Verified solution · Exam FM

SOA Exam FM Sample Question #405 Solution

This Exam FM sample reference tests Loan Amortization. Because payment growth equals the 1% monthly interest rate, every payment has the same discounted value; the first payment is 202, containing 120 interest and 82 principal, choice A.

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 Amortization question

Setup

The monthly loan rate is 1%, exactly equal to the payment growth rate.

12000=\sum_{t=1}^{60}\frac{P(1.01)^{t-1}}{(1.01)^t}=\frac{60P}{1.01}

Model

After discounting, each of the sixty growing payments contributes the same present value.

P=202

Compute

The first payment is 202; subtracting first-month interest of 120 leaves 82.

I_1=12000(0.01)=120,\qquad PR_1=202-120=82

Answer

Principal repaid in the first payment is 82, corresponding to choice A.

\boxed{PR_1=82\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 is inconsistent with the equal-growth growing-annuity loan equation and first-payment split; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the equal-growth growing-annuity loan equation and first-payment split; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the equal-growth growing-annuity loan equation and first-payment split; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the equal-growth growing-annuity loan equation and first-payment split; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: advance supported by growing monthly payments

A loan is repaid over 48 months. The first month-end payment is 250, and each later payment is 0.5% larger. Interest is 1% effective per month. Determine the loan amount.

A
9,598.29
B
10,098.29
C
10,598.29
D
11,098.29
E
11,598.29

Variant answer in brief

Discounting the 48 growing payments at 1% gives an advance of 10,598.29, choice C.

Setup

Place every growing payment at its actual month-end date.

L=\sum_{t=1}^{48}\frac{250(1.005)^{t-1}}{1.01^t}

Model

The discounted sequence is geometric with ratio 1.005 divided by 1.01.

L=\frac{250}{1.01}\frac{1-(1.005/1.01)^{48}}{1-1.005/1.01}

Compute

Summing 48 terms gives 10,598.2939.

L=10598.2939

Answer

The supported loan amount is 10,598.29, corresponding to choice C.

\boxed{L=10598.29\quad\text{(C)}}

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ActuaryProof. “SOA Exam FM Sample Question #405 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Loan Amortization question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: advance supported by growing monthly payments

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution