Introduction to Present Value A choice between money now and money later.
Introduction to Present Value
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- 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.
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