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:
PVhigh=E×1−(1+g1)−n1r−g1PV_{\text{high}} = E \times \frac{1 – (1 + g_1)^{-n_1}}{r – g_1}
Where EE is the earnings in Year 1.
Earnings in Year 1: 4,000,000×(1+0.23)=4,920,0004,000,000 \times (1 + 0.23) = 4,920,000
PVhigh=4,920,000×1−(1+0.23)−70.11−0.23PV_{\text{high}} = 4,920,000 \times \frac{1 – (1 + 0.23)^{-7}}{0.11 – 0.23} PVhigh≈4,920,000×4.0296≈19,832,632PV_{\text{high}} \approx 4,920,000 \times 4.0296 \approx 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)=5,220,0004,920,000 \times (1 + 0.06) = 5,220,000
Using the Gordon Growth Model:
PVperpetuity=E×(1+g2)r−g2PV_{\text{perpetuity}} = \frac{E \times (1 + g_2)}{r – g_2} PVperpetuity=5,220,0000.11−0.06=104,400,000PV_{\text{perpetuity}} = \frac{5,220,000}{0.11 – 0.06} = 104,400,000
Discounting this back to Year 0:
PVtotal=PVperpetuity×(1+r)−n1PV_{\text{total}} = PV_{\text{perpetuity}} \times (1 + r)^{-n_1} PVtotal=104,400,000×(1+0.11)−7≈104,400,000×0.484≈50,500,800PV_{\text{total}} = 104,400,000 \times (1 + 0.11)^{-7} \approx 104,400,000 \times 0.484 \approx 50,500,800
Total Present Value:
PVtotal=PVhigh+PVtotal discounted≈19,832,632+50,500,800=70,333,432PV_{\text{total}} = PV_{\text{high}} + PV_{\text{total discounted}} \approx 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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