Worked solution · Exam FM
This Exam FM sample reference tests Macaulay Approximation for a Geometric Perpetuity. At 8% the price is 10.8 and Macaulay duration is 9.8. The Macaulay rate-ratio estimate at 7% is 11.83084 versus exact price 11.88889, a −0.4883% error, choice D.
Independent solution
Setup
For a due perpetuity with first payment 1 and growth minus 2%, use its closed-form price.
Model
Its Macaulay duration is one plus the growth rate divided by yield minus growth.
Compute
Apply the Macaulay rate-ratio estimate and compare it with exact repricing at 7%.
Answer
The percentage error is −0.49%, choice D.
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Error diagnosis
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Original practice · fully worked
A geometric perpetuity-due makes its first payment today and each later annual payment is 1% smaller than the preceding payment. At an annual effective yield of 6%, determine its Macaulay duration.
For a geometric perpetuity-due, Macaulay duration is one plus growth divided by yield minus growth. With growth −1%, the duration is 0.99 divided by 0.07 = 14.1429, choice C.
Setup
Represent the annual payment growth rate as r = −0.01.
Model
Use the present-value-weighted timing formula for a geometric perpetuity-due.
Compute
Insert the decline and yield.
Answer
The Macaulay duration is 14.14 years, choice C.
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