# Present Value of Periodic Payments Calculator

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Imagine you have come into an inheritance of \$20,000. However, you are faced with a choice. You can either receive the whole amount as a lump sum today or in annual installments of \$4000 over five years. Which one would you choose? The answer to this question may differ from person to person, depending on their monetary circumstances or personality. From a mathematical standpoint, though, there is no ambiguity. Experts will tell you that it makes more financial sense to receive the whole amount today than in installments. This preference is because, even though you receive the same amount in both scenarios, the purchasing power of the \$20,000 will be much less in the future. For a detailed explanation of why that is so, please read our article on inflation rate calculations.

## What Is Present Value And Why Is It Important?

So how do we arrive at the above conclusion mathematically? This situation is where Present Value comes in. Built on the concept that time adds value to money, Present Value is defined as the current or present-day value of payments made or received in the future. It lets you know how much the money you are receiving in the future is worth today.

In essence, it allows you to compare the purchasing power of a dollar from tomorrow to that of a dollar from today. It is one of the most useful concepts in finance as it helps you make sound investment decisions, plan for the future, and budget yourself. Shareholders and investors use this metric regularly to determine if a business, product, or share is worth investing their money in or not. Besides, it is also commonly used by lawyers to calculate the value of a structured cash settlement so they can negotiate a better deal for their clients.

## How To Calculate Present Value

### Components of Present Value

There are three main components to a Present Value calculation. They are:

• Annuity or Payment: This is the fixed amount that you are giving or receiving
• Number of periods: The total number of payment installments needed to be made or received
• Interest rate: This reflects the risk-free interest you could expect to receive if you choose to invest your periodical payments.

As with many financial calculations, Present Value calculations also make a few assumptions. For one, it presumes that the periodical payments are equal in value and, for another, that the interest rate remains the same throughout. If any of those components change, you will need to tweak the calculations. Even with this limitation, it remains one of the most accurate ways to evaluate your investment.

### Present Value Formula

The formula to calculate the Present Value of your money changes slightly according to when you receive the payment. Payments received at the end of a payment period is called ‘ordinary annuity’ (Example: interest payments from a bond are generally received at the end of a quarter). Conversely, payments received at the beginning of a period is called ‘annuity due’ (Example: rent paid at the beginning of a month). An ordinary annuity is the more common of the two. The reason for the discrepancy is that there is less money to discount with an ‘annuity due’ since you collect your payment earlier. Correspondingly, an annuity due’s PV, will be worth more.

Here is the formula for calculating PV for ordinary annuity:

PV = Pmt x 1 – ((1 / (1 + r n)) / r

And, here is the formula for calculating PV for annuity due:

PV = (Pmt x 1 – ((1 / (1 + r n))/r) x (1 + r)

In both cases,

• PV is Present Value
• Pmt is a periodical payment
• r is the rate of interest
• n in number of installments

### Examples

#### Example 1

Let us stick with the first example of your coming into an inheritance. Imagine you chose the second option of receiving the \$20,000 in annual payments of \$4000 over five years. Let us assume that you have deposited the \$4000 in a bank at an interest rate of 5%. If this is an ordinary annuity payment, this is how the calculation would go:

PV = 4000 x 1 – ((1 / (1 + 0.05 5)) / 0.05 = \$17,317.91

If this were an annuity due payment, the calculation would look like this:

PV = (4000 x 1 – ((1 / (1 + 0.05 5)) / 0.05) x (1 + 0.05) = \$18,182.85

As you can see, in both cases, the amount you get with staggered annual payments is less than \$20,000. Which is why it is a better idea to receive the amount in a lump sum.

#### Example2

Here is another example of the utility of calculating PV: Imagine an alternate scenario where you are given the option of claiming your full inheritance of \$20,000 today or waiting for a year, in which case, you will receive \$21,000. Which option would you choose?

In this scenario, we need to calculate the present value of \$21,000 to see if it is more than the original amount of \$20,000. Since there are no periodical payments to account for here, the formula for calculating PV changes to:

PV = Future Value / (1 + r) n

Here,

• FV is the future value of your money. This amount is \$21,000 in the above example.
• r is the rate of interest you would have received if you had invested \$20,000 in a bank. Let us again assume it is 5%
• n is number of years, which in our case is 1

Therefore, applying the formula, we get:

PV of \$21,000 = 21000 / (1 + 0.05) 1 = \$20,000

According to our calculations, the present value of \$21,000 is only \$20,000; it makes better sense to wait a year and pocket the higher amount rather than to take the \$20,000 and invest it in a bank.

We are talking about risk-free interest rates and guaranteed payments here. If there is any ambiguity in either, that changes things completely. For example, if there is no guarantee that you will get the \$21,000 after a year, then it is probably a better option to take \$20,000 in the first year. As the saying goes, a bird in hand is worth two in the bush.

## How To Use Our Present Value Calculator

The formula for calculating PV is a slightly complicated one. If math is not your forte, you can use our Present Value calculator. It has all the necessary components marked out in an easy-to-understand manner. What’s more, It allows you to make calculations in both ‘ordinary annuity’ and ‘annuity due’ scenarios. Input the payment amount, number of payments, and interest rate. Then hit calculate, and you will have your Present Value in no time.

## Conclusion

Present value is a vital and oft-used metric in the financial world. It has a plethora of functionalities, chief of which is that you no longer need to choose your investments or make business decisions based on instinct. On the contrary, you now have a mathematical basis for them.