The Present Value of Annuity Due formula is used to calculate the present value of a series of cash flows, or periodic payments, that are generated by an investment in the future.

These payments are expected to be made on predetermined future dates and in predetermined amounts. Present Value of Annuity is based on Time Value of Money, according to which, the same amount of money available at the present time is worth more than the same amount in the future, because money available at the present time can be used to create more money.

So, the PV of Annuity Due formula can be very useful while deciding between taking a lump sum payment now or to instead receiving a series of cash payments in the future.

### How to Calculate Present Value of Annuity Due?

$$P + P \bigg[\frac{1 - (1 + r)^{-(n-1)}}{r}\bigg]$$

Where P = periodic payment, r = rate per period and n = number of periods.

This formula is similar to Present Value of an Ordinary Annuity formula, but the difference is in adding one immediate payment since the payments start immediately at the beginning of a period.

Present Value of Annuity Due (payments are made at the beginning of each period) can also be described as Present Value of an Ordinary Annuity (the payments are made at the end of each period) multiplied by (1+interest rate).

This formula assumes that the rate and periodic payments do not change. If the interest rate or periodic payments are different at different time periods, the Present Value of Annuity Due must be calculated as a sum of future values of each payment.

$$PV = P_{1} + \frac{P_{2}}{(1 + r)} + \frac{P_{3}}{(1 + r)^{2}} + \frac{P_{4}}{(1 + r)^{3}} + ... + \frac{P_{n}}{(1 + r)^{n-1}}$$

### Alternative formula for present value of annuity due

$$(1 + r) \times P\bigg[\frac{1 - (1 + r)^{-n}}{r}\bigg]$$

The formula can also be stated as in the above example, which is 1 +r times the present value of an annuity.

### Example of Present Value of Annuity Due Calculation

A company is paying its supplier $11,000 at the beginning of each month for the next 6 months for the supplies. The supplier gave a company 3 options: to pay$11,000 at the beginning of each month for the next 6 months, to pay $11,500 at the end of each month or to pay$60,000 for all the supplies in advance. The interest rate is 2%.

• Option 1 – to pay $11,000 every month at the beginning of each month for a half a year. • Option 2 – to pay$11,000 every month at the end of each month for a half a year.
• Option 3 – to pay $60,000 today. Option 1. The Present Value of the prize is$60,000 since the company can pay it now.

Option 2. In this case, we deal with Annuity Due when the payments are made at the beginning of each month.

$$(1 + 2\%) \times \11,000\bigg[\frac{1 - (1 + 2\%)^{6}}{2\%}\bigg] = \62,848.05$$

Option 3. In this case, we deal with an Ordinary Annuity when the payments are made at the end of each month.

$$\11,000 \times \bigg[\frac{1 - (1 + 2\%)^{-6}}{2\%}\bigg] = \61,615.74$$

The best option for the company is to pay $60,000 in advance (Option 1), then the Present Value of the supply payments is the lowest. However, if the company doesn’t have$60,000 to pay at the moment, it should choose the third option, where the payments are made at the end of each month.