Welcome to Coverage Modifications: LER, ILF, and Deductible Factors!

Hello there, future Actuary! In this chapter, we are diving into the heart of how insurance policies are actually "priced" and adjusted. Think of a standard insurance policy as a plain vanilla cake. Coverage modifications are like adding frosting, sprinkles, or taking a slice out of it. They change how much the insurance company pays and how much the policyholder keeps.

We will focus on three main tools: Loss Elimination Ratios (LER), Increased Limits Factors (ILF), and Deductible Factors. These help actuary determine how much a premium should change when the terms of the contract change. Don't worry if these terms sound a bit intimidating—we will break them down piece by piece!

1. Loss Elimination Ratio (LER)

Imagine you have a phone insurance policy with a \$50 deductible. If you crack your screen and it costs \$150 to fix, you pay the first \$50, and the insurer pays \$100. That \$50 you paid was "eliminated" from the insurer's responsibility. The Loss Elimination Ratio tells us, on average, what percentage of total losses the insurance company saves by having a deductible in place.

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The Formula

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To calculate the LER for a deductible \( d \), we use the Limited Expected Value. If \( X \) is the total loss amount, the LER is:

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\( \text{LER} = \frac{E[X \wedge d]}{E[X]} \)

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Where:
\n- \( E[X \wedge d] \) is the expected value of losses capped at the deductible \( d \).
\n- \( E[X] \) is the total expected loss without any modifications.

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Why This Matters

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If an insurer knows that a \$1,000 deductible results in an LER of 0.20, it means the deductible is expected to eliminate 20% of the total claim costs. This is a huge deal when deciding how much of a discount to give a customer for choosing a higher deductible!

Example Analogy: The Coffee Shop

Imagine a coffee shop owner who gives away free coffee but has a "spill allowance." If a customer spills their coffee, the shop owner will replace it, but only the amount above 2 ounces. If most spills are small (under 2 ounces), the shop owner "eliminates" most of their loss. The LER represents that "saved" coffee.

Quick Review:
- LER measures the "savings" for the insurer due to a deductible.
- It is always a ratio between 0 and 1 (or 0% and 100%).
- The numerator is the expected value of the loss up to the deductible.

2. Increased Limits Factors (ILF)

In many lines of insurance (like auto liability), policies have a "Limit." This is the maximum the insurer will pay. But what if a customer wants more protection? What if they want to move from a \$100,000 limit to a \$500,000 limit? How much more should they pay? This is where the Increased Limits Factor (ILF) comes in.

The Formula

The ILF calculates the ratio of the expected cost of a higher limit \( u \) compared to a basic limit \( b \):

\( \text{ILF}(u) = \frac{E[X \wedge u]}{E[X \wedge b]} \)

Sometimes, insurers include a "Risk Load" to account for the extra danger of high-limit claims. If a risk load \( \lambda \) is included, the formula might look like this:
\( \text{ILF}(u) = \frac{E[X \wedge u] \times (1 + \lambda_u)}{E[X \wedge b] \times (1 + \lambda_b)} \)

Important Concept: The "Base"

The Basic Limit (the denominator) is the standard level of coverage. Every higher limit is expressed as a multiple of that base. If the ILF for a \$1M limit is 1.50, it means the coverage for \$1M is expected to cost 50% more than the basic limit coverage.

Did you know?
As the limit \( u \) increases, the ILF also increases, but it usually does so at a decreasing rate. This is because massive losses (like a \$10 million car accident) are much rarer than small ones.

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Common Mistake to Avoid:
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Don't confuse the limit with the deductible. A deductible is what the customer pays first (at the bottom of the loss), while a limit is the maximum the insurer pays at the top.

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Key Takeaway:
\nThe ILF tells us how much to scale the base premium to cover a higher maximum payout. It is the ratio of two Limited Expected Values.

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3. Deductible Factors

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While the LER looks at how much is eliminated, Deductible Factors are used to adjust the premium based on what is remaining. If you have a "base" deductible (let's say \$500) and you want to know the relative cost of a policy with a \$1,000 deductible, you use a deductible factor.

How it works

The cost to the insurer for a policy with a deductible \( d \) is the expected payment per loss:
\( E[(X-d)_+] = E[X] - E[X \wedge d] \)

A Deductible Factor (sometimes called a "Relativity") compares the cost of a policy with deductible \( d \) to the cost of a policy with a base deductible \( b \):

\( \text{Factor} = \frac{E[X] - E[X \wedge d]}{E[X] - E[X \wedge b]} \)

Simplified View

Think of it as: (What the insurer pays with the new deductible) / (What the insurer pays with the base deductible).

Step-by-Step Calculation:

1. Calculate the total expected loss \( E[X] \).
2. Find the Limited Expected Value at the base deductible \( E[X \wedge b] \).
3. Find the Limited Expected Value at the new deductible \( E[X \wedge d] \).
4. Subtract the LEVs from the total to find the insurer's expected "net" payment.
5. Divide the new net payment by the base net payment.

Pro-Tip: If the question asks for the factor relative to no deductible, the denominator is simply \( E[X] \), and the factor becomes \( 1 - \text{LER} \)!

4. Summary and Memory Aids

Don't worry if these formulas feel a bit "math-heavy." The logic is always about comparing Limited Expected Values (LEVs). Here is a quick summary table to keep them straight:

1. Loss Elimination Ratio (LER): Focuses on the Savings.
Formula: \( \frac{\text{LEV at deductible}}{\text{Total Expected Loss}} \)

2. Increased Limits Factor (ILF): Focuses on the Ceiling.
Formula: \( \frac{\text{LEV at High Limit}}{\text{LEV at Base Limit}} \)

3. Deductible Factor: Focuses on the Payment.
Formula: \( \frac{\text{Expected Payment with New Deductible}}{\text{Expected Payment with Base Deductible}} \)

Memory Trick: "The Top and Bottom"

- LER is about the bottom portion of the loss (what the customer pays).
- ILF is about the top portion of the loss (how high the insurer goes).
- Both rely on the Limited Expected Value, which is the most important function to master for Exam ASTAM!

You're doing great! These concepts are the bread and butter of actuarial pricing. Practice a few problems using the Pareto or Exponential distributions for \( X \), as those are very common on the exam. You've got this!