Reinsurance Impact on Ruin Probability

Introduction

Welcome to the final lesson of our module on Collective Risk and Ruin Theory. In our previous lessons, we've built a solid understanding of how to quantify an insurer's risk of insolvency. We started with the adjustment coefficient and its use in Lundberg's inequality to find an upper bound for the ruin probability, $ψ (u) \leq e^{- R u}$ . Most recently, we explored a recursive method for calculating the exact probability of ruin for discrete claim models.

Throughout these discussions, we've treated the insurer's risk profile as a given. In reality, insurers actively manage their risk. One of the most fundamental tools for this is reinsurance. This lesson will bridge the gap between theory and practice by addressing a critical question: how does reinsurance impact the probability of ruin?

Today, you will learn to assess the effect of different reinsurance arrangements on the adjustment coefficient, and by extension, on the probability of ruin. We will see that the choice of reinsurance structure and retention level is not arbitrary; it's a careful optimization problem aimed at maximizing the insurer's stability.

Reinsurance and the Surplus Process

As we covered briefly in the first lesson of this module, reinsurance involves one insurer (the ceding company) transferring part of its risk to another (the reinsurer). In return, the ceding company pays a reinsurance premium. This has two immediate effects on the ceding insurer's surplus process:

Reduced Claim Payments: The insurer's outgo for large or frequent claims is reduced.
Reduced Premium Income: The net premium income is lower because of the premium paid to the reinsurer.

Let's consider how this affects the classical risk model $U (t) = u + c t - S (t)$ . With reinsurance, the aggregate claims process $S (t)$ is replaced by a retained claims process $S_{r e} (t)$ , and the premium rate $c$ is replaced by a net premium rate $c_{r e}$ . The new surplus process is $U_{r e} (t) = u + c_{r e} t - S_{r e} (t)$ .

Consequently, the Lundberg equation, which we use to find the adjustment coefficient $R$ , must be modified to reflect this new reality.

The Adjustment Coefficient with Reinsurance

Recall the Lundberg equation for a compound Poisson process with claim arrival rate $λ$ , claim severity distribution $X$ , and premium rate $c$ :

λ + c R = λ M_{X} (R)

With reinsurance, the claim severity distribution changes from $X$ to a new retained claim distribution $Y$ , and the premium rate changes from $c$ to $c_{r e}$ . The adjustment coefficient after reinsurance, which we'll denote $R_{r e}$ , is the smallest positive root of the new equation:

λ + c_{r e} R_{r e} = λ M_{Y} (R_{r e})

Here, $M_{Y}$ is the moment generating function of the retained claims. Since the retained claim distribution $Y$ and the net premium $c_{r e}$ depend on the reinsurance arrangement (e.g., the retention level), the adjustment coefficient $R_{r e}$ also becomes a function of that arrangement.

A rational insurer would aim to structure its reinsurance to maximize the value of $R_{r e}$ . A higher adjustment coefficient leads to a lower Lundberg inequality bound $e^{- R_{r e} u}$ , indicating greater financial stability.

To see this in action, let's explore how to model this for the main types of reinsurance. The following resource from the documentation of R's actuar package provides the essential formulas.

adjCoef function - Adjustment Coefficient - RDocumentation

This document describes a function for computing the adjustment coefficient. The "Details" section provides the precise mathematical form of the Lundberg equation under different reinsurance schemes.

Please read the section titled Details. Focus on the three equations presented for h(x). Note that in this context, x represents the adjustment coefficient (our R), B is the claim severity, W is the inter-arrival time, and y is the retention level. Observe how the term e^{xB} in the "no reinsurance" case is modified to reflect proportional reinsurance and excess-of-loss reinsurance. This change in the MGF of claims is the key to our analysis.

As you can see, for both proportional and excess-of-loss reinsurance, the equation for the adjustment coefficient incorporates a parameter for the retention level. This means we can analyze the adjustment coefficient as a function of the retention and find the optimal level that maximizes it.

Case Study: Optimal Excess of Loss Reinsurance

Let's delve deeper into excess of loss (XL) reinsurance, where the insurer retains claims up to a limit $M$ and the reinsurer covers the excess. How do we find the optimal retention limit $M^{*}$ ?

The paper "Excess of Loss Reinsurance and the Probability of Ruin in Finite Horizon" by Maria de Lourdes Centeno provides an excellent analysis. While its main focus is the finite-time horizon, it lays out the infinite-horizon case clearly.

[PDF] EXCESS OF LOSS REINSURANCE AND THE PROBABILITY OF ...

This paper investigates how to choose an optimal reinsurance retention limit.

First, read Section 2, starting from the paragraph introducing the XL arrangement. Understand how the insurer's surplus U(M;t) is defined. Next, focus on equation (8), which defines the adjustment coefficient after reinsurance, R(M), and the text immediately following it. This is the Lundberg equation for the retained risk. Finally, read the crucial result starting from "We know that the value of M...". This presents a famous formula for the optimal retention level M that maximizes the adjustment coefficient.

The paper establishes two key points:

The adjustment coefficient $R (M)$ is the positive root of the equation for the retained surplus process.
When the reinsurance premium is calculated using the expected value principle with a loading coefficient $ξ > 0$ , the value of $M$ that maximizes the adjustment coefficient is given by: $M = \frac{1}{R ( M )} ln (1 + ξ)$ Here, $R (M)$ is the adjustment coefficient corresponding to that optimal retention $M$ , and $ξ$ is the reinsurer's safety loading. Notice that $M$ appears on both sides of the equation, meaning it must be solved numerically. This elegant result connects the optimal retention level directly to the resulting adjustment coefficient and the reinsurer's pricing.

Example: Exponential Claims

The paper proceeds to give an example with exponential claims. Let's look at the results. With a primary insurer's safety loading of $θ = 0.2$ and a reinsurer's loading of $ξ = 0.4$ , the value of $M$ that maximizes the adjustment coefficient (and minimizes Lundberg's bound) is found to be $M \approx 1.49$ .

The following graph, taken from Figure 1 in the paper, visualizes the effect of the retention level $M$ on different measures of risk for a specific scenario ( $u = 30, t = 200$ ).

Figure 1 from Centeno (1997), showing the calculated ruin probability, Gerber's bound, and Lundberg's bound as functions of the excess of loss retention level M. Note how Lundberg's bound reaches its minimum at a specific retention level (M≈1.5), which corresponds to the maximum adjustment coefficient.

The key line for our current discussion is "Lundberg's Bound" (the top-most curve). You can see that this bound for the ruin probability is very high for low retention levels (when the insurer retains very little risk but pays high reinsurance premiums) and also for high retention levels (when the insurer is essentially self-insuring). The bound reaches a minimum at a specific point, which the paper identifies as $M \approx 1.49$ . This is the optimal retention level from the perspective of minimizing the ruin probability bound, achieved by maximizing the adjustment coefficient.

Practical Implementation in R

Your strong preference for applied problem-solving will be satisfied by seeing how these concepts are implemented in practice. The actuar package in R provides the adjCoef function to compute the adjustment coefficient, even for complex reinsurance scenarios.

Let's look at the examples provided in the documentation for this function.

adjCoef function - Adjustment Coefficient - RDocumentation

This section provides R code demonstrating how to use the adjCoef function.

Please read the Examples section. You don't need to run the code, but try to understand its structure. Focus on the chunk titled "Proportional reinsurance". Notice how it defines a premium function p(x) and an MGF mgfx(x, y) that depend on the retention level. The adjCoef function is then called with reins = "proportional", and the result R1 is a function that can calculate the adjustment coefficient for any retention level. plot(R1) would create a graph of the adjustment coefficient vs. retention. Next, look at the chunk for "Excess-of-loss reinsurance". The logic is the same: the premium and MGF are functions of the retention limit, and adjCoef calculates the adjustment coefficient across a range of retention limits.

These examples show how an actuary could use software to solve for the optimal retention level. They would generate the adjustment coefficient for a range of retentions, plot the result, and identify the retention that gives the maximum adjustment coefficient. This provides a direct, practical method for optimizing a reinsurance program to enhance solvency.

Conclusion

This lesson concludes our journey through risk modeling and ruin theory. We have seen that reinsurance is not just a simple transfer of risk; it is a sophisticated tool that reshapes an insurer's entire risk profile. By altering the claim distribution and premium flow, reinsurance directly impacts the adjustment coefficient.

Key Takeaways:

Reinsurance modifies the insurer's surplus process by changing the claim severity distribution and the net premium rate.
This leads to a new Lundberg equation, and the adjustment coefficient becomes a function of the reinsurance retention level, $R_{r e} (retention)$ .
A primary goal in designing a reinsurance program is to find the optimal retention level that maximizes the adjustment coefficient.
Maximizing the adjustment coefficient is equivalent to minimizing the upper bound on the ultimate ruin probability given by Lundberg's inequality, $ψ (u) \leq e^{- R_{r e} u}$ .
Practical tools, like the adjCoef function in R, allow actuaries to compute and visualize the adjustment coefficient across different retention levels to find this optimal point.

This module has equipped you with the fundamental theory of collective risk, from calculating aggregate claim distributions to estimating the probability of ruin and now, to assessing the impact of reinsurance. This completes the syllabus for the CS2 exam. You have covered a vast and challenging curriculum, from stochastic processes and survival models to the intricacies of ruin theory. Well done. I wish you the very best of luck in your exam preparation and the exam itself.

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