Verified solution · Exam FM

SOA Exam FM Sample Question #314 Solution

This Exam FM sample reference tests Loan Balances and Amortization. The six-month accumulation factor is 1.045, so a month-n cash flow is discounted by 1.045 to the n/6. Payment n is 500 + (n−1)X for n from 1 through 60. This timing and payment expression is choice E.

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 semiannual periodic rate is 9% divided by 2, giving six-month accumulation factor 1.045.

1+j=1.045

Model

A payment at month n is n/6 half-year periods from origination.

v_n=(1.045)^{-n/6}

Compute

The first payment is 500, so payment n contains n minus 1 increments.

30000=\sum_{n=1}^{60}\frac{500+(n-1)X}{(1.045)^{n/6}}

Answer

This is exactly the equation displayed in choice E.

\boxed{\text{choice E}}

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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 (30, 000 = ∑ n) does not satisfy the month-1 through month-60 timing under a six-month accumulation factor and n−1 increments; no distinct standard one-step error is identifiable.
BChoice B (30, 000 = ∑ n) does not satisfy the month-1 through month-60 timing under a six-month accumulation factor and n−1 increments; no distinct standard one-step error is identifiable.
CChoice C (30, 000 = ∑ n) does not satisfy the month-1 through month-60 timing under a six-month accumulation factor and n−1 increments; no distinct standard one-step error is identifiable.
DChoice D (30, 000 = ∑ n) does not satisfy the month-1 through month-60 timing under a six-month accumulation factor and n−1 increments; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: monthly payment increment required under a semiannual nominal rate

A 20,000 loan is repaid with 36 monthly payments. The first payment is 400 and each later payment is X larger. The annual nominal interest rate is 8% convertible semiannually. Calculate X.

A
11.41
B
12.09
C
12.76
D
13.43
E
14.10

Variant answer in brief

Discounting the 36 base payments and the month-index increment stream using the sixth root of 1.04 gives increment X = 13.43, choice D.

Setup

Convert the six-month factor 1.04 to an equivalent one-month factor.

q=(1.04)^{1/6}

Model

Separate the payment stream into 400 each month plus X times the index 0 through 35.

20000=400\sum_{n=1}^{36}q^{-n}+X\sum_{n=1}^{36}(n-1)q^{-n}

Compute

The base-payment present value is 12789.21 and the increment factor is 536.994277, giving X = 13.4280.

X=13.42804840

Answer

Each payment rises by 13.43, selecting choice D.

\boxed{X\approx13.43\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #314 — 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: monthly payment increment required under a semiannual nominal rate

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution