**Definition of annuity**

Annual constant cashflows for a definite period of time.

**Formula – Present value of annuity**

**Explanation**

Suppose you have $50,000 and you want to invest in a profitable project. There is an option that you can buy a machine for $50,000 and you rent it out at annual rental income of $15,000. The life of the machine is 5 years so you will receive annual constant cashflows for 5 years. The cost of capital is 8%. As an investor, you want to calculate the NPV of the project to find out whether you should invest in the project or not.

Normally, we can calculate the NPV of that project by discounting each year’s cashflow into present value and then deduct the outflows from the inflows. But if we have constant cash flows, then we can use annuity formula to calculate the PV of constant cash flows instead of discounting each year’s cash flows into present value terms separately. The formula is given above. However, we can also use annuity tables from which we can extract the annuity factor for a particular year and rate of interest. This is then multiplied with the annual cashflow to calculate the PV of annuity. So, it will save the time.

PV of Annuity = Annual cashflow x Annuity Factor (A.F)

Now, to calculate the NPV of the project:

Since all the inflows are constant, we can calculate their PV in two ways:

**Calculating present value of annuity – Using annuity formula**

**Calculating present value of annuity – Using annuity tables**

From tables, the Annuity Factor at the rate of 8% for 5 years is 3993.

PV of annuity = Annual Cashflow * Annuity Factor

PV of annuity = $15,000*3.993

**PV of annuity = $59,895**

Now to calculate the NPV, deduct cash outflows from cash inflows as follows:

NPV = PV of inflows – PV of outflows

NPV = $59,895 – $50,000

**NPV = $9,895**

In order to explain further, here is another example.

**Example – Simple Annuity**

What is the present value of 6 annual lease payments of $30,000, if cost of capital is 7%?

**Solution**

PV of annuity = Annual Cashflow * Annuity Factor

PV of annuity = $30,000*4.767

**PV of annuity = $143,010**

**Calculating present value of advance annuity**

Normally, we assume that cash flows relating to the project starts at Year 1 i.e., at the end of first year. However, in some cases, the cash flows from the project start at time 0. This is called **Advanced Annuity.** Let us tale the same example with the only difference that first payment is due at time 0. To calculate the PV of advance annuity, a little adjustment to the Annuity Formula is made.

**Formula**

PV of Annuity = Annual Cashflow x (1+ Annuity Factor)

**Example – Advance Annuity**

What is the present value of 10 annual rental payments of $22,500, if cost of capital is 9% and 1^{st} payment is due now?

**SOLUTION**

In this example first payment is starting at year 0, second payment will be at year 1, and the last payment will be at year 9. Now we can take present value of all 10 payments by taking nine year’s annuity factor and multiplying with $22,500. The answer will be the present value of nine payments. We need to $22,500 to the answer as we need to calculate present value of 10 payments and the amount at time 0 remains same.

There is simpler approach for this. Take the annuity factor of nine years and add 1 into it, then multiply with annuity, we will get the present value of all 10 payments.

PV of Annuity = Annual Cashflow x (1+ Annuity Factor)

PV of Annuity = $22,500 x (1+ 5.995)

PV of Annuity = $22,500 x 6.995

**PV of Annuity = $157,388**

** Calculating present value of delayed annuity**

There can be a case in which the first cash flows relating to the project occurs after year 1. This is called Delayed Annuity. To calculate the PV of delayed annuity, there

**Example – Delayed Annuity**

What is the present value of 9 annual payments of $20,000 at the rate of 8%, if 1^{st} payment starts in year 4?

**PV of Annuity = $99,180**