1. **Problem Statement:** Leah wants to set up a fund that pays $1500 every month forever (a perpetuity). The fund earns 3.00% interest compounded monthly. We need to find: a. The amount of money required to set up this fund. b. How much less money is required compared to if the interest was compounded semi-annually. 2. **Formula for Perpetuity with Monthly Compounding:** The present value $PV$ of a perpetuity paying $P$ per period with interest rate $i$ per period is: $$PV = \frac{P}{i}$$ Here, $P = 1500$ (monthly payment), and $i$ is the monthly interest rate. 3. **Calculate monthly interest rate for 3.00% annual rate compounded monthly:** $$i = \frac{0.03}{12} = 0.0025$$ 4. **Calculate the present value for monthly compounding:** $$PV = \frac{1500}{0.0025} = 600000$$ 5. **Calculate equivalent monthly interest rate for 3.00% compounded semi-annually:** Annual nominal rate = 3.00% compounded semi-annually means 2 compounding periods per year. Semi-annual rate $r_{sa} = \frac{0.03}{2} = 0.015$ Equivalent monthly rate $i_{m} = (1 + r_{sa})^{\frac{1}{6}} - 1 = (1.015)^{\frac{1}{6}} - 1$ Calculate: $$i_{m} = e^{\frac{\ln(1.015)}{6}} - 1 \approx e^{0.002481} - 1 \approx 0.002484$$ 6. **Calculate present value for semi-annual compounding using equivalent monthly rate:** $$PV_{sa} = \frac{1500}{0.002484} \approx 603982.68$$ 7. **Calculate how much less money is required with monthly compounding:** $$\text{Difference} = PV_{sa} - PV = 603982.68 - 600000 = 3982.68$$ **Final answers:** - a. The amount required to set up the fund with monthly compounding is **600000.00**. - b. The amount less required compared to semi-annual compounding is **3982.68**.