Future Value of an Annuity Calculator

Inch Calculatorin this video we're going to talk about how to calculate the present value of an annuity or a stream of income payments so let's use this example problem to illustrate this how much money do you need to invest now to generate a cash flow of a thousand dollars every year for the next five years given an annual interest rate of six percent

well

well let's find the answer so this is a formula that we could use to get the answer the present value of an annuity is equal to the cash flow times 1 minus 1 plus r r being the interest rate raised to the negative n divided by r let's begin by writing down what we know so the cash flow is the amount of money that we're receiving for each time period in this case every year so the cash flow is a thousand dollars r is the annual interest rate which is six percent but we need to convert that into a decimal so six divided by a hundred is point zero six and the time period or rather the number of time periods that's going to be five we're going to be credited with interest five times in this problem so n is five so let's replace c with a thousand and let's replace r with .06 so 1 plus 0.06 is 1.06 that's going to be raised to the negative 5. we're going to divide that by .06 1 minus 1.06 raised to the negative 5 that's 0.2527418271 let's divide that by .06

and then you&#39;ll get four point two one

and then you'll get four point two one two three six three seven eight six so multiplying that by a thousand this will give us the present value so the present value for this annuity is four thousand two hundred twelve dollars and thirty six cents so that's how much money we need to invest now to generate a cash flow of a thousand dollars every year for the next five years so we're putting about 4 200 in and we're gonna get over five thousand over the next of five over the the next five years now let's talk about some other ways in which we can get the same answer it's important to understand that the present value of money is equal to the future value divided by 1 plus r raised to the n so let's create a number line

so this is 0 which

so this is 0 which corresponds to the present this is one two three four five so a thousand dollars received one year from now is worth how much today so using the formula that we see here 1000 is the future value so if you plug in a thousand and divided by one plus r or 1.06 raised to the first power because we're trying to find the present value one year from now it's a thousand divided by 1.06 and that's 949 dollars and 39 actually 40 cents it's 0.396 i'm gonna get a few numbers after the decimal so a thousand dollars one year from now is worth 949 dollars and 40 cents today what about a thousand dollars two years from now how much is that worth today at the present so to get that it's going to be a thousand dollars or a thousand divided by 1.06 raised to the second power

so a thousand dollars two years from now

so a thousand dollars two years from now is worth eight hundred eighty nine dollars well you have to round it to eight hundred ninety dollars but it's eight eight nine point nine nine six four now what about a thousand dollars three years from now how much is that worth so that's going to be a thousand divided by 1.06 raised to the third power and so a thousand dollars three years from that will be worth 839.62 now let's continue the process if we take a thousand divided by 1.

now let&#39;s continue the process

6 raised to the fourth power we'll get this answer 792 dollars and nine cents now what about a thousand dollars five years from now how much will that be worth today at an interest rate of six percent so take a thousand divided by 1.06 raised to the fifth power and you'll get 747 dollars and 26 cents so this is point two five eight two now what we're gonna do is we're gonna take the sum total of these five numbers now i do need to make one small correction a thousand divided by 1.06 raised to the first power is not 949 dollars but it's 943 dollars so let me just correct that so now let's go ahead and add these five numbers feel free to pause the video as you add them as well and this will give you the same answer four thousand two hundred twelve dollars and thirty six cents so that's another way in which you can calculate the present value of an annuity if you don't want to draw the picture you can also calculate it this equivalent way you can take the cash flow which is a thousand and then multiply it by 1.06 raised to the minus 1 plus 1.06 raised to the minus 2 plus 1.06 raised to the minus 3 and then do this all the way to 5. this will give you the same answer

you&#39;ll get the same thing four thousand

you'll get the same thing four thousand two hundred twelve dollars and thirty six cents so now you have three different ways in which you can calculate the present value of an annuity now let's work on another example problem timothy wishes to buy an immediate annuity that offers a fixed interest rate of seven percent he wants to receive a cash flow of five thousand dollars per month for the next thirty years how much money does he need to put into an annuity to generate this cash flow now we're going to ignore any fees charge by the insurance company or any bonuses credited to them sometimes when you buy an annuity the insurance company will give you a premium bonus of five percent or 10 percent it really depends on what's specified in the contract and you have to look at any fees that they charge for maintaining the annuity contract but let's ignore that extra information for now

so we&#39;re going to use

so we're going to use the same formula but it's going to be modified a bit to take into account the fact that we're receiving a monthly cash flow instead of yearly payments the present value of the annuity is going to equal that monthly cash flow times 1 minus now this is going to be 1 plus r raised to the n i mean r divided by n raised to the negative and t and then all of that will be divided by r divided by n c is going to be well let's write down everything that we know c is gonna be five thousand he wants to receive five thousand dollars every month the interest rate is seven percent seven divided by a hundred is 0.

c

7 n is the number of times that interest is credited to the account in the year so we're dealing with monthly payments so we're assuming that the interest is credited on a monthly basis so n is going to be 12 there's 12 months per year and t is going to be the number of years which is 30 years so nt would represent the total number of payments if you multiply 12 by 30 you're going to get 360 payments and then r over n that is the monthly interest rate if you take .07 and divided by 12 you'll get .00583 repeating but i'm going to leave it as .07 over 12. so keep in mind that we're assuming that interest is credited to this account on a monthly basis if the interest is credited on an annual basis then we would need to use the same formula that we did in the last problem but the only difference here would be that the cash flow wouldn't be a thousand dollars per month but rather it would be 12 000 per year which will still equate to a thousand dollars per month r would still be 0.

but the only difference here would be

6 and n would be 30 years if the interest is credited on an annual basis so this problem is equivalent to the insurance company making 30 payments once every year with each payment being 12 000 a year but if interest is credited monthly then the cash flow needs to be the monthly cash flow and we need to use r over n we need to divide r by the 12 months in a year so just keep that in mind so the cash flow is five thousand dollars and that's going to be one plus r which is .

and that&#39;s going to be one plus r which

7 divided by n and then raised to the negative 12 times 30 or negative 360. it's going to be 360 payments in total and then we're going to divide this by 0.07 divided by 12. so let's take this one step at a time just to be on the safe side so first let's take .07 divided by 12 and then let's add 1 to that and then let's raise it to the negative 360. so that's going to be point one two three two zero five eight five three six and on the bottom point zero seven divided by twelve that's point zero whoa i forgot a zero point zero zero five eight three repeating so you can write a few threes to get an accurate answer so one minus point one two oh that's 0.

that&#39;s

76794 and then divide that by .0058 and then 3 repeating you should get 150.307568 if you multiply that by 5000 you'll get the present value which is 751 537 dollars and 84 cents so if john wishes to generate a cash flow of 5 000 per month for retirement for the next 30 years he needs to roll over this amount of money from his ira into an annuity and that insurance company will have to pay him five thousand dollars per month for the next 30 years now let's see the total amount of money that the insurance company is paying john and let's compare it to the amount of money that he's putting into this insurance contract so the insurance company is paying him five thousand dollars per month

and there&#39;s 12 months

and there's 12 months per year so the unit months will cancel and the insurance company will be paying him for a total of 30 years so we can cancel the time unit years so it's 5 000 times 12 so that's 60 000 per year times thirty so in total over the course of thirty years the insurance company will be paying him a million dollars and one million eight hundred thousand dollars let's calculate his net profit from for putting his money into this contract so keep in mind we're not taking into account any fees charged by the insurance company so if we take this number and subtract it by 751 thousand 537 dollars and 84 cents

he&#39;s receiving

he's receiving a million dollars a million forty eight thousand and four hundred sixty two dollars and sixteen cents so this is the amount of money that he's receiving in interest over the course of 30 years and so by stretching out this contract over a long period of time this money is allowed to collect interest over time and thus give him an extra boost in retirement so that's how you can calculate the present value of an annuity which is this number and you could also calculate the amount of interest that you receive over the course of the annuity thanks for watching

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