Question

Scheduled payments of $1400 due today and $1600 due with interest at 11.5% compounded annually in five years are to be replaced by two equal payments. The first replacement payment is due in 18 months and the second payment is due in 4 years. Determine the size of the two replacement payments if interest is 11% compounded quarterly and the focal date is 18 months from now.

Answer 1

Assuming current time to be t=0, the focal date of 18 months from now is t=18 months or 1.5 years (18/12)

The replacement payments of equal magnitude should have a focal date present value (present value on the stated focal date of t=18) equal to that of the scheduled payments' focal date present value. The only difference being that while replacement payments are compounded quarterly, the scheduled payments are compounded annually.

Scheduled Payment:

$ 1400 at t=0 (current time) and $ 1600 due five years from t=0

Interest Rate = 11.5 % compounded annually

Future Value of $1400, 18 month from now = FV1 = 1400 x (1.115)^(1.5) = $ 1400 x 1.177368 = $ 1648.315

Present Value of $ 1600, 18 months from now = PV1 = 1600 / (1.115)^(3.5) = $ 1093.095

Total Value of Scheduled Payments 18 months from now = FV1 + PV1 = 1648.315 + 1093.095 = $ 2741.41

Replacement Payment:

Let the required replacement payment be $ K and Interest Rate = 11. 5 % compounded quarterly

Applicable Quarterly Interest Rate = (1.115)^(1/4) - 1 = 0.027587 or 2.7587 %

Therefore, Total Present Value of Replacement Payments 18 months from now = TPV = K + K / (1.027587)^(10) (the time period is 10 because between t=18 months (1.5 years) and t= 4 years, the number of quarters (3 month period) is 10)

TPV = K + 0.761751 K

TPV = (1.761751) K

Now TPV = FV1 + PV1

1.761751 K = 2741.41

K = 2741.41 / 1.761751 = $ 1556.072