Verified solution · Exam FM

SOA Exam FM Sample Question #246 Solution

This Exam FM sample reference tests Loan Balances and Amortization. The principal in payment n−4 equals R times v to the fifth, while the balance after payment n−1 equals Rv. Their ratio gives v to the fourth, hence i = 7.2506%. Since first-year interest is Xi, the loan is 10999.02, 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

Relate the late principal portion and penultimate balance to the level payment.

865=Rv^5

1144.50=Rv

Model

Divide the equations to recover the discount factor and annual interest rate.

v^4=\frac{865}{1144.50}

i=v^{-1}-1

Compute

The rate is 7.250647%. First-year interest equals Xi, so X = 10999.02.

X=\frac{797.50}{i}=10999.01783381

Answer

The original loan is approximately 11,000, selecting choice D.

\boxed{X\approx11000\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 (9,500) does not satisfy the ratio of the stated late principal portion to the penultimate balance and the first-year interest equation; no distinct standard one-step error is identifiable.
BChoice B (10,000) does not satisfy the ratio of the stated late principal portion to the penultimate balance and the first-year interest equation; no distinct standard one-step error is identifiable.
CChoice C (10,500) does not satisfy the ratio of the stated late principal portion to the penultimate balance and the first-year interest equation; no distinct standard one-step error is identifiable.
EChoice E (11,500) does not satisfy the ratio of the stated late principal portion to the penultimate balance and the first-year interest equation; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: loan rate inferred from two separated principal portions

In a level-payment annual loan, the principal portion of payment 3 is 700 and the principal portion of payment 8 is 893.40. Calculate the annual effective interest rate.

A
4.75%
B
5.00%
C
5.25%
D
5.50%
E
5.75%

Variant answer in brief

Successive principal portions grow by 1 + i. Five progression steps separate payments 3 and 8, so taking the fifth root of their ratio gives 5.00%, choice B.

Setup

Principal portions of a level-payment loan form a geometric progression with ratio 1 + i.

P_{k+1}=P_k(1+i)

Model

Five progression steps link the two observed principal portions.

893.40=700(1+i)^5

Compute

The fifth root of the ratio gives annual rate 5.000000%.

i=\left(\frac{893.40}{700}\right)^{1/5}-1=0.0500000000

Answer

The annual effective loan rate is 5.00%, which is choice B.

\boxed{i=5.00\%\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #246 — 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: loan rate inferred from two separated principal portions

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution