Welcome to Severity Models!

Hello future Actuaries! Welcome to one of the most practical and interesting parts of the ASTAM curriculum. While frequency tells us how often claims happen, severity tells us how much those claims actually cost. Think of it this way: frequency is how many times you drop your phone, and severity is the cost of the repair bill!

In this chapter, we will learn how to describe and compare the "shapes" of different claim costs. Some claims are small and predictable (like a broken windshield), while others can be massive and rare (like a hurricane). Understanding these characteristics helps us price insurance fairly and ensure companies have enough money to pay for those "big" surprises.

Don't worry if this seems tricky at first! We'll break it down step-by-step, from basic definitions to the advanced "tail weight" concepts that the SOA loves to test.

1. The Fundamental Functions

Before we can analyze a distribution, we need our basic tools. You might remember these from Exam FAM, but they are vital here too.

The Cumulative Distribution Function (CDF): \(F(x)\)
This tells us the probability that a claim \(X\) is less than or equal to a certain amount \(x\).
\(F(x) = P(X \le x)\)

The Survival Function: \(S(x)\)
In severity, we often care more about the probability that a loss exceeds a certain amount (like a deductible).
\(S(x) = 1 - F(x) = P(X > x)\)

The Probability Density Function (PDF): \(f(x)\)
The derivative of the CDF. It represents the "likelihood" of a claim being exactly a certain value.
\(f(x) = F'(x) = -S'(x)\)

The Hazard Rate (Failure Rate): \(h(x)\)
This is a key ASTAM concept. It represents the "instantaneous" probability of a claim occurring right now, given it hasn't occurred yet. In severity, it helps us see how the "danger" of a larger claim changes as the claim size increases.
\(h(x) = \frac{f(x)}{S(x)}\)

Quick Review: The Hazard Rate

If \(h(x)\) is decreasing, it means that as the claim gets larger, the "rate" of the claim ending (the cost stopping) slows down. This is a sign of a heavy-tailed distribution (very large claims are possible).

2. Moments and Their Relatives

Moments help us summarize a distribution with just a few numbers.

Raw Moments (\(k\)-th moment)

The \(k\)-th raw moment is denoted as \(E[X^k]\).
\(E[X^k] = \int_{0}^{\infty} x^k f(x) dx\)

Actuarial Trick: For severity distributions that are only positive (which is most of them!), you can also calculate the mean using the survival function:
\(E[X] = \int_{0}^{\infty} S(x) dx\)

Variance, Skewness, and Kurtosis

  • Variance: Measures the "spread." High variance means claim sizes are all over the place.
  • Skewness: Measures asymmetry. Most severity distributions are positively skewed (skewed to the right), meaning there is a long tail of very large claims.
  • Kurtosis: Measures "peakedness" and tail weight. Higher kurtosis generally means more "extreme" values (fat tails).

Did you know? In the insurance world, we almost never see a "Normal Distribution" for severity because claim costs can't be negative and they often have huge "outlier" claims that the Normal distribution can't handle!

3. Tail Weight: How "Dangerous" is the Distribution?

This is the most important concept in this chapter for ASTAM. Tail weight describes how likely we are to see extremely large claims.

Comparing Distributions

The SOA will often ask you which of two distributions is "heavier-tailed." A heavier-tailed distribution is riskier for an insurer. Here are three ways to check:

Method 1: Existence of Moments
If distribution A has all its moments (like the Exponential or Gamma), but distribution B only has its first two moments (like some Pareto distributions), then distribution B is heavier-tailed. If the moments don't exist for large \(k\), the tail is "fat."

Method 2: Ratio of Survival Functions
Compare two distributions by looking at the limit of the ratio of their survival functions:
\(\lim_{x \to \infty} \frac{S_1(x)}{S_2(x)}\)
If this limit goes to \(\infty\), Distribution 1 has a heavier tail. If it goes to 0, Distribution 2 is heavier.

Method 3: Hazard Rate Behavior
A decreasing hazard rate (DHR) indicates a heavy tail. If \(h(x) \to 0\) as \(x \to \infty\), it’s a very heavy tail (like the Pareto). If \(h(x) \to c\) (a constant), it’s medium (Exponential). If \(h(x) \to \infty\), it’s a light tail.

Summary of Tail Weight (from Lightest to Heaviest)
  1. Light Tails: Normal, Gamma (with \(\alpha > 1\)). These are "safe."
  2. Medium Tails: Exponential. Our "baseline."
  3. Heavy Tails: Lognormal, Pareto, Burr. These keep actuaries awake at night!

4. Mean Residual Life (MRL)

The Mean Residual Life function, \(e(x)\), is the expected additional cost of a claim, given that the claim has already exceeded amount \(x\).

Formula: \(e(x) = E[X - x | X > x] = \frac{\int_{x}^{\infty} S(t) dt}{S(x)}\)

Why MRL Matters:

  • If \(e(x)\) is constant: This is the Exponential distribution (the "memoryless" property). No matter how big the claim is, the expected additional cost remains the same.
  • If \(e(x)\) is increasing: This is a heavy-tailed distribution (like Pareto). The larger the claim gets, the larger we expect the additional cost to be!
  • If \(e(x)\) is decreasing: This is a light-tailed distribution.

Analogy: Imagine you are repairing a very old house. If the "repair severity" has an increasing MRL, it means that every time you find a problem and fix it, you actually expect the remaining hidden problems to be even more expensive! (That's a heavy-tailed nightmare).

5. Important Severity Distributions to Know

While you have a formula sheet, you should recognize the "personality" of these distributions:

1. Exponential: The "middle ground." Constant hazard rate and constant MRL.
2. Gamma: Flexible. If the shape parameter \(\alpha > 1\), it’s light-tailed. If \(\alpha < 1\), it's heavier than Exponential.
3. Pareto: The classic heavy-tail. It is used for large catastrophe losses. It has a decreasing hazard rate and an increasing MRL.
4. Weibull: Also flexible. Its tail weight depends on its shape parameter \(\tau\).
5. Lognormal: A common choice for many insurance lines. It is heavier than Gamma but generally lighter than Pareto.

6. Common Pitfalls and Mistakes

  • Confusion between \(f(x)\) and \(h(x)\): Remember that the hazard rate \(h(x)\) is a ratio. Even if the PDF \(f(x)\) is going to zero, the hazard rate might be going to infinity.
  • Ignoring the Support: Always check if the distribution starts at 0 or a different value (like a truncated distribution).
  • Mixing up Tail Comparisons: When using the ratio \(\frac{S_1(x)}{S_2(x)}\), make sure you are taking the limit as \(x \to \infty\), not \(x \to 0\).

Key Takeaways for the Exam

1. Hazard Rate: Decreasing hazard rate = Heavy tail. Constant = Exponential.
2. Mean Residual Life: Increasing MRL = Heavy tail. Useful for pricing excess-of-loss reinsurance.
3. Moment Existence: If the higher moments don't exist, you're dealing with a "dangerous" heavy-tailed distribution.
4. Comparisons: Use limits of survival functions or hazard rates to rank distributions by riskiness.

Keep going! Severity characteristics are the building blocks for the more complex models you'll see later in ASTAM like coverage modifications (deductibles and limits). Master these "shapes" now, and the rest of the course will be much smoother!