**What Is an Annuity?**

*An annuity is a series of equal payments made at equal intervals.* Annuity payments don’t have to be made annually but can be made monthly, weekly, or even daily. The critical factors are:

- The payments equal to each other and;
- The interval between the payments is the same.

**An ordinary annuity** is the one in which payments are made at the end of each period. The formula for calculating present value of ordinary annuity can be written as

\[\begin{matrix}P{{V}_{annuity}}=PMT\times \frac{1-\frac{1}{{{\left( 1+i \right)}^{n}}}}{i} & {} & \left( 1 \right) \\\end{matrix}\]

Where

PMT=the periodic payment in the annuity

i=the interest rate

n= the number of payments

The term after the multiplication sign in **Eq. 1** is the **present value interest factor for an annuity (FVIFA)**.

\[\begin{matrix}PVIF{{A}_{i,n}}=\frac{1-\frac{1}{{{\left( 1+i \right)}^{n}}}}{i} & {} & \left( 2 \right) \\\end{matrix}\]

**Example** PV of Ordinary Annuity

Say you have won the lottery! The prize is $100 paid at the end of each of the next four years. What would be the present value of the lottery annuity if the interest rate is 10%?

**Solution**

\[\begin{align} & PMT=\$100\\&i=0.1\\&n=4\\\end{align}\]

This question asks for the future value of an ordinary annuity, so

\[\begin{align}& P{{V}_{annuity}}=PMT\times PVIF{{A}_{i,n}} \\& PVIF{{A}_{i,n}}=\frac{1-\frac{1}{{{\left( 1+0.10 \right)}^{4}}}}{0.10}=3.16987 \\& P{{V}_{annuity}}=100\times 3.16987=\$316.99\\\end{align}\]

We’ve just discovered that the present value of a four-period stream of $100 cash flows is $316.99. This is the present value of the lottery prize. You would be indifferent between the four payments of $100 or a 316.99 to invest today, then it has just enough money to fund each of the $100 payouts. The government treasurer could invest the $316.99 at 10% and have $348.69 1 year later. The first $100 prize is then subtracted from the balance, which is reinvested.

One year later, the balance is $273.56 and an additional $100 is subtracted. This process continues as shown in **Figure 1** until the final $100 is subtracted and the remaining balance is zero.

**Figure 1**

This example provides an important clue as to when you’ll want to use present value equations. Anytime you want to know how much you need today to create a future cash flow stream, find the present value of the cash flows.