Optimal Investment for an Insurer in Two Currency Markets: A Stochastic Control Analysis

Table of Contents

1. Introduction

This paper addresses a critical gap in insurance risk management literature: optimal investment strategies for insurers operating in multiple currency markets. While traditional models focus on single-currency environments, globalized insurance operations necessitate understanding cross-currency risk dynamics. The research combines actuarial science with financial mathematics to develop a comprehensive framework for insurers investing in both domestic and foreign markets.

The fundamental challenge lies in managing three interconnected risks: insurance claim risk, financial market risk, and foreign exchange risk. Previous works by Browne (1995), Yang and Zhang (2005), and Schmidli (2002) established foundations for insurer investment problems but neglected the multi-currency dimension that becomes increasingly relevant in today's global economy.

2. Model Framework

2.1 Surplus Process

The insurer's surplus process follows the diffusion approximation of the classical Cramér-Lundberg model:

$dX(t) = c dt - dS(t)$

where $c$ represents the premium rate and $S(t)$ is the aggregate claims process. Under diffusion approximation, this becomes:

$dX(t) = \mu dt + \sigma dW_1(t)$

where $\mu$ is the safety loading adjusted drift and $\sigma$ represents claim volatility.

2.2 Foreign Exchange Rate Model

The exchange rate between domestic and foreign currencies follows:

$dE(t) = E(t)[\theta(t)dt + \eta dW_2(t)]$

where the instantaneous mean growth rate $\theta(t)$ follows an Ornstein-Uhlenbeck process:

$d\theta(t) = \kappa(\bar{\theta} - \theta(t))dt + \zeta dW_3(t)$

This mean-reverting specification captures the empirical behavior of exchange rates influenced by fundamental economic factors like inflation differentials and interest rate spreads.

2.3 Investment Portfolio

The insurer allocates wealth across:

• Domestic risk-free asset with rate $r_d$
• Foreign risky asset with return dynamics in foreign currency
• Currency conversion through exchange rate $E(t)$

The total wealth process $W(t)$ evolves according to the investment strategy $\pi(t)$, representing the proportion invested in the foreign risky asset.

3. Optimization Problem

3.1 Exponential Utility Objective

The insurer aims to maximize expected exponential utility of terminal wealth:

$\sup_{\pi} \mathbb{E}[U(W(T))] = \sup_{\pi} \mathbb{E}[-\frac{1}{\gamma}e^{-\gamma W(T)}]$

where $\gamma > 0$ is the constant absolute risk aversion coefficient. This utility function is particularly suitable for insurers due to its constant risk aversion property and analytical tractability.

3.2 Hamilton-Jacobi-Bellman Equation

The value function $V(t,w,\theta)$ satisfies the HJB equation:

$\sup_{\pi} \{V_t + \mathcal{L}^\pi V\} = 0$

with terminal condition $V(T,w,\theta) = -\frac{1}{\gamma}e^{-\gamma w}$, where $\mathcal{L}^\pi$ is the infinitesimal generator of the wealth process under strategy $\pi$.

4. Analytical Solution

4.1 Optimal Investment Strategy

The optimal investment strategy in the foreign risky asset takes the form:

$\pi^*(t) = \frac{\mu_F - r_f + \eta\rho\zeta\phi(t)}{\gamma\sigma_F^2} + \frac{\eta\rho}{\sigma_F}\frac{V_\theta}{V_w}$

where $\mu_F$ and $\sigma_F$ are the foreign asset's return parameters, $r_f$ is the foreign risk-free rate, $\rho$ is the correlation between exchange rate and foreign asset returns, and $\phi(t)$ is a function of the exchange rate drift process.

4.2 Value Function

The value function admits an exponential affine form:

$V(t,w,\theta) = -\frac{1}{\gamma}\exp\{-\gamma w e^{r_d(T-t)} + A(t) + B(t)\theta + \frac{1}{2}C(t)\theta^2\}$

where $A(t)$, $B(t)$, and $C(t)$ satisfy a system of ordinary differential equations derived from the HJB equation.

5. Numerical Analysis

5.1 Parameter Sensitivity

Numerical experiments demonstrate:

• Risk Aversion Impact: Higher $\gamma$ reduces optimal foreign investment proportion from approximately 60% to 25% across tested scenarios
• Exchange Rate Volatility: Optimal strategy decreases by 15-20% when $\eta$ increases from 0.1 to 0.3
• Mean Reversion Speed: Faster mean reversion (higher $\kappa$) reduces hedging demand against exchange rate drift changes

5.2 Strategy Performance

Comparative analysis shows the multi-currency strategy outperforms single-currency approaches by 8-12% in certainty equivalent wealth across various parameter configurations, particularly during periods of exchange rate trend persistence.

6. Core Insight & Analysis

Core Insight: This paper delivers a crucial but narrowly focused advancement—it successfully extends insurer investment theory to two currencies but does so within restrictive assumptions that limit immediate practical application. The real value lies not in the specific solution but in demonstrating that the HJB framework can handle this complexity, opening doors for more realistic extensions.

Logical Flow: The authors follow a classic stochastic control template: 1) Model setup with diffusion approximations, 2) HJB formulation, 3) Guess-and-verify solution with exponential affine form, 4) Numerical verification. This approach is mathematically rigorous but pedagogically predictable. The inclusion of an Ornstein-Uhlenbeck process for exchange rate drift adds sophistication, reminiscent of Vasicek-type models in fixed income, but the treatment remains theoretically neat rather than empirically grounded.

Strengths & Flaws: The primary strength is technical completeness—the solution is elegant and the separation of variables technique is expertly applied. However, three critical flaws undermine practical relevance. First, the diffusion approximation of insurance claims washes away jump risk, which is fundamental to insurance (as emphasized in the seminal work of Schmidli (2002, "On Minimizing the Ruin Probability by Investment and Reinsurance")). Second, the model assumes continuous trading and perfect frictionless markets, ignoring liquidity constraints that plague currency markets during crises. Third, the numerical analysis feels like an afterthought—it verifies rather than explores, lacking the robustness tests seen in contemporary computational finance papers like those from the Journal of Computational Finance.

Actionable Insights: For practitioners, this paper offers a benchmark, not a blueprint. Risk managers should extract the qualitative insight—that exchange rate drift predictability (via the OU process) creates hedging demand—but should implement it using more robust estimation techniques for the OU parameters. For researchers, the clear next steps are: 1) Incorporate jump-diffusion claims following the approach of Kou (2002, "A Jump-Diffusion Model for Option Pricing"), 2) Add stochastic volatility to the exchange rate process, acknowledging the well-documented volatility clustering in FX markets, and 3) Introduce transaction costs, possibly using impulse control methods. The field doesn't need more variations on this exact model; it needs this model's elegance combined with the empirical realism found in the best work of Jarrow (2018, "A Practitioner's Guide to Stochastic Finance").

7. Technical Details

The key mathematical innovation involves solving a system of Riccati-type ODEs:

$\frac{dC}{dt} = 2\kappa C - \frac{(\eta\rho\zeta C + \zeta^2 B)^2}{\sigma_F^2} + \gamma\eta^2 C^2 e^{2r_d(T-t)}$

$\frac{dB}{dt} = \kappa\bar{\theta} C + (\kappa - \gamma\eta^2 e^{2r_d(T-t)} C) B - \frac{(\mu_F - r_f)(\eta\rho\zeta C + \zeta^2 B)}{\sigma_F^2}$

with terminal conditions $C(T)=B(T)=0$. These equations govern the dependence of the value function on the stochastic exchange rate drift $\theta(t)$.

The optimal strategy decomposes into three components:

1. Myopic Demand: $\frac{\mu_F - r_f}{\gamma\sigma_F^2}$ – standard mean-variance term
2. Exchange Rate Hedge: $\frac{\eta\rho}{\sigma_F}\frac{V_\theta}{V_w}$ – hedges changes in investment opportunity set
3. Drift Adjustment: $\frac{\eta\rho\zeta\phi(t)}{\gamma\sigma_F^2}$ – accounts for predictability in exchange rate drift

8. Analysis Framework Example

Case Study: Global P&C Insurer

Consider a property & casualty insurer with liabilities in both USD and EUR. Using the paper's framework:

1. Parameter Estimation:
   • Estimate OU parameters for EUR/USD drift using 10-year rolling regression
   • Calibrate claim process parameters from historical loss data
   • Estimate risk aversion γ from company's historical investment patterns
2. Strategy Implementation:
   • Calculate optimal EUR-denominated investment proportion daily
   • Monitor hedge ratio $\frac{V_\theta}{V_w}$ for rebalancing signals
   • Implement with 5% tolerance bands to reduce transaction costs
3. Performance Attribution:
   • Separate returns into: (a) myopic component, (b) exchange rate hedge, (c) drift timing
   • Compare to naive 60/40 domestic/foreign fixed allocation

This framework, while simplified, provides a structured approach to multi-currency insurer asset allocation that is more rigorous than typical ad hoc methods.

9. Future Applications & Directions

Immediate Applications:

• Dynamic Currency Overlay Programs: Insurers can implement the strategy as a currency overlay, dynamically adjusting hedge ratios based on exchange rate drift predictions
• Solvency II Optimization: Incorporate the framework into ORSA (Own Risk and Solvency Assessment) processes for European insurers
• Multi-National Corporate Treasury: Extend to corporate risk management beyond insurance

Research Directions:

1. Regime-Switching Extensions: Replace OU process with Markov regime-switching model to capture structural breaks in exchange rate behavior
2. Machine Learning Integration: Use LSTM networks to estimate the exchange rate drift process θ(t) rather than assuming parametric OU dynamics
3. Decentralized Finance Applications: Adapt framework for crypto-insurance products with multiple cryptocurrency exposures
4. Climate Risk Integration: Incorporate climate transition risk into exchange rate dynamics for long-term insurer investments

10. References

1. Browne, S. (1995). Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20(4), 937-958.
2. Schmidli, H. (2002). On Minimizing the Ruin Probability by Investment and Reinsurance. The Annals of Applied Probability, 12(3), 890-907.
3. Yang, H., & Zhang, L. (2005). Optimal Investment for Insurer with Jump-Diffusion Risk Process. Insurance: Mathematics and Economics, 37(3), 615-634.
4. Kou, S. G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101.
5. Jarrow, R. A. (2018). A Practitioner's Guide to Stochastic Finance. Annual Review of Financial Economics, 10, 1-20.
6. Zhou, Q., & Guo, J. (2020). Optimal Control of Investment for an Insurer in Two Currency Markets. arXiv:2006.02857.
7. Bank for International Settlements. (2019). Triennial Central Bank Survey of Foreign Exchange and OTC Derivatives Markets. BIS Quarterly Review.
8. European Insurance and Occupational Pensions Authority. (2020). Solvency II Statistical Report. EIOPA Reports.