Anderson Manufacturing​ Co., a small fabricator of​ plastics, needs to purchase an extrusion molding machine for ​$200,000. Kersey will borrow money from a bank at an interest rate of 13​% over five years. Anderson expects its product sales to be slow during the first​ year, but to increase subsequently at an annual rate of 11​%. Anderson therefore arranges with the bank to pay off the loan on a​ “balloon scale,” which results in the lowest payment at the end of the first year and each subsequent payment being just 11​% over the previous one. Determine the five annual payments.（1,2,3,4,5）

EXPERT ANSWER

The annual loan payments increasing at a rate of 11% shall form a growing annuity with growth rate of 11%. The annual payments shall be calculated in such a manner that the present value of all the five payments discounted at the interest rate of 13% shall add up to the loan amount.

Loan amount = $200,000

Interest rate = r = 13% = 0.13

Growth rate = g = 11% = 0.11

Number of payments = n = 5

Let the first payment be P

Present value of growing annuity = P/(r-g) * {1 – (1+g/1+r)ⁿ}

$200,000 = P/(0.13-0.11) * {1-(1.11/1.13)⁵}

$200,000 = P/0.02 * (1-0.91458)

$200,000 = P*4.27090

P = $200,000/4.27090 = $46,828.57

Hence, the first payment is $46,828.57

The five annual payment shall be as below:

Year	Payment
1	$46,828.57
2	$51,979.71
2	$57,697.48
3	$64,044.20
4	$71,089.07
5	$78,908.86