Samuel H. Cox

Natural Hedging of Life and Annuity Mortality Risks

Abstract The values of life insurance and annuity liabilities move in opposite directions in response to a change in the underlying mortality. Natural hedging utilizes this to stabilize aggregate liability cash flows. We find empirical evidence that suggests that annuity writing insurers who have more balanced business in life and annuity risks also tend to charge lower premiums than otherwise similar insurers. This indicates that insurers who have a natural hedge have a competitive advantage. In addition, we show how a mortality swap might be used to provide the benefits of natural hedging.

Modeling Mortality With Jumps: Applications to Mortality Securitization

In this article, we incorporate a jump process into the original Lee–Carter model, and use it to forecast mortality rates and analyze mortality securitization. We explore alternative models with transitory versus permanent jump effects and find that modeling mortality via transitory jump effects may be more appropriate in mortality securitization. We use the Swiss Re mortality bond in 2003 as an example to show how to apply our model together with the distortion measure approach to value mortality-linked securities. Pricing the Swiss Re mortality bond is challenging because the mortality index is correlated across countries and over time. Cox, Lin, and Wang (2006) employ the normalized multivariate exponential tilting to take into account correlations across countries, but the problem of correlation over time remains unsolved. We show in this article how to account for the correlations of the mortality index over time by simulating the mortality index and changing the measure on paths.

Mortality Regimes and Pricing

Abstract Mortality dynamics are characterized by changes in mortality regimes. This paper describes a Markov regime-switching model that incorporates mortality state switches into mortality dynamics. Using the 1901-2005 U.S. population mortality data, we illustrate that regime-switching models can perform better than well-known models in the literature. Furthermore, we extend the 1992 Lee-Carter model in such a way that the time-series common risk factor to all cohorts has distinct mortality regimes with different means and volatilities. Finally, we show how to price mortality securities with this model.

Insurance Futures and Hedging Insurance Price Risk

This article discusses the Chicago Board of Trades proposal to establish a market for futures contracts (and options on futures) written on insurance business. A European call option on an insurance future is equivalent to a stop-loss reinsurance contract on the portfolio of insurance policies underlying the futures contract. Put options correspond to the discounted expected value of retained losses. After a credible experience record is developed, an insurer can determine the correlation of its own portfolio of policies with the market portfolio and use the futures options market as a partial substitute for traditional stop-loss reinsurance. The advantages of such a market are that it provides a tool for hedging business risk, it allows an entity to participate in the market portfolios profitability without being a licensed insurer, it may have lower transactions costs than traditional reinsurance, and it provides a mechanism for price discovery. But, despite the advantages of an insurance futures market, there are serious barriers to its success, which are discussed briefly.

Economic Aspects of Securitization of Risk

This paper explains securitization of insurance risk by describing its essential components and its economic rationale. We use examples and describe recent securitization transactions. We explore the key ideas without abstract mathematics. Insurance-based securitizations improve opportunities for all investors. Relative to traditional reinsurance, securitizations provide larger amounts of coverage and more innovative contract terms. This paper explains securitization of risk with an emphasis on risks that are usually considered insurable risks. We discuss the economic rationale for securitization of assets and liabilities and we provide examples of each type of securitization. We also provide economic axguments for continued future insurance-risk securitization activity. An appendix indicates some of the issues involved in pricing insurance risk securitizations. We do not develop specific pricing results. Pricing techniques are complicated by the fact that, in general, insurance-risk based securities do not have unique prices based on axbitrage-free pricing considerations alone. The technical reason for this is that the most interesting insurance risk securitizations reside in incomplete markets. A market is said to be complete if every pattern of cash flows can be replicated by some portfolio of securities that are traded in the market. The payoffs from insurance-based securities, whose cash flows may depend on

Mortality Portfolio Risk Management

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 insurers 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.

Managing Capital Market and Longevity Risks in a Defined Benefit Pension Plan

This paper proposes a model for a defined benefit pension plan to minimize total funding variation while controlling expected total pension cost and funding downside risk throughout the life of a pension cohort. With this setup, we first investigate the plan’s optimal contribution and asset allocation strategies, given the projection of stochastic asset returns and random mortality evolutions. To manage longevity risk, the plan can use either the ground-up hedging strategy or the excess-risk hedging strategy. Our numerical examples demonstrate that the plan transfers more unexpected longevity risk with the excess-risk strategy due to its lower total hedge cost and more attractive structure.

Insurance Versus Self-insurance: A Risk Management Perspective

The scientific risk-retention or self-insurance decision from a utility theoretic point of view is examined under the assumption that the risk manager has only partial stochastic information about the loss severity distribution. When only the range and a few central moments are known about the loss severity distribution, it is shown how to obtain maximally tight bounds on the expected utility of self insurance. Significantly, the extremal probability distributions derived do not depend upon the particular decision makers utility function and, therefore, should be applicable to a wide variety of financial decisions. Moreover, financial and/or risk managers in a business can make decisions without assessing the preference structure of the firms owners.

Longevity Risk and Capital Markets: The 2009-2010 Update

As populations in countries around the world age, governments, corporations, and individuals face increasing longevity risk. Pay-as-you-go state pensions and corporate pension plans are putting severe financial pressures on governments and companies; IBM and Verizon are just two of many recent corporate examples in the United States; British Airways and the Co-op are two current examples in the United Kingdom. Fertility rates have also fallen in many countries around the world and this, in conjunction with increased longevity, has caused the inversion of some countries’ age distributions and so increased the severity of the longevity risk problem for pay-asyou-go government pension plans by both reducing the tax base and extending the payout period. Mortality improvements at older ages have also increased the severity and make it ever more likely that individuals with inadequate pension arrangements will require other tools to manage their longevity risk. Longevity risk exists at an individual and aggregate level. For the individual, it is the risk of outliving one’s accumulated wealth. In the aggregate, it is the risk that the average member of a birth cohort will live longer than expected. Tools have long existed for the management of individual longevity risk; some of the modern means include social security systems provided by governments, defined benefit plans provided by corporations through pension funds, and life annuities 1 provided to individuals by insurers. 2 The Law of Large Numbers would suffice to make longevity risk manageable for pension funds and insurers in the absence of the aggregate longevity risk

Portfolio Risk Management with CVaR-Like Constraints

Abstract A current research stream in the portfolio allocation literature develops models that take into account the asymmetric nature of asset return distributions. Our paper contributes to this research stream by extending the Krokhmal, Palmquist, and Uryasev approach. We add CVaR-like constraints in the traditional portfolio optimization problem to reshape the tails of the portfolio return distribution while not significantly affecting its mean and variance. We illustrate how to apply this approach, called the “MV + CVaR approach,” to manage tail risk of an insurer’s asset-liability portfolio. Finally, we compare the MV + CVaR approach with the traditional Markowitz method and a method recently introduced by Boyle and Ding. Our numerical analysis provides empirical support for the effectiveness of the MV + CVaR approach in controlling downside risk. Moreover, we find that the MV + CVaR approach may improve skewness of mean-variance portfolios, especially for high-variance portfolios.