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To solve this problem using **annuities** and **present value calculations**, we need to reverse the previous problem and find the minimum salary at age 27 that would justify the cost of $100,000 for graduate school. Here’s how we can approach the solution step by step.

### 1. Problem Summary:
- You have to pay $100,000 **today** for graduate school.
- After completing graduate school, you will start working at age 27.
- Your salary grows at a rate of 3% per year (annual wage growth).
- The discount rate used for calculating the present value is 5%.
- You retire at age 65, which means you will work for 38 years (from age 27 to 65).

The goal is to find the minimum salary $ w_{27} $ that makes the present value of your post-graduate earnings, minus the $100,000 cost of schooling, **equal** to the present value of pre-graduate earnings (calculated previously as **$1,341,604**).

### 2. Formula for Present Value of a Growing Annuity:
The formula for the present value of a growing annuity is:
\[
PV = \frac{w}{r - g} \left[ 1 - \left( \frac{1 + g}{1 + r} \right)^n \right]
\]
Where:
- $ PV $ is the present value of the growing annuity.
- $ w $ is the salary at age 27.
- $ r = 5\% $ is the discount rate (valuation rate).
- $ g = 3\% $ is the wage growth rate.
- $ n = 38 $ is the number of years from age 27 to retirement (65 years old).

### 3. Setting up the Equation:
To justify the $100,000 cost of graduate school, the present value of post-graduate earnings, minus the $100,000, must equal the present value of pre-graduate earnings. This gives us the equation:

\[
PV_\text{post-school} - 100,000 = PV_\text{pre-school}
\]
\[
\frac{w_{27}}{r - g} \left[ 1 - \left( \frac{1 + g}{1 + r} \right)^n \right] - 100,000 = 1,341,604
\]

### 4. Substituting Known Values:
Substitute the known values into the equation:
- $ r = 0.05 $
- $ g = 0.03 $
- $ n = 38 $
- $ PV_\text{pre-school} = 1,341,604 $

\[
\frac{w_{27}}{0.05 - 0.03} \left[ 1 - \left( \frac{1.03}{1.05} \right)^{38} \right] - 100,000 = 1,341,604
\]

Let’s now calculate the values inside the brackets:

\[
\left( \frac{1.03}{1.05} \right)^{38} \approx 0.456
\]

\[
1 - 0.456 = 0.544
\]

Now, the equation becomes:

\[
\frac{w_{27}}{0.02} \times 0.544 - 100,000 = 1,341,604
\]

### 5. Solving for $ w_{27} $:
First, isolate $ w_{27} $:

\[
\frac{w_{27} \times 0.544}{0.02} = 1,341,604 + 100,000
\]
\[
\frac{w_{27} \times 0.544}{0.02} = 1,441,604
\]
\[
w_{27} \times 0.544 = 1,441,604 \times 0.02
\]
\[
w_{27} \times 0.544 = 28,832.08
\]
\[
w_{27} = \frac{28,832.08}{0.544} \approx 52,990
\]

### 6. Conclusion:
The minimum salary you must earn at age 27 to justify the $100,000 expense for graduate schooling is approximately **$52,990** per year.

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This solution shows that after paying for graduate school, you would need to start with a salary of at least **$52,990** per year, growing at 3% annually, in order for the investment in education to be financially justified compared to your pre-graduate school earnings.

Let me know if you need further details or explanations!

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Here are 5 related questions:
1. How would the result change if the discount rate increased to 6%?
2. What happens if the wage growth rate is reduced to 2% instead of 3%?
3. How would extending the retirement age to 70 impact the required salary?
4. Can you calculate the total earnings after graduate school using the minimum salary found?
5. How does paying for schooling using loans (with interest) affect the calculation?

**Tip:** Always consider both the long-term financial and personal benefits when making decisions about higher education investments!

Assume that you have to pay $100,000 for graduate schooling. What is the minimum salary you must earn at age 27 to justify the expense?

Learn how to calculate the minimum salary needed at age 27 to justify a $100,000 graduate school expense using present value calculations and annuities. This problem covers key concepts such as wage growth, discount rate, and present value of earnings.

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