Worked solution · Exam FM

SOA Exam FM Sample Question #100 Solution

This Exam FM sample reference tests Duration and Convexity. The first duration equation has the unique positive solution v = 0.9; using that discount factor for the four-payment annuity gives duration 1.369, choice B.

Independent analysis by ActuaryProof · reviewed October 6, 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

Index the first beginning-of-year payment by time zero, so only the later payments contribute to the duration numerator.

0.93=\frac{v+2v^2}{1+v+v^2}

Model

Cross-multiplying the known duration equation produces a quadratic in the annual discount factor.

1.07v^2+0.07v-0.93=0

Compute

The admissible root is 0.9. Adding the fourth payment at time three and recomputing the present-value-weighted time gives 1.369.

v=0.9,\qquad D_B=\frac{v+2v^2+3v^3}{1+v+v^2+v^3}=1.369

Answer

The four-payment annuity therefore has Macaulay duration 1.369 years, which is choice B.

\boxed{D_B=1.369\quad\text{(B)}}

Test the method on unseen problems

Financial Mathematics Duration & Immunization Workbook

30 original general cash flows, portfolios, and asset liability management problems with complete solutions and wrong-choice analysis. Written for Exam FM, not copied from the SOA set. One-time $19 PDF.

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Original practice · fully worked

Original variant: duration of community-theatre royalty stream

A community-theatre royalty stream pays 90 at year 0, 60 at year 2, 140 at year 5. The valuation rate is 5.5% effective annually. Determine the Macaulay duration of these promised cash flows.

A
2.113
B
2.363
C
2.563
D
2.813
E
3.113

Answer and calculation

Discounted time weights give a Macaulay duration of 2.563 years, choice C.

Setup

Place each promised amount at its payment year and use the same annual discount factor for every term.

v=(1+0.0550)^{-1}

Model

Macaulay duration is the present-value-weighted average payment time.

D_M=\frac{\sum_t tC_tv^t}{\sum_t C_tv^t}

Compute

The denominator is the cash-flow present value; dividing the time-weighted numerator by it gives 2.563.

D_M=\frac{0(90)v^{0}+2(60)v^{2}+5(140)v^{5}}{90v^{0}+60v^{2}+140v^{5}}=2.563

Answer

Thus the center of present value occurs 2.563 years after valuation, corresponding to choice C.

\boxed{D_M=2.563\text{ years}\quad\text{(C)}}

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How to solve this Duration and Convexity question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Original variant: duration of community-theatre royalty stream

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

Next steps for Exam FM General Cash Flows, Portfolios, and Asset Liability Management