Academic journal article Journal of Risk and Insurance

Living with Ambiguity: Pricing Mortality-Linked Securities with Smooth Ambiguity Preferences

Academic journal article Journal of Risk and Insurance

Living with Ambiguity: Pricing Mortality-Linked Securities with Smooth Ambiguity Preferences

Article excerpt

ABSTRACT

Mortality is a stochastic process. We have imprecise knowledge about the probability distribution of mortality rates in the future. Mortality risk, therefore, can be defined in a broader term of ambiguity. In this article, we investigate the effects of ambiguity and ambiguity aversion on prices of mortality-linked securities. Ambiguity may arise from parameter uncertainty due to a finite sample of data and inaccurate old-age mortality rates. We compare the price of a mortality bond in three scenarios: (1) no parameter uncertainty, (2) parameter uncertainty with Bayesian updates, and (3) parameter uncertainty with the smooth ambiguity preference. We use the indifference pricing approach to derive the minimum ask price and the maximum bid price, and adopt the economic pricing method to compute the equilibrium price that clears the market. We reveal the connection between the indifference pricing approach and the economic pricing approach and find that ambiguity aversion has a much smaller effect on prices of mortality-linked securities than risk aversion in our example.

INTRODUCTION

Mortality risk is twofold. On the one hand, longevity risk from uncertain mortality improvements in the future has been significant over many years. Life expectancy for men aged 60 was 5 years longer in 2005 than anticipated in mortality projections made in the 1980s (Hardy, 2005). Looking forward, possible changes in lifestyle, medical advances, and new discoveries in genetics make future improvements to life expectancy highly unpredictable. On the other hand, adverse mortality risk, which is usually caused by catastrophic events, results in much higher mortality rates than would normally be experienced. The 1918 Spanish flu killed up to 50 million people worldwide and 500,000 in the United States (Rasmussen, 2005). The massive 9.0 earthquake/tsunami that struck Japan on March 11, 2011 brought about 15,698 deaths and 4,666 missing. (1) To sum up, mortality is a stochastic process: it is improving in an unpredictable way. We have imprecise knowledge about the probability distribution of future mortality rates. Therefore, it is appropriate to define mortality/longevity risk in a more general term of ambiguity in the sense of Knight (1921). (2)

In this article, we explore the effects of risk aversion and ambiguity aversion on mortality risk modeling and pricing. We introduce the smooth ambiguity preference into the indifference pricing approach and the economic pricing approach to compute a possible price range and the equilibrium price of a mortality-linked security. We elucidate how these two pricing methods are intrinsically connected. Our comparative analysis illustrates that ambiguity aversion impacts prices in a similar way to risk aversion but in a much smaller magnitude in our example.

Mortality models are fundamental to quantify mortality/longevity risk and provide the basis of pricing and reserving. A wide variety of stochastic mortality models have been proposed. (3) The prices of mortality-linked securities based on these mortality models are subject to two types of uncertainties. (4) The first is the uncertainty from model misspecification. The stochastic mortality model itself may be inaccurate or even incorrect, so it cannot perfectly describe the evolution of mortality dynamics. The second is the uncertainty from parameter estimation. Even if the model itself is correct, parameter estimates may be imprecise due to a finite sample size and missing old-age mortality data. Parameter uncertainty is unavoidable in any model-based approach (Li and Ng, 2011).

A common way to deal with parameter uncertainty is the standard Bayesian approach. Basically, one regards the unknown parameter as a random variable and updates his/her prior belief about the distribution of the parameter with observed signals to form a posterior distribution. The individual then makes decisions by maximizing his/her expected utility under this posterior distribution. …

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