This Exam FM sample reference tests Perpetuities with Different Frequencies. Substitution of the recovered rate in the second pricing equation gives R = 1.74. The result agrees with the published answer key, choice D.
How to solve this Perpetuities with Different Frequencies question
Setup
Setup
Use the first perpetuity price to recover its effective annual yield. Because payments are every two years, its geometric ratio is based on the two-year accumulation factor.
7.21=1+(1+i)2−11⟹i=0.0775
Model
Model
The second perpetuity has a different payment interval and yield. Discount its first payment to time 0 and value the remaining three-year-spaced payments as a geometric perpetuity.
7.21=R(1.0875)−1(1+(1.0875)3−11)
Compute
Compute
Substitution of the recovered rate in the second pricing equation gives R = 1.74.
R=1.74
Answer
Answer
The calculation gives 1.74 for perpetuities with different frequencies, matching published choice D.
R=1.74(D)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A (1.23) does not match the checked perpetuities with different frequencies result (1.74); no distinct standard one-step error is identifiable.
BChoice B (1.56) does not match the checked perpetuities with different frequencies result (1.74); no distinct standard one-step error is identifiable.
CChoice C (1.60) does not match the checked perpetuities with different frequencies result (1.74); no distinct standard one-step error is identifiable.
EChoice E (1.94) does not match the checked perpetuities with different frequencies result (1.74); no distinct standard one-step error is identifiable.
Original practice · fully worked
Original variant: annual perpetuity inferred from a triennial quote
One endowment distributes 30 at times 3, 6, 9, and so on; its time-0 value is 150. A second endowment distributes 10 at times 1, 2, 3, and so on under the same yield. Determine the second endowment’s time-0 value.
A 145.60
B 150.00
C 159.60
D 169.85
E 180.00
Variant answer in brief
The annual discount factor is 0.941036, giving present value 159.60. The annual perpetuity is worth 159.60, so choice C is correct.
Setup
Setup
Use the triennial perpetuity to determine the three-year discount factor.
150=1−v330v3⟹v3=65
Model
Model
Take its cube root to obtain the annual discount factor, then apply the annual perpetuity-immediate formula.
v=(5/6)1/3,PV=1−v10v
Compute
Compute
The annual discount factor is 0.941036, giving present value 159.60.
PV=159.595090
Answer
Answer
The annual perpetuity is worth 159.60, so choice C is correct.
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