Question

You have a loan outstanding. It requires making eight annual payments of $3,000 each at the end of the next eight years. Your bank has offered to restructure the loan so that instead of making the eight payments as originally​ agreed, you will make only one final payment in eight years. If the interest rate on the loan is 9%​, what final payment will the bank require you to make so that it is indifferent to the two forms of​ payment?

Answer 1

Given Annual payment = 3000

Interest rate on loan= 9%

Number of payments = 8

Present value of all the payments = Annual Payment/(1+ Interest rate)^1+Annual Payment/(1+ Interest rate)^2+Annual Payment/(1+ Interest rate)^3+Annual Payment/(1+ Interest rate)^4+Annual Payment/(1+ Interest rate)^5+Annual Payment/(1+ Interest rate)^6+Annual Payment/(1+ Interest rate)^7+Annual Payment/(1+ Interest rate)^8

=3000/(1+9%)^1+3000/(1+9%)^2+3000/(1+9%)^3+3000/(1+9%)^4+3000/(1+9%)^5+3000/(1+9%)^6+3000/(1+9%)^7+3000/(1+9%)^8

=3000/(1.09)^1+3000/(1.09)^2+3000/(1.09)^3+3000/(1.09)^4+3000/(1.09)^5+3000/(1.09)^6+3000/(1.09)^7+3000/(1.09)^8

=3000/1.09+3000/1.1881+3000/1.295029+3000/1.411582+3000/1.538624+3000/1.6771+3000/1.828309+3000/1.992563

=2752.294+2525.04+2316.55++2125.276+1949.794+1788.802+1641.103+1505.599

=16604.457

To be indifferent on one time payment made at the 8th year the amount should be equal to future value of the present value of equal payments

Hence One time payments= 16604.457*(1+9%)^8

= 16604.457*(1.09)^8

=16604.457*1.992563=33085.42

Hence to be indiffferent between both the payment structure the one time payment in the 8 th year should be equal to 33085.42

(Note alternatively the one Payment made in the 8th year can be computed by calculating the future value of the equal payment which is shown below=

=3000*(1+ 9% )^7+3000*(1+ 9%)^6+3000*(1+ 9%)^5+3000*(1+ 9%)^4+3000*(1+ 9%)^3+3000*(1+ 9%)^2+3000*(1+ 9%)^1+3000*(1+ 9%)^0

=3000*1.828039+3000*1,6771+3000*1.538624+3000*1.411582+3000*1.295029+3000*1.881+3000*1.09+3000

= 5484.11736+ 5031.30033+4615.87186+4234.74483+3885.087+3564.3+3270+3000

=33085.4214 or 33085.42)