Worked solution · Exam FM

SOA Exam FM Sample Question #428 Solution

This Exam FM sample reference tests Perpetuities with Different Frequencies. Value the semiannual perpetuity-due at time zero and equate it to the two-year-payment perpetuity beginning at time one. This gives k = 12149.94, choice A.

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 Perpetuities with Different Frequencies question

Setup

Let v be the annual discount factor and let its square root discount six months.

v=(1.0494)^{-1}

v_{1/2}=v^{1/2}

Model

Perpetuity X pays at times zero, one-half, one, and so on. Perpetuity Y pays at times one, three, five, and so on.

P_X=\frac{3{,}000}{1-v^{1/2}}

P_Y=\frac{kv}{1-v^2}

Compute

Equate the two present values and isolate k.

k=P_X\frac{1-v^2}{v}

k=12{,}149.944

Answer

The two-year payment is approximately 12150, choice A.

\boxed{k\approx12{,}150\quad\text{(A)}}

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.

BChoice B does not match the exact frequency-converted factors for payments at times zero, one-half, one and times one, three, five.
CChoice C approximates four semiannual payments by 12000 and then adds interest for the wrong one-year interval.
DChoice D places the first payment of the two-year perpetuity at time two rather than time one.
EChoice E converts 4.94% by simple division across payment frequencies instead of using effective discount factors.

Original practice · fully worked

Original variant: annual equivalent of a deferred quarterly stream

A trust pays 1,200 every quarter forever, with its first payment six months from today. At an annual effective yield of 5%, it is replaced by a level annual perpetuity-due beginning today. Determine the annual payment that gives the same present value.

A
4,371.09
B
4,486.21
C
4,599.82
D
4,830.00
E
4,942.47

Answer and calculation

The deferred quarterly stream is worth 96596.25. Multiplying by one minus the annual discount factor gives the equivalent annual due payment 4599.82, choice C.

Setup

Convert the annual effective discount factor to a quarterly discount factor.

v=(1.05)^{-1}

v_q=v^{1/4}

Model

The quarterly payments occur at half a year and every quarter thereafter. The annual replacement begins immediately.

P_Q=\frac{1{,}200v_q^2}{1-v_q}

P_A=\frac{Y}{1-v}

Compute

Set the two prices equal and solve for Y.

P_Q=96{,}596.253

Y=P_Q(1-v)=4{,}599.822

Answer

The annual due payment is 4599.82, choice C.

\boxed{Y=4{,}599.82\quad\text{(C)}}

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How to solve this Perpetuities with Different Frequencies question

Setup

Model

Compute

Answer

Financial Mathematics Practice Examination I

Why the other letters tempt

Original variant: annual equivalent of a deferred quarterly stream

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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