Welcome to Loss Distributions and Risk Sharing!

Hello there! Welcome to one of the most practical chapters in the CS2 curriculum. In the world of risk modelling, we don't just care about how often things happen (frequency); we care deeply about how much they cost when they do (severity). This chapter is all about finding the right mathematical "ruler" to measure those costs and understanding how insurance contracts—like deductibles and limits—change the size of the bill. Don't worry if the formulas look a bit intimidating at first; we'll break them down step-by-step!

1. Modeling Claim Severity: The Basics

In insurance, most claims are small, but a few can be massive. Because of this, we need distributions that are non-negative (you can't have a negative claim!) and positively skewed (a long tail to the right).

Common Distributions in the Curriculum

Gamma: A flexible distribution often used for medium-sized claims. It’s the "reliable workhorse" of insurance modelling.
Lognormal: Used when claim amounts vary across a wide range of magnitudes. If you take the log of the claim and it looks like a Normal distribution, you’ve got a Lognormal.
Pareto: This is the "Heavy Hitter." It has a heavy tail, meaning it predicts a higher probability of very large, catastrophic claims compared to other distributions.
Exponential: The simplest case, where the probability of a claim decreases steadily as the size increases.

Key Concept: The Tail of the Distribution

The "tail" refers to the right-hand side of the probability density function (PDF). A heavy-tailed distribution (like the Pareto) means there is a significant risk of extreme losses. As an actuary, ignoring the tail is like ignoring the iceberg while sailing the Titanic!

Quick Review:
- Claims are always \( \ge 0 \).
- Most claim distributions are skewed to the right.
- Pareto = Heavy tail (Big risks).
- Exponential/Gamma = Thinner tails (More predictable risks).

2. Risk Sharing: Policy Alterations

In the real world, an insurer rarely pays the exact amount of the loss \( X \). Contracts usually include "risk-sharing" mechanisms. Think of this as the "rules of the game" for who pays what.

A. Proportional Reinsurance (Quota Share)

This is the simplest form. The insurer and reinsurer agree to split every claim by a fixed percentage \( k \).
The insurer pays: \( Y = kX \)
Analogy: Sharing a pizza and the bill with a friend 50/50. If the bill goes up, you both pay more proportionally.

B. Deductibles (Excess of Loss)

A deductible \( d \) is the amount the policyholder (or the primary insurer) must pay before the insurance kicks in. The insurer only pays the "excess."
The amount paid by the insurer is: \( Y = \max(0, X - d) \)
This is often written as \( (X - d)_+ \).
If the claim \( X \) is less than \( d \), the insurer pays zero.

C. Policy Limits (Capping)

A limit \( L \) is the maximum amount an insurer will pay for a single claim. Anything above \( L \) is the responsibility of the policyholder.
The amount paid by the insurer is: \( Y = \min(X, L) \)
This is often written as \( X \wedge L \).

Did you know? There is a very important mathematical relationship between these two:
\( X = (X - L)_+ + (X \wedge L) \)
This means the total loss is equal to the "amount above the limit" plus the "amount capped at the limit."

3. Calculating Expected Values with Risk Sharing

One of the most common tasks in CS2 is calculating the expected payment when there is a deductible or a limit.

The "Limited Expected Value" Shortcut

To find the expected value of a claim capped at \( L \), we use:
\( E[X \wedge L] = \int_0^L x f(x) dx + L \times P(X > L) \)
Or, an even easier way using the survival function \( S(x) = 1 - F(x) \):
\( E[X \wedge L] = \int_0^L (1 - F(x)) dx \)

Finding the Expected Payment with a Deductible

If you have a deductible \( d \), the expected cost to the insurer is:
\( E[(X - d)_+] = E[X] - E[X \wedge d] \)
This formula is a lifesaver in exams! Instead of doing a complex integration for the "excess," just subtract the limited expected value from the total mean.

Common Mistake to Avoid: Don't forget that if there is a deductible, the number of claims the insurer pays might change. Some claims will be smaller than the deductible and "disappear" from the insurer's perspective. This is called data truncation.

4. The Impact of Inflation

Inflation doesn't just make things more expensive; it has a weird "leverage effect" when deductibles are involved.

The Leverage Effect

Imagine a claim of 1,100 and a deductible of 1,000. The insurer pays 100.
Now, imagine 10% inflation. The claim becomes 1,210. The deductible stays at 1,000.
The insurer now pays 210!
Wait! Even though inflation was only 10%, the insurer's payment increased by 110% (from 100 to 210).
Key Takeaway: Inflation hits the insurer harder when there is a fixed deductible.

Mathematical Trick: If every claim \( X \) is multiplied by an inflation factor \( (1+i) \), the new expected payment with a deductible \( d \) is:
\( (1+i) E[X - \frac{d}{1+i}]_+ \)

5. Summary and Final Tips

Key Takeaways:
1. Severity is modeled using skewed, non-negative distributions like Pareto or Gamma.
2. Deductibles \( (X-d)_+ \) reduce the insurer's cost and can eliminate small claims.
3. Limits \( X \wedge L \) protect the insurer from extreme "tail" events.
4. Inflation has a disproportionate impact on the insurer's cost when deductibles are fixed.

Memory Aid: Think of a Deductible as a "floor" (you pay above it) and a Limit as a "ceiling" (you pay below it).
Final Encouragement: This chapter is very "formula-heavy," but once you master the relationship between \( E[X] \), \( E[X \wedge d] \), and \( E[(X-d)_+] \), you'll find these questions become much more manageable. Keep practicing those integrals!