Verified solution · Exam FM

SOA Exam FM Sample Question #261 Solution

This Exam FM sample reference tests Annuities. There are 18 withdrawals from ages 65 through 82 inclusive. Accumulating the original fund for 17 years and subtracting the accumulated withdrawals while restoring the 1,000,000 ending fund gives P = 88915.43, choice B.

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

Use the 82nd birthday immediately after withdrawal as the comparison date; 17 years have elapsed since age 65.

t=17

Model

The withdrawals at times 0 through 17 accumulate as an 18-payment annuity-immediate shifted to the terminal date.

1000000(1.10)^{17}-Ps_{\overline{18}|0.10}=1000000

Compute

Solving the terminal balance gives withdrawal 88915.4343.

P=\frac{1000000[(1.10)^{17}-1]}{s_{\overline{18}|0.10}}=88915.43434343

Answer

The annual birthday withdrawal is approximately 88,915, selecting choice B.

\boxed{P\approx88915\quad\text{(B)}}

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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 (81,655) does not satisfy the terminal fund after all 18 withdrawals from ages 65 through 82; no distinct standard one-step error is identifiable.
CChoice C (90,909) does not satisfy the terminal fund after all 18 withdrawals from ages 65 through 82; no distinct standard one-step error is identifiable.
DChoice D (98,879) does not satisfy the terminal fund after all 18 withdrawals from ages 65 through 82; no distinct standard one-step error is identifiable.
EChoice E (109,729) does not satisfy the terminal fund after all 18 withdrawals from ages 65 through 82; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: ending reserve after a fixed series of anniversary withdrawals

A reserve contains 500,000 immediately before an initial withdrawal of 40,000. It earns 6% annually, and another 40,000 is withdrawn after each of the next ten years. Calculate the balance immediately after the tenth-year withdrawal.

A
252,074.42
B
266,902.33
C
281,730.24
D
296,558.14
E
311,386.05

Variant answer in brief

The original reserve accumulates ten years, while the eleven withdrawals from time 0 through time 10 accumulate as an eleven-term series. The resulting balance is 296,558.14, choice D.

Setup

Use the date immediately after the time-10 withdrawal as the comparison date.

F_{10}=500000(1.06)^{10}-40000\sum_{k=0}^{10}(1.06)^k

Model

Recognize the withdrawal accumulation as an eleven-payment accumulated annuity.

F_{10}=500000(1.06)^{10}-40000s_{\overline{11}|0.06}

Compute

The resulting ending reserve is 296558.14.

F_{10}=296558.14271448

Answer

The balance after the final withdrawal is 296,558.14, selecting choice D.

\boxed{F_{10}\approx296558.14\quad\text{(D)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #261 — 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: ending reserve after a fixed series of anniversary withdrawals

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution