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SOA Exam FM Sample Question #461 Solution

This Exam FM sample reference tests Macaulay Approximation for a Geometric Perpetuity. At 8% the price is 10.8 and Macaulay duration is 9.8. The Macaulay rate-ratio estimate at 7% is 11.83084 versus exact price 11.88889, a −0.4883% error, choice D.

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 Macaulay Approximation for a Geometric Perpetuity question

Setup

For a due perpetuity with first payment 1 and growth minus 2%, use its closed-form price.

P(i)=\frac{1+i}{i+0.02}

M=P(0.08)=10.8

Model

Its Macaulay duration is one plus the growth rate divided by yield minus growth.

D_{\mathrm{Mac}}=\frac{0.98}{0.08+0.02}=9.8

Compute

Apply the Macaulay rate-ratio estimate and compare it with exact repricing at 7%.

E=10.8\left(\frac{1.08}{1.07}\right)^{9.8}=11.830839

P=\frac{1.07}{0.09}=11.888889

\frac{E-P}{P}=-0.00488272

Answer

The percentage error is −0.49%, choice D.

\boxed{-0.49\%\quad\text{(D)}}

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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 uses modified duration inside the Macaulay rate-ratio exponent and then compares with exact price.
BChoice B applies a linear Macaulay-duration change rather than the specified rate-ratio approximation.
CChoice C treats payment decline as zero when computing duration.
EChoice E rounds both the 9.8 exponent and the exact price before forming a sub-one-percent error.

Original practice · fully worked

Original variant: Macaulay duration of a declining due perpetuity

A geometric perpetuity-due makes its first payment today and each later annual payment is 1% smaller than the preceding payment. At an annual effective yield of 6%, determine its Macaulay duration.

A
13.14
B
14.00
C
14.14
D
14.29
E
15.14

Variant answer in brief

For a geometric perpetuity-due, Macaulay duration is one plus growth divided by yield minus growth. With growth −1%, the duration is 0.99 divided by 0.07 = 14.1429, choice C.

Setup

Represent the annual payment growth rate as r = −0.01.

r=-0.01,\qquad i=0.06

Model

Use the present-value-weighted timing formula for a geometric perpetuity-due.

D_{\mathrm{Mac}}=\frac{1+r}{i-r}

Compute

Insert the decline and yield.

D_{\mathrm{Mac}}=\frac{0.99}{0.06-(-0.01)}=14.142857

Answer

The Macaulay duration is 14.14 years, choice C.

\boxed{D_{\mathrm{Mac}}\approx14.14\quad\text{(C)}}

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ActuaryProof. “SOA Exam FM Sample Question #461 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Macaulay Approximation for a Geometric Perpetuity question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: Macaulay duration of a declining 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