Optimal debt ratio and dividend payment strategies with reinsurance

Abstract

This paper derives the optimal debt ratio and dividend payment strategies for an insurance company. Taking into account the impact of reinsurance policies and claims from the credit derivatives, the surplus process is stochastic that is jointly determined by the reinsurance strategies, debt levels, and unanticipated shocks. The objective is to maximize the total expected discounted utility of dividend payment until financial ruin. Using dynamic programming principle, the value function is the solution of a second-order nonlinear Hamilton–Jacobi–Bellman equation. The subsolution–supersolution method is used to verify the existence of classical solutions of the Hamilton–Jacobi–Bellman equation. The explicit solution of the value function is derived and the corresponding optimal debt ratio and dividend payment strategies are obtained in some special cases. An example is provided to illustrate the methodologies and some interesting economic insights.

Introduction

Since the collapse of US housing market in 2007, which led the initial sub-prime mortgage crisis into a global financial crisis in 2008, financial/insurance institutions and regulators have drawn increasing attention to evaluate and monitor risk so as to avoid insolvency. In this paper we analyze the failure of American International Group (AIG) in the global financial crisis. Our work focuses on AIG that sold Credit Default Swaps (CDSs), a form of insurance, against the financial risks that were based upon debt from the real estate market. At its peak, AIG was one of the largest and most successful companies in the world boasting a Triple-A credit rating, over $1 trillion in assets, and 76 million customers in more than 130 countries. AIG occupied an important role in the financial system. However, due to poor risk management structure, combined with a lack of regulatory oversight, AIG accumulated substantial amounts of risk and unsustainable insurance liabilities by issuing large amount of CDSs, which led to the crash of the insurance giant.

During 2001–2006, the low interest rates and rises in housing prices induced a substantial demand for mortgages. However, as financial institutions were chasing for higher returns, the leverage tools were overused and quality of mortgages declined. Mortgage originators such as Countrywide sell packages of mortgages to the major banks. The latter securities firms in turn structure the packages and tranche them into senior, mezzanine and equity tranches. The securities firms then sell the collateralized debt obligations (CDOs) to international investors, hedge funds and investment banks such as Merrill Lynch, Citi-group and Goldman–Sachs. If the mortgagors are unable to service their debts, the income from the mortgages declines. The cash flows all along the line will suffer. Securities firms and hedge funds may buy CDSs from companies such as AIG as insurance against depreciation in the values of the CDOs. When the market is highly leveraged, the financial system becomes vulnerable since the small change in asset values will significantly influence the net wealth. After the housing price peaked in early 2006, the bursting of housing bubble resulted in the credit and liquidity crisis and the recession thereafter. AIG Financial Product (AIGFP), a subsidiary of AIG, entered the credit derivatives market in 1998 when it underwrote its first CDS with JP Morgan. Over time AIGFP became a central player in the fast-growing CDS market. AIGFP’s corporate arbitrage CDS portfolio was comprised of CDS contracts written on CDOs. The collateral pools backing the multi-sector CDOs included prime, Alt-A, and subprime residential mortgage-backed securities (RMBS), commercial mortgage-backed securities (CMBS), other asset-backed securities (ABS). In many cases non-agency CDOs are required to carry insurance in order to obtain a high credit rating. The CDSs are privately negotiated contracts that perform in a similar manner to insurance contacts, but their payoff function is similar to a put option. The CDS requires that the insurer put up more collateral if the market value of the securities insured falls below the predetermined level. Claims are the required payments to the insured holders of CDSs, due to either defaults of the obligors or for collateral calls when the prices of the insured securities decline. However, AIG did not set sufficient surplus aside to cope with the collateral claims, which led to the catastrophe of AIG and the biggest corporate bailout in US history.

The CDSs insured by AIG were ultimately related to the systemic risk from the inability of the mortgagors to service their debts. AIG made a series of serious mistakes in the risk management. Risk was underestimated because AIG ignored the negative correlation between the investment income and the claims. The estimate of the drift of the capital gain was based on the unsustainable growth of the housing price index during the era of booming house market. Taking into account the financial leverage effect, a collapse would occur when the unsustainable capital gain sunk below the interest rate. The CDS claims surged when the value of the insured securities declined. This triggered additional collateral requirements, and the stability and credit rating of AIG was undermined. From AIG’s case, we can see that it is of great importance to discuss the optimal debt level and amount of insurance liabilities an insurance company could offer. Meng et al. (2013) considered an optimal dividend problem with nonlinear insurance risk processes attributed to internal competition factors, and incorporated other important features such as the presence of debts and transaction costs. On the other hand, in insurance companies, insurers tend to accumulate relatively large amounts of cash, cash equivalents, and pursue capital gains in order to pay future claims and avoid financial ruin because of the nature of their insurance product. The payment of dividends to shareholders may reduce an insurer’s ability to survive adverse investment and underwriting experience. The study of optimizing the stream of dividend payments and management of surplus is a high priority task. Initiated in the work of De Finetti (1957), there have been increasing efforts on using advanced methods of stochastic control to study the optimal dividend policy; see Asmussen and Taksar (1997), Gerber and Shiu (2004), Gerber and Shiu (2006), Kulenko and Schimidli (2008), Yao et al. (2011) and Jin et al. (2013a). Moreover, to protect insurance companies against the impact on various risks, reinsurance is a standard tool with the goal of reducing and eliminating risk. The primary insurance carrier pays the reinsurance company a certain part of the premiums. In return, the reinsurance company is obliged to share the risk of large claims. Some recent work can be found in Asmussen et al. (2000), Bai and Guo (2008), Bai et al. (2008), Choulli et al. (2001), Pang (2006), Jin et al. (2013b), Wei et al. (2010), Zhang and Siu (2012) and Meng and Siu (2011) and references therein. A practitioner manages the reserve and dividend payment against future risks arising from the written CDSs by taking into account reinsurance tools.

The Cramér–Lundberg process (Lundberg, 1903) is inadequate to model the risk and return in our formulation for several reasons. First, classical Cramér–Lundberg process did not consider the surplus changes coming from the assets, which are held by insurance companies against the liabilities. The assets make income from the investment return and capital gains or losses that are represented in the second term in (2.4). Second, the correlation between the value of the claims against insurers that provide protections for CDSs and the value of the insured securities cannot be ignored. When the market value of the insured securities decline, the insurers either compensate the policyholders for the value difference or put up more collateral as requested, both of which will lead to surplus decrease. Hence, the value of the claims are highly negatively correlated with the value of the insured securities. Third, the assets in insurers’ portfolio are quite closely correlated to the insured securities. The dependence will increase complexity of the formulation in our problem. In addition, unlike the classical ruin problem or the Cramér–Lundberg approach, our criterion does not focus solely upon the probability of ruin. The criterion in our problem is to maximize the expectation of the discounted value of the utility of dividend until financial ruin under optimal liabilities and dividend strategies.

In this study, we choose different criteria and take into account the risk aversion level for different types of insurers to find the optimal capital requirement or leverage that balances risk against expected growth and return. The value function considered in this stochastic control problem has two variables, which represent the surplus and claim rate. The two control variables are debt ratio and dividend payment rate, respectively. By dynamic programming principle, the value function obeys a second order nonlinear partial differential equation (PDE) generally. Due to the nonlinearity, explicit solutions are generally not able to be obtained for this type of PDE. Fleming and Pang (2004) introduced a subsolution–supersolution method to obtain existence of classical solutions of the Hamilton–Jacobi–Bellman (HJB) equation. In our formulation, the stochastic control problem can be solved analytically. Under general assumptions, we prove the existence of classical solution. Moreover, we obtain the explicit form of the value function and corresponding optimal strategies in some special cases. An example is provided to illustrate the ideas and methodologies. The impact of reinsurance strategies on the debt management and dividend payment policies are clearly obtained from the analytical solutions of optimal controls.

The rest of the paper is organized as follows. A general formulation of asset value, debt, surplus, insurance liabilities, claim rates, dividend strategies, and assumptions are presented in Section  2. Section  3 deals with optimal debt ratio and dividend payment strategies in logarithm utilities. The subsolution–supersolution method are introduced, and the existence of classical solution of HJB equation is proved in Section  3.1. The verification theorem of optimal value function is presented in Section  3.2. Section  4 deals with optimal debt ratio and dividend payment strategies in power utilities. An example is provided in Section  5 and the impact of reinsurance strategies is considered. Finally, additional remarks are provided in Section  6.