Runoff triangles are a statistical tool used in the insurance industry to analyze and estimate the development of claims over time. They are essentially matrices that show the changes in the amounts of outstanding insurance claims over successive periods, typically by year.

The rows of the triangle represent the accident years in which claims occurred, while the columns represent the development years during which the claims were reported and settled. The cells of the triangle represent the amount of outstanding claims at a given point in time.

By analyzing the data in a runoff triangle, insurers can gain insights into the patterns and trends in their claims experience, such as the average time it takes for a claim to be reported and settled or how quickly the severity of claims increases over time. This information can help insurers better manage their risks and plan for future claims payments.

There are several types of runoff triangles that are commonly used in the insurance industry, including:

1. Paid Triangle: This triangle shows the actual amount of claims that have been paid out over time. The rows of the triangle represent the accident years, while the columns represent the years in which the claims were paid.
2. Incurred Triangle: This triangle shows the estimated amount of claims that will ultimately be paid out, including both paid and outstanding claims. The rows of the triangle represent the accident years, while the columns represent the years in which the claims were incurred.
3. Ultimate Triangle: This triangle shows the estimated ultimate amount of claims that will be paid out, taking into account all past and future development. The rows of the triangle represent the accident years, while the columns represent the ultimate development years.
4. Excess of Loss Triangle: This triangle shows the estimated amount of claims that exceed a certain threshold or deductible level. The rows of the triangle represent the accident years, while the columns represent the years in which the claims were incurred.
5. Frequency Triangle: This triangle shows the number of claims reported over time. The rows of the triangle represent the accident years, while the columns represent the years in which the claims were reported.

Each of these triangles provides a different perspective on an insurer’s claims experience and can be used to analyze different aspects of the business.

The purpose of runoff triangles is to help insurance companies analyze and understand their historical claims data and to use this information to make better business decisions. By using runoff triangles, insurance companies can:

1. Monitor claims development: Runoff triangles allow insurance companies to track the development of claims over time, which can help them understand trends and patterns in their claims experience. This information can be used to make better underwriting decisions and to improve the accuracy of claims reserving.
2. Forecast future claims payments: By analyzing historical claims data, insurance companies can develop models to forecast future claims payments. This information is critical for setting reserves and pricing insurance policies.
3. Manage risk: Runoff triangles can help insurance companies identify areas of the business that are high-risk and take steps to mitigate these risks. This might include adjusting pricing or coverage, or taking steps to improve loss prevention measures.
4. Evaluate performance: By comparing actual claims experience to expected claims experience, insurance companies can evaluate their performance and make adjustments as necessary.

Overall, runoff triangles provide insurance companies with a valuable tool for managing their business and improving their bottom line. By using historical claims data to forecast future claims payments and manage risk, insurance companies can make more informed business decisions and better serve their policyholders.