### Present Value of Future Earnings Calculation for Growing Solar Panel Business

**Question:** You own a business that sells solar panels. Analysts predict its earnings will grow at 23% per year for the next 7 years. After that, as competition increases, earnings growth is expected to slow to 6% per year and continue at that level forever. Your company has just announced earnings of $4, 000, 000.

A) What is the present value of all future earnings if the interest rate is 11%?

To determine the present value of all future earnings for the business, you can use a two-stage dividend discount model (DDM) approach. Here’s how you can approach the calculation:

**Steps for Calculation:**

**Calculate the Present Value of Earnings During the High-Growth Period:**

Earnings grow at 23% per year for 7 years.

Use the formula for the present value of a growing annuity for this period.

**Calculate the Present Value of Earnings Beyond the High-Growth Period:**

Earnings grow at a constant rate of 6% per year thereafter.

This is calculated using the Gordon Growth Model (a perpetuity formula) at the end of the high-growth period.

**Combine the Two Values:**

Add the present value of the high-growth period earnings to the present value of the earnings beyond the high-growth period.

**Detailed Calculation:**

**Present Value of Earnings During High-Growth Period:**

- Earnings at Year 0: $4,000,000
- Growth Rate (g1): 23%
- Number of Years (n1): 7
- Discount Rate (r): 11%

Using the formula for the present value of a growing annuity:

$PV_{high}=E×r−g−(+g)−n1 $

Where $E$ is the earnings in Year 1.

Earnings in Year 1: $4,000,000×(1+0.23)=4,920,000$

$PV_{high}=4,920,000×−−(+)−7 $ $PV_{high}≈4,920,000×4.0296≈19,832,632$

**Present Value of Earnings Beyond High-Growth Period:**

- Growth Rate (g2): 6%
- Earnings in Year 8: $4,920,000×(1+0.06)=5,220,000$

Using the Gordon Growth Model:

$PV_{perpetuity}=r−gE×(+g) $ $PV_{perpetuity}=−,, =104,400,000$

Discounting this back to Year 0:

$PV_{total}=PV_{perpetuity}×(1+r_{n1}$ $PV_{total}=104,400,000×(1+0.11_{−7}≈104,400,000×0.484≈50,500,800$

**Total Present Value:**

$PV_{total}=PV_{high}+PV_{total discounted}≈19,832,632+50,500,800=70,333,432$

**Summary:**

The present value of all future earnings, given the specified growth rates and interest rate, is approximately $70,333,432.

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