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  • 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.

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

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A2 初級

現在価値の紹介 (Introduction to Present Value)

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    kellylin007 に公開 2021 年 01 月 14 日
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