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We'll now learn about what is arguably the most useful concept in finance

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and that is called the present value.

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And if you know the present value

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then it's very easy to understand

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the net present value and the discounted cash flow

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and the internal rate of return

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and we'll eventually learn all of those things.

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But the present value, what does that mean?

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Present value.

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So let's do a little exercise.

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I could pay you a hundred dollars today.

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So let's say today

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I could pay you one hundred dollars.

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Or (and it's up to you) in one year, I will pay you

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I don't know, let's say in a year I agree to pay you $110.

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And my question to you

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and this is a fundamental question of finance

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everything will build upon this

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is which one would you prefer?

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and this is guaranteed.

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I guarantee you, I'm either going to pay you $100 today

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and there's no risk, even if I get hit by a truck or whatever.

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This is going to happen, if the Earth exists, I will pay you $110 in one year.

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It is guaranteed, so there's no risk here.

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So it's just a notion of

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You're definitely gonna get $100 today, in your hand

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or you're definitely gonna get $110 one year from now.

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So how do you compare the two?

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And this is where present value comes in.

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What if there were a way

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to say, well what is $110

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a guaranteed $110 in the future?

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What if there were a way to say

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How much is that worth today?

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How much is that worth in today's terms?

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So let's do a little thought experiment.

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Let's say that you could put money

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in some, let's say you could money in the bank.

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And these days, banks are kind a risky.

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But let's say you could put it in the safest bank in the world.

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Let's say you could put it in government treasuries

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which are considered risk free

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because the US government, the treasury

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can always indirectly print more money.

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We'll one day do a whole thing on the money supply.

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But at the end of the day

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the US government has the rights on the printing press, etc.

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It's more complicated than that, but for these purposes, we assume

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that the US treasury, which essentially is

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you lending money to the US government

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that it's risk free.

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So let's say that

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you could lend money

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Let's say today, I could give you $100

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and that you could invest it

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at 5% risk free.

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So you could invest it 5% risk free.

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And then a year from now, how much would that be worth?

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In a year.

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That would be worth $105 in one year.

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Actually let me write $110 over here.

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So this is a good way of thinking about it.

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You're like, okay. Instead of taking the money

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from Sal a year from now

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and getting $110 dollars,

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If I were to take $100 today and put it in something risk free

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in a year I would have $105.

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So assuming I don't have to spend the money today

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This is a better situation to be in. Right?

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If I take the money today and risk-free

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invest it at 5%, I'm gonna end up at

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$105 in a year.

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Instead, if you just tell me

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Sal, just give me the money in a year and give me $110

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you're gonna end up with more money in a year.

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You're gonna end up with $110.

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And that is actually the right way to think about it.

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And remember, everything is risk-free.

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Once you introduce risk,

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And we have to start introducing different interest rates and

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probabilities, and we'll get to that eventually.

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But I want to just give the purest example right now.

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So already you've made the decision.

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We still don't know what present value is.

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So to some degree

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when you took this $100 and you

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said, well if I lend it to the government

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or if I lend it to some risk-free bank at 5%

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in a year they'll give me $105

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This $105 is a way of saying, what is the one-year value of $100 today?

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So what if we wanted to go in the other direction?

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If we have a certain amount of money

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and we want to figure out today's value

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what could we do?

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Well to go from here to here, what did we do?

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We essentially took $100

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and we multiplied by 1+5%.

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So that's 1,05

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So to go the other way,

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to say how much money

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if I were to grow it by 5%

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would end up being $110, we'll just divide by 1,05

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And then we will get the present value

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And the notation is PV

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We'll get the present value of $110 a year from now.

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So $110 year from now.

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So the present value of $110 in 2009

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It's currently 2008

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I don't know what year you're watching this video in.

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Hopefully people will be watching this in the next millenia.

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But the present value of $110 in 2009

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— assuming right now is 2008— a year from now, is equal to $110

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divided by 1,05.

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Which is equal to— let's take out this calculator

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which is probably overkill for this problem— let me clear everything.

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OK, so I want to do 110 divided by 1,05

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is equal to 104 (let's just round) ,76.

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So it equals $104,76.

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So the present value of $110 a year from now

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if we assume that we could invest money risk-free at 5%— if we would get it today—

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the present value of that is— let me do it in a different color, just to fight the monotony—

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the present value is equal to $104,76.

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Another way to kind of just talk about this is to get

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the present value of $110 a year from now, we discount the value by a discount rate.

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And the discount rate is this.

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Here we grew the money by— you could say—

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our yield, a 5% yield, or our interest.

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Here we're discounting the money 'cause we're backwards in time—

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we're going from a year out to the present.

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And so this is our yield. To compound the amount of money we invest

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we multiply the amount we invest times 1 plus the yield.

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Then to discount money in the future to the present,

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we divide it by 1 plus the discount rate— so this is

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a 5% discount rate.

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To get its present value.

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So what does this tell us?

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This tells us if someone is willing to pay $110— assuming this 5%, remember

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this is a critical assumption— this tells us that if I tell you

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I'm willing to pay you $110 a year from now

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and you can get 5%, so you can kind of say

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that 5% is your discount rate, risk-free.

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Then you should be willing to take today's money if

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today I'm willing to give you more than the present value.

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So, if this compares in— let me clear all of this,

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let me just scroll down— so let's say

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that one year— so today, one year—

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so we figured out that $110 a year from now, its

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present value is equal to— so the present value of $110—

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is equal to $104,76.

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So— and that's 'cause I used a 5% discount rate (and that's the key assumption)—

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what this tells you is— this is a dollar sign, I know it's hard to read—

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what this tells you is, is that if your choice was between

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$110 a year from now and $100 today,

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you should take the $110 a year from now.

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Why is that?

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Because its present value is worth more than $100.

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However, if I were to offer you $110 a year from now or

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$105 today, this— the $105 today— would be the better choice,

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because its present value— right, $105 today

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you don't have to discount it, it's today— its present value

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is itself.

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$105 today is worth more than the present value of $110, which

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is $104.76.

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Another way to think about it is, I could take this $105 to the bank,

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get 5% on it, and then I would have— what would

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I end up with?— I would end up with 105 times 1,05, it's equal to $110,25.

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So a year from now, I'd be better off by a quarter.

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And I'd have the joy of being able to touch my money for a year,

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which is hard to quantify, so we leave it out of the equation.