Welcome, ladies and gentlemen.
So what I'd like to do is show you
how to determine the final value in compound interest.
And one of the common formulas that we
have for compound interest is final value, which is FV,
is equal to the present value, PV,
times one plus r, divided by n, raised to the n times t.
And I wrote up what everything represents.
FV represents future value, which
is your final amount which we're going to find,
we'll determine in this video.
PV represents your present value which
is like your initial amount.
R represents your interest rate, but make sure
that it's in decimal form.
T represents the time.
How long you're going to invest.
N represents the number of times you're
going to compound per year.
Now I wrote FV and PV because Present Value, Future Value,
that kind of makes sense.
But a lot of times will also just shorten it to F
equals P times one plus r over n raised to the n times t.
And the reason why it's probably sometimes preferred to just
to use F and P is because we want
you to understand that future value is just one value.
FV is not two different variables.
FV represents the same for one variable.
So when I rewrite these down, I'm just
going to use in the F and the P rather than the FV and PV
because I don't want you to think that F and V
and P and V are different, two variables.
They represent one variable.
All right.
So basically, in each of these problems
are always determining the future value.
So basically what we're just going to do
is just plug-in the information and then use a calculator
to determine it.
Now, you can plug-in the information all at once
and calculate it, but usually a lot of times that's
where students may mistakes.
So what I like to do is just kind of compute things
one after the other, but I'll also
show you how to use parentheses to make
sure you're doing everything correctly,
if you are going to plug-in each in your calculator.
All right, but first of all we're finding future value.
So we don't know that, so I'm going to leave that as F,
is equal to my present value says is 2,500
and that's going to be times one plus my interest rate which
is 8%.
So that's going to be 0.08, right?
You've got to make sure you put it into decimal form.
So to decimal form is just be dividing that by 100.
So 0.08 divided by n as how many times
we're going to compound a year.
In this case, we have monthly, so I'm
going to divide that by 12.
And then it's going to be 12 times
the number of years, which in this case is 10.
Now, obviously, we can simplify that to 10 times 12,
you know is 120, but I'm just entering everything
in at one time.
Also, another thing is I wrote out the problems.
I wrote out what each and everything is,
but you'll also probably commonly find,
you'll see problems like this as word problems.
So, if you had a word problem, the main important thing
is just to list out all the information you're given
and then plug it in just like we're doing.
Now, just like following the order of operations,
what we want to do is when we're plugging something
in our calculator and make sure we're doing everything correct.
The number one mistake students usually
make a mistake is not plugging in everything correctly.
They either make [INAUDIBLE] as monthly as far
as their n and not our t.
They usually forget to convert your percentages
to decimal form and that's usually the main kind
of issues students have.
So once we've got everything down, written now correctly,
we want to make sure that we're entering everything
in correctly.
So I'm going to do this one kind of step-by-step, and then
the other ones I kind of do a little bit quicker.
So following the order of operations,
we want to do everything inside the parentheses first, right?
PEMDAS.
So we do everything inside the parentheses first.
So inside the parentheses I have one plus 0.08 divided by 12.
Well, inside the parentheses, I again
have to follow order of operations.
So there's no further parentheses.
There's no exponents inside the parentheses,
but I have to do multiplication and division before I
do addition and subtraction.
So I need to do 0.08 divided by 12.
So I'm just going to take my calculator and dew 0.08 divided
by 12 and that gives me 0.00667.
Now it's very important when using a calculator
to not round your answers.
And I have a problem with this because I can't
write an irrational number.
That 0.66 is going on repeating.
So I'm going to keep that answer, though,
inside my calculator.
Now I'm going to use the second answer
key to help me reuse that.
Because, if you round before the end of the answer,
you're going to have rounding errors at the end
and you're not going to get the exact answer we're looking for.
Then I'm going to take this value
and add it to one, which fortunately for me that's
pretty easy, that's going to be 1.0066 repeating, raised
to the 12 times 10 is 120.
OK?
Now, again, another big mistake for students.
They don't follow the order of operations
They want to multiply these two numbers before raising 1.066
repeating to the 120th power.
We have to do that first because PEMDAS, parentheses, which
we've already completed, exponents, then multiplication
and division.
So I got to add one to it.
So I got to take 1.066 repeating which I'm just
leaving in my calculator.
I'm just going to use the raise that answer to the 120th power
and I get 2.219640235.
And obviously my calculator is only
going to one, two, three, four, five, six, seven, eight,
nine decimal places, which I think is pretty good.
But, again, I'm not going to round an answer.
I'm going to keep it in my calculator.
Even though nine decimal places is probably not
going to affect my answer because I'm only
going to round it to the hundredth decimal place.
It's probably not going to affect it.
It's better safe than sorry.
Obviously, this could be a longer answer.
It just depends on how many digits
your calculator is going to approximate to.
But then I'll take that answer and then
I multiply that by 2,500.
And my final answer, again, since we're dealing with money,
I'm going round to the nearest 100th, 549.10.
So therefore, if you have an initial investment $2,500,
earning the interest rate of 8%, and you
invest that money for 10 years, compounded monthly,
at the end of 10 years you will now have $5,549.10.
Not a bad return on your investment, right?
For not doing anything.
Compound interest is doing all your work.
Let's go and do the next one.
So I have F equals my present value is now 5,000.
I have a lower interest rate, 0.06.
Did I not do any of these?
No, I kept these simple.
OK.
Point 0 6 divided by compounds per yearly.
Yearly, is just one times one times 30.
Now you realize that 0.06 divided by one,
we don't really need the one, right?
So that actually brings us to another formula.
It's when it's only compounded yearly.
We don't need to use this format.
We can simply just use F equals P times one plus r raised
to the T. Because what happens here
is you can see that one is not really-- right?
We're not multiplying.
30 times one is just 30.
And 0.6 divided by one is just 0.06.
So that's a little bit of a shortcut.
You don't need to really plug-in those ones.
You can figure that out.
All right.
So now I'll just do-- what I'm going
to do is one plus 0.06 is 1.06.
And then I'll take 1.06, raise it to the 30th power,
and, again, my calculator is approximating
to the ninth decimal place, which
is pretty good for accuracy.
But, again, do not round it.
Do not write in 5.7 and then multiply that.
Because I'll actually show you, times 5,000.
Then I'll take that answer, the whole answer
that my calculator gave me and multiply that by 5,000.
And what I'll get is 28,00-- oh-- 1.06, oh for 30 years,
yeah, 20,707.46 as I round.
So, if you invest $5,000 for 30 years, you'll have $28,000.
Now, just a minute.
I want to show you something.
If you rounded that, let's say you did,
instead of that you did 5000, and you rounded it to 5.7,
right?
You didn't use the whole decimal place in your calculator.
Let's see what happens there.
5,000 times 5.7.
You now get 28,500.
Which, yeah, is close, right?
I mean it's both 28,000s, but you're missing out that $217
extra and 46 cents extra, right?
So if it was a multiple choice, you
might be able to guesstimate what
is going to be the correct answer,
but you might not be able to.
And I would mark this wrong as well because you're not
using the exact value.
So you've got to be very careful.
Make sure you do not approximate to the end.
All right.
So for the next one, I'm going to show you
how to type it into your calculator.
So again, we still want to write it out.
Final value, we don't know.
Present value is 1,000 times one plus 0.03 divided by daily.
So how many days do we have in a year?
365, or at least that's what we're going to use, 365 times
five years.
OK.
So, if I was going to type this in my calculator,
this is how I would type it out.
And the main important thing is we
want to make sure we are using parentheses correctly.
So you are going to type in your calculator just like this.
You don't need to write 1,000.
So I will do 1,000 times one, plus parentheses 0.03 divided
by 365 parentheses, and double parentheses
raised caret parentheses, 365 times parentheses five.
OK.
So that's how you'd want to type in your calculator.
The main important thing is putting those parentheses.
So let's go and try that and see what are value is.
So I'll do 1,000 times one parenthesis, one
plus parentheses, 0.03 divided by 365,
end parentheses, end parentheses, caret parentheses,
365 end parentheses, parentheses five.
Now I'm sure probably the calculator you don't need
to use maybe the parentheses of 365 times five,
but I just want to make sure that everything they know
exactly what's-- actually, I'm going to do 365 times, yeah,
365 times five.
OK?
And then I just go ahead and hit enter and I get 1,161.83.
OK?
So, therefore, if you invest $1,000
at a 3% interest for five years daily,
you're going to have a final value of $1,161.83.
I'm just going to do this individually, 365,
just to make sure I did this correctly
because I just want to make sure that I typed it in correctly.
[SIGH]
Hmm.
That's interesting.
OK.
Oh, I forgot to-- plus one raised to.
OK.
Perfect.
All right.
Now the last one, you can see that we have continuously.
Now what is continuously?
We talked about-- we dealt with monthly was 12 times a year.
Yearly was one time a year.
Daily is 365.
What is continuously?
Well, that's where our e is going to come into play
and the video is not going to be explaining e.
Yeah, but we have a different formula
for continuous interest.
And the continuous interest formula
is final value equals present value times e, which, again,
is an irrational number, 2.71 approximately,
raise to the r times t.
So I have final value, which I don't know.
My present value is 7000, times e raised to the r,
which is 0.05 times t which is 20.
Now for this, there is an e button in your calculator,
as well as an e raised to a power
that you're simply going to use.
Now, you can do these all together,
but I like just to plug this all in together.
So I'll just type in 7,000 times my e raised
to caret which is my second LN and then I'm just going to type
in 0.05 times 20.
And when I hit enter, I get $19,027.97.
So there you go, ladies and gentlemen.
That is how you determine the final value
for compound interest.
Thanks.