Verified solution · Exam FM

SOA Exam FM Sample Question #225 Solution

This Exam FM sample reference tests Spot and Forward Rates. Discounting each dated cash flow at its matching spot rate gives present-value components of 9.62, 10.99, 12.77, and 15.26. Their sum is 48.64, 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 Spot and Forward Rates question

Setup

Use a different discount factor for each maturity because the table gives a term structure, not one flat yield.

PV=\sum_{t=1}^{4}\frac{C_t}{(1+s_t)^t}

Model

Insert the four cash flows and their maturity-specific spot rates.

PV=\frac{10}{1.04}+\frac{12}{(1.045)^2}+\frac{15}{(1.055)^3}+\frac{20}{(1.07)^4}

Compute

The four discounted amounts sum to 48.6363.

PV=48.636253

Answer

The investment is worth approximately 48.64 at the start, so choice B is correct.

\boxed{PV\approx48.64\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 (47.33) does not satisfy the sum of four maturity-specific discounted cash flows; no distinct standard one-step error is identifiable.
CChoice C (49.50) does not satisfy the sum of four maturity-specific discounted cash flows; no distinct standard one-step error is identifiable.
DChoice D (50.04) does not satisfy the sum of four maturity-specific discounted cash flows; no distinct standard one-step error is identifiable.
EChoice E (51.14) does not satisfy the sum of four maturity-specific discounted cash flows; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: missing three-year spot rate recovered from a quoted portfolio value

A portfolio pays 500 at year 1, 800 at year 2, and 1,000 at year 3. The one- and two-year spot rates are 3% and 4%. Its present value is 2088.92. Calculate the three-year annual effective spot rate.

A
4.75%
B
5.00%
C
5.25%
D
5.50%
E
5.75%

Variant answer in brief

The first two payments contribute 1225.08 of present value. The remaining value assigned to the year-3 payment implies a three-year spot rate of 5.00%, choice B.

Setup

Remove the known present values of the first two payments from the quoted total price.

PV_3=2088.92-\frac{500}{1.03}-\frac{800}{(1.04)^2}

Model

The residual must equal the discounted year-3 payment at the unknown three-year spot rate.

PV_3=1000(1+s_3)^{-3}

Compute

Solving the residual discount factor gives annual spot rate 5.0000%.

s_3=\left(\frac{1000}{PV_3}\right)^{1/3}-1=0.05000000

Answer

The three-year spot rate is 5.00%, selecting choice B.

\boxed{s_3\approx5.00\%\quad\text{(B)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #225 — 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

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: missing three-year spot rate recovered from a quoted portfolio value

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution