1. Problem Statement: We have an ordinary annuity with an interest rate of 7.92% compounded monthly. (A) A person deposits 100 monthly for 30 years and then withdraws equal amounts monthly for 15 years until the balance is zero. Find the monthly withdrawal amount and the total interest earned. (B) If the monthly withdrawal is 1500 for 15 years, find the monthly deposit amount for the first 30 years. --- 2. Key Information and Formulas: - Monthly interest rate: $i = \frac{7.92\%}{12} = 0.0066$ (approx) - Number of months depositing: $n_1 = 30 \times 12 = 360$ - Number of months withdrawing: $n_2 = 15 \times 12 = 180$ For an ordinary annuity (deposits or withdrawals at end of each month): - Future value after deposits: $$FV = P \times \frac{(1+i)^{n_1} - 1}{i}$$ - Present value needed to withdraw $W$ for $n_2$ months: $$PV = W \times \frac{1 - (1+i)^{-n_2}}{i}$$ --- 3. Part (A) Find the monthly withdrawal $W$ - Calculate $FV$ after 30 years of deposits: $$FV = 100 \times \frac{(1+0.0066)^{360} -1}{0.0066}$$ Calculate $(1+0.0066)^{360}$: $$ (1.0066)^{360} \approx 10.8736 $$ So, $$FV = 100 \times \frac{10.8736 - 1}{0.0066} = 100 \times \frac{9.8736}{0.0066} \approx 100 \times 1496.61 = 149661\ $$ - Set this $FV$ to the present value needed for withdrawals $PV$: $$149661 = W \times \frac{1-(1.0066)^{-180}}{0.0066}$$ - Calculate $(1.0066)^{-180}:$ $$ (1.0066)^{-180} = \frac{1}{(1.0066)^{180}} \approx \frac{1}{4.5012} = 0.2221 $$ - Substitute: $$149661 = W \times \frac{1 - 0.2221}{0.0066} = W \times \frac{0.7779}{0.0066} = W \times 117.89$$ - Solve for $W$: $$W = \frac{149661}{117.89} \approx 1270.17$$ 4. Interest earned during entire 45 years: - Total deposited: $$100 \times 360 = 36000$$ - Total withdrawn: $$1270.17 \times 180 = 228630.6$$ - Total interest earned is total withdrawn minus total deposited: $$228630.6 - 36000 = 192630.6$$ --- 5. Part (B) Find monthly deposits $P$ if monthly withdrawal $W=1500$ - The present value needed at withdrawal start is: $$PV = 1500 \times \frac{1-(1.0066)^{-180}}{0.0066} = 1500 \times 117.89 = 176835$$ - Solve for $P$ from future value after deposits: $$176835 = P \times \frac{(1.0066)^{360} -1}{0.0066} = P \times 1496.61$$ - Thus, $$P = \frac{176835}{1496.61} \approx 118.13$$ --- Final Answers: (A) Monthly withdrawals are $1270.17$. Interest earned is $192630.60$. (B) Monthly deposits must be $118.13$ to allow $1500$ withdrawals monthly for 15 years.