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  • So far, we've been assuming that the discount rate is the

  • same thing, no matter how long of a period

  • we're talking about.

  • But we know if you go to the bank and you say, hey, bank, I

  • want to essentially invest in a one-year CD, they'll say,

  • oh, OK, one-year CD will give you 2%.

  • And you're like, well, what if we give you the

  • money for two years?

  • So you can keep our money, locked in for even longer.

  • They'll say, oh, then we'll give you a little bit more

  • interest, because we have more flexibility.

  • For two years, we don't have to worry about paying you.

  • So instead of giving you 2%, we'll give you 7%, because we

  • get to keep your money for two years.

  • And maybe if you say, well, you know, I actually don't

  • even need my money for 10 years, so let me give you the

  • money for 10 years.

  • They'll say oh, 10 years, if we get to keep your money,

  • we'll give you 12%.

  • So in general-- and this tends to be the case, although it's

  • not always the case-- the longer that you defer your

  • money, or the longer you lock up the money, the higher an

  • interest rate you get.

  • So the same thing is true when you're doing a discount rate.

  • Oftentimes you want to discount a payment two years

  • out by a higher value than something that's

  • only one year out.

  • So how do you do that?

  • So let's say the risk-free rate, if you were to go out

  • and get a government bond-- the one-year rate, let's say

  • that they're only giving you 1%.

  • But let's say that the two-year rate,

  • they'll give you 5%.

  • So what does that mean?

  • Well, let's take the example.

  • So that means you could take that $100 and essentially lend

  • it to the federal government, and in a year they'll

  • give you 1% on it.

  • So that these are annual rates.

  • So 1%, 1.01 times 100, that's just $101, right?

  • Fair enough.

  • Now your other option is, you could lock it in.

  • You could lend it to the federal government for two

  • years and not see your money.

  • And they say, oh, then we're going to give you 5% a year.

  • So then you're going to go 5% a year.

  • So how much do you end up with in two years?

  • Well, remember, this is an annual rate.

  • These are always quoted in annual rates.

  • So if you're getting 5% a year, that's going to be equal

  • to-- let's do it on the calculator.

  • That's going to be 100-- after one year you're going to get

  • 1.05, and after two years you're going to get 1.05.

  • Or you can view that as 100 times 1.05 squared.

  • So you'd have $110.25.

  • So you already see, not even doing any present value, this

  • is actually-- you can almost view this as a future value

  • calculation.

  • If you take a future value, you already know that this

  • option is better than this option, when you have these

  • varying interest rates.

  • But anyway, the whole topic of this is to talk about present

  • value, so let's do that.

  • So in this circumstance, what is the present

  • value of the $110?

  • Well, actually, what is the present value of the $100?

  • Well, we always know that.

  • That's easy.

  • That is $100.

  • Present value of $100 today is $100.

  • What is the present value of the $110?

  • So we take $110, and we're going to use the two-year

  • rate, and discount twice.

  • And that makes sense, because essentially you're deferring

  • your money for two years.

  • You're not going to get anything,

  • even a year from now.

  • So you're deferring your money for two years.

  • So you divide it by 1-- so it's a 5% rate, 1.05 squared.

  • And then that is equal to-- I think that was our first

  • problem, right?

  • So I'll just do it again.

  • 110 divided by 1.05 squared.

  • That's equal to $99.77, right?

  • That was our first problem.

  • And now this one is interesting.

  • The $20 you get today-- and this is a side note.

  • It's very important when you're doing this, when they

  • talk about year one, or year zero, just make sure-- is that

  • today, is that a year from now?

  • Because if it's a year from now, you'd have to discount it

  • by the one-year interest rate.

  • If it's today, you don't discount it.

  • So anyway, I clarified that.

  • I was a little ambiguous about that in the last two videos,

  • but I clarified it.

  • The $20 is now.

  • So the present value of something given you today, is

  • the value of it.

  • So it's $20 plus $50.

  • Now $50, what do we use?

  • Do we use the one-year rate or the two-year rate?

  • Well of course, we use the one-year rate, because you're

  • not deferring the pleasure of that $50 for two years.

  • You're actually getting it in one year.

  • So plus $50 divided by the one-year rate.

  • Divided by 1.01.

  • Plus $35 divided by the two-year rate-- but this is an

  • annual rate, so you have to discount it twice-- divided by

  • 1.05 squared.

  • Let's get the TI-85 out.

  • So you get 20 plus 50 divided by 1.01, plus 35 divided by

  • 1.05 squared, is equal to $101.25.

  • So notice, the actual payment streams I did not change in

  • any of the three scenarios.

  • And let me just draw a line between them, because I got a

  • little bit messy.

  • So that was scenario one.

  • This is scenario two.

  • And this is scenario three.

  • But in scenario one, because we used a 5% discount rate for

  • all-- you could say, I don't want to use fancy words-- but

  • for all durations out we used a 5% discount rate.

  • We saw that choice number one was the best.

  • But then if the discount rate were to change-- if we were to

  • change our assumption.

  • If we had a 2% rate, for whatever reason, we could lend

  • money to the federal government in the form of

  • buying bonds from them-- we could lend the federal

  • government two years over any time period at 2%.

  • Then all of a sudden, choice two became the best option.

  • And then finally, if we had this kind of-- and this is the

  • most realistic scenario, and even though the math is fairly

  • simple, we're actually doing something fairly

  • sophisticated here.

  • When I had a different discount rate for my one year

  • out cash flows and my two year out cash flows, and it was

  • these exact numbers.

  • I had to play with the numbers to get the right result.

  • Then all of a sudden choice three was the best option.

  • I'll leave it to you-- I want you to think about why this

  • was better for choice three than it was for choice two.

  • And if you really understand that, then I think you are

  • starting to have a lot of intuition

  • about present values.

  • And frankly, what we're learning here is a

  • discounted cash flow.

  • What is a discounted cash flow?

  • I'm giving you a stream of cash flows.

  • $20 now, $50 a year from now, $35 in two years.

  • And you are essentially discounting them back to get

  • today's present value.

  • So when someone says, you know, I can use Excel to do a

  • discounted cash flow, that's all they're doing.

  • They're making some assumption about the discount rates.

  • And they're just using this fairly straightforward

  • mathematics to get the present value of

  • those future cash flows.

  • But it's a very powerful technique.

  • Because if you were to take-- if you're good at Excel, and

  • you were to say, oh, I have a business.

  • And based on my assumptions, in year one, right now, this

  • business gives me $20.

  • The next year it's going to give $50.

  • The year after that it's $35.

  • And this risk-free is the big assumption.

  • But if it was risk-free, you could discount it like that.

  • You'd say, if these are the interest rates, this business

  • is worth $101.25.

  • That's what I'm willing to pay for it.

  • Or, I'm neutral.

  • If I could get it for $90, that's a good deal for me.

  • That's all a discounted cash flow is.

  • But the big learning from this is how dependent the present

  • value of future payments are on your discount rate

  • assumption.

  • The discount rate assumption is everything in finance.

  • And this is where finance really diverges from a lot of

  • other fields, especially the sciences.

  • There really is no correct answer.

  • It's all assumption driven.

  • All of these discounted cash flows, and all these models,

  • they're really just to help you understand

  • the dynamics of things.

  • And frankly-- and this happens a lot in the real world of

  • finance-- if you ever become an analyst at an investment

  • bank, you'll probably do this yourself.

  • But you can almost justify any present value, by picking the

  • right discount rate.

  • And actually the whole topic of, how do you decide on the

  • right discount rate?

  • Because we assumed risk-free.

  • Everything is risk-free.

  • You're guaranteed these payments.

  • But we know in the real world, if you're investing in

  • pets.com and they tell you that they're going to pay

  • these cash flows to you, that's not risk-free.

  • There's some risk implicit in that.

  • So actually, most of finance, and most of portfolio theory,

  • and modern finance, is based on figuring out

  • that discount rate.

  • And that is the crux of everything, because as we see,

  • that completely changes which of these options is the best.

  • But anyway, I don't want to confuse you too much.

  • What you have already is a very powerful tool.

  • If you can think of a discount rate, you can make a very

  • rational comparison between three, or ten, or whatever

  • different types of payments.

  • And this is actually really useful.

  • You don't realize how many things in the

  • world are like this.

  • These college payment schemes where you pay some company $25

  • a year for 20 years, and then in year 21 they're willing to

  • pay for your college tuition, or your kids' college tuition.

  • You could figure out with that really is worth, how much

  • money are they making off of you, by taking a

  • discounted cash flow.

  • And of course if you're paying out, these

  • become negative numbers.

  • And when they pay you, it becomes a positive number.

  • Anyway.

  • Maybe I'll do that in a couple of videos, because I think

  • that's a fairly useful thing to be able to analyze.

  • See you in the next video.

So far, we've been assuming that the discount rate is the

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