SOA Exam P Sample Question #686. Official problem wording is withheld; follow the source link to read it.

Verified solution · Exam P

SOA Exam P Sample Question #686

The expected unreduced loss is 30(0.01)+10(0.05)=0.80. A target profit of 0.25 from a premium of 1 leaves 0.75 for expected reimbursement, so the reimbursement share is 0.75/0.80=0.9375, choice E.

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Independent solution

How to solve this Expected Value question

Answer in brief

The expected unreduced loss is 30(0.01)+10(0.05)=0.80. A target profit of 0.25 from a premium of 1 leaves 0.75 for expected reimbursement, so the reimbursement share is 0.75/0.80=0.9375, choice E.

Setup

Let F and L count the two covered event types, and let q be the constant share reimbursed.

\mathbb{E}[F]=0.01,\qquad \mathbb{E}[L]=0.05

Model

Linearity of expectation gives the expected gross loss without requiring any dependence assumption between the event counts.

\mathbb{E}[30F+10L]=30\mathbb{E}[F]+10\mathbb{E}[L]

Compute

Set premium less expected reimbursement equal to the target profit and solve for q.

\mathbb{E}[30F+10L]=30(0.01)+10(0.05)=0.80

1-q(0.80)=0.25

q=\frac{0.75}{0.80}=0.9375

Answer

The company should reimburse 93.75% of each covered loss.

\boxed{q=0.9375\quad\text{(E)}}

Error diagnosis

Why the other letters tempt

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

AThe value 0.20 is the expected profit if all of the 0.80 expected gross loss were reimbursed; it is a profit amount, not the reimbursement share.
BThe ratio 0.25/0.80=0.3125 divides target profit by expected loss, but q must fund the 0.75 reimbursement budget left after target profit.
CThe value 0.46875 results from halving the required reimbursement budget while using a gross-loss total that already includes both covered causes.
DThe value 0.80 is the expected gross loss per policyholder, expressed in premium units, rather than a proportion selected to meet the profit target.

Original practice · fully worked

Original variant: equipment protection cooperative

An equipment protection cooperative collects an annual fee of 100 dollars per member. A member is expected to incur 0.30 sensor failures costing 80 dollars each and 0.04 power-surge failures costing 400 dollars each during the year. The cooperative reimburses the same fraction of every covered loss and wants expected surplus of 75 dollars per member, with no other expenses. Calculate that reimbursement fraction.

A
0.250
B
0.375
C
0.400
D
0.625
E
0.750

Variant answer in brief

Expected covered loss is 0.30(80)+0.04(400)=40 dollars. The surplus target leaves 25 dollars for reimbursement, so the common fraction is 25/40=0.625, choice D.

Setup

Let q denote the common fraction of each loss paid by the cooperative.

\text{fee}=100,\qquad \text{target surplus}=75

Model

Add frequency times severity across the two failure sources to obtain expected covered loss.

\mathbb{E}[L]=0.30(80)+0.04(400)

Compute

Equate fee less expected reimbursement to the target surplus.

\mathbb{E}[L]=24+16=40

100-40q=75

q=\frac{25}{40}=0.625

Answer

The cooperative should reimburse 62.5% of every covered loss.

\boxed{q=0.625\quad\text{(D)}}

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ActuaryProof. “SOA Exam P Sample Question #686 — Verified Solution.” Verified 2026-07-30.

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