ReviewThe concept of comonotonicity in actuarial science and finance: applications
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,…,Xn) has the comonotonic distribution, which means that each two possible outcomes (x1,x2,…,xn) and (y1,y2,…,yn) of (X1,X2,…,Xn) 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,…,Xn, identically distributed, with cdf F and finite variance σ2, say. If the risks are mutually independent, it is well known that as n goes to infinity. If the risks are comonotonic, then 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,…,Xn) 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.
Section snippets
Comonotonic rv’s
In this section, we will describe several situations in an actuarial or financial context where comonotonic rv’s emerge.
Convex bounds for sums of rv’s
In this section, we will illustrate the technique of deriving convex lower and upper bounds for sums of rv’s, as explained in Dhaene et al. (2002), by some numerical examples. Especially, we will consider sums of normal or lognormal rv’s (see also Kaas et al., 2001).
Recall that a random vector (Y1,Y2,…,Yn) has the multivariate normal distribution if and only if every linear combination of its variates has a univariate normal distribution. Now assume that (Y1,Y2,…,Yn) has a multivariate normal
Approximate evaluation of provisions
Consider a series of deterministic payments α1,α2,…,αn, of arbitrary sign, that are due at times 1,2,…,n, respectively. We want to find an answer to the following question: “What is the amount of money required at time 0 in order to be able to meet these future obligations (α1,α2,…,αn)?” We will call this amount the provision, or depending on the situation at hand, the (prospective) reserve or the required capital. Of course, the level of the provision will strongly depend on the way how this
Definitions and some theoretical results
Assume that we are currently at time 0. Consider a risky asset (a non-dividend paying stock) with prices described by the stochastic process {A(t),t≥0}, and a risk-free continuously compounded rate δ that is constant through time. In this section, all probabilities and expectations have to be considered as conditional on the information available at time 0, i.e. the prices of the risky asset up to time 0. Note that in general, the conditional expectation (with respect to the physical
Conclusions
In this paper, we demonstrated the usefulness of the concept of comonotonicity for describing dependencies between rv’s in several financial and actuarial applications. We showed that very tight upper bounds as well as lower bounds can be obtained using the techniques described in Dhaene et al. (2002) and in Kaas et al. (2000). It is shown how the techniques can be used to determine provisions for future payment obligations, taking into account the stochastic nature of the return process.
We
Acknowledgements
Michel Denuit, Jan Dhaene and Marc Goovaerts would like to acknowledge the financial support of the Committee on Knowledge Extension Research of the Society of Actuaries for the project “Actuarial Aspects of Dependencies in Insurance Portfolios”. The current paper and also Dhaene et al. (2002) result from this project.
Marc Goovaerts and Jan Dhaene also acknowledge the financial support of the Onderzoeksfonds K.U. Leuven (GOA/02: Actuariële, financiële en statistische aspecten van
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