1. **Problem Statement:** Calculate the present value (investment today) needed to reach a future value of 10,000 after 20 years with different quarterly compounded interest rates. 2. **Formula Used:** The formula for compound interest is: $$FV = PV \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $FV$ is the future value - $PV$ is the present value (investment today) - $r$ is the annual interest rate (decimal) - $n$ is the number of compounding periods per year - $t$ is the number of years We want to find $PV$, so rearranging: $$PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}$$ 3. **Given:** - $FV = 10000$ - $t = 20$ years - $n = 4$ (quarterly compounding) 4. **Calculations:** **a. For 5% rate:** - $r = 0.05$ - Calculate denominator: $$\left(1 + \frac{0.05}{4}\right)^{4 \times 20} = \left(1 + 0.0125\right)^{80} = 1.0125^{80}$$ - Calculate $1.0125^{80}$: $$1.0125^{80} \approx 2.71264$$ - Calculate $PV$: $$PV = \frac{10000}{2.71264} \approx 3687.75$$ **b. For 7% rate:** - $r = 0.07$ - Calculate denominator: $$\left(1 + \frac{0.07}{4}\right)^{80} = 1.0175^{80}$$ - Calculate $1.0175^{80}$: $$1.0175^{80} \approx 3.86968$$ - Calculate $PV$: $$PV = \frac{10000}{3.86968} \approx 2583.68$$ **c. For 9% rate:** - $r = 0.09$ - Calculate denominator: $$\left(1 + \frac{0.09}{4}\right)^{80} = 1.0225^{80}$$ - Calculate $1.0225^{80}$: $$1.0225^{80} \approx 5.60441$$ - Calculate $PV$: $$PV = \frac{10000}{5.60441} \approx 1784.04$$ 5. **Final Answers:** - a. Investment today = $3687.75$ - b. Investment today = $2583.68$ - c. Investment today = $1784.04$