Present value 2 More choices as to when you get your money.
Present value 2
- Now I'll give you a slightly more complicated choice
- between two payment options.
- Both of them are good, because in either case
- you're getting money.
- So choice one.
- Today I will give you $100.
- I'll circle the payment when you get it in magenta.
- So today you get $100.
- Choice two.
- And I'll try to write this choice a little bit neater.
- Choice two is that not in 1 year, but in 2 years.
- So let's say this is year 1.
- And now this is year 2.
- Actually I'm going to give you three choices.
- That'll really hopefully hit things home.
- So actually let me scoot this choice two over to the left.
- Back to green.
- So now I'm back in business.
- So choice two, I am willing to give you, let's say, oh I
- don't know, $110 in 2 years.
- So not in 1 year.
- In 2 years I'm going to give you $110.
- And so I'll circle in magenta when you
- actually get your payment.
- And then choice 3.
- And choice 3 is going to be fascinating.
- I've done it in a slightly different shade of green.
- Choice 3, I am going to pay you-- I'm making this up
- on the fly as I go-- I'm going to pay you $20 today.
- I'm going to pay you $50 in 1 year.
- That's $70.
- Let me make this so it's close.
- And then I'm going to pay you – I don't know – $35 in year 3.
- So all of these are payments.
- I want to differentiate between the actual dollar
- payments and the present values.
- And just for the sake of simplicity, let's assume that
- I am guaranteed.
- I am the safest person available.
- If the world exists, if the sun does not supernova, I will
- be paying you this amount of money.
- So I'm as risk-free as the federal government.
- And I had a post on the previous present value, where
- someone talked about, well is the federal
- government really that safe?
- And this is the point.
- The federal government, when it borrows from you, say, $100 –
- Let's say it borrows $100 and it promises
- to pay it in a year.
- It'll give you that $100.
- The risk is, what is that $100 worth?
- Because they might inflate the currency to death.
- Anyway, I won't go into that right now.
- Let's just go back to this present value problem.
- And actually sometimes governments
- do default on debt.
- But the U.S. government has never defaulted.
- It has inflated its currency.
- So that's kind of a roundabout way of defaulting.
- But it has never actually said, "I will not pay you" –
- because if that happened, our entire financial system would
- blow up, and we would all be living off the land again.
- Anyway, back to this problem.
- Enough commentary from Sal.
- So let's just compare Choice 1 and Choice 2 again.
- And once, again let's say that – risk-free – I could put money –
- I could lend it to the federal government at 5%.
- And it doesn't matter over what –
- WRITING: Risk-free rate is 5%.
- And for the sake of simplicity-- in the next video
- I will make that assumption less simple-- but for the sake
- simplicity, the government will pay you 5% whether you
- give them the money for 1 year, whether you give them
- the money for 2 years, or whether you give them the
- money for 3 years, right?
- So if I had $100, what would that be worth in 1 year?
- We figured that out already.
- It's 100 times 1.05.
- So that's $105.
- And then if you got another 5%?
- So the government is giving you 5% per year.
- It would be 105 times 1.05.
- And what is that?
- So I have 105 times 1.05, which equals $110.25.
- So that is the value in 2 years.
- So immediately, without even doing any present value, we
- see that you'll actually be better off in 2 years if you
- were to take the money now and just lend it to the
- Because the government, risk-free, will give you
- $110.25 in 2 years, while I'm only willing to give you $110.
- So that's all fair and good.
- But the whole topic, what we're trying to solve, is
- present value.
- So let's take everything in today's money.
- And to take this $110 and say what is that worth today, we
- can just discount it backwards by the same method, right?
- So $110 in 2 years, what is its 1-year value?
- Well, you take $110 and you divide it by 1.05.
- You're just doing the reverse.
- And then you get some number here.
- Well that number you get is 110 divided by 1.05.
- And then to get its present value, its value today, you
- divide that by 1.05 again.
- So you get 110 divided.
- If I were to divide by 1.05 again what do I get?
- I divide by 1.05, and then I divide by 1.05 again.
- I'm dividing by 1.05 squared.
- And what does that equal?
- And I'm writing this on purpose, because I want to get
- you used to this notation.
- Because this is what all of our present values and our
- discounted cash flow, this type of dividing by 1 plus the
- discount rate to the power of however many years out, this
- is what all of that's based on.
- And that's all we're doing though, we're just dividing by
- 1.05 twice because we're 2 years out.
- So let's do that.
- 110 divided by 1.05 squared is equal to $99.77.
- So once again we have verified, by taking the
- present value of $110 in 2 years to today, that its
- present value-- if we assume a 5% discount rate.
- And this discount rate, this is where all of the fudge
- factor occurs in finance.
- You can tweak that discount rate and make a few
- assumptions in discount rate and
- pretty much assume anything.
- But right now, for simplification, we're assuming
- a risk-free discount rate.
- But when the present value is based on that, you get $99.77.
- You say, wow, yeah, this really isn't as good as this.
- I would rather have $100 today than $99.77 today.
- Now this is interesting.
- Choice number three.
- How do we look at this?
- Well what we do is, we present value each of
- the payments, right?
- So the present value of $20 today, well that's just $20.
- What's the present value of $50 in 1 year?
- Well the present value of that is going to be-- so plus $50
- divided by 1.05, right-- that's the present value of
- the $50, because it's 1 year out.
- And then I want the present value of the $35.
- So that's plus $35 divided by what-- it's 2 years out,
- right, so you have to discount it twice--
- divided by 1.05 squared.
- Just like we did here.
- So let's figure out what that present value is.
- Notice I'm just adding up the present values of each of
- those payments.
- Get out my virtual TI-85.
- Let's see, so the present value of the $20 payment is
- $20, plus the present value of the $50 payment.
- Well that's just 50 divided by 1.05, plus the present value
- of our $35 payment.
- 35 divided by-- and it's 2 years out, so we discount by
- our discount rate twice-- so it's divided by 1.05 squared.
- And then that is equal to-- we'll round it-- $99.37.
- So now we can make a very good comparison
- between the three options.
- This might have been confusing before.
- You know, you have this guy coming up to you.
- And this guy is usually in the form of some type of
- retirement plan or insurance company, where they say, hey,
- you pay me this for years a, b, and c, and I'll pay you
- that in years b, c, and d.
- And you're like, boy, how do I compare if that's really a
- good value?
- Well this is how you compare it.
- You present value all of the payments and you say well what
- is that worth to me today.
- And here we did that.
- We said well actually choice number one is the best deal.
- And it just depended on how the mathematics work out.
- If I lowered the discount rate, if this discount rate is
- lower, it might have changed the outcomes.
- And maybe I'll actually do that in the next video, just
- to show you how important the discount rate is.
- Anyway I'm out of time, and I'll see
- you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
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This is great, I finally understand quadratic functions!
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At 2:33, Sal said "single bonds" but meant "covalent bonds."
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