Introduction to Present Value

A choice between money now and money later.

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?

Present value.

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

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,

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.