Verified solution · Exam FM

SOA Exam FM Sample Question #423 Solution

This Exam FM sample reference tests Perpetuity Duration. The ten-year zero has Macaulay duration 10, so the perpetuity-immediate has Macaulay duration 20. This fixes the yield at 1/19 and gives modified duration 19, choice A.

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 Perpetuity Duration question

Setup

A zero-coupon bond's Macaulay duration equals its maturity. Apply the stated two-to-one duration ratio.

D_{\mathrm{Mac},B}=10

D_{\mathrm{Mac},A}=20

Model

For a level perpetuity-immediate, Macaulay duration is one plus the reciprocal of the annual yield.

D_{\mathrm{Mac},A}=\frac{1+i}{i}=20

Compute

Solving gives i = 1/19. Divide Macaulay duration by one plus i to obtain modified duration.

i=\frac1{19}

D_{\mathrm{mod},A}=\frac{20}{1+1/19}=19

Answer

The modified duration is 19.0 years, choice A.

\boxed{D_{\mathrm{mod},A}=19.0\quad\text{(A)}}

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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.

BChoice B neither equals the 20-year Macaulay duration nor that duration divided by the implied accumulation factor.
CChoice C reports the perpetuity's Macaulay duration without converting it to modified duration.
DChoice D adds one year when converting duration even though modified duration must be smaller than Macaulay duration.
EChoice E treats infinite payment horizon as infinite duration; discounting makes the present-value-weighted time finite.

Original practice · fully worked

Original variant: zero maturity matched to a due perpetuity

A level perpetuity-due is valued at an annual effective yield of 5%. A zero-coupon asset is sought whose Macaulay duration equals the perpetuity-due's modified duration. Determine the zero-coupon maturity in years.

A
18.10
B
19.00
C
19.05
D
20.00
E
21.00

Variant answer in brief

A level perpetuity-due has modified duration 1 divided by i times 1 plus i. At 5% this is 19.0476 years, so C is correct.

Setup

A zero-coupon asset has Macaulay duration equal to its maturity n.

D_{\mathrm{Mac,zero}}=n

Model

For the perpetuity-due, differentiate its price to obtain modified duration.

P(i)=\frac{1+i}{i}

D_{\mathrm{mod,due}}=\frac{1}{i(1+i)}

Compute

At 5%, set the zero maturity equal to the computed modified duration.

n=\frac{1}{0.05(1.05)}=19.047619

Answer

The matching maturity is about 19.05 years, choice C.

\boxed{n\approx19.05\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #423 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Perpetuity Duration question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: zero maturity matched to a due perpetuity

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution