Verified solution · Exam FM

SOA Exam FM Sample Question #198 Solution

This Exam FM sample reference tests Bond Valuation. Because the bond sells at a premium, the earliest permitted call is worst for the investor; solving the six-year call price with i = 0.96r gives i = 14.32%, 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 Bond Valuation question

Setup

A premium bond gives the investor the lowest yield when the premium is returned over the shortest allowed period.

1023=1000r\,a_{\overline{6}|\,i}+1000(1+i)^{-6},\qquad i=0.96r

Model

Use the year-six call date and replace coupon rate r by i divided by 0.96.

1023=\frac{1000}{0.96}\left[1-(1+i)^{-6}\right]+1000(1+i)^{-6}

Compute

The resulting price equation isolates the six-year discount factor and gives annual yield 0.1432.

(1+i)^{-6}=0.448,\qquad i=0.1432

Answer

The worst-case yield is 14.32%, corresponding to choice E.

\boxed{i=14.32\%\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 is inconsistent with the earliest-call premium-bond price equation; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the earliest-call premium-bond price equation; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the earliest-call premium-bond price equation; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the earliest-call premium-bond price equation; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: identify a callable bond's yield-to-worst date

A 1,000 redemption bond pays 140 annually and sells for 1,050. It may be called at par after years 5, 7, or 9 and otherwise matures after year 12. Which redemption date produces the investor's yield to worst?

A
Year 5
B
Year 7
C
Year 9
D
Year 12
E
All dates give the same yield

Variant answer in brief

The corresponding yields rise from 12.59% at year 5 to 13.15% at year 12, so the year-5 call produces yield to worst, choice A.

Setup

Price the bond separately to each possible redemption date using its promised coupons through that date.

1050=140a_{\overline{n}|\,i_n}+1000(1+i_n)^{-n}

Model

Solving the year-five and year-seven equations shows yield increasing with the call horizon.

i_5=12.5925\%,\quad i_7=12.8739\%

Compute

The year-nine and maturity yields are higher still, so the minimum is the year-five yield.

i_9=13.0248\%,\quad i_{12}=13.1494\%

Answer

The investor's yield to worst occurs if the bond is called in year 5, corresponding to choice A.

\boxed{\text{yield to worst occurs at year 5}\quad\text{(A)}}

Calculator reproduction

BA II Plus keystrokes

2nd CLR TVM; 2nd I/Y; 1 ENTER; ↓; 1 ENTER; 2nd CPT; 2nd PMT; if BGN is displayed, 2nd ENTER; 2nd CPT; 5 N; 1050 +/- PV; 140 PMT; 1000 FV; CPT I/YI/Y = 12.5925END mode; I/Y is the annual yield if called in year 5.
2nd CLR TVM; 2nd I/Y; 1 ENTER; ↓; 1 ENTER; 2nd CPT; 2nd PMT; if BGN is displayed, 2nd ENTER; 2nd CPT; 7 N; 1050 +/- PV; 140 PMT; 1000 FV; CPT I/YI/Y = 12.8739END mode; I/Y is the annual yield if called in year 7.
2nd CLR TVM; 2nd I/Y; 1 ENTER; ↓; 1 ENTER; 2nd CPT; 2nd PMT; if BGN is displayed, 2nd ENTER; 2nd CPT; 9 N; 1050 +/- PV; 140 PMT; 1000 FV; CPT I/YI/Y = 13.0248END mode; I/Y is the annual yield if called in year 9.
2nd CLR TVM; 2nd I/Y; 1 ENTER; ↓; 1 ENTER; 2nd CPT; 2nd PMT; if BGN is displayed, 2nd ENTER; 2nd CPT; 12 N; 1050 +/- PV; 140 PMT; 1000 FV; CPT I/YI/Y = 13.1494END mode; I/Y is the annual yield if held to year 12.

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 Bonds practice problems Free Exam FM formula sheet

Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #198 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Bond Valuation question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: identify a callable bond's yield-to-worst date

Variant answer in brief

Setup

Model

Compute

Answer

BA II Plus keystrokes

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution