Verified solution · Exam FM

SOA Exam FM Sample Question #298 Solution

This Exam FM sample reference tests Annuities. Value the first five payments under 4% growth, then value at time 5 the perpetuity whose next payment reflects the fifth 4% increase and grows 8% thereafter. The two components are 3.87 and 17.26, totaling 21.13; choice A.

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 five payments from the later perpetuity because the growth rate changes after the fifth 4% increase.

PV_1=\sum_{t=1}^{5}\frac{(1.04)^{t-1}}{(1.12)^t}

Model

At time 5, the tail begins with payment 1.04 to the fifth and then grows at 8%.

PV_2=\frac{(1.04)^5}{0.12-0.08}(1.12)^{-5}

Compute

The finite block is 3.870481 and the tail is 17.259038, giving total 21.129519.

PV=PV_1+PV_2=21.12951914

Answer

The perpetuity present value is approximately 21.13, selecting choice A.

\boxed{PV\approx21.13\quad\text{(A)}}

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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.

BChoice B (23.51) does not satisfy the finite 4%-growth block plus the time-5 value of the 8%-growth tail; no distinct standard one-step error is identifiable.
CChoice C (25.95) does not satisfy the finite 4%-growth block plus the time-5 value of the 8%-growth tail; no distinct standard one-step error is identifiable.
DChoice D (28.87) does not satisfy the finite 4%-growth block plus the time-5 value of the 8%-growth tail; no distinct standard one-step error is identifiable.
EChoice E (31.23) does not satisfy the finite 4%-growth block plus the time-5 value of the 8%-growth tail; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: growth-transition date inferred from a perpetuity price

A perpetuity pays 1 at year 1. Payments initially grow 3% annually; after N early payments, all later payments grow 6% annually. At a 10% yield, the price is 22.5222. Determine N.

A
3 payments
B
4 payments
C
5 payments
D
6 payments
E
7 payments

Variant answer in brief

Testing the finite early-growth block together with the later 6%-growth tail shows the quoted price is produced by N = 4 early payments, choice B.

Setup

For a candidate N, value the first N payments under 3% growth.

PV_1(N)=\sum_{t=1}^{N}\frac{(1.03)^{t-1}}{(1.10)^t}

Model

At time N, append the perpetuity tail beginning with the next 3%-grown payment and then growing 6%.

PV_2(N)=\frac{(1.03)^N}{0.10-0.06}(1.10)^{-N}

Compute

The sum matches the quoted price at N = 4.

PV_1(4)+PV_2(4)=22.5221892289

Answer

There are 4 early-growth payments, selecting choice B.

\boxed{N=4\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #298 — 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: growth-transition date inferred from a 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