Mortality Portfolio Risk Management
Cox, Samuel H., Lin, Yijia, Tian, Ruilin, Zuluaga, Luis F., Journal of Risk and Insurance
We provide a new method, the "MV + CVaR approach," for managing unexpected mortality changes underlying annuities and life insurance. The MV + CVaR approach optimizes the mean-variance trade-off of an insurer's mortality portfolio, subject to constraints on downside risk. We apply the method of moments and the maximum entropy method to analyze the efficiency of MV + CVaR mortality portfolios relative to traditional Markowitz mean-variance portfolios. Our numerical examples illustrate the superiority of the MV + CVaR approach in mortality risk management and shed new light on natural hedging effects of annuities and life insurance.
Life insurance companies sell a wide variety of life insurance and annuity products. The insurers' liabilities for these products depend on future mortality rates. During recent years, economic and demographic changes have made mortality projection and risk management more important than ever. On the one hand, life expectancy for ages 60 and older in the past two decades has improved at a much higher rate than what pension plans and annuity providers expected. Cowling and Dales (2008) find that companies in the United Kingdom FTSE100 index underestimated their aggregate pension liabilities by more than 40 billion pounds sterling]. If the firms do not take measures to control mortality downside risk, such longevity shocks are likely to cause serious financial consequences. For example, unanticipated mortality improvement was an important factor accounting for the failure of Equitable Life, once a highly regarded U.K. life insurer (Ombudsman, 2008). On the other hand, population growth, urbanization, and increased global mobility may lead to a more rapid and widespread disease. Genetic analysts recently confirmed that today's "bird flu" is similar to the 1918 "Spanish flu" that killed more than 40 million people. This finding spurs fears of a worldwide epidemic (Juckett, 2006). According to Toole (2007), losses due to a severe pandemic could amount to 25 percent of the U.S. life insurance industry's statutory capital. While the great majority of U.S. life insurance companies would weather such a pandemic, it is clear that these companies should be interested in mitigating the risk.
We propose a method that life insurance companies can use to alleviate extreme mortality outcomes while maintaining a relatively efficient mean-variance relationship for their mortality portfolios of life insurance and/or annuities. This method, the "MV+CVaR approach," combines Markowitz mean-variance (MV) portfolio theory and conditional value at risk (CVaR) by optimizing the trade-off between mean and variance subject to an upper bound on CVaR. Variance measures both positive and negative deviations of portfolio values from its expected level, while CVaR focuses on the portfolio tail loss caused by extreme events. Although the MV+CVaR portfolios are suboptimal relative to the Markowitz counterparts in terms of the mean-variance efficiency, they are attractive to insurers since the MV+CVaR portfolios have lower downside risk while achieving desirable risk-return trade-offs. In practice, life insurers are keenly interested in searching for an optimal risk-return relationship for their business. At the same time they are required to meet various solvency requirements for possible catastrophes such as flu epidemics. Therefore, incorporating both variance and CVaR as risk measures in business optimization such as the MV+CVaR approach should be appealing to life insurance companies. The risk control framework adopted in this article closely follows that of Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010). In their framework, firms minimize portfolio losses subject to CVaR constraints. In our context, we consider both variance and CVaR as risk measures by incorporating a CVaR constraint into the classical mean-variance setup.
We extend Rockafellar and Uryasev (2000) and Tsai, Wang, and Tzeng (2010) in one important dimension by applying the well-developed moments method to validate the quality of MV+CVaR portfolios. …