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..