Time value of money | Interest and debt | Finance & Capital Markets | Khan Academy

Narrator: 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 bit 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. 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 throw 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 one year,
instead of giving you the $100 immediately, in one year I could give you $109 and then in 2 years, this is kind of option 3,
I'd be willing to give you $120, so your choice is,
someone walks up to you off the street. I could give you $100
bill now, $109 bill … (laughing) $109 bill, $109 in
a year, $120, 2 years from now and you know in the back of your mind you could 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 would say, "Hey, look. $120, that's the
biggest amount of money." "I'm going to 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's maybe something I'm losing out there?" And you'd be right. You'd be losing out on
the opportunity to get the 10% risk free interest if you
were to get the money earlier.

And if you wanted to
compare them directly, the thought process would be, "Well, let's see. If I took
option 1. If I got the $100." And 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 is $10. So, you would get $10 in interest. So, after one year, you're entire savings in the bank will now be $110. So, just doing that little exercise we actually see that $100 given now, put it in the bank at 10% risk free, will actually turn into
$110 in a year from now, which is better than the
$109 one year from now. So, given this scenario, or
given this kind of situation or this option, you would rather do this than do this. A year from now you're better off by $1. 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. So, this idea that not
just the amount matters, but when you get it, this idea is called the time value of money. 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
can 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, present value. Maybe I'll talk about
present and future value. So, present and future
value, future value. So, given this assumption,
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 or what amount of money would
you have to put into the bank risk free for the next
2 years to get $121? 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? So, what is the future value
of this $100 in 1 year? So, in 1 year. Well, 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 $121. 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 that someone
says they're willing to give us $65 in 1 year and we were to ask ourselves, "What is the present value of this?" So, what is the present value of this.

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 2 are equivalent to you? 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 by 10%, that's literally, I'm taking X+10%X+ …
let me write it this way. +10%xX … Let me write it …
Let me make it clear this way. X+10%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 $65. This is the same thing as 1X or we can say that
1X+10% is the same thing as 0.10X is equal to
65, or you add these 2. 1.10X = 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.

You get X is equal to …
let me do it this way. It will be a little bit
more clear about it. So, let's divide both sides by 1.0 and really that trailing
zero doesn't matter. We're not really too worried
about the precision here because this actually exactly 10%. So, this is going to
be … these cancel out and X is going to be equal to, let me get the calculator out, X is going to be equal
to 65 divided by 1.1, $59.09, rounding it. So, X=59.09, which was the present value of $65 in one year, or another way to think about it is if you wanted to know
what the future value of $59.09 is in 1 year,
assuming the 10% interest, you would get the $65..

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