This Exam FM sample reference tests Duration and Convexity. Because Macaulay duration exceeds modified duration at positive yield, c must pair with modified duration a; hence 1 + i = c/a and the other Macaulay duration is bc/a, choice A.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThe ratio ac/b pairs c with the larger modified duration, contradicting c < b at positive yield.
CThe product ab/c does not apply the common factor c/a to the second modified duration.
DAdding and subtracting durations is inconsistent with the multiplicative conversion by one plus yield.
EThis additive expression likewise ignores the common yield scale.
Original practice · fully worked
Original variant: infer the common yield from duration measures
A restoration bond has Macaulay duration 4.24 years and modified duration 4.00 years. A second bond at the same annual effective yield has modified duration 6.00 years. Determine their common yield.
A 4.00%
B 5.00%
C 6.00%
D 7.00%
E 8.00%
Variant answer in brief
The ratio of Macaulay to modified duration is 1.06, so the common yield is 6.00%, choice C.
Setup
Setup
At an annual effective yield, Macaulay duration equals modified duration multiplied by one plus yield.
DM=(1+i)Dmod
Model
Model
The first bond supplies both measures, so their ratio identifies that multiplier directly.
4.24=(1+i)(4.00)
Compute
Compute
Dividing 4.24 by 4.00 gives 1.06 and therefore yield 0.06.
i=4.004.24−1=0.06
Answer
Answer
The common annual effective yield is 6.00%, corresponding to choice C.
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