Worked solution · Exam FM

SOA Exam FM Sample Question #339 Solution

This Exam FM sample reference tests Full Immunization with Zero-Coupon Assets. At the liability date, the five-year zero contributes 388888 of value and the other zero contributes 1361112. The first-derivative condition makes its timing displacement two years, so its maturity is year 14, 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 Full Immunization with Zero-Coupon Assets question

Setup

Move all values to the liability date at year 12. Let N be the maturity value of the n-year zero.

242{,}180(1.07)^7+N(1.07)^{-(n-12)}=1{,}750{,}000

Model

Full immunization also sets the first timing moment about year 12 to zero.

242{,}180(7)(1.07)^7-N(n-12)(1.07)^{-(n-12)}=0

Compute

The first equation assigns 1361112 of year-12 value to the later zero. Substitute it into the moment equation.

n-12=\frac{242{,}180(7)(1.07)^7}{1{,}361{,}112}=2

Answer

The second zero matures in year 14, choice B.

\boxed{n=14\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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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 places the later zero only one year beyond the liability and cannot offset the seven-year displacement of the early zero.
CChoice C uses the face amount 242180 directly as a year-12 weight instead of accumulating it seven years.
DChoice D matches present value but leaves a nonzero first timing moment around year 12.
EChoice E assumes symmetric maturity distances even though the two asset values at year 12 are unequal.

Original practice · fully worked

Original variant: maturity value of the long immunizing zero

A liability has present value 500,000 and Macaulay duration 10 years. It is fully immunized at a 6% annual yield with zero-coupon assets maturing in 6 and 14 years. Determine the maturity value of the 14-year zero.

A
424,173
B
500,000
C
530,000
D
565,226
E
625,000

Answer and calculation

Duration 10 is midway between 6 and 14, so each zero receives 250000 of present value. Accumulating the long allocation for 14 years gives maturity value 565225.99, choice D.

Setup

Let w be the present-value share in the 14-year zero.

6(1-w)+14w=10

Model

The duration equation gives equal present-value shares.

w=\frac12

PV_{14}=250{,}000

Compute

Convert the time-zero allocation to its year-14 maturity amount.

F_{14}=250{,}000(1.06)^{14}=565{,}225.989

Answer

The long zero matures for about 565226, choice D.

\boxed{F_{14}\approx565{,}226\quad\text{(D)}}

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How to solve this Full Immunization with Zero-Coupon Assets question

Setup

Model

Compute

Answer

Financial Mathematics Duration & Immunization Workbook

Why the other letters tempt

Original variant: maturity value of the long immunizing zero

Answer and calculation

Setup

Model

Compute

Answer

Related Exam FM solutions

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