Optimizing dividends and capital injections limited by bankruptcy, and practical approximations for the Cramér-Lundberg process

This paper concerns the approximate optimization of a new type of boundary mechanism, which emerged recently in the actuarial literature [APY18, GK17, AGLW20], in the context of the optimal control of dividends and capital injections.

The objective is to maximize the profit:

The value function is

Motivation. The recent papers [GK17, AGLW20] investigated the above control problem of optimizing dividends and capital injections for processes with jumps, when bankruptcy is allowed as well. The second paper works under the Cramér-Lundberg model with exponential jumps, while the first works under the spectrally negative Lévy model, allowing also for the presence of Brownian motion and infinite activity jumps. It turns out that the optimal policy belongs to the class of $(-a,0,b),a>0,b\geq 0$ “bounded buffer policies", which consist in allowing only capital injections smaller than a given $a$ and declaring bankruptcy at the first time when the size of the overshoot below 0 exceeds $a,$ and in paying dividends when the reserve reaches an upper barrier $b$. These will briefly be described as $(-a,0,b)$ policies from now on. Furthermore, the optimal $(a^{*},b^{*})$ are the roots of one variable equations with explicit solutions related to the Lambert-W(right) function (ProductLog in Mathematica).

Below, our goal is to show numerically that exponential approximations provide quite reasonable results (as the de Vylder approximation provides for the ruin problem). We will focus in our examples on the case of matrix exponential jumps(which are known to be dense in the class of general nonnegative jumps, with even error bounds for completely monotone jumps being available [VAVZ14]), for two reasons. One is in order to highlight certain exact equations which are similar to their exponential versions, and which may at their turn be used to produce even more accurate approximations in the future, and, secondly, since numerical Laplace inversion for this class may easily tuned to have arbitrarily small errors.

History of the problem: The case of no capital injections (also characterized by $k=\infty$ or absorption below 0) is the dividend problem posed by De Finetti [DF57, Ger69] where dividends are paid above barrier $b^{*}$ and $a^{*}=0$ is imposed. “The challenge is to find the right compromise between paying early in view of the discounting or paying late in order not to reach ruin too early and thus profit from the positive safety loading for a longer time" [AAM20].

Forced injections and no bankruptcy at $0$ (also characterized by a reflection at $0$) is studied in Shreve [SLG84] where dividends are paid above barrier $b^{*}$ and $a^{*}=\infty$ is imposed.

From Lokka, Zervos, [LZ08] we know that in the Brownian motion case, it is optimal to either always inject, if $k\leq k_{c}$, for some critical cost $k_{c}$ (i.e. use Shreve), or, stop at $0$ (use De Finetti). We propose to call this the Lokka, Zervos alternative. The “proof" of this alternative starts by largely assuming it via a heuristically justified border Ansatz [LZ08, (5.2)]: $\max\left\{-V(0),\ V^{\prime}(0)-k\right\}=0\Longrightarrow\text{ either }V(0)=0\text{ or }V^{\prime}(0)=k.$

Extensive literature on SLG forced bailouts (no bankruptcy) can be found at Avram et al., (2007) [APP07], Kulenko and Schmidli, (2008) [KS08], Eisenberg and Schmidli, (2011) [ES11], Pérez et al., (2018) [PYB18], Lindensjo, Lindskog (2019) [LL19], Noba et al., 2020 [NPY20].

Articles [GK17, AGLW20] are the only papers which relate declaring bankruptcy to the size of jumps, with general and exponential jumps, respectively. [GK17] deals also with the presence of Brownian motion and infinite activity jumps, by conditioning at the first draw-down time; the optimality proof is quite involved.

In [AGLW20], it is also shown that neither $V(0)=0$ nor $V^{\prime}(0)=k$ are possible: the Lokka-Zervos alternative disappears, but another interesting alternative holds. Above a certain critical $k_{c}$ the optimal dividends  barrier switches from strictly positive to $0$, and $k_{c}$ is related in (26) to the Lambert-W function .

The results of [GK17, AGLW20] may be divided in three parts:

1. 1.

   Compute the value of bounded buffer policies. The key result is (5) below, an explicit determination of the objective $J_{0}=J_{0}^{a,b}$, which allows optimizing it numerically.

   REMARK 1.

   Computing the value function is considerably simplified by the use of first passage recipes available for spectrally negative Lévy processes [AKP04, Kyp14, KKR13, AGVA19], which are built around two ingredients: the $W_{q}$ and $Z_{q}$ $q$-scale functions, defined respectively for $x\geq 0,q\geq 0$ as:

   1. (a)

      the inverse Laplace transform of $\frac{1}{\kappa(s)-q}$, where $\kappa(s)$ is the Laplace exponent (which characterizes a Lévy process) and

   2. (b)

      $Z_{q}(x)=1+q\int_{0}^{x}W_{q}(y)dy$

   – see the papers [Sup76, Ber98, AKP04] for the first appearance of these functions. The name $q$-scale/harmonic functions is justified by the fact that these functions are harmonic for the process $X$ killed upon entering $(-\infty,0)$, in the sense that

   $$\{e^{-q\min[t,T_{0}]}\;W_{q}(X_{\min[t,T_{0}]}),e^{-q\min[t,T_{0}]}\;Z_{q}(X_{\min[t,T_{0}]})\},t\geq 0$$

   are martingales, as shown in [Pis04, Prop. 3] (in the case of $Z_{q}$, there is also a penalty of $1$ at ruin, generalizing to other penalties produces the so-called Gerber-Shiu function).

2. 2.

   Equations determining candidates for the optimal $a^{*},b^{*}$ are obtained by differentiating the objective (which is expressed in terms of the scale functions $W_{q},Z_{q}$), and the optimal pair $(a^{*},b^{*})$ is identified. As a result, the critical $k_{c}$ is related in (26) to the Lambert-W function.

3. 3.

   The optimality of the $(-a^{*},0,b^{*})$ policy is established.

Note that the last step is quite non-trivial and is achieved by different methods in [GK17] and [AGLW20]. The latter paper starts by formulating a (new) HJB equation associated to this stochastic control problem – see (8).

Exponential approximations may also be used, which are similar in spirit with the de Vylder-type approximations. Recall that the philosophy of the de Vylder approximation  is to approximate a Cramèr-Lundberg process by a simpler process with exponential jumps, with cleverly chosen exponential rate $\mu$, and the parameters $\lambda,c$ may also be modified, if one desires to make the approximations exact at $x=0$ –see for example [AHPS19] for more details)

The efficiency of the de Vylder approximation for approximating ruin probabilities is well documented [DV78]. The natural question of whether this type of techniques may work for other objectives, like for example  for optimizing dividends and/or reinsurance  was already discussed in [Høj02, DD05, BDW07, GSS08, AHPS19]. In this paper, following on previous works [ACFH11, AP14, ABH18], we draw first the attention to the fact that we have not one, but three de Vylder-type approximations for $W_{q}(x)$ (as for the ruin probability). The best approximation in our experiments when the loading coefficient $\theta$ is large turn out to be the classic de Vylder approximation. However, for approximating near the origin, the two point Padé approximation which fixes both the values $W_{q}(0)=\frac{1}{c},W_{q}^{\prime}(0)=\frac{q+\lambda}{c^{2}}$ works better. The end result here is simply replacing the inverse rate $\mu^{-1}$ by $m_{1},$ in the formula for the scale function of the Cramèr-Lundberg process  with exponential jumps. In between $x=0$ and $x\to\infty$, the winner is sometimes the “Renyi approximation" which replaces the inverse rate by $\frac{m_{2}}{2m_{1}}$, and modifies $\lambda$ as well (for the de Vylder approximation, $\mu^{-1}$ is replaced by $\frac{m_{3}}{3m_{2}}$, and both $\lambda,c$ are modified).

We end this introduction by highlighting in figure 1 the fact that for exponential jumps, the limited capital injections objective function $J_{0}$ given by (14) for arbitrary $b$ but optimal $a=s(b)$ (via a complicated formula) improves the value function with respect to de Finetti and Shreve, Lehoczky and Gaver, for any $b$.

Contents and contributions. Section 2 offers a conjectured profit formula for $(-a,0,b)$ policies, where we include also a final penalty $P$. The theoretical result of the section 1 revisits [GK17, AGLW20] by linking the two formulations together and emphasizing the impact of the bankruptcy penalty $P$ (via the scale function $G$). Its proof is beyond the scope of the present (already lengthy enough) paper and it can be inferred from either one of [GK17] and [AGLW20] through a three step argument:

1. 1.

   express the cost by conditioning on the reserve ($J_{x}$) starting from $0\leq x\leq b$ hitting either 0 or b;

2. 2.

   get a further relationship on costs $J_{b}$ and $J_{0}$ by conditioning on the first claim;

3. 3.

   finally, mix these conditions together in order to obtain the explicit formula for $J_{x}$.

We also wish to point out (in section 2.1) the link to an appropriate HJB variational inequality equation8 and the definition 1 specifying the action regions and their computation starting from the regimes in the HJB system. To the best of our knowledge, this is new and the preliminary studies conducted on more complicated problems (involving reinsurance and reserve-dependent premium) seem to reinforce the relevance of this tool.

In section 7 we provide an alternative matrix exponential form of the exact cost, in the case of matrix exponential jumps.

An explicit determination of $a^{*},b^{*}$ and an equity cost dichotomy when dealing with exponential jumps are given in section 3, taking also advantage of properties of the Lambert-W function, which were not exploited before. The two main novelties of the section are:

• •

  emphasizing the computations of the optimal buffer/barrier (from [AGLW20]) in relation with the scale-like quantities appearing in [GK17];

• •

  making explicit use of the (computation-ready) Lambert-W function to describe the dependency of optimal $a^{*}b$ (in equation 14) and of the dichotomy-triggering cost $k_{c}$ in equation 26.

Again, a further novelty is the presence of the bankruptcy cost $P$.

Section 4 reviews, for completeness, the de Vylder approximation-type approximations. Section 4.1 recalls, for warm-up, some of the oldest exponential approximations for ruin probabilities. Section 4.2 recalls in Proposition 5, following [AP14, AHPS19] three approximations of the scale function $W_{q}(x)$ ⁴⁴4essentially, this is the “dividend function with fixed barrier”, which had been also extensively studied in previous literature before the introduction of $W_{q}(x)$ , obtained by approximating its Laplace transform. These amount finally to replacing our process by one with exponential jumps  and cleverly crafted parameters based on the first three moments of the claims.

In section 5, we consider particular examples and obtain very good approximations for two fundamental objects of interest: the growth exponent $\Phi_{q}$ of the scale function $W_{q}(x)$, and the (last) global minimum of $W_{q}^{\prime}(x)$, which is fundamental in the de Finetti barrier problem. Proceeding afterwards to the problem of dividends and limited capital injections, concepts in section 3 are used to compute a straightforward exponential approximation based on an exponential approximation of the claim density, and a new “correct ingredients approximation" which consists of plugging into the objective function (10) for exponential claims the exact "non-exponential ingredients" (scale functions and, survival and mean functions) of the non-exponential densities. Both methods are observed to yield reasonable values in approximating the objective.

This leads us to our conclusion that from a practical point of view, exponential approximations are typically sufficient in the problems discussed in this paper.

In this section, we allow $\bar{\xi}_{t}$ to be a spectrally positive Lévy process, with a Lévy measure admitting a density $\nu(dy)=\nu^{\prime}(y)dy$. The simplest example is that of the perturbed Cramèr-Lundberg risk model with

where $N_{t}$ is a Poisson process of intensity $\lambda>0$, $\left(\xi_{i}\right)_{i\geq 1}$ is an independent family of i.i.d.r.v. with density $f(y)$, and $B_{t}$ is an independent Brownian motion.

We revisit here the problem of optimizing the value of "bounded buffer $(-a,0,b)$ policies", following [GK17, AGLW20] (in order to relate the results, one needs to replace $\gamma$ in the objective of [GK17] by $1/k$), while taking into account also the bankruptcy penalty $P$.

An important role in the results will be played by the expected scale after a jump

where $\overline{\nu}(y)=\int_{y}^{\infty}\nu(u)du$ is the tail of the Lévy measure and $\sigma$ is the Brownian volatility (the identity above follows easily from the $q$-harmonicity of $Z_{q}$, after an integration by parts of the convolution term and a division by $q$).

The problem of limited reflection requires introducing a new "scale function $S_{a}(x)$ and Gerber-Shiu function $G_{a,\sigma}(x)$"– see Remark 4 for further comments on this terminology:

where

We state now a generalization  of [GK17, Thm. 4] for the value function $J_{0}^{a,b}$ of $(-a,0,b)$ policies, in terms of $S_{a}(x),G_{a}(x)$. In the Cramèr-Lundberg case illustrated below, the proof is straightforward, following [AGLW20]. In the other case, one needs to adapt the proof of [GK17].

Proof: Recalling the expressions of $J_{0}^{a,b}$, $G_{a}(x)$, in (6), in (2), and from [GK17, Lem. A.4]

where $C_{a^{\prime}}(x),M_{a^{\prime}}(x)$ denote derivatives with respect to the subscript $a$, Whenever $b>0$, if $a$ achieves the maximum in $J^{a,b}_{0}$, it is straightforward (think of the economic interpretation) that $a$ achieves the maximum of $a\mapsto J^{a,b}_{x}$ for every $x\in[0,b]$. Therefore, we find

∎

In this section we turn to the exponential case, where explicit formulas for the optimizers $a^{*},b^{*}$ are available. In particular, we will take advantage of properties of the Lambert-W function, which were not exploited in [AGLW20]. Subsequently, in sections 5, 6 we will show that exponential approximations work typically excellently in the general case. Although these results have already been established in [AGLW20], the present formulations have two achievements:

1. 1.

   allow an unified formulation of [AGLW20] and [GK17] (via the previously introduced scale functions);

2. 2.

   make use of a numerical tool (Lambert-W function) to express the optimal quantities of interest $a^{*},b^{*}$.