[Paper Review] Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions

Helena Jasiulewicz, Wojciech Kordecki|arXiv (Cornell University)|Jun 14, 2013

Probability and Risk Models10 references3 citations

TL;DR

This paper analyzes the finite-horizon ruin probability in a discrete-time risk process with proportional reinsurance and random investment returns, focusing on exponential (light-tailed) and Pareto (heavy-tailed) loss distributions. It derives recurrence equations for ruin probabilities, provides a Lundberg upper bound for exponential claims, and establishes an asymptotic approximation for Pareto claims as initial capital increases, demonstrating fast convergence and practical utility for risk management decisions under varying reinsurance levels and interest rates.

ABSTRACT

In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.

Motivation & Objective

• To analyze the finite-horizon ruin probability in a discrete-time risk process with proportional reinsurance and investment of surplus.
• To evaluate the performance of the Lundberg upper bound for ruin probability under exponential claims, particularly when reinsurer loading exceeds insurer loading.
• To develop and validate an asymptotic approximation for ruin probability under heavy-tailed Pareto claims as initial capital increases.
• To provide practical guidance on optimal reinsurance retention levels and initial capital requirements to achieve acceptable ruin probability thresholds.
• To compare numerical results across light-tailed (exponential) and heavy-tailed (Pareto) distributions under identical geometric moments.

Proposed method

• Derives recurrence equations for finite-horizon ruin probabilities in a discrete-time risk model with proportional reinsurance and random interest rates on surplus.
• Applies the Lundberg adjustment coefficient method to derive an upper bound for ruin probability under exponential claims.
• Establishes a theoretical asymptotic approximation for ruin probability under heavy-tailed Pareto claims using regularly varying tail theory.
• Uses numerical computation to solve recurrence equations and generate ruin probability tables and graphs for various combinations of initial capital, retention levels, and interest rates.
• Compares results across exponential and Pareto distributions by matching their geometric means and variances for fair numerical comparison.
• Employs graphical visualization of ruin probability as a function of initial capital and retention level to illustrate trends and convergence.

Experimental results

Research questions

• RQ1 Does the discrete-time risk process exhibit the same optimal retention behavior as the continuous-time model, where adjustment coefficient maximization determines optimal retention?
• RQ2 How accurate is the Lundberg upper bound for ruin probability when the reinsurer's loading exceeds the insurer's loading (ξ > θ), particularly in the context of proportional reinsurance?
• RQ3 Is the asymptotic approximation for ruin probability under heavy-tailed Pareto claims both accurate and rapidly convergent as initial capital increases?
• RQ4 What is the relationship between initial capital, reinsurance retention level, and the resulting ruin probability for both light-tailed and heavy-tailed loss distributions?
• RQ5 Can numerical tables and graphs be used to determine the minimal initial capital or optimal retention level to keep ruin probability below a specified threshold (e.g., 0.05)?

Key findings

• For exponential claims, the Lundberg upper bound remains valid but becomes less precise when the reinsurer's loading exceeds the insurer's loading (ξ > θ), as the adjustment coefficient is no longer convex.
• The adjustment coefficient is convex when ξ = θ, which improves the quality of the upper bound, but this convexity is lost when ξ > θ, reducing the bound's reliability.
• For Pareto-distributed claims, the asymptotic approximation of ruin probability converges quickly to the limit value as initial capital increases, confirming its practical usefulness.
• Numerical results show that ruin probability increases with higher retention levels for fixed initial capital, indicating that minimal retention minimizes ruin probability in the discrete model.
• For a 5-year horizon and 5% interest rate, initial capital of u=1 is insufficient to keep ruin probability below 0.05 for any retention level b ∈ (0.2,1] under Pareto claims, as shown by 'lack' in Table 4.
• Graphs in Figures 3 and 4 illustrate that ruin probability decreases with increasing initial capital and decreases with decreasing retention level, with steeper declines for lower retention and higher capital.