Question

You borrow $10,000 on 1/1/2020, at the annual interest rate of 4%, and will repay in 10 annual installments, beginning on 12/31/2020, and continuing at the end of each year for subsequent years.  The installments are not level, but will increase at an annual rate of 3% with the first payment of $x.  Thus, the second payment will be $x(1.03), the third payment will be $x(1.03)², etc.

(a)        Calculate $x.

(b)       What is the total amount of payments?  (Just add the payments, without interest.)

(c)        What is the total amount of interest?

(d)       Assume that all the calculations are repeated without the 3% annual increase, i.e., assume level repayments.  Withoutdoing the actual calculations, do you expect the total amount of interest to be higher than, the same as, or lower than your answer in part (c)?

Answer 1

(a) Borrowing = $ 10000, Growing Annuity Repayments begin with value of $ x and grow at a constant rate of 3% per annum, Total Repayment Tenure = 10 years and Annual Interest Rate = 4 %

Therefore, 10000 = x /(0.04-0.03) x [1-{(1.03)/(1.04)}^(10)}]

10000 = x * 9.20982

x = 10000 / 9.20982 = $ 1085.79698 ~ $ 1085.8

(b) Total Payments = x * Number of Payments = 1085.8 x 10 = $ 10857.9698 ~ $ 10858

(c) Total Interest = Total payments - Borrowing = 10858 - 10000 = $ 858

(d) If annual repayments are equal in magnitude, then let them be $ s each

Therefore, 10000 = s x (1/0.04) x [1-{1/(1.04)^(10)}]

As is observable, the discount factor expression on the RHS (the expression being multiplied to $ s) has a lower value as compared to the previous growing annuity expression.This implies that the magnitude of $ s is higher as compared to $ x. Consequently, the total interest paid in this case will be higher.