423.4

4.Let’s say you graduate from the Jedi Academy with $14,400 in student loans. Assume that these loans were disbursed as $300 every month over the previous 4 years. The stated rate of interest on these loans is 6%, compounded monthly. However, as is typical with student loans, the loans did not accrue interest during your schooling (even though you were loaned the first $300 4 years ago, you have not been charged interest on it over the past 4 years. Thus the total amount you owe is only 48 times $300 instead of that amount plus interest).

1. How much did this interest deferral save you? That is, if your regular loans of $300 every month had been earning interest at the stated rate, what total amount would you owe now? (8)

   The amount you would owe if you'd been paying interest is the FV of the annuity of your loan disbursements:

   \( \Large 300 \left [ \frac{1}{\frac{.06}{12}} - \frac{1}{\frac{.06}{12} \left (1+\frac{.06}{12} \right )^{48} } \right ] = 12{,}744.10 = PV \) of interest payments.

   The amount at the end is the FV:

   \( \Large 12{,}744.10 \left ( 1+ \frac{.06}{12}\right)^{48} = 16{,}229.35 \)

   So, 16,229.35 - 14,400 (what you owe without interest) = 1829.35. The difference is 1829.35

2. If you owe $14,400 now and you intend to pay it back in monthly installments over the next 6 years, plus one lump sum of $5000 along with the 72^nd payment, how much will you have to pay per month (assume an interest rate of 6%, compounded monthly)? (8)

0	1	…	72
	CF	CF	CF+5000

The PV of the lump payment is:

\( \Large \frac{5000}{\left( 1 + \frac{.06}{12} \right)^{72}} \) = 3491.51

The rest of the payments must cover for the rest of the PV: 14400 - 3491.51 = 10908.49

\( \Large CF = \frac{10{,}908.49} { \left [ \frac{1}{\frac{.06}{12}} - \frac{1}{\frac{.06}{12} \left (1+\frac{.06}{12} \right )^{72}} \right ] } = 180.79 \)

Back to Practice Problems