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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