Time Value of Money Why when you get your money matters as much as how much money. Present and future value also discussed.
Time Value of Money
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- Whenever we talk about money, the amount of money
- is not the only thing that matters
- what also matters is when you have to get, or
- when you have to give the money.
- So to think about this or to make it a little more
- concrete, let's assume that we live in a world
- that if you put money in a bank, you are guaranteed
- 10% interest.
- 10% risk free interest in a bank.
- And this is high by historical standards,
- but it will make our math easy.
- So let's just assume that you can always get
- 10% risk free interest in the bank.
- Now, given that, let me through out scenarios
- and have you think about which of these
- that you would most want.
- So I could give you $100 right now
- that's option 1,
- I could in 1 year, instead of giving you the
- $100 immediately,
- in 1 year I could give you $109
- and then in 2 years, this is kind of option 3
- I'd be willing to give you
- So your choice is, someone walks up to you in the street
- "I could give you $100 bill now,
- "...$109 in a year, or a $120 2 years from now"
- And you know in the back of your mind
- you can get 10% risk free interest.
- So given that you don't have an immediate
- need for money
- we're assuming that this money you will save.
- That you don't have a bill to pay immediately.
- Which of these things are the most desirable?
- which of these would you most want to have?
- Well, if you just cared about the absolute value
- or the absolute amount of the money
- you'd say, "hey look! $120! that's the biggest
- amount of money. I'm gonna take that one"
- because that's just the biggest number,
- but you probably have in the back of your mind
- "well I'm getting that later"
- so there may be something I'm losing out on there
- and you're right, you would be losing out on
- the opportunity to get the 10% risk free interest
- if you were to get the money earlier.
- And if you want to
- if you wanted to compare them
- directly, the thought process would be
- well let's see
- if I got the, if I took option 1, if I got the $100
- if you were to put it in the bank, what would that grow to
- based on that 10% risk free interest?
- Well after 1 year, 10% of $100 dollars is $10
- So you would get $10 in interest
- so after 1 year, your entire savings in the bank
- will now be $110.
- So just doing that little exercise,
- we actually see that $100 given now
- put in the bank at 10% risk free
- will actually turn into $110 a year from now
- which is better than the $109 a year from now
- So, given this scenario, or this kind of situation
- or this option, you would rather do this than do this
- A year from now you're better off by a dollar
- What about 2 years from now?
- Well if you take that $100 after 1 year
- it becomes $110, then 10% of $110
- is $11, you want to add $11 to it
- so it becomes $121
- so once again, you're better off taking the $100
- investing it in the bank risk free
- 10% per year. It turns into $121
- that is a better situation than just someone
- guaranteeing you to give the $120 in 2 years
- once again you are better off by $1.
- And so this idea, not just the amount matters
- but when you get it,
- this idea is called "the time value of money,"
- Or another way to think about it is,
- think about what the value of this money is
- over time, given some expected interest rate
- and when you do that, you could compare this
- money to equal amounts of money at some future date.
- Now another way of thinking about the time value,
- or I guess another related concept
- to the time value of money, is the idea of
- present value.
- and maybe I'll talk about present value and
- future value.
- So, present and future value.
- So, given this assumption, this $110, this 10% assumption
- if someone were to ask you,
- "what is the present value of $121 2 years in the future?"
- They're essentially asking you
- So what is the present value?
- PV stands for Present Value
- So what is the present value of $121 2 years in the future?
- That's equivalent to asking,
- what type of money, what amount of money
- would you have put into the bank, risk free
- over the next 2 years to get $121?
- Now we know that, if you put $100 in the bank
- for 2 years, at 10% risk free
- you would get $121. So the present value here
- The present value of $121 is the $100
- or another way to think about present and future value
- if someone were to ask, what is the future value?
- What is the future value of this $100 in 1 year?
- So, in 1 year, if you put
- if you get 10% in the bank that's guaranteed
- it's future value is $110,
- after 2 years, it's 2 year future value is
- and so with that in mind let me give you one
- slightly more interesting problem:
- so let's say that I have
- Let's say, we're going to assume this the whole time,
- that makes our math easy at 10% risk free interest,
- and let's say someone says that they're willing to give us
- $65 in 1 year, and we were to ask ourselves,
- "what is the present value of this?"
- So, remember, the present value is
- just asking you, what amount of money
- that if you were to put it in the bank
- at this risk free interest
- would be equivalent to this $65
- which of these two are equivalent to you?
- And so you would say, well look
- whatever amount of money that is, let's call that "x"
- whatever amount of money that is,
- times, if I grow it at 10%
- That's literally, I'm taking x plus 10% x
- let me write it this way,
- plus 10% times x
- or let me write it, let me make it clearer this way
- x plus 10% of x should be equal to our $65
- If I take the amount, I get 10% of that amount
- over the year, that should be equal to
- And this is the same thing as 1x
- or we could say that 1x plus 10% is the same thing as
- 0.10x is equal to 65, or you add these two
- 1.10x is equal to 65
- and if you want to solve for the actual amount
- of the present value here
- you would just divide both sides by the 1.10
- So you'd get x is equal to
- Let me do it this way, be a little bit more clear about it
- So let's divide both sides by 1.1
- and really that trailing zero doesn't matter
- we're not really too worried about the precision here
- because this is actually exactly 10%
- So this is going to be, these cancel out
- and x is going to be equal to
- get the calculator out
- x is going to be equal to
- 65 divided by 1.1
- $59.09 rounding it
- So, x is equal to $59.09
- which was the present value
- of $65 in 1 year
- or another way to think about it is
- if you wanted to know what the future value is of $59.09 is in 1 year
- assuming that 10% interest
- you would get $65.00.
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