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how comes for the calculations we dont roud off!!! for example if the answer is $133.239 whay cant e use 133.24 anstead of 133.23? however the lessons are clear and interesting.
So far, we've been assuming that the discount rate is the
same thing, no matter how long of a period
we're talking about.
But we know if you go to the bank and you say, hey, bank, I
want to essentially invest in a one-year CD, they'll say,
oh, OK, one-year CD will give you 2%.
And you're like, well, what if we give you the
money for two years?
So you can keep our money, locked in for even longer.
They'll say, oh, then we'll give you a little bit more
interest, because we have more flexibility.
For two years, we don't have to worry about paying you.
So instead of giving you 2%, we'll give you 7%, because we
get to keep your money for two years.
And maybe if you say, well, you know, I actually don't
even need my money for 10 years, so let me give you the
money for 10 years.
They'll say oh, 10 years, if we get to keep your money,
we'll give you 12%.
So in general-- and this tends to be the case, although it's
not always the case-- the longer that you defer your
money, or the longer you lock up the money, the higher an
interest rate you get.
So the same thing is true when you're doing a discount rate.
Oftentimes you want to discount a payment two years
out by a higher value than something that's
only one year out.
So how do you do that?
So let's say the risk-free rate, if you were to go out
and get a government bond-- the one-year rate, let's say
that they're only giving you 1%.
But let's say that the two-year rate,
they'll give you 5%.
So what does that mean?
Well, let's take the example.
So that means you could take that $100 and essentially lend
it to the federal government, and in a year they'll
give you 1% on it.
So that these are annual rates.
So 1%, 1.01 times 100, that's just $101, right?
Now your other option is, you could lock it in.
You could lend it to the federal government for two
years and not see your money.
And they say, oh, then we're going to give you 5% a year.
So then you're going to go 5% a year.
So how much do you end up with in two years?
Well, remember, this is an annual rate.
These are always quoted in annual rates.
So if you're getting 5% a year, that's going to be equal
to-- let's do it on the calculator.
That's going to be 100-- after one year you're going to get
1.05, and after two years you're going to get 1.05.
Or you can view that as 100 times 1.05 squared.
So you'd have $110.25.
So you already see, not even doing any present value, this
is actually-- you can almost view this as a future value
If you take a future value, you already know that this
option is better than this option, when you have these
varying interest rates.
But anyway, the whole topic of this is to talk about present
value, so let's do that.
So in this circumstance, what is the present
value of the $110?
Well, actually, what is the present value of the $100?
Well, we always know that.
That is $100.
Present value of $100 today is $100.
What is the present value of the $110?
So we take $110, and we're going to use the two-year
rate, and discount twice.
And that makes sense, because essentially you're deferring
your money for two years.
You're not going to get anything,
even a year from now.
So you're deferring your money for two years.
So you divide it by 1-- so it's a 5% rate, 1.05 squared.
And then that is equal to-- I think that was our first
So I'll just do it again.
110 divided by 1.05 squared.
That's equal to $99.77, right?
That was our first problem.
And now this one is interesting.
The $20 you get today-- and this is a side note.
It's very important when you're doing this, when they
talk about year one, or year zero, just make sure-- is that
today, is that a year from now?
Because if it's a year from now, you'd have to discount it
by the one-year interest rate.
If it's today, you don't discount it.
So anyway, I clarified that.
I was a little ambiguous about that in the last two videos,
but I clarified it.
The $20 is now.
So the present value of something given you today, is
the value of it.
So it's $20 plus $50.
Now $50, what do we use?
Do we use the one-year rate or the two-year rate?
Well of course, we use the one-year rate, because you're
not deferring the pleasure of that $50 for two years.
You're actually getting it in one year.
So plus $50 divided by the one-year rate.
Divided by 1.01.
Plus $35 divided by the two-year rate-- but this is an
annual rate, so you have to discount it twice-- divided by
Let's get the TI-85 out.
So you get 20 plus 50 divided by 1.01, plus 35 divided by
1.05 squared, is equal to $101.25.
So notice, the actual payment streams I did not change in
any of the three scenarios.
And let me just draw a line between them, because I got a
little bit messy.
So that was scenario one.
This is scenario two.
And this is scenario three.
But in scenario one, because we used a 5% discount rate for
all-- you could say, I don't want to use fancy words-- but
for all durations out we used a 5% discount rate.
We saw that choice number one was the best.
But then if the discount rate were to change-- if we were to
change our assumption.
If we had a 2% rate, for whatever reason, we could lend
money to the federal government in the form of
buying bonds from them-- we could lend the federal
government two years over any time period at 2%.
Then all of a sudden, choice two became the best option.
And then finally, if we had this kind of-- and this is the
most realistic scenario, and even though the math is fairly
simple, we're actually doing something fairly
When I had a different discount rate for my one year
out cash flows and my two year out cash flows, and it was
these exact numbers.
I had to play with the numbers to get the right result.
Then all of a sudden choice three was the best option.
I'll leave it to you-- I want you to think about why this
was better for choice three than it was for choice two.
And if you really understand that, then I think you are
starting to have a lot of intuition
about present values.
And frankly, what we're learning here is a
discounted cash flow.
What is a discounted cash flow?
I'm giving you a stream of cash flows.
$20 now, $50 a year from now, $35 in two years.
And you are essentially discounting them back to get
today's present value.
So when someone says, you know, I can use Excel to do a
discounted cash flow, that's all they're doing.
They're making some assumption about the discount rates.
And they're just using this fairly straightforward
mathematics to get the present value of
those future cash flows.
But it's a very powerful technique.
Because if you were to take-- if you're good at Excel, and
you were to say, oh, I have a business.
And based on my assumptions, in year one, right now, this
business gives me $20.
The next year it's going to give $50.
The year after that it's $35.
And this risk-free is the big assumption.
But if it was risk-free, you could discount it like that.
You'd say, if these are the interest rates, this business
is worth $101.25.
That's what I'm willing to pay for it.
Or, I'm neutral.
If I could get it for $90, that's a good deal for me.
That's all a discounted cash flow is.
But the big learning from this is how dependent the present
value of future payments are on your discount rate
The discount rate assumption is everything in finance.
And this is where finance really diverges from a lot of
other fields, especially the sciences.
There really is no correct answer.
It's all assumption driven.
All of these discounted cash flows, and all these models,
they're really just to help you understand
the dynamics of things.
And frankly-- and this happens a lot in the real world of
finance-- if you ever become an analyst at an investment
bank, you'll probably do this yourself.
But you can almost justify any present value, by picking the
right discount rate.
And actually the whole topic of, how do you decide on the
right discount rate?
Because we assumed risk-free.
Everything is risk-free.
You're guaranteed these payments.
But we know in the real world, if you're investing in
pets.com and they tell you that they're going to pay
these cash flows to you, that's not risk-free.
There's some risk implicit in that.
So actually, most of finance, and most of portfolio theory,
and modern finance, is based on figuring out
that discount rate.
And that is the crux of everything, because as we see,
that completely changes which of these options is the best.
But anyway, I don't want to confuse you too much.
What you have already is a very powerful tool.
If you can think of a discount rate, you can make a very
rational comparison between three, or ten, or whatever
different types of payments.
And this is actually really useful.
You don't realize how many things in the
world are like this.
These college payment schemes where you pay some company $25
a year for 20 years, and then in year 21 they're willing to
pay for your college tuition, or your kids' college tuition.
You could figure out with that really is worth, how much
money are they making off of you, by taking a
discounted cash flow.
And of course if you're paying out, these
become negative numbers.
And when they pay you, it becomes a positive number.
Maybe I'll do that in a couple of videos, because I think
that's a fairly useful thing to be able to analyze.
See you in the next video.