Present Value of an Annuity Calculator

Inch CalculatorNow, in this short video, I'm going to show you how to calculate the present value of an annuity before we worked on the present value of a payment, so we're dealing with one particular figure and then discounting that to present day dollars. In this case, we're going to deal with annuities, which is, we know, are a series of equal payments that usually span a certain period of time and obviously that is a little more intricate, because we're dealing with money that we're going to receive not necessarily at one time, but over a period of time and as a result, they have to be discounted at different periods of time, of course, which just adds to the complexity. So what I'm going to do is I'm going to show you two different ways to do this problem? First graphically and just so, you kind of see visually how it works, and then I'm going to show you a quick equation that you can use to actually solve the problem and work through things a lot faster, similar to how we calculated the present

value

value or future value of annuities using the graphical form it is time consuming if you have a number of different periods, so I am going to keep things to just three periods so that we can get through things fairly quickly, but I just want you to see ultimately how this works. So you understand how the equation functions here so, first off we have to assume a couple things uh: let's assume that we're going to be receiving a series of three equal payments of 100, and so we are going to be receiving 100 per year for three years and at the end of those three years, the payments will essentially stop. Now, if you were to simply add up three payments of 100, you would think well, I'm going to earn 300, but, as we now know, because of the time value of money, that three hundred dollars is not worth the same in today's dollars because of inflationary risks.

you were to

Opportunity costs and different things so, just like we did previously when discounting the present value of a one lump sum. We have to do the same thing with the annuity, so first off, let's go ahead and once again identify what are the variables that we have- and obviously we know what we have to solve for so the variables that we have first of all, are we know we have the payment which we abbreviate as pmt, and this is our series of equal payments we're going to state. This is 100. Next, we need to know.

next we need to know obviously the

Obviously the number of periods which I identified before is going to be three years, and we also need to know roughly the interest rate that we could earn on this particular type investment. So let's say for this example: we think we could have earned seven percent, which is what we're going to discount these equal payments of 100 as and, of course, we're trying to identify what the I'm sorry, what the present value of our investment is not the future value but the present value, of course, and so how you would work this out first kind of longhand is you want to obviously identify and be really organized in what you're working with here? So let's draw just one straight line and we're gonna do a couple of hash marks here for the different periods and let's go ahead and add some numbers to these, so that's zero, one two and three and we're receiving one hundred dollars the first period, the second period and the third period, and so what we need to do is we need to take these from the times that we receive them and then discount them back to today, essentially which has identified as period 0. That would be essentially this particular point in time, and so we start with the closest to the particular time frame, we're working with here being the first period, and we discount that back to today's dollars here and so what we have to do is we take that period that payment and then we divide that by 1 plus the interest rate, which is 1.

period

7, of course, and that actually gets us 93. And 46 cents. And so let's do the same exact thing with period two so we'll take this we'll carry it over and we'll take 100 divided by 1.07, but the difference here is: we have to actually use this and we take that to the nth power and so for the first example, obviously we're in the first period, so n would be one so 1.07 to the first power is 1.07, so very, very simple to do. In the second period, it's actually going to be 100, divided by 1.07 to the second power, and so that'll look a little different. Obviously, and so if you work that problem out, what you end up is with 87 and 34 cents and then we do the same thing. We take the last 100 move this over and we take 100 and divide it by 1.07 to the third power. We would actually get 81 dollars and 63 cents, and so if we take those three numbers that we had identified before both 93 46, 87, 34 and 8163- and if we add those together, we actually get 262 dollars and 43 cents. So if we were to actually receive a series of equal payments for three years in the amount of one hundred, that would actually be worth in today's dollars: 262 dollars and 43 cents, due to obviously not the time value of money.

so if we were to actually receive a

Given that that money is not going to be worth 300 in the present form, because we obviously aren't going to have it for some period of time you know after year, one we'll have a hundred dollars after year, two we'll have 200 and then finally, after three years, we'll have all three hundred dollars and so discounted that would be 262.43. So how do we work this particular problem out simply using an equation as opposed to doing this by hand?

so how do we work this particular

If you add, let's say you had 20 periods, this would obviously take a lot of time because you'd have to take the payment and ultimately divide it by 1, plus the interest rate to the nth power and then do that potentially 20 times to be very, very cumbersome so so the equation to solve this type of problem is the present value equals pmt, which stands for payment. Of course, I'm going to put a bracket here and it's going to be multiplied by 1 minus and then 1, divided by 1, plus your interest rate to the nth power and you're going to divide everything by once again the interest rate again so a little bit of a lengthy equation, but far better than actually doing everything by hand, especially if you have many many years quarters or periods that you're actually working with here and so let's go ahead and simply plug in the existing information so we know what we're looking at here we know the payment of course is going to be 100 so we can plug that in and obviously we have one still minus 1 divided by I'm going to go ahead and show this as 1.07 since we know our interest rate is 7 percent to the third power and divide that by .07 once again now 1.07 to the third power is actually 1.225043 and so if we take that number and if we divide 1 which is this number right here and we divide it by the result of 1.

and so if we take

7 to the third power what we would actually get is point eight one six two nine seven nine and then of course we have to go ahead and take one minus that number and then ultimately divide that by our interest rate and that is still going to get multiplied by our payment of 100 here and so 1 minus 0.8162975 is actually 0.18370 two one this is zero right there and then we still have to divide that by point zero seven and that equals two point six two 4 3 1 5 7. And the last thing that we have to do of course is multiply that by 100 and so 100 multiplied by 2.6 two four three one five seven actually equals coincidentally two hundred and sixty two dollars and forty three cents and so ultimately we got to the same point same place that we did previously this was much quicker of course and it obviously gets faster when you have multiple periods because the other method gets obviously much slower and so you can use either of these methods of course if you only have a couple of periods maybe you want to work things out simply by hand but if you have anything more than maybe three or so it's probably your best bet to utilize the present value of an annuity equation to go ahead and work this out now let me just say if you had maybe a annuity plus a future lump sum so

let&#39;s just say

let's just say at the end of the third year we would also receive in addition to our 100 of monthly payments we would get an extra 1500 well you know now because you know how to calculate the present value of a future investment or a future payment you know that you would then discount that by three years and then simply add that to your existing cash flows here being 262.43

and that would be essentially your

and that would be essentially your present value so there are multiple different ways you can do this knowing not only the present value of an annuity but also the present value of a simply a payment or lump sum you can combine the two equations or combine the results from the two equations to ultimately find out what you're looking for as well

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