Introductory Corporate Finance: Time Value of Money Perpetuity PV C r Growing Perpetuity

What This Is

A perpetuity is a stream of cash flows that continues indefinitely. It's essential in corporate finance for valuing companies, estimating future cash flows, and determining the cost of capital. For example, consider a company that generates $100 million in annual cash flows. If we assume a 10% discount rate, the present value of this perpetuity is $1 billion (PV = C / r = $100M / 0.10).

Key Formulas & Models

• PV = C / r – present value of a perpetuity; C = annual cash flow, r = discount rate.
• C = PV × r – annual cash flow; PV = present value, r = discount rate.
• G = r × (1 + g) – growth rate of a growing perpetuity; r = discount rate, g = growth rate.
• PV = C / (r - g) – present value of a growing perpetuity; C = annual cash flow, r = discount rate, g = growth rate.
• C = PV × (r - g) – annual cash flow of a growing perpetuity; PV = present value, r = discount rate, g = growth rate.
• Sustainable Growth Rate (SGR) = ROE × (1 - Retention Ratio) – sustainable growth rate; ROE = return on equity, Retention Ratio = percentage of earnings retained.
• Terminal Value (TV) = FCF × (1 + g) / (r - g) – terminal value; FCF = free cash flow, r = discount rate, g = growth rate.

Step-by-Step Calculation

1. Estimate the annual cash flow (C) of the perpetuity.
2. Determine the discount rate (r) or growth rate (g) of the perpetuity.
3. Apply the formula PV = C / r or PV = C / (r - g) to calculate the present value of the perpetuity.
4. If the perpetuity is growing, calculate the growth rate (g) using the formula g = r × (1 + g) or g = ROE × (1 - Retention Ratio).
5. Use the growth rate (g) to calculate the annual cash flow (C) of the growing perpetuity using the formula C = PV × (r - g).
6. Calculate the terminal value (TV) of the growing perpetuity using the formula TV = FCF × (1 + g) / (r - g).

Common Mistakes

• Mistake: Using the wrong discount rate or growth rate.
• Correction: Ensure the discount rate or growth rate is accurate and relevant to the perpetuity.
• Example: A company has a perpetuity with a 10% discount rate, but the analyst uses a 5% discount rate, resulting in an overvaluation of $1.1 billion instead of $1 billion.
• Mistake: Ignoring the growth rate of a growing perpetuity.
• Correction: Calculate the growth rate (g) using the formula g = r × (1 + g) or g = ROE × (1 - Retention Ratio).
• Example: A company has a growing perpetuity with a 10% discount rate and 5% growth rate, but the analyst ignores the growth rate, resulting in an undervaluation of $900 million instead of $1 billion.

Exam / CFA Tips

• Tip: Be careful when using the formula PV = C / (r - g) for a growing perpetuity, as it assumes the growth rate is constant and the perpetuity will continue indefinitely.
• Tip: When calculating the sustainable growth rate (SGR), ensure the retention ratio is accurate and relevant to the company's business model.
• Tip: Be prepared to distinguish between the perpetuity growth rate (g) and the company's actual growth rate.

Quick Practice Problem

A company has a perpetuity with an annual cash flow of $50 million and a 10% discount rate. What is the present value of the perpetuity?

Answer: $500 million (PV = C / r = $50M / 0.10)

Last-Minute Cram Sheet

• Perpetuity: a stream of cash flows that continues indefinitely.
• PV = C / r: present value of a perpetuity.
• C = PV × r: annual cash flow of a perpetuity.
• G = r × (1 + g): growth rate of a growing perpetuity.
• PV = C / (r - g): present value of a growing perpetuity.
• C = PV × (r - g): annual cash flow of a growing perpetuity.
• SGR = ROE × (1 - Retention Ratio): sustainable growth rate.
• TV = FCF × (1 + g) / (r - g): terminal value.
• ⚠️ In M&M Proposition I (no taxes), firm value is independent of capital structure – but with taxes, value increases with debt due to the interest tax shield.
• ⚠️ The perpetuity growth rate (g) is not the same as the company's actual growth rate.
• ⚠️ The sustainable growth rate (SGR) assumes the retention ratio is constant and the company will continue to grow indefinitely.