9.5 B14

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 Scheduled payments of $800 due two years ago and $1000 due in five years are to be replaced by two equal payments. The first replacement payment is due in four years and the second payment is due in eight years. Determine the size of the two replacement payments if interest is 12% compounded semi-annually and the focal date is four years from now.

Draw a timeline:

800 NOW X 1000 X
Two debts, 800 and 1000. The 800 is a FV problem..it will have increased by the focal date. The time is 2 + 4 = 6.

The 800
PY 2
PV 800
IY 12
N 2 * 6 = 12
CPT FV 1609.76

The 1000 is a PV problem, the time is 5 – 4 = 1
N = 1 * 2 = 2
FV 1000
CPT PV 890 So on the focal date your debts are 1609.76 + 890 = 2499.76


2499.76 = X + PV(X). How do we get the PV of an unknown quantity, X? Call X = $1. The time is 8 – 4 = 4
FV = 1
N = 4 * 2 = 8
CPT PV 0.63. Whatever X is, the PV of X will be 0.63 * X
So 2499.76 = X + 0.63 X
2499.76 = 1.63 X, so X = 2499.76/1.63 = 1533.6

Finding the PV of the second X by the formula method...some folks asked me about this, but note you do NOT need to know this.

First, we are using compound interest, so we need to divide the nominal annual interest rate by the number of payments per year. This gives us the periodic interest rate, in this example 0.12/2=0.06. So the PV of $1 will be 1/(1+0.06)^2*4 = 0.63.