Question

The RBC Students Leading Change Scholarship is awarded to eight students graduating high

school and seven students currently enrolled in a post-secondary program each year. Each of the
fifteen scholarships is worth $10,000, totaling $150,000 worth of scholarships.
In order to continue to offer these scholarships every year in perpetuity, RBC was looking into
setting up an endowment fund, growing at 5.25% per annum, compounded daily. The first set of
scholarships is to be offered today.
a) What should be the size of the endowment fund set up by RBC today in order to fully
cover the costs of offering the scholarships every year in perpetuity?
b) RBC currently has $3,900,000 that they are able to invest in the endowment fund. If RBC
invested this amount in the fund today, what is the maximum amount that may be
withdrawn every year to put towards the scholarships?
c) What should be the size of the endowment fund set up by RBC today in order to fully
cover the costs of offering the scholarships every year in perpetuity, if the first set of
scholarships is offered in five years instead of today?
d) Instead of investing in an endowment fund today, RBC contemplated setting up a
high-growth fund using their monthly profits to save up for the cost of the endowment
fund. If they make payments at the end of every month, what would be the size of RBC’s
first monthly payment in order to accumulate to the value calculated in (a) in five years?
The fund has an interest rate of 10.5% per annum, compounded monthly.
e) Does RBC earn more interest if it follows the payment plan laid out in (c) or (d)? By how
much?

Answer 1

a) The size of the endowment fund should be $2,857,143. b) The maximum amount that may be withdrawn every year is $204,375. c) The size of the endowment fund should be $2,123,272.

Step-by-step explanation:

a) To fully cover the costs of offering the scholarships every year in perpetuity, RBC needs to set up an endowment fund today with a size equal to the present value of the perpetuity stream of $150,000 scholarship payments. We can use the formula:

PV = PMT / r, where PV is the present value, PMT is the annual payment, and r is the interest rate.

In this case, PMT = $150,000 and r = 5.25%.

Plugging these values into the formula, we get

PV = $150,000 / 0.0525 = $2,857,143.

The size of the endowment fund should be $2,857,143.

b) If RBC invests $3,900,000 in the endowment fund today, they can withdraw a maximum amount each year equal to the interest earned on the fund. The interest earned can be calculated using the formula:

Interest = Principal * (1 + r)^n - Principal, where Principal is the initial investment, r is the interest rate, and n is the number of years.

Plugging in the values, we get

Interest = $3,900,000 * (1 + 0.0525)^1 - $3,900,000 = $204,375

Therefore, the maximum amount that may be withdrawn every year is $204,375.

c) If the first set of scholarships is offered in five years instead of today, the size of the endowment fund needs to be adjusted to the present value of the perpetuity stream of $150,000 scholarship payments starting in year five. Using the same formula as in (a), but with only 10 years left in perpetuity, we get

PV = $150,000 / 0.0525 = $2,857,143.

However, this value needs to be discounted to its present value in year five.

Using the formula:

PV5 = PV / (1 + r)^n, where PV5 is the present value in year five, PV is the present value in year zero, r is the interest rate, and n is the number of years, we can calculate

PV5 = $2,857,143 / (1 + 0.0525)^5 = $2,123,272.

Therefore, the size of the endowment fund should be $2,123,272.

d) If RBC makes monthly payments to accumulate to the value calculated in (a) in five years, they need to calculate the monthly payment using the formula for annuity:

P = PV * r * (1 + r)^n / ((1 + r)^n - 1), where P is the monthly payment, PV is the present value in year zero, r is the interest rate, and n is the number of months. Plugging in the values, we get

P = $2,857,143 * (0.105 / 12) * (1 + 0.105 / 12)^60 / ((1 + 0.105 / 12)^60 - 1) = $43,385.

Therefore, the size of RBC's first monthly payment should be $43,385.

e) RBC earns more interest if it follows the payment plan laid out in (c) because the interest rate for the high-growth fund is higher than the interest rate for the endowment fund. The difference in interest earned can be calculated by subtracting the amount of interest earned in (d) from the amount of interest earned in (b).

Using the same formula as in (b), we get

Interest = $3,900,000 * (1 + 0.0525)^1 - $3,900,000 = $204,375.

Therefore, RBC earns $204,375 - $43,385 = $160,990 more interest by following the payment plan laid out in (c).