Present Value 4 (and discounted cash flow) Lets change the discount rates depending on how far out the payments are.
Present Value 4 (and discounted cash flow)
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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
- 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
- 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.
- 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.
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