Verified solution · Exam FM

SOA Exam FM Sample Question #293 Solution

This Exam FM sample reference tests Duration and Convexity. Modified duration is negative P prime divided by P. Differentiating the year-3 and year-4 discounted terms gives the numerator 1,500 times (1+i) to the −4 minus 4,000 times (1+i) to the −5, which is the expression in choice A.

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 Duration and Convexity question

Setup

Use the definition of modified duration for the given price function.

D_{\mathrm{mod}}=-\frac{P'(i)}{P(i)}

Model

Differentiate each discounted term; the time-0 amount has zero derivative.

P'(i)=-1500(1+i)^{-4}+4000(1+i)^{-5}

Compute

Negating the derivative gives the displayed numerator over the original price function.

D_{\mathrm{mod}}=\frac{1500(1+i)^{-4}-4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}

Answer

This derivative ratio is the expression in choice A.

\boxed{\text{choice A}}

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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.

BChoice B does not satisfy negative price derivative divided by the original cash-flow price function; no distinct standard one-step error is identifiable.
CChoice C does not satisfy negative price derivative divided by the original cash-flow price function; no distinct standard one-step error is identifiable.
DChoice D does not satisfy negative price derivative divided by the original cash-flow price function; no distinct standard one-step error is identifiable.
EChoice E does not satisfy negative price derivative divided by the original cash-flow price function; no distinct standard one-step error is identifiable.

Original practice · fully worked

Original variant: modified duration of a three-date positive cash-flow stream

A portfolio pays 200 today, 300 at year 2, and 500 at year 5. At a 4% annual effective yield, calculate its modified duration.

A
2.26 years
B
2.40 years
C
2.54 years
D
2.68 years
E
2.82 years

Variant answer in brief

The portfolio price is 888.33. Dividing the negative price derivative 2509.18 by that price gives modified duration 2.8246 years, choice E.

Setup

Price the three cash flows at 4%; the time-0 cash flow is included in price but has no duration numerator.

P=200+300(1.04)^{-2}+500(1.04)^{-5}

Model

Compute the negative derivative with respect to annual yield.

-P'(i)=2(300)(1.04)^{-3}+5(500)(1.04)^{-6}

Compute

The price is 888.330417 and derivative magnitude is 2509.184130, giving 2.82460679 years.

D_{\mathrm{mod}}=\frac{-P'(i)}{P}=2.8246067912

Answer

The modified duration is 2.8246 years, selecting choice E.

\boxed{D_{\mathrm{mod}}\approx2.8246\text{ years}\quad\text{(E)}}

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Cite this solution

ActuaryProof. “SOA Exam FM Sample Question #293 — Verified Solution.” Verified 2026-07-31.

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

How to solve this Duration and Convexity question

Setup

Model

Compute

Answer

All 461 Exam FM sample solutions in syllabus order

Why the other letters tempt

Original variant: modified duration of a three-date positive cash-flow stream

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam FM solutions

Turn this solution into a complete study sequence

Cite this solution