Betty and Bob buy a membership on the WORLDEX, costing 
$650,000, by making payments to the bank at the end of June and at the end of December for 11 years. interest rates are 12% per annum compounded semiannually. What should these equal payments be? Your final answer should be correct to 2 places after the decimal point.


To determine the equal payments Betty and Bob should make at the end of each semiannual period, we need to apply the formula for the present value of an ordinary annuity, since they are making payments every six months (semiannually).

The formula for the present value of an ordinary annuity is:

\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ PV $ is the present value (cost of the membership), which is $650,000.
- $ P $ is the semiannual payment (what we are solving for).
- $ r $ is the semiannual interest rate.
- $ n $ is the total number of periods.

### Step 1: Determine the semiannual interest rate and the number of periods.
Since the annual interest rate is 12%, compounded semiannually, the semiannual interest rate is:

\[
r = \frac{12\%}{2} = 6\% = 0.06
\]

They are making payments for 11 years, and there are 2 payments per year. Thus, the total number of periods is:

\[
n = 11 \times 2 = 22 \, \text{semiannual periods}.
\]

### Step 2: Rearrange the formula to solve for $ P $ (the payment):
\[
P = \frac{PV \times r}{1 - (1 + r)^{-n}}
\]

Substitute the known values:

\[
P = \frac{650,000 \times 0.06}{1 - (1 + 0.06)^{-22}}
\]

Now, let's calculate this.The equal semiannual payments Betty and Bob should make are approximately **$53,979.62**.

Would you like more details on the calculation, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would the payment change if the interest rate were 10% instead of 12%?
2. What would the total amount paid over the 11 years be?
3. How much of the total payment goes towards interest?
4. What if the payments were made annually instead of semiannually?
5. How would the payment be affected if the membership cost increased to $750,000?

**Tip:** Always break down financial problems into small steps, identifying key parameters like interest rate and compounding periods for clarity.

Betty and Bob buy a membership on the WORLDEX, costing $650,000, by making payments to the bank at the end of June and at the end of December for 11 years. Interest rates are 12% per annum compounded semiannually. What should these equal payments be?

Learn how to calculate equal semiannual payments for a $650,000 WORLDEX membership using the Present Value of an Ordinary Annuity formula. This problem involves calculating payments over 11 years with 12% annual interest compounded semiannually. Suitable for undergraduate students in finance or economics.

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