Solving Compound Interest Problems PLEASE READ DESCRIPTION:) - YouTube

HELLO, Mr. Tarrou

Now, if you are watching many of my lessons you would know if any day I should be wearing a money shirt

it would be a day when I am talking about compound money growth or compound interest.

Or money questions

I made another video with a $100 bill t shirt on...anyway.

This used to be one of my favorite topics until the economy went to crap and my 401k went down.

But I am sure that will not last forever and this video will last for who knows how long.

Maybe today stocks are really high up, I don't know:)

So compound growth and sometimes with the economy it feels like decay,

but compound growth formulas

as far as money is concerned, the two that we are going to work with today

is the final amount of our investment is equal to

P (which is equal to Principa: the starting value)

times 1 plus r (that is the interest rate)

We almost always work with interest rates, or percents, in terms of decimals.

So 5 percent would be .05

N is the number of compounding periods.

In the examples I am going to take a look at today are going to be compounded monthly

like most bank accounts are.

So, n will equal to 12.

T is the number of years.

Then if you have continuous compounding which is

always in textbooks as money questions

ahhh...not many money question are going to be compound continuously every single second.

That is more for like bacteria growth and in nature where things are constantly multiplying on themselves, and growth & decay is not set by patterns like per day or per month.

Ok, so lets take a look at a couple of examples.

Lets say that you have got 10,000 dollars

and that is going to get compounded monthly.

We are using monthly examples because like savings accounts give you your interest every month.

It gets deposited in your account, and unless you pull it out, it starts to grow on that slowly growing base.

Therefore it compounds monthly because your interest comes back monthly and grows on itself.

This money is going to earn 7% interest though banks are not paying anywhere near that much at this time.

Maybe your money is in a stock investment

and we are looking over a long run of 20 years.

We are going to attempt to make and average of 7% on our money over a 20 year period.

So...monthly, 7 percent, for 20 years, how much is your money going to be worth?

We are going to use this formula right here.

Which is the final amount of the investment

is going to be equal to P, your initial amount of $10,000

times 1 plus your interest rate of 7% written as a decimal of .07

divided by the number of compounded periods in a 1 year period which is monthly so this is 12.

Raised to the N...12...times t power which in this example will be 20.

Now this one is a pretty straight forward question.

We are looking for a final amount and as long as you have a decent scientific calculator

you will be able to solve this without the use of logarithms

or any kind of fancy roots.

Let's see what happens. This is 10,000.

Now I can't do this math in my head so I am going to cheat.

.07 divided by 12 plus 1 comes out to be 1.00583

Yes, I have worked this out ahead of time. Usually I just do this on the fly.

Of course sometimes I will make little sign errors then and you see voice bubbles over my problems.

12 times 20 is going to be 240.

This is why you need a scientific calculator, so you can raise this to the 240th power.

You don't want to hit times this value 240 times!

So 1.00583 raised to the 240 power. Remember you need to do your exponents before you multiply.

We get 10,000 times this raised to the 240th power which is 4.036

Then you multiply those two together and you find out if you can maintain that 7% average over a 20 year span, you are going to have $40,387.39

..over that 20 year period.

Just by not spending this $10,000

and letting it sit over a 20 year period,

that will gain you an addition $30,000 of interest.

You don't even have to work for it, all you need to do is not spend this money.

As make this recording the economy has been very bad.

So, 7% seems like maybe an unrealistic value

but hopefully over 20 years that will not be too hard to achieve.

Even three or four years ago a bank would have a CD that would pay 5 percent.

Now it seems like they will not give you anything.

But that is today's economy and it will not stay that way forever.

Ok, lets take a look at another example.

Lets take a look at the fact

that we want 15,000 to become 25,000

and we have ten years

in which we need that to happen.

If we have this and we want it to be $25,000

and we have 10 years to allow that money to grow,

what interest rate to we need to make?

What is our goal? What interest rate do we need to achieve to have this $10,000 of interest over 10 years?

Again we are going to compound that monthly.

This means we are once again going to use this equation like in our first example.

So, 25,000 equals 15,000 times one plus our interest rate over the number of compounding periods per year

raised to the n*t power or 12 times 10

We do not have a variable in the exponent so again we are not going to need logarithms

we just need a scientific calculator.

The first thing we are going to is divide both sides by 15,000.

We are isolating the variable base with the high exponent.

And 25,000 divided by 15,000 is going to be

1.67, that is rounded off. It is actually 1.6 repeating

Equals 1 plus r over 12 raise to the 120th power.

Now we are not going to be able to solve for r unless we can get rid of this 120th power.

This involves raising both sides of the equation by

1 over 120 power, or effectively taking the 120th root of both sides of the equation.

So, again a scientific calculator is going to be needed to complete this problem.

If you are watching this lesson hopefully you have a graphing or scientific calculator.

Raising both sides by 1 over 120

sets up a power to power property and these multiply together so this becomes 1.

1.67 raised to the 1 over 120 power gives us 1.0043 equals 1 plus r over 12.

We are going to subtract both sides by 1.

This gives us .0043 equals r over 12.

Multiply both sides by 12.

We get an interest rate of approximately .0516

or 5.16 percent.

With this interest rate we can have $15,000 become $25,000 in a ten year span.

AWESOME! BAM!!! Moving on to the next problem.

The next example is going to be continuous compound interest. I am going to do the continuous compounding first but I am going to try and to this example twice.

Lets say that we want our money to double.

We have continuous compounding.

Continous compounding means we are going to use the PERT formula.

We are going to attempt to make 5 percent on our money

and the question is how long is that going to take.

If you can do continuous compounding which is a little unrealistic because most banks will only do monthly compounding

or at the most daily compounding for investments.

But, at any rate...

We want our money to double, continuous compounding, 5 percent. How long is that going to take?

Well, it does not matter if you are talking about a dollar becoming two dollars,

$50,000 becoming $100,000, or whatever.

Doubling is doubling.

So why don't we pick an easy value.

Again, because of the continuous compounding we are going to use the PERT formula.

Doubling is doubling is doubling. $50,000 becoming $100,000

would make me a lot happier if I had $50,000 to begin with

but it will take the same period of time as making $1 into $2.

So we want whatever our initial investment is to double.

See, no dollar amounts were mention because all we need is an initial value and a final value which is double what you started with.

So, 1 and 2...why not?!

e which is approximately 2.718

Interest rate is 5% so .05

and the time period is what we are looking for.

Now when your variable is in the exponent you are probably going to need logarithms to solve the equation.

Since this is base e, we are going to use Natural Logs to save a step.

We do not need to divide by 1 because that is not going to do anything.

2 equals e to the .05t power.

To get rid of that base e we are going to apply the natural log to both sides.

The natural log of two equals

the natural log of e to the .05t power.

That natural log of e is going to cancel and the exponent of .05t is just going to fall down.

If the natural log did not have the same base of e, which of course it does,

but if they did not have the same base you could pull that exponent down in front of the logarithm.

Use that Power Property of Logarithms to bring the variable down out of the exponent and thus allow us to solve the equation.

Here however, this will cancel out.

We get the natural log of 2 equals .05t

Divide both sides by .05t, excuse me, .05

The time period required is going to be 13.8 years.

Now continuous compounding is going to make you money a little bit faster than monthly compounding.

I am going to attempt to do another example and we are going to end up with the same decimal answer due to round off error, if I have time.

I actually am not going to have time. I am going to just write as much as I can.

If we did this monthly...lets see if I can do this in two minutes.

Don't need to worry about the 1.

The variable is in the exponent so we are going to take the natural log of both sides. (Common log would work too)

Using the Power Rule this becomes the natural log of 2 equals 12t times ln(1.0092)

Natural log of 2 divided by 12 times natural log of 1.0042 equals t.

If you get this entered into your calculator properly

with a little bit of round off error you will get approximately 13.8.

BAM!!! That is compound interest. Go do your homework.

Thanks for watching and allowing me to help you learn some math.