**Annuity due payment with PV = PV × [r ÷ {1 – (1 + r)**^{-n}** } ]**

PV = present value

r = rate for period

n = # of periods

Annuity due payment with present value formula helps tocalculate each installment for a series of payments or cash flows where firstcash flow is instant. This formula can only be used in cases where the presentvalue of an investment is known. Furthermore, as it can be seen in the equationabove the formula uses present value and without this figure, it cannot besolved.

The decision to use annuity due payment with FV or useannuity due payment with PV varies with case to case. It only depends onincreasing or decreasing the balance of annuity over a period of time. Tounderstand this concept let’s consider an example;

To understand the increasing and decreasing balance concept let’s consider an example. An individual is looking to calculate the cash outflow per annum from an interest account having $25,000 balance for the next 7 years. The balance will earn up to 6% per year. Moreover, the individual wants to increase cash outflows with 3% per annum to consider inflation.

As the balance is decreasing with time and we know thepresent value (i.e. $25,000) therefore, we will use the annuity due paymentwith present value formula.

Similarly, if an individual is looking to determine theamount to save per year in a saving account in order to get an ending balanceof $25,000 after 7 years. The deposit amount will increase by 3% per annum and therate of return will be 6% per annum. In this scenario, the balance isincreasing with time and future value (i.e. $25,000 is known) therefore theannuity due payment formula will be used.

## Annuity due payment using present value example

Let’s consider an example, where an individual is looking todetermine the amount he can withdraw from his savings account each year for upto 7 years. The balance in his account $25,000 and interest rate is 4% on hisaccount.

The balance must be $0 at the end of 7^{th} year orsay at the start of 8^{th} year and the amount will be withdrawn at theend of each year. Now putting values in the annuity due payment using presentvalue formula to get;

Annuity due Payment = $25,000 [(0.04) ÷ {1 – (1 + 0.04)^{-7}}]× [1 ÷ (1 + 0.04)]

Annuity due payment = $4014.4

### How annuity due payment using PV formula induced

To reach the annuity due payment with PV formula, first onemust have to consider the present value of annuity due formula which isas below;

PV of annuity due = (1 + r) × P [{1 – (1 + r)^{-n}}÷ r]

As you know that we have to solve for payment ** P,**for that we will factor out all other values to solve the equation for paymentto get;

Annuity due payment with PV = PV × [1 ÷ {(1 – (1 + r)^{-n})÷ r } × 1 ÷ (1 + r)]

Lastly, at this step, we will multiply the whole equationwith the inverse of the denominator from the middle portion of the equation.This will then give us formula which is available at the top of this page.