Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefèvre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.     

Fourier Inversion Formulas in Option Pricing and Insurance

Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max (S – K, 0) also arises in excess-of-loss or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for payoffs which occur in insurance. Numerical examples are provided.      

On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums

The paper develops a method for the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums.     

Third-order extensions of Lo’s semiparametric bound for European call options

Computing semiparametric bounds for option prices is a widely studied pricing technique. In contrast to parametric pricing techniques, such as Monte-Carlo simulations, semiparametric pricing techniques do not require strong assumptions about the underlying asset price distribution. We extend classical results in this area. Specifically, we derive closed-form semiparametric bounds for the payoff of a European call option, given up to third-order moment (i.e., mean, variance, and skewness) information on the underlying asset price. We analyze how these bounds tighten the corresponding bounds, when only second-order moment (i.e., mean and variance) information is provided. We describe applications of these results in the context of option pricing; as well as in other areas such as inventory management, and actuarial science.     

Solvency II,regulatory capital,and optimal reinsurance: How good are Conditional Value-at-Risk and spectral risk measures?

We study the problem of optimal reinsurance as a means of risk management in the regulatory framework of Solvency II under Conditional Value-at-Risk and, as its natural extension, spectral risk measures. First, we show that stop-loss reinsurance is optimal under both Conditional Value-at-Risk and spectral risk measures. Spectral risk measures thus constitute a more general class of suitable regulatory risk measures than specific Conditional Value-at-Risk. At the same time, the established type of stop-loss reinsurance can be maintained as the optimal risk management strategy that minimizes regulatory capital. Second, we derive the optimal deductibles for stop-loss reinsurance. We show that under Conditional Value-at-Risk, the optimal deductible tends towards restrictive and counter-intuitive corner solutions or “plunging”, which is a serious objection against its use in regulatory risk management. By means of the broader class of spectral risk measures, we are able to overcome this shortcoming as optimal deductibles are now interior solutions. Especially, the recently discussed power spectral risk measures and the Wang risk measure are shown to avoid any plunging. They yield a one-to-one correspondence between the risk parameter and the optimal deductible and, thus, provide economically plausible risk management strategies.     

Bounds on compound distributions and stop-loss premiums

In the actuarial literature a lot of attention is given to the approximation of aggregate claims distributions by compound Poisson and Polya distributions and their subsequent numerical evaluation. The present contribution derives bounds for the tail of compound distributions and stop-loss premiums. The bounds are obtained in an elementary manner based on a version of the Chebyshev inequality. The main point of this contribution consists in deriving bounds with explicit dependence on the distribution function itself as well as on some partial moments of it.     

Optimal risk retention under partial insurance

This paper examines the situation where a risk-averse insured determines the optimal amount of deductible (or stop-loss) insurance. The insurer uses two different premium principles, the expected value principle and the exponential principle. The insured has an exponential utility function. Specific numerical results are obtained for the optimal stop-loss limit in the case of a group life insurance plan. The exact results are contrasted with those obtained by using the normal approximation instead of the exact distribution of aggregate claims.     

Error bounds for the compound poisson approximation

Explicit error bounds in terms of probabilities and stop-loss premiums are given for two kinds of compound Poisson approximations: the first concerns the difference between the individual and the collective model; the second is about the difference of the compound negative binomial and the compound Poisson distribution.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Characterizing a comonotonic random vector by the distribution of the sum of its components

In this article, we characterize comonotonicity and related dependence structures among several random variables by the distribution of their sum. First we prove that if the sum has the same distribution as the corresponding comonotonic sum, then the underlying random variables must be comonotonic as long as each of them is integrable. In the literature, this result is only known to be true if either each random variable is square integrable or possesses a continuous distribution function. We then study the situation when the distribution of the sum only coincides with the corresponding comonotonic sum in the tail. This leads to the dependence structure known as tail comonotonicity. Finally, by establishing some new results concerning convex order, we show that comonotonicity can also be characterized by expected utility and distortion risk measures.