 0 is the dividend barrier from the model without tax.
The proof is postponed to Appendix 4.
Theorem 4 implies that the optimal strategy is, as in [7][8][9], a strategy with two barriers. If an immediate dividend of size one is not taxed, the dividend barrier x 0 is used. That is, all capital above x 0 is paid as dividend. If tax has to be paid for an immediate dividend, the barrier x 1 is used. That means, if starting with an initial capital below x 0 , then as long as Z t > 0 the premium is paid as dividend until the next claim occurs. Whenever Z t reaches zero, no dividend is paid until the surplus reaches x 1 . Then the premium is paid as dividend until the next time of a claim.

Corollary 1
The optimal strategy is given as ◻

All surplus as dividend
We next consider the strategy u(x, z) = x , that is, to pay all surplus as dividend.

Lemma 6 The value of the strategy paying all surplus as dividends has the value
Proof The value of the dividend at zero is min{x, z} + (x − z) + . Thus we can assume x = 0 . The second term is the value of the dividends if no tax had to be paid. We therefore need to calculate the value of the dividends for which tax is paid. Note that S n = ∑ n k=1 (U k − L k ) . It is easy to see that T n = (max k≤n S k − z) + . The discounting of the first time point with Z n = 0 is IE[ z ] , where z = inf{n ≥ 0 ∶ S n = z} . We Thus v 0 = ( − 1) −1 . This yields the last term. ◻ We next look for conditions under which it is optimal to pay all surplus as dividend. That is, we have to verify that (10) is fulfilled. For simplicity, we let f (x, z) = min{x, z} + (x − z) + + 1 − 2 −(z−x) + for x ≥ 0 , where 1 and 2 are the constants from Lemma 6, and f (x, z) = f (0, z − x) + x for x < 0 . Then for z = 0, Then g(0) = 0 and This function is increasing in x. Therefore, (10) where we used the definition of . Analogous to the case z = 0 it follows that (10) is fulfilled if g(1, z) ≥ 0 or which is always fulfilled if the condition for z = 0 is fulfilled. Suppose now x ≥ z . Then The condition for g(x, 0) ≥ 0 was obtained above. Further, This shows that g(x + z, z) is increasing in z and therefore (10) is fulfilled if (12) holds.
x 2 with the constants determined in the case without tax.
We look for The first order condition for the minimum is which is indeed a minimum if C 2 > 0 or equivalently, C 1 >C . This is Thus, From the latter equation, C can be determined. If C is known, also x 1 can be determined. Since x 1 ∈ IN , we have to choose ⌊x 1 ⌋ or ⌊x 1 + 1⌋ , and then to calculate C corresponding to x 1 .

Conclusion
We considered a risk model in discrete time where dividends may be paid and capital has to be injected in order to keep the surplus positive. We determined the optimal dividend strategy in order to maximise the expected discounted value of dividends minus penalised injections both for the case without and with tax. In the problem with tax and premia of size one we could determine explicitly the form of the optimal strategy. The value function considered has to be interpreted as a stability measure. The dividend payments are a measure for the profits, the injections a measure for the losses. The penalising factor reflects that large losses cause administrative action and therefore lead to additional costs. Further, the choice of the penalising factor gives the possibility to weight the profits and the losses differently. This technical tool may be used to measure the influence of strategic decisions on the profit and the risk. The dividend barriers give a hint how large fluctuation reserves are reasonable to be kept. Too low fluctuation reserves means one needs to raise money for large losses. Too large reserves means that one loses money because of regulative restrictions for investments. Tax payments occur (C 1 −C) log 1x at times where the process without dividends and injections reaches a new maximum. In practice, a company has then to pay taxes, or the share holders require dividends. Taking this into account, reserves should not increase too much in such a situation.
The difference between the first dividends is also x 0 − x if it is paid at time . After time the dividend and injection processes coincide. Thus, We conclude that IE x 0 [( X >x 0 + X =0 ) ] ≥ 1 . For x 0 < x < ⌊x 0 + 1⌋ , we follow the (maybe not optimal) strategy not paying a dividend before and then to follow the optimal strategy. The we obtain analogously,

Proof of Lemma 3
(i) The lower bound is the value of the strategy if all surplus is paid as dividend and tax has to be paid for all the dividends except for the amount z.
For the upper bound note that The right hand side can be made maximal if all capital is paid as dividend but no injections are made, and no tax has to be paid for all dividends after the first dividend. Concavity follows now similarly as in [5]. (iv) It suffices to show Lipschitz continuity in each of the variables. Let x < y and z ≥ 0 . Choose > 0 . There is a strategy U, such that V(y, z) < V U (y, z) + . Using the strategy U for initial capital x, additional injections may have to be made (or less dividends may be paid if the surplus would become negative otherwise). This shows that Let x ≥ 0 and z 1 < z 2 . Similarly, as for the first argument we obtain V(x, z 2 ) − V(x, z 1 ) ≤ (1 − )(z 2 − z 1 ) , since using the same strategy may lead to additional tax payments. ◻

Proof of Lemma 4
Suppose the assertion does not hold. Let x 2 be a point where We consider now the model without tax. Note that x 2 > x 0 by the results of Section 2. Consider the strategy U n = X n =x 2 +1 and suppose x = X 0 ≤ x 0 + 1 . Then Thus Because {X n } is bounded by x 2 + 1 , we can let n → ∞ obtaining This is the value of the proposed strategy but with all dividends taxed with the rate 1 − . Thus, the value function of the model without tax is strictly larger than f 0 (x) , contradicting the fact that f 0 (x) is the value function for x ≤ x 0 + 1.

Proof of Lemma 5
Let a > x ≥ 0 . We consider the process S n = x + ∑ n k=1 Y k and define the stopping times a = inf{n > 0 ∶ S n = a} and − = inf{n > 0 ∶ S n < 0} . We This implies that the second term converges. Since the net profit condition is fulfilled under IP * , that is IP * [ − = ∞] > 0 , the limit of f 0 (a) −a must be finite. We claim that the limit is nonzero. If the latter was not the case, we had which cannot hold for all x by Lemma 4. Thus, lim x→∞ f 0 (x) −x > 0 . This shows the assertion.  (4) where we used that the upper bound also holds for x < 0 . This implies Thus u(y) ≠ 0 for y larger than the right hand side.

Thus V(x) = IE[V(x + Y)] and V(x) − V(x) = u(x) . Note that
Because u(x) is the largest value at which the supremum is taken, we have u(x) = 0 . Suppose that u(x) > 0 and choose y > x . We have by the concavity or equivalently, V(y) ≤ y −x + V(x) . By Lemma 1, we have equality. This implies that u(y) ≥ y −x . Now, Thus V(x) =x − (y − u(y)) + V(y − u(y)) . Since u(x) = 0 we find u(y) = y −x . This shows that there is x 0 such that u(x) = (x −x 0 ) + . By the definition of x 0 , we get x 0 = x 0 . (iii) follows like the corresponding statement in [6, Theorem 1.10]. (iv) Let ũ(x) be the optimiser if we replace V by f. The considerations above apply also to f implying that ũ(x) exists. As in [6,Cor. 1.3] it follows that f (x) = V(x).
with f 0 (x) = f 0 (0) + x for x < 0 . Because f 0 (x) is the value function for the case without tax for x ≤ x 0 , it is concave. Further f 0 (x 0 + 1) − f 0 (x 0 ) = 1 . Thus f 0 (x + 1) − f 0 (x) is decreasing and the result follows as in the case z = 0 and x > x 1 .
Next consider x 0 + z < x ≤ x 1 + z . Then