Verified solution · Exam FM

SOA Exam FM Sample Question #313 Solution

This Exam FM sample reference tests Annuities. The first 30 due payments grow 3% and are discounted at 7%. From payment 31 onward the level amount equals payment 30. The finite block plus the time-29 level-perpetuity value totals 8033.38, so choice E is correct.

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

Value payments 1 through 30 at times 0 through 29 as a growing annuity-due.

PV_1=\sum_{k=0}^{29}350(1.03)^k(1.07)^{-k}

Model

At time 29, the future level perpetuity begins one year later with amount equal to payment 30.

P_{30}=350(1.03)^{29}

PV_2=\frac{P_{30}}{0.07}(1.07)^{-29}

Compute

The finite segment and tail sum to 8033.377503.

X=PV_1+PV_2=8033.37750310

Answer

The perpetuity-due price is approximately 8,033, selecting choice E.

\boxed{X\approx8033\quad\text{(E)}}

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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 (7508) does not satisfy the 30-payment growing due segment plus the level tail beginning after payment 30; no distinct standard one-step error is identifiable.
BChoice B (7855) does not satisfy the 30-payment growing due segment plus the level tail beginning after payment 30; no distinct standard one-step error is identifiable.
CChoice C (7925) does not satisfy the 30-payment growing due segment plus the level tail beginning after payment 30; no distinct standard one-step error is identifiable.
DChoice D (7971) does not satisfy the 30-payment growing due segment plus the level tail beginning after payment 30; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: early payment growth rate inferred from a capped perpetuity-due price

A perpetuity-due pays 200 immediately. Payments grow at rate g through payment 10, and every payment from number 11 onward remains equal to payment 10. At a 6% annual yield, the price is 4050.31. Calculate g.

A
1.90%
B
2.00%
C
2.10%
D
2.20%
E
2.30%

Variant answer in brief

Valuing the first ten due payments and the level tail as functions of g, then solving the quoted price, gives growth rate 2.00%, choice B.

Setup

Value the first ten due payments under the unknown growth rate.

PV_1(g)=\sum_{k=0}^{9}200(1+g)^k(1.06)^{-k}

Model

At time 9, append the level perpetuity beginning one year later at payment-10 amount.

PV_2(g)=\frac{200(1+g)^9}{0.06}(1.06)^{-9}

Compute

Solving PV1 plus PV2 equal to the quoted price gives g = 2.000000%.

g=0.0200000000

Answer

The early payment growth rate is 2.00%, selecting choice B.

\boxed{g=2.00\%\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #313 — 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: early payment growth rate inferred from a capped perpetuity-due price

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution