Present Value of an Annuity

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). Finding out the present value of future cash flows helps investors to understand how much money they will receive over the period of time in today’s dollar’s term and make informed investment decisions.

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

Formula 1

Here,

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

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

Formula 2

Here,

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

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For eg:
Source: Present Value of an Annuity (wallstreetmojo.com)

Example #1

Mr. ABC is a 60 years old, retired Government servant.  He has been paying into his retirement account per month for the last 30 years, and now, after his retirements, he 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.

Formula 3
present value ex3
present value ex4

By using the above present value of annuity formula calculation, we can see now, annuity payments are worth about $ 400,000 today, assuming the 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 the present value changes drastically.  The discount factorDiscount FactorDiscount Factor is a weighing factor most often used to find the present value of future cash flows, i.e., to calculate the Net Present Value (NPV). It is determined by, 1 / {1 * (1 + Discount Rate) Period Number}read more 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 %.

present value ex4-1

Here,

i – Frequency of occurrences

present value ex4-2
present value ex4-3

Present value Annuity Factor

present value ex4-4

Here,

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

For the sake of simplicity and ease of using financial models, professionals usually calculate present value annuity factors, which helps them to keep an 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.

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.

present value ex4-5

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.

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

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