Questions: You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find how much the account needs to hold to make this possible. Round your answer to the nearest dollar. Regular withdrawal: 2600 Interest rate: 4.5% Frequency: weekly Time: 21 years Account balance:

You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find how much the account needs to hold to make this possible. Round your answer to the nearest dollar.

Regular withdrawal: 2600
Interest rate: 4.5%
Frequency: weekly
Time: 21 years

Account balance:

Transcript text: You want to be able to withdraw the specified amount periodically from a payout annuity with the given terms. Find how much the account needs to hold to make this possible. Round your answer to the nearest dollar. \begin{tabular}{|r|r|} \hline Regular withdrawal: & $\$ 2600$ \\ \hline Interest rate: & $4.5 \%$ \\ \hline Frequency & weekly \\ \hline Time: & 21 years \\ \hline \end{tabular} Account balance: $\$$ $\square$

Solution

Solution Steps

Step 1: Define Given Values

We are given the following values:

Regular withdrawal: $ \$2600 $
Annual interest rate: $ 4.5\% $
Frequency of withdrawals: weekly ($ 52 $ times per year)
Time period: $ 21 $ years

Step 2: Convert Annual Interest Rate to Weekly Interest Rate

The weekly interest rate is calculated using the formula: \[ \text{weekly\_interest\_rate} = \left(1 + \frac{4.5}{100}\right)^{\frac{1}{52}} - 1 \] Substituting the values, we get: \[ \text{weekly\_interest\_rate} \approx 0.0008468369297969236 \]

Step 3: Calculate Total Number of Withdrawals

The total number of withdrawals over $ 21 $ years is: \[ \text{total\_withdrawals} = 52 \times 21 = 1092 \]

Step 4: Calculate Present Value of the Annuity

The present value of the annuity is calculated using the formula: \[ \text{present\_value} = \frac{2600 \times \left(1 - (1 + 0.0008468369297969236)^{-1092}\right)}{0.0008468369297969236} \] Substituting the values, we get: \[ \text{present\_value} \approx 1852012.6348100351 \]

Step 5: Round to the Nearest Dollar

The account balance needed is: \[ \text{account\_balance} = \text{round}(1852012.6348100351) = 1852013 \]