To find the present value of an ordinary annuity, we use the formula:

$PV = P \times \frac{1 - (1 + r)^{-nt}}{r}$

Where:

• $PV$ is the present value of the annuity.
• $P$ is the payment amount per period.
• $r$ is the interest rate per period.
• $n$ is the number of periods per year.
• $t$ is the number of years.

Given:

• $P = 2000$ (payment per semiannual period)
• $r = \frac{5\%}{2} = 0.025$ (interest rate per semiannual period)
• $n = 2$ (since it is compounded semiannually)
• $t = 7$ years

The total number of periods $nt$ is:

$nt = 2 \times 7 = 14 \text{ periods}$

Now, substitute the values into the formula:

$PV = 2000 \times \frac{1 - (1 + 0.025)^{-14}}{0.025}$

Let's calculate this step by step:

1. Calculate $(1 + r)^{-nt}$:

$(1 + 0.025)^{-14} \approx (1.025)^{-14} \approx 0.69832$

2. Calculate $1 - (1.025)^{-14}$:

$1 - 0.69832 = 0.30168$

3. Calculate the fraction:

$\frac{0.30168}{0.025} \approx 12.0672$

4. Finally, multiply by the payment amount $P$:

$PV = 2000 \times 12.0672 = 24134.40$

So, the present value of the ordinary annuity is $24,134.40.

Would you like to explore further details or have any questions?

Here are 5 related questions:

1. How does the present value change if the interest rate increases?
2. What would be the present value if the annuity were compounded quarterly instead of semiannually?
3. How is the present value formula derived?
4. How does the term "ordinary annuity" differ from "annuity due"?
5. What would happen to the present value if the payments were made annually instead of semiannually?

Tip: Always double-check the number of periods and the rate per period to ensure accurate annuity calculations.