Emily Jacob is


45


years old and has saved nothing for retirement.​ Fortunately, she just inherited


$ $77,000.


Emily plans to put a large portion of that money into an investment account earning​ a(n)


11% 


return. She will let the money accumulate for


20


​years, when she will be ready to retire. She would like to deposit enough money today so she could begin making withdrawals of


​$40,000


per year starting at age 66​ (21 years from​ now) and continuing for 24 additional​ years, when she will make her last withdrawal at age 90. Whatever remains from her​ inheritance, Emily will spend on a shopping spree. Emily will continue to earn


11%


on money in her investment account during her retirement​ years, and she wants the balance of her retirement account to be​ $0 after her withdrawal on her ninetieth birthday.

This is a two-part retirement planning problem where Emily wants to:

1. **Invest a portion of her inheritance for 20 years** at an 11% annual return.
2. **Withdraw $40,000 per year for 25 years** starting from age 66 (21 years from now), and have the account fully depleted by age 90. 

We can solve this step-by-step:

### Part 1: Determine the Present Value (PV) of Withdrawals

Emily wants to start making annual withdrawals of $40,000 per year for 25 years, starting in 21 years (at age 66) and ending in 46 years (at age 90). The present value of these withdrawals at age 66 is equivalent to a **25-year annuity**. 

The formula for the present value of an annuity is:

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

Where:
- $ P = 40,000 $ (the annual withdrawal),
- $ r = 0.11 $ (the annual interest rate),
- $ n = 25 $ (the number of years).

Let's calculate the **present value of the withdrawals** Emily will need at the start of her retirement (age 66).

### Part 2: Determine How Much She Needs to Invest Today

Once we know how much money Emily will need in her account at age 66, we will calculate how much she needs to invest **today** to grow to that amount in 20 years. The formula for the future value of an investment is:

\[
FV = PV \times (1 + r)^n
\]

Where:
- $ FV $ is the future value at age 66,
- $ PV $ is the amount Emily needs to invest today,
- $ r = 0.11 $ (the annual interest rate),
- $ n = 20 $ (the number of years).

Let me do the calculations.### Solution:

1. **Present Value of Withdrawals at Age 66**: Emily will need approximately **$336,870** in her retirement account when she turns 66 to support $40,000 in annual withdrawals for 25 years.

2. **Amount to Invest Today**: To have $336,870 by the time she's 66, Emily needs to invest approximately **$41,783** today at an 11% annual return.

This means Emily should invest **$41,783** of her $77,000 inheritance today to meet her retirement goal. She can spend the remaining **$35,217** on her shopping spree.

---

Do you need further details or have any questions?

Here are some related questions to explore:

1. What happens if Emily delays her investment by a few years?
2. How much more would Emily need to invest if the interest rate were lower, say 8%?
3. What if Emily wanted to withdraw more or less than $40,000 annually?
4. How would inflation impact Emily’s retirement savings and withdrawals?
5. What if Emily planned to continue working part-time during her retirement?

**Tip:** Start saving for retirement early to maximize the benefits of compound interest!

Emily Jacob is 45 years old and has saved nothing for retirement. Fortunately, she just inherited $77,000. Emily plans to put a large portion of that money into an investment account earning an 11% return. She will let the money accumulate for 20 years, when she will be ready to retire. She would like to deposit enough money today so she could begin making withdrawals of $40,000 per year starting at age 66 (21 years from now) and continuing for 24 additional years, when she will make her last withdrawal at age 90. Whatever remains from her inheritance, Emily will spend on a shopping spree. Emily will continue to earn 11% on money in her investment account during her retirement years, and she wants the balance of her retirement account to be $0 after her withdrawal on her ninetieth birthday.

Explore how Emily can invest her $77,000 inheritance to support $40,000 annual withdrawals during retirement, with an 11% return. This problem covers core financial concepts like present value, future value, and annuities. Ideal for college-level finance students or advanced high school learners.

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