Verified solution · Exam FM

SOA Exam FM Sample Question #418 Solution

This Exam FM sample reference tests Interest Rate Valuation. At time zero, include the initial charge, discounted annual fees, all 24 discounted monthly payments, and the discounted remaining balance; the resulting equation is 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 Interest Rate Valuation question

Setup

Use the card-opening date as the comparison date and preserve signs of charges and repayments.

v=e^{-0.015}

1500+20v^{12}+20v^{24}-X\sum_{k=1}^{24}v^k=200v^{24}

Model

Continuous annual force 18% gives monthly exponent 0.015 and annual-fee exponents 0.18 and 0.36.

\delta_m=0.18/12=0.015

Compute

The remaining 200 at time two is an ending liability and is discounted to time zero.

X=\frac{1500+20v^{12}+20v^{24}-200v^{24}}{\sum_{k=1}^{24}v^k}

Answer

Only choice A contains all four components with consistent discounting.

\boxed{\text{equation in choice 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.

BThis treats the year-two remaining balance as if it were due at time zero.
CIt accumulates charges and payments rather than discounting them to the chosen time-zero date.
DThis replaces continuous compounding with an annual effective 18% convention.
EIt also uses the wrong compounding convention and omits proper discounting of the residual balance.

Original practice · fully worked

Original variant: monthly payment under continuous card interest and fees

A card is charged 1,000 immediately, charges a fee of 15 at each year-end, and uses a 12% annual force of interest. Equal payments occur at the end of each month for two years and leave zero balance at year 2. Determine the monthly payment.

A
38.28
B
43.28
C
48.28
D
53.28
E
58.28

Variant answer in brief

Discounting the initial charge, two fees, and 24 payments at monthly force 0.01 gives X = 48.28, choice C.

Setup

Value all card charges and payments at the card-opening date.

1000+15e^{-0.12}+15e^{-0.24}=X\sum_{k=1}^{24}e^{-0.01k}

Model

Monthly payments discount with exponent 0.01 per month, while fees occur at years one and two.

\sum_{k=1}^{24}e^{-0.01k}=21.230706

Compute

Dividing the charge value by the 24-payment discount sum gives 48.28399.

X=48.28399

Answer

The monthly payment is 48.28, corresponding to choice C.

\boxed{X=48.28\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #418 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Interest Rate Valuation question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: monthly payment under continuous card interest and fees

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution