Verified solution · Exam P

SOA Exam P Sample Question #543 Solution

This Exam P sample reference tests Deductible Payments. Because the loss is already above the fixed deductible, the payment is the affine transform p(X-d). Subtracting d does not change variance and multiplying by p multiplies variance by p², so the result is p²v and choice B is correct.

Independent analysis by ActuaryProof · reviewed August 4, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

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Independent solution

How to solve this Deductible Payments question

Setup

Write the payment as a constant shift followed by a proportional scaling.

Y=p(X-d)=pX-pd

Model

Variance is invariant under an additive constant and is multiplied by the square of a scale factor.

\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)

Compute

Apply the affine-variance rule to the payment.

\operatorname{Var}(Y)=p^2\operatorname{Var}(X)=p^2v

Answer

The deductible changes the mean but not the variance; the proportional factor supplies the square.

\boxed{p^2v\quad\text{(B)}}

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Error diagnosis

Why the other letters tempt

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

AThis scales variance by p rather than p², overlooking that standard deviation, not variance, scales linearly.
CThis both subtracts the deductible from a variance and uses a linear p factor; neither operation follows the affine-variance rule.
DThis correctly squares p but incorrectly subtracts the deductible d from the variance v.
EThis treats 1-p as the payment scale, confusing the retained share with the paid share.

Original practice · fully worked

Original variant: recover raw measurement variance

A laboratory converts a raw measurement L into a quality score Q=0.6(L-20). The standard deviation of Q is 9. Calculate Var(L).

A
15
B
20
C
81
D
135
E
225

Variant answer in brief

The converted score has variance 81, while Var(Q)=0.6²Var(L). Therefore Var(L)=81/0.36=225 and choice E is correct.

Setup

Square the reported score standard deviation.

\operatorname{Var}(Q)=9^2=81

Model

The shift by 20 has no variance effect, and the multiplier 0.6 is squared.

\operatorname{Var}(Q)=(0.6)^2\operatorname{Var}(L)

Compute

Invert the scale relationship.

\operatorname{Var}(L)=\frac{81}{0.36}=225

Answer

The raw measurement variance is 225.

\boxed{225\quad\text{(E)}}

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ActuaryProof. “SOA Exam P Sample Question #543 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Deductible Payments question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: recover raw measurement variance

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution