Future Value of an Ordinary Annuity - YouTube

Moving along in our discussion about the time value of money we've already talked

about future value and present value and finding those numbers as well as finding

the periods and the interest rate but instead of talking just about lump sum

amounts we also need to move our discussion toward annuities. Annuities

are periodic payments or receipts which may be called rents of the same amount

of the same length interval between these payments and compounding of

interest once each interval. So the important part here is to remember that

the difference in a lump sum and an annuity is lump sum is a one-time

payment. An annuity is the same amount paid over the life of the instrument

that we're talking about so every five years or every ten years we're paying

the exact same amount and interest is compounded over those payments. There's

three types of annuities we're going to talk about and the one that we're going

to talk about in this video will be the first one which is ordinary annuities.

This is where the rents occur at the end of each period and in the next video

we'll talk about annuity due which is where the rents occur at the beginning

of each period because they are different and they're treated

differently and the last one we'll talk about will be the deferred annuity which

is actually a combination of both lump sum amounts and annuities. So let's start

our discussion with the future value of an ordinary annuity. This is where again

the rents occur at the end of each period so if you look at our timeline

here you see that we're going to make our first payment at the end of year 1

and then our second at the end of year 2 and so on. So what happens here is

there's no interest earned or incurred depending on what side of the annuity

your on during that first year. Okay and then the last payment is made at the end

of the last year. So let's look at an example of the future value of an

ordinary annuity. What is the future value of 5 $5,000 deposits made at the

end of each of the next 5 years earning interest of 12%?

Now if we did not know due to the title of this slide that this is an ordinary

annuity we would be able to figure that out because it tells us that we're going

to make these payments at the end of each of the next five years and because

they're at the end and they're equal payments we know that this is an

ordinary annuity and since they're asking us what will we have at the end

of these five years we know it's a future value question. Looking at our

timeline here we want to know if we deposit $5,000 at the end

of the next five years what will we have at 12% what will we have at the

end of those five years? So the first thing we have to ask ourself is what

table do we use? Well we've decided this the future value of an ordinary annuity.

So therefore we can use the future value of ordinary annuity tables. So we take

our table and we've got to find that factor that we've been talking about in

previous videos with present value and future value. We know it's 12% rate and

we know it's five periods so we go to our 12% column and we scroll down to

five periods and we find the factor to be 6.35285.

We multiply that times the amount of each deposit that we're going to make.

Remember this factor incorporates everything: the interest rate, the number

of periods, and the number of payments. All that's factored into that. We

multiply that factor times the number of deposits and we find that if we make a

deposit at the end of the next five years of $5,000 at 12% we will end up

with $31,764. Okay so let's let

you try one. Here we have the Gomez Incorporated company will deposit

$30,000 in a 12% fund at the end of each year for eight years beginning December

31st. What amount will be the will be in the fund immediately after the last

deposit? So again the first question you have to ask yourself is it a future value

or present value? Well they're wanting to know what will we have at the end of

these eight years so that's a future value question. We have to know if it's a

lump sum or an annuity question and because it says that we will

deposit 30,000 at the end of each year that makes it an annuity and it isn't an

ordinary annuity or an annuity due and it tells us that we're gonna make these

deposits at the end of each year. That makes this an ordinary annuity problem.

So therefore we're gonna use our future value of an ordinary annuity tables so I

would like for you to calculate what we will have in eight years making a

$30,000 deposit at the end of each year at 12%. So press pause on your

player. Go find out the answer and come back and we'll look at look at it

together. Okay so you should have your answer by now. We're gonna go to our

future value of an ordinary annuity tables and we know it the interest rate

is 12% and this time there's eight periods so we find our 12% column and we

scroll down until the eighth period and we find our factor to be

12.29969. Again your table may be slightly different, less decimal

places, they may round differently so as long as you've got the same place in

your chart we should be fine. Taking that factor times the $30,000 deposit

each year you don't sum all those deposits up it's just you remember that

factor incorporates all that information in it so you only multiply the factor

times the deposit amount that you're going to make each year. So the $30,000

times the 12.29969 shows us that at the end of

eight years at 12% depositing $30,000 at the end of each of those eight years we

will end up with $368,991

$368,991. So just as we saw with the future value and present value questions

of a lump sum sometimes we want to find other amounts. For example if I know

what I want to have at the end of so many years and I know what my interest

rate would be, can I find out how much I would need to deposit each year to end

up with that amount? Well assume that you plan to accumulate $14,000 for a down

payment on a condominium apartment five years from now. For the next five years

you earn an annual return of 8% compounded semi-annually. How

much did you deposit at the end of each six-month period? So here's our timeline

here and we know what we want to have at the end of these five years but keep in

mind we are compounded semi-annually now so because we're compounded twice a year

we don't have five periods we have ten periods. So we have two periods every

year and the annual interest rate is 8% that means we have to divide that in

half because now remember 8% is an annual return. If we got two periods per

year that means it's only 4% per period. So that's how we get that that

semi-annually compounding there. When we use this information to figure out

what that payment would be we're just going to set up an algebraic expression.

For the future value of an ordinary annuity is equal to your rents which we

can call payments times that future value factor of an ordinary annuity at a

certain number of periods and an interest rate. Well we know that the

future value that we want is $14,000. We're looking for the R which is our

rents or our payments whatever you want to call it and we've got a future value

factor that we need to look up in our table at 10 periods and 4% interest. Okay

so we doubled our periods because of semiannual compounding and we halved

our interest rate because of that semiannual compounding. So if we look in

our chart at the 4% column go all the way down to 10 periods we find the

factor to be 12.00611. We want those rents or payments

on one side by themselves so we divide both sides by that factor and we find out

that our payment is $1,166.07

per period which again is twice a year now every six months. We would need to

deposit that amount earning 4% or 8% semiannual compounding to be able to end

up with $14,000 at the end of five years or ten periods. Well similarly to this

question we may want to know well if I know how much

I want to end up with and I know how much I can deposit each month and I know

what my interest rate is maybe I want to know how many payments I'll actually

have to make or how many deposits I'll actually have to make to end up with

that amount. That's the computation of number of payments or periodic rents.

Suppose that a company's goal is to accumulate $117,000

$117,332

dollars by making periodic deposits of $20,000 at the end of

each year which will earn 8% compounded annually while accumulating.

How many deposits must it make? Well again we can use a algebraic expression

to solve for this. So now we have our future value of an ordinary annuity

which we know is equal to our payments times a future value factor of an

ordinary annuity. Well we know the future value is $117,332

$117,332 and we know the amount of the payments is

$20,000. What we don't know is that annuity factor which we've

got to calculate then we can go to our table. So if we solve for that annuity

factor that unknown amount we'll divide both sides by $20,000 or

the payments to get the annuity factor on one side by itself and we find the

factor to be 5.86660. Now that we've got that

factor we can go into our future value of an ordinary annuity tables and we'll

go to the 8% because that's what we're earning so we'll go to the

8% column and we're gonna look for that 5.86660

5.86660 and then scroll over to find the number of periods and we find that

we would need to deposit $20,000 for five year at the end of each

year for five years at 8% interest to accumulate $117,332

$117,332 by the end of that period.

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