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By getting acquainted with this process,

it's easy to learn how to calculate the value of money as it changes over time.

So before we start a few calculations,

let's think about how money grows in value over time.

We know that money may not always be worth more in the future

because this is going to depend on how much money was invested,

how much was earned, whether there was inflation, and a host of other factors.

But we can all agree on how to calculate the way money is compounded,

which in fact, grow exponentially into the future.

Similarly, it's just as important to understand the process in reverse.

That is we need to convert the future amount

to their equivalent present amount today.

This process is known as discounting which reflects a loss in value.

In fact, most financial decisions try to look into the future

with respect to money flows but since this decisions are being made in the present

we must convert the future amounts into their equivalent present values.

So let me summarize this, the value of money changes over time.

Think of it as planting seeds to produce grapes which you might use

to make and sell juice or jelly or wine or whatever you make out of this fruit.

The seeds have the potential to grow and bear fruit and

gain in value, but not all of them will.

What we can do today is to imagine the possibilities and

this involves agreeing on how to measure and calculate the possible future values.

And it's corollary of converting those future amounts into their present

values today, so let me work through an example to help you to quickly understand

compounding and future values as well as discounting and present values.

3:35

And as a practical matter, most people are going to think about providing money for

their futures and most often for their children and for people they care about.

Financial planners of course advise us about how much money we could have in

the future through wise investing so let's keep it simple.

Lets say you have $100 today which the present value.

Also assume a interest rate of 10% and

ask what is the future value of this $100 three years from now?

4:11

We will start with $100 and let me show you how that will grow.

So we have various time periods,

today, one year, two years, let's say three years.

We have an amount to work with which will grow.

These are amounts.

We're going to start of course with $100.

I'm assuming an interest rate in this case to be 10%.

And of course then I can compute the future values.

This is where I'm going with this problem.

So $100 on a 10% interest obviously gives me

interest of $10 by the end of the year right?

So I took 100 and multiplied it by 10%.

Now what's the future value?

The future value is 100 plus the interest earned which is $110.

So this time I'm going to start with $110 which will again on 10% interest 10%

of 110 now is going to be of course $11 and

the future amount 110 plus 11 gives me the 121 which again

Is my beginning amount for my last year, the third period.

Again, I'm going to earn on 121, 10% which is $12.10.

And so if I add up the future value, it works out to be 133.10.

5:45

So this is the future value of $100 that

growing at an interest rate of 10% compounded annually.

So note the interest rate of 10% also

we can refer this to as the the annual percentage rate.

APR, that is the rate that we are applying to the original investment,

and of course to any of the subsequent interests that we've earned.

In other words, we're running interest on interest.

And this is the process of compounding,

since the interests will compound over time.

What had the interest rate of 10% earned the same amount, the same interest

based on the original investment of 100, this would be known as simple interest.

So if we had earned simple interest of $10 each year, of course our future value

would be lower, it would be $130 instead of $133.10.

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So for compound interest if we had to depict this over time,

if we draw our timeline here for three years, 0, 1,

2, and 3, we can see we started with 100.

I'll use some different colors here.

So that you can clearly see the impact of compounding 100 grew to 110.

110 grew to 121 and we finally ended up with 13310, right?

What I want to show you is that in fact this process, this growth.

We can derive a very powerful and useful formula from it.

So we started with 100, the 100 grew at 10% which became 110.

The 110 of course then grew again at 10%.

This became 121.

I'll do it one more time.

121 grew again.

That became 133 10.

In other words, to calculate the future value of

13310 you must know the present value.

You must know the interest rate, and you must know the time period.

The rest is really easy, right?

But what's important in deriving the formula is what we did was

we took the present value of 100, we multiplied it by the interest

rate three times, to get our future value of 133.10.

Notice what we've done here is we've come up with a very important formula for

future value.

Which is present value, this amount here, multiplied by 1 plus the interest rate.

Which is what you see here.

And for three periods we denoted for time period t.

And that is our fundamental future value equation, okay?

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As long as we know one, two, three values,

we can compute the the fourth.

Now I also mentioned to you that it's really important

we do the reverse process, and that's what most financial decisions makers do,

they work with future amounts, and convert them into the present.

So we want the present value isolated on the left-hand side.

9:22

So once we rearrange the equation for our future value into present value,

what we're doing now, of course, is we're discounting these amounts.

And in the above example, we can of course plug the numbers in, and

we have a future amount already of 133.10.

If we divide that by the present value factor,

in this case it would be 1/1 plus the interest rate raised to the power 3.

Not surprisingly, we're going to get the result

of the original amount we started with, which is $100.

Now let's introduce a couple of variations.

First, we're going to look at how the frequency of

compounding can dramatically change values over time.

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In this example we assumed an annual percentage rate, or

APR, that corresponded to 10%.

Now 10% is per year, per annum, so it implies it has a frequency of one.

But what happens if that frequency changes?

If interest was compounded every six months, or semiannually?

The frequency would be two because there are two six month periods in a year.

If interest was compounded each day the frequency would be 365 since

there are 365 discrete periods in a year.

So if M denotes this frequency, how would this change our future value equation?

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There's a rule, whenever M is greater than one, T is going to increase and

R is going to decrease by a factor of M.

In our case, in this example, T is going to be, it was a three year problem,

12 months in each year, and that gives us 36 periods.

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However, for the interest rate, R, which was 10%,

we now want to show how this can be divided up into 12 periods.

So we must divide this by 12, which works

out to be 0.0083.

So notice all we have to do now is to reflect a different interest rate and

a different time period.

And if we do that in our example,

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we see the answer of course is going to be different.

The future value, this time,

is going to equal to the present value which is still 100.

But this time we're going to multiply this by 1+R 0.0083

raised to the power, not 3 years but 36 periods, and

that's going to work out to a higher number, about $135 if I round up.

So we do earn more interest on interest.

12:23

So this is the impact of frequency.

So I'm going to summarize.

Whenever compounding is greater than 1, meaning whenever there's M, whenever M

is greater than 1, then we're going to have to introduce this within the formula.

So if I denote that here future value equals to present value

multiplied by 1 plus adjust the interest rate by M, multiply the time period by M.

That becomes our adjusted formula for frequencies greater than once a year.

13:03

Now this helps us to understand a second important variation of trying to

understand the relationship between nominal rates and effective rates.

So here we had the annual percentage rate.

But if that rate is being compounded monthly,

what does that do effectively to the annual rate?

Now this conversion is very important because it's going to help you to

understand how compounding can make or break your financial health.

Take for example something as common as your credit cards or your retail cards.

Now these of course are backed up by agreements,

the small fine print where you know the advertised rate, but

in fact, if you read that small print, they're charging you a different rate.

14:07

And in addition to that, they will legally charge up to 30%

in terms of penalty rates if you don't pay on time.

So this appears to be really exorbitant in times when

you are earning close to 0% on your deposits in the bank.

But that disparity is another question and

we'll take that up later on in one of our activities.

But for now, in terms of calculations, let's assume your credit card has

an annual percentage rate of 18.5% compounded daily.

So let's put that information down.

We've read our credit card agreement.

It says rate is 18.5% compounded daily.

What does that mean?

Compounded daily means a frequency of 365 periods within a year.

So what does this frequency do in terms of converting this annual percentage rate?

This is an annual percentage rate we want to convert this into

an effective annual rate.

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So in our particular example here, all we have to do is plug the numbers in.

We have the effective annual rate equal to 1.

Plus the nominal rate, in this case 1 + 0.185.

That's 18.5%.

We have to now divide this by M, which is 365.

Raise it to the power m 365 and then subtract 1 to get the answer,

and it does work out to 20.32%, okay?

So what's the relationship between 20.32% and 18.5%?

So what this means is that compounding 18.5% daily

is effectively equal to paying 20.32% annually.

And you might say well, that's not a big deal.

It's not a huge difference between the two.

16:34

But in fact, this discrepancy is going to get worse and

worse, especially if you don't have access to credit,

you don't have a credit card, and you're not as well off.

So people still need money, and when they need money and

they cannot go through financial institutions to get it,

they try and access whatever source there is available.

16:59

So let's take a practical example to see how that might work and how these

nominal rates can translate into dramatically high effective rates.

Let's say it's okay to pay back $5 for using $100 for a week.

So if you're going to pay, $5 over $100 for

money you desperately need for something, so

that would be 5 over 100 and that is 5%.

You might say, hey, that's not bad.

If I can use the money for a week, I'll pay this person back $105.

What does the 5% mean, though, in terms of an APR?

The 5% of course is for a week, and

we know here m is equal to there are 52 weeks in a year.

So we must multiply this weekly rate and convert it into an annual percentage rate,

which is going to be 5 multiplied by 52 weeks.

And that works out to you're in fact paying 260% for this particular loan.

And if we use our formula here for the effective

annual rate and convert the APR into an EAR,

you're going to get an astounding 1,164%.

Wow, suddenly the 5% loan, which is in fact 1,164%,

is something that looks what you might pay a loan shark,

or at least it should be considered illegal.

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What's really hard for the borrower, of course, is wonderful for the lender.

So if you were earning this $5 interest from several clients each week and

earning an annual rate of over 1,000%, this would be too good to be true.

And for lenders and investors, the magic of compounding,

it's going to remain extremely significant even if you earn a very low rate.

As long as we have long periods of time ahead of us,

we can see the impact of compounding in the next video on time value of money.

All right, so let's come back to our basic future value and

present value equations and explore the concept of opportunity cost,

a term that's used all the time in financial decision-making.

20:03

Let's say you're working in an engineering firm negotiating a price with

the government.

Let's say it's for

an infrastructure project that is going to be ready in five years.

The price tag ranges anywhere from $10 million to $14 million, and

the government is offering you $12 million right now or,

it's giving you a choice, $15 million if you complete the project.

20:28

What are you going to choose, the $12 million now or

the $15 million in the future?

So the opportunity cost is not having the money right now and

getting a lower amount in five years.

But think of it as taking the money now, investing it, and

generating a higher value in five years.

20:48

Clearly the key here is going to be the interest rate that

converts these values over time.

So we can use our trusted equation of future value,

which equals present value into (1+r) raised to the power t.

And we have all the information in this problem to plug it in.

Future value of 15 million if we wait for five years, or

take the money now, 12 million.

And everything is going to depend on that interest rate,

because we know the time period.

So if you plug this into the financial calculator and get the value for r,

it actually works out to 4.56%.

In other words, if you think you can add more than 4.56% over the next five years,

take the 12 million now,

because its future value is going to be higher that 15 million.

If you think you'll earn less than 4.56%, wait until five years.

So this shows us by understanding and

determining interest rates, it's really key to good financial decision making.

Which is why we're going to spend a whole lecture on exploring just

how interest rates are set, why they are continuously changing, and

what impact they can have on the many decisions that we make.

So let's summarize what we've learned.

Time value is a process that explains how money is valued over time.

And to compute these values over time,

you can work with the basic future value equation and the present value formula.

So all the formulas we need are right here.

Future value is present value into (1+r) raised to the power t.

And then we're going to see a lot of applications where we have to compute

the present value, so

we just cross-multiply future value divided by (1+r) raised to the power t.

[COUGH] Now each of the formulas can be adjusted for

the frequency of compounding, as we mentioned, with this variable m.

So [COUGH] we can actually adjust these by dividing the interest rates by m,

so r would be divided by m and t would be multiplied by m, right?

[COUGH] This adjustment, of course, helps us to understand the relationship

between annual percentage rates and effective annual rates.

And here's our third formula to remember.

Effective annual rate is equal to 1 plus

the nominal annual percentage rate divided by m,

raised to the power m minus 1.

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So it's actually just a couple of formulas that are in your calculators.

You don't actually have to remember them.

All you have to do is assign values over time.

And so as you practice and as you will see in some of the upcoming video lectures,

we'll learn more about using these financial calculators

that are really going to make this a piece of cake, make this