Verified solution · Exam FM

SOA Exam FM Sample Question #120 Solution

This Exam FM sample reference tests Annuities and Perpetuities. Writing the payments as a level perpetuity of 4 plus twice an increasing perpetuity gives 4/0.06 + 2(1.06)/0.06² = 655.56, 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 and Perpetuities question

Setup

Separate each payment into a constant four-unit layer plus twice the sequence one, two, three, and so on.

6,8,10,\ldots=4+2(1,2,3,\ldots)

Model

The first layer is a level perpetuity-immediate and the second is a scaled increasing perpetuity-immediate.

PV=\frac4i+2\frac{1+i}{i^2}

Compute

Applying the two closed forms at 6% produces 66.67 plus 588.89, totaling 655.56.

PV=\frac4{0.06}+2\frac{1.06}{0.06^2}=655.56

Answer

The required present value is approximately 656, which is choice B.

\boxed{PV=655.56\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 is inconsistent with the level-plus-increasing decomposition of payments 6, 8, 10, …; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the level-plus-increasing decomposition of payments 6, 8, 10, …; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the level-plus-increasing decomposition of payments 6, 8, 10, …; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the level-plus-increasing decomposition of payments 6, 8, 10, …; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: perpetual maintenance grants with a fixed annual step

A district pays 10 at the end of year 1, 13 at the end of year 2, 16 at the end of year 3, and continues increasing by 3 each year forever. At 5% effective, find the present value.

A
1,000
B
1,200
C
1,400
D
1,600
E
1,800

Variant answer in brief

A level perpetuity of 10 plus a three-unit arithmetic increase has value 1,400, choice C.

Setup

Retain the first-payment level of ten and isolate the three-unit step that begins with the second payment.

PV=\frac{10}{0.05}+3\sum_{k=1}^{\infty}(k-1)v^k

Model

The step component has discounted value one over i squared for each unit of annual increase.

\sum_{k=1}^{\infty}(k-1)v^k=\frac1{i^2}

Compute

At 5%, the level layer is 200 and the growth layer is 1,200.

PV=200+\frac3{0.05^2}=1400

Answer

Their sum is 1,400, which appears in choice C.

\boxed{PV=1400\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #120 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Annuities and Perpetuities question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: perpetual maintenance grants with a fixed annual step

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution