We present an inventory of non-exponential bounds for ruin probabilities and stop-loss premiums in the general Sparre-Andersen model (renewal model) of risk theory. Various additional bounds are given if one assumes that the ladder height distribution F associated with the risk process belongs to a certain class of distributions, in particular if it is concave or it exhibits a (positive or negative) aging property. In most cases, these bounds are shown to improve existing ones in the literature and/or possess the correct asymptotic behaviour when the distribution F is subexponential. Since in the classical (compound Poisson) risk model the ladder height distribution is always concave, all the bounds given in the paper are also valid for this model. Finally, in many cases the results of the paper are also valid for any compound geometric distribution.

In the literature, most of the bounds for the renewal function U(x) corresponding to a lifetime distribution F are given in terms of the first two moments of F only. The best general upper bound of this type is the one given in Lorden (1970). In the present article, we show that improved bounds can be obtained if one exploits the specific form of the distribution F. We derive a bound that improves upon Lorden's, at least on an interval [0,a) with a ≤ ∞, and we give both sufficient and necessary conditions for this improvement to hold uniformly for x ≥ 0. Refined upper as well as lower bounds are given for the case where F belongs to a class of distributions with monotone aging or when the renewal density is monotone.

In the classical risk model with Poisson arrivals, we study a functional approach which can be used to obtain new approximation formulae for the probability of ultimate ruin. In particular, we consider a map Φ between appropriate function spaces with Φ(f) = ψ, where f denotes the density of claim sizes in the model and ψ is the function that gives the associated infinite-time ruin probability, ψ(x), for a given initial surplus x. Suppose then that , represent two claim size densities in the model, with corresponding outputs ψ⁽⁰⁾ and ψ⁽¹⁾ respectively, and that only ψ⁽⁰⁾ is known analytically. By establishing that the map Φ has derivatives of any order, we can approximate ψ⁽¹⁾ by using ψ⁽⁰⁾ and the derivatives of Φ, provided that are not “too far” from each other. Since the explicit formulae for high order derivatives of Φ are typically complicated, we explain how computer algebra can be used for the implementation of the method. We also discuss how the functional approach can be used to obtain approximations for other quantities of interest in risk theory and we carry out some numerical illustrations to investigate the accuracy of the approximations.