This Exam FM sample reference tests First-Order Duration Approximation. At 6.25% the perpetuity-due price is 17 and its modified duration is 15.05882. The first-order estimate at 5.75% is 18.28 versus the exact 18.39130, an error of −0.6052%, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A uses Macaulay duration in the linear estimate without converting it to yield sensitivity.
BChoice B rounds the initial duration and both prices before forming the small relative error.
DChoice D compares the price change with the initial price rather than comparing the approximation error with the exact new price.
EChoice E applies only half of the 50-basis-point yield change in the duration estimate.
Original practice · fully worked
Original variant: approximation error for a zero-coupon bond
A ten-year zero-coupon bond redeems for 1,000 and initially yields 5% effective annually. Its yield falls to 4.5%. Use modified duration for a first-order price estimate and determine the percentage error of that estimate relative to the exact new price.
A −0.50%
B −0.24%
C −0.12%
D +0.12%
E +0.50%
Variant answer in brief
The first-order estimate is 643.1472 and the exact new price is 643.9277. The relative error is −0.1212%, choice C.
Setup
Setup
Price the zero at the initial yield and compute its modified duration.
P0=1.05101,000=613.913254
Dmod=1.0510=9.523810
Model
Model
Apply the negative 0.5% yield change to the linear duration formula.
E=P0[1−Dmod(−0.005)]=643.147218
Compute
Compute
Calculate the exact price at 4.5% and compare the estimate with it.
P1=1.045101,000=643.927682
P1E−P1=−0.00121204
Answer
Answer
The first-order estimate is low by about 0.12%, choice C.
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