A foundation announces that it will be offering one MIT scholarship every year for an indefinite number of years. The first scholarship is to be offered exactly one year from now. When the scholarship is offered, the student will receive $20,000 annually for a period of four years, beginning from the date the scholarship is offered. This student is then expected to repay the principal amount received ($80,000) in 10 equal annual installments, interest-free, starting one year after the expiration of her scholarship. This implies that the foundation is really giving an interest-free loan under the guise of a scholarship. The current interest is 6% for all maturities and is expected to remain unchanged.

The foundation invests a lump sum to fund all future scholarships. Determine the size of the investment today.

• $421,717
• $447,020
• $452,300
• $455,220
• $457,500

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.

Solution: B

The PV of the first scholarship from the foundations point of view is

[[math]]-(20,000 / 0.06)\left[1-\left(1 /(1.06)^4\right)\right]+\left(1 /(1.06)^5\right)(8,000 / 0.06)\left[1-\left(1 /(1.06)^{10}\right)\right]=-25,303[[/math]]

So it loses $25,303 every year beginning from t=0. The PV of this perpetuity is $25,303 / 0.06-25,303=-447,020

This implies that the investment needed to fund this is $447,020.

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.