Shaun Wang

PREMIUM CALCULATION BY TRANSFORMING THE LAYER PREMIUM DENSITY

This paper examines a class of premium principles which are (i) comonotonic additive and (ii) preserving stochastic dominance. The representation for this class is a transform on the decumulative distribution function. It has close connections with the recent developments in economic decision theory (e.g. Yaari, 1987). The proportional hazard transform may provide an alternative to the variance as a risk measure.

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.

Insurance pricing and increased limits ratemaking by proportional hazards transforms

Abstract This paper proposes a new premium principle, where risk loadings are imposed by a proportional decrease in the hazard rates. This premium principle is scale invariant and additive for layers. It is shown that this principle will generate stop-loss contracts as optimal reinsurance arrangements in a competitive market when the reinsurer is less risk-averse than the direct insurer. Finally, increased limits factors are calculated based on this principle.

A Universal Framework for Pricing Financial and Insurance Risks

This paper presents a universal framework for pricing financial and insurance risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can be either traded or not traded. The paper also outlines an application of the framework to prescribe capital allocations within insurance companies, and to determine fair values of insurance liabilities.

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.

On the Stability of Recursive Formulas

Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by PANJER (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

Exponential and scale mixtures and equilibrium distributions

Abstract In this article we discuss mixed exponential distributions and, more generally, scale mixtures with specific consideration the purpose of insurance modeling. Results are derived for equilibrium distributions (defined via stop-loss transforms) of mixed distributions. Some recursive relations are identified for the stop-loss transforms and moments of mixed exponential distributions. Explicit expressions are obtained for equilibrium gamma distributions with arbitrary shape parameter.

Equilibrium Pricing Transforms: New Results Using Buhlmann’s 1980 Economic Model

In this paper we revisit an economic model of Buhlmann (ASTIN Bulletin, 1980) and derive equilibrium pricing transforms. We obtain the Esscher Transform and the Wang Transform under different sets of assumptions on the aggregate economic environment. We show that the Esscher Transform and the Wang Transform exhibit very different behaviors when used in pricing insurance risks.

Risk-adjusted credibility premiums using distorted probabilities

Abstract Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.

Normalized Exponential Tilting: Pricing and Measuring Multivariate Risks

Abstract This article discusses methods of risk-neutralizing multivariate probability distributions by applying exponential tilting to the joint probability density function with respect to a set of reference risks. To ensure consistent interpretations of the exponential tilting parameters, a normalization procedure is performed on the reference risks via percentile mapping to standard normal variables. The article establishes links between normalized exponential tilting and multivariate probability distortions. It provides efficient methods for computing risk-neutralized multivariate probability distributions, and it gives illustrative examples in pricing contingent claims on multiple risks.Abstract This article discusses methods of risk-neutralizing multivariate probability distributions by applying exponential tilting to the joint probability density function with respect to a set of reference risks. To ensure consistent interpretations of the exponential tilting parameters, a normalization procedure is performed on the reference risks via percentile mapping to standard normal variables. The article establishes links between normalized exponential tilting and multivariate probability distortions. It provides efficient methods for computing risk-neutralized multivariate probability distributions, and it gives illustrative examples in pricing contingent claims on multiple risks.