Verified solution · Exam FM

SOA Exam FM Sample Question #242 Solution

This Exam FM sample reference tests Annuities. Payments at years 5 through 10 are discounted at 1%, while the remaining perpetuity is valued at time 10 using 5% and then discounted through the first regime. The total factor is 23.675078, giving X = 4223.85 and choice B.

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 Annuities question

Setup

Separate the six payments at times 5 through 10 from the payments beginning at time 11.

PV_{5:10}=Xa_{\overline6|0.01}(1.01)^{-4}

Model

At time 10, payments from year 11 onward form a perpetuity-immediate at 5%.

PV_{10,\mathrm{tail}}=\frac{X}{0.05}

PV_0=PV_{5:10}+\frac{X}{0.05}(1.01)^{-10}

Compute

The combined present-value factor is 23.67507807, so X = 4223.8509.

100000=X(23.6750780728)

X=4223.85090737

Answer

The annual perpetuity payment is approximately 4,224, which is choice B.

\boxed{X\approx4224\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.

AChoice A (4100) does not satisfy the split present value of six early payments and the time-10 perpetuity tail; no distinct standard one-step error is identifiable.
CChoice C (4357) does not satisfy the split present value of six early payments and the time-10 perpetuity tail; no distinct standard one-step error is identifiable.
DChoice D (4401) does not satisfy the split present value of six early payments and the time-10 perpetuity tail; no distinct standard one-step error is identifiable.
EChoice E (5696) does not satisfy the split present value of six early payments and the time-10 perpetuity tail; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: long-run rate inferred from a deferred perpetuity price

A perpetuity pays 3,000 annually beginning at year 3. Interest is 2% annually through year 5 and a constant rate j thereafter. Its present value is 62659.54. Calculate j.

A
5.00%
B
5.25%
C
5.50%
D
5.75%
E
6.00%

Variant answer in brief

Remove the year-3 through year-5 payments from the quoted value. Accumulating the residual to time 5 gives the value of a 3,000 perpetuity, so j = 5.00%, choice A.

Setup

Discount the three payments occurring under the 2% regime separately.

PV_{3:5}=3000(1.02)^{-3}+3000(1.02)^{-4}+3000(1.02)^{-5}

Model

The residual price is the discounted time-5 value of payments beginning at year 6.

P-PV_{3:5}=\frac{3000}{j}(1.02)^{-5}

Compute

Solving the tail-value equation gives long-run rate 5.0000%.

j=0.05000000

Answer

The later annual rate is 5.00%, selecting choice A.

\boxed{j=5.00\%\quad\text{(A)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #242 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Annuities question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: long-run rate inferred from a deferred perpetuity price

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution