Money subject to a rate of interest will grow over time. Thus, the value of the amount of money changes over time. This change is known as the time value of money. For example, if $1000 is invested today at 1% p.a. compounded annually. In one year, this value is $1010, $1020.10 in two years, and $1030.30 in three years.
The values are called equivalent values as they represent the same investment with the same interest rate at different times.
This is important because in order to compare choices, we must make a rational choice on a specific date called the focal date. The focal date can be set on any date.
The choice of calculating the Present Value, P, compared to the Future Value, S, depends on the relation of the due date/payment date in comparison to the focal date.
1. If the due date falls before the focal date, calculate the future value, S.This means with a positive interest rate, you should get a larger value after calculation. For simple interest, you will be calculating S from \(S=P(1+i)^n\).
2. If the due date falls after the focal date, calculate the present value, P.This means with a positive interest rate, you should get a smaller value after calculation. For simple interest, you will be calculating P from \(P=S\left(1+i\right)^{-n}\).
A payment of $500 is due in six months with interest at 12% compounded quarterly. A second payment of $800 is due in 18 months with interest at 10% compounded semi-annually. These two payments are to be replaced by a single payment nine months from now. Determine the size of the replacement payment if interest is 9% compounded monthly and the focal date is nine months from now.
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