Verified solution · Exam FM

SOA Exam FM Sample Question #222 Solution

This Exam FM sample reference tests Spot and Forward Rates. Equating the five-year spot accumulation to the four-year spot accumulation followed by the fifth-year forward factor gives 11.52%, which corresponds to 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.

Read the solution Open official source

Independent solution

How to solve this Spot and Forward Rates question

Setup

Use only the four- and five-year spot rates because they bracket the one-year forward period being requested.

s_4=0.09,\qquad s_5=0.095

Model

No-arbitrage requires the direct five-year accumulation and the rolled four-year investment to agree.

(1+s_4)^4(1+f_{4,5})=(1+s_5)^5

Compute

Isolating the final one-year factor gives forward rate 11.52%.

f_{4,5}=\frac{(1.095)^5}{(1.09)^4}-1=0.11523041

Answer

The implied fifth-year rate is approximately 11.5%, so choice E is correct.

\boxed{f_{4,5}\approx11.5\%\quad\text{(E)}}

Continue without hunting through PDFs

All 461 Exam FM sample solutions in syllabus order

The searchable 2210-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.

See inside the Exam FM study manual PDF

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 (9.0%) does not satisfy the no-arbitrage equality between direct five-year and rolled four-plus-one-year accumulation; no distinct standard one-step error is identifiable.
BChoice B (9.5%) does not satisfy the no-arbitrage equality between direct five-year and rolled four-plus-one-year accumulation; no distinct standard one-step error is identifiable.
CChoice C (10.0%) does not satisfy the no-arbitrage equality between direct five-year and rolled four-plus-one-year accumulation; no distinct standard one-step error is identifiable.
DChoice D (10.5%) does not satisfy the no-arbitrage equality between direct five-year and rolled four-plus-one-year accumulation; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: four-year spot rate reconstructed from a final-year forward rate

The three-year annual effective spot rate is 4%. The one-year forward rate applying only during year 4 is 6%. Under no arbitrage, calculate the four-year annual effective spot rate.

A
4.50%
B
4.72%
C
4.95%
D
5.17%
E
5.40%

Variant answer in brief

The four-year accumulation factor is the three-year spot factor multiplied by 1.06. Taking its fourth root gives a four-year spot rate of 4.50%, choice A.

Setup

Build the four-year accumulation by rolling the three-year spot investment through the known fourth-year forward period.

(1+s_4)^4=(1.04)^3(1.06)

Model

Take the fourth root to convert the total four-year factor to an annual effective spot rate.

s_4=\left[(1.04)^3(1.06)\right]^{1/4}-1

Compute

The annualized four-year spot rate is 4.4964%.

s_4=0.04496434

Answer

The four-year spot rate is 4.50%, selecting choice A.

\boxed{s_4\approx4.50\%\quad\text{(A)}}

Keep practicing

Turn this solution into a complete study sequence

The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.

See the Exam FM study manual PDF — $49 More Exam FM General Cash Flows, Portfolios, and Asset Liability Management practice problems Free Exam FM formula sheet

Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #222 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Spot and Forward Rates question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: four-year spot rate reconstructed from a final-year forward rate

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution