Verified solution · Exam FM

SOA Exam FM Sample Question #387 Solution

This Exam FM sample reference tests Duration and Convexity. The decreasing perpetuity-due has Macaulay duration 0.99 divided by 0.01 plus i; setting this to 12 gives i = 7.25%, and modified duration is 12/1.0725 = 11.19 years, choice C.

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 Duration and Convexity question

Setup

Sum the immediate first payment and the geometric tail to express price as a function of yield.

P(i)=1+\frac{0.99}{0.01+i}

Model

Differentiating the price gives a compact Macaulay-duration formula; the stated duration determines yield.

D_M=\frac{0.99}{0.01+i}=12\Longrightarrow i=0.0725

Compute

Convert the twelve-year Macaulay duration using the derived 7.25% yield.

D_{\mathrm{mod}}=\frac{12}{1.0725}=11.1888

Answer

The modified duration is 11.19 years, corresponding to choice C.

\boxed{D_{\mathrm{mod}}\approx11.19\quad\text{(C)}}

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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 is inconsistent with the geometrically decreasing perpetuity price derivative; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the geometrically decreasing perpetuity price derivative; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the geometrically decreasing perpetuity price derivative; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the geometrically decreasing perpetuity price derivative; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: decline rate implied by perpetuity duration

A perpetuity-due pays 1 immediately and then decreases by a fixed percentage each year. At 6% effective annually its Macaulay duration is 14 years. Determine the annual percentage decrease.

A
0.57%
B
0.82%
C
1.07%
D
1.32%
E
1.57%

Variant answer in brief

Solving the geometric-duration relation gives payment ratio 0.989333, hence annual decrease 1.07%, choice C.

Setup

Let q be the ratio of each payment to the preceding payment.

14=\frac{q}{1.06-q}

Model

The perpetuity-due Macaulay duration is q divided by the difference between the yield factor and q.

14(1.06-q)=q

Compute

Solving gives ratio 0.9893333, so the missing fraction is 0.0106667.

q=0.9893333,\qquad 1-q=0.0106667

Answer

Payments decrease by approximately 1.07% annually, corresponding to choice C.

\boxed{\text{decrease}=1.07\%\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #387 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Duration and Convexity question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: decline rate implied by perpetuity duration

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution