1. **Problem 1: Future Value of an Ordinary Annuity (Simple Annuity)** Maria deposits 5,000 every month for 5 years at 6% annual interest compounded monthly. We need to find the future value (FV) of this annuity. - The formula for the future value of an ordinary annuity is: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $P = 5000$ (monthly deposit) - $r = \frac{0.06}{12} = 0.005$ (monthly interest rate) - $n = 5 \times 12 = 60$ (total number of payments) Calculate: $$FV = 5000 \times \frac{(1 + 0.005)^{60} - 1}{0.005}$$ First, compute $(1 + 0.005)^{60}$: $$ (1.005)^{60} \approx 1.34885 $$ Then: $$FV = 5000 \times \frac{1.34885 - 1}{0.005} = 5000 \times \frac{0.34885}{0.005} = 5000 \times 69.77 = 348,850$$ **Final answer:** $FV = 348,850$ 2. **Problem 2: Present Value of a Deferred Annuity (General Annuity)** A company promises to pay 50,000 every quarter for 10 years, with an interest rate of 10% compounded monthly. We are to find the present value (PV) of this annuity. - Quarterly payment, so frequency of payment is 4/year. - Monthly compounding means monthly rate $i_m = 0.10/12 = 0.0083333$. - We need an effective quarterly rate $r$ over 3 months: $$r = (1 + i_m)^3 - 1 = (1 + 0.0083333)^3 - 1$$ Calculate: $$r = 1.0252 - 1 = 0.0252$$ - Number of quarters $n = 10 \times 4 = 40$ - Present value formula for ordinary annuity: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ Substitute: $$PV = 50000 \times \frac{1 - (1 + 0.0252)^{-40}}{0.0252}$$ Calculate $(1 + 0.0252)^{-40}$: $$ (1.0252)^{-40} = \frac{1}{(1.0252)^{40}} \approx \frac{1}{2.718} = 0.368$$ Then: $$PV = 50000 \times \frac{1 - 0.368}{0.0252} = 50000 \times \frac{0.632}{0.0252} = 50000 \times 25.08 = 1,254,000$$ **Final answer:** $PV = 1,254,000$