Severity model moments and percentiles

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 5 min read

Welcome to the World of Severity!

In the "Severity, Frequency, and Aggregate Models" section of Exam FAM, we aren't just looking at how many claims happen (that’s Frequency); we are looking at how much those claims cost. This is what we call Severity.

In this chapter, we will learn how to describe the "behavior" of these claim amounts using Moments and Percentiles. Think of these as the DNA of a probability distribution—they tell us where the center is, how much the values spread out, and whether we should expect some massive "shocks" to the system. Don't worry if this seems a bit abstract right now; we’ll break it down piece by piece!

1. Raw Moments: Measuring "Size"

The simplest way to describe a distribution is through its raw moments. A raw moment is just the expected value of the claim amount raised to a power.

Definition: The $k$-th raw moment of a random variable $X$ is denoted as $E[X^k]$ (sometimes written as $\mu'_k$).

The First Raw Moment: The Mean

The 1st raw moment ($k=1$) is the most famous: the Mean ($\mu$). It represents the "average" claim amount you expect to see over a long period.
$E[X] = \int_{0}^{\infty} x \cdot f(x) \, dx$ (for continuous variables)
$E[X] = \sum x \cdot p(x)$ (for discrete variables)

Higher Raw Moments

As $k$ gets larger ($E[X^2], E[X^3]$), we are giving more "weight" to larger claim amounts. This is why higher moments are very sensitive to the "tail" of the distribution (the really big claims).

Quick Review:
- 1st Moment = Center (Mean)
- 2nd Moment = Helps measure spread (Variance)

2. Central Moments: Measuring "Spread"

While raw moments look at the distance from zero, central moments look at the distance from the mean. This tells us how much the actual claims fluctuate around the average.

Definition: The $k$-th central moment is denoted as $E[(X - \mu)^k]$ (sometimes written as $\mu_k$).

The Variance ($\sigma^2$)

The 2nd central moment is the Variance. It tells us how "unpredictable" the claim amounts are.
$\text{Var}(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2$

Common Mistake to Avoid: Students often confuse $E[X^2]$ and $(E[X])^2$. Remember: The average of the squares is NOT the same as the square of the average!

Standard Deviation ($\sigma$)

Since variance is in "squared units" (like dollars squared), we take the square root to get back to the original units.
$\sigma = \sqrt{\text{Var}(X)}$

Coefficient of Variation (CV)

This is a great "real-world" metric. It measures relative risk.
$CV = \frac{\sigma}{\mu}$
Analogy: A $100 variation on a $1,000 claim (CV = 0.1) is "stable." A $100 variation on a $50 claim (CV = 2.0) is "wildly unpredictable!"

Key Takeaway: Central moments help us understand the risk and volatility of claim amounts.

3. Skewness and Kurtosis: The "Shape" of Risk

Insurance claims are rarely "normal." They usually have a few very large claims and many small ones. We use Skewness and Kurtosis to describe this "lopsidedness."

Skewness

Skewness measures the lack of symmetry.
- Positive Skew (Right-skewed): Most insurance distributions are here! There is a long tail of very large claims on the right.
- Zero Skew: Perfect symmetry (like a Normal distribution).

Kurtosis

Kurtosis measures how "fat" the tails are. High kurtosis means that "extreme events" (black swans) happen more often than a normal distribution would suggest.

Did you know? Actuaries care deeply about skewness because a single massive hurricane or wildfire claim can outweigh thousands of small claims. That "tail risk" is what makes insurance challenging!

4. Percentiles and Quantiles

Sometimes, we don't want the average; we want to know, "What is the claim amount that we will stay below 95% of the time?" This is where percentiles come in.

Definition: The $p$-th percentile (or $100p$-th percentile) is the value $x_p$ such that the probability of a claim being less than or equal to $x_p$ is $p$.
$F(x_p) = P(X \le x_p) = p$

VaR (Value at Risk): In the actuarial and finance world, the $p$-th percentile is often called VaR. For example, the 99% VaR is the claim amount that we only expect to exceed 1% of the time.

Example: If the 95th percentile of car insurance claims is $10,000, it means 95% of claims are $10,000 or less, and only 5% are larger than $10,000.

5. Limited Expected Value (LEV)

This is a CRITICAL concept for Exam FAM. In the real world, insurance companies often have a "limit" ($u$) on how much they will pay for a single claim.

Definition: The Limited Expected Value, $E[X \land u]$, is the expected value of the claim amount if we cap every claim at the value $u$.

$E[X \land u] = \int_{0}^{u} x \cdot f(x) \, dx + u \cdot [1 - F(u)]$

How to think about it:
1. For claims smaller than the limit, the insurer pays the actual amount ($x$).
2. For claims larger than the limit, the insurer pays exactly the limit ($u$).

Memory Trick: Think of LEV as "The average amount the insurance company is actually responsible for if they have a ceiling on payouts."

6. Summary Table for Quick Review

Mean ($\mu$): $E[X]$ - The average claim.
Variance ($\sigma^2$): $E[X^2] - \mu^2$ - The spread of claims.
CV: $\sigma / \mu$ - Relative volatility.
Percentile ($x_p$): $F(x_p) = p$ - The "cutoff" for a certain probability.
LEV ($E[X \land u]$): The average claim when capped at $u$.

Final Encouragement

Severity models can feel heavy on calculus and formulas at first. Don't worry if this seems tricky! The secret to mastering this chapter is recognizing that moments and percentiles are just different ways to describe the same "pile of claims." Keep practicing with the specific distributions in your Exam FAM Formula Sheet (like the Gamma, Pareto, and Exponential distributions), as they have pre-derived formulas for these moments that will save you tons of time!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does the term "Severity" refer to in actuarial modeling?

Severity refers to the magnitude or cost of insurance claims.

How is the $k$-th raw moment of a random variable $X$ defined?

$E[X^k]$ (or $\mu'_k$), which is the expected value of the claim amount raised to the power of $k$.

What is the first raw moment and what does it represent?

The 1st raw moment is the Mean ($\mu$), representing the average expected claim amount.

How do central moments differ from raw moments?

Central moments measure the distance from the mean ($E[(X - \mu)^k]$) rather than the distance from zero.

What is the relationship between Variance, the 2nd raw moment, and the mean?

$\text{Var}(X) = E[X^2] - (E[X])^2$

What is the Coefficient of Variation (CV) and why is it used?

$CV = \frac{\sigma}{\mu}$; it measures the relative risk or volatility of a distribution.

What does positive skewness indicate in a severity distribution?

It indicates a lack of symmetry where there is a long tail of very large claims on the right.

What does high Kurtosis suggest about a distribution's tails?

High kurtosis indicates "fat tails," meaning extreme events happen more frequently than in a normal distribution.

How is the $p$-th percentile (or quantile) defined?

It is the value $x_p$ such that $F(x_p) = P(X \le x_p) = p$.

What is the formula for Limited Expected Value (LEV) at limit $u$?

$E[X \land u] = \int_{0}^{u} x \cdot f(x) \, dx + u \cdot [1 - F(u)]$

How is the payout determined under a Limited Expected Value scenario?

The insurer pays the actual claim amount $x$ if it is below the limit, and pays exactly $u$ if the claim exceeds the limit.

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