Verified solution · Exam FM

SOA Exam FM Sample Question #355 Solution

This Exam FM sample reference tests Arithmetic-Increasing Perpetuity-Due. The two receipts are worth 1821927.11 today. An arithmetic perpetuity-due has value X times 26 plus the 1000 increments' value 650000. Solving gives X = 45074.12, 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 Arithmetic-Increasing Perpetuity-Due question

Setup

Value the two original receipts at time zero.

PV=1{,}000{,}000+\frac{1{,}000{,}000}{1.04^5}=1{,}821{,}927.1068

Model

A level perpetuity-due of X has factor one plus i divided by i. The 1000 arithmetic increments use the increasing-perpetuity factor.

PV_{\infty}=X\frac{1.04}{0.04}+1{,}000\frac{1.04}{0.04^2}

Compute

Equate values and isolate the first payment.

1{,}821{,}927.1068=26X+650{,}000

X=45{,}074.1195

Answer

The first payment is approximately 45074, choice A.

\boxed{X\approx45{,}074\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 values the arithmetic increments as an increasing annuity-immediate and shifts the level component inconsistently.
CChoice C omits discounting of the second million-dollar receipt.
DChoice D treats the 1000 increase as a 1000 level perpetuity rather than an arithmetic gradient.
EChoice E values only the arithmetic increments and allocates the remainder over too few payments.

Original practice · fully worked

Original variant: level replacement for a finite arithmetic annuity

Five annual payments occur at times 0 through 4. The first is 2,000 and each later payment is 100 larger. At a 5% yield, replace them by five equal payments on the same dates. Determine the level payment.

A
2,100.00
B
2,150.67
C
2,190.25
D
2,224.81
E
2,250.00

Variant answer in brief

The arithmetic stream has present value 9956.78. Dividing by the five-payment annuity-due factor gives a level payment of 2190.25, choice C.

Setup

Discount the five arithmetic payments at their actual dates.

PV=\sum_{t=0}^{4}\frac{2{,}000+100t}{1.05^t}=9{,}956.7773

Model

Let L be the equal payment on dates zero through four.

PV=L\ddot a_{\overline5|0.05}

Compute

Divide by the due-annuity factor.

L=\frac{9{,}956.7773}{4.5459505}=2{,}190.2520

Answer

The equivalent level payment is 2190.25, choice C.

\boxed{L=2{,}190.25\quad\text{(C)}}

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ActuaryProof. “SOA Exam FM Sample Question #355 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Arithmetic-Increasing Perpetuity-Due question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: level replacement for a finite arithmetic annuity

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution