Worked solution · Exam FM

SOA Exam FM Sample Question #436 Solution

This Exam FM sample reference tests Arithmetic Increasing Perpetuity. The original first payment is 80909.09. After the 2020 payment, the next amount is 101909.09; valuing that arithmetic-increasing perpetuity-immediate gives 1119090.91, choice B.

Independent analysis by ActuaryProof · reviewed October 6, 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 question

Setup

Let P be the payment on January 1, 2000. Payments at times t = 0, 1, 2 and so on are P plus 1000t.

1{,}000{,}000=P\ddot a_{\infty}+1{,}000\sum_{t=0}^{\infty}tv^t

Model

At 10%, the due level factor is 11 and the increasing component has value 110000.

\ddot a_{\infty}=\frac{1.10}{0.10}=11

\sum_{t=0}^{\infty}tv^t=\frac{1.10}{0.10^2}=110

Compute

Thus P = 80909.09. Immediately after the time-20 payment, the next payment is P plus 21000.

P=80{,}909.09

V_{20+}=\frac{P+21{,}000}{0.10}+\frac{1{,}000}{0.10^2}=1{,}119{,}090.91

Answer

The prospective value rounds to 1119091, choice B.

\boxed{V_{20+}\approx1{,}119{,}091\quad\text{(B)}}

Test the method on unseen problems

Financial Mathematics Practice Examination I

30 original Exam FM questions across the syllabus, to be worked in 150 timed minutes, with a full worked solution for each one. Written for Exam FM, not copied from the SOA set. One-time $39 PDF.

See the cover and sample pages

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 advances only the original level component and misses part of the future arithmetic increases.
CChoice C values a level perpetuity of the 2020 payment and omits all increases after 2020.
DChoice D treats the next payment as the just-paid 2020 amount and adds the gradient at the wrong time.
EChoice E includes the payment made on January 1, 2020 even though value is requested immediately afterward.

Original practice · fully worked

Original variant: recover the annual increase

Immediately before its first payment, an annual perpetuity-due is worth 500,000 at 6% effective. Its first payment is 20,000 today, and every later payment exceeds the preceding payment by a constant amount Q. Determine Q.

A
320.75
B
452.83
C
498.11
D
600.00
E
1,698.11

Answer and calculation

The level 20000 due perpetuity uses 353333.33 of value. Equating the remaining value to the arithmetic-gradient component gives Q = 498.11, choice C.

Setup

Payments at time t are 20000 plus tQ for nonnegative integer t.

v=(1.06)^{-1}

Model

Separate the level perpetuity-due from the gradient series.

500{,}000=20{,}000\frac{1.06}{0.06}+Q\frac{1.06}{0.06^2}

Compute

Solve the linear equation for the annual increase.

Q=\left(500{,}000-353{,}333.333\right)\frac{0.06^2}{1.06}

Q=498.113

Answer

The annual increase is 498.11, choice C.

\boxed{Q=498.11\quad\text{(C)}}

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How to solve this Arithmetic Increasing Perpetuity question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: recover the annual increase

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

Next steps for Exam FM Annuities and Non-contingent Cash Flows