Review
The concept of comonotonicity in actuarial science and finance: theory

https://doi.org/10.1016/S0167-6687(02)00134-8Get rights and content

Abstract

In an insurance context, one is often interested in the distribution function of a sum of random variables. 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 realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent 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 traditional risk theory, the individual risks of a portfolio are usually assumed to be mutually independent. Standard techniques for determining the distribution function of aggregate claims, such as Panjer’s recursion, De Pril’s recursion, convolution or moment-based approximations, are based on the independence assumption. Insurance is based on the fact that by increasing the number of insured risks, which are assumed to be mutually independent and identically distributed, the average risk gets more and more predictable because of the Law of Large Numbers. This is because a loss on one policy might be compensated by more favorable results on others. The other well-known fundamental law of statistics, the Central Limit Theorem, states that under the assumption of mutual independence, the aggregate claims of the portfolio will be approximately normally distributed, provided the number of insured risks is large enough. Assuming independence is very convenient since the mathematics for dependent risks are less tractable, and also because, in general, the statistics gathered by the insurer only give information about the marginal distributions of the risks, not about their joint distribution, i.e. the way these risks are interrelated.

A trend in actuarial science is to combine the (actuarial) technical risk with the (financial) investment risk. Assume that the nominal random payments Xi are due at fixed and known times ti, i=1,2,…,n. Let Yt denote the nominal discount factor over the interval [0,t], t≥0. This means that the amount one needs to invest at time 0 to get an amount 1 at time t is the random variable Yt. By nominal we mean that there is no correction for inflation. In this case, a random variable of interest will be the scalar product of two random vectors: S=∑i=1nXiYti.If the payments Xi at time ti are independent of inflation, then the vectors X=(X1,X2,…,Xn) and Y=(Yt1,Yt2,…,Ytn) can be assumed to be mutually independent. On the other hand, if the payments are adjusted for inflation, the vectors X and Y are not mutually independent anymore. Denoting the inflation factor over the period [0,t] by Zt, the random variable S can, in this case, be rewritten as S=∑i=1nXiYti′, where the real payments Xi′ and the real discount factors Yti′ are given by Xi′=Xi/Zti and Yti′=YtiZti, respectively. Hence, in this case S is the scalar product of the two mutually independent random vectors (X1′,X2′,…,Xn′) and (Yt1′,Yt2′,…,Ytn′).

In general, however, each vector on its own will have dependent components. Especially, the factors of the discount vector will possess a strong positive dependence.

Introduction of the stochastic financial aspects in actuarial models immediately reveals the necessity of determining distribution functions of sums of dependent random variables. Hereafter we describe some situations where random variables, which are scalar products of two vectors, arise.

First, consider the random variable S=∑i=1nXiYi, where the Xi represent the claim amounts of one policy (or one portfolio) at different times i, i=1,2,…,n. Even if the discount factors Yi are deterministic, S will often be a sum of dependent random variables in this case. An example is a life annuity on a single life (x) which pays an amount equal to 1 at times 1,2,…,n provided the insured (x) is alive at that time. It is clear that the stochastic payments Xi possess a strong positive dependence in this case. Another example is the case of an individual automobile insurance policy where Xi represents the loss in year i of the policy under consideration. High values of X1 and X2 might indicate that the insured is a bad risk with high claim frequencies and/or severities also in the coming years.

In case of stochastic discount factors Yi, the sum S=∑i=1nXiYi will be a sum of strongly positive dependent random variables, where the dependence is also caused by the dependence between the Yi. Consider for instance Yi and Yi+j, with j small. Discounting over the period [0,i+j] is equal to discounting over the period [0,i]∪(i,i+j]. Hence, in any realistic model these discount factors Yi and Yi+j will possess a strong positive dependence.

Intuitively, in the presence of positive dependencies, large values of one term in a sum of random variables tend to go hand in hand with large values of the other terms. The Law of Large Numbers will not hold and the aggregate risk S will exhibit greater variation than in the case of a sum of mutually independent random variables. So in this case, the independence assumption tends to underestimate the tails of the distribution function of S.

Second, consider the case where the Xi represent the claims or gains/losses of the different policies in an insurance portfolio and that all ti are identical and equal to t. The random variable S=∑i=1nXiYt can then be interpreted as the aggregate claims of the portfolio over a certain reference period, for instance 1 year.

If the discount factor Yt is stochastic, then S is a sum of strongly positive dependent random variables as each individual random variable XiYt contains the same discount factor Yt.

If the discount factor Yt is assumed to be deterministic, then the independence assumption will often be not too far from reality, and can be used for determining the distribution of S. Moreover, one can force a portfolio of risks to satisfy the independence assumption as much as possible by diversifying, not including too many related risks like the fire risks of different floors of the same building or the risks concerning several layers of the same large reinsured risk.

In certain situations, however, the individual risks Xi will not be mutually independent because they are subject to the same claim generating mechanism or are influenced by the same economic or physical environment. The independence assumption is then violated and just is not an adequate way to describe the relations between the different random variables involved. The individual risks of an earthquake or flooding risk portfolio which are located in the same geographic area are correlated, since individual claims are contingent on the occurrence and severity of the same earthquake or flood. On a foggy day all cars of a region have higher probability to be involved in an accident. During dry hot summers, all wooden cottages are more exposed to fire. More generally, one can say that if the density of insured risks in a certain area or organization is high enough, then catastrophes such as storms, explosions, earthquakes, epidemics and so on can cause an accumulation of claims for the insurer. As a financial example, consider a bond portfolio. Individual bond default experience may be conditionally independent for given market conditions. However, the underlying economic environment (for instance interest rates) affects all individual bonds in the market in a similar way. In life insurance, there is ample evidence that the lifetimes of husbands and their wives are positively associated. There may be certain selection mechanisms in the matching of couples (“birds of a feather flock together”): both partners often belong to the same social class and have the same life style. Further, it is known that the mortality rate increases after the passing away of one’s spouse (the “broken heart syndrome”). These phenomena have implications on the valuation of aggregate claims in life insurance portfolios. Another example in a life insurance context is a pension fund that covers the pensions of persons working for the same company. These persons work at the same location, they take the same flights. It is evident that the mortality of these persons will be dependent, at least to a certain extent.

As a theoretical example, consider an insurance portfolio consisting of n risks. The payments to be made by the insurer are described by a random vector (X1,X2,…,Xn), where Xi is the claim amount of policy i during the insurance period. We assume that all payments have to be done at the end of the insurance period [0,1]. In a deterministic financial setting, the present value at time 0 of the aggregate claims X1+X2+⋯+Xn to be paid by the insurer at time 1 is determined by S=(X1+X2+⋯+Xn)v,where v=(1+r)−1 is the deterministic discount factor and r the technical interest rate. This will be chosen in a conservative way (i.e. sufficiently low), if the insurer does not want to underestimate his future obligations. To demonstrate the effect of introducing random interest on insurance business, we look at the following special case. Assume all risks Xi to be non-negative, independent and identically distributed, and let X=dXi, where the symbol =d is used to indicate equality in distribution. The average payment S/n has mean and variance ESn=vE[X],VarSn=v2nVar[X].The stability necessary for both insured and insurer is maintained by the Law of Large Numbers, provided that n is indeed ‘large’ and that the risks are mutually independent and rather well behaved, not describing for instance risks of catastrophic nature for which the variance might be very large or even infinite.

Now let us examine the consequences of introducing stochastic discounting. Replacing the fixed discount factor v by a random variable Y, representing the stochastic amount to be invested at time 0 with value 1 at the end of the period [0,1], the present value of the aggregate claims becomes S=(X1+X2+⋯+Xn)Y.If we assume that the discount factor is independent of the payments, we find that the average payment per policy S/n has mean and variance ESn=E[X]E[Y],VarSn=Var[X]nE[Y2]+(E[X])2Var[Y].Assuming that E[X] and Var[Y] are positive, the Law of Large Numbers no longer eliminates the risk involved. This is because for n→∞, Var[S/n] converges to its second term. So to evaluate the total risk, both the distributions of insurance risk and financial risk are needed. Risk pooling and large portfolios are no longer sufficient tools to eliminate or reduce the average risk associated with a portfolio. This observation implies that the introduction of stochastic financial aspects in actuarial models immediately leads to the necessity of determining distribution functions of sums of dependent random variables.

Under the assumption that the vectors X=(X1,X2,…,Xn) and Y=(Yt1,Yt2,…,Ytn) are mutually independent and that the marginal distributions of the Xi and the Yti are given, the problem of determining bounds for the distribution function of S=∑i=1nXiYti can be reduced to determining bounds for the distribution function of a sum S=Z1+Z2+⋯+Znof random variables Z1,Z2,…,Zn with given marginal distributions, but of which the joint distribution is either unspecified or too cumbersome to work with. The unknown or complex nature of the dependence between the random variables Zi is the reason why it is impossible to derive the distribution function of S exactly.

Recently, several authors in the actuarial literature have derived stochastic lower and upper bounds for sums S of this type. These bounds are bounds in the sense of convex order. The concept of convex order is closely related to the notion of stop-loss order which is more familiar in actuarial circles. Both stochastic orders express which of two random variables is the “less dangerous” one. Replacing S by a less attractive random variable S′ will be a safe strategy from the viewpoint of the insurer. Considering also “more attractive” random variables will help to give an idea of the degree of overestimation of the real risk.

In this paper, we will describe how to make safe decisions in case we have a sum of random variables with given marginal distribution functions but of which the stochastic-dependent structure is unknown. We will give an overview of the recent actuarial literature on this topic. This paper is partly based on the results described in Dhaene and Goovaerts, 1996, Dhaene and Goovaerts, 1997, Wang and Dhaene, 1998, Goovaerts and Redant, 1999, Goovaerts and Dhaene, 1999, Goovaerts and Kaas (2002), Dhaene et al., 2000b, Goovaerts et al., 2000, Simon et al., 2000, Vyncke et al., 2001, Kaas et al., 2000, Kaas et al., 2001, Denuit et al. (2001a) and De Vijlder and Dhaene (2002). It is the first text integrating these results in a consistent way. The paper also contains several new results and simplified proofs of existing results. Actuarial–financial applications, demonstrating the practical usability of this theory, are considered in Dhaene et al. (2002). Dependence in portfolios and related stochastic orders are also considered in Denuit and Lefèvre, 1997, Müller, 1997, Bäuerle and Müller, 1998, Wang and Young, 1998, Denuit et al., 1999a, Denuit et al., 1999b, Denuit et al., 2001b, Denuit et al., 2002, Denuit and Cornet, 1999, Dhaene and Denuit, 1999, Embrechts et al., 2001, Cossette et al., 2000, Cossette et al., 2002 and Dhaene et al. (2000a), amongst others.

Section snippets

Ordering random variables

In the sequel, we will always consider random variables with finite mean. This implies that for any random variable X we have that limx→∞x(1−FX(x))=limx→−∞xFX(x)=0, where FX(x)=Pr[Xx] is used to denote the cumulative distribution function (cdf) of X. Using the technique of integration by parts on both terms of the right-hand side in E[X]=∫−∞0xdFX(x)−∫0xd(1−FX(x)), we find the following expression for E[X]: E[X]=−∫−∞0FX(x)dx+∫0(1−FX(x))dx.In the actuarial literature it is common practice to

Inverse distribution functions

The cdf FX(x)=P[Xx] of a random variable X is a right-continuous (further abbreviated as r.c.) non-decreasing function with FX(−∞)=limx→−∞FX(x)=0,FX(+∞)=limx→+∞FX(x)=1.The usual definition of the inverse of a distribution function is the non-decreasing and left-continuous (l.c.) function FX−1(p) defined by FX−1(p)=inf{x∈R|FX(x)≥p},p∈[0,1]with inf∅=+∞ by convention. For all x∈R and p∈[0,1], we have FX−1(p)≤x⇔p≤FX(x).In this paper, we will use a more sophisticated definition for inverses of

Comonotonic sets and random vectors

As mentioned in Section 1, quite often in financial actuarial situations one encounters random variables of the type S=∑i=1nXi, where the terms Xi are not mutually independent, but the multivariate distribution function of the random vector X=(X1,X2,…,Xn) is not completely specified because one only knows the marginal distribution functions of the random variables Xi. In such cases, to be able to make decisions, it may be helpful to find the dependence structure for the random vector (X1,…,Xn)

The comonotonic upper bound for ∑i=1nXi

In this section we will derive bounds for sums S=X1+X2+⋯+Xn of random variables X1,X2,…,Xn of which the marginal distributions are given. The bounds are random variables that are larger (or smaller) than S in the sense of convex order. Therefore, we will call these bounds convex bounds. The reason we will resort to convex bounds is that the joint distribution of the random vector (X1,X2,…,Xn) is either unspecified or too cumbersome to work with.

The upper bound that we will derive in this

Conclusions

In this paper, we presented some simple yet powerful techniques to deal with sums of dependent random variables whose marginal distributions are known but with an unknown or complicated joint distribution. The central idea consists in replacing the original sum by another one, with a simpler dependence structure, and which is considered to be less favorable by all risk-averse decision makers. This extremal sum involves the components of the comonotonic version of the original random vector.

The

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

References (38)

  • N. Bäuerle et al.

    Modeling and comparing dependencies in multivariate risk portfolios

    ASTIN Bulletin

    (1998)
  • Cossette, H., Denuit, M., Dhaene, J., Marceau, E., 2002. Stochastic approximations of present value functions. Bulletin...
  • Denneberg, D., 1994. Non-additive Measure and Integral. Kluwer Academic Publishers, Boston, 184...
  • M. Denuit et al.

    Premium calculation with dependent time-until-death random variables: the widow’s pension

    Journal of Actuarial Practice

    (1999)
  • M. Denuit et al.

    Stochastic product orderings, with applications in actuarial sciences

    Bulletin Français d’Actuariat

    (1997)
  • Denuit, M., Dhaene, J., Lebailly De Tilleghem, C., Teghem, S., 2001a. Measuring the impact of a dependence among...
  • Denuit, M., Dhaene, J., Ribas, C., 2001b. Does positive dependence between individual risks increase stop-loss...
  • Denuit, M., Genest, Ch., Marceau, E., 2002. A criterion of the stochastic ordering of random sums. Scandinavian...
  • De Vijlder, F., Dhaene, J., 2002. An introduction to the theory of comonotonic risks. Working...
  • Cited by (475)

    • Monotonicity of equilibria in nonatomic congestion games

      2024, European Journal of Operational Research
    • Reinsurance games with two reinsurers: Tree versus chain

      2023, European Journal of Operational Research
    • Pairwise counter-monotonicity

      2023, Insurance: Mathematics and Economics
    View all citing articles on Scopus
    View full text