Optimal investment and risk control policies for an insurer: Expected utility maximization
Introduction
The financial crisis of 2007–2008 caused a significant recession in global economy, considered by many economists to be the worst financial crisis since the Great Depression of the 1930s. It resulted in the threat of bankruptcy of large financial institutions, the bailout of banks, and downturns in stock markets around the world (see more on Wikipediahttp://en.wikipedia.org/wiki/Financial_crisis_of_2007-08). American International Group, Inc. (AIG), once the largest insurance company in the United States with a triple-A credit rating, collapsed within a few months in 2008. The stock price of AIG was traded at over $50 per share in February 2008, but plunged down to less than $1 per share when AIG was on the brink of bankruptcy. The severity of AIG’s liquidity crisis led to an initial rescue of $85 billion and a total of $182 billion bailout by the U.S. government, the largest government bailout in history (see Stein (2012, Chapter 6) for more statistical data of AIG during the past financial crisis and Sjostrom (2009) for detailed discussions on the AIG bailout case). According to Stein (2012, Chapter 6), AIG made several major mistakes which together contributed to its sudden collapse. First, AIG underpriced the risk of writing Credit Default Swap (CDS) contracts since it ignored the negative correlation between its liabilities and the capital gains in the financial market. Second, AIG applied a problematic model for risk management and misunderstood the impact of derivatives trading on the capital structure. To address these issues, we propose a jump-diffusion process to model an insurer’s risk (per policy risk) and consider optimal investment and risk control problem from an insurer’s view. Our research has two roots in the literature: optimal consumption and investment problem and optimal reinsurance (risk control) problem.
Merton (1969) was the first to apply stochastic control theory to solve consumption and investment problem in continuous time. Most major generalizations to Merton’s work can be found in the books of Karatzas (1996), Karatzas and Shreve (1998), Sethi (1997), et cétera. Zhou and Yin (2004), and Sotomayor and Cadenillas (2009) considered consumption/investment problem in a financial market with regime switching. They obtained explicit solutions under the mean–variance criterion and the utility maximization criterion, respectively. Moore and Young (2006) incorporated an external risk process (which can be insured against by purchasing insurance policy) into Merton’s framework and studied optimal consumption, investment and insurance problem. Following Moore and Young (2006), Perera (2010) revisited the same problem in a more general Levy market. Along the same vein, many researchers added an uncontrollable risk process to Merton’s model. They considered a stochastic control problem for optimal investment strategy (mostly without consumption) under certain criteria. For instance, Browne (1995) modeled the risk by a continuous diffusion process and studied optimal investment problem under two different criteria: maximizing expected exponential utility of terminal wealth and minimizing the probability of ruin. Wang et al. (2007) applied a jump-diffusion model for the risk process and considered optimal investment problem under the utility maximization criterion.
The second root of our research is optimal reinsurance problem, which studies an insurer who wants to control the reinsurance payout for certain objectives. Reinsurance is an important tool for insurance companies to manage their risk exposure. The classical model for risk in the insurance literature is Cramér–Lundberg model, which uses a compound Poisson process to measure risk. The Cramér–Lundberg model was introduced by Lundberg in 1903 and then republished by Cramér in 1930s. Since the limiting process of a compound Poisson process is a diffusion process, see, e.g., Taksar (2000), recent research began to model risk by a diffusion process or a jump-diffusion process, see, e.g., Wang et al. (2007). Hojgaard and Taksar (1998) assumed the reserve of an insurance company is governed by a diffusion process and considered optimal proportional reinsurance problem under the criterion of maximizing expected utility of running reserve up to bankruptcy. Kaluszka (2001) studied optimal reinsurance in discrete time under mean–variance criterion for both proportional reinsurance and step-loss reinsurance. Schmidli (2001) considered both the Cramér–Lundberg model and the diffusion model and obtained optimal proportional reinsurance policy when the insurer’s objective is to minimize the probability of ruin. Recent generalizations in modeling for optimal reinsurance problem include incorporating regime switching, see Zhuo et al. (2013), and interest rate risk and inflation risk, see Guan and Liang (2014).
In mathematics, there are two main tools for solving stochastic control problems. The first tool is dynamic programming and maximum principle, see, for instance, Fleming and Soner (1993) and Cadenillas and Karatzas (1995). The second tool is martingale approach, based on equivalent martingale measures and martingale representation theorems. The martingale approach and its application in continuous time finance were developed by Harrison and Kreps (1979). Thereafter, the martingale method has been applied to solve numerous important problems in economics and finance, such as option pricing problem in Harrison and Pliska (1981), optimal consumption and investment problem in Karatzas (1996, Chapter 2), Karatzas and Shreve (1998, Chapter 6), optimal consumption, investment and insurance problem in Perera (2010) and optimal investment problem in Wang et al. (2007). In this paper, we also apply the martingale approach to solve our stochastic control problem.
Our model and optimization problem are different from the existing ones in the literature in several directions. Comparing with Merton’s framework and its generalizations, we add a controllable jump-diffusion process into the model, which will be used to model an insurer’s risk. We then regulate the insurer’s risk by controlling the number of polices. So our model is also different from the ones considered in optimal reinsurance problem and its variants, which control risk by purchasing reinsurance policies from another insurer. As suggested in Stein (2012, Chapter 6), we assume there exists negative correlation between the insurer’s risk and the capital gains in the financial market. Stein (2012, Chapter 6) considered a similar risk regulation problem as ours, but in his model, the investment strategy is fixed, and the insurer’s risk is modeled by a diffusion process. To generalize Stein’s work, we use a jump-diffusion process to model risk and allow the insurer to select investment strategy continuously. Stein (2012, Chapter 6) considered the problem only with logarithmic utility function, which can be easily solved using classical stochastic method. We obtain explicit solutions to optimal investment and risk control problem for various utility functions, including hyperbolic absolute risk aversion (HARA) utility function (logarithmic function and power function), constant absolute risk aversion (CARA) utility function (exponential function), and quadratic utility function.
The structure of this paper is organized as follows. We describe the model and formulate optimal investment and risk control problem in Section 2. We obtain explicit solutions of optimal investment and risk control strategies for logarithmic utility function in Section 3, power utility function in Section 4, exponential utility function in Section 5, and quadratic utility function in Section 6, respectively. In Section 7, we conduct an economic analysis to study the impact of the market parameters on the optimal policies. We conclude our study in Section 8.
Section snippets
The model
In the financial market, there are two assets available for investment, a riskless asset with price process and a risky asset (stock) with price process . On a filtered probability space , the dynamics of and are given by where and are positive bounded functions and is a standard Brownian motion. The initial conditions are and .
For an insurer like AIG, its main liabilities (risk) come from
The analysis for
We first consider Problem 2.1 when the utility function is given by , which belongs to the class of hyperbolic absolute risk aversion (HARA) utility functions.
We choose as control and denote as the set of all admissible controls when . For every is progressively measurable with respect to the filtration and , satisfies the following conditions, Furthermore, to avoid the possibility of bankruptcy
The analysis for
The second utility function we consider is power function, which also belongs to HARA class. Here, we choose as the admissible set for Problem 2.1.
Since , we define as where is a stopping time and almost surely. With the help of , we define a new probability measure by .
From the SDE (2), we obtain
Thanks to Remark 3.1, also bears
The analysis for
In this section, we consider Problem 2.1 for exponential utility function, which is of constant absolute risk aversion (CARA) class. In this case, we use control and define the admissible set as follows: for any admissible control is progressively measurable with respect to the filtration , and satisfies the integrability conditions and .
By Lemma 3.1, optimal control should satisfy the following
The analysis for
As pointed in Wang et al. (2007), to find a mean–variance portfolio strategy is equivalent to maximize expected utility for a quadratic function. So in this section, we consider a quadratic utility function, and solve Problem 2.1 with admissible set . Notice that the quadratic utility function considered in this section is not strictly increasing for all , but rather has a maximum point at . This means if investor’s wealth is greater than the maximum point, he/she will experience a
Economic analysis
In this section, we analyze the impact of the market parameters on the optimal policies in three cases: logarithmic utility function, power utility function and exponential utility function. To conduct the economic analysis, we assume the coefficients in the financial market are constants and select the market parameters as given in Table 1. Notice that in Table 1, three variables: and , have not been assigned fixed values. We shall analyze the impact of those three variables on the
Conclusions
Motivated by the bailout case of AIG in the financial crisis of 2007–2008 and the increasing demand on risk management in the insurance industry, we consider optimal investment and risk control problem for an insurer (like AIG). In our model, the insurer’s risk follows a jump-diffusion process, which can be controlled proportionally by the insurer. As discussed in Stein (2012, Chapter 6), one major mistake AIG made is ignore the negative correlation between its liabilities (risk) and the
Acknowledgments
The work of B. Zou and A. Cadenillas was supported by the Natural Sciences and Engineering Research Council of Canada (Grant 194137-2010). We are grateful to the referees for suggestions to improve our paper. Existing errors are our sole responsibility.
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