To find the annuity payment at the end of every 6 months, we can use the formula for the future value of an ordinary annuity, which is given by:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \(FV\) is the future value of the annuity (in this case, $2000),
- \(P\) is the annuity payment per period,
- \(r\) is the interest rate per period,
- \(n\) is the total number of payments.
Given data:
- Future Value \(FV = 2000\)
- Annual interest rate = 6%, so the semi-annual interest rate \(r = \frac{6%}{2} = 3% = 0.03\)
- Total period = 4 years, so the number of payment periods \(n = 4 \times 2 = 8\) (since payments are made every 6 months).
Plugging in the values into the formula:
\[ 2000 = P \times \frac{(1 + 0.03)^8 - 1}{0.03} \]
Now we calculate \((1 + 0.03)^8\):
\[ (1 + 0.03)^8 \approx 1.26677 \]
Then:
\[ (1 + 0.03)^8 - 1 \approx 1.26677 - 1 = 0.26677 \]
Now substitute into the equation:
\[ 2000 = P \times \frac{0.26677}{0.03} \]
Calculating \(\frac{0.26677}{0.03}\):
\[ \frac{0.26677}{0.03} \approx 8.89233 \]
Now, substituting back into the equation:
\[ 2000 = P \times 8.89233 \]
To find \(P\):
\[ P = \frac{2000}{8.89233} \approx 224.53 \]
Conclusion
The annuity payment at the end of every 6 months for 4 years, which amounts to $2000 at an interest rate of 6% compounded semi-annually, is approximately $224.53.