Verified solution · Exam FM

SOA Exam FM Sample Question #393 Solution

This Exam FM sample reference tests Annuities and Perpetuities. The first five due payments are worth 94,341.97; discounting the later 50,000 perpetuity back through the first rate regime leaves an equation whose solution is i = 5%, choice C.

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

Value the first five beginning-of-year payments under the initial 3% rate.

1000000=20000\ddot a_{\overline{5}|\,0.03}+(1.03)^{-5}\left(50000\ddot a_{\infty|\,i}\right)

Model

At time five, the continuing payments form a perpetuity-due under the later rate.

905658.03=(1.03)^{-5}\frac{50000(1+i)}{i}

Compute

Move the remaining value to time five and solve the due-perpetuity equation.

\frac{50000(1+i)}{i}=1049905.88

Answer

The later annual effective rate is 5.00%, corresponding to choice C.

\boxed{i=5.00\%\quad\text{(C)}}

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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 five-payment due annuity plus deferred perpetuity equation; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the five-payment due annuity plus deferred perpetuity equation; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the five-payment due annuity plus deferred perpetuity equation; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the five-payment due annuity plus deferred perpetuity equation; no distinct standard single-step error producing it is identifiable.

Original practice · fully worked

Original variant: price of a two-regime perpetuity

A perpetuity-due pays 15,000 for its first four payments and 40,000 thereafter. The first four years use 4% effective interest and later valuation uses 6%. Determine the present value.

A
560,688
B
610,688
C
660,688
D
710,688
E
760,688

Variant answer in brief

The four temporary payments plus the time-four value of the continuing due perpetuity total 660,688, choice C.

Setup

Value the first four beginning-of-year payments at the initial rate.

P=15000\ddot a_{\overline{4}|\,0.04}+(1.04)^{-4}\left(40000\ddot a_{\infty|\,0.06}\right)

Model

At time four, value the continuing level stream as a perpetuity-due at 6%.

40000\ddot a_{\infty|\,0.06}=40000(1.06/0.06)

Compute

Discount that continuation through the first four-year regime and add both components.

P=660687.9938

Answer

The present value is approximately 660,688, corresponding to choice C.

\boxed{P\approx660688\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #393 — 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: price of a two-regime perpetuity

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution