**1. Introduction**

The Sparre Andersen model is a classical object of study in insurance risk theory, see e.g., Labbé and Sendova (2009); Li and Garrido (2005); Temnov (2004, 2014); Willmot (2007); and Asmussen and Albrecher (2010) for an overview. In this model, claims occur according to a renewal process, which generalises the Cramér–Lundberg model, where claims arrive according to a Poisson process. Ruin probabilities in such a general setting are typically expressed as solutions of defective renewal equations, differential equations, the so-called Wiener–Hopf factorisation, etc., but the latter are typically inadequate to be used for numerical computations. However, if either the interclaim times or the claim sizes belong to the class of phase-type distributions, then ruin-related quantities can be found in an explicit form; see, e.g., Albrecher and Boxma (2005); Dickson (1998); Li and Garrido (2005) and Landriault and Willmot (2008), respectively.

However, in many relevant situations in practice, the behaviour of the claim sizes is better captured by heavy-tailed distributions (Embrechts et al. 1997); however, in that case, explicit expressions are hard or impossible to evaluate even in terms of Laplace transforms. Under a heavy-tailed setting, a standard approach is hence to seek for asymptotic approximations (Albrecher et al. 2012; Dong and Liu 2013; Wei et al. 2008), for initial capital levels being very large. At the same time, this capital level typically has to be very large, so as to be reasonably accurate, when actual magnitudes matter. One mathematically appealing solution is then to look for higher-order approximations (see e.g., Albrecher et al. 2010); but, then an actual error bound for fixed values also cannot be given. Another alternative is to approximate the actual heavy-tailed claim distribution by a tractable light-tailed one and control the introduced error in some way. Spectral approximations in this spirit were recently developed in Vatamidou et al. (2014) for the classical Cramér–Lundberg model.

The present paper proposes an extension of techniques in Vatamidou et al. (2014) to the more general Sparre Andersen model, and at the same time improves the bound derived there and the efficiency of the algorithm to establish it. Using the geometric compound tail representation of the ruin probability, we derive our error bound in terms of the ladder height distribution, which is explicitly available when the distribution of the interclaim times has a rational Laplace transform. We focus on

heavy-tailed claim sizes, where numerical evaluations of ruin probabilities are typically challenging, and we develop an algorithm for the class of completely monotone distributions. Concretely, we approximate the ladder height distribution by a hyperexponential distribution, and we are able to prescribe the number of required phases for a desired resulting accuracy for the ruin probability.

The rest of the paper is organised as follows. In Section 2, we introduce the model and provide the exact formula for the ladder height distribution. As a next step, we derive, in Section 3, the error bound for the ruin probability, and we construct our approximation algorithm. In Section 4, we compare our approximations with existing asymptotic approximations. In Section 5, we then perform an extensive numerical analysis to check the tightness of the bound and the quality of the derived approximations. Finally, we conclude in Section 6.
