Verified solution · Exam FM

SOA Exam FM Sample Question #287 Solution

This Exam FM sample reference tests Loan Balances and Amortization. After payment 40, twenty payments remain. Payment 41 equals 1,000 times 0.98 to the fortieth, and the remaining declining annuity discounted at 0.75% monthly is worth 6889.11, so choice A 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 Loan Balances and Amortization question

Setup

The monthly rate is 9% divided by 12, and the first remaining payment is the original payment reduced forty times.

j=0.09/12

P_{41}=1000(0.98)^{40}

Model

The balance immediately after payment 40 is the present value of payments 41 through 60.

B_{40}=\sum_{k=1}^{20}\frac{1000(0.98)^{39+k}}{(1.0075)^k}

Compute

Evaluating the finite geometric present value gives 6889.114798.

B_{40}=6889.11479801

Answer

The outstanding balance is approximately 6,889, selecting choice A.

\boxed{B_{40}\approx6889\quad\text{(A)}}

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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.

BChoice B (7289) does not satisfy the present value after payment 40 of exactly twenty remaining 2%-declining payments; no distinct standard one-step error is identifiable.
CChoice C (7344) does not satisfy the present value after payment 40 of exactly twenty remaining 2%-declining payments; no distinct standard one-step error is identifiable.
DChoice D (7407) does not satisfy the present value after payment 40 of exactly twenty remaining 2%-declining payments; no distinct standard one-step error is identifiable.
EChoice E (7862) does not satisfy the present value after payment 40 of exactly twenty remaining 2%-declining payments; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: original payment inferred from a later declining-payment balance

A 48-payment monthly loan has payments that decline 1% each month and charges 6% nominal interest convertible monthly. Immediately after payment 24, the balance is 3,000. Calculate the first payment.

A
179.60
B
189.05
C
198.51
D
207.96
E
217.41

Variant answer in brief

Payments 25 through 48 are the original payment times successive powers beginning with 0.99 to the twenty-fourth. Their unit present-value factor at 0.5% monthly gives first payment 189.05, choice B.

Setup

Express every remaining payment in terms of the unknown first payment P.

P_{25+k}=P(0.99)^{24+k},\qquad k=0,\ldots,23

Model

Value the 24 remaining payments immediately after payment 24.

3000=P\sum_{k=0}^{23}\frac{(0.99)^{24+k}}{(1.005)^{k+1}}

Compute

Solving the finite geometric equation gives P = 189.05.

P=189.05404456

Answer

The original first payment is 189.05, selecting choice B.

\boxed{P\approx189.05\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #287 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Loan Balances and Amortization question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: original payment inferred from a later declining-payment balance

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution