The concept of comonotonicity in actuarial science and finance: applications

Abstract

In an insurance context, one is often interested in the distribution function of a sum of random variables (rv’s). Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio, at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not a realistic one. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined approximations for sums of rv’s, when the distributions of the components are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. Practical applications of this theory will be considered in this paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

Introduction

In Dhaene et al. (2002), we presented an overview of the actuarial literature on the problem how to make decisions in case we have a sum of random variables (rv’s) with given marginal distribution functions but of which the stochastic dependence structure is unknown or too cumbersome to work with. We proved that the convex-largest sum of the components of a random vector with given marginals is obtained in case the random vector (X₁,X₂,…,X_n) has the comonotonic distribution, which means that each two possible outcomes (x₁,x₂,…,x_n) and (y₁,y₂,…,y_n) of (X₁,X₂,…,X_n) are ordered componentwise.

In this paper, we will present several applications of the concept of comonotonicity in the field of actuarial science and finance. The notations, assumptions and results used throughout this paper are presented in the above mentioned companion paper and will not be repeated here. References to equations and theorems presented in the first paper will be denoted by adding a “T” to the relevant equation or theorem number.

As a theoretical example of the concept of comonotonicity in an insurance context, consider a portfolio of n risks X₁,X₂,…,X_n, identically distributed, with cdf F and finite variance σ², say. If the risks are mutually independent, it is well known that

$$\text{Var}\text{1}\text{n}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{=}\text{σ}{}^2\text{n}\text{→0}$$

as n goes to infinity. If the risks are comonotonic, then

$$\text{Var}\text{1}\text{n}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{=}\text{Var}\text{1}\text{n}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{F}{}^{\mathrm{-1}}\text{(U)}\text{=σ}{}^2\text{,}$$

where U is uniformly distributed on [0,1]. Hence, in case of comonotonic risks, risk pooling has completely no risk reducing effect: adding an additional risk to the portfolio will not reduce the variance of the average risk.

In general, the risks of an insurance portfolio (X₁,X₂,…,X_n) will not exhibit the extreme comonotonic dependence structure. However, in the presence of positive dependencies between the individual risks, assuming independence might lead to an underestimation of the probability of large total claims for the portfolio. In this case, the technique of risk pooling might not be as effective as expected. On the other hand, resorting to comonotonicity is a conservative approach in case the structure of dependence is unknown to the actuary.

In Section 2, we will give examples of comonotonic rv’s occurring in an actuarial or financial environment. In Section 3, we give some numerical examples how to construct convex lower and upper bounds for sums of rv’s. The evaluation of cash-flows in case of a lognormal discount process is considered in Section 4. In Section 5, we derive lower and upper bounds for the price of arithmetic Asian options.