Full solution

Q. You want to be able to withdraw

\$ 35,000

each year for

30

years. Your account same

6 \%

interest.

\newline

a) How much do you need in your account at the beginning?

Formula Explanation: To solve this problem, we will use the formula for the present value of an annuity, which calculates the amount needed initially to be able to withdraw a fixed amount each year for a certain number of years at a given interest rate. The formula is: $\newline$ $PV = PMT \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]$ $\newline$ where $PV$ is the present value (initial amount needed), $PMT$ is the annual withdrawal amount, $r$ is the annual interest rate (expressed as a decimal), and $n$ is the number of years.
Interest Rate Conversion: First, we need to convert the interest rate from a percentage to a decimal. The interest rate is $6\%$ , so as a decimal, it is $0.06$ .
Formula Application: Next, we will plug the values into the formula: $\newline$ $PMT = \$35,000$ (annual withdrawal) $\newline$ $r = 0.06$ (interest rate as a decimal) $\newline$ $n = 30$ (number of years) $\newline$ $PV = \$35,000 \times \left[\frac{1 - (1 + 0.06)^{-30}}{0.06}\right]$
Calculate $(1.06)^{-30}$ : Now, we calculate the value inside the brackets: $\newline$ $(1 + 0.06)^{-30} = (1.06)^{-30}$ $\newline$ We need to calculate the value of $(1.06)^{-30}$ using a calculator.
Calculate Value Inside Brackets: After calculating $(1.06)^{-30}$ , we get approximately $0.17411$ . $\newline$ Now we can continue with the calculation: $\newline$ $PV = \$35,000 \times \left[\frac{1 - 0.17411}{0.06}\right]$
Subtract and Divide: Subtract $0.17411$ from $1$ : $\newline$ $1 - 0.17411 = 0.82589$
Final Calculation: Now, divide $0.82589$ by $0.06$ : $\newline$ $0.82589 / 0.06 \approx 13.76483$
Final Calculation: Now, divide $0.82589$ by $0.06$ : $\newline$ $0.82589 / 0.06 \approx 13.76483$ Finally, multiply $\$$ $35$ , $000$ by $13.76483$ to find the present value: $\newline$ PV = $\$ $35$ , $000$ \times $13$ . $76483$ \approx $\$$ $481$ , $769$ . $05$ $