Worked solution · Exam FM

SOA Exam FM Sample Question #8 Solution

This Exam FM sample reference tests Geometric Annuity Exchange. Equating 25X/1.08 to 1,250 gives X = 54. This cancellation is specific to equal growth and yield rates. The result agrees with the published answer key, choice A.

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.

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Independent solution

How to solve this Geometric Annuity Exchange question

Setup

At time 5, the original perpetuity is worth 100/0.08 = 1,250. Move the replacement stream to that same date before comparing it.

V_5=100/0.08=1250

Model

Each replacement payment grows at the same 8% rate used for discounting. Consequently every discounted term has the same time-5 value X/1.08, and the 25-term sum is 25X/1.08.

1250=X\sum_{t=1}^{25}1.08^{t-1}v^t

Compute

Equating 25X/1.08 to 1,250 gives X = 54. This cancellation is specific to equal growth and yield rates.

1250=X(25/1.08)\Longrightarrow X=54

Answer

The calculation gives 54 for geometric annuity exchange, matching published choice A.

\boxed{X=54\quad\text{(A)}}

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.

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Original practice · fully worked

Original variant: ten-payment growing service stream

A service contract pays 50 one year from now, with each of the next nine annual payments 3% larger. At 5% interest, find its present value.

A
400.00
B
420.50
C
437.38
D
450.00
E
475.00

Answer and calculation

The series totals 437.3798, so value is 437.38. This selects choice C.

Setup

Payment t is 50(1.03) to the power t−1 and is discounted t years at 5%.

PV=\sum_{t=1}^{10}\frac{50(1.03)^{t-1}}{1.05^t}

Model

The present values form a ten-term geometric series with first term 50/1.05 and ratio 1.03/1.05.

PV=\frac{50}{1.05}\frac{1-(1.03/1.05)^{10}}{1-1.03/1.05}

Compute

The series totals 437.3798, so value is 437.38.

PV=437.3798

Answer

The requested value is 437.38, so the correct choice is C.

\boxed{PV=437.38\quad\text{(C)}}

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How to solve this Geometric Annuity Exchange question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Original variant: ten-payment growing service stream

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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