Generalized Extreme Value and Generalized Pareto distributions

Welcome to the Extremes: Understanding GEV and GPD!

Hello, future Actuary! Welcome to one of the most exciting (and slightly intimidating) parts of the ASTAM syllabus. We are diving into Severity Models, but specifically, we are looking at the "monsters" under the bed: Extreme Events.

In your previous studies, you might have focused on average claims. But in the real world, one massive hurricane or a giant liability lawsuit can sink an insurance company. That’s where Generalized Extreme Value (GEV) and Generalized Pareto (GP) distributions come in. They help us model the "tails" of a distribution—the rare but huge events.

Don't worry if this seems heavy on the math at first. We’re going to break it down into simple pieces with plenty of analogies!

1. The Big Picture: Two Ways to Look at Extremes

Think of extreme events like giant waves in the ocean. There are two main ways scientists (and actuaries) study them:

1. Block Maxima (GEV): You look at the biggest wave that happened each year for the last 100 years. You are only interested in the maximums.

2. Peaks Over Threshold (GPD): You pick a high wall (a threshold) and only study the waves that were tall enough to crash over it. You are interested in the excess.

Quick Review: Why do we need these?

Just like the Central Limit Theorem tells us that averages tend to follow a Normal Distribution, Extreme Value Theory tells us that maximums and tail-excesses tend to follow GEV and GPD. They are the "limit laws" for the scary stuff!

2. The Generalized Extreme Value (GEV) Distribution

The GEV distribution is a "3-in-1" distribution. Depending on a specific parameter, it can take three different shapes. It is used to model Block Maxima.

The Math Bit

The Cumulative Distribution Function (CDF) for GEV is:
\( F(x) = \exp\left[ -\left( 1 + \xi \frac{x - \mu}{\sigma} \right)^{-1/\xi} \right] \)

Where:
• \(\mu\) (mu) is the location parameter (where is the center?).
• \(\sigma\) (sigma) is the scale parameter (how spread out is it?).
• \(\xi\) (xi) is the Shape Parameter. This is the most important one!

The Three Faces of GEV

The value of \(\xi\) (xi) tells you what kind of "tail" you are dealing with:

1. \(\xi > 0\) (Fréchet Type): Heavy-tailed. This is what we usually see in insurance (like huge catastrophe losses). The tail decays slowly.
2. \(\xi = 0\) (Gumbel Type): Light-tailed. The tail decays exponentially (like a Normal or Gamma distribution).
3. \(\xi < 0\) (Weibull Type): Short-tailed. The distribution actually has a hard upper limit. (Think of the maximum possible human age—it can't go to infinity).

Memory Aid: Think of \(\xi\) as the "Danger Level."
• Positive \(\xi\) = Positive Danger (Huge losses possible).
• Zero \(\xi\) = Moderate Danger.
• Negative \(\xi\) = Limited Danger.

3. The Generalized Pareto Distribution (GPD)

While GEV looks at the maximum of a group, the GPD looks at the excess over a threshold. If a claim exceeds $1 million, how much more than $1 million will it be?

The Math Bit

The CDF of the GPD is:
\( G_{\xi, \beta}(x) = 1 - \left( 1 + \xi \frac{x}{\beta} \right)^{-1/\xi} \)

(Note: This formula is for \(x \ge 0\) when \(\xi \ge 0\)).

Where:
• \(\beta\) (beta) is the scale parameter.
• \(\xi\) (xi) is the same shape parameter we saw in GEV!

Why is GPD so useful?

Did you know? There is a famous theorem (Pickands-Balkema-de Haan) that says as you raise your threshold higher and higher, the distribution of the "excess" almost always starts looking like a GPD. This makes it the "go-to" model for reinsurance, where companies only pay for the part of a claim that exceeds a certain limit.

Key Takeaway: GEV vs. GPD

• GEV models the maximum of a sample (e.g., the largest claim of the year).
• GPD models the amount over a threshold (e.g., the part of a claim above $500,000).
• They are siblings! If your block maxima follow a GEV, your exceedances follow a GPD with the same shape parameter \(\xi\).

4. Working with the Shape Parameter (\(\xi\))

Since \(\xi\) is the star of the show, ASTAM questions often focus on its impact.

Common Mistake: Don't confuse \(\xi\) with its reciprocal. In some older textbooks, they used \(\alpha = 1/\xi\). In ASTAM, stick to the \(\xi\) notation provided in the official tables!

The Mean and Variance of GPD

You don't need to memorize the whole world, but keep these in mind for a GPD:

• The Mean only exists if \(\xi < 1\).
• The Variance only exists if \(\xi < 0.5\).

If \(\xi\) is very large (e.g., \(\xi = 1.2\)), the "mean" is technically infinite because the tail is so heavy. In insurance terms, that means the losses are incredibly volatile!

5. Step-by-Step: Using GPD for Severity

If you are asked to calculate the probability of a claim exceeding a value \(y\), given it has already exceeded a threshold \(u\):

Step 1: Identify the threshold \(u\).
Step 2: Calculate the excess \(x = y - u\).
Step 3: Use the GPD formula \(G_{\xi, \beta}(x)\).
Step 4: Remember that \(1 - G(x)\) gives you the survival probability (the chance the excess is even larger).

6. Summary and Quick Tips

Quick Review Box:
• Block Maxima \(\rightarrow\) GEV
• Threshold Excess \(\rightarrow\) GPD
• \(\xi > 0\): Heavy tail (Fréchet / Pareto). Most common in ASTAM.
• \(\xi = 0\): Exponential-like tail (Gumbel / Exponential).
• \(\xi < 0\): Bounded tail (Weibull).

Final Encouragement

The math here can look scary because of the exponents and the Greek letters, but the logic is simple: we are just fitting a curve to the very end of a distribution. Practice identifying whether a problem is asking for a Maximum (GEV) or an Excess (GPD), and you'll be halfway to the right answer!

You're doing great—keep pushing through those severity models!