# Claire needs money to set up a website. She predicts she can afford to pay \$17,000 for the loan in 2 years. The plan she is arranging offers a loan at 9.8%, compounded monthly. How much can she borrow?

If Claire waits to pay all \$17,000 at the end of 2 years, she can borrow about \$13,985.34. If Claire makes monthly payments, she can borrow as much as \$15,380.87. 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:

`17,000=P(1+.098/12)^24`

`P=17,000/(1+.098/12)^24~~13,985.33592`

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.

So `708.33=PV((.098/12)/(1-(1+.098/12)^(-24)))`

Then `PV=(708.33[1-(1+.098/12)^(-24)])/(.098/12)~~15,380.87`

Thus with monthly payments of \$708.33 over 2 years (for a total payment of \$17,000), Claire can borrow approximately \$15,380.87.

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