Verified solution · Exam FM

SOA Exam FM Sample Question #220 Solution

This Exam FM sample reference tests Spot and Forward Rates. The contractual annual payment is 6,505.14. Reinvesting all 30 payments at 4% produces 364,840.57, equivalent to a 30-year yield of 4.41%, matching 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 Spot and Forward Rates question

Setup

First determine the level annual payment from the loan's contractual 5% rate.

R=\frac{100000}{a_{\overline{30}|0.05}}

Model

The bank's terminal proceeds are the accumulated value of those payments at the 4% reinvestment rate.

F_{30}=Rs_{\overline{30}|0.04}

Compute

The payment is 6,505.14 and the terminal fund is 364,840.57. Equating that fund to accumulation of the original advance gives the realized yield.

100000(1+j)^{30}=364840.568707

j=0.04408720

Answer

The bank earns an annual effective yield of about 4.4%, so choice D is correct.

\boxed{j\approx4.4\%\quad\text{(D)}}

Calculator reproduction

BA II Plus keystrokes

Check END/BGN, period, sign, TVM, and cash-flow setup

2ND · CLR TVM · 30 · N · 5 · I/Y · 100000 · +/- · PV · 0 · FV · CPT · PMTPMT = 6505.14END mode and annual periods.
30 · N · 4 · I/Y · 0 · PV · 6505.14 · +/- · PMT · CPT · FVFV = 364840.57
30 · N · 100000 · +/- · PV · 0 · PMT · 364840.57 · FV · CPT · I/YI/Y = 4.41

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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 (4.0%) does not satisfy the 30-year terminal-value equation at the contractual and reinvestment rates; no distinct standard one-step error is identifiable.
BChoice B (4.1%) does not satisfy the 30-year terminal-value equation at the contractual and reinvestment rates; no distinct standard one-step error is identifiable.
CChoice C (4.2%) does not satisfy the 30-year terminal-value equation at the contractual and reinvestment rates; no distinct standard one-step error is identifiable.
EChoice E (4.5%) does not satisfy the 30-year terminal-value equation at the contractual and reinvestment rates; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: initial advance supported by a reinvested payment stream

A development fund will receive 7,000 at each year-end for 20 years and can reinvest every receipt at 3% annually. What initial advance would earn the fund a 4.5% annual effective return over the full 20 years?

A
74,091.70
B
77,991.26
C
81,890.83
D
85,790.39
E
89,689.95

Variant answer in brief

The receipts accumulate to 188,092.62 at year 20. Discounting that terminal amount at the required 4.5% return gives an initial advance of 77,991.26, choice B.

Setup

Accumulate the twenty receipts at the rate actually available for reinvestment.

F_{20}=7000s_{\overline{20}|0.03}

Model

The initial advance must grow at the required realized return to the same terminal amount.

A(1.045)^{20}=F_{20}

Compute

The reinvested receipts total 188,092.62 at year 20, so the supported advance is 77,991.26.

A=\frac{7000s_{\overline{20}|0.03}}{(1.045)^{20}}=77991.262432

Answer

The maximum initial advance is 77,991.26, which is choice B.

\boxed{A\approx77991.26\quad\text{(B)}}

Calculator reproduction

BA II Plus keystrokes

2ND · CLR TVM · 20 · N · 3 · I/Y · 0 · PV · 7000 · +/- · PMT · CPT · FVFV = 188092.62
20 · N · 4.5 · I/Y · 0 · PMT · 188092.62 · FV · CPT · PVPV = -77991.26

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #220 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Spot and Forward Rates question

Setup

Model

Compute

Answer

BA II Plus keystrokes

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: initial advance supported by a reinvested payment stream

Variant answer in brief

Setup

Model

Compute

Answer

BA II Plus keystrokes

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution