As actuaries, we often find ourselves at the intersection of mathematics and finance, tasked with managing risk and ensuring the financial stability of insurance companies. One crucial tool in our arsenal is ruin theory, a set of mathematical models designed to assess an insurer’s vulnerability to insolvency. Ruin theory has its roots in the early 20th century, notably with the work of Filip Lundberg and later Harald Cramér, who laid the foundation for what is now known as the Cramér–Lundberg model. This model is pivotal in understanding how an insurance company can avoid financial ruin by balancing premiums with potential claims.

When preparing for the SOA Exam C or CAS Exam MAS-I, understanding compound Poisson processes is essential because these exams test your ability to model aggregate losses—a fundamental skill in actuarial science. The compound Poisson process elegantly captures the randomness in both the number of claims and their sizes, making it a cornerstone for modeling insurance claims and risk.

At its core, a compound Poisson process models the total claim amount as the sum of a random number of individual claims. The number of claims follows a Poisson distribution, reflecting the frequency of claims over a fixed period, while each claim size is an independent random variable drawn from the same distribution, representing severity. This setup aligns well with real-world insurance scenarios, where both how many claims happen and how big they are vary unpredictably.