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
Answer and calculation
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.
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.
PV≈29531(C) 