A couple is saving for retirement with three different accounts. The table below shows the current balances in their accounts, along with their yearly contribution, and the yearly return on each account. The couple will retire in 24.00 years and pool the money into a savings account that pays 3.00% APR. They plan on living for 28.00 more years and making their yearly withdrawals at the beginning of the year. What will be their yearly withdrawal?

Account	Balance	Yearly Contribution	APR
Fidelity Mutual Fund	$24,722.00	$1,000.00	6.00%
Vanguard Mutual Fund	$182,542.00	$10,000.00	7.00%
Employer 401k	$303,060.00	$15,000.00	5.00%


This is a multi-step financial problem where the couple is saving for retirement using three accounts, each with different contributions and interest rates. We need to calculate their annual withdrawal amount once they retire, assuming that their pooled money earns a 3.00% APR over their retirement period.

Here’s how we can approach the problem:

### Step 1: Calculate the future value of each account at the time of retirement.
The future value ($FV$) of an account can be found using the **future value of an annuity formula** combined with the **future value of a lump sum**:

\[
FV = P \times (1 + r)^n + C \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- $ P $ = Current balance of the account
- $ C $ = Yearly contribution
- $ r $ = APR (annual interest rate)
- $ n $ = Number of years until retirement (24 years in this case)

### Step 2: Sum up the future values of all three accounts to get the total balance at retirement.

### Step 3: Determine the yearly withdrawal amount using the **present value of annuities formula**:
The present value ($PV$) of the pooled retirement fund is calculated assuming they make withdrawals for 28 years, at the start of each year. We can use the following formula to determine the withdrawal amount $W$ that they can take:

\[
PV = W \times \frac{1 - (1 + r)^{-t}}{r}
\]
Where:
- $ W $ = Yearly withdrawal amount
- $ r $ = APR (3% or 0.03)
- $ t $ = Number of years of withdrawals (28 years)

Let me calculate these values for you.The total amount the couple will have at retirement, after 24 years of saving and earning interest, is approximately **$3,303,528.83**.

Given that they plan to withdraw money over 28 years, and the savings account earns 3.00% APR, their yearly withdrawal amount will be approximately **$176,055.73**.

Would you like further details on how these calculations were performed, or do you have any other questions?

Here are 5 related questions to consider:
1. How would changes in the APR during withdrawals affect the yearly withdrawal amount?
2. What happens if they contribute more or less to one of the accounts?
3. How does inflation impact the real value of their yearly withdrawal?
4. How does starting withdrawals one year later or earlier impact their retirement plan?
5. What is the effect of choosing different types of investment vehicles during retirement?

**Tip**: Diversifying investments can help manage risk, especially when nearing retirement.

A couple is saving for retirement with three different accounts. The table below shows the current balances in their accounts, along with their yearly contribution, and the yearly return on each account. The couple will retire in 24.00 years and pool the money into a savings account that pays 3.00% APR. They plan on living for 28.00 more years and making their yearly withdrawals at the beginning of the year. What will be their yearly withdrawal?

Account Balance Yearly Contribution APR
Fidelity Mutual Fund $24,722.00 $1,000.00 6.00%
Vanguard Mutual Fund $182,542.00 $10,000.00 7.00%
Employer 401k $303,060.00 $15,000.00 5.00%

Learn how to calculate the future value of retirement savings and determine yearly withdrawals using compound interest and annuity formulas. This problem covers key financial concepts like future value, present value, and APR, suitable for students in Grades 11-12 and college-level finance.

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