Verified solution · Exam FM

SOA Exam FM Sample Question #276 Solution

This Exam FM sample reference tests Annuities. The first ten level payments contribute 147.20. The 19-payment decreasing segment and the final level-1 perpetuity contribute discounted values 72.98 and 3.08. Total is 223.26, choice D.

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 question

Setup

Split the stream into ten level payments of 20, a 19-payment sequence 19 down to 1, and a level perpetuity of 1 beginning afterward.

PV_1=20a_{\overline{10}|0.06}

Model

Value the decreasing block at time 10 and the remaining perpetuity at time 29, then discount both to time 0.

PV_2=(Da)_{\overline{19}|0.06}(1.06)^{-10}

PV_3=\frac{1}{0.06}(1.06)^{-29}

Compute

The three present-value components sum to 223.258800.

PV=PV_1+PV_2+PV_3=223.25879988

Answer

The perpetuity value is approximately 223, selecting choice D.

\boxed{PV\approx223\quad\text{(D)}}

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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 (208) does not satisfy the sum of the level, decreasing, and final perpetuity segments at their correct deferral dates; no distinct standard one-step error is identifiable.
BChoice B (213) does not satisfy the sum of the level, decreasing, and final perpetuity segments at their correct deferral dates; no distinct standard one-step error is identifiable.
CChoice C (218) does not satisfy the sum of the level, decreasing, and final perpetuity segments at their correct deferral dates; no distinct standard one-step error is identifiable.
EChoice E (228) does not satisfy the sum of the level, decreasing, and final perpetuity segments at their correct deferral dates; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: annual decrement inferred from a piecewise perpetuity price

A perpetuity pays 30 in each of years 1 through 5. Payments then decline by K each year for ten years; from year 16 onward they remain level at the year-15 amount. At a 5% annual yield, the price is 345.89. Calculate K.

A
2.00
B
2.10
C
2.20
D
2.30
E
2.40

Variant answer in brief

Valuing the first level block, the ten declining payments, and the level tail as functions of K gives a linear price equation. Solving it yields annual decrement 2.00, choice A.

Setup

Separate the five level payments from the ten-step declining block and the permanent level tail.

PV=30a_{\overline5|0.05}+\sum_{k=1}^{10}(30-kK)(1.05)^{-(5+k)}+\frac{30-10K}{0.05}(1.05)^{-15}

Model

Set the piecewise present value equal to the quoted market price.

345.89=PV(K)

Compute

The resulting linear equation gives K = 2.000000.

K=2.00000000

Answer

The annual decrement is 2.00, selecting choice A.

\boxed{K=2.00\quad\text{(A)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #276 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Annuities question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: annual decrement inferred from a piecewise perpetuity price

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution