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Present Value of Annuity Due

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Present Value of Annuity Due Formula

The formula for the present value of an annuity due, sometimes referred to as an immediate annuity, is used to calculate a series of periodic payments, or cash flows, that start immediately.

How is the Present Value of an Annuity Due Derived?

The present value of an annuity due formula uses the same formula as an ordinary annuity, except that the immediate cash flow is added to the present value of the future periodic cash flows remaining. The number of future periodic cash flows remaining is equal to n - 1, as n includes the first cash flow.

To show this visually, the extended version of the present value of annuity due formula of

Extended Version of PV of Annuity Due Formula

shows that the first cash flow is not discounted and that the discounted cash flows start at period 2. After factoring out the first immediate payment, the additional payments consist of an ordinary annuity with n - 1 payments remaining.

The formula shown on the top of the page can be shown as P + PV of ordinary
annuityn-1.

Alternative Formula for the Present Value of an Annuity Due

The present value of an annuity due formula can also be stated as

Alternative PV of Annuity Due Formula

which is (1+r) times the present value of an ordinary annuity. This can be shown by looking again at the extended version of the present value of an annuity due formula of

PV of Annuity Due restated

This formula shows that if the present value of an annuity due is divided by (1+r), the result would be the extended version of the present value of an ordinary annuity of

PV of Annuity restated

If dividing an annuity due by (1+r) equals the present value of an ordinary annuity, then multiplying the present value of an ordinary annuity by (1+r) will result in the alternative formula shown for the present value of an annuity due.


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