Verified solution · Exam FM

SOA Exam FM Sample Question #309 Solution

This Exam FM sample reference tests Annuities. After payment 11, nine annual payments of 10 remain, worth 68.02. The equivalent half-year rate is 2.9563%, and a semiannual perpetuity growing 0.5% has value K divided by j − 0.005. Solving gives K = 1.67, choice B.

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 the nine remaining annual payments immediately after payment 11.

V=10a_{\overline9|0.06}

Model

Convert 6% annual effective to a half-year effective rate and value the growing perpetuity-immediate.

j=(1.06)^{1/2}-1

V=\frac{K}{j-0.005}

Compute

The exchange value is 68.016923, giving first semiannual payment 1.670701.

K=V(j-0.005)=1.67070063

Answer

The first perpetuity payment is approximately 1.67, selecting choice B.

\boxed{K\approx1.67\quad\text{(B)}}

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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 (1.53) does not satisfy the equality of the nine-payment annual annuity value and semiannual growing perpetuity value; no distinct standard one-step error is identifiable.
CChoice C (2.37) does not satisfy the equality of the nine-payment annual annuity value and semiannual growing perpetuity value; no distinct standard one-step error is identifiable.
DChoice D (3.42) does not satisfy the equality of the nine-payment annual annuity value and semiannual growing perpetuity value; no distinct standard one-step error is identifiable.
EChoice E (3.74) does not satisfy the equality of the nine-payment annual annuity value and semiannual growing perpetuity value; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: remaining level-annuity term inferred from an exchange perpetuity

At a 6% annual effective yield, a level annuity paying 10 annually is exchanged immediately after a payment for a semiannual perpetuity whose first payment is 1.5253 and grows 0.5% each half-year. Calculate the number of annual payments remaining in the original annuity.

A
6 payments
B
7 payments
C
8 payments
D
9 payments
E
10 payments

Variant answer in brief

The semiannual perpetuity is worth 62.10 at the exchange date. Equating that to 10 times an n-year annual annuity factor gives n = 8, choice C.

Setup

Convert the annual yield to a half-year rate and value the growing perpetuity.

j=(1.06)^{1/2}-1

V=\frac{1.5253}{j-0.005}

Model

Equate that exchange value to the remaining level annual annuity.

V=10a_{\overline n|0.06}

Compute

The exchange value is 62.097938, and solving the annuity term gives n = 8.000000.

n=8.00000000

Answer

There are 8 annual payments remaining, selecting choice C.

\boxed{n=8\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #309 — 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: remaining level-annuity term inferred from an exchange perpetuity

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution