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Present value is simply the best, I know that money today is worth more than that of the future.
We'll now learn about what is arguably the most useful concept in finance
and that is called the present value.
And if you know the present value
then it's very easy to understand
the net present value and the discounted cash flow
and the internal rate of return
and we'll eventually learn all of those things.
But the present value, what does that mean?
So let's do a little exercise.
I could pay you a hundred dollars today.
So let's say today
I could pay you one hundred dollars.
Or (and it's up to you) in one year, I will pay you
I don't know, let's say in a year I agree to pay you $110.
And my question to you
and this is a fundamental question of finance
everything will build upon this
is which one would you prefer?
and this is guaranteed.
I guarantee you, I'm either going to pay you $100 today
and there's no risk, even if I get hit by a truck or whatever.
This is going to happen, if the Earth exists, I will pay you $110 in one year.
It is guaranteed, so there's no risk here.
So it's just a notion of
You're definitely gonna get $100 today, in your hand
or you're definitely gonna get $110 one year from now.
So how do you compare the two?
And this is where present value comes in.
What if there were a way
to say, well what is $110
a guaranteed $110 in the future?
What if there were a way to say
How much is that worth today?
How much is that worth in today's terms?
So let's do a little thought experiment.
Let's say that you could put money
in some, let's say you could money in the bank.
And these days, banks are kind a risky.
But let's say you could put it in the safest bank in the world.
Let's say you could put it in government treasuries
which are considered risk free
because the US government, the treasury
can always indirectly print more money.
We'll one day do a whole thing on the money supply.
But at the end of the day
the US government has the rights on the printing press, etc.
It's more complicated than that, but for these purposes, we assume
that the US treasury, which essentially is
you lending money to the US government
that it's risk free.
So let's say that
you could lend money
Let's say today, I could give you $100
and that you could invest it
at 5% risk free.
So you could invest it 5% risk free.
And then a year from now, how much would that be worth?
In a year.
That would be worth $105 in one year.
Actually let me write $110 over here.
So this is a good way of thinking about it.
You're like, okay. Instead of taking the money
from Sal a year from now
and getting $110 dollars,
If I were to take $100 today and put it in something risk free
in a year I would have $105.
So assuming I don't have to spend the money today
This is a better situation to be in. Right?
If I take the money today and risk-free
invest it at 5%, I'm gonna end up at
$105 in a year.
Instead, if you just tell me
Sal, just give me the money in a year and give me $110
you're gonna end up with more money in a year.
You're gonna end up with $110.
And that is actually the right way to think about it.
And remember, everything is risk-free.
Once you introduce risk,
And we have to start introducing different interest rates and
probabilities, and we'll get to that eventually.
But I want to just give the purest example right now.
So already you've made the decision.
We still don't know what present value is.
So to some degree
when you took this $100 and you
said, well if I lend it to the government
or if I lend it to some risk-free bank at 5%
in a year they'll give me $105
This $105 is a way of saying, what is the one-year value of $100 today?
So what if we wanted to go in the other direction?
If we have a certain amount of money
and we want to figure out today's value
what could we do?
Well to go from here to here, what did we do?
We essentially took $100
and we multiplied by 1+5%.
So that's 1,05
So to go the other way,
to say how much money
if I were to grow it by 5%
would end up being $110, we'll just divide by 1,05
And then we will get the present value
And the notation is PV
We'll get the present value of $110 a year from now.
So $110 year from now.
So the present value of $110 in 2009
It's currently 2008
I don't know what year you're watching this video in.
Hopefully people will be watching this in the next millenia.
But the present value of $110 in 2009
— assuming right now is 2008— a year from now, is equal to $110
divided by 1,05.
Which is equal to— let's take out this calculator
which is probably overkill for this problem— let me clear everything.
OK, so I want to do 110 divided by 1,05
is equal to 104 (let's just round) ,76.
So it equals $104,76.
So the present value of $110 a year from now
if we assume that we could invest money risk-free at 5%— if we would get it today—
the present value of that is— let me do it in a different color, just to fight the monotony—
the present value is equal to $104,76.
Another way to kind of just talk about this is to get
the present value of $110 a year from now, we discount the value by a discount rate.
And the discount rate is this.
Here we grew the money by— you could say—
our yield, a 5% yield, or our interest.
Here we're discounting the money 'cause we're backwards in time—
we're going from a year out to the present.
And so this is our yield. To compound the amount of money we invest
we multiply the amount we invest times 1 plus the yield.
Then to discount money in the future to the present,
we divide it by 1 plus the discount rate— so this is
a 5% discount rate.
To get its present value.
So what does this tell us?
This tells us if someone is willing to pay $110— assuming this 5%, remember
this is a critical assumption— this tells us that if I tell you
I'm willing to pay you $110 a year from now
and you can get 5%, so you can kind of say
that 5% is your discount rate, risk-free.
Then you should be willing to take today's money if
today I'm willing to give you more than the present value.
So, if this compares in— let me clear all of this,
let me just scroll down— so let's say
that one year— so today, one year—
so we figured out that $110 a year from now, its
present value is equal to— so the present value of $110—
is equal to $104,76.
So— and that's 'cause I used a 5% discount rate (and that's the key assumption)—
what this tells you is— this is a dollar sign, I know it's hard to read—
what this tells you is, is that if your choice was between
$110 a year from now and $100 today,
you should take the $110 a year from now.
Why is that?
Because its present value is worth more than $100.
However, if I were to offer you $110 a year from now or
$105 today, this— the $105 today— would be the better choice,
because its present value— right, $105 today
you don't have to discount it, it's today— its present value
$105 today is worth more than the present value of $110, which
Another way to think about it is, I could take this $105 to the bank,
get 5% on it, and then I would have— what would
I end up with?— I would end up with 105 times 1,05, it's equal to $110,25.
So a year from now, I'd be better off by a quarter.
And I'd have the joy of being able to touch my money for a year,
which is hard to quantify, so we leave it out of the equation.