Virginia R. Young

Axiomatic characterization of insurance prices

Abstract In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability (Yaari, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.

Ordering risks: Expected utility theory versus Yaari's dual theory of risk

Abstract We introduce a class of partial orderings of risks that are dual to stochastic dominance orderings. These arise as “distortion-free” orderings in Yaaris dual theory of risk (1987). We show that these dual orderings are equivalent to inverse stochastic dominance orderings (Muliere and Scarsini, 1989). We motivate third dual stochastic dominance via insurance economics, while providing an alternative interpretation for second (dual) stochastic dominance. We apply dual stochastic dominance to actuarial science and show how the dual ordering of risks is related to ordering income distributions in the economics of income inequality.

Optimal insurance under Wang’s premium principle

Abstract Wang et al. (1997) [Axiomatic characterization of insurance prices. Insurance: Mathematics & Economics 21(2), 173–183] propose axioms for pricing insurance that characterize the premium principle of Wang (1996) [Premium calculation by transforming the layer premium density. ASTIN Bulletin 26, 71–92]. Under this premium principle, the price to insure a given risk is the expectation of the risk with respect to a distorted probability. In this paper, we assume that prices are given by Wang’s premium principle. We determine the optimal indemnity contract for a risk-averse buyer who acts to maximize expected utility. Deprez and Gerber (1985) [On convex principles of premium calculation. Insurance: Mathematics & Economics 4, 179–189] describe the optimal insurance for convex premium principles that are Gâteaux differentiable. Wang’s premium principle is convex, but it is not Gâteaux differentiable; thus, we extend the work of Deprez and Gerber (1985) to this special case.

Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin

Abstract I study the problem of how individuals should invest their wealth in a risky financial market to minimize the probability that they outlive their wealth, also known as the probability of lifetime ruin. Specifically, I determine the optimal investment strategy of an individual who targets a given rate of consumption and seeks to minimize the probability of lifetime ruin. Two forms of the consumption function are considered: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of his or her wealth. The first is arguably more realistic, but the second has a close connection with optimal consumption in Merton’s model of optimal consumption and investment under power utility. For constant force of mortality, I determine (a) the probability that individuals outlive their wealth if they follow the optimal investment strategy; (b) the corresponding optimal investment rule that tells individuals how much money to invest in the risky asset for a given wealth level; (c) comparative statics for the functions in (a) and (b); (d) the distribution of the time of lifetime ruin, given that ruin occurs; and (e) the distribution of bequest, given that ruin does not occur. I also include numerical examples to illustrate how the formulas developed in this paper might be applied.

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preferences and yields two reservation prices - one each for the underwriter and buyer of the contract. The framework is rather general and applies to a number of applications that we extensively analyze.

Asset Allocation and Annuity-Purchase Strategies to Minimize the Probability of Financial Ruin

In this paper, we derive the optimal investment and annuitization strategies for a retiree whose objective is to minimize the probability of lifetime ruin, namely the probability that a fixed consumption strategy will lead to zero wealth while the individual is still alive. Recent papers in the insurance economics literature have examined utility-maximizing annuitization strategies. Others in the probability, finance, and risk management literature have derived shortfall-minimizing investment and hedging strategies given a limited amount of initial capital. This paper brings the two strands of research together. Our model pre-supposes a retiree who does not currently have sufficient wealth to purchase a life annuity that will yield her exogenously desired fixed consumption level. She seeks the asset allocation and annuitization strategy that will minimize the probability of lifetime ruin. We demonstrate that because of the binary nature of the investors goal, she will not annuitize any of her wealth until she can fully cover her desired consumption with a life annuity. We derive a variational inequality that governs the ruin probability and the optimal strategies, and we demonstrate that the problem can be recast as a related optimal stopping problem which yields a free-boundary problem that is more tractable. We numerically calculate the ruin probability and optimal strategies and examine how they change as we vary the mortality assumption and parameters of the financial model. Moreover, for the special case of exponential future lifetime, we solve the (dual) problem explicitly. As a byproduct of our calculations, we are able to quantify the reduction in lifetime ruin probability that comes from being able to manage the investment portfolio dynamically and purchase annuities.

A Longitudinal Data Analysis Interpretation of Credibility Models

Abstract In this paper, we develop links between credibility theory in actuarial science and longitudinal data models in statistics. Our primary contribution to actuarial science is to demonstrate that many additive credibility models can be expressed as special cases of the longitudinal data model. We, thereby, unify the many existing credibility models with this framework. In addition, a longitudinal data interpretation suggests additional models and techniques that actuaries can use in credibility ratemaking. We also apply standard statistical software, which has been developed to analyze longitudinal data models, to the private passenger automobile data of Hachemeister [Hachemeister, C.A., 1975. Credibility for regression models with applications to trend. In: Kahn, P.M. (Ed.), Credibility: Theory and Applications. Academic Press, New York, pp. 129–163].

Insurance Rate Changing: A Fuzzy Logic Approach

This article describes how fuzzy logic can be used to make insurance pricing decisions that consistently consider supplementary data, including vague or linguistic objectives of the insurer. The theory of fuzzy logic was developed in the 1970s to improve the accuracy and efficiency of expert systems, and through it, one can account for vague notions whose boundaries are not clearly defined. Using group health insurance data from an insurance company, I show how to build and fine-tune fuzzy logic models for changing rates to reflect supplementary data.

Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio

We provide an overview of how the law of large numbers breaks down when pricing life-contingent claims under stochastic as opposed to deterministic mortality (probability, hazard) rates. In a stylized situation, we derive the limiting per-policy risk and show that it goes to a non-zero constant. This is in contrast to the classical situation when the underlying mortality decrements are known with certainty, per policy risk goes to zero. We decompose the standard deviation per policy into systematic and non-systematic components, akin to the analysis of individual stock (equity) risk in a Markowitz portfolio framework. Finally, we draw upon the financial analogy of the Sharpe Ratio to develop a premium pricing methodology under aggregate mortality risk.