Applying GEV and GPD to tail risk estimation

Welcome to the Extremes! An Introduction to Tail Risk

Hello there! If you’ve made it this far into Exam ASTAM, you already know that actuarial science isn't just about the "average" claim. It’s often about those rare, catastrophic events—the "black swans"—that can threaten an insurer's solvency. In the Severity Models section, we focus on the "tail" of the distribution. This chapter focuses on Extreme Value Theory (EVT), which gives us the mathematical tools (GEV and GPD) to model these scary, high-impact events with precision. Don't worry if this seems a bit "extreme" at first—we’re going to break it down step-by-step!

1. The Two Pillars of Extreme Value Theory

When we look at extreme losses, we generally use two different mathematical lenses. Think of these as two different ways to look at a mountain range: Block Maxima (looking at the highest peak in each mountain range) and Peaks Over Threshold (looking at every peak that is higher than a certain altitude).

• Block Maxima (The GEV Approach): We divide our data into "blocks" (like years) and only look at the single largest claim in each block.
• Peaks Over Threshold (The GPD Approach): We set a very high "bar" (threshold) and look at every claim that crosses it.

2. The Generalized Extreme Value (GEV) Distribution

The GEV distribution is the limit distribution of the maximum of a large number of independent and identically distributed (i.i.d.) random variables. Just like the Normal distribution is the "go-to" for sums (Central Limit Theorem), the GEV is the "go-to" for maximums.

The Formula

The cumulative distribution function (CDF) for the GEV is:
\( H_{\xi, \mu, \sigma}(x) = \exp\left( -\left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right) \)

Breaking Down the Parameters

• \(\mu\) (Location): Where the distribution is centered (similar to the mean).
• \(\sigma\) (Scale): How spread out the distribution is (must be \(> 0\)).
• \(\xi\) (Shape Index): This is the most important one! It tells us how "heavy" the tail is.

The Three "Sub-Types" of GEV

Depending on the value of \(\xi\), the GEV takes on three famous names:
1. Fréchet (\(\xi > 0\)): Heavy-tailed. This is the one we see most often in general insurance (catastrophes, large liability).
2. Gumbel (\(\xi = 0\)): Light-tailed (exponential-like). Used for things like normal life insurance risks.
3. Weibull (\(\xi < 0\)): Short-tailed with a finite upper bound. (Rarely used for loss severity as claims usually don't have a hard cap).

Quick Review: Remember, \(\xi\) (xi) is the key to the "thickness" of the tail. Higher \(\xi\) means a scarier, heavier tail!

3. The Generalized Pareto Distribution (GPD)

The GPD is often the "star of the show" in ASTAM. While GEV looks at the maximum, the GPD looks at the excess over a high threshold. This is called the POT (Peaks Over Threshold) method.

The Logic (The Pickands-Balkema-de Haan Theorem)

This theorem essentially says: If you set a threshold \(u\) high enough, the distribution of the amounts above that threshold (\(X - u | X > u\)) will eventually look like a GPD.

The Formula

The CDF for the GPD is:
\( G_{\xi, \beta}(x) = 1 - \left( 1 + \frac{\xi x}{\beta} \right)^{-1/\xi} \)
(Where \(x \ge 0\) for \(\xi \ge 0\))

• \(\beta\) (Scale): Represents the scale of the exceedances.
• \(\xi\) (Shape): This is the same \(\xi\) as in the GEV! It determines the tail thickness.

Real-World Analogy: Imagine you are a bouncer at a club. You only care about people taller than 6 feet. The GPD doesn't care about everyone in the club; it only models how much taller than 6 feet those specific people are.

4. Estimating Tail Risk: VaR and TVaR

The whole point of using GPD is to calculate risk measures like Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) at very high confidence levels (like 99.9%).

Calculating VaR with GPD

If we know the number of total observations \(n\), the number of observations above the threshold \(n_u\), and our target probability \(p\), the VaR formula is:
\( \text{VaR}_p = u + \frac{\beta}{\xi} \left( \left( \frac{n}{n_u} (1-p) \right)^{-\xi} - 1 \right) \)

Calculating TVaR with GPD

TVaR is the expected value given that the loss exceeds the VaR. For the GPD (when \(\xi < 1\)):
\( \text{TVaR}_p = \frac{\text{VaR}_p + \beta - \xi u}{1 - \xi} \)

Did you know? If \(\xi \ge 1\), the mean of the distribution becomes infinite! This means the tail is so "fat" that the TVaR is technically undefined (infinite). This is a nightmare scenario for an actuary!

5. Practical Implementation: Choosing the Threshold

How do we pick the threshold \(u\)? This is a balancing act:
• Too high: We don't have enough data points to get a good estimate (high variance).
• Too low: The theorem doesn't apply yet, and our model will be biased.

The Mean Excess Plot

To find the "sweet spot," acts often use the Mean Excess Plot. We plot the threshold \(u\) against the average amount by which claims exceed \(u\).
• Key Trick: If the data follows a GPD, the Mean Excess Plot should look like a straight line. Look for the point where the graph starts to look linear—that's your threshold \(u\)!

6. Summary and Key Takeaways

Common Mistakes to Avoid:
1. Mixing up GEV and GPD: GEV is for "Block Maxima" (one per year); GPD is for "Peaks Over Threshold" (everything over a limit).
2. Parameter Confusion: Always check if \(\xi\) is positive or negative. A positive \(\xi\) in GPD indicates a heavy-tailed distribution (Pareto-like), which is standard for ASTAM severity.
3. The \(n/n_u\) term: In the VaR formula, don't forget that \(\frac{n_u}{n}\) is the empirical probability of being above the threshold.

Key Takeaway Quick Review:
• EVT helps us model the "un-modellable" extremes.
• GEV = Maximums. GPD = Excesses.
• \(\xi\) (Shape) tells us how likely "mega-losses" are.
• Mean Excess Plot = Linearity helps us choose the threshold \(u\).
• VaR/TVaR formulas are the primary tools for setting capital requirements based on these models.

Don't worry if the formulas look intimidating. Practice plugging in the values for \(\beta\), \(\xi\), and \(u\), and you'll find that GPD is actually very consistent and predictable once you get the hang of it. You've got this!