Suppose you want to have $500,000
for retirement in 30 years.
Your retirement account earns 4% interest
compounded monthly.
A, how much would you need to deposit
in the account each month and b, how much
interest would you earn over the 30 years?
To answer this question we will use the annuity
formula shown here below where A is the account
balance after two years, PMT is the regular
deposit amount, r is the annual interest rate
as a decimal, n is the number of compounds
per year, and t is the time in years.
Because you want $500,000 in the account
after 30 years, A is $500,000.
This must be equal to the regular payment or PMT.
And then we have times, and then in parentheses
we have one plus r divided by n raised
to the power of nt and then minus one.
So we have one plus r is equal to 4%
which as a decimal is 0.04, which is divided by n.
Because the interest is compounded monthly
and there are 12 months a year, n is 12.
And this is raised to the power of n times two
which is 12 times, t is the time in years
and therefore t is 30.
So the exponent is 12 times 30.
And then we have minus one, and then close parentheses.
All of this is divided by r divided by n
which is 0.04 divided by 12.
And now, to solve the equation for PMT
we will now evaluate this quotient on the calculator.
We have open parentheses, open parentheses,
and then one plus 0.04 divided by 12,
close parentheses, raise to the power
of 12 times 30, which is 360, right arrow,
minus one, close parentheses.
So there's the numerator.
And then we have divided by, in parentheses,
0.04 divided by 12.
And enter.
So on the right side of the equation
we would have PMT times this value here.
Let's go ahead and write that down.
The right side is going to be PMT times 694.0494044.
And now to solve for PMT, we divide both sides
of the equation by 694.0494044.
Notice on the right side of the equation
this quotient simplifies to one, giving us
PMT times one, which is PMT.
So we have PMT, which is the monthly payment,
is equal to the quotient on the left
which we will now find and round to the nearest cent.
So going back to the calculator
we have 500,000 divided by 694.0494044, enter.
To the nearest penny, we have $720.41.
So now we know you would need to deposit $720.41
each month to reach the financial goal.
And now for the second question, we're asked
to determine how much interest you would earn
over the 30 years.
This would be equal to, this would be equal
to the account balance of $500,000 and then minus
all the monthly deposits you put into the account.
So the interest earned is equal to 500,000
minus the number of monthly payments of $720.41.
So that would be 720.41 times the number
of months in 30 years, which is 30 times 12
which gives us 500,000 minus 720.41 times 360.
And now let's go back to the calculator.
We have 500,000 minus the monthly payment
of 720.41 times the number of payments
which is 360, which gives us the interest of 240,652.40.
So for part b, the interest earned would be $240,652.40.
I hope you found this helpful.