Annuity Present Values

Present Value of Ordinary Annuity:

Present value of of an ordinary annuity (payments at the end of the periods) with  $10,000 payments for two periods at 10% = $17,355.  You can prove the answer by calculating the growth of the present value amount over the two periods.
 
Period
Beg.  P. Value
Interest*
Payment
Ending** 
Balance
1
17,355
1,736
10,000
9,091
2
9,09
909
10,000
0
2,645
20,000
* 10% times beginning present value
** Beginning balance + interest - payment

The schedule shows the growth in interest required to realize two $10,000 payments.
 

Present Value of Annuity Due:

Present value of of an annuity due (payments at the beginning of the periods) with  $10,000 payments for two periods at 10% = $19,091.  You can prove the answer by calculating the growth of the present value amount over the two periods.
 
Period
Beg. P. Value
Payment
Interest#
Ending##
Balance
1
19,091
10,000
909
10,000
2
10,000
10,000
0
0
   
20,000
909
  
# 10% of beginning present value less the payment
##  Beginning balance - payment + interest

The schedule shows the growth in interest required to realize two $10,000 payments.

Methods of Determining Annuity Present Values:

1.  With Excel, use the PV function.
2.  Use annuity tables.
3.  Finncial Calculator.
 

Excel

 Present value function:  PV(rate, nper, pmt, fv, type)

Rate    is the interest rate per period. For example, if you obtain an automobile loan at a 10% annual interest rate and make monthly payments, your interest rate per month is 10%/12, or 0.83%. You would enter 10%/12, or 0.83%, or 0.0083, into the formula as the rate.
Nper    is the total number of payment periods in an annuity. For example, if you get a four-year car loan and make monthly payments, your loan has 4*12 (or 48) periods. You would enter 48 into the formula for nper.

Pmt    is the payment made each period and cannot change over the life of the annuity. Typically, pmt includes principal and interest but no other fees or taxes. For example, the monthly payments on a $10,000, four-year car loan at 12% are $263.33. You would enter -263.33 into the formula as the pmt.

Fv    is the future value, or a cash balance you want to attain after the last payment is made. If fv is omitted, it is assumed to be 0 (the future value of a loan, for example, is 0). For example, if you want to save $50,000 to pay for a special project in 18 years, then $50,000 is the future value. You could then make a conservative guess at an interest rate and determine how much you must save each month.

Type    is the number 0 or 1 and indicates when payments are due.

Set type equal to 0 (or omit) if payments are made at the end of the period.
Set type eual to 1 if payments are made at the beginning of the period.
 
 

Annuity Tables

Ordinary annuity:

10,000 X 1.7355 = 17,355  (using table D on page 1210)

Annuity due:

10,000 X 1.9091 = 19,091 (using table E on page 1212)

Financial Calculator

With the financial calculator, choose the ordinary annuity or annuity due.  If you have problems with the annuity due solution, you can reduce the number of periods by one and then add one payment amount because the present value of the first payment is the same as the payment amount.  That is, the present value and the future value both are the same for the first payment.  For example, the present value of an ordinary annuity of one $10,000 payment at 10% is $9,091 plus another payment of $10,000 equals the $19,091 present value of an annuity due for two payments.