Question

A couple have a 3-year-old child; they decided to make annual deposits into a savings account to fund his 4-year university education. With the first deposit being made on his fourth birthday and the last deposit being made on his 15th birthday. Then, starting on his 18th birthday, 4 withdrawals are required, starting at $x1 and increasing at a rate of 11%. If the effective annual interest rate is x2% during the whole period of time, what are the annual deposits in years 4 through 15?

X1=5400$
X2=20%​

Answer 1

To calculate the annual deposits in years 4 through 15, we need to consider the future value of the deposits and withdrawals. Using the formula for future value of an annuity, we can calculate the annual deposits based on the given withdrawal rate and effective annual interest rate. Therefore, the annual deposits in years 4 through 15 are $x1 x 13.3109.

Step-by-step explanation:

To calculate the annual deposits in years 4 through 15, we need to consider the future value of the deposits and withdrawals. We know that the withdrawals start at $x1 and increase at a rate of 11% per year, and the effective annual interest rate is x2%. Using the formula for future value of an annuity, we can calculate the annual deposits:

FV = Deposit x ((1 + x2%)^n - 1) / x2%

Where FV is the future value, Deposit is the annual deposit amount, and n is the number of years. The future value of the deposits must be equal to the future value of the withdrawals.

Let's solve for Deposit:

$x1 x ((1 + 0.11)^4 - 1) / 0.11% = Deposit x ((1 + x2%)^12 - 1) / x2%

By substituting the given values:

$x1 x 1.4641 / 0.11% = Deposit x 1.1333523 / x2%

We can simplify further:

$x1 x 13.3109 = Deposit x 1.1333523 / x2%

Therefore, the annual deposits in years 4 through 15 are $x1 x 13.3109.