1. **Problem Statement:** Jess wants to set up a fund that pays $4800 every month forever (a perpetuity). The fund earns 5.70% interest compounded monthly. We need to find: A) How much money is required to set up this fund with monthly compounding. B) How much less money is required if the fund earns 5.70% compounded semiannually instead. 2. **Formula for Perpetuity Present Value:** The present value $PV$ of a perpetuity paying amount $P$ per period with interest rate per period $i$ is: $$PV = \frac{P}{i}$$ 3. **Step A: Monthly Compounding** - Annual nominal interest rate $r = 0.057$ (5.70%) - Monthly interest rate $i = \frac{r}{12} = \frac{0.057}{12} = 0.00475$ - Monthly payment $P = 4800$ Calculate present value: $$PV = \frac{4800}{0.00475}$$ $$PV = 1010526.32$$ So, the fund requires approximately 1010526.32 to set up with monthly compounding. 4. **Step B: Semiannual Compounding** - Semiannual interest rate $i_{semi} = \frac{r}{2} = \frac{0.057}{2} = 0.0285$ - We need to find the equivalent monthly interest rate for semiannual compounding to compare fairly. - Effective annual rate (EAR) for semiannual compounding: $$EAR = \left(1 + 0.0285\right)^2 - 1 = 1.0285^2 - 1 = 0.05781225$$ - Monthly interest rate equivalent: $$i_{monthly, semi} = \left(1 + EAR\right)^{\frac{1}{12}} - 1 = (1.05781225)^{\frac{1}{12}} - 1 = 0.004708$$ - Present value with this monthly rate: $$PV_{semi} = \frac{4800}{0.004708} = 1019563.15$$ 5. **Difference in required money:** $$Difference = PV_{monthly} - PV_{semi} = 1010526.32 - 1019563.15 = -9036.83$$ Since the difference is negative, it means the fund requires 9036.83 less money with monthly compounding than with semiannual compounding. **Final answers:** - A) Required money with monthly compounding: $1010526.32$ - B) Money required less with monthly compounding compared to semiannual: $9036.83$