Question

Mr. Happy won the jackpot at the Bloemfontein casino. He decides to invest the money at 13% per annum compounded half yearly for the following 10 years. This investment must provide him with an income of R12000 every 6 months. The payment of the income takes place in advance. What is the original amount Mr. Happy won?

Answer 1

To find the original amount that Mr. Happy won, we need to calculate the present value of an annuity due, since the income payment of R12,000 occurs at the beginning of each period (every 6 months).

Given:
- Payment (PMT) = R12,000
- Interest rate (i) = 13% per annum compounded half-yearly = 13% / 2 = 6.5% = 0.065
- Number of payments (n) = 10 years × 2 = 20 (since payments are made every 6 months)

Since this is an annuity due, we will first calculate the present value of an ordinary annuity and then adjust for it being an annuity due.

The present value \( PV \) of an ordinary annuity can be calculated using the formula:

\[
PV = PMT \times \frac{(1 - (1 + i)^{-n})}{i}
\]

Calculating \( PV \) for an ordinary annuity first:

1. Plug in the values:
- \( PMT = 12000 \)
- \( i = 0.065 \)
- \( n = 20 \)

\[
PV_{\text{ordinary}} = 12000 \times \frac{(1 - (1 + 0.065)^{-20})}{0.065}
\]

2. Let's calculate \( (1 + 0.065)^{-20} \):

\[
(1 + 0.065)^{-20} \approx (1.065)^{-20} \approx 0.2277
\]

3. Now we can calculate:

\[
PV_{\text{ordinary}} = 12000 \times \frac{(1 - 0.2277)}{0.065}
\]
\[
PV_{\text{ordinary}} = 12000 \times \frac{0.7723}{0.065}
\]
\[
PV_{\text{ordinary}} \approx 12000 \times 11.89 \approx 142,674.24
\]

4. Since this is an ordinary annuity, now we adjust it to find the present value of an annuity due:

\[
PV_{\text{due}} = PV_{\text{ordinary}} \times (1 + i)
\]
\[
PV_{\text{due}} = 142,674.24 \times (1 + 0.065)
\]
\[
PV_{\text{due}} = 142,674.24 \times 1.065 \approx 151,485.72
\]

Thus, the original amount Mr. Happy won is approximately **R151,485.72**.