SOA Exam P Sample Question #727. Official problem wording is withheld; follow the source link to read it.

SOA Exam P Sample Question #727 — Verified Solution

Independent solution

How to solve this Exponential Distribution question

Answer in brief

For an exponential loss, the expected excess above a deductible equals the mean times the survival probability at that deductible. Applying that identity gives a mean of approximately 1790.190, which matches choice B.

Setup

Represent the insurer's payment as the positive part of the loss after the deductible.

d=0.4\beta,\qquad Y=(X-d)_+

Model

Use the exponential survival function and memoryless property to factor the expected payment into the chance of crossing the deductible and the mean residual loss.

\mathbb{E}[Y]=\Pr(X>d)\,\mathbb{E}[X-d\mid X>d]

\Pr(X>d)=e^{-d/\beta}=e^{-0.4},\qquad \mathbb{E}[X-d\mid X>d]=\beta

Compute

Equate the stop-loss expectation to the supplied expected benefit and solve for the exponential mean.

1200=\beta e^{-0.4}

\beta=1200e^{0.4}=1790.189637\ldots

Answer

The nearest listed amount is 1790.

\boxed{\beta\approx 1790.190\quad\text{(B)}}

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AMultiplying 1200 by the survival factor gives 1200e^{-0.4}=804.38; the factor must divide the expected payment when solving for the mean.
CUsing 1-0.4=0.6 as though it were an exponential survival probability gives 1200/0.6=2000, but the correct survival value is e^{-0.4}.
DDividing the correct 1790.19 result by a further 0.6 gives 2983.65; that applies an unsupported retained-loss fraction after the deductible has already been handled.
EUsing the probability of falling below the deductible gives 1200/(1-e^{-0.4})=3639.89, which conditions on the wrong side of the threshold.

Original practice · fully worked

Original variant: environmental cleanup reimbursement

An environmental response fund reimburses 80% of the portion of a chemical-cleanup cost above a trigger. The cleanup cost C is exponential with mean theta, the trigger is 0.2 theta, and the expected reimbursement per incident is 1000 dollars. Determine theta.

A
819
B
1250
C
1527
D
1908
E
6896

Variant answer in brief

Coinsurance multiplies the exponential excess-loss expectation by 0.8. Solving the resulting equation gives theta approximately 1526.753, so choice C is correct.

Setup

Write the fund's reimbursement using the trigger and the 80% reimbursement rate.

R=0.8(C-0.2\theta)_+

Model

For an exponential cost, the expected excess above 0.2 times the mean is the mean multiplied by the corresponding survival factor.

\mathbb{E}[R]=0.8\theta e^{-0.2}

Compute

Set the modeled reimbursement equal to 1000 and isolate the mean.

1000=0.8\theta e^{-0.2}

\theta=\frac{1000e^{0.2}}{0.8}=1526.753448\ldots

Answer

Rounding to the listed whole-dollar values selects choice C.

\boxed{\theta\approx 1527\quad\text{(C)}}

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ActuaryProof. “SOA Exam P Sample Question #727 — Verified Solution.” Verified 2026-07-30.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.