A college has a scholarship fund that pays a sum of money twice a year. The scholarship will pay out 500 at the end of six months and another 500 at the end of one year. Every year thereafter, the two semi-annual payments will be increased by 10. For example, in year two, both payments will be 510 and in year three both payments will be 520. The scholarship fund earns interest at an annual effective interest rate of 7.5%.

Calculate the fund balance needed today to provide this scholarship indefinitely.

• 16,000
• 16,589
• 16,889
• 17,134
• 17,200

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

Split this into two perpetuities. One starts at time 0.5 at 500 increasing by 10 every year. The other starts at time 1 at 500 with payments increasing by 10 every year. The semiannual interest rate is

[[math]] 1.075^{0.5}-1=0.0368221 [[/math]]

The present value of an increasing perpetuity immediate is found using the formula:

[[math]]\frac{P}{i} + \frac{Q}{i^2}[[/math]]

where P is the initial amount and Q is the increase amount. The first perpetuity, valued at time 0:

[[math]]\left(\frac{500}{0.075}+\frac{10}{0.075^{2}}\right)\!\left(1.0368221\right)=875.39 [[/math]]

The second perpetuity, valued at time 0:

[[math]]\left(\frac{500}{0.075}+\frac{10}{0.075^{2}}\right)=8444.444 [[/math]]

The total is 8755.39 + 8444.44 = 17,199.83.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.