1. **State the problem:** Edward wants to set up a fund that pays his family 4500 at the beginning of every month forever (a perpetuity). The fund earns 3% interest compounded semi-annually. We need to find the size of the initial investment. 2. **Identify the formula:** For a perpetuity paying at the beginning of each period (an annuity due), the present value is given by: $$PV = \frac{P}{i} \times (1 + i)$$ where $P$ is the payment per period and $i$ is the effective interest rate per period. 3. **Convert the nominal interest rate to the effective monthly rate:** - The nominal annual interest rate is 3% compounded semi-annually. - Semi-annual rate: $\frac{3\%}{2} = 1.5\% = 0.015$ per 6 months. - Effective annual rate: $$ (1 + 0.015)^2 - 1 = 1.015^2 - 1 = 0.030225 = 3.0225\% $$ - To find the effective monthly rate $i$, solve: $$ (1 + i)^{12} = 1.030225 $$ $$ i = (1.030225)^{\frac{1}{12}} - 1 $$ Calculate: $$ i = e^{\frac{\ln(1.030225)}{12}} - 1 \approx e^{0.00248} - 1 \approx 0.00248 $$ So, $i \approx 0.00248$ or 0.248% per month. 4. **Calculate the present value of the perpetuity (annuity due):** Given $P = 4500$, and $i = 0.00248$, $$ PV = \frac{4500}{0.00248} \times (1 + 0.00248) $$ $$ PV = 1814516.13 \times 1.00248 = 1818970.7 $$ 5. **Interpretation:** Edward needs to invest approximately 1818971 in the fund to pay 4500 at the beginning of every month forever at 3% interest compounded semi-annually.