Verified solution · Exam FM

SOA Exam FM Sample Question #236 Solution

This Exam FM sample reference tests Annuities. The first 20 growing payments have present value 157337.48. The perpetuity beginning with payment 21 contributes 49202.44, giving total 206539.93 and choice C.

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 Annuities question

Setup

Split the payment stream after payment 20 because the growth rate changes at payment 21.

P_t=14000(1.04)^{t-1},\quad1\le t\le20

Model

Value the finite 4%-growth segment directly and value the later 1%-growth perpetuity at time 20.

PV_1=\sum_{t=1}^{20}\frac{14000(1.04)^{t-1}}{(1.10)^t}

PV_{20}=\frac{14000(1.04)^{19}(1.01)}{0.10-0.01}

Compute

The finite segment is 157337.48 and the discounted tail is 49202.44, totaling 206539.93.

PV=PV_1+PV_{20}(1.10)^{-20}=206539.925220

Answer

The perpetuity price is approximately 206,540, so choice C is correct.

\boxed{PV\approx206540\quad\text{(C)}}

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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 (185,542) does not satisfy the sum of the 20-payment growing segment and the time-20 value of the later growing perpetuity; no distinct standard one-step error is identifiable.
BChoice B (191,834) does not satisfy the sum of the 20-payment growing segment and the time-20 value of the later growing perpetuity; no distinct standard one-step error is identifiable.
DChoice D (208,508) does not satisfy the sum of the 20-payment growing segment and the time-20 value of the later growing perpetuity; no distinct standard one-step error is identifiable.
EChoice E (212,823) does not satisfy the sum of the 20-payment growing segment and the time-20 value of the later growing perpetuity; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: long-run growth rate inferred from a piecewise perpetuity price

A perpetuity pays 2,000 at year 1. Payments grow 2% annually through payment 10. Beginning with payment 11 they grow at a constant rate g forever. At a 7% annual yield, the price is 35666.33. Calculate g.

A
0.85%
B
0.90%
C
0.95%
D
1.00%
E
1.05%

Variant answer in brief

The first ten payments contribute 15213.00. Removing that value and bringing the residual to time 10 identifies the growing-perpetuity denominator, giving g = 1.00%, choice D.

Setup

Value the first ten payments using their known 2% growth pattern.

PV_{1:10}=\sum_{t=1}^{10}\frac{2000(1.02)^{t-1}}{(1.07)^t}

Model

The residual price is the time-0 value of a perpetuity whose first tail payment is the year-10 payment grown once at g.

P-PV_{1:10}=\frac{2000(1.02)^9(1+g)}{0.07-g}(1.07)^{-10}

Compute

Using the quoted price and the first-segment value 15213.00, solving the tail equation gives g = 1.0000%.

g=0.01000000

Answer

The long-run growth rate is 1.00%, which is choice D.

\boxed{g=1.00\%\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #236 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Annuities question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: long-run growth rate inferred from a piecewise perpetuity price

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution