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 What is the amount that should be deposited now so that R250 can be withdrawn at the end of every six months for the next 10 years?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. [5] TOTAL MARKS: 25 TOT

We are given that the account earns \( 5\% \) per year compounded semi‑annually. This means that the interest rate per six‐month period is \[ i = \frac{5\%}{2} = 2.5\% = 0.025. \] There will be a withdrawal of R250 at the end of every six‑month period for 10 years, i.e., a total of \[ n = 10 \times 2 = 20 \text{ withdrawals}. \] The amount that must be deposited now is the present value (at time 0) of an annuity with periodic payment \( \text{PMT} = 250 \), interest rate per period \( i = 0.025 \), and \( n = 20 \) periods. The present value \( PV \) of an annuity immediate is given by: \[ PV = \text{PMT} \times \frac{1 - (1+i)^{-n}}{i}. \] Substitute the given values: \[ PV = 250 \times \frac{1 - (1.025)^{-20}}{0.025}. \] ### Step 1. Compute \((1.025)^{-20}\) First, compute \( (1.025)^{20} \). (For a rough calculation, you might use logarithms or a calculator.) Assuming \[ (1.025)^{20} \approx 1.63862, \] then \[ (1.025)^{-20} \approx \frac{1}{1.63862} \approx 0.60997. \] ### Step 2. Substitute back into the formula: \[ PV = 250 \times \frac{1 - 0.60997}{0.025} = 250 \times \frac{0.39003}{0.025}. \] ### Step 3. Simplify the fraction: \[ \frac{0.39003}{0.025} \approx 15.6012. \] ### Step 4. Multiply by 250: \[ PV \approx 250 \times 15.6012 \approx 3900.3. \] Thus, the amount that should be deposited now is approximately **R3900**. --- ### Timeline Below is the timeline corresponding to the deposits and withdrawals: ``` Time (Years): 0 0.5 1.0 ... 9.5 10.0 | | | | | Event: Initial Deposit -> R250 Withdrawals at end of each 6-month period ``` - At time \( t = 0 \): Deposit approximately R3900. - At time \( t = 0.5,\,1.0,\,1.5,\,\ldots,\,10.0 \): Withdraw R250 each period (total of 20 withdrawals). --- Thus, the required deposit amount now is approximately R3900.