Verified solution · Exam FM

SOA Exam FM Sample Question #302 Solution

This Exam FM sample reference tests Annuities. Payment k is 100 times 0.9 to the k−1 and earns 3% from its receipt date through year 20. Accumulating all twenty payments gives 1295.80, so the balance is 1,296 and 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.

Read the solution Open official source

Independent solution

How to solve this Annuities question

Setup

Write the declining payment and its accumulation time to the final receipt date.

P_k=100(0.9)^{k-1}

F_{20}=\sum_{k=1}^{20}P_k(1.03)^{20-k}

Model

Recognize the terminal-value expression as a finite geometric sum.

F_{20}=100\sum_{k=1}^{20}(0.9)^{k-1}(1.03)^{20-k}

Compute

Evaluating the twenty accumulated contributions gives 1295.795831.

F_{20}=1295.79583083

Answer

The savings balance is approximately 1,296, selecting choice D.

\boxed{F_{20}\approx1296\quad\text{(D)}}

Continue without hunting through PDFs

All 461 Exam FM sample solutions in syllabus order

The searchable 2210-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam FM study manual PDF

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 (696) does not satisfy the year-20 accumulated value of all twenty 10%-declining payments; no distinct standard one-step error is identifiable.
BChoice B (717) does not satisfy the year-20 accumulated value of all twenty 10%-declining payments; no distinct standard one-step error is identifiable.
CChoice C (739) does not satisfy the year-20 accumulated value of all twenty 10%-declining payments; no distinct standard one-step error is identifiable.
EChoice E (1335) does not satisfy the year-20 accumulated value of all twenty 10%-declining payments; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: first deposit required for a declining contribution target

A saver makes twelve year-end deposits. The first is P and every later deposit is 5% smaller. The account earns 4% annually. Immediately after the twelfth deposit, the fund must equal 10,000. Calculate P.

A
678.81
B
721.24
C
763.67
D
806.09
E
848.52

Variant answer in brief

Accumulating twelve unit-scaled deposits under 5% decline and 4% account growth gives factor 11.785246. Dividing 10,000 by this factor gives first deposit 848.52, choice E.

Setup

Express deposit k and its accumulation to the time-12 target.

D_k=P(0.95)^{k-1}

F_{12}=\sum_{k=1}^{12}D_k(1.04)^{12-k}

Model

Factor out the unknown first deposit from the finite geometric sum.

10000=P\sum_{k=1}^{12}(0.95)^{k-1}(1.04)^{12-k}

Compute

The unit accumulation factor is 11.78524590, so P = 848.52.

P=848.51857023

Answer

The first deposit is 848.52, selecting choice E.

\boxed{P\approx848.52\quad\text{(E)}}

Keep practicing

Turn this solution into a complete study sequence

The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

See the Exam FM study manual PDF — $49 More Exam FM Annuities and Non-contingent Cash Flows practice problems Free Exam FM formula sheet

Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #302 — 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: first deposit required for a declining contribution target

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution