Calculating the Impact of Coverage Modifications

An actuary models losses as exponential with mean $5,000 and estimates payment above a $1,000 ordinary deductible by subtracting the deductible from the mean. That gives $4,000, but the correct expected payment per loss is $4,093.65. Losses below the deductible produce zero payment, so the payment transformation must be applied before taking the expectation.

To perform calculations assessing a coverage modification, begin with the payment per loss. Conditional payment amounts then divide by the probability that a payment occurs.

Every coverage modification calculation reduces to limited expected values. From 1a:

$$E[(X-d)_+] = E[X] - E[X \wedge d]$$

$$E[\min((X-d)_+, u)] = E[X \wedge (d+u)] - E[X \wedge d]$$

Your task is to evaluate $E[X \wedge c]$ for specific distributions using:

$$E[X \wedge c] = \int_0^{c} S(x)\,dx$$

For $X \sim \text{Exponential}(\theta)$ with mean $\theta$:

$$E[X \wedge c] = \theta(1 - e^{-c/\theta})$$

The expected excess above deductible $d$:

Common mistakes

• Applying memorylessness after other modifications. For exponential losses with only an ordinary deductible, the per-payment mean is $\theta$. With payment limit $u$ and no coinsurance, it is $\theta(1-e^{-u/\theta})$. Check all policy terms before using the shortcut.
• Using the wrong Pareto LEV formula. Two-parameter Pareto has $(\theta + c)$ in the denominator. Single-parameter Pareto has a different formula. Mixing them changes the answer entirely.
• Omitting the limit's effect on per-payment mean. Without a limit, Pareto per-payment is $(\theta + d)/(\alpha - 1)$. With a limit, it is smaller. The unlimited value is a trap distractor.

Bottom line

• Exponential: $E[(X-d)_+] = \theta e^{-d/\theta}$; per-payment mean is $\theta$ with only an ordinary deductible
• Pareto: $E[Y^P] = (\theta + d)/(\alpha - 1)$, per-payment severity increases with $d$
• Uniform on (0,b): $E[(X-d)_+] = (b-d)^2/(2b)$, valid only while $d < b$
• Layer payment: $E[X \wedge (d+u)] - E[X \wedge d]$ works for all distributions

Exam shortcut

For exponential, skip the LEV, jump straight to $\theta e^{-d/\theta}$. For Pareto, compute the survival ratio $(\theta/(\theta+c))^{\alpha-1}$ first and evaluate it numerically. Most errors creep in when computing this ratio. With only an ordinary deductible, exponential per-payment mean equals $\theta$. For Pareto with shape above one, it is $(\theta + d)/(\alpha - 1)$. Apply any limit or coinsurance before taking the payment expectation.

The full lesson (about 3,186 words, 21 min read) adds 6 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1b

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