Example: Calculate the present value of a series of $10,000 annual payments to be made at the beginning of each year for the next three years, assuming a discount rate of 9% (an

annuity due is a series of equal payments made at the beginning of each period).

One way to calculate the amount required derives from realizing that the present value of a 4-year

annuity due commencing at the end of 5 years is equal to the value of a 9-year

annuity due commencing today less the value of a 5-year

annuity due commencing today.

[Using Valuation Tables--The future value of a level ordinary annuity due can also be calculated using the tables for the future value of an ordinary annuity contained in Appendix F.

Analogous to the derivation of the present value of an annuity due formula, the future value of an annuity due formula is just the future value of an ordinary annuity formula multiplied by (1 + r):

[Using Valuation Tables--The future value of a level annuity due can also be calculated using the tables for the future value of an annuity due contained in Appendix E.

The Pmt for an annuity due given the future value is derived in a manner analogous to the annuity due given the present value by rearranging equation FV4 to solve for Pmt in terms of the other variables.

Number of Periods Present Value Will Provide Annuity Due Payments

This 2-step calculation derives exactly the same result as was determined by using the 3-step process of (1) inflating the $36,000 payment for 5 years of inflation at 6% to derive the first-year college cost figure, (2) calculating the present value in 5 years of a 4-year inflation-adjusted annuity due, and (3) computing the present value today (5 years earlier) of the amount determined in step 2.

The future value of an inflation-adjusted annuity due can be determined by simply multiplying the future value of the ordinary annuity by (1 + r):

Equation Pmt3' shows the formula for computing the initial payment amount for an inflation-adjusted annuity due based upon a present value:

The present before-tax value one needs to invest to generate level after-tax annuity due payments is equal to equation PVAEat1 multiplied by (1 + r):

The formula to determine the present before-tax value one needs to invest in a nondeductible currently fully taxable investment to generate level after-tax annuity due payments is: