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## Macroeconomics

### Unit 4: Lesson 8

Interest rates and the time value of money# Introduction to present value

Present value is the value

*right now*of some amount of money in the future. For example, if you are promised $110 in one year, the present value is the current value of that $110 today. Present value is one of the foundational concepts in finance, and we explore the concept and calculation of present value in this video. Created by Sal Khan.## Video transcript

We'll now learn about what is
arguably the most useful concept in finance, and that's
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? Present value. So let's do a little exercise. I could pay you $100 today. So let's say today, I
could pay you $100. 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. The U.S. government, if the
earth exists, we will pay you $110 in one year. It is guaranteed. So there's no risk here. So it's just the notion of
you're definitely going to get $100 today in your hand, or
you're definitely going to 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 the bank. And these days banks
are kind of risky. But let's say you could
put it in the safest bank in the world. Let's say you , although someone
would debate, you put it in government treasuries. Which are considered risk-free,
because the U.S. 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
U.S. government has the rights on the printing press,
et cetera. It's more complicated
than that. But for those purposes, we
assume that with the U.S. Treasury, which essentially is
you're lending money to the U.S. government, that
it's risk-free. So let's say today I could give
you $100 and that you could invest it at
5% risk-free. And then in 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
the $110 over here. So this was a good way
of thinking about it. You're like, OK, instead of
taking the money from Sal a year from now and getting $110,
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 going to end up with
$105 in a year. Instead, if you just tell me,
Sal, just give me the money in a year-- give me $110-- you're
going to end up with more money in a year, right? You're going to end
up with $110. And that is actually the right
way to think about it. And remember, and I keep saying
it over and over again, everything I'm talking about,
it's critical that we're talking about risk-free. Once you introduce risk, then
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. But we still don't know what
the present value was. 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 a 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? What is the one-year-out
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- what did we multiply by-- 1 plus 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 the present value of $110,
let's say 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 millennia. But the present value of $110
in 2009, assuming right now it's 2008, a year from now, is
equal to $110 divided by 1.05. And 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-- let's just round-- 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 were to get it today -- 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 discounted the value by a discount rate. And the discount rate is this. Right here we grew the money by,
you could say, our yield. A 5% yield or our interest.
Here we're discounting the money, because we're going
backwards in time. We're going from 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 divided 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's
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
could get 5%-- so you could kind of say that 5% is your
discount rate risk-free-- that 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 comparison were-- let
me clear all of this, let me just scroll down-- so let's
say that today, 1 year. So we figured out that $110 a
year from now, its present value is equal to-- so the
present value of that $110-- is equal to $104.76. And that's because I used a 5%
discount rate, and that's a key assumption. This is a dollar sign. I know it's hard to read. What this tells you 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 is itself. $105 today is worth more than
the present value of $110, which is $104.76. Another way to think about it
is, I could take this $105 to the bank-- let's assume
I have a risk-free bank-- get 5% on it. And then I would have-- what
would I end up with-- I'd end up with 105 times 1.05. Equal to $110.25. So a year from now, I'd be
better off by $0.25. And I'd have the joy of being
able to touch my money for a year, which is hard to quantify,
so we leave out of the equation. Anyway, I'll see you
in the next video.