Worked solution · Exam FM

SOA Exam FM Sample Question #201 Solution

This Exam FM sample reference tests Growing Perpetuities. Separating the first 15 level payments from the deferred growing perpetuity and solving the value equation gives k = 1.889%, 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.

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Independent solution

How to solve this Growing Perpetuities question

Setup

Use time 0 as the comparison date. The level payments occur at times 3 through 17, and the growing tail begins at time 18.

i=0.035

v=(1.035)^{-1}

Model

Value the level block directly and value the growing perpetuity one period before its first payment, then discount that tail from time 17.

115000=2500v^2a_{\overline{15}|0.035}+2500v^{17}\frac{1+k}{0.035-k}

Compute

The first block is 26,879.07 and v to the 17th power is 0.557204. Solving the resulting linear equation in k gives 0.018893.

2500v^2a_{\overline{15}|}=26879.07

k=0.0188934

Answer

Expressed as a percentage, the annual growth rate is 1.89%, which matches choice E.

\boxed{k=1.89\%\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.

AUsing 1.66% understates the growing tail. The first growing payment is 2500 times 1 plus k, not 2500.
BThe value 1.74% is not obtained when the tail is valued at time 17 and then discounted by v to the 17th power.
CThe value 1.78% results from premature rounding of the deferred tail or annuity factor. Retaining v to the 17th power gives 1.889%.
DThe value 1.83% is too low because it effectively shifts the growing perpetuity one period farther from the valuation date.

Original practice · fully worked

Original variant: growing grants beside a capital outlay

An endowment has 50,000 today. It must make a one-time capital outlay of 12,000 at time 6. It will also fund annual grants forever, with the first grant at time 3 and each later grant 1.2% larger. At a 4% annual effective return, determine the first grant.

A
1,052
B
1,138
C
1,227
D
1,315
E
1,402

Answer and calculation

After reserving for the time-6 outlay, the remaining fund supports a first growing grant of 1,227.03, choice C.

Setup

Let X be the grant at time 3. Reserve the present value of the separate capital outlay before valuing the growing grants.

i=0.04

g=0.012

v=(1.04)^{-1}

Model

The growing perpetuity is worth X divided by i minus g at time 2, one period before the first grant.

50000=12000v^6+v^2\frac{X}{0.04-0.012}

Compute

The capital outlay requires 9,483.77 today. Solving for the first grant gives 1,227.03.

PV_{\mathrm{outlay}}=12000v^6=9483.77

X=(50000-9483.77)(0.028)v^{-2}=1227.03

Answer

The first perpetuity grant is about 1,227, which is choice C.

\boxed{X\approx1227\quad\text{(C)}}

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How to solve this Growing Perpetuities question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: growing grants beside a capital outlay

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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