## Present Value of an Annuity Definition

Present value of annuity is the present value of future cash flows adjusted to time value of money considering all the relevant factors like discounting rate (specific rate) and it is calculated by adjusting equated annual payments to discounting rate considering time period which helps to find out present value of annuity which will be received in future.

Because of inflation, the purchasing power of money gets diminished so because of the time value of money concept, money received today has more value than money which will be received tomorrow. In simple terms, we can say that if one has money now he can invest that money and enjoy returns on that money so automatically the value of money gets appreciated. By the same logic, $ 10,000 money received today is more worthy than $ 10,000 received tomorrow.

**Formula to Calculate Present Value of Annuity**

Here,

- p1, p2 – Annuity payments,
- r – Discount rate
- n – Time Period in years

After simplifying this Present value of annuity formula, we can get

Here,

- p – Equated annual payments
- r – Discount rate
- n – a time period in years

Finding out the present value of a sequential payments helps investors to understand how much money he or she will be actually receiving over the period of time in today’s dollar’s term and allows him or her to make informed investment decisions.

Let us now understand the calculation of present value of an annuity.

**Example #1**

Mr. ABC is a 60 years old retired Government servant. He has been paying into his retirement account per month from the last 30 years and now after his retirements, he is can start withdrawing funds from the retirement account. As per the agreement, the retirement company is giving him to pay $ 30,000 on the 1st of each year for the next 25 years, or another option is a one-time payment of $ 500,000. Now Mr. ABC wants to know what is the value of the $30,000 yearly payments made to him compared to a one-time payment. He has the option to choose and he wants to choose which gives him more money.

By using the above present value of annuity formula calculation we can see now, annuity payments are worth about $ 400,000 today assuming interest rate or the discount rate at 6 %. So Mr. ABC should take off $ 500,000 today and invest by himself to get better returns.

Using the present value formula above, we can see that the annuity payments are worth about $400,000 today assuming an average interest rate of 6 percent. Thus, Mr. Johnson is better off taking the lump sum amount today and investing in himself.

Here, if we change the discount rate then present value changes drastically. Choosing a discount factor is one of the crucial things while calculating the present value of an annuity. The discount factor can be taken based on the interest rates or cost of funds for the company, it depends upon the usage of the discount factor. Thus, the lower the discount rate, the higher the present value.

**Example #2**

Find out the annuity of $ 500 paid at the end of each month of the calendar years for one year. The annual interest rate is 12 %.

Here,

i – Frequency of occurrences

### Present value Annuity Factor

Here,

- r – Discount rate
- n – the time period in years

For the sake of simplicity and ease of using into financial models, professionals usually calculate present value annuity factor which helps them to keep eye on discount rates as well as total annuity factors.

This factor is maintained into tabular forms to find out the present value per dollar of cash flow based on the periods and the discount rate period. Once the value of dollar cash flows is known, the actual period cash flows are multiplied by the annuity factor to find out the present value of the annuity.

**Formula to Calculate Present Value of an Annuity Due **

Until now, we have seen annuity payment was done at the end of each period. What if payment is made at the starting of the period then the above formula will misguide us. Annuity due formula can help us in finding out the present value of annuity whose payment is made at the starting date of the period.

Here,

- p – Equated annual payments
- r – Discount rate
- n – the time period in years

### Conclusion

The present value of the annuity is one of the very important concepts to figure out the actual value of the future cash flows. The same formula can be used for cash inflows as well as cash outflows. For cash inflows, you can use the term discount rate whereas, for cash outflows, you can use the term interest rate. By using the same concept, you can find out the present value of the future cash flows either incoming or outgoing. The normal formula can help us finding the present value of an annuity if cash flows are at the end of the period. But if cash flows are at the beginning of the period then annuity due formula will help.

### Recommended Articles

This has been a guide to what is Present Value of an Annuity. Here we discuss the formulas to calculate Present Value of an Annuity along with a practical example. You can learn more from the following articles –

- What is the Present Value of an Annuity Formula?
- Put-Call Parity Formula
- Components of the NPV Profile
- Annuity Formula
- Purchasing Power Parity Formula
- Time vs Money – Compare
- Formula of Discount Factor
- Use NPV in Excel
- Formula of Present Value Factor

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