### What is Present Value of Annuity?

The present value of annuity formula calculates the value of a series of payments at a given time. It relies on the concept of the time value of money (i.e one dollar today is worth more than one dollar at a future date).

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

Where P is Periodic Payment, r is rate per period and n is number of periods.

The Annuity is the financial instrument that gives regular payments to the holder till the end of the contract. The holder receives his payment each period. The Present Value is the amount that the investor invests today at a defined interest rate. The total investment of the holder equates the total amount of future payments and this amount is discounted by the pre-defined interest rate.

How to calculate Present Value of Annuity?

When calculating the Present Value of Annuity, it is important that the rate should be consistent with the rest of variables stated in the formula. For month-wise payments, the rate needs to be month-wise as well.

The formula of Present Value of Annuity gets derived on the assumption that the period payments do not undergo any change.  The second assumption is that the rate does not change as well. The Present Value of Annuity gets derived on a third assumption as well. The assumption is that the third payment is a period away.

We will first look at the formula for a series of payments whether they are same or different.

#### Formula 1

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

Now the above-mentioned shows that the dividends are the same for the entire series. Well, this is why the above-mentioned series is a geometric series. We can apply the geometric series formula to the above-mentioned series (1) and you can rewrite the formula as below.

#### Formula 2

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

Multiply formula (2) by (1+r)/(1+r). As a result, (1+r) gets cancelled through the equation. The formula now looks like this:

#### Formula 3

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

#### Final Present Value of Annuity Formula

Now, you can factor out the P in the formula (3) and they become 1. The 1’s that exist in the denominator formula get subtracted from each other. The final formula looks like the below mentioned:

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

### Example of Present Value of Annuity

Mr. Michaels is 70 years old. He is a retired Government Officer who funded his account for the past 30 years. He is ready to withdraw the funds now. The retirement company will pay him about $30,000 for the next 25 years as part of the agreement. Mr. Michael will get this amount on the 1st of each month. Mr. Michael also has the option to go for a one-time payment of about$500,000. However, he is eager to know existing worth of yearly payments $30,000. The reason Mr. Michael wishes to know the worth is that he wants to identify which option best suits his needs. If we use the formula of Present Value of Annuity, then we can identify that the annuity payments will be worth$40,000 at an interest rate of 6%. Well, in this situation Mr. Michael should think along the lines to opt for lump sum amount, and he can invest this amount on his own.

### Alternate Present Value of Annuity Formula

Now, we will look at the alternate formula for Present Value of Annuity as well.

P = PMT x ((1 – (1 / (1 + r) ^ n)) / r)

Where:

P = the existing value of the annuity stream

PMT = Each annuity payment’s dollar amount

r = Discount rate or interest rate

n= The Total period allocated for the payments

Let us assume that an individual pay about $50,000 each year for the next 25 years. The discount rate is about 6%. The individual also has the option to make a lump sum payment of about$650,000.

Here

PMT= $50,000 R=0.06 n=25 Present Value of Annuity=$ 639, 168

After calculating the Present Value of Annuity, the individual should choose the lump sum payment over annuity.

You can use the Present Value of Annuity Formula for calculating lottery winnings. The reason is that you have the option to take the lump sum amount for lottery winning or you can get a series of payments.