Worked solution · Exam FM
This Exam FM sample reference tests Perpetuity Duration. The ten-year zero has Macaulay duration 10, so the perpetuity-immediate has Macaulay duration 20. This fixes the yield at 1/19 and gives modified duration 19, choice A.
Independent solution
Setup
A zero-coupon bond's Macaulay duration equals its maturity. Apply the stated two-to-one duration ratio.
Model
For a level perpetuity-immediate, Macaulay duration is one plus the reciprocal of the annual yield.
Compute
Solving gives i = 1/19. Divide Macaulay duration by one plus i to obtain modified duration.
Answer
The modified duration is 19.0 years, choice A.
Test the method on unseen problems
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.
Error diagnosis
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
A level perpetuity-due is valued at an annual effective yield of 5%. A zero-coupon asset is sought whose Macaulay duration equals the perpetuity-due's modified duration. Determine the zero-coupon maturity in years.
A level perpetuity-due has modified duration 1 divided by i times 1 plus i. At 5% this is 19.0476 years, so C is correct.
Setup
A zero-coupon asset has Macaulay duration equal to its maturity n.
Model
For the perpetuity-due, differentiate its price to obtain modified duration.
Compute
At 5%, set the zero maturity equal to the computed modified duration.
Answer
The matching maturity is about 19.05 years, choice C.
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