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The definition of an annuity includes the words constant payment - in other words, annuities involve payments that are the same in every period. Although many financial decisions do involve constant payments, other important decisions involve uneven, or non constant, cash flows. For example, common stocks typically pay an increasing stream of dividends over time, and fixed asset investments such as new equipment normally do not generate constant cash flows. Consequently, it is necessary to extend the time value discussion to include uneven cash flow streams.

The Present Value ( PV ) of an uneven cash flow stream is found as the sum of the PVs of the individual cash flows of the stream. For example, suppose we need to find the PV of the following cash flow stream, discounted at 6%.

The PV will be found by applying this general present value equation:

EXAMPLE 1

Suppose we need to find the Present Value of Uneven Cash Flows discounted at 6% 
(100 , 200 , 200 ,  200 ,  200 , 0 , 1000)

SOLUTION

The present value of the series of uneven payments can be found by using the formula mentioned above 

Using MS Excel to compute Present Value of Annuities

In MS Excel you can use the financial function PV(rate,nper,pmt,fv,type) to compute the present value of an investment. The present value is the total amount that a series of future payments is worth now. For example, when you borrow money, the loan amount is the present value to the lender.

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 percent are $263.33. You would enter -263.33 into the formula as the pmt. If pmt is omitted, you must include the fv argument.

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. If fv is omitted, you must include the pmt argument.

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