Answer :
Certainly! Let's solve this step by step to determine the monthly withdrawal amount from your retirement savings.
### Step-by-Step Solution:
1. Principal Amount (P):
- You have a principal amount of [tex]$500,000 saved for retirement. 2. Annual Interest Rate (r_annual): - The annual interest rate earned on the account is 7%, which we can represent as 0.07. 3. Number of Years (t): - You want to make withdrawals for 15 years. 4. Convert Annual Interest Rate to Monthly Interest Rate: - The annual interest rate needs to be converted to a monthly interest rate because withdrawals are made monthly. \[ r_{monthly} = \frac{r_{annual}}{12} = \frac{0.07}{12} \approx 0.005833 \] 5. Calculate the Total Number of Withdrawals: - Since withdrawals are made monthly over 15 years, we need to find the total number of months: \[ n = t \times 12 = 15 \times 12 = 180 \] 6. Using the Annuity Formula to Calculate Monthly Withdrawals: - We can use the annuity formula to determine the monthly withdrawal amount (PMT): \[ PMT = \frac{P \times r_{monthly} \times (1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1} \] - Plugging in the known values: \[ PMT = \frac{500,000 \times 0.005833 \times (1 + 0.005833)^{180}}{(1 + 0.005833)^{180} - 1} \] 7. Calculate the Monthly Withdrawal Amount: - Substituting the calculated values into the formula gives us the monthly withdrawal amount: \[ PMT \approx 4494.14 \] ### Conclusion: If you have $[/tex]500,000 saved for retirement with an annual interest rate of 7%, you will be able to withdraw approximately $4494.14 each month for 15 years.
### Step-by-Step Solution:
1. Principal Amount (P):
- You have a principal amount of [tex]$500,000 saved for retirement. 2. Annual Interest Rate (r_annual): - The annual interest rate earned on the account is 7%, which we can represent as 0.07. 3. Number of Years (t): - You want to make withdrawals for 15 years. 4. Convert Annual Interest Rate to Monthly Interest Rate: - The annual interest rate needs to be converted to a monthly interest rate because withdrawals are made monthly. \[ r_{monthly} = \frac{r_{annual}}{12} = \frac{0.07}{12} \approx 0.005833 \] 5. Calculate the Total Number of Withdrawals: - Since withdrawals are made monthly over 15 years, we need to find the total number of months: \[ n = t \times 12 = 15 \times 12 = 180 \] 6. Using the Annuity Formula to Calculate Monthly Withdrawals: - We can use the annuity formula to determine the monthly withdrawal amount (PMT): \[ PMT = \frac{P \times r_{monthly} \times (1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1} \] - Plugging in the known values: \[ PMT = \frac{500,000 \times 0.005833 \times (1 + 0.005833)^{180}}{(1 + 0.005833)^{180} - 1} \] 7. Calculate the Monthly Withdrawal Amount: - Substituting the calculated values into the formula gives us the monthly withdrawal amount: \[ PMT \approx 4494.14 \] ### Conclusion: If you have $[/tex]500,000 saved for retirement with an annual interest rate of 7%, you will be able to withdraw approximately $4494.14 each month for 15 years.
