 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) 

111% 

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.

To solve Emily Jacob's retirement plan problem, we need to calculate how much money she should invest today (from the $77,000 inheritance) so that she can withdraw $40,000 annually for 25 years (from age 66 to 90), while earning an 11% annual return during both the accumulation and withdrawal phases.

### Step 1: Calculate the Present Value of Withdrawals (from age 66 to 90)
Emily wants to withdraw $40,000 per year for 25 years, starting at age 66, with a return rate of 11% per year. To find the present value of these withdrawals at age 65 (when she retires), we use the formula for the present value of an annuity:

\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
where:
- $PMT = 40,000$ (annual withdrawal),
- $r = 11\%$ or 0.11 (annual return rate),
- $n = 25$ years (number of withdrawals).

\[
PV = 40,000 \times \left( \frac{1 - (1 + 0.11)^{-25}}{0.11} \right)
\]

Let’s compute the value for this present value (the amount needed at age 65).

### Step 2: Calculate the Required Investment Today (at age 45)
Emily wants to invest enough today to accumulate the present value of the withdrawals calculated in Step 1 by age 65. Since the money will earn an 11% return for 20 years, we use the future value formula to find how much she should invest today:

\[
FV = PV \times (1 + r)^n
\]
where:
- $FV = $ the future value needed at age 65 (calculated in Step 1),
- $r = 11\%$ or 0.11,
- $n = 20$ years.

Finally, we rearrange the future value formula to solve for the present value (investment today):

\[
PV_0 = \frac{FV}{(1 + r)^n}
\]

This will tell us how much Emily needs to invest today. Let’s calculate both the present value of the withdrawals and the required initial investment.

### Step 3: Shopping Spree
Whatever is left from the $77,000 inheritance after the required investment will be spent on a shopping spree. Therefore, we subtract the investment from $77,000 to find the remaining amount for the shopping spree.

Let's calculate these values.### Results:

1. **Present Value of Withdrawals**: Emily will need **$336,869.79** at age 65 to fund her $40,000 annual withdrawals for 25 years, earning an 11% return during retirement.
   
2. **Required Investment Today**: To accumulate that amount by age 65, Emily needs to invest **$41,783.28** today.

3. **Amount for Shopping Spree**: After investing, Emily will have **$35,216.72** left from her $77,000 inheritance for a shopping spree.

Would you like further clarification or more details on any step?

Here are five related questions to consider:
1. How does the investment required today change if the return rate is lower?
2. What happens if Emily wants to make withdrawals for more or fewer years?
3. How much more will Emily need if she wants to start withdrawals earlier than age 66?
4. What is the total interest earned during the accumulation phase?
5. How would the calculations change if the withdrawal amount was higher or lower?

**Tip**: Regularly reviewing and adjusting your retirement plan is essential to account for changes in life circumstances or market conditions.

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 111% 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.

Learn how to calculate the investment required today for Emily to make $40,000 annual withdrawals over 25 years, starting at age 66, with an 11% return. Explore present value of annuities, future value of investments, and the impact of compound interest. Ideal for finance students and retirement planners.

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