Verified solution · Exam FM

SOA Exam FM Sample Question #250 Solution

This Exam FM sample reference tests Loan Balances and Amortization. The 3% level-payment option totals 1344.31. Under constant principal, interest is i times the sum of balances 1,000 through 50, totaling 10500. Equating totals gives i = 3.28%, choice C.

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

Compute the level annual payment and its total over 20 years.

R=\frac{1000}{a_{\overline{20}|0.03}}

T_1=20R

Model

The second option repays 50 principal each year, so total interest is i times the arithmetic sequence of opening balances.

T_2=1000+i(10500)

Compute

Equating T1 = 1344.3142 to T2 gives i = 3.279182%.

i=\frac{1344.31415194-1000}{10500}=0.0327918240

Answer

The constant-principal option uses an annual rate of approximately 3.28%, selecting choice C.

\boxed{i\approx3.28\%\quad\text{(C)}}

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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 (2.86%) does not satisfy the equality of total payments under the level-payment and constant-principal schedules; no distinct standard one-step error is identifiable.
BChoice B (3.00%) does not satisfy the equality of total payments under the level-payment and constant-principal schedules; no distinct standard one-step error is identifiable.
DChoice D (3.44%) does not satisfy the equality of total payments under the level-payment and constant-principal schedules; no distinct standard one-step error is identifiable.
EChoice E (4.76%) does not satisfy the equality of total payments under the level-payment and constant-principal schedules; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: total-payment penalty of a higher-rate constant-principal schedule

A 12,000 loan can be repaid over ten years either by level annual payments at 4% or by 1,200 principal each year plus 5% interest on the opening balance. Calculate how much larger the total payments are under the constant-principal option.

A
404.07
B
429.32
C
454.58
D
479.83
E
505.09

Variant answer in brief

The 4% level-payment schedule totals 14,794.91. Principal plus 5% interest on the ten opening balances totals 15,300.00, an excess of 505.09, choice E.

Setup

Compute the annual level payment and ten-payment total under option A.

R=\frac{12000}{a_{\overline{10}|0.04}}

T_A=10R

Model

Under option B, add 12,000 principal to 5% of the ten opening balances.

T_B=12000+0.05\sum_{k=0}^{9}(12000-1200k)

Compute

The totals are 14794.91 and 15300.00, so the excess is 505.09.

T_B-T_A=505.08668038

Answer

The constant-principal option costs 505.09 more, selecting choice E.

\boxed{T_B-T_A\approx505.09\quad\text{(E)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #250 — 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: total-payment penalty of a higher-rate constant-principal schedule

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution