This Exam FM sample reference tests Arithmetically Increasing Perpetuities. Splitting the semiannual stream into two annual increasing perpetuities gives present values 8,755.39 and 8,444.44, totaling 17,199.83; choice E.
How to solve this Arithmetically Increasing Perpetuities question
Setup
Setup
Separate the half-year and year-end payments into two annual sequences. Each sequence begins at 500 and increases by 10 each year.
i=0.075
j=(1.075)1/2−1=0.0368221
Model
Model
An annual perpetuity with first payment P at time 1 and annual increase Q has value P divided by i plus Q divided by i squared. The half-year sequence is shifted six months earlier.
PVyear=0.075500+0.075210
PVhalf=(1.075)1/2PVyear
Compute
Compute
The year-end sequence is worth 8,444.44, and shifting the other sequence earlier raises its value to 8,755.39.
PVyear=8444.44
PVhalf=8755.39
PV=17199.83
Answer
Answer
The required fund is about 17,200, which is choice E.
PV≈17200(E)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A values two level perpetuities and omits the annual 10-unit increases.
BChoice B discounts the half-year sequence by an extra half-year instead of shifting it earlier.
CChoice C does not include the correct increasing-perpetuity term 10 divided by 0.075 squared for both sequences.
DChoice D treats both sequences as if their first payments occur at the same date. One begins half a year earlier.
Original practice · fully worked
Original variant: quarterly museum stipends increasing by year
A museum pays four stipends each year, at the end of each quarter. All four first-year stipends are 300. Every stipend in year two is 308, every stipend in year three is 316, and this annual pattern continues forever. At a 6% annual effective rate, find the required fund.
A 27,889
B 28,642
C 29,531
D 30,118
E 31,204
Variant answer in brief
Splitting the quarterly dates into four shifted annual increasing perpetuities gives a total present value of 29,531.00, choice C.
Setup
Setup
Create four annual sequences, one for each quarter position. Every sequence starts at 300 and increases by 8 each year.
i=0.06
B=0.06300+0.0628=7222.22
Model
Model
B is the value when the first payment occurs at time 1. Shift the four sequences to times 0.25, 0.50, 0.75, and 1.00.
PV=Bq=1∑4(1.06)1−q/4
Compute
Compute
The four shift multipliers sum to 4.08983, producing a total value of 29,531.00.
PV=7222.22(4.08983)=29530.999
Answer
Answer
The museum needs approximately 29,531, so choice C is correct.
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