Worked solution · Exam P
This Exam P sample reference tests Expected Value. 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.
Independent solution
Setup
Let F and L count the two covered event types, and let q be the constant share reimbursed.
Model
Linearity of expectation gives the expected gross loss without requiring any dependence assumption between the event counts.
Compute
Set premium less expected reimbursement equal to the target profit and solve for q.
Answer
The company should reimburse 93.75% of each covered loss.
Test the method on unseen problems
30 original multivariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
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.
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.
Model
Add frequency times severity across the two failure sources to obtain expected covered loss.
Compute
Equate fee less expected reimbursement to the target surplus.
Answer
The cooperative should reimburse 62.5% of every covered loss.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 140 Exam P multivariate random variables practice problems.
Drill the method: Exam P Tail Integral Formula and Expected Value problems
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