Verified solution · Exam FM

SOA Exam FM Sample Question #279 Solution

This Exam FM sample reference tests Loan Balances and Amortization. The monthly rate equivalent to 12% annually is 0.9489%. Principal portions grow by 1 + j, and payment 33 is 30 steps after payment 3, giving 1194.78 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 Loan Balances and Amortization question

Setup

Convert the annual effective rate to a monthly effective rate.

j=(1.12)^{1/12}-1

Model

Use the geometric progression of principal portions in a level-payment loan.

P_{33}=P_3(1+j)^{30}

Compute

Advancing 900 through 30 monthly steps gives 1194.778960.

P_{33}=900(1+j)^{30}=1194.77896006

Answer

The principal portion is approximately 1,195, selecting choice D.

\boxed{P_{33}\approx1195\quad\text{(D)}}

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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 (668) does not satisfy the 30-step principal-portion progression at the monthly rate equivalent to 12% annually; no distinct standard one-step error is identifiable.
BChoice B (900) does not satisfy the 30-step principal-portion progression at the monthly rate equivalent to 12% annually; no distinct standard one-step error is identifiable.
CChoice C (1008) does not satisfy the 30-step principal-portion progression at the monthly rate equivalent to 12% annually; no distinct standard one-step error is identifiable.
EChoice E (1213) does not satisfy the 30-step principal-portion progression at the monthly rate equivalent to 12% annually; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: first payment whose principal portion crosses a target

A level-payment monthly loan has annual effective rate 10%. The principal portion of payment 5 is 700. Find the first payment number whose principal portion is at least 900.

A
36th payment
B
37th payment
C
38th payment
D
40th payment
E
41th payment

Variant answer in brief

Convert 10% annually to a monthly rate and solve the geometric progression inequality. Rounding the exact step count upward gives payment 37, whose principal portion is 902.57; choice B.

Setup

Convert the annual effective rate to monthly rate j.

j=(1.10)^{1/12}-1

Model

Principal portions after payment 5 follow 700 times the monthly accumulation factor.

P_k=700(1+j)^{k-5}

P_k\ge900

Compute

The required step count is rounded upward, giving k = 37; that payment's principal portion is 902.57.

k=5+\left\lceil\frac{\ln(900/700)}{\ln(1+j)}\right\rceil=37

Answer

The first qualifying payment is number 37, selecting choice B.

\boxed{k=37\quad\text{(B)}}

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The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #279 — 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: first payment whose principal portion crosses a target

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution