Verified solution · Exam FM

SOA Exam FM Sample Question #123 Solution

This Exam FM sample reference tests Annuities and Perpetuities. The decreasing payments form a geometric perpetuity with ratio 0.996v; at force 0.06 its value factor is 15.189, so the first payment is 10,000/15.189 = 658.37, choice D.

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

The first payment occurs after one year; each later amount is 99.6% of the preceding payment.

v=e^{-0.06}

Model

Combining payment decline and force-of-interest discounting gives a geometric ratio of 0.996 times exp(−0.06).

10000=X(v+0.996v^2+0.996^2v^3+\cdots)

Compute

The infinite-series factor is 15.189, and dividing the purchase price by it gives 658.37.

10000=X\frac{v}{1-0.996v}=15.189X

Answer

Thus the first annual payment is approximately 658, choice D.

\boxed{X=658.37\quad\text{(D)}}

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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 geometric ratio combining 0.4% payment decline and continuous discounting; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the geometric ratio combining 0.4% payment decline and continuous discounting; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the geometric ratio combining 0.4% payment decline and continuous discounting; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the geometric ratio combining 0.4% payment decline and continuous discounting; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: value of a gradually shrinking watershed grant

A watershed grant pays 100 at the end of year 1. Every later annual payment is 1% smaller than the preceding payment. Money is discounted at annual force 0.04. Calculate the present value of the grant stream.

A
1,700.00
B
1,850.00
C
1,968.09
D
2,050.00
E
2,200.00

Variant answer in brief

The combined payment-and-discount ratio gives present value 1,968.09, choice C.

Setup

Each step from one payment to the next combines a 1% amount reduction with one further year of discounting.

v=e^{-0.04},\qquad q=0.99v

Model

The year-one discounted payment is the first term of a geometric series with ratio 0.99e to the negative 0.04.

PV=100v\sum_{k=0}^{\infty}q^k

Compute

Summing that series gives 1,968.0865.

PV=\frac{100e^{-0.04}}{1-0.99e^{-0.04}}=1968.0865

Answer

The grant stream is worth 1,968.09 today, so choice C is correct.

\boxed{PV=1968.09\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #123 — 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: value of a gradually shrinking watershed grant

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution