Calculating the Impact of Coverage Modifications

An actuary models losses as exponential with mean $5,000 and computes the expected payment above a $1,000 deductible as $4,000. That shortcut only works for some distributions. Apply it blindly to a Pareto or uniform loss and the premium is materially wrong.

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 $d$:

$$E[(X-d{)}_{+}]=\theta e^{-d/\theta }$$

HIGH-FREQUENCY: The exponential per-payment mean is always $\theta$, regardless of the deductible. This is the memoryless property. The deductible only affects frequency, not severity.

For a layer from $d$ to $d+u$:

Common mistakes

• Forgetting the memoryless property. For exponential, the per-payment mean is always $\theta$. If your per-payment answer differs from $\theta$, you have an error.
• 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 always $\theta$ (memoryless)
• 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. "EXP stays the same" (exponential per-payment equals $\theta$, always. "Shift theta by d") Pareto per-payment is $(\theta +d)/(\alpha -1)$.

The full lesson (about 2,659 words, 18 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1b

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