Perpetuity Calculator
Calculate the present value of an infinite series of cash flows. Ideal for valuing preferred stock and dividend models.
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By Prof. David Anderson
Finance Professor | CFA Charterholder
“In my 20 years of teaching finance, students often struggle with the concept of ‘infinity.’ How can something that pays you forever have a finite price tag? But Perpetuity is not just a theoretical concept. From the historic British Consols to the Dividend Discount Model used by Warren Buffett, valuing infinite cash flows is a cornerstone of modern finance. Whether you are valuing a share of Preferred Stock or estimating the terminal value of a business, the math of infinity is simpler than you think.”
Perpetuity Calculator & Guide: Valuing Infinite Cash Flows & The Gordon Growth Model
Mastering the Present Value of Forever: Formulas, History & Real-World Application
1. What is a Perpetuity? (The Price of “Forever”)
A Perpetuity is a financial instrument that pays a fixed stream of cash flows for an infinite amount of time. There is no maturity date.
It sounds like it should cost an infinite amount of money to buy, right? Wrong. Because of the Time Value of Money, a dollar received 100 years from now is worth almost nothing today. Therefore, the sum of all future payments converges to a finite number.
2. The Mathematical Engine: Two Formulas
Depending on whether the cash flow stays the same or grows over time, we use one of two elegant formulas.
A. Standard Perpetuity (Constant Cash Flow)
Used for Preferred Stock or flat-rate bonds.
$$ PV = \frac{C}{r} $$
- $PV$: Present Value (What it’s worth today).
- $C$: Cash Flow per period (Dividend or Interest).
- $r$: Discount Rate (Required Rate of Return).
B. Growing Perpetuity (Gordon Growth Model)
Used for Common Stock valuation or Real Estate with rent increases.
$$ PV = \frac{C_1}{r – g} $$
- $C_1$: Cash Flow expected next year.
- $g$: Constant Growth Rate.
- Constraint: $r$ must be greater than $g$.
3. History Lesson: The British Consol
To truly understand perpetuity, we must look at financial history. In 1751, the British Government issued “Consolidated Annuities” (Consols).
The Bond That Never Dies
Consols had no maturity date. They promised to pay interest forever.
For over two centuries, investors bought and sold these bonds. Even though the principal was never repaid, the bond had value because of the reliable stream of interest payments.
Example: If a Consol paid £2.50 per year and the market interest rate was 4%, the price of the bond would be:
$$ Price = \frac{£2.50}{0.04} = £62.50 $$
Although the UK government finally redeemed them in 2015 (after 264 years!), they remain the textbook definition of a perpetuity.
4. Application: Valuing Stocks (Gordon Growth Model)
How does Warren Buffett value a company like Coca-Cola? He often uses the Gordon Growth Model, which treats the company’s dividends as a Growing Perpetuity.
| Variable | Value | Description |
|---|---|---|
| Next Dividend ($D_1$) | $2.00 | Expected cash payout next year. |
| Required Return ($r$) | 8.0% | Risk-adjusted return investor demands. |
| Growth Rate ($g$) | 3.0% | Long-term sustainable growth. |
| Intrinsic Value | $40.00 | Calculated as $2.00 / (0.08 – 0.03)$. |
If the stock is trading at $30 on the market, but your Growing Perpetuity formula says it’s worth $40, it is Undervalued (a buy signal).
5. Real Estate: The Hidden Perpetuity
Real estate investors use Perpetuity math every day without realizing it. It’s called the Cap Rate.
$$ \text{Property Value} = \frac{\text{NOI (Net Operating Income)}}{\text{Cap Rate}} $$
Look familiar? It is exactly $PV = C / r$.
If an apartment building generates $100,000 in net income (NOI) and the market Cap Rate is 5%, the building is worth $100,000 / 0.05 = $2,000,000.
6. The “Infinity Trap”: When g > r
Students often ask: “Professor, what if the growth rate is higher than the discount rate?”
⚠️ Mathematical Explosion
If $g \ge r$, the denominator becomes zero or negative.
$$ PV = \frac{C}{Negative Number} $$
This implies the asset has an Infinite Value. In economics, this is impossible. No company can grow faster than the entire economy forever. If your inputs result in $g > r$, your assumptions are flawed.
7. Professor’s FAQ Corner
Q: What is “Terminal Value” in DCF?
When investment bankers value a company using a Discounted Cash Flow (DCF) model, they usually project 5 years of specific cash flows. For everything after Year 5, they assume the company runs forever. This chunk is called the Terminal Value, and it is calculated using the Perpetuity Growth formula.
Q: Why are Preferred Stocks considered perpetuities?
Preferred stocks pay a fixed dividend (e.g., $2/share) and typically do not have a maturity date. Since the payment is fixed and indefinite, they are valued using the standard $PV = C / r$ formula.
Q: Can I use this for my retirement planning?
Yes! The “4% Rule” is a reverse perpetuity. If you want a perpetuity income of $40,000/year, and you assume a safe withdrawal rate of 4%, you need $40,000 / 0.04 = $1,000,000 saved.
References
- Gordon, M. J. (1959). “Dividends, Earnings, and Stock Prices”. Review of Economics and Statistics.
- Damodaran, A. (2012). Investment Valuation. Wiley Finance.
- Investopedia. “Perpetuity Definition and Formula”.