Worked solution · Exam FM

SOA Exam FM Sample Question #460 Solution

This Exam FM sample reference tests Geometric Perpetuity Approximation. The initial price is 14.2857 and modified duration is also 14.2857. The first-order estimate at 5.5% is 16.3265, while exact price is 16.6667; the relative error is −2.0408%, choice B.

Independent analysis by ActuaryProof · reviewed October 6, 2026

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Independent solution

How to solve this Geometric Perpetuity Approximation question

Setup

For first payment 1 and growth rate minus 0.5%, the immediate-perpetuity price is one divided by i plus 0.005.

P(i)=\frac1{i+0.005}

M=P(0.065)=14.285714

Model

Differentiate the price to obtain modified duration at the original yield.

D_{\mathrm{mod}}=-\frac{P'(i)}{P(i)}=\frac1{i+0.005}=14.285714

Compute

Use the one-percentage-point yield decline for the estimate and also compute exact price.

E=M[1-14.285714(-0.01)]=16.326531

P=P(0.055)=16.666667

\frac{E-P}{P}=-0.0204082

Answer

The approximation error is −2.04%, choice B.

\boxed{-2.04\%\quad\text{(B)}}

Test the method on unseen problems

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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 uses the exact price change divided by the original price rather than the estimate's error at the new price.
CChoice C rounds modified duration to 14 before calculating both the estimate and relative error.
DChoice D uses a level perpetuity duration based only on the 6.5% yield and ignores the negative growth rate.
EChoice E divides approximation error by initial price M instead of exact new price P.

Original practice · fully worked

Original variant: infer a new yield from a duration estimate

A geometric perpetuity-immediate has first payment 2, annual payment growth of 1%, and initial yield 6%. A first-order modified-duration estimate says its price will rise by 10% after an immediate yield change. Determine the new yield implied by that estimate.

A
5.00%
B
5.25%
C
5.50%
D
5.75%
E
6.50%

Answer and calculation

Modified duration at the initial yield is 1 divided by 0.06 minus 0.01, or 20. A 10% price rise therefore corresponds to a yield change of −0.005, giving a new yield of 5.50%, choice C.

Setup

For a geometric perpetuity-immediate, modified duration is the reciprocal of yield minus growth.

D_{\mathrm{mod}}=\frac1{0.06-0.01}=20

Model

Relate the stated positive 10% price estimate to the yield change.

0.10\approx-D_{\mathrm{mod}}\Delta i

Compute

Solve for the yield change and add it to the initial yield.

\Delta i=-\frac{0.10}{20}=-0.005

i_1=0.06-0.005=0.055

Answer

The implied new yield is 5.50%, choice C.

\boxed{i_1=5.50\%\quad\text{(C)}}

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How to solve this Geometric Perpetuity Approximation question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Why the other letters tempt

Original variant: infer a new yield from a duration estimate

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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