Worked solution · Exam FM
This Exam FM sample reference tests Duration and Convexity. A perpetuity beginning in year 5 has Macaulay duration 15 years at 10%; dividing by 1.10 gives modified duration 13.64, choice A.
Independent solution
Setup
A level perpetuity with its first payment in year one has Macaulay duration one plus i divided by i.
Model
Deferring the first payment from year one to year five shifts every payment time four years later.
Compute
The resulting Macaulay duration is 15; converting to modified duration divides by 1.10.
Answer
The modified duration is 13.64, corresponding to choice A.
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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.
Original practice · fully worked
A level annual perpetuity is valued at 8% effective annually. Its modified duration is 18.9815. At the end of which year is its first payment?
Modified duration 18.9815 implies Macaulay duration 20.5; subtracting the base perpetuity duration 13.5 gives a seven-year shift, so payment starts in year 8, choice D.
Setup
Convert the observed modified duration back to Macaulay duration.
Model
Represent a perpetuity first paid in year k as the ordinary perpetuity duration plus a k-minus-one timing shift.
Compute
The observed duration exceeds the base 13.5 years by seven years, so k equals eight.
Answer
The first payment occurs at the end of year 8, corresponding to choice D.
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Drill the method: Exam FM Duration and Convexity problems
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