Verified solution · Exam FM

SOA Exam FM Sample Question #45 Solution

This Exam FM sample reference tests Piecewise Geometric Cash Flows. The first segment is worth 13,136.41 and the second 7,552.22. The loan value is 20,688.63, or 20,689. The result agrees with the published answer key, 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 Piecewise Geometric Cash Flows question

Setup

Split the payment contract at its turning point. Payments 1–8 grow at 3%, while payments 9–16 decline from the year-8 level at 3%.

PV_1=\sum_{t=1}^{8}2000(1.03)^{t-1}v^t=13136.41

Model

Discount each segment term by term at 6%. Keeping the year-8 level in the second segment prevents restarting the declining series at an incorrect base payment.

PV_2=\sum_{t=9}^{16}2000(1.03)^7(0.97)^{t-8}v^t=7552.22

Compute

The first segment is worth 13,136.41 and the second 7,552.22. The loan value is 20,688.63, or 20,689.

L=PV_1+PV_2=20688.63

Answer

The calculation gives 20,689 for piecewise geometric cash flows, matching published choice A.

\boxed{L=20689\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 (20,716) does not match the checked piecewise geometric cash flows result (20,689); no distinct standard one-step error is identifiable.
CChoice C (20,775) does not match the checked piecewise geometric cash flows result (20,689); no distinct standard one-step error is identifiable.
DChoice D (21,147) does not match the checked piecewise geometric cash flows result (20,689); no distinct standard one-step error is identifiable.
EChoice E (22,137) does not match the checked piecewise geometric cash flows result (20,689); no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: rise-then-fall payment contract

A ten-year contract pays 1,000 in year 1, rises 2% annually through year 5, then falls 2% annually for years 6 through 10. At 6%, find its present value.

A
7200
B
7400
C
7589.84
D
7800
E
8000

Variant answer in brief

The cash-flow vector has present value 7,589.8442, or 7,589.84. This selects choice C.

Setup

Define payments explicitly: 2% annual growth through year 5, then 2% annual decline from that attained level.

P_t=1000(1.02)^{t-1}\ (t\le5)

Model

Discount the ten dated payments one by one at 6%.

P_t=1000(1.02)^4(0.98)^{t-5}\ (t>5)

Compute

The cash-flow vector has present value 7,589.8442, or 7,589.84.

PV=\sum_{t=1}^{10}P_t1.06^{-t}=7589.8442

Answer

The requested value is 7589.84, so the correct choice is C.

\boxed{PV=7589.84\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #45 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Piecewise Geometric Cash Flows question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: rise-then-fall payment contract

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution