Classes of severity distributions and their relationships

Welcome to Severity Distributions!

Welcome to one of the most important chapters in your Exam FAM journey! In the world of insurance, we care about two main things: Frequency (how often things happen) and Severity (how much each event costs). In this section, we are focusing entirely on Severity.

Think of severity distributions as a "menu" of shapes. Some insurance claims, like broken windshields, are usually small and predictable. Others, like hurricane damage, can be massive. Choosing the right distribution is like picking the right tool for the job. Don't worry if the math looks intimidating at first—we're going to break down these "families" of distributions so you can see how they are all related!

1. What Exactly is a Severity Distribution?

A severity distribution is a probability distribution used to model the size of a single loss. Since claims can't be negative, these distributions are defined for \(x > 0\).

Quick Review: - PDF \(f(x)\): The likelihood of a claim being exactly a certain size. - CDF \(F(x)\): The probability that a claim is less than or equal to \(x\). - Survival Function \(S(x)\): The probability that a claim exceeds \(x\). In insurance, we love \(S(x)\) because it tells us about the big, scary losses!

2. The "Family Tree" of Distributions

Most severity distributions used in FAM belong to specific families. Understanding these relationships helps you memorize formulas more easily. If you know the "parent," you can often figure out the "children."

The Transformed Beta Family

This is like the "Grandparent" of many distributions. By changing the parameters, you can turn a Transformed Beta distribution into a Pareto, Burr, or Loglogistic distribution. Tip: You don't usually need to memorize the massive Transformed Beta formula, but you should know it's the "umbrella" for many heavy-tailed distributions.

The Transformed Gamma Family

This family includes the Gamma, Exponential, and Weibull distributions. These are generally used for "lighter" tails (meaning massive claims are less likely compared to the Beta family).

Key Takeaway: Distributions aren't just random formulas; they are related through mathematical transformations. If you see a formula that looks like another one but with an extra exponent, it's likely a "transformed" version!

3. Meet the "Celebrity" Distributions

Let's look at the specific distributions you'll see most often on the exam.

The Exponential Distribution (The "Baseline")

The simplest model. It assumes the probability of a claim stays constant over time. - Memory Aid: Think of it as the "boring" distribution. It has no "memory" and a very simple tail. - Formula: \(F(x) = 1 - e^{-x/\theta}\)

The Gamma Distribution

Think of the Gamma distribution as the sum of several Exponential distributions. - If you have 3 independent claims that each follow an Exponential distribution with mean \(\theta\), their total sum follows a Gamma distribution with shape \(\alpha = 3\) and scale \(\theta\).

The Pareto Distribution (The "Heavy Hitter")

This is the superstar of severity models. It is heavy-tailed, meaning it predicts a higher chance of huge, "catastrophic" claims. - Real-World Example: Medical malpractice or liability claims often follow a Pareto distribution because most claims are small, but a few can be multi-million dollar settlements. - Mistake to Avoid: Don't confuse the two parameters. Usually, \(\alpha\) (alpha) is the shape and \(\theta\) (theta) is the scale. A smaller \(\alpha\) means a "heavier" tail (more risk!).

The Lognormal Distribution

If you take the natural log of your claim sizes (\(ln(X)\)) and the result looks like a Normal (Bell Curve) distribution, then your original claims were Lognormal. - Did you know? This is very common in property insurance. Most houses cost a medium amount to fix, with very few being extremely cheap or extremely expensive.

4. Tail Weight: Who is the "Riskiest"?

A major part of Exam FAM is comparing "tail weights." This is just a fancy way of asking: "Which distribution is more likely to produce a giant claim?"

The Hierarchy of Heaviness (from Lightest to Heaviest): 1. Normal (Very light - giant claims are almost impossible) 2. Exponential (Medium-light) 3. Gamma (Depends on shape, but generally light) 4. Lognormal (Medium-heavy) 5. Pareto (Very heavy!) 6. Cauchy (Extremely heavy - though less common on FAM)

How to test tail weight on the exam: If you are asked to compare two distributions, look at the ratio of their survival functions: \[ \lim_{x \to \infty} \frac{S_1(x)}{S_2(x)} \] If the limit goes to infinity, then Distribution 1 has a heavier tail than Distribution 2. Analogy: Imagine a race to zero. The "heavier" distribution is the one that stays above zero the longest as we move toward the right side of the graph.

5. Relationships and Transformations

You can create new distributions from old ones using these three tricks:

1. Multiplication by a Constant (Scale Transformation): If \(X\) is a distribution, then \(Y = cX\) just changes the scale parameter (\(\theta\)). It doesn't change the "shape" of the family. - Example: Changing a claim from Dollars to Euros.

2. Raising to a Power (Power Transformation): If \(X\) is Exponential, then \(X^{1/\tau}\) becomes a Weibull distribution. - Mnemonic: "Power to the Weibull!"

3. Exponentiation: If \(X\) is Normal, then \(e^X\) is Lognormal. (This one is in the name!)

Quick Review Box: - Light Tails: Exponential, Gamma. - Heavy Tails: Pareto, Lognormal, Burr. - To make a tail heavier: Decrease the shape parameter (\(\alpha\)) or use a power transformation.

6. Summary and Key Takeaways

- Why we care: Different insurance products need different models. You wouldn't use a "light" Exponential model for a "heavy" Liability product.

- The "Scale" (\(\theta\)): Usually relates to the currency or size. Changing the scale doesn't change the fundamental "riskiness" of the shape.

- The "Shape" (\(\alpha\)): This is the most important parameter. It determines how the tail behaves.

- Comparing Tails: Use the limit of the ratio of survival functions. The one that "stays larger" at infinity is the heavier one.

Don't worry if this seems tricky at first! The more you practice identifying these distributions in the SOA tables, the more these relationships will feel like second nature. You've got this!