Worked solution · Exam FM

SOA Exam FM Sample Question #209 Solution

This Exam FM sample reference tests Reinvestment of Interest. Separating the 50,000 of deposited principal from reinvested interest gives i = 15.592%, so the answer is choice B.

Independent analysis by ActuaryProof · reviewed October 6, 2026

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Independent solution

How to solve this Reinvestment of Interest question

Setup

Ten beginning-of-year deposits contribute 50,000 of principal. The remaining 50,000 of the final fund must come from accumulated interest.

100000-10(5000)=50000

Model

Interest earned in each year is 5000i on every deposit then on deposit. Reinvest those interest amounts to time 10 at 5%.

5000i\frac{\ddot{s}_{\overline{10}|0.05}-10}{0.05}=50000

Compute

The accumulated-value annuity-due factor is 13.206787. Substitution isolates i.

\ddot{s}_{\overline{10}|0.05}=13.206787

i=\frac{10(0.05)}{13.206787-10}=0.155919

Answer

The annual rate is 15.6% to the listed precision, which is choice B.

\boxed{i\approx15.6\%\quad\text{(B)}}

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 understates the interest contribution by treating the deposits as end-of-year rather than beginning-of-year cash flows.
CChoice C fails to subtract the 50,000 of deposited principal before solving for the earned-interest rate.
DChoice D reinvests the principal as well as the interest at 5%, contrary to the stated separation.
EChoice E omits the division by the 5% reinvestment rate in the accumulated-interest series.

Original practice · fully worked

Original variant: terminal fund with separately reinvested earnings

A cooperative deposits 3,000 at the beginning of each year for eight years. Each deposit earns annual interest equal to 12% of its original principal, and every interest credit is reinvested at 4% annually. Determine the fund at the end of year eight.

A
33,518
B
36,804
C
38,245
D
40,000
E
42,617

Answer and calculation

The 24,000 of principal plus accumulated reinvested interest produces 38,245.16 at time 8, which is choice C.

Setup

The deposits contribute 24,000 of principal. Treat the annual interest credits as a separate cash-flow stream.

P=8(3000)=24000

Model

Each active deposit earns 360 per year. Accumulating the triangular pattern of interest credits at 4% gives the standard annuity-due difference factor.

I_8=3000(0.12)\frac{\ddot{s}_{\overline{8}|0.04}-8}{0.04}

Compute

The annuity-due accumulated factor is 9.582795, so reinvested interest contributes 14,245.16.

I_8=360\frac{9.582795-8}{0.04}=14245.16

F_8=24000+14245.16=38245.16

Answer

The ending fund is approximately 38,245, so the correct choice is C.

\boxed{F_8\approx38245\quad\text{(C)}}

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How to solve this Reinvestment of Interest question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: terminal fund with separately reinvested earnings

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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