Verified solution · Exam FM

SOA Exam FM Sample Question #452 Solution

This Exam FM sample reference tests Yield Uniqueness and Bounds. All outlays precede all receipts, so the cash-flow signs change only once and the yield is unique. At 50% the last receipt would need to occur at time 2.71; its specified time is later than 3, so the true yield is below 50%, choice B.

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 Yield Uniqueness and Bounds question

Setup

Order the project cash flows by time. Their signs move from negative outlays to positive receipts exactly once.

(-,-,+,+)

Model

A single sign change makes the positive yield root unique. Evaluate the 50% benchmark by asking when the final receipt would make net present value zero.

-30-\frac{40}{1.5}+\frac{60}{1.5^2}+\frac{90}{1.5^n}=0

Compute

The benchmark equation reduces to 1.5 to the n equals 3, so n would be 2.71. The actual final receipt occurs later than year 3 and is worth less at 50%.

n=\frac{\ln3}{\ln1.5}=2.7095<3

Answer

A lower discount rate is needed to restore zero NPV; the unique yield is below 50%, choice B.

\boxed{\text{unique yield below }50\%\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 has the correct uniqueness conclusion but reverses the bound implied by negative NPV at 50%.
CChoice C asserts multiple roots despite the single transition from outflows to inflows.
DChoice D gets the below-50% direction but ignores the one-sign-change uniqueness result.
EChoice E also requires multiple yields on both sides of 50%, which the ordered cash-flow signs exclude.

Original practice · fully worked

Original variant: classify and bound a project yield

A project requires 100 at time 0 and returns 40 at time 1 and 80 at time 3. Determine whether its positive annual effective yield is unique and whether it is above or below 10%.

A
Unique and above 10%
B
Unique and below 10%
C
Two yields, both below 10%
D
Two yields, one on each side of 10%
E
No positive yield

Variant answer in brief

There is one sign change, so the positive yield is unique. NPV at 10% is −3.531, so the zero occurs at the lower rate 8.259%, choice B.

Setup

The time-ordered signs are one negative amount followed by positive receipts.

(-,+,+)

Model

Define net present value as a function of annual effective yield.

N(i)=-100+\frac{40}{1+i}+\frac{80}{(1+i)^3}

Compute

The single sign change ensures at most one positive root. At 10%, N is negative; solving gives 8.259%.

N(0.10)=-3.53118

N(0.0825914)=0

Answer

The project has one positive yield and it is below 10%, choice B.

\boxed{\text{unique and below }10\%\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #452 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Yield Uniqueness and Bounds question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: classify and bound a project yield

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution