Verified solution · Exam FM

SOA Exam FM Sample Question #316 Solution

This Exam FM sample reference tests Annuities. Value the first ten level payments of k, then at time 10 value a perpetuity beginning with k + 200 and increasing by 200 annually. Solving the linear 50,000 price equation gives k = 162.83, so choice C 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 Annuities question

Setup

Separate the first ten level payments from the increasing perpetuity beginning with payment 11.

PV_1=ka_{\overline{10}|0.052}

Model

At time 10, value the tail as a level perpetuity beginning at k + 200 plus an arithmetic-increase perpetuity.

PV_{10}=\frac{k+200}{0.052}+\frac{200}{(0.052)^2}

Compute

Discounting the tail ten years and solving gives k = 162.834657.

50000=PV_1+PV_{10}(1.052)^{-10}

k=162.83465738

Answer

The first ten payments are approximately 163, selecting choice C.

\boxed{k\approx163\quad\text{(C)}}

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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 (34) does not satisfy the first ten level payments plus the deferred arithmetic-increasing perpetuity; no distinct standard one-step error is identifiable.
BChoice B (86) does not satisfy the first ten level payments plus the deferred arithmetic-increasing perpetuity; no distinct standard one-step error is identifiable.
DChoice D (283) does not satisfy the first ten level payments plus the deferred arithmetic-increasing perpetuity; no distinct standard one-step error is identifiable.
EChoice E (409) does not satisfy the first ten level payments plus the deferred arithmetic-increasing perpetuity; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: perpetual annual increment inferred after an initial level block

A perpetuity-immediate pays 500 in each of years 1 through 8. Beginning with year 9, every payment is D larger than the previous payment. At a 6% annual yield, the price is 17570.24. Calculate D.

A
42.50
B
45.00
C
47.50
D
50.00
E
52.50

Variant answer in brief

Value the first eight payments separately and express the time-8 increasing perpetuity in terms of D. Solving the quoted price gives annual increment 50.00, choice D.

Setup

Value the first eight level payments at 6%.

PV_1=500a_{\overline8|0.06}

Model

At time 8, the tail begins with 500 + D and increases by D forever.

PV_8=\frac{500+D}{0.06}+\frac{D}{(0.06)^2}

Compute

Solving the quoted price equation gives D = 50.000000.

17570.24=PV_1+PV_8(1.06)^{-8}

D=50.00000000

Answer

The perpetual annual increment is 50.00, selecting choice D.

\boxed{D=50.00\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #316 — 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: perpetual annual increment inferred after an initial level block

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution