4Year Annuity Time Line
0  1  2  3  4  
PV  C  C  C  C 
This time line is for an ordinary annuity, in which the cash payments are made at the end of each year. For example, the first payment is made exactly one year from the present. The present value of this cash flow is calculated by: PV = C / ( 1 + i ) + C / ( 1 + i )2 + C / ( 1 + i )3 + C / ( 1 + i )4 In general, for a t year annuity: PV = C / ( 1 + i ) + C / ( 1 + i )2 + ... + C / ( 1 + i )t From this potentially long series, a present value formula can be derived. First, multiply each side by 1 / ( 1 + i ). PV / ( 1 + i ) = C / ( 1 + i )2 + C / ( 1 + i )3 + ... + C / ( 1 + i )t+1 In order to eliminate most of the terms in the series, subtract the second equation from the first equation: PV  PV / ( 1 + i ) = C / ( 1 + i )  C / ( 1 + i )t+1 Solving for PV, the present value of an ordinary annuity is given by:  
 
This equation assumes that the first payment of the annuity is made at the end of the first time period. If instead the payments are made at the beginning of each time period, then the present value calculation would be similar to the above, except that all payments would be shifted forward by one year. This shift can be accomplished by multiplying the entire present value expression by ( 1 + i ). Such an annuity with the payments occurring at the beginning of each time period is called an annuity due.Annuity Factor TableThe factor for calculating the present value of an ordinary annuity may be calculated for a range of time periods and interest rates and tabulated for quick reference. The annuity factor is the value of the following expression:
Table of Present Value Annuity Factors

No comments:
Post a Comment