Joseph is 21 years old, he plans to retire at age 63, and he expects to live to age 96. He expects that his investments can earn an average return of 10.4% until retirement, and after retirement, he plans to earn 5.6%. If he wants to be able to withdraw $2,500 per month after retirement,

(a) How much will he need to have in the account when he retires in order to meet his goal? $


(b) How much will he have to deposit each month, starting now, in order to get this balance at retirement? $


To solve this problem, we need to approach it in two parts: 

### Part (a): The amount Joseph needs in his account at retirement to meet his withdrawal goals.

Joseph wants to withdraw \$2,500 per month during his retirement. He expects to live for $96 - 63 = 33$ years after retirement, so the total number of withdrawals will be $33 \times 12 = 396$ months.

Given that his investments will earn 5.6% per year after retirement, we need to find how much money he will need at the time of retirement to sustain these withdrawals.

To solve this, we use the **Present Value of an Annuity** formula:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ PV $ = Present Value (amount needed at retirement),
- $ P $ = Monthly withdrawal (\$2,500),
- $ r $ = Monthly interest rate (5.6% annually = $\frac{5.6\%}{12} = 0.00467$ per month),
- $ n $ = Total number of withdrawals (396 months).

Substituting the values:

\[
PV = 2500 \times \frac{1 - (1 + 0.00467)^{-396}}{0.00467}
\]

Let's calculate this.

### Part (b): The monthly deposit Joseph needs to make to reach his retirement goal.

Joseph wants to retire in $63 - 21 = 42$ years, so he has $42 \times 12 = 504$ months to save. His investments are expected to earn 10.4% per year until retirement, or $\frac{10.4\%}{12} = 0.00867$ per month.

We can use the **Future Value of an Annuity** formula to find out how much Joseph needs to deposit monthly:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ FV $ = Future Value (the amount he needs at retirement, which we will find in part (a)),
- $ P $ = Monthly deposit (which we need to find),
- $ r $ = Monthly interest rate (10.4% annually = 0.00867 per month),
- $ n $ = Total number of months (504 months).

We can rearrange the formula to solve for $ P $:

\[
P = \frac{FV \times r}{(1 + r)^n - 1}
\]

Let's compute both parts.It seems like I can’t do more advanced data analysis right now. Please try again later. If you want, I can walk you through how to calculate these values manually or try again when the tools are available. Let me know how you'd like to proceed!

Would you like to:
1. Go over the formulas step-by-step to do the calculations yourself?
2. Learn more about the concept of annuities and their applications in finance?
3. Discuss how different interest rates can impact retirement savings?
4. Explore alternative investment strategies for retirement planning?
5. Ask about how changes in retirement age or life expectancy affect savings needs?

**Tip:** When calculating future or present values, always check whether you are working with monthly or annual rates to avoid mistakes.

Joseph is 21 years old, he plans to retire at age 63, and he expects to live to age 96. He expects that his investments can earn an average return of 10.4% until retirement, and after retirement, he plans to earn 5.6%. If he wants to be able to withdraw $2,500 per month after retirement, (a) How much will he need to have in the account when he retires in order to meet his goal? (b) How much will he have to deposit each month, starting now, in order to get this balance at retirement?

Learn how to calculate the amount needed for retirement based on annuities, interest rates, and monthly withdrawals. This problem covers the Present Value of an Annuity and Future Value of an Annuity formulas. Perfect for finance and economics students.

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