Optimal reinsurance and investment under common shock dependence between financial and actuarial markets

The insurance business and the overall social economy are closely related and play crucial roles in the financial system. Basically, insurance promotes economic development and social stability via its own deepening and functions of transferring risks, financial intermediary, and loss compensation. Indeed, insurance companies safeguard the financial stability of households and firms by insuring their risks. Moreover, there are growing links between banks and insurance companies, which act as large investors and improve the economic efficiency of the system by spreading individual risks in the financial market. For instance, the policyholder transfers the risk to insurers by paying premiums and insurers employ premiums to make investments.
It is important to realize that, similarly to any other business, insurance companies require protection against risk. Reinsurance helps insurers to manage their risks by absorbing some of their losses. Legally, reinsurance is an insurance contract in which one insurance company indemnifies, for a premium, another insurance company against all or part of the losses that it may incur. Additionally, the management of the surplus of an insurance company may also involve investments in the financial market. Optimal reinsurance and investment problems for various risk models have gained a lot of interest in financial and insurance literature. They have been widely investigated under different criteria, especially via expected utility maximization, mean-variance criterion and ruin probability minimization (see e.g. Irgens and Paulsen [16], Liu and Ma [21], Shen and Zeng [26], Gu et al. [13], Brachetta and Ceci [6], Cao et al. [10] and references therein). Despite the richness of contributions, only a few papers on optimal investment-reinsurance address the problem in relation to dependent risks. Nevertheless, more and more natural and manmade disasters in recent years have brought great damage to the safety of lives and properties for people and have demonstrated that the insurance businesses and the financial market are not independent to each other. Therefore, considering dependent risk models in the actuarial literature has become necessary.
Pandemics, such as COVID-19, as well as other economically destructive natural phenomena such as hurricanes, earthquakes and wildfires, may have serious consequences for insurance companies, see e.g. some empirical studies as Born and Viscusi [4], Benali and Feki [1], Richter and Wilson [23]. Reinsurance allows insurance companies to manage potential claim shocks by transferring a specific portion or category of risk out of their portfolios to reinsurers’ portfolios, and hence, to stabilize their earnings and to increase their capacity to issue more business.
The main novelty of this paper is to consider an optimal investment-reinsurance problem for an insurance company with two lines of business in a stochastic-factor model which accounts for a common shock dependence between the financial and the insurance markets. Precisely, the two lines of business correspond to ordinary claims and catastrophic claims, whose arrival intensities are affected by environmental (stochastic) factors, as well as the insurance and reinsurance premia; furthermore, we assume that the arrival of catastrophic claims affects risky asset prices by inducing downward jumps. This modeling framework implies that the financial and the insurance frameworks are correlated by means of a common shock. Optimal reinsurance and investment problems in dependent financial and non-life insurance markets are also studied e.g. in Hainaut [14], Brachetta and Schmidli [8]. In Hainaut [14] a potential contagion risk between financial and insurance activities due to catastrophic events is modeled by time-changed processes; in Brachetta and Schmidli [8], instead, a diffusion risk model with an external driver modeling a stochastic environment is considered. It is important to stress that in most of the contributions on optimal investment-reinsurance problems the expression common shock dependence refers to the assumption of dependent classes of insurance business, see e.g. Yuen et al. [27], Liang and Yuen [18], Han et al. [15], where only the optimal proportional reinsurance problem is addressed, and see e.g. Bi et al. [2, 3], Zhang and Zhao [28] which also include the optimal investment problem. To the best of our knowledge, there are only a few papers, where the modeling framework allows for a common shock affecting both the stock market and the insurance market, see Liang et al. [19, 20]. However, in this literature, the insurance risk model is modulated by a compound Poisson process with constant arrival intensity and risky asset jump sizes (induced by the common shock) are independent of the claim sizes; moreover, only classical premium calculation principles are considered, which considerably simplify the analysis. Instead, in our setting, claim arrival intensities of both business lines are modeled as functions of additional exogenous stochastic factors; therefore, intensities of the claims arrival process are stochastic and evolve according to changing environmental conditions. Stochastic intensity models can be also found in non-life insurance settings by means of shot-noise Cox processes, see e.g. Dassios and Jang [12], Schmidt [25], Cao et al. [10], and more generally, stochastic risk factor models in insurance are further considered in Liang and Bayraktar [17], Brachetta and Ceci [6, 5, 7]. To disentangle the effect of catastrophic and ordinary claims on the financial market we assume that they may produce jumps of different sizes on asset prices and these jumps depend on the claims size as well. In addition, we apply an extended version of the classical expected value and variance premium calculation principles.
The portfolio of the insurance company consists of the retention levels of proportional reinsurance agreements and the investment strategy in the financial market, where a riskless asset and a risky asset are traded. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we provide a verification result for the value function (see Theorem 4.1) via classical solutions to two backward partial differential equations, one of which accounts for dependence between the insurance and the financial markets, and to the best of our knowledge, has not been previously analyzed in the literature. We characterize the optimal strategy (see Proposition 5.1, Proposition 5.5 and Proposition 6.1) and discuss its properties by considering a comparison with a corresponding scenario without a common shock effect.
The paper is organized as follows. Section 2 introduces the mathematical framework for the financial-insurance market model. In Section 3 we formulate the main assumptions and describe the optimization problem. Section 4 includes the derivation of the Hamilton-Jacobi-Bellman equation and the Verification Theorem. In Section 5 we solve the resulting minimization problems, discussing in Section 5.1 how the results apply under the expected value premium, and we characterize the optimal investment-reinsurance strategy in Section 6. The effects of the common shock on the optimal strategy are investigated in Section 6.1. Moreover, some sufficient conditions for existence and uniqueness of the solution to the Hamilton-Jacobi-Bellman equation are provided in Section 7. Finally, most technical proofs are collected in Appendix A.

Let $(\Omega,\mathcal{F},\mathbf{P};\mathbb{F})$ be a filtered probability space and assume that the filtration $\mathbb{F}=\{\mathcal{F}_{t},\ t\in[0,T]\}$ satisfies the usual hypotheses. The time $T$ is a finite time horizon that may represent the maturity of a reinsurance contract. We consider two independent counting processes $N^{(1)}=\{N_{t}^{(1)},\ t\in[0,T]\}$, $N^{(2)}=\{N_{t}^{(2)},\ t\in[0,T]\}$ that indicate the number of ordinary claims and catastrophic claims, respectively. In the sequel, we refer to ordinary and catastrophic claims as two business lines for the same insurance company. The jump times of $N^{(1)}$ and $N^{(2)}$ correspond to claim arrival times and are denoted by $\{T^{(1)}_{n}\}_{n\in\mathbb{N}}$ and $\{T^{(2)}_{n}\}_{n\in\mathbb{N}}$. We assume that the claim size of each line of business is described by two sequences of independent and identically distributed $\mathbb{R}^{+}$-valued random variables denoted by $\{Z^{(1)}_{n}\}_{n\in\mathbb{N}}$ and $\{Z^{(2)}_{n}\}_{n\in\mathbb{N}}$, respectively, with cumulative distribution functions $F^{(1)},F^{(2)}:\mathbb{R}\to[0,1]$, satisfying the condition

$$\int_{0}^{+\infty}z^{2}F^{(i)}(\mathrm{d}z)<\infty,\ \quad i=1,2.$$

For any $n,m\in\mathbb{N}$, $Z^{(1)}_{n}$ and $Z^{(2)}_{m}$ indicate the claim amount at time $T^{(1)}_{n}$ and $T^{(2)}_{m}$ for each business line. Processes $N^{(1)}$, $N^{(2)}$, and the families of variables $\{Z^{(1)}_{n}\}_{n\in\mathbb{N}}$ and $\{Z^{(2)}_{n}\}_{n\in\mathbb{N}}$ are assumed to be mutually independent.

We now introduce two stochastic processes $Y^{(1)}=\{Y_{t}^{(1)},\ t\in[0,T]\}$, $Y^{(2)}=\{Y_{t}^{(2)},\ t\in[0,T]\}$ representing environmental factors that affect arrival intensities of ordinary and catastrophic claims respectively. To model claim arrival intensities of the processes $N^{(1)}$ and $N^{(2)}$, we consider two strictly positive measurable functions $\lambda^{(i)}:[0,T]\times\mathbb{R}\to(0,+\infty)$, with $i=1,2$, and define the processes $\{\lambda^{(i)}(t,Y_{t}^{(i)}),\ t\in[0,T]\}$, for $i=1,2$, satisfying

$$\mathbb{E}\left[\int_{0}^{T}\lambda^{(i)}(t,Y^{(i)}_{t})\mathrm{d}t\right]<\infty,\quad i=1,2.$$

(2.1)

According to the classical construction of a Cox process we assume that for $i=1,2$,

$$\mathbb{E}[N^{(i)}_{t}-N^{(i)}_{s}=k\mid\mathcal{F}^{Y^{(i)}}_{T}\lor\mathcal{F}_{s}]=\frac{\bigl{(}\int_{s}^{t}\lambda^{(i)}(u,Y^{(i)}_{u})\,\mathrm{d}u\bigr{)}^{k}}{k!}e^{-\int_{s}^{t}\lambda^{(i)}(u,Y^{(i)}_{u})\,\mathrm{d}u},\quad 0\leq s<t\leq T,$$

for every $k=0,1,\dots$, where $\mathcal{F}^{Y^{(i)}}_{T}:=\sigma\{Y^{(i)}_{t},\ 0\leq t\leq T\}$. Then, the process $\{\lambda^{(i)}(t,Y_{t}^{(i)}),\ t\in[0,T]\}$ provides the intensity of the counting process $N^{(i)}$, for $i=1,2$, and condition (2.1) guarantees that $N^{(1)}$ and $N^{(2)}$ are non-explosive (see Brémaud [9]).

We define the cumulative claim process $C=\{C_{t},\ t\in[0,T]\}$ as

$$C_{t}=C_{t}^{(1)}+C_{t}^{(2)}:=\sum_{j=1}^{N_{t}^{(1)}}Z_{j}^{(1)}+\sum_{j=1}^{N_{t}^{(2)}}Z_{j}^{(2)},\quad t\in[0,T],$$

(2.2)

where for any $t\in[0,T]$, $C_{t}^{(1)}$ and $C_{t}^{(2)}$ represent the cumulative claim amount corresponding to each line of business in the time interval $[0,t]$.

We model the dynamics of the stochastic factors $Y^{(1)}$, $Y^{(2)}$ by diffusion processes

	$\displaystyle\mathrm{d}Y^{(1)}_{t}=b^{(1)}(t)\mathrm{d}t+a^{(1)}(t)\mathrm{d}W^{(1)}_{t},\quad Y^{(1)}_{0}=y^{(1)}\in\mathbb{R},$		(2.3)
	$\displaystyle\mathrm{d}Y^{(2)}_{t}=b^{(2)}(t)\mathrm{d}t+a^{(2)}(t)\mathrm{d}W^{(2)}_{t},\quad Y^{(2)}_{0}=y^{(2)}\in\mathbb{R},$		(2.4)

where $a^{(i)},b^{(i)}:[0,T]\to\mathbb{R}$, for $i=1,2$, are measurable functions such that

$\displaystyle\int_{0}^{T}\left(\left(a^{(i)}(t)\right)^{2}+|b^{(i)}(t)|\right)\mathrm{d}t<\infty$

(2.5)

and $W^{(1)}=\{W_{t}^{(1)},\ t\in[0,T]\}$, $W^{(2)}=\{W_{t}^{(2)},\ t\in[0,T]\}$ are independent Brownian motions also independent of $C^{(1)},C^{(2)}$.

In the sequel, we will make use of the notation $m^{(1)}(\mathrm{d}t,\mathrm{d}z)$ and $m^{(2)}(\mathrm{d}t,\mathrm{d}z)$ to indicate the random counting measures associated with $C^{(1)}$ and $C^{(2)}$ respectively, and which are defined by

$\displaystyle m^{(i)}(\mathrm{d}t,\mathrm{d}z)=\sum_{n\in\mathbb{N}}\delta_{\{T^{(i)}_{n},Z^{(i)}_{n}\}}(\mathrm{d}t,\mathrm{d}z){\mathbf{1}}_{\{T^{(i)}_{n}\leq T\}},\quad i=1,2,$

(2.6)

where $\delta_{\{t,z\}}$ denotes the Dirac measure at point $(t,z)$. The dual predictable projections of the random measure are computed in the lemma below.

The insurance company invests premia received by the policyholders in a financial market where a riskless asset and a risky asset are negotiated and subscribes reinsurance contracts for each business line. Securities are traded continuously on the time interval $[0,T]$ and their price processes $B=\{B_{t},\ t\in[0,T]\}$ and $P=\{P_{t},\ t\in[0,T]\}$ follow the dynamics

$$\mathrm{d}B_{t}=r(t)B_{t}\mathrm{d}t,\quad B_{0}=1,$$

(2.9)

where $r:[0,T]\to\mathbb{R}$ is an integrable function (i.e. $\int_{0}^{T}|r(s)|\mathrm{d}s<\infty$) representing the interest rate, and

$$\mathrm{d}P_{t}=P_{t-}\left(\mu(t)\mathrm{d}t+\sigma(t)\mathrm{d}W_{t}-\int_{0}^{+\infty}K(t,z)m^{(2)}(\mathrm{d}t,\mathrm{d}z)\right),\quad P_{0}=p_{0}>0,$$

(2.10)

where $W=\{W_{t},\ t\in[0,T]\}$ is a standard Brownian motion independent of $W^{(1)},W^{(2)},C^{(1)}$ and $C^{(2)}$, and the measurable functions $\mu:[0,T]\to\mathbb{R}$ and $\sigma:[0,T]\to(0,+\infty)$ are such that

$$\int_{0}^{T}\left(|\mu(t)|+\sigma^{2}(t)\right)\mathrm{d}t<\infty,$$

and $K:[0,T]\times[0,+\infty)\to[0,1)$. The latter, i.e. $K(t,z)\in[0,1)$, ensures that $P$ stays positive. Moreover, because of condition (2.1) and the fact that $K(t,z)\in[0,1)$, it holds that

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{0}^{+\infty}K(t,z)\lambda^{(2)}(t,Y^{(2)}_{t})F^{(2)}(\mathrm{d}z)\mathrm{d}t\right]<\infty.$

The model for the price dynamics has the following interpretation: when a catastrophic claim arrives, it instantaneously affects the asset price which jumps downwards of an amount that depends on the size of the claim. Common shocks naturally induce dependence between the financial and the pure insurance frameworks.

The gross risk premium rate of each business line is described by a nonnegative predictable process $\{c^{(i)}(t,Y^{(i)}_{t}),\ t\in[0,T]\}$, for $i=1,2$, where $c^{(i)}:[0,T]\times\mathbb{R}\to(0,+\infty)$ are measurable functions such that

$$\mathbb{E}\left[\int_{0}^{T}c^{(i)}(t,Y^{(i)}_{t})\,\mathrm{d}t\right]<\infty,\quad i=1,2.$$

(3.1)

To partially cover for the losses, the insurance company buys proportional reinsurance contracts with retention level $u\in[0,1]$, that is, retained losses are modeled via functions $g^{(i)}(z,u)=zu$, for $i=1,2$, where $1-u$ represents the percentage of losses covered by the reinsurance¹¹1It is well known that proportional reinsurance satisfies the properties of self-insurance functions (see, e.g. Schmidli [24, Chapter 4]).. We assume that the insurance company continuously buys reinsurance agreements whose reinsurance premium processes are defined as follows.

We also assume

$$\mathbb{E}\biggl{[}\int_{0}^{T}q^{(i)}(t,Y^{(i)}_{t},0)\,\mathrm{d}t\biggr{]}<\infty,\quad i=1,2,$$

(3.2)

which means that the total expected premium payment in case of full reinsurance does not explode.

The total surplus (or total reserve) process is $R=\{R_{t}:=R^{{(1)},u^{(1)}}_{t}+R^{{(2)},u^{(2)}}_{t},\ t\in[0,T]\}$, where for each $i=1,2$, $R^{{(i)},u^{(i)}}$ is the surplus process associated to each line of business with a given reinsurance strategy $\{u^{(i)}_{t},\ t\in[0,T]\}$ satisfying

$$\mathrm{d}R^{{(i)},u^{(i)}}_{t}=\left[c^{(i)}(t,Y^{(i)}_{t})-q^{(i)}(t,Y^{(i)}_{t},u^{(i)}_{t})\right]\mathrm{d}t-\int_{0}^{+\infty}zu_{t}^{(i)}\,m^{(i)}(\mathrm{d}t,\mathrm{d}z),$$

(3.10)

and $R_{0}\in\mathbb{R}^{+}$ is the initial wealth of the insurance company.

Let $\{u^{(i)}_{t},\ t\in[0,T]\}$, for $i=1,2$, be the dynamic retention levels corresponding to each reinsurance contract and let $\{w_{t},\ t\in[0,T]\}$ be the total amount invested in the risky asset. We assume that for every $t\in[0,T]$, $w_{t}$ takes values in $\mathbb{R}$, which means that short-selling and borrowing from the bank account are allowed. Then, the wealth of the insurance company associated with the reinsurance-investment strategy $\alpha=\{\alpha_{t}:=(u^{(1)}_{t},u^{(2)}_{t},w_{t}),\ t\in[0,T]\}$ satisfies

$$\mathrm{d}X^{\alpha}_{t}=\mathrm{d}R^{{(1)},u^{(1)}}_{t}+\mathrm{d}R^{{(2)},u^{(2)}}_{t}+w_{t}\frac{\mathrm{d}P_{t}}{P_{t}}+(X^{\alpha}_{t}-w_{t})\frac{\mathrm{d}B_{t}}{B_{t}},\qquad X^{\alpha}_{0}=R_{0}\in\mathbb{R}^{+}.$$

(3.11)

The goal of the insurance company is to maximize the expected utility from its terminal wealth, that is, to solve the optimization problem

$\displaystyle\sup_{\alpha\in\mathcal{A}}{\mathbb{E}\bigl{[}U(X_{T}^{\alpha})\bigr{]}},$

(3.18)

where $U(x)=1-e^{-\gamma x}$ represents the exponential preferences of the insurance company, with risk aversion parameter $\gamma>0$, and $\mathcal{A}$ denotes the class of admissible reinsurance-investment strategies defined as follows.

Condition (3.20) guarantees that the Verification theorem (see Theorem 4.1 below) applies. Nevertheless, this condition is satisfied by the optimal strategy under suitable assumptions on the model coefficients.

Now, we assume that the following assumptions are in force throughout the paper; they provide a set of sufficient conditions for a strategy $\alpha$ to be admissible (see Proposition 3.6).

The proof of Proposition 3.6 is postponed to Appendix A.

We define the value function corresponding to the optimization problem (3.18) as

$\displaystyle V(t,y^{(1)},y^{(2)},x):=\inf_{\alpha\in\mathcal{A}}{\mathbb{E}^{t,y^{(1)},y^{(2)},x}\bigl{[}e^{-\gamma X_{T}^{\alpha}}\bigr{]}},\quad(t,y^{(1)},y^{(2)},x)\in[0,T)\times\mathbb{R}^{3},$

(4.1)

where the notation $\mathbb{E}^{t,y^{(1)},y^{(2)},x}[\cdot]$ stands for the conditional expectation given $Y^{(1)}_{t}=y^{(1)},Y^{(2)}_{t}=y^{(2)}$ and $X^{\alpha}_{t}=x$.

If $V$ is sufficiently smooth, by classical control theory results it can be characterized as the solution of the Hamilton-Jacobi-Bellman (in short HJB) equation

$$\frac{\partial V}{\partial t}(t,y^{(1)},y^{(2)},x)+\inf_{\alpha\in[0,1]\times[0,1]\times\mathbb{R}}\mathcal{L}^{\alpha}_{t}V(t,y^{(1)},y^{(2)},x)=0,\quad(t,y^{(1)},y^{(2)},x)\in[0,T)\times\mathbb{R}^{3},\\$$

(4.2)

with the final condition

$$V(T,y^{(1)},y^{(2)},x)=e^{-\gamma x},\quad(y^{(1)},y^{(2)},x)\in\mathbb{R}^{3},$$

(4.3)

where, for any constant control $\alpha=(u^{(1)},u^{(2)},w)\in[0,1]\times[0,1]\times\mathbb{R}$, the operator $\mathcal{L}_{t}^{\alpha}$ denotes the infinitesimal generator of the process $(Y^{(1)},Y^{(2)},X^{\alpha})$ which satisfies

	$\displaystyle\mathcal{L}^{\alpha}_{t}f(y^{(1)},y^{(2)},x)$		(4.4)
	$\displaystyle\ =\frac{\partial f}{\partial y^{(1)}}b^{(1)}(t)+\frac{\partial f}{\partial y^{(2)}}b^{(2)}(t)+\frac{\partial f}{\partial x}\mu^{X}(t,x,y^{(1)},y^{(2)},\alpha)+\frac{1}{2}\frac{\partial^{2}f}{\partial{y^{(1)}}^{2}}({a^{(1)}})^{2}(t)+\frac{1}{2}\frac{\partial^{2}f}{\partial{y^{(2)}}^{2}}({a^{(2)}})^{2}(t)$		(4.5)
	$\displaystyle\ +\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}w^{2}\sigma^{2}(t)+\int_{0}^{+\infty}\left(f(y^{(1)},y^{(2)},x-zu^{(1)})-f(y^{(1)},y^{(2)},x)\right)\lambda^{(1)}(t,y^{(1)})F^{(1)}(\mathrm{d}z)$		(4.6)
	$\displaystyle\ +\int_{0}^{+\infty}\left(f(y^{(1)},y^{(2)},x-zu^{(2)}-wK(t,z))-f(y^{(1)},y^{(2)},x)\right)\lambda^{(2)}(t,y^{(2)})F^{(2)}(\mathrm{d}z),$		(4.7)

for every function $f$ which is $\mathcal{C}^{2}$ in $(y^{(1)},y^{(2)},x)\in\mathbb{R}^{3}$.