Claire needs to find the maximum loan she can take out if (a) she can pay back $17,000 in 2 years (b) with an annual interest rate of 9.8% (c) compounded monthly.
(1) Suppose Claire intends to make no payments until the end of 2 years.
She can use the compound interest formula:
`A=P(1+r/n)^(nt)` where A is the total amount of the loan paid back (including interest), P is the amount of the loan, r is the annual interest rate expressed as a decimal, n is the number of compounding periods per year, and t is the number of years.
Here we know A=17,000; r=.098; n is 12; and t=2. We want to know P:
So the loan amount can be as much as $13,985.34
(2) If Claire arranges to make monthly payments, from the point of view of the lender this is an annuity. Then
`PMT=PV(r/n)/(1-(1+r/n)^(-nt))` where PMT is the monthly payment, PV (present value) is the initial loan amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. Here we are looking for the loan amount PV. The payment amount can be found by taking 17,000/24 to get $708.33; at the end of 2 years, Claire will have paid a total of $17,000.
Thus with monthly payments of $708.33 over 2 years (for a total payment of $17,000), Claire can borrow approximately $15,380.87.