A wealthy art collector has decided to endow her favorite art museum by establishing funds for an endowment which would provide the museum with $1,000,000 per year for acquisitions into perpetuity. The art collector will give the endowment upon her fiftieth birthday 11 years from today. She plans to accumulate the endowment by making annual end-of-year deposits into an account. The rate of interest is expected to be 6 percent in all future periods. How much must the art collector deposit each year to accumulate to the required amount?

Question

Quizplus · Accepted Answer

To accumulate $1,000,000 each year in perpetuity at an interest rate of 6%, we can use the perpetuity formula like this:
PV = PMT / r
where PV is the present value of the perpetuity, PMT is the annual payment, and r is the interest rate.
PV = $1,000,000 / 0.06 = $16,666,667
So the art collector needs to accumulate $16,666,667 in 11 years. We can use the future value formula to find out how much she needs to deposit each year:
FV = PV x (1 + r)^n
where FV is the future value, r is the interest rate, and n is the number of periods.
FV = $16,666,667, PV = 0, r = 0.06, n = 11.
$16,666,667 = PMT x (1 + 0.06)^11
PMT = $16,666,667 / 17.9088
PMT = $930,586.93
So the art collector needs to deposit $930,586.93 at the end of each year for 11 years to accumulate $16,666,667, which will provide the museum with $1,000,000 per year in perpetuity. The closest answer choice is C) $1,264,400, which is not exactly equal to $930,586.93 but is the closest to it.

A Wealthy Art Collector Has Decided to Endow Her Favorite