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Deductibles, policy limits and risk measures

Insurance modifies a loss before it is paid. Each modification is an integral of the survival function over the relevant range.

The formulas

Ordinary deductible

$$E[(X-d{)}_{+}]={\int }_d^{\infty }S(x)dx$$

Policy limit

$$E[X\wedge u]={\int }_0^{u}S(x)dx$$

Decomposition

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

Exponential deductible

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

memorylessness

Coinsurance

$$E[\alpha (X-d{)}_{+}]=\alpha E[(X-d{)}_{+}]$$

Where it comes from

1. Writing $E[(X-d{)}_{+}]={\int }_d^{\infty }(x-d)f(x)dx$ and integrating by parts gives ${\int }_d^{\infty }S(x)dx$ — the area under the survival curve past $d$.
2. The same argument on $[0,u]$ gives the limited expected value.
3. The two pieces reassemble the whole loss, which is the identity worth checking every answer against.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Per LOSS averages the zeros in; per PAYMENT conditions on exceeding the deductible and divides by S(d).
• For the exponential the excess above any deductible is exponential with the same mean — memorylessness does all the work.

Traps

• Reporting the expected payment per loss when the question asked per payment (they differ by the factor S(d)).
• Applying memorylessness to a distribution that is not exponential.

Related

• Exponential distribution
• Pareto distribution
• Expectation, variance and moments

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