# Pete Morton is planning to go to graduate school, in a program of study that will take three...

## Question:

Pete Morton is planning to go to graduate school, in a program of study that will take three years. Pete wants to have $8,000 available each year, for various school and living expenses. If he earns 4% on his money, how much must he deposit, at the start of his studies, to be able to withdraw$8,000 a year, for three years?

## Present Value of Annuity Due:

An annuity due is a series of equal payments received or made at the start of each consecutive period (e.g., year, month). The present value of such annuity refers to the current amount equivalent to this stream of payments given an appropriate discount rate.

We suppose that Pete will withdraw $8,000 at the start of each year, so that he can pay for different school and living expenses during the year. Therefore, his withdrawals have a form of an annuity due, and the amount that must be deposited at the start of his studies can be calculated using the following formula: {eq}PV_{ANNUITY\ DUE} = PMT \times \dfrac{1-(1+r)^{-n}}{r} \times (1+r) {/eq} This is where: • PMT = payment amount =$8,000
• r = discount rate or rate of return = 0.04
• n = number of periods = 3 years

We will now plug in the above values and calculate the value of the withdrawals today.

{eq}PV_{ANNUITY\ DUE} = \$8,000 \times \dfrac{1-(1+0.04)^{-3}}{0.04} \times (1+0.04) = \$23,088.76 {/eq}

The present value of the withdrawals is $23,088.76, so Pete must deposit$23,088.76 at the start of his studies. 