To find the present value of all future earnings, we need to calculate the present value of two separate stages of earnings growth:

1. Stage 1 (High Growth Period): For the first 8 years, earnings grow at 30% per year.
2. Stage 2 (Stable Growth Period): After 8 years, earnings grow at a constant 2% per year indefinitely.

We will use the formula for the Present Value (PV) of a growing annuity for the first stage and the Present Value of a perpetuity with growth for the second stage. The steps are as follows:

Stage 1: High Growth Period (30% growth for 8 years)

The formula for the present value of a growing annuity is:

$PV_{\text{growing annuity}} = \sum_{t=1}^{8} \frac{E_t}{(1+r)^t}$

Where:

• $E_t = E_0(1 + g)^t$ is the earnings in year $t$,
• $E_0 = 3 \, \text{million dollars}$,
• $g = 0.30$ (30% growth),
• $r = 0.10$ (10% discount rate).

The earnings for each year are calculated, then discounted back to the present using the formula above.

Stage 2: Stable Growth Period (2% growth forever after 8 years)

After 8 years, the earnings grow at a constant rate of 2% forever. For the perpetuity, we first calculate the earnings in year 9, which will be the starting point for the perpetuity:

$E_9 = E_8 \times (1 + g) = E_0 \times (1 + 0.30)^8$

Then, the present value of the perpetuity is:

$PV_{\text{perpetuity}} = \frac{E_9}{r - g} = \frac{E_9}{0.10 - 0.02}$

Finally, we discount this perpetuity value back to the present (year 0), using the 10% discount rate:

$PV_{\text{total}} = \frac{PV_{\text{perpetuity}}}{(1+r)^8}$

Let me calculate both stages and the total present value.The present value of all future earnings for your company, given the growth assumptions and discount rate, is approximately $197.41 million.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How would changes in the discount rate affect the present value?
2. What if the high growth period lasted longer than 8 years?
3. How would the present value change with a different stable growth rate (e.g., 3%) after 8 years?
4. How is the perpetuity formula used in valuing long-term investments?
5. What factors influence the selection of the discount rate in such analyses?

Tip: The perpetuity formula is a powerful tool for valuing stable cash flows that grow indefinitely at a constant rate, often used in financial models.