This Exam FM sample reference tests Duration and Convexity. For a perpetuity-due, Macaulay duration 30 implies i = 1/30; dividing by 1 + i gives modified duration 29.032, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A is inconsistent with the perpetuity-due duration identity and the Macaulay-to-modified conversion; no distinct standard single-step error producing it is identifiable.
BChoice B is inconsistent with the perpetuity-due duration identity and the Macaulay-to-modified conversion; no distinct standard single-step error producing it is identifiable.
DChoice D is inconsistent with the perpetuity-due duration identity and the Macaulay-to-modified conversion; no distinct standard single-step error producing it is identifiable.
EChoice E is inconsistent with the perpetuity-due duration identity and the Macaulay-to-modified conversion; no distinct standard single-step error producing it is identifiable.
Original practice · fully worked
Original variant: yield implied by a library endowment's price sensitivity
A library endowment pays one unit immediately and at every later year-end forever. Its modified duration is measured as 20 years. Assuming a level annual effective yield, determine that yield.
A 4.00%
B 4.50%
C 4.77%
D 5.00%
E 5.25%
Variant answer in brief
The perpetuity-due identity 20 = 1/[i(1+i)] gives i = 4.77%, choice C.
Setup
Setup
The immediate payment makes this a perpetuity-due, whose Macaulay duration is one divided by the annual yield.
Dmod=1+i1/i=i(1+i)1
Model
Model
Converting that duration to modified duration produces a quadratic relationship in the unknown yield.
20i(1+i)=1
Compute
Compute
Only the positive quadratic root is economically admissible, and it equals 0.0477226.
i=2−1+1.2=0.0477226
Answer
Answer
Thus the yield consistent with the measured sensitivity is about 4.77%, choice C.
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