Verified solution · Exam FM

SOA Exam FM Sample Question #20 Solution

This Exam FM sample reference tests Perpetuity Share Decomposition. Thus the requested tail share is (0.60)² = 0.36, or 36%. The result agrees with the published answer key, 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 Perpetuity Share Decomposition question

Setup

For a level perpetuity, the fraction of value contributed by the first n payments is 1 − vⁿ. The given 40% share therefore determines the tail discount factor.

a_{\overline n|}i=0.40\Longrightarrow1-v^n=0.40

Model

Solving 1 − vⁿ = 0.40 gives vⁿ = 0.60. Moving two complete n-payment blocks into the future multiplies value by v²ⁿ.

v^n=0.60

Compute

Thus the requested tail share is (0.60)² = 0.36, or 36%.

K=v^{2n}=0.36

Answer

The calculation gives 36% for perpetuity share decomposition, matching published choice D.

\boxed{K=36\%\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 (24%) does not match the checked perpetuity share decomposition result (36%); no distinct standard one-step error is identifiable.
BChoice B (28%) does not match the checked perpetuity share decomposition result (36%); no distinct standard one-step error is identifiable.
CChoice C (32%) does not match the checked perpetuity share decomposition result (36%); no distinct standard one-step error is identifiable.
EChoice E (40%) does not match the checked perpetuity share decomposition result (36%); no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: three generations of perpetuity payments

The first n payments of a level perpetuity have 30% of its present value, the next n payments go to a second recipient, and all later payments go to a foundation. Find the foundation's share.

A
42%
B
45%
C
49%
D
51%
E
70%

Variant answer in brief

The requested share is 0.70² = 0.49, or 49%. This selects choice C.

Setup

The first n payments supply 30% of perpetuity value, so vⁿ = 0.70.

1-v^n=0.30

Model

Value remaining after two n-payment blocks is multiplied by v²ⁿ.

v^n=0.70

Compute

The requested share is 0.70² = 0.49, or 49%.

K=v^{2n}=0.49

Answer

The requested value is 49%, so the correct choice is C.

\boxed{K=49\%\quad\text{(C)}}

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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.

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #20 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Perpetuity Share Decomposition question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: three generations of perpetuity payments

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution