To find the present value (PV) of a growing perpetuity that starts at a future date, we follow these steps:

Given Data:

• First Payment ($C$) = $2,000
• Growth Rate ($g$) = 4% = 0.04
• Discount Rate ($r$) = 8% = 0.08
• Start Year = 4

Step 1: Compute the Present Value at Year 3

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

$PV_{\text{year 3}} = \frac{C}{r - g}$

Substituting the values:

$PV_{\text{year 3}} = \frac{2000}{0.08 - 0.04} = \frac{2000}{0.04} = 50,000$

This gives the value of the perpetuity at year 3.

Step 2: Discount Back to Year 0

Since the perpetuity value is at year 3, we discount it back to year 0 using the present value formula:

$PV_0 = \frac{PV_{\text{year 3}}}{(1 + r)^3}$

$PV_0 = \frac{50,000}{(1.08)^3}$

$PV_0 = \frac{50,000}{1.2597} \approx 39,694.42$

Final Answer:

$\mathbf{PV_0 \approx 39,694.42}$

Would you like a step-by-step breakdown of the calculations or a different approach?

Related Questions:

1. What happens if the growth rate is higher than the discount rate?
2. How would the present value change if payments started in year 6 instead of year 4?
3. How do you derive the formula for growing perpetuity?
4. What is the difference between a growing perpetuity and an annuity?
5. How does inflation affect the valuation of growing perpetuities?

Tip:

When discounting future values to the present, always ensure that the exponent matches the number of years being discounted.