A couple is saving for their child's college. They decided they would like to have $50,000 in 18 years. If they can earn 4.5% compounded semiannually, how much should they deposit at the beginning of each period?
To calculate how much they should deposit at the beginning of each period, we can use the future value of an ordinary annuity formula.
The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = payment per period
r = interest rate per period
n = number of periods
In this case, we want to find the payment per period. We know that the future value (FV) is $50,000, the interest rate (r) is 4.5% compounded semiannually, and the number of periods (n) is 18 years.
First, we need to convert the interest rate from annual compounding to semiannual compounding. We can use the formula:
r = (1 + (annual interest rate / m)) ^ m - 1
Where:
r = interest rate per period
m = number of compounding periods per year
For semiannual compounding, m = 2 (since there are two periods per year).
r = (1 + (0.045 / 2))^2 - 1
r = (1 + 0.0225)^2 - 1
r = (1.0225)^2 - 1
r = 1.045625 - 1
r = 0.045625
Now we can substitute the values into the formula for the future value of an ordinary annuity:
50000 = P * ((1 + 0.045625)^18 - 1) / 0.045625
To find P, we can rearrange the formula:
P = 50000 * 0.045625 / ((1 + 0.045625)^18 - 1)
Using a calculator, we can solve for P:
P = 50000 * 0.045625 / 1.967913
P ≈ 1145.62
Therefore, the couple should deposit approximately $1,145.62 at the beginning of each period to reach their goal of $50,000 in 18 years, with an interest rate of 4.5% compounded semiannually.
To determine how much the couple should deposit at the beginning of each period, we can use the formula for the future value of an investment with compound interest:
\[FV = P \times \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
FV = Future Value (desired amount, $50,000)
P = Principal (amount to be deposited at the beginning of each period)
r = Annual interest rate (4.5%)
n = Number of times interest is compounded per year (semiannually, so n = 2)
t = Number of years (18)
We can rearrange the formula to solve for P:
\[P = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}\]
Plugging in the values:
\[P = \frac{50000}{\left(1 + \frac{0.045}{2}\right)^{(2 \times 18)}}\]
Now we can calculate this using a calculator or a spreadsheet.