Calculating Present Value of Growing Perpetuities

The question asks us to calculate the present value of a growing perpetuity under different scenarios and then arrange these present values in descending order. A growing perpetuity is a series of payments that grow at a constant rate and continue indefinitely.

The formula for the present value (PV) of a growing perpetuity is given by:

$$PV = \frac{C_1}{r - g}$$

Where:

• \(C_1\) = The payment amount at the end of the first period (next year). In this question, \(C_1 = \text{Rs. } 3,000\).
• \(r\) = The discount rate or required rate of return.
• \(g\) = The constant growth rate of the payments.

It is important to note that this formula is valid only when the discount rate (\(r\)) is greater than the growth rate (\(g\)), i.e., \(r > g\).

Let's calculate the present value for each given scenario:

Scenario A: Present value at 8% growth and 10% discount rate

• \(C_1 = \text{Rs. } 3,000\)
• \(r = 10\% = 0.10\)
• \(g = 8\% = 0.08\)

Since \(r > g\) (0.10 > 0.08), the formula is applicable.

$$PV_A = \frac{3000}{0.10 - 0.08} = \frac{3000}{0.02} = \text{Rs. } 150,000$$

Scenario B: Present value at 3% growth and 9% discount rate

• \(C_1 = \text{Rs. } 3,000\)
• \(r = 9\% = 0.09\)
• \(g = 3\% = 0.03\)

Since \(r > g\) (0.09 > 0.03), the formula is applicable.

$$PV_B = \frac{3000}{0.09 - 0.03} = \frac{3000}{0.06} = \text{Rs. } 50,000$$

Scenario C: Present value at 6% growth and 11% discount rate

• \(C_1 = \text{Rs. } 3,000\)
• \(r = 11\% = 0.11\)
• \(g = 6\% = 0.06\)

Since \(r > g\) (0.11 > 0.06), the formula is applicable.

$$PV_C = \frac{3000}{0.11 - 0.06} = \frac{3000}{0.05} = \text{Rs. } 60,000$$

Scenario D: Present value at 5% growth and 6% discount rate

• \(C_1 = \text{Rs. } 3,000\)
• \(r = 6\% = 0.06\)
• \(g = 5\% = 0.05\)

Since \(r > g\) (0.06 > 0.05), the formula is applicable.

$$PV_D = \frac{3000}{0.06 - 0.05} = \frac{3000}{0.01} = \text{Rs. } 300,000$$

Scenario E: Present value at 1% growth and 4% discount rate

• \(C_1 = \text{Rs. } 3,000\)
• \(r = 4\% = 0.04\)
• \(g = 1\% = 0.01\)

Since \(r > g\) (0.04 > 0.01), the formula is applicable.

$$PV_E = \frac{3000}{0.04 - 0.01} = \frac{3000}{0.03} = \text{Rs. } 100,000$$

Summary of Present Values

Arranging Present Values in Descending Order

Now, we arrange the calculated present values from highest to lowest:

1. Rs. 300,000 (Scenario D)
2. Rs. 150,000 (Scenario A)
3. Rs. 100,000 (Scenario E)
4. Rs. 60,000 (Scenario C)
5. Rs. 50,000 (Scenario B)

Therefore, the present values in descending order are D, A, E, C, B.

Revision Table: Growing Perpetuity Calculations

Additional Information on Perpetuities and Valuation

A perpetuity is a type of annuity that lasts forever. A growing perpetuity is a stream of cash flows that occurs at regular intervals and grows at a constant rate indefinitely.

• Perpetuity: If the payments are fixed (zero growth, \(g=0\)), the formula simplifies to \(PV = \frac{C_1}{r}\).
• Annuity: A series of fixed payments for a finite period.
• Growing Annuity: A series of payments that grow at a constant rate for a finite period.
• Discount Rate (\(r\)): This represents the required rate of return or the cost of capital used to discount future cash flows back to their present value. A higher discount rate leads to a lower present value.
• Growth Rate (\(g\)): This is the rate at which the payments are expected to grow each period. For the growing perpetuity formula to work, the growth rate must be less than the discount rate. If \(g \ge r\), the present value approaches infinity or is undefined, indicating the model is not suitable or the required return is insufficient relative to growth.

Understanding these concepts is crucial for valuing assets that provide a perpetual stream of income, such as preferred stock dividends or rental income from real estate, assuming certain conditions are met.