Verified solution · Exam FM

SOA Exam FM Sample Question #398 Solution

This Exam FM sample reference tests Annuities and Perpetuities. The rise to 10, fall to 5, and level tail can be represented by shifted increasing perpetuities; at 9% the resulting factor is 57.83, 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 Annuities and Perpetuities question

Setup

Start with an indefinitely increasing payment sequence.

(Ia)_\infty=\frac{1+i}{i^2}=134.5679

Model

Subtract two increasing sequences beginning after the peak to reverse the slope, then add one later sequence to stop the decline at five.

X=(Ia)_\infty-2v^{10}(Ia)_\infty+v^{15}(Ia)_\infty

Compute

All three components share the increasing-perpetuity factor and differ only by their shifts.

X=134.5679(1-2v^{10}+v^{15})=57.83

Answer

The present value is 57.83, nearest choice D.

\boxed{X\approx58\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 is inconsistent with the shifted increasing-perpetuity decomposition; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the shifted increasing-perpetuity decomposition; no distinct standard single-step error producing it is identifiable.
CChoice C is inconsistent with the shifted increasing-perpetuity decomposition; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the shifted increasing-perpetuity decomposition; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: level tail implied by a target perpetuity value

A perpetuity pays 1, 2, 3, 4, 5, and 6 in years 1 through 6, then pays level amount L from year 7 onward. At 9% effective, its present value is 50. Determine L.

A
4.35
B
4.85
C
5.35
D
5.85
E
6.35

Variant answer in brief

Removing the first six increasing payments leaves a deferred level perpetuity, giving L = 5.35, choice C.

Setup

Value the six explicitly increasing payments first.

50=\sum_{t=1}^{6}tv^t+Lv^6a_{\infty|\,0.09}

Model

At time six, the remaining level payments beginning in year seven form a perpetuity-immediate.

\sum_{t=1}^{6}tv^t=14.5783

Compute

Solving the residual-value equation gives 5.3465.

L=\frac{(50-14.5783)(0.09)}{v^6}=5.3465

Answer

The level tail payment is 5.35, corresponding to choice C.

\boxed{L=5.35\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #398 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Annuities and Perpetuities question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: level tail implied by a target perpetuity value

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution