Verified solution · Exam FM

SOA Exam FM Sample Question #451 Solution

This Exam FM sample reference tests Two-Phase Deposit Accumulation. The first 20 quarterly deposits accumulate to 48138.59. The following 60 deposits grow 2% each quarter and accumulate to 161639.07. Their total is 209777.66, which rounds to choice E.

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 Two-Phase Deposit Accumulation question

Setup

The quarterly yield is 1.25%. Separate the level first five years from the geometrically increasing final fifteen years.

j=0.05/4=0.0125

Model

Accumulate the twenty beginning-quarter deposits at time 5 and then for sixty more quarters. Index the later deposits from k = 1 through 60.

S_1=1{,}000\ddot s_{\overline{20}|j}(1+j)^{60}

S_2=\sum_{k=1}^{60}1{,}000(1.02)^k(1+j)^{61-k}

Compute

The two accumulated components are 48138.59 and 161639.07.

S_1=48{,}138.591

S_2=161{,}639.068

S_1+S_2=209{,}777.659

Answer

The time-20 fund rounds to 209778, choice E.

\boxed{209{,}778\quad\text{(E)}}

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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 accumulates only a portion of the sixty growing deposits after year 5.
BChoice B treats the 2% change as annual rather than applying it to each successive quarterly deposit.
CChoice C omits the first twenty level deposits after valuing the growing sequence.
DChoice D makes the first post-year-5 deposit equal to 1000 instead of 1020, shifting the entire geometric block down one step.

Original practice · fully worked

Original variant: first deposit for a growing savings plan

A saver deposits money at the beginning of each year for five years, at times 0 through 4. Each deposit is 3% larger than the previous one. The account earns 5% effective annually and is worth 6,000 at time 5. Determine the first deposit.

A
900.14
B
945.71
C
976.74
D
1,018.27
E
1,200.00

Variant answer in brief

Accumulating the five geometrically increasing deposits to time 5 gives a coefficient of 6.142893 times the first deposit. The first deposit is 976.74, choice C.

Setup

Let X be the deposit at time zero. Deposit k is X times 1.03 to the k.

D_k=X(1.03)^k,\qquad 0\le k\le4

Model

Accumulate each beginning-year deposit from time k to time 5.

6{,}000=X\sum_{k=0}^{4}(1.03)^k(1.05)^{5-k}

Compute

The accumulation coefficient is 6.142893.

X=\frac{6{,}000}{6.14289313}=976.7385

Answer

The first deposit is 976.74, choice C.

\boxed{X=976.74\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #451 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Two-Phase Deposit Accumulation question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: first deposit for a growing savings plan

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution