# Present Value of an Annuity Calculator

"Present value of an annuity" is finance jargon meaning **present value with a cash flow**. The cash flow may be an investment, payment or savings cash flow, or it may be an income cash flow.

The present value (PV) is what the cash flow is worth today. Thus this present value of an annuity calculator calculates today's value of a future cash flow. **The annuity may be either an ordinary annuity or an annuity due (see below).**

The PV will always be less than the future value, that is, the sum of the cash flows (except in the rare case when interest rates are negative).

Why?

Because there must be compensation made to the party who has to wait for the money. Think of it in reverse. Would you rather have $100 today, or $100 one year from now?

Of course, you would rather have $100 today since there is risk in not receiving the money if you wait, and further, if you receive the payment today, you can invest it today and earn a return on the capital.

The present value of an annuity calculation considers these things and discounts the cash flow. More below

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## What is present value used for?

At a high level, there are two scenarios when you may want to know the present value of a cash-flow.

- when someone or some entity owes you money
- when you want to make an investment

Perhaps you have won a court settlement payable as an annuity, or maybe you've been lucky enough to win a state lottery, and you want to receive the proceeds at once. How much should you expect?

Use this PV of an annuity calculator to tell you. Since an annuity is a regular, periodic cash-flow, and because this calculator allows you to set a specific first cash-flow date, it is capable of calculating the present value for any cash flow. The calculator is also particularly suitable for calculating the PV of a legal settlement, such as one involving alimony.

For the same reasons, this calculator can be used to calculate the PV of an investment cash-flow. Perhaps you want to invest in a mortgage? You'll need to calculate the PV of the said mortgage before you can make an offer or know if the offering price allows you to meet your investment objective.

## What is the correct discount rate?

A discount rate is a personal number. That is, there is no absolute right or wrong value one can use.

When determining the discount rate, you could use several approaches. If you invest in the stock market, and for you, you earn on average 8% per year, you can use 8% for the discount rate to compare the present value with the return you earn from the market.

If you want to compare PV to something safer, you might use the US Treasury ten-year rate, which currently is at about 2.5%.

### Example

Additionally, **buyers and sellers are very likely to use different discount rates**. For example, a commercial building's owner is selling the property, and a tenant has ten years remaining on the lease. What is the value of the contract to the prospective buyer?

The buyer may feel that mutual funds and the lease have similar risks (mutual funds loss of value and the lessee not paying). In that case, the buyer can use their average mutual fund return rate, say 7%, to calculate the PV of the lease. After all, why would they pay more to purchase the contract if they can earn 7% in mutual funds? The buyer will always want to use the highest discount rate they can justify because the higher the discount rate, the lower the PV – or the lower the cost of the asset. In other words, **for the buyer, using a higher discount rate is the more conservative approach**.

On the other hand, the seller may feel the tenants are reliable, and the cash flow is safe. They'll ask themselves why take a risk and put the money into the market where there is the risk of losing principal? In that case, the seller might want to park the money in a 2% CD, so they'll use 2% as their discount rate. Lower rates result in a higher PV. Thus, **for the seller, the lower rate is more conservative**. They'll need to be paid a higher price so they can put the proceeds from the sale in a lower yielding CD to reduce the investment risk.

With this example, it looks as if no deal would ever get done. The buyer will want to pay to little, and the seller will want to receive too much.

However, all deals depend on each participant's perspective. Perhaps the seller thinks that they have an opportunity to reinvest the money and earn not 2% but instead 20%. In that case, the seller might be willing to sell the lease at a 10% or 12% discount to have the funds available to take advantage of the more profitable opportunity.

See how what discount rate to use is a matter of personal choice and perspective?

## PV Calculator for Either Ordinary Annuity or Annuity Due

You may have heard of the terms "ordinary annuity" or "annuity due". This calculator will calculate the present value for either type of annuity.

First, what's the difference between an ordinary annuity and an annuity due? These two terms are a bit of financial jargon for an easy to understand financial concept.

**An ordinary annuity will have its first cash flow scheduled for a future date**. Textbooks frequently explain this concept by saying the cash flow gets paid at the end of the period.

**An annuity due will have its first cash flow scheduled on the as-of-date**, that is, the date for which the present value is calculated. Textbooks explain this concept by stating the cash flow gets paid at the beginning of the period.

The present value formula needs to be slightly modified depending on the annuity type.

Since this calculator prompts the user for the present value date (today's date) and the first cash flow date, it will work equally as well for either annuity type. **If you set the dates to the same day, then the calculator will use the annuity due formula; otherwise, it will use the ordinary annuity formula.**

Note, if you are calculating the present value for a deal that closes in the future, then you should set today's date to the day the contract is scheduled to close.

## Present Value of an Annuity Help

An "annuity" is a fixed sum of money paid someone each period, typically for the rest of their life. More loosely, it means any regular cash flow stream which may or may not have an explicit declared term. If an annuity is scheduled for 10 annual payments of $10,000 each, the sum of the payments is $100,000. However, if instead of being paid in 10 annual installments you wanted to receive a single sum, you would not receive $100,000. Why? Because if you receive a single sum today, there is no future risk of not receiving the amount due. Therefore, you would take less today to eliminate the risk of not collecting all payments.

If you are scheduled to receive a series of regular fixed payments of $2,500 for 20 years, what is today's cash value, assuming a 5.5% annual discount rate? The "annual discount rate" is the rate of return that you expect to receive on your investments. This is a personal number. There is no "right" answer, though you want to use a realistic number based on your investment history. The discount rate will vary from individual to individual.

Enter $2,500 in the "Cash Flow Amount" field (never type the currency symbol or commas). The cash flow frequency will be monthly. Enter 240 for the "Number of Cash Flows" (240 months is 20 years). Assume monthly compounding. Since the first payment isn't due until a month from now, set the "First Cash Flow Date" to one month from "Today's Date".

The PV is $363,431.62. Thus, you could accept $363,431.62 today in lieu of receiving $2,500 a month for twenty years. For you, the two are equal.

A note or two about "Compounding Frequency". The "Exact/Simple" option is actually exact day simple interest. When you make this selection, the calculator uses no compounding and the exact number of days between cash flow dates are used. The "Daily" option uses the exact number of days between dates, but daily compounding is assumed. If you are considering receiving a single amount in lieu of a cash flow stream, the "Exact/Simple" compounding option is the most conservative setting. That is, it will result in the highest present value calculation.

The prior version of this calculator provided you with an option to set the "Cash Flow Timing". Since you can enter "Today's Date" and the "First Cash Flow Date" this option is no longer necessary because the calculator will calculate the exact dates the cash flow is due.

One additional point about "Today's Date". This input does not have to be set to the current date. The date you use is the date you want to know the present value. If you were closing on a deal to buy a mortgage and the deal is expected to close in a week, then you would want to use the date of the closing for "Today's Date" so you'll know the present value on the closing date.

## Charles says:

Just a few thoughts –

Previously I mentioned that the verbal predicted FMV always matched, within a penny & sometimes exact, the letter when it arrived about 11 months later.

One time the insurance company said it took a while to calculate this because they had to compute it month by month. Maybe this accounts for their values being different than the calculator ones. But I can’t imagine them doing this at the year end for all their customers. They wouldn’t give me the formula so who knows how they’re doing it.

That being said, 2017 was different. The verbal was $113,463.26 and the letter was $113,470.70. That is way more than a penny difference. What’s interesting is that the verbal is only 46 cents different than your calculator’s value of $113,463.72. It’s as if they used your calculator but with only 5 decimal place rate accuracy. But then what did they use for the year end? For my purposes all the values are close enough.

Do you know the inner workings of your calculator & how it uses dates? Or how any formula would do that?

## Karl says:

Do I know the inner workings? I should, I programmed the calculators on this site, but alas, this site has been a 4 year project (even as an upgrade from an older site) and details are fuzzy. 🙂 That’s one of the reasons why I don’t discuss the equations themselves. Another reason is, such a discussion, by necessity, gets into the weeds too much and I only have so much time to answer questions.

That said, this calculator, as I recall, calculates PV for "X" periods, adjusts for any fractional period and then rounds once.

I do have another idea about why there are differences. Your insurance company telling you that they compute "month by month" made me realize that they may have a schedule of when the distributions are due and they may take true due dates into account. This could come into play if a due date falls on a Saturday or Sunday, they may pay on Friday and adjust the FMV accordingly. We’ve already seen how one day difference in the dates causes the FMV to change by a few dollars.

See what I mean by getting into the weeds?

If this is of interest to you, I do have a calculator that will let users calculate the PV month by month and it will round after each calculation (because it lets the user, if they want to, to set the date each annuity payment is due). Please see the Ultimate Financial Calculator. If you check it out, scroll down the page and there are a number of tutorials. #20 deals specifically with PV. (All users should read #1 to get started.)

## bw says:

Does anyone know the exact formula this calculator uses?

I am trying to calculate the present value of an annuity using the formula c(1-(1+r)^-n)/r formula, but the numbers I am getting are absurd

This calculator provides the exact answer, but the formula, when I do it by hand, does not. Is my bedmas that bad or is there a tweak to the formula to get the results that this calculator does.

## David says:

What exactly is the definition of the

discount rate?

## Karl says:

The discount rate is the rate used to find the present value.

The more important question perhaps is "What rate should I use for the discount rate?"

And that depends. If you want to know the present value of a future cash flow that would be derived from investing, then you want to use a rate you think you can earn if you were investing the money.

If you were to borrow the money then you should use the interest rate you would have to pay on a loan.

Does this help?

## Jim Lequin says:

At 16% discount rate, what is the present value of Php 36,000 which is due at the end of 90 days? What is the discount?

## Karl says:

This present value calculator is a better one to use for your particular need. It solves for the PV of a single future amount. (You’ll need to do the subtraction to learn the discount. I’ll add that calculation in the next update.

## Rebecca says:

Do you have any suggestions for calculating PV when there’s a balloon at maturity?

## Rebecca says:

Just found your ultimate financial calculator link which looks like may the solution for this.

## Karl says:

Yes, I was just going to recommend that calculator.

Basically, you’ll create 3 rows. The 1st for an "Unknown" amount. This will be the PV.

The 2nd row will be for the amount and number of cash flows.

The 3rd row will be for the single balloon on the maturity date expected.

Scroll down the page and you see some tutorials that should be useful.