Question 5 An investment account pays interest at the rate of $$ 5 \% $$ per year, compounded semi-annually. What is the amount that should be deposited now so that R250 can be withdrawn at the end of every six months for the next 10 years? Draw an appropriate timeline showing all the given values.

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**Step 1. Determine the Interest Rate per Period and the Number of Periods**

- The annual interest rate is $$ 5\% $$. Since the account is compounded semi-annually, the interest rate per six-month period is  
  $$
  i = \frac{5\%}{2} = 2.5\% = 0.025.
  $$
- The withdrawals are made for 10 years with two periods per year. Hence, the total number of periods is  
  $$
  n = 10 \times 2 = 20.
  $$

---

**Step 2. Set Up the Present Value Formula for an Annuity**

We need to determine the present value (the amount to deposit now) that will allow for a withdrawal of $$ R250 $$ at the end of each period for $$ 20 $$ periods. The formula for the present value of an annuity (where payments start one period from now) is  
$$
PV = R \cdot \frac{1 - (1+i)^{-n}}{i},
$$
where:
- $$ R = 250 $$,
- $$ i = 0.025 $$,
- $$ n = 20 $$.

Substitute the values into the formula:
$$
PV = 250 \cdot \frac{1 - (1.025)^{-20}}{0.025}.
$$

---

**Step 3. Compute the Value**

1. Calculate $$(1.025)^{20}$$. (Using a calculator or approximation)
   $$
   (1.025)^{20} \approx 1.63862.
   $$
2. Then compute $$(1.025)^{-20}$$:
   $$
   (1.025)^{-20} \approx \frac{1}{1.63862} \approx 0.61027.
   $$
3. Now substitute into the present value formula:
   $$
   PV = 250 \cdot \frac{1 - 0.61027}{0.025} = 250 \cdot \frac{0.38973}{0.025}.
   $$
4. Compute the fraction:
   $$
   \frac{0.38973}{0.025} \approx 15.5892.
   $$
5. Finally, multiply by $$ 250 $$:
   $$
   250 \cdot 15.5892 \approx 3897.30.
   $$

Thus, the present value (the amount that should be deposited now) is approximately $$ R3897.30 $$.

---

**Step 4. Timeline Diagram**

Below is a timeline illustrating the cash flows:

$$
\begin{array}{ccccccccccccc}
\text{Time:} & 0 & \quad & 1 & \quad & 2 & \quad & \cdots & \quad & 19 & \quad & 20 \
\
\text{Amount:} & \boxed{-PV} & \quad & \boxed{+250} & \quad & \boxed{+250} & \quad & \cdots & \quad & \boxed{+250} & \quad & \boxed{+250} \
\end{array}
$$

- At time $$ 0 $$: Deposit $$ -PV $$ (an outflow, the initial deposit).
- At times $$ 1, 2, \ldots, 20 $$: Withdrawals of $$ R250 $$ at the end of each period.

---

**Final Answer**

The amount that should be deposited now is approximately  
$$
R3897.30.
$$
The amount to deposit now is approximately R3897.30.

Question 5 An investment account pays interest at the rate of \( 5 \% \) per year, compounded semi-annually. What is the amount that should be deposited now so that R250 can be withdrawn at the end of every six months for the next 10 years? Draw an appropriate timeline showing all the given values.

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