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..