Lin Lowe plans to deposit $1,800 at the end of every 6 months for the next 15 years at 8% interest compounded semiannually. What is the value of Lin's annuity at the end of 15 years? (p. 317)
To find the value of Lin's annuity at the end of 15 years, we can use the formula for the future value of an ordinary annuity:
\(FV = P \times \left( \left(1 + \frac{r}{n} \right)^{nt} - 1 \right) \div \left(\frac{r}{n} \right)\)
Where:
FV = Future Value of the annuity
P = Payment amount
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case, Lin deposits $1,800 at the end of every 6 months, so the payment amount (P) is $1,800. The annual interest rate (r) is 8%, which can be expressed as 0.08 as a decimal. Since the interest is compounded semiannually, the number of compounding periods per year (n) is 2. And the duration of the annuity (t) is 15 years.
Plugging these values into the formula, we get:
\(FV = 1800 \times \left( \left(1 + \frac{0.08}{2} \right)^{(2 \times 15)} - 1 \right) \div \left(\frac{0.08}{2} \right)\)
Now, let's simplify the equation and calculate the value:
\(FV = 1800 \times \left(1.04^{30} - 1 \right) \div 0.04\)
Using a calculator, we find that \(1.04^{30} \approx 5.187379\).
\(FV = 1800 \times (5.187379 - 1) \div 0.04\)
Simplifying further:
\(FV = 1800 \times 4.187379 \div 0.04\)
\(FV \approx 187,938.48\)
Therefore, the value of Lin's annuity at the end of 15 years is approximately $187,938.48.