Worked 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 October 6, 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 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)}}

Test the method on unseen problems

Financial Mathematics Practice Examination I

30 original Exam FM questions across the syllabus, to be worked in 150 timed minutes, with a full worked solution for each one. Written for Exam FM, not copied from the SOA set. One-time $39 PDF.

See the cover and sample pages

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

Answer and calculation

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)}}

Keep practicing

Next steps for Exam FM Annuities and Non-contingent Cash Flows

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam FM solution in one searchable PDF.

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

How to solve this Two-Phase Deposit Accumulation question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: first deposit for a growing savings plan

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

Next steps for Exam FM Annuities and Non-contingent Cash Flows