## PV of Cash Flows — Infinite Period (Perpetuity)

A perpetuity is a stream of cash flows expected to continue forever, with no end date. Infinite cash flows are classified into two types:

```

Cash Flows for an Infinite Period (Perpetuity)

├── Equal CFs up to Perpetuity (Level Perpetuity)

└── Growing CFs up to Perpetuity (Growing Perpetuity)

```

### A. Equal Cash Flows up to Perpetuity (Level Perpetuity)

When the same cash flow recurs forever:

$$PV = \frac{\text{Annual CF}}{\text{Discount Rate}}$$

Even though the cash flows never stop, the present value is finite — because flows in distant years are discounted so heavily they add almost nothing.

### B. Growing Cash Flows up to Perpetuity (Growing Perpetuity)

When the cash flow grows at a constant rate forever:

$$PV = \frac{CF_1}{(\text{Discount Rate} - \text{Growth Rate})}$$

Where $CF_1$ is the cash flow of the next period (year 1), not the current period.

### Important Conditions

The growing perpetuity formula is only valid when Discount Rate > Growth Rate. If growth ≥ discount rate, the value is infinite and the formula breaks down.
Use $CF_1$ (the first future cash flow). If you are given the current cash flow $CF_0$, grow it first: $CF_1 = CF_0 \times (1 + g)$.

This growing-perpetuity logic is the foundation of the dividend growth (Gordon) model used later in valuation and cost of equity.

Worked example

### Example 1

Level perpetuity: A bond pays ₹600 every year forever. Required return = 8%.

$$PV = \frac{600}{0.08} = ₹7{,}500$$

### Example 2

Growing perpetuity: A share is expected to pay a dividend of ₹10 next year (CF₁), growing at 5% p.a. forever. Required return = 12%.

$$PV = \frac{10}{0.12 - 0.05} = \frac{10}{0.07} = ₹142.86$$

### Example 3

Growing perpetuity with current cash flow given: Current dividend (CF₀) = ₹10, growth = 5%, discount rate = 12%. First grow the dividend:

$$CF_1 = 10 \times (1.05) = 10.50$$

$$PV = \frac{10.50}{0.12 - 0.05} = \frac{10.50}{0.07} = ₹150$$

⚠️ Common exam mistakes

Using the current period's cash flow (CF₀) directly in the growing-perpetuity formula instead of next year's CF₁ = CF₀ × (1+g).
Applying the formula when the growth rate equals or exceeds the discount rate — it gives a negative or infinite (meaningless) value.
Forgetting that a perpetuity's present value is finite even though the cash flows are infinite.
Confusing the level-perpetuity denominator (just the discount rate) with the growing-perpetuity denominator (discount rate minus growth rate).

Reference:

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PV of Cash Flows for an Infinite Period (Perpetuity) — Level and Growing

Worked example

⚠️ Common exam mistakes