Worked solution · Exam FM

SOA Exam FM Sample Question #420 Solution

This Exam FM sample reference tests Stepped and Level Perpetuities. Discount the five scheduled payments individually and value the level tail from its first payment date. Their sum is 5917.19, which selects D.

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 Stepped and Level Perpetuities question

Setup

Let v be the annual discount factor. The first five payments occur at times zero through four because the contract is a perpetuity-due.

v=(1.07)^{-1}

P_t=350(1.03)^t,\quad 0\le t\le4

Model

After the fifth payment, the amount no longer grows. The remaining level perpetuity begins at time five with payment equal to the fifth payment.

X=\sum_{t=0}^{4}350(1.03)^t v^t+350(1.03)^4\frac{v^5}{1-v}

Compute

The finite stepped portion and the discounted level tail are evaluated at time zero.

X=1{,}638.376+4{,}278.810

X=5{,}917.186

Answer

The price rounds to 5917, the amount in choice D.

\boxed{X\approx5{,}917\quad\text{(D)}}

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.

AThe value 5249 is too small to include both the five due payments and the level tail beginning after the fifth payment.
BThe value 5530 does not equal the sum of the separately discounted stepped segment and the fifth-payment-level tail.
CThe value 5617 is consistent with shortening or misdating the tail; the tail actually starts at time five.
EThe value 6046 overstates the tail by allowing the payment to keep growing after the fifth amount has been reached.

Original practice · fully worked

Original variant: determine the level reserve payment

A maintenance reserve is worth 8,000 today. It pays 600 at the end of year 1, 660 at the end of year 2, and 726 at the end of year 3. Beginning at the end of year 4, it pays the same amount T forever. At an annual effective rate of 8%, determine T.

A
588.08
B
617.13
C
635.12
D
685.93
E
726.00

Answer and calculation

The first three payments have present value 1697.72. Equating the remaining value to a perpetuity whose first payment is at time 4 gives T = 635.12.

Setup

Value every cash flow at time zero and reserve the symbol T for the new level tail.

v=(1.08)^{-1}

Model

The first level payment is four years away, so the tail value is T times v to the fourth divided by one minus v.

8{,}000=600v+660v^2+726v^3+T\frac{v^4}{1-v}

Compute

The known payments account for 1697.721 of present value. Solve the remaining linear term.

T=(8{,}000-1{,}697.721)\frac{1-v}{v^4}

T=635.124

Answer

The required permanent payment is 635.12, which is choice C.

\boxed{T=635.12\quad\text{(C)}}

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How to solve this Stepped and Level Perpetuities question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: determine the level reserve payment

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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