Verified solution · Exam FM

SOA Exam FM Sample Question #357 Solution

This Exam FM sample reference tests Equivalent Quarterly and Sparse Annual Annuities. The quarterly annuity-due has present value 19407.20 at the exact quarterly rate corresponding to 8%. The three annual payment dates have unit-value factor 2.400341, so X = 8085.18 and choice D.

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 Equivalent Quarterly and Sparse Annual Annuities question

Setup

Convert the annual effective yield to an exact quarterly effective rate.

q=1.08^{1/4}-1=0.01942655

Model

Value twenty-four quarterly beginning payments.

PV_Q=1{,}000\ddot a_{\overline{24}|q}=19{,}407.2010

Compute

Value one unit paid at years 1, 3, and 5.

f=1.08^{-1}+1.08^{-3}+1.08^{-5}=2.40034136

Answer

Equating present values gives 8085.18, choice D.

X=\frac{19{,}407.2010}{2.40034136}=8{,}085.1837

\boxed{X\approx8{,}085\quad\text{(D)}}

Calculator reproduction

BA II Plus keystrokes

Check END/BGN, period, sign, TVM, and cash-flow setup

24 · N; 1.9426547 · I/Y; 1000 · PMT; 0 · FV; BGN mode; CPT · PV−19,407.20Use the magnitude as the quarterly annuity present value.

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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 treats the quarterly payments as end-of-quarter rather than beginning-of-quarter.
BChoice B divides the 8% annual rate by four instead of taking the exact quarterly equivalent.
CChoice C places the sparse annual payments at years 0, 2, and 4.
EChoice E discounts the quarterly stream at 8% per quarter.

Original practice · fully worked

Original variant: present-value difference between payment patterns

At a 6% annual effective yield, compare two four-year plans. Plan Q pays 1,000 at the beginning of every quarter; Plan A pays 4,000 at each year-end. How much greater is Plan Q's present value?

A
328.14
B
421.77
C
515.98
D
604.22
E
721.35

Variant answer in brief

The quarterly beginning payments are worth 14376.40, while the four annual end payments are worth 13860.42. Their difference is 515.98, choice C.

Setup

Use the quarterly rate exactly equivalent to 6% annually.

q=1.06^{1/4}-1

Model

Price the sixteen beginning-of-quarter payments.

PV_Q=1{,}000\ddot a_{\overline{16}|q}=14{,}376.4027

Compute

Price the four year-end payments and subtract.

PV_A=4{,}000a_{\overline4|0.06}=13{,}860.4225

PV_Q-PV_A=515.9803

Answer

Plan Q is worth 515.98 more, choice C.

\boxed{515.98\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #357 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Equivalent Quarterly and Sparse Annual Annuities question

Setup

Model

Compute

Answer

BA II Plus keystrokes

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: present-value difference between payment patterns

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution