To calculate the semi-annual annuity payment required to accumulate $5000 in 5 years at an interest rate of 6% per annum, we can use the Future Value of Annuity formula:
\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]
where:
- \( FV \) is the future value of the annuity (which is $5000 in this case),
- \( PMT \) is the payment made per period (this is what we are solving for),
- \( r \) is the interest rate per period, and
- \( n \) is the total number of payments.
Step 1: Identify the parameters
- Future Value (FV): $5000
- Annual Interest Rate: 6% or 0.06
- Period: semi-annually, so we calculate the rate per period and the number of periods:
- Semi-annual interest rate (r): \( 0.06 / 2 = 0.03 \)
- Total number of periods (n): \( 5 \text{ years} \times 2 = 10 \)
Step 2: Substitute values into the formula
We can rearrange the formula to solve for \( PMT \): \[ PMT = \frac{FV \cdot r}{(1 + r)^n - 1} \]
Substituting in our values: \[ PMT = \frac{5000 \cdot 0.03}{(1 + 0.03)^{10} - 1} \]
Step 3: Calculate \((1 + r)^n\)
\[ (1 + 0.03)^{10} = (1.03)^{10} \]
Calculating \((1.03)^{10}\): \[ (1.03)^{10} \approx 1.3439 \]
Step 4: Continue with the formula
Now substitute back into the formula for \( PMT \): \[ PMT = \frac{5000 \cdot 0.03}{1.3439 - 1} \] \[ PMT = \frac{150}{0.3439} \approx 435.51 \]
Conclusion
The semi-annual annuity payment required to accumulate $5000 in 5 years at an interest rate of 6% per annum is approximately $435.51.