Deferred Annuities - YouTube

So this will be our final video in the time value of money series and in this

video we're going to be talking about the third type of annuity which is a

deferred annuity. Remember a deferred annuity incorporates both lump sum

amounts as well as annuity amounts so therefore we call it a deferred annuity.

So let's look at an example. Assume a contract involving payments of different

amounts each year for a three year period and then we're also going to add an

annuity at the end of that but the first thing we have to do is to discount each

one of these back to the present because we're looking for the present value of

this deferred annuity and this in the first part of this is the lump sum

amounts. We have to treat each one of these as single lump sums because

they're not the same amount. Therefore they're not an annuity. These are each

single lump sums. So for example in period one we made a payment of $1,000

or we're receiving a payment of $1,000 either way. What that what we need to

find out is what would I need to invest today or at the beginning to be able to

have $1,000 at the end of year 1? The second payment is $2,000. What would I

have to invest today to have $2,000 at the end of the second year? The third

payment is $3,000. What would I have to invest today to have

$3,000 at the end of three years? So we've got to look into our present value

factor tables and when you do that at 8% you find your factors. For

year one is 0.9259 or 0.926 whatever your table might be

it depends on the rounding. Second year would be 0.8573

and year three would be 0.79388. You multiply that

times the lump sums for each period and you find the present value of these

three lump sums to be $5,022. Now up to this point

this is simply just three different lump sums it's not really a deferred annuity.

However if we tack on individual equal payments to that as in this timeline

here you see years 4, 5, 6, 7, & 8 are all $1,000-dollar payments. That makes the end

of this income stream or payment stream and annuity. Therefore we have a deferred

annuity because the first several payments are lump sum amounts they are

differing amounts and the ending payments are equal amounts which means

there's a portion of this that's an annuity. Well we found the present value

of the first three lump sums to be $5,022. Now

we've got to find the present value of the annuity portion. Well to do that we

would need to go into our present value of an ordinary annuity tables because as

we see in the second bullet up here that these are paid at the end of each year

so that makes this an ordinary annuity. So if we go to our ordinary annuity

tables at 8% we're looking for five periods so years four five six

seven and eight so at 8% in five periods we find the factor to be

3.99271 or there abouts again depending on what table

you're using. So if we take that factor and multiply it times our $1,000

dollars we find that the present value at the end of the third year and the

beginning of the fourth year for this stream of annuity payments is

$3,993. Well let's look at that in a

timeline. Here we have our present value of our annuity stream at the end

of the third period or the beginning of the fourth period that's the same time

period is $3,993. But we don't want

to know what that annuity is is valued at at the end of the third year. We want

to know what it's valued at today. So in other words what would I have to invest

today to be able to start withdrawing or what would I be what would I have to

accrue to be able to pay $1,000 a period at the beginning of the

fourth period? We're starting in year four making the payment at the end of

your four end of year five end of year six etc. What would I need

that amount to be? So now that we've found that lump sum what it's valued at

at the end of the third period we can now discount that amount back to today

as a lump sum, so we would go into our present

value of lump sum tables look for three periods at an interest rate of 8%

discounting that $3,993 dollars back to today and we find that we would need to

invest or deposit $3,170 today at 8% to

be able to have $3,993 dollars at the end

of the third period. Now that still doesn't really answer the question. We

need to know what is the present value of this deferred annuity? Well that

includes the annuity portion and the three lump sums that we had in years one

two and three which we found the present value of those to be $5,022

$5,022. If we add the two numbers together, the

5,022 and the 3,170, we find that the total present

value of this deferred annuity to be $8,192

$8,192. So now I'd like for you to try one. Here we have Del Monty will

receive the following payments at the end of the next three years: $2,000,

$3500, and $4500. Then from the end of the fourth

through the end of the tenth year he will receive an annuity of $5,000

dollars per year. At a discount rate of 9% what is the present value

of all future benefits? So using whatever tables you have figure out what the net

present value of all this deferred annuity will be. We know it's a deferred

annuity because there are lump sum portions and there's actually an annuity

portion. So press pause on your player now. Figure out the answer to this

question then we'll come back and look at it together. Okay so the first thing

you had to do was find the present value of the first three lump sums. So going

into your present value tables you will get the present value factor of each one

of these as you go down the line and your your present value factor table at

9%. Multiplying those factors times the lump sums we find that the

present value of the lump sum amounts in years one, two, and three to be

$8,255. The second step is to find the

present value of the annuity portion.

Well the value of the annuity at the beginning of the fourth year would be 9% at

seven periods using your present value of an ordinary annuity tables because

the problem tells us that the payments are made at the end of each year that

makes this an ordinary annuity. So we take the payment amount of 5,000

multiply that times the factor that we find in our tables 5.033

should have been approximately the number you found and multiplying those

through we get $25,165. This does

not answer the question. This simply gives me the amount of the annuity at

the end of the third year or the beginning of the fourth year. I want to

know what the total net benefit or present value of all this cash flows are

at the beginning here. Okay so the value at the beginning we would need to now

discount this $25,165 back to

today. Well that's over three periods at 9%. So if you go to your

present value of lump sum tables we find that factor to be 0.772 or there abouts.

Multiplying that through you get $19,427.38

$19,227.38. Now that we know the present value of the annuity

the deferred annuity portion we can add that amount back to the present value of

the first three lump sum payments that we found to be $8,225

$8,225 and we get a total net present value of the deferred

annuity and the three lump sum payments of $27,682.38.

$27,682.38.

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