Welcome to a lesson
on determining the regular savings amount
needed to reach a financial goal.
In this video, we'll use the value of an annuity
formula to achieve a financial goal
through a regular savings plan.
If we solve the formula here,
used to determine the value of an annuity A
for the value P,
where P is the regular deposit amount,
we would have this formula here,
where P will give us the regular savings amount
needed to reach our financial goal of A.
Let's quickly show how we can solve this equation for P.
If you multiply both sides by r over n
this would simplify out.
We'd be left with A times r over n
equals P times a quantity of one plus r over n
to the nt power, minus one.
And now to solve for P we can just
divide by this quantity here
and the right side simplifies nicely
so now we have P equals this fraction here,
which is the formula that we can use
to determine our regular savings amount
to reach our financial goal of A.
Let's go ahead and give it a try.
Let's say you want to purchase a car
in four years and you want to pay
cash for the car, and have determined
that it will cost $15,500.
If you are going to make monthly deposits
into an account that pays 6% interest compounded monthly,
what would be the amount of the monthly deposits?
And how much interest would you earn over this period?
So in this case, our monthly payment P
required to reach our financial goal of $15,500,
which is A, multiply this by r over n,
where r is our interest rate, so point zero six,
divided by n, which is the number of compounds per year,
it's monthly so n is 12,
and we'll divide this by one plus
point zero six divided by 12, all raised to the
n times t power, we just said n was 12,
t is time in years, and we're saving for four years,
so it's 12 times four minus one.
Let's go ahead and evaluate this on our calculator.
So our numerator is going to be 15,500
times point zero six divided by 12,
so there's our numerator, divided by
our denominator,
one plus point zero six divided by 12,
this will be raised to the power of
12 times four, that will be 48
minus one, and then enclose parenthesis for our denominator.
So we'll have to save $286 and approximately
52 cents per month if we want to pay cash for this car.
Now the second part asks us how much
interest would be earned over this four year period.
We'll pay this amount 12 times a year
for four years, so that value would represent
the amount paid under the account,
so to figure this out we'll take
the ending account balance, which is 15,500
but we deposited $286.52
every month, so that will be times 12
for the number of months per year
times four for the number of years.
So we would earn $1,747.04
in interest over this four year period.
Now for the second example I want to look at
the same problem, but just change
the time frame for the savings.
So the only difference on this problem here
is that you're going to save for two years
instead of four years.
So the formula will be exactly the same
except now t will be equal to two.
So we'll have 12 times two as our exponent here.
Now let's go back to our calculator
and see how much more we're going to have to save
if we only save for two years.
There's our numerator
and our exponent here is going to be
12 times two, that will be 24
and there's our denominator.
So now if we only save for two years
we have to save $609.47 per month,
which will obviously be a lot more difficult to do.
Now let's go and determine how much
interest would be earned over the
two year period compared to the four year period.
So the ending account balance is still going to be $15,500,
but our payments were only over two years now,
so we'll have the monthly savings amount
times 12 payments per year,
times two years.
So over the two year period we only earned $872.72.
So this really begins to illustrate
the power of compounded interest.
As you probably all know, cars
are not a very good investment.
Cars depreciate on average of 15% per year
according to carsdirect.com.
So really you could save quite a bit of money
by purchasing a used car instead of a brand new car.
Maybe not this car pictured here,
but you probably can save a considerable amount of money.
I hope you found this video helpful.
Thank you for watching.