Upper and lower bounds for sums of random variables.

In this contribution, the upper bounds for sums of dependent random variables X1 + X2 + … + X n derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the Xi, given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of lognormal random variables is given.Random variable; Variables; Distribution; Functions;

Economic capital allocation derived from risk measures.

We examine properties of risk measures that can be considered to be in line with some 'best practice' rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violate best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability and the risk distribution.Risk; Insurance; Solvency; Margin; Dependency; Structure;

The hurdle-race problem.

We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numerical examples illustrating the presented approximations.Processes; Value;

A unified approach to generate risk measures.

Markov inequality; Premium; Premium principle; Principles; Probability; Recall; Risk; Risk measure;

Risk measurement with the equivalent utility principles.

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable at- tention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker. In incomplete financial markets, prices are no more unique but depend on the agents' attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities.Actuarial; Coherence; Decision; Expected; Market; Markets; Measurement; Preference; Premium; Prices; Pricing; Principles; Random variables; Research; Risk; Risk measure; Risk measurement; Space; Studies; Theory; Uncertainty; Utilities; Variables;

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

n 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.Risk; Actuarial; Science; Theory;

Comonotonic approximations for optimal portfolio selection problems.

We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskless and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of 'constant mix' portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari's dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically.Approximation; Choice; Comonotonicity; Criteria; Decision; Market; Optimal; Optimal portfolio selection; Portfolio; Problems; Risk; Selection; Simulation; Theory; Time;

Bounds for present value functions with stochastic interest rates and stochastic volatility.

The distribution of the present value of a series of cash flows under stochastic interest rates has been investigated by many researchers. One of the main problems in this context is the fact that the calculation of exact analytical results for this type of distributions turns out to be rather complicated, and is known only for special cases. An interesting solution to this difficulty consists of determining computable upper bounds, as close as possible to the real distribution.In the present contribution, we want to show how it is possible to compute such bounds for the present value of cash flows when not only the interest rates but also volatilities are stochastic. We derive results for the stop loss premium and distribution of these bounds.Distribution; Value; Cash flow; Interest rates; Researchers; Problems;