Ok, we talked about present value as far as the bonds.
Let's now look at this and apply it.
Now, if you come back over here, we said how do you figure out the proceeds on the bond?
We said face, par, million dollar face times the present value of the lump sum, 10 percent,
boom.
Plus, 80,000 present value of an ordinary annuity, 5 years, 10 percent, boom.
So the question is, what does this mean, where do these factors come from?
Those are called present value.
If you look in your notes you will see present value of an amount.
That is present value of a lump sum.
That's the amount you need to invest today at a certain interest rate for so many years
to get back a dollar in the future.
So that's called the present value of a lump sum.
That's going to be the present value as we go through the million dollars present value
in five years, ten percent.
We will look at that.
Present value in ordinary annuity.
What is an ordinary annuity?
That is an annuity.
Where is it?
Your rear, it's at the end of the year.
That's when you're getting it, at the end of the year.
That is present value ordinary annuity.
I just thought of a joke but I won't say it.
Present value of an annuity due now.
Due now is up front, that's in advance.
My example was like a lease.
If you pay rent, you have to pay at the beginning of the month, not the end of the month.
We have future values.
This is how much the amount will grow into in the future or accumulate into.
If you look in your notes, future value of a dollar.
If I put a dollar in the bank at ten percent, in one year it will be a dollar ten.
In two years, it will earn interest on interest.
Compounding interest that will be $1.21.
If you look at the ten percent, two years, 1.210.
That would be future value.
We don't use that as much, we use present value, because we want to say "how much is
that money "in the future worth today?"
Present value of a dollar.
Now, how much do I have to put in the bank and it will grow?
In other words, a dollar in a year is worth what?
So the present value of a dollar is worth how much?
So if I get a dollar in a year, what do I have to put in the bank today at ten percent
for it to grow to a dollar?
If I put in .909, if I put in 90.9 cents, at ten percent it will grow to a dollar.
If it's two years, if I put money in at .826, then in two years it will grow to a dollar.
Then in three years, and in four years, and in five years.
So when we're talking about present valuing, what we are looking at is the present value
of a lump sum is how much today.
Basically, if you put, let's look at, ten percent, five years, .621.
If I come back over here and I say a million dollars, and the present value of a lump sum
in five years at ten percent, well let's see what that is.
The factor for the present value of a lump sum, five years, ten percent, is what?
It is .621, I think it was. .621?
Yes. .621 times a million dollars equals 621,000 dollars.
If I put 621,000 dollars in the bank today at ten percent, it will grow in five years
to a million dollars.
That means that if the person I am loaning money to says, "Hey Rog, I'm going to pay
you back "a million dollars in five years."
How much is that really worth today?
Its worth, at ten percent, if I want to earn ten, its 621,000, because if you put 621,000
in the bank today at ten percent, it will grow to what?
A million dollars.
Then we have future value of an ordinary annuity.
Again, not frequently used for our purpose here.
Present value of an ordinary annuity that is what we are looking at.
What is the ordinary annuity?
Ordinary, a rear, is at the end.
When do you earn the interest?
At the end of the year.
So, what this says is present value of an ordinary annuity.
If you look down, how much would I have to put in the bank at ten percent in order to
get a dollar in a year?
.909.
If I want a dollar in a year, and a dollar in two years, and a dollar in three years,
so that's the present value of an ordinary annuity.
If I want a dollar at the end of the year, and another dollar at the end of two years,
what is that worth today?
1.736, a dollar, seventy-three point six.
How much dollar, a dollar, a dollar?
Three dollars.
So you're going to say, "I'm going to give you three dollars", it's really not worth
three dollars, and it�s worth 2.487.
Why is it worth 2.487?
Because, a dollar a year from today is not worth a dollar, it's worth 90 cents.
Then a dollar two years from today is worth 82 cents.
A dollar three years from today is worth 75 cents.
A dollar four years from today is 68 cents.
A dollar five years is 62 cents.
Because, remember, what's a lump sum?
The lump sum is 62.1.
So, if I put 62 cents in the bank five years from now, it will grow to a dollar.
Another way to look at this, is when you look at the annuity, let's look back at the present
value of a dollar.
What is the present value of a dollar?
Well, let's see.
The present value, this is ten percent.
The present value of a lump sum.
So present value of a dollar, in one year, .909.
Two years, eight, two six.
We're only rounding, it's really eight, two six, five, or nine zero nine one, but they
cut if off here. .751, four years .683, five years .621.
If you add these up, this is how much I need to put, if I put 90 cents in, in a year it
grows to a dollar.
82, in two years, three years, four years, five years.
When you add those up, that equals 3.791.
What is 3.791?
Look at the present value of an ordinary annuity of a dollar, ten percent, and five years.
That is really the present value of a dollar in one, two, three, four, five, if you add
them all up that is the present value of an annuity, ordinary annuity that is 3.791.
If you take that 3.791, that would be the present value of what, the interest payments.
Remember, I'm getting 80,000 times the present value of an ordinary annuity for five years
at ten percent.
Well, the factor for that is what?
It's going to be 80,000 dollars times this 3.791.
Of course, we all know that equals 303,280.
So, when you add this plus this, you end up with 924,280.
That's how much we should charge in order for you to earn ten percent when the bond
is paying eight percent.
That is how much we should charge.
In my example, I just said 900,000 remember for my discount example?
Because I didn't want to go through all of this.
But the real number would be 924,280.
That is a formula.
So this is present value of a lump sum, present value of an ordinary annuity.