Insurance Surplus Modeling with Poisson Processes

Overview

This project investigates how an insurance company's financial reserve evolves over time when claims arrive randomly—a fundamental problem in actuarial science and applied probability. By modeling claim arrivals as a Poisson process and analyzing the resulting jump process, we explore the critical question: What is the probability that the company eventually runs out of money?

Key Focus: Understanding when classical risk theory works, when it fails, and what that failure costs in terms of required capital.

Motivation

Real-World Context

These events expose a critical gap in classical insurance theory: traditional models assume claim sizes follow light-tailed distributions, but reality often produces heavy-tailed catastrophic events.

The Central Questions

1. When does classical risk theory provide accurate predictions?
2. When does it fail catastrophically?
3. How much more capital is needed to account for heavy-tailed risks?

Part I: The Mathematical Model

The Cramér-Lundberg Model

The surplus (reserve) of an insurance company at time $t$ is modeled as:

where:

• $u$ = initial capital (starting reserve)
• $ct$ = cumulative premium income at rate $c$
• $N(t)$ = number of claims by time $t$
• $X_i$ = size of the $i$-th claim
• $\sum_{i=1}^{N(t)} X_i = S(t)$ = total claims paid (compound Poisson process)

Model Assumptions

Part II: Ruin Theory

Defining Ruin

Time of Ruin:

The first time the surplus becomes negative. If $U(t)$ never goes negative, then $\tau = \infty$.

Ultimate Ruin Probability:

The probability of eventual ruin starting with initial capital $u$.

Key Properties

• $\lim_{u \to \infty} \psi(u) = 0$ (more capital = safer)
• $\lim_{u \to 0} \psi(u) = 1$ if $c \leq \lambda\mu$ (certain ruin without net profit condition)
• Question: How fast does $\psi(u) \to 0$ as $u \to \infty$?

Part III: Lundberg's Inequality

The Classical Result

Theorem (Lundberg): Assuming the net profit condition $c > \lambda\mu$ holds, if there exists $R > 0$ satisfying:

then:

where $C$ is a constant depending on $R$.

Interpretation

• $R$ is called the adjustment coefficient
• Quantifies company safety: higher premium rate → larger $R$ → safer
• Exponential decay: Ruin probability decreases exponentially with initial capital
• This assumes light-tailed distributions (exponential safety)

Proof Sketch: Martingale Method

The proof uses the Optional Stopping Theorem:

1. Given the adjustment coefficient equation: $\lambda E[e^{RX}] = \lambda + cR$
2. Define the exponential martingale: $M(t) = e^{RU(t)} \cdot e^{-\lambda t(E[e^{RX}] - 1)}$
3. Apply Optional Stopping Theorem at time $\tau$: $E[M(\tau)] = M(0) = e^{Ru}$
4. At ruin: $U(\tau) \leq 0 \implies e^{RU(\tau)} \leq 1$
5. Therefore: $e^{Ru} \geq P(\tau < \infty)$, which gives $\psi(u) \leq e^{-Ru}$

Part IV: When Theory Fails - Heavy Tails

The Critical Assumption

Lundberg's inequality requires:

This assumes the moment generating function exists.

Heavy-Tailed Distributions

When $E[e^{RX}] = \infty$ for all $R > 0$:

• No adjustment coefficient exists
• MGF does not exist
• Tails decay polynomially (much slower than exponential)
• Examples: Pareto, Log-Normal distributions

Exponential vs Polynomial Decay

Key Insight: Exponential decay is dramatically faster than polynomial decay.

Part VI: Analysis & Interpretation

Key Findings

The "Tail Risk Premium"

The extra capital needed for heavy-tailed risks:

• At 1% ruin: Only 7% more capital
• At 0.1% ruin: Difference would be much larger
• The gap widens at higher safety levels

Light-Tailed vs Heavy-Tailed: Complete Comparison

Limitations & Future Work

Current Limitations

1. Simple assumptions: Pure compound process (no investment returns)
2. No risk management: No reinsurance, capital injections, or dividends
3. Single tail type: Most insurers face both light-tailed and sudden catastrophic claims
4. Parameter certainty: In reality, limited data makes $\alpha$, $\lambda$, $c$ hard to estimate

Proposed Extensions

Acknowledgments

This project was completed as part of the Directed Reading Program (DRP) in Fall 2025. Special thanks to Daniel Viray for his invaluable guidance and mentorship throughout this research.

Resources