Different kinds of renewal equations repeatedly arise in connectionwith renewal risk models and variations. It is often appropriate toutilize bounds instead of the general solution to the renewalequation due to the inherent complexity. For this reason, as a firstapproach to construction of bounds we employ a general Lundberg-typemethodology. Second, we focus specifically on exponential boundswhich have the advantageous feature of being closely connected tothe asymptotic behavior (for large values of the argument) of therenewal function. Finally, the last section of this paper includesseveral applications to risk theory quantities.

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

In this paper we investigate multivariate risk portfolios, where the risks are dependent. By providing some natural models for risk portfolios with the same marginal distributions we are able to compare two portfolios with different dependence structure with respect to their stop-loss premiums. In particular, some comparison results for portfolios with two-point distributions are obtained. The analysis is based on the concept of the so-called supermodular ordering. We also give some numerical results which indicate that dependencies in risk portfolios can have a severe impact on the stop-loss premium. In fact, we show that the effect of dependencies can grow beyond any given bound.

The approximation of the individual risk model by a compound Poisson model plays an important role in computational risk theory. It is thus desirable to have sharp lower and upper bounds for the error resulting from this approximation if the aggregate claims distribution, related probabilities or stop-loss premiums are calculated.

The aim of this paper is to unify the ideas and to extend to a more general setting the work done in this connection by Bühlmann et al. (1977), Gerber (1984) and others. The quality of the presented bounds is discussed and a comparison with the results of Hipp (1985) and Hipp & Michel (1990) is made.

It is shown how the upper bounds for stop-loss premiums (and approximations to tail probabilities) obtained by replacing the individual model for a portfolio of risks by the collective model can be improved upon at the cost of only slightly more computer time. The method used is simply to keep a restricted number of large risks as they are instead of approximating them by a compound Poisson distribution. In a real-life example, the relative error in the stop-loss premium is shown to be reduced drastically by keeping only 10 out of 743 risks unchanged.

In Industrial Fire insurance an aggregate limit for the amount retained by the policyholder under a deductible policy has been agreed upon more frequently in recent times. This agreement is equivalent to a stop-loss cover on the retained loss amount. For the Poisson-lognormal model the corresponding stop-loss net premium is calculated using various methods (normal power, translated gamma, various discretisations) and the methods are compared. Finally, the influence of the model parameters is examined and it is demonstrated how a variety of parameter value combinations can be reduced to only a few rating curves.