Question

For a child born in 2018 , the cost of a 4 -year college education at a public university is projected to be \(\$ 185,000 .\) Assuming a \(4.75 \%\) per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have \(\$ 185,000\) in 18 years when the child begins college?

Short answer

Approx. \$519.48 must be contributed monthly.

Step-by-step solution

Identify the variables

We need to find the monthly contribution required to save \$185,000\ for college. The rate of return is \(4.75\%\) per annum, compounded monthly, and the time period is 18 years. Let's denote the variables as follows:\( FV = 185,000 \) (future value)\( r = 4.75\% \) per annum compounded monthly\( = \frac{4.75}{100 \times 12} \) per month\( n = 18 \times 12 \) (total number of months)

Convert the annual interest rate to a monthly rate

The monthly rate \( r \) is given by: \[ r = \frac{4.75}{100 \times 12} = \frac{0.0475}{12} \approx 0.003958 \]

Calculate the total number of monthly contributions

The total number of months \( n \) is:\[ n = 18 \times 12 = 216 \]

Use the Future Value of an Ordinary Annuity Formula

The formula for the future value of an ordinary annuity is:\[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \( P \) is the monthly contribution. Rearranging to solve for \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \]

Substitute the variables into the formula

Substitute \( FV = 185,000 \), \( r = 0.003958 \), and \( n = 216 \) into the rearranged formula:\[ P = \frac{185,000 \times 0.003958}{(1 + 0.003958)^{216} - 1} \]

Compute the value

Calculate the denominator first:\[ (1 + 0.003958)^{216} \approx 2.4075 \]Now, calculate the value of \( P \):\[ P = \frac{185,000 \times 0.003958}{2.4075 - 1} \approx \frac{731.23}{1.4075} \approx 519.48 \]

Key concepts

Compound Interest

Compound interest is the interest calculated on the initial principal, which also includes all the accumulated interest from previous periods on a deposit or loan. It is more effective than simple interest because it allows money to grow at a faster rate.

The formula to calculate compound interest, considering it is compounded monthly, is: \[ A = P \times (1 + \frac{r}{n})^{n \times t} \] Here, \( A \) is the amount of money accumulated after \( n \) years, including interest. \( P \) is the principal amount (the initial amount of money). \( r \) is the annual interest rate (decimal). \( n \) is the number of times that interest is compounded per year. \( t \) is the time the money is invested for in years.

In our example, we need to convert the annual interest rate to a monthly rate and compound it accordingly over 216 months (18 years). This step is crucial for our further calculations.

Monthly Contributions

To afford a future expense such as college tuition, making regular monthly contributions is a practical strategy. These contributions, when combined with the power of compound interest, can grow into a substantial amount over a planned period.

The notion of setting aside money monthly towards saving goals highlights the principle of consistency and discipline in financial planning. By breaking down a large future sum into manageable smaller payments, saving for significant expenses becomes feasible.

Consistent monthly contributions are particularly effective when combined with a favorable interest rate, as each contribution not only helps accumulate the desired amount but also earns interest over time.

Future Value Calculation

The future value (FV) represents what a current investment will grow to over a specified period, at a particular interest rate and compounding frequency. This value helps to understand how much you'd have in the future, based on today's contributions and periodic interest accruals.

In our context, the future value is the projected cost of a 4-year college education for a child starting college 18 years from now, calculated to be \$185,000. It's computed using the future value of an ordinary annuity formula for regular monthly contributions.

To calculate FV, taking into account compound interest, the following formula is used:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Here, \( P \) is the monthly contribution, \( r \) is the monthly interest rate, and \( n \) is the total number of payments (in months). By using this formula, we can compute how much needs to be set aside each month to achieve the future value goal.

Annuity Formula

The annuity formula is essential for calculating the future value of multiple periodic payments. An ordinary annuity involves payments made at the end of each period. For our calculation, it helps determine the monthly contribution needed to reach a future amount when compounded monthly.

The formula for the future value of an ordinary annuity is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Rearranging it to solve for the monthly payment \( P \):
\[ P = \frac{FV \times r}{(1 + r)^n - 1} \]
In our example, \( P \) is the monthly contribution, \( FV \) is the future value of \$185,000, \( r \) is the monthly interest rate (0.003958), and \( n \) is the total number of payments (216 months). Plugging these values into the formula allows us to find that the required monthly contribution is approximately \$519.48.