Types of coverage modifications for short-term insurance

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

Welcome to the World of Coverage Modifications!

Hello future actuaries! Today, we are diving into a crucial part of the FAM curriculum: Coverage Modifications. In short-term insurance (like car or home insurance), the insurer rarely pays the exact amount of the loss. Instead, they use "modifications" to decide how much of the bill they will pick up and how much you (the policyholder) will pay.

Why do they do this? It helps keep premiums affordable, encourages people to be more careful, and filters out tiny claims that cost more to process than they are worth. If you’ve ever had a car accident and had to pay a "deductible," you’ve already experienced coverage modifications firsthand! Let’s break these concepts down into simple, manageable pieces.

1. Ordinary Deductibles: The "Entry Fee"

An ordinary deductible is the most common type of modification. It is a fixed amount, let's call it $d$, that the policyholder must pay before the insurance company pays a single cent.

How it works:
If the loss $X$ is less than or equal to the deductible $d$, the insurer pays nothing.
If the loss $X$ is greater than the deductible $d$, the insurer pays the amount that exceeds $d$.

The payment variable $Y$ (the amount the insurer pays) is defined as:
$Y = (X - d)_+ = \max(0, X - d)$

Real-World Analogy:
Imagine you have a \$500 deductible on your phone insurance. You drop your phone, and the repair costs \$700. You pay the first \$500, and the insurance company pays the remaining \$200. If the repair only cost \$300, you pay the whole thing, and the insurance company pays $0$.

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Key Distinction: Cost per Loss vs. Cost per Payment

This is a favorite topic for exam questions! Pay close attention here:

Cost per Loss ($Y_L$): This considers every single accident, even the tiny ones where the insurer pays nothing. The mean is $E[Y_L] = E[(X-d)_+]$.
Cost per Payment ($Y_P$): This only looks at the cases where the insurer actually wrote a check (i.e., when $X > d$). The mean is $E[Y_P] = E[X-d | X > d]$.

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Quick Review: The relationship between them is:
\n$E[Y_L] = E[Y_P] \times P(X > d)$

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Common Mistake: Don't confuse these two! If a question asks for the "expected payment per claim," check if they mean for every claim or only claims that exceed the deductible.

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2. Franchise Deductibles: The "Magic Threshold"

A franchise deductible is a bit sneakier. It acts like a gatekeeper. If your loss is below the deductible $d$, the insurer pays nothing. But the moment your loss exceeds $d$, the insurer pays the entire loss!

The Formula:
\n$Y = 0$ if $X \le d$
\n$Y = X$ if $X > d$

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Analogy:
\nImagine a "Free Shipping" rule. If you spend under \$50, you pay for shipping. But if you spend \$51, the store covers the entire shipping cost. That \$50 is like a franchise deductible.

Did you know? Franchise deductibles are rare in personal car insurance but are sometimes used in marine or commercial insurance to discourage very small claims while providing full protection for larger ones.

3. Policy Limits: The "Safety Cap"

A policy limit (denoted as $u$) is the maximum amount an insurance company will pay for a single loss. It protects the insurer from "catastrophic" losses that could bankrupt them.

How it works:
If the loss $X$ is less than the limit $u$, the insurer pays the whole loss.
If the loss $X$ is more than the limit $u$, the insurer pays exactly $u$ and nothing more.

The payment is: $Y = X \wedge u = \min(X, u)$

Memory Aid: Think of a policy limit like a "ceiling." No matter how high the floor (the loss) goes, the insurer's payment can't go through the ceiling.

4. Coinsurance: The "Partnership"

Coinsurance (denoted by $\alpha$) is when the insurer and the policyholder share the loss proportionally. The insurer pays a percentage (like 80%), and the policyholder pays the rest (20%).

The Formula:
$Y = \alpha \times X$

Step-by-Step Example:
1. A loss occurs for \$1,000.
\n2. Coinsurance $\alpha = 0.80$.
\n3. Insurer pays $0.80 \times 1,000 = \$800$.
4. Policyholder pays the remaining \$200.

5. Putting It All Together: The Master Formula

In the real world (and on Exam FAM), you will often see all these modifications at once. The order of operations matters! Usually, you apply the limit first, then the deductible, then the coinsurance.

The "Master" Payment Formula:
$Y = \alpha \times [(X \wedge u) - d]_+$

This means: Take the loss, cap it at the limit $u$, subtract the deductible $d$, and then multiply by the coinsurance percentage $\alpha$.

Important Note: Sometimes the limit $u$ is applied to the loss (as shown above), but sometimes it is applied to the payment. Read the question carefully! If the limit is on the payment, the $\alpha$ might be applied before the limit.

6. The Impact of Inflation

Inflation is a major theme in FAM. When prices go up, losses ($X$) go up. Let's say inflation is $r$. The new loss becomes $X' = (1+r)X$.

How inflation affects your math:
If there is a fixed deductible $d$, inflation actually makes the insurer pay more than just the inflation rate! This is called leveraged effect of inflation.

Example:
Loss $X = 100$, Deductible $d = 50$. Insurer pays $100 - 50 = 50$.
Now, imagine 10% inflation. Loss becomes 110.
New payment = $110 - 50 = 60$.
Even though the loss only went up by 10%, the insurer's payment went from 50 to 60 (a 20% increase!).

Key Takeaway: Deductibles amplify the effect of inflation for the insurer. Limits do the opposite—they protect the insurer from inflation by capping the maximum payout.

Summary Checklist for Success

Ordinary Deductible: $Y = (X-d)_+$. Insurer pays only the excess.
Franchise Deductible: $Y = X$ if $X > d$. It's all or nothing!
Policy Limit: $Y = X \wedge u$. The insurer's "ceiling."
Coinsurance: $Y = \alpha X$. Sharing the bill.
Cost per Payment: Remember to divide the expected payment by the probability of exceeding the deductible: $E[Y_P] = \frac{E[Y_L]}{P(X > d)}$.
Inflation: Always apply the inflation factor to the loss $X$ before calculating the effect of deductibles and limits.

Don't worry if this seems tricky at first! These formulas become second nature once you practice applying them to different probability distributions (like the Exponential or Pareto). Keep practicing, and you'll be calculating these expected values in your sleep!

Quick check

Can you answer these now?

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

What is the mathematical definition of a payment under an ordinary deductible $d$?

$Y = (X - d)_+ = \max(0, X - d)$

How is the 'Cost per Loss' ($Y_L$) defined in the context of deductibles?

It is the payment amount considered across all losses, including those that fall below the deductible (where the payment is zero).

How is the 'Cost per Payment' ($Y_P$) defined in the context of deductibles?

It is the expected payment amount given that a payment actually occurred (i.e., when the loss exceeded the deductible).

What is the formula relating the expected cost per loss to the expected cost per payment?

$E[Y_L] = E[Y_P] \times P(X > d)$

What is the payment rule for a franchise deductible $d$?

$Y = 0$ if $X \le d$, and $Y = X$ if $X > d$.

What is the payment formula for a policy limit $u$?

$Y = X \wedge u = \min(X, u)$

How is a coinsurance factor $\alpha$ applied to a loss?

$Y = \alpha \times X$

What is the 'Master Formula' for insurance payments involving a limit, deductible, and coinsurance?

$Y = \alpha \times [(X \wedge u) - d]_+$

What is the 'leveraged effect' of inflation in the presence of a fixed deductible?

Inflation increases the insurer's expected payment by a higher percentage than the inflation rate itself.

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