Optimal Investment Strategy for an Insurer in Two Currency Markets

1.   Introduction

In order to increase profits, insurers are allowed to invest their surpluses in financial markets. In recent years, the studies on optimal investment problems with various objectives have gained significant interest. For example, Hipp and Plum^[1], Schmidli^[2–3], Azcue and Muler^[4], and Chen et al.^[5] studied the optimization problem of minimizing ruin probability under different assumptions. Bäuerle^[6], Bi et al.^[7], Bai and Zhang^[8], Sun et al.^[9], and Shen and Zeng^[10] investigated the optimal investment or the optimal investment and optimal reinsurance problems with mean-variance criteria.

In this paper our objective is to maximize the expected utility of terminal wealth, which is another important criterion for various optimization problems in finance and modern risk theory. For example, Browne^[11] obtained an optimal investment strategy for an insurer whose surplus process is modeled by a drifted Brownian motion. Bai and Guo^[12] discussed the optimal proportional reinsurance and investment problem for an insurer who was allowed to invest into risk-free asset and multiple risky assets. Zhang et al.^[13] investigated the optimal investment and reinsurance strategies for insurers with a generalized mean-variance premium principle. Yang and Zhang^[14] studied the same optimal investment problem for an insurer with the risk process modeled by a jump-diffusion process, in which the diffusion term stands for the uncertainty associated with the surplus of the insurer. Irgens and Paulsen^[15] examined the optimal control problem for an insurer whose risk process follows a jump-diffusion process which represents the additional small claims.

However, most existing literature addresses optimization problems within a single currency market. In this paper, we investigate the optimal investment problem for an insurer with access to multiple currency markets, including the domestic risk-free asset and a foreign risky asset.

Geometric Brownian motion is the most commonly used model to describe the price of exchange rate. However, taking into account of a variety of factors which affect exchange rate price, the geometric Brownian motion can't well reflect the changes of exchange rate price. Thus other models have been created to describe exchange rate price more precisely. One of the most popular model is the one in which interest-rate spread is incorporated. For more details one can refer to Ahlip^[16], and Ahlip and Rutkowski^[17–18].

As we all know, there are many factors that influence exchange rate price, such as inflation, balance of international payment, interest-rate spread and so on. Inflation is the most important fundamental factor affecting the price of exchange rate. If the inflation rate of a domestic country is higher than that of a foreign country, then the competitiveness of the domestic country's exports is weakened which increases the competitiveness of foreign goods in domestic country's market. It would cause the domestic country's trade balance of payments deficit and the demand of foreign exchange is larger than its supply. Thus the foreign exchange rate price is increasing. Conversely, the price of foreign exchange rate declines. The balance of payments is the direct factor that affects exchange rate price. For example, when a country has a large balance of payments surplus, i.e., the country's imports are less than its exports, then its currency demand will increase and this will lead to an increase in foreign exchange flowing into the country. In this way, in the foreign exchange market, the supply of foreign exchange is greater than its demand. Then the price of foreign exchange rate goes down. But, if a country has a large balance of payments deficit, i.e., the country's imports are more than its exports, then the foreign exchange rate price goes up. Under certain conditions, interest rates have a great short-term impact on exchange rate price. The effect is caused by the difference of interest rate in different countries. In general, if interest-rate spread is increasing then the demand for domestic currency will increase. This leads to the increase of the domestic currency price. Thus the foreign exchange rate price declines. Conversely, the price of foreign exchange rate is increasing.

Although interest-rate spread has a certain impact on exchange rate prices and the models with interest-rate spread are excellent, from the perspective of the basic factors determining the price of foreign exchange rate, the effect of interest-rate spread is limited. Moreover, from the above descriptions it can be seen that the supply and demand of domestic and foreign currencies is the most paramount and direct factor that affects the price of exchange rate. When the demand for foreign currency exceeds the supply, the foreign exchange rate price increases, and we call it as the "bull foreign exchange rate". Conversely, when the supply exceeds the demand, the price of foreign exchange rate goes down, and we call it as the "bear foreign exchange rate".

Therefore, inspired by Rishel^[19], in this paper the price of foreign exchange rate $Q_t$ is described by the following differential equation:

where $a(t)=u_Q+m(t)$ and $m(t)$ is given by Ornstein-Uhlenbeck equation

Here, $u_Q, \sigma_Q, q_0, \alpha, \beta, m_0$ are known constants and all are positive except $\alpha,$ $\beta$ and $m_0.$ In (1), $u_Q$ is the target mean growth rate of exchange rate price. When $m(t)>0$ the growth rate of the exchange rate price is increasing. Conversely, if $m(t)<0$, the growth rate is decreasing. In particular, if $a(t)<0$, the exchange rate price goes down. $W^{2}_{t}$ and $W^3_t$ are standard Brownian motions.

The rest of this paper is organized as follows. Section 2 presents the model formulation, and Section 3 summarizes the main results. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations, we derive the optimal strategies and value functions. In particular, we find that, if the insurer only invests in the foreign risky asset and the price of exchange rate is modeled by geometric Brownian motion then the optimal investment strategy is a constant, regardless of the level of wealth the insurer has. Section 4 provides numerical examples and analysis. We find that in some cases investing in two currency markets can produce a higher value function than investing in a single currency market.

2.   Model formulation

Throughout this paper, let $(\Omega, \mathcal{F}, P)$ be a probability space with filtration $\{\mathcal{F}_t\}_{t\in [0, T]}$ for any given $T>0.$ The filtration $\{\mathcal{F}_t\}_{t\in [0, T]}$ is generated by the four standard Brownian motions $W_t, W^{1}_t, W^{2}_t, W^{3}_t.$ All of the processes introduced below are well defined and adapted in this space. We also assume that all tradings in this paper are continuous and without any taxes and transaction costs, and all the assets are infinitely divisible.

It is started from the classical Cramer-Lundberg model in which the surplus of the insurance company is modeled as

where $p$ is the premium rate, $N(t)$ is a Poisson process with intensity $\lambda_0$ representing the number of claims until time $t$, and $\{Z_{i}\}_{i\ge 1}$ is a sequence of positive, independent and identically distributed random variables representing the size of each claim. Without loss of generality, we assume that the intensity $\lambda_0=1.$ Denote the mean and second order moment of $Z_i$ by $\mu_1=\mathrm{E}[Z_i]$ and $\mu_2=\mathrm{E}[Z^2_i],$ respectively. Further, we suppose that the claim sequences $\{Z_i\}_{i\ge 1}$ are independent from $N(t)$ and the premium rate $p=(1+\eta_1)\mu_1$ according to the expected value principle, where $\eta_1>0$ is the safety loading. Then the dynamics of $X_t$ can be approximated by

where $u=p- \mathrm{E}[Z_1]=\eta_1\mu_1>0, \sigma^{2}=\mu_2,$ and $W_{t}$ is a standard Brownian motion. For more details of the diffusion approximation of the surplus process, one can refer to Emanuel et al.^[20], Garrido^[21] and Iglehart^[22].

The insurer is allowed to invest its surplus in a financial market, in which the financial market consists of domestic risk-free assets and foreign risky assets. The price of the domestic risk-free asset is given by

The foreign risky asset price $S^{f}_t$ is modeled by means of geometric Brownian motion such that

We adopt the convention that the price $S_{t}^{f}$ is denominated by foreign currency and the exchange rate is denominated in units of domestic currency per unit of foreign currency. This means that $Q_{t}$ represents the domestic price at time $t$ of one unit of the foreign currency. Thus, let $g_{t}:=g(S_{t}^{f}, Q_{t})=Q_{t}S_{t}^{f},$ and then $g_{t}$ is the price of the foreign risky asset denominated by domestic currency. By Itô's formula, from (3) and (1), it can be seen that $g_{t}$ satisfies the following stochastic differential equation:

For mathematical convenience, in the rest of this paper, all the assets are valued by domestic currency. Let $\pi_t$ be the total amount of money invested in foreign risky asset at time $t$. The rest of the surplus is invested in domestic risk-free asset. Under the strategy $\pi_t,$ the surplus of the insurer is as follows:

where $A_1=u_f+u_{Q}-r_d$ and the initial surplus is $X_0^{\pi}=x.$ Following the assumption of Browne^[11], we allow that $\pi_t<0$ and $\pi_t>X_t$, which means that the insurer is allowed to short sell the foreign risky asset and borrow money for investment in foreign risky asset.

A strategy $\pi_{t}$ is said to be admissible, if $\pi_{t}$ is $\mathcal{F}_t$-progressively measurable and for any $T>0,$ $\mathrm{E}[\int_{0}^{T}\pi^{2}_t{\text{d}}t]<\infty.$ The set of all admissible strategies is denoted by $\Pi.$ Here we also assume that $W_t, W^1_t, W^2_t$ and $W^3_t$ are independent Brownian motions.

3.   The main results

The insurer is interested in maximizing the exponential utility of terminal wealth, say at time $T.$ Denote the utility function as $u(x)$ with $u^{'}(x)>0$ and $u^{''}(x)<0.$ Thus, our optimization problem can be described by

Define the associated value function

We are going to find an optimal strategy $\pi^{*}$ satisfying $J^{\pi^{*}}(t, x, m)=V(t, x, m).$

Assume now that the insurer has an exponential utility function

where $\gamma>0$ and $\theta>0.$ This utility function has constant absolute risk aversion parameter $\theta.$ Such a utility function plays a remarkable part in insurance mathematics and actuarial practice, since it is the only utility function under which the principle of "zero utility" gives a fair premium that is independent of the level of reserve of an insurance company (see [23]).

Applying the dynamic programming approach described by Fleming and Soner^[24], from standard arguments, we see that if the value function $V(t, x, m)$ and its partial derivatives $V_{t}, V_{x}, V_{xx}, V_{m}, V_{mm}$ are continuous on $[0, T]\times R^{1}\times R^{1},$ and then $V(t, x, m)$ satisfies the following HJB equation:

with boundary condition $V(T, x, m)=u(x).$

By [24] the following verification theorem exists.

Theorem 1 Let $W\in C^{1, 2, 2}([0, T]\times R^2)$ be a classical solution to the HJB equation (8) with boundary condition $W(T, x, m)=u(x),$ and then the value function $V$ given by (6) coincides with $W$ such that

In addition, let $\pi^{*}$ be the optimizer of (8), that is for any $(t, x, m)\in [0, T]\times R^2$

Then $\pi^{*}(t, X^{*}_t, m(t))$ is the optimal strategy with

where $X^{*}_t$ is the surplus process under the optimal strategy $\pi^{*}.$

3.1   Optimal investment strategy

In this subsection, we investigate the optimization problem (5) subject to (4) and derive explicit expressions for the optimal strategy and value function.

In order to solve HJB equation (8), we first find the value $\pi(x, m)$ which maximizes the function

Differentiating with respect to $\pi$ in (9), the optimizer

is obtained.

Assume that HJB equation (8) has a classical solution $V$ with $V_x > 0$ and $V_{xx} < 0.$ We try to find a solution of (8) as the form

where $h(t, m)$ is a suitable function such that (11) is a solution of (8). The boundary condition $V(T, x, m)=u(x)$ implies that $h(T, m)=0.$

From (11) it can be calculated that

where $V_t, V_{x}, V_{xx}, V_{m}, V_{mm}$ are the partial derivatives of $V(t, x, m)$ and $h_t, h_m, h_{mm}$ are the partial derivatives of $h(t, m).$ Substituting (12) into (8) and (10) yields

and

Put $\pi^{*}_t$ into (13) and calculate

It can be shown that (11) is a solution to (13) if $h(t, m)$ is a solution to (14).

Lemma 2 With the terminal condition $h(T, m)=0,$ the partial differential equation (PDE) (14) has a solution of the form

where $K(t)$ is a solution to

$L(t)$ is a solution to

and $J(t)$ is a solution to

Proof Substituting (15) into (14) and combining like terms with respect to the powers of $m,$ we have

Then, it is obvious that (15) is a solution to (14) if $K(t), L(t)$ and $J(t)$ are solutions to the differential equations (16), (17) and (18), respectively. □

Next, we will solve the differential equations (16), (17) and (18), respectively. Let

and then the Riccati equation (16) simplifies to

If $B\neq0,$ i.e., $\beta\neq 0,$ integrating

on both sides with respect to $t$, we obtain

where $E$ is a constant. Since $\Delta=C^2-4BD=4\alpha^2+\frac{4\beta^2}{\sigma^2_f+\sigma^2_Q}>0,$ the quadratic equation $BK^2(t)+CK(t)+D=0$ has two different real roots given by

Then we get

Substituting (20) into (19) and taking into consideration the boundary condition $K(T)=0,$ we obtain

If $B=0,$ i.e., $\beta=0,$ then

With the value of $K(t)$ defined in (21) or (22), the linear ordinary differential equation (17) has a solution of the form

Upon the knowing of $K(t)$ and $L(t),$ a solution of (18) is given by

From the above statements we have the following results.

Theorem 3 With the utility function (7), the optimal strategy for the optimization problem (5) subject to (4) is

Moreover, the value function is given by

with $h(t, m)=K(t)m^2+L(t)m+J(t),$ where $K(t), L(t)$ and $J(t)$ are given by (21)–(24).

Remark 1 From the result in Theorem 3, we observe that the optimal investment strategy $\pi^{*}_t$ depends only on financial market parameters $u_f, u_Q, r_d, \sigma_f, \sigma_Q, m$ and risk aversion parameter $\theta$, and does not depend on other market parameters. In particular, $\pi^{*}_t$ is decreasing with respect to $\theta$, which means that the more risk averse the insurer is, the less it invests in risky assets. Moreover, $\pi^{*}_t$ is increasing with respect to $m$. When the parameter $m$ is increasing, the price of foreign risky asset valued by domestic currency will become larger. Then the insurer prefers to invest more wealth in foreign risky asset.

If $m(t)=0$ in (1), the exchange rate price $Q_t$ is degenerated into the process which is modeled by means of geometric Brownian motion

In this case, under the control of $\pi,$ the wealth process $X^{\pi}_t$ satisfies the following stochastic differential equation:

Then the HJB equation in (8) takes the following form:

By solving the above HJB equation (26) the following corollary is obtained.

Corollary 4 With $X^{\pi}_t$ in (25) the optimal investment strategy is given by

Further, the value function has the form

where

Let $S^d_t$ be the price of domestic risky asset described by the following stochastic differential equation:

where $u_d$ and $\sigma_d$ are constants and $W^4_t$ is the standard Brownian motion which is independent to $W_t.$ Assume the insure invests its wealth only in domestic currency market, i.e., domestic risk-free asset and domestic risky asset, and then the surplus under control $\pi$ is

Corollary 5 In domestic currency market, the optimal strategy for the optimization problem (5) subject to (28) is

and the corresponding value function has the form

where

Remark 2 By comparing the results in Corollaries 4 and 5, it can be seen that

(i) If the parameters satisfy that $\sigma^2_d(u_f+u_Q-r_d)=(\sigma^2_f+\sigma^2_Q)(u_d-r_d),$ it is easy to see that the optimal investment strategies in Corollaries 4 and 5 are the same. Further, if $u_f+u_Q\ge u_d,$ the value function subject to (25) is always larger than that subject to (28). Thus, it is better for the insurer to invest in foreign risky asset. Conversely, if $u_f+u_Q<u_d,$ it is better for the insure to invest in domestic risky asset.

(ii) If $\sigma^2_d\left(u_f+u_Q-r_d\right)^2=(\sigma^2_f+\sigma^2_Q)\left(u_d-r_d\right)^2$, this implies that the value functions in Corollaries 4 and 5 are the same. In addition, if $u_f+u_Q\ge u_d,$ then in order to get the same value functions in the two cases, the amount of wealth invested in foreign risky asset is less than that invested in domestic risky asset. Conversely, if $u_f+u_Q<u_d,$ in order to get the same value functions, the insure should invest more in foreign risky asset.

Remark 3 From Corollaries 4 and 5, it is not difficult to see that if no domestic risk-free asset is traded and only risky asset is considered even domestic or foreign risky asset, the optimal strategies are always constants, regardless of the level of wealth the insurer has.

3.2   Optimal reinsurance-investment strategy

Except for investing in the financial market, in this subsection, reinsurance is also allowed. Here, we consider the proportional reinsurance. Denote the reinsurance proportion by $q(t)\in[0, +\infty).$ That is, for any claim $Z_i$ the insurer only pays $q(t)Z_i$ while the reinsurer pays the rest. However, based on the expected value principle, the insurer has to pay a premium at the rate of $(1+\eta_2)(1-q(t))\mu_1$ to the reinsurer due to the reinsurance business. In general, $\eta_2>\eta_1,$ otherwise, arbitrage will exist. $q(t)>1$ implies that the insurer can take new reinsurance business from the insurance market. In order to ensure that the insurance company do not go bankrupt, we suppose that

i.e., $q_t>\overline{q}=1-\eta_1/\eta_2.$

Further, as presented in Section 2, the insurer invests its wealth into foreign risky asset and domestic risk-free asset. Let $\pi_t$ be the total amount of money invested in the foreign risky asset at time $t$. The rest of the surplus is invested into domestic risk-free asset. Then, after incorporating policy $(q_t, \pi_t)$ into (2), the surplus of the insurer is as follows:

with $X^{q, \pi}_0=x.$

Remark 4 A reinsurance-investment strategy $(q_t, \pi_t)$ is said to be admissible, if $(q_t, \pi_t)$ is $\mathcal{F}_t$-progressively measurable and for any $T > 0$, $\mathrm{E}[\int_{0}^{T}\pi^{2}_t {\text{d}}t]<\infty.$ The set of all admissible strategies is denoted by $\widetilde{\Pi}$.

In this subsection, the optimization problem is to maximize (5) subject to (29) over strategy $\widetilde{\Pi}$. Thus, its associated value function is

Then, the corresponding HJB equation is

with the boundary condition $V(T, x, m)=u(x).$

In order to solve the HJB equation (30), we first find the value $(q, \pi)$ which maximizes the function

Differentiating with respect to $q, \pi$ in (31), the optimizer

is obtained.

Again, we try to find a solution with the form of (11). Substituting (12) and (32) into (30), we obtain

with the boundary condition $h(T, m)=0.$

Lemma 6 The PDE (33) has a solution of the form

where

$K(t)$ and $L(t)$ are defined as in Theorem 2.

Substituting (12) into (32) we get

Consider that $q^{*}_t>\overline{q}=1-\eta_1/\eta_2$, which implies that $\theta<\frac{\eta^2_2\mu_1}{\sigma^2(\eta_2-\eta_1)}\text{e}^{-r_d(T-t)}.$

Finally, we have the following result.

Theorem 7 Under the condition that $0<\theta<\frac{\eta^2_2\mu_1}{\sigma^2(\eta_2-\eta_1)}\text{e}^{-r_d(T-t)},$ the optimal reinsurance-investment strategy for the optimization problem (5) subject to (29) is $(q^{*}_t, \pi^{*}_t)$ defined by (34) for any $t\in [0, T]$. Further, the corresponding value function is

where $\hat{h}(t, m)$ is defined in Lemma 6.

Remark 5 From (34), we see that the optimal reinsurance strategy $q^{*}_t$ depends not only on insurance market parameters $\eta_2, \mu_1$ and $\sigma$, but also on financial market parameter $r_d$ and risk aversion parameter $\theta.$ It is easy to see that $q^{*}_t$ is increasing with respect to $\eta_2,$ where $\eta_2$ is the safety loading of reinsurance premium. It means that the higher reinsurance premium is, the more insurance the insurer keeps, that is, the less reinsurance business it buys. In addition, $q^{*}_t$ is decreasing with respect to $\theta$ which leads to that the risk-averse insurer prefers to buy more reinsurance business to reduce its risk.

4.   Numerical examples and analysis

In order to demonstrate our results, numerical examples are presented for the optimal investment strategies and value functions in Corollaries 4 and 5. Our objective is to study the effect of exchange rate on optimal investment strategy and value function. The particular numbers of basic parameters are given in the following tables.

The numbers in Table 1 satisfy

$T$	$r_{d}$	$\lambda$	$\theta$	$\gamma$	$u$	$\sigma$	$u_{f}$	$\sigma_{f}$	$u_{Q}$	$\sigma_{Q}$	$x$	$u_d$	$\sigma_d$
4	0.1	1	1	1	0.4	0.1	0.3	$\sqrt{0.1}$	0.2	$\sqrt{0.3}$	2	0.3	$\sqrt{0.2}$

 | Show Table

DownLoad: CSV

Figure  1. 

DownLoad: Full-Size Img PowerPoint

Figure  2. 

DownLoad: Full-Size Img PowerPoint

and

The graphs of the optimal investment strategies and value functions according to the data in Table 1 are shown in (a) and (b). By graphs (a) and (b), we see that, under the condition that $\sigma_d^2(u_f+u_Q-r_d)=(\sigma^2_f+\sigma^2_Q)(u_d-r_d),$ the two optimal investment strategies in Corollaries 4 and 5 are the same. But, the value function with exchange rate is larger than that without exchange rate. Thus, in this case, it is better for the insurer to invest in foreign risky asset which coincides with the conclusion (i) in Remark 2.

The numbers in Table 2 satisfy

$T$	$r_{d}$	$\lambda$	$\theta$	$\gamma$	$u$	$\sigma$	$u_{f}$	$\sigma_{f}$	$u_{Q}$	$\sigma_{Q}$	$x$	$u_d$	$\sigma_d$
4	0.1	1	1	1	0.4	0.1	0.3	0.2	0.2	$\sqrt{0.12}$	2	0.3	0.2

 | Show Table

DownLoad: CSV

Figure  3. 

DownLoad: Full-Size Img PowerPoint

Figure  4. 

DownLoad: Full-Size Img PowerPoint

and

According to Table 2, the graphs of optimal investment strategies and value functions are given in (c) and (d). It can be seen that the value functions are the same. But, the insurer should invest more in domestic risky asset than in foreign risky asset. The results showed in graphs (c) and (d) coincide with that presented in (ii) of Remark 2.