What is the present value of the following annuity?
$4,486 every half year at the beginning of the period for the next 12 years, discounted back to the present at 9.63 percent per year, compounded semiannually.
Round the answer to two decimal places.
To calculate the present value of an annuity, we can use the formula for the present value of an ordinary annuity and adjust it for the fact that the payments are made at the beginning of each period, rather than at the end. In this case, we have an annuity of $4,486 every half year at the beginning of the period for the next 12 years, discounted back to the present at an annual interest rate of 9.63 percent, compounded semiannually.
The formula for the present value of an ordinary annuity is:
\[PV = PMT \times \frac{1 – (1 + r)^{-n}}{r}\]
Where:
– \(PV\) is the present value of the annuity.
– \(PMT\) is the periodic payment (in this case, $4,486).
– \(r\) is the periodic interest rate (in this case, semiannual rate of 9.63% divided by 2, which is 0.0963/2 = 0.04815).
– \(n\) is the total number of periods (in this case, 12 years, which is 12 * 2 = 24 semiannual periods).
Now, let’s plug these values into the formula:
\[PV = 4486 \times \frac{1 – (1 + 0.04815)^{-24}}{0.04815}\]
Calculating this equation step by step:
1. Calculate the term inside the parentheses:
\[1 + 0.04815 = 1.04815\]
2. Calculate the exponent:
\[(1.04815)^{-24} ≈ 0.543790\]
3. Subtract the result from 1:
\[1 – 0.543790 ≈ 0.456210\]
4. Now, divide by the interest rate:
\[PV ≈ 4486 \times \frac{0.456210}{0.04815}\]
5. Calculate the final present value:
\[PV ≈ 4486 \times 9.475633 ≈ $42,512.50\]
So, the present value of the annuity is approximately $42,512.50 when rounded to two decimal places.
In summary, the present value of the given annuity, which consists of $4,486 every half year at the beginning of the period for the next 12 years, discounted back to the present at an annual interest rate of 9.63 percent, compounded semiannually, is approximately $42,512.50.
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