This Exam FM sample reference tests Continuous Perpetuity. The first component is 7.58 and the tail is 19.45. Their sum is 27.03. The result agrees with the published answer key, choice A.
Split the continuous payment stream at time 10, where its growth pattern changes. The first component is level; the second grows continuously at 3%.
PV1=∫010(1.06)−tdt=7.58
Model
Model
Discount both integrals at the 6% annual-effective accumulation convention. For the deferred tail, first discount ten years and then integrate the net growth-discount exponent.
PV2=(1.06)−10∫0∞(1.03)s(1.06)−sds=19.45
Compute
Compute
The first component is 7.58 and the tail is 19.45. Their sum is 27.03.
PV=27.03
Answer
Answer
The calculation gives 27.03 for continuous perpetuity, matching published choice A.
PV=27.03(A)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BChoice B (30.29) does not match the checked continuous perpetuity result (27.03); no distinct standard one-step error is identifiable.
CChoice C (34.83) does not match the checked continuous perpetuity result (27.03); no distinct standard one-step error is identifiable.
DChoice D (38.64) does not match the checked continuous perpetuity result (27.03); no distinct standard one-step error is identifiable.
EChoice E (42.41) does not match the checked continuous perpetuity result (27.03); no distinct standard one-step error is identifiable.
Original practice · fully worked
Original variant: continuous growing perpetuity under a constant force
A data license pays continuously at rate 100exp(0.02t) per year at time t, for all nonnegative t. The force of interest is a constant 0.05. Calculate the present value.
A 2,000.00
B 2,500.00
C 3,000.00
D 3,333.33
E 5,000.00
Variant answer in brief
The integral is 100/0.03 = 3,333.33. The continuous perpetuity is worth 3,333.33, which is choice D.
Setup
Setup
Discount the continuous payment rate by exp(−0.05t).
c(t)=100e0.02t,δ=0.05
Model
Model
Combining payment growth and discounting leaves a decaying exponential with net force 0.03.
PV=∫0∞100e0.02te−0.05tdt
Compute
Compute
The integral is 100/0.03 = 3,333.33.
PV=100∫0∞e−0.03tdt=100/0.03=3333.3333
Answer
Answer
The continuous perpetuity is worth 3,333.33, which is choice D.
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