# Present Value of an Annuity Calculator

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## 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?