1. **State the problem:** You want to find the amount of money needed at retirement to fund withdrawals of $5833.33 made 24 times per year from age 55 to 82, with an annual interest rate of 3.6%. Withdrawals are at the beginning of each period, and no withdrawal is made on the final birthday. 2. **Identify the variables:** - Withdrawal amount per period: $P = 5833.33$ - Number of withdrawals per year: $m = 24$ - Total years of withdrawals: $T = 82 - 55 = 27$ - Total number of withdrawals: $n = m \times T = 24 \times 27 = 648$ - Annual interest rate: $r = 0.036$ - Periodic interest rate: $i = \frac{r}{m} = \frac{0.036}{24} = 0.0015$ 3. **Formula for present value of an annuity due:** Since withdrawals are at the beginning of each period, use the annuity due formula: $$ PV = P \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i) $$ 4. **Calculate each part:** Calculate $(1 + i)^{-n}$: $$ (1 + 0.0015)^{-648} = (1.0015)^{-648} $$ Calculate the power: $$ (1.0015)^{648} = e^{648 \times \ln(1.0015)} \approx e^{648 \times 0.001499} = e^{0.971} \approx 2.641 $$ So, $$ (1.0015)^{-648} = \frac{1}{2.641} \approx 0.3787 $$ 5. **Calculate the fraction:** $$ \frac{1 - 0.3787}{0.0015} = \frac{0.6213}{0.0015} = 414.2 $$ 6. **Calculate present value:** $$ PV = 5833.33 \times 414.2 \times 1.0015 \approx 5833.33 \times 414.82 = 2,419,000.5 $$ 7. **Final answer:** You need approximately **2,419,000.50** saved at retirement to fund these withdrawals.