Verified solution · Exam FM

SOA Exam FM Sample Question #257 Solution

This Exam FM sample reference tests Interest Rate Measures. Writing the present-value equality as a quadratic in v produces discount factors 0.970874 and 0.952381, corresponding to 3% and 5% annual rates. Their absolute difference is 2%, so choice B is correct.

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 Measures question

Setup

Let v be the one-year discount factor and write the common present-value equation.

10000+10815v^2=20800v

Model

Solve the quadratic for its two positive discount-factor roots.

v_1=0.9708737864

v_2=0.9523809524

Compute

Converting by i = 1/v − 1 gives rates 3.00% and 5.00%, whose difference is 2.00%.

|i_1-i_2|=0.02000000

Answer

The rates differ by 2.0%, selecting choice B.

\boxed{|x-y|=2.0\%\quad\text{(B)}}

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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 (1.8%) does not satisfy both positive roots of the present-value quadratic in the one-year discount factor; no distinct standard one-step error is identifiable.
CChoice C (3.0%) does not satisfy both positive roots of the present-value quadratic in the one-year discount factor; no distinct standard one-step error is identifiable.
DChoice D (4.0%) does not satisfy both positive roots of the present-value quadratic in the one-year discount factor; no distinct standard one-step error is identifiable.
EChoice E (5.0%) does not satisfy both positive roots of the present-value quadratic in the one-year discount factor; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: sum of two yields satisfying one nonconventional value equation

The present value of 8,550 now plus 10,000 at year 2 equals the present value of 18,500 at year 1 at two positive annual effective rates. Calculate the sum of those two rates.

A
15.56%
B
16.37%
C
17.19%
D
18.01%
E
18.83%

Variant answer in brief

The value equation factors into discount roots 0.95 and 0.90. These correspond to rates 5.26% and 11.11%, whose sum is 16.37%, choice B.

Setup

Write the equality as a quadratic in the annual discount factor.

8550+10000v^2=18500v

Model

Factor the quadratic to obtain both positive discount roots.

10000(v-0.95)(v-0.90)=0

Compute

The corresponding rates are 5.263158% and 11.111111%, summing to 16.374269%.

i_1=1/0.95-1

i_2=1/0.90-1

Answer

The sum of the two rates is 16.37%, selecting choice B.

\boxed{i_1+i_2\approx16.37\%\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #257 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Interest Rate Measures question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: sum of two yields satisfying one nonconventional value equation

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution