Worked solution · Exam FM

SOA Exam FM Sample Question #219 Solution

This Exam FM sample reference tests Macaulay Duration. The liability duration is 2.64516 years and the equal-payment asset duration is 2.62331 years. Their absolute difference is 0.02185, which rounds to choice C.

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 Macaulay Duration question

Setup

Let v be the 10% annual discount factor. Macaulay duration is the present-value-weighted average payment time.

v=(1.10)^{-1}

Model

Compute the duration of the two liability payments and the two equal asset payments separately. The common asset amount X cancels.

D_L=\frac{2(100v^2)+3(200v^3)}{100v^2+200v^3}

D_A=\frac{1(Xv)+5(Xv^5)}{Xv+Xv^5}

Compute

The weighted payment times are 2.64516 and 2.62331 years.

D_L=2.645161

D_A=2.623311

|D_L-D_A|=0.021851

Answer

The duration difference rounds to 0.022, so the answer is choice C.

\boxed{|D_L-D_A|\approx0.022\quad\text{(C)}}

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.

See the cover and sample pages

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 results from rounding both duration numerators and denominators before division.
BChoice B rounds the discount factor to one decimal place, which is too coarse for a difference of durations.
DChoice D uses undiscounted cash-flow weights for one side and discounted weights for the other.
EChoice E omits the time multiplier from one payment in a dollar-duration numerator.

Original practice · fully worked

Original variant: asset ratio that matches liability duration

A reserve discounts at 5% annually and owes 150 at time 3 and 250 at time 6. It will buy one asset paying X at time 2 and another paying Y at time 8. Determine the ratio Y divided by X that makes the asset and liability Macaulay durations equal.

A
0.842
B
0.996
C
1.150
D
1.327
E
1.516

Answer and calculation

The liability duration is 4.77036 years. Solving the two-asset duration equation gives Y divided by X = 1.14952, choice C.

Setup

Compute the liability's present-value-weighted payment time at the 5% discount rate.

v=(1.05)^{-1}

Model

Let r equal Y divided by X. Divide the asset duration numerator and denominator by X.

D_L=\frac{3(150v^3)+6(250v^6)}{150v^3+250v^6}

D_A=\frac{2v^2+8rv^8}{v^2+rv^8}

Compute

The liability duration is 4.770355. Set D_A equal to that value and solve the linear equation in r.

r=\frac{v^2(D_L-2)}{v^8(8-D_L)}

r=1.149520

Answer

The required payment ratio is approximately 1.150, so choice C is correct.

\boxed{Y/X\approx1.150\quad\text{(C)}}

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How to solve this Macaulay Duration question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Why the other letters tempt

Original variant: asset ratio that matches liability duration

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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