Present Value of Growing Perpetuity Calculator

Growing Perpetuity Present Value Estimator

Present Value of Growing Perpetuity Formula

Formula

\[ PV = \frac{C}{r - g} \]

Where:

$PV$ — present value of the growing perpetuity
$C$ — cash flow (payment) in the first period
$r$ — annual discount or interest rate, as a decimal
$g$ — annual growth rate of the cash flow, as a decimal

This formula gives the value today of a payment stream that continues forever, with each payment larger than the previous one by a factor $(1 + g)$. It only works when the discount rate exceeds the growth rate $(r > g)$; otherwise the series never converges.

Growing Perpetuity – Calculation Example

Example: an investment pays $1,000 at the end of the first year, and the payment grows by 3% every year thereafter. With a discount rate of 8%, what is the stream worth today?

\[ PV = \frac{C}{r - g} = \frac{1000}{0.08 - 0.03} \]

\[ PV = \frac{1000}{0.05} = 20000 \]

Result: the present value of the growing perpetuity is $20,000.

How to Use the Calculator

Enter the Annual Payment — the cash flow you expect to receive at the end of the first period (e.g. $1,000).
Enter the Annual Interest Rate as a percentage — the discount rate that reflects the time value of money and the risk of the cash flows (e.g. 8%).
Enter the Annual Growth Rate as a percentage — the constant rate at which the payment increases each year (e.g. 3%).
Read the Present Value (PV) — the value in today's dollars of the entire infinite payment stream.

Note that the growth rate must be lower than the interest rate. If $g \geq r$, the calculator shows an error, because the present value would be infinite or undefined.

What Is a Growing Perpetuity?

A growing perpetuity is a series of cash flows that never ends and grows at a constant rate each period. The first payment $C$ arrives at the end of period 1; the second is $C(1+g)$, the third $C(1+g)^2$, and so on forever. Although the payments continue indefinitely, their present value is finite as long as the discount rate is higher than the growth rate, because each payment is shrunk by discounting faster than it is enlarged by growth.

This concept is one of the cornerstones of finance. It underlies the Gordon Growth Model used to value stocks that pay steadily rising dividends, the terminal value in a discounted cash flow (DCF) analysis, and the valuation of real estate, preferred shares, and endowments that are expected to generate income far into the future.

How you interpret the result depends on your inputs. A higher discount rate — reflecting greater risk or a higher opportunity cost — lowers the present value, while a higher growth rate raises it. Small changes in either input can move the result a lot, because the denominator $(r - g)$ is often a small number. For instance, with $1,000 growing at 3%, raising the discount rate from 8% to 9% drops the value from $20,000 to $16,666.67.

Assumptions, Limitations and Practical Tips

Constant growth forever: the formula assumes the payment grows at exactly $g$ every year, indefinitely. In practice, growth rates change over time, so the result is only as reliable as this assumption.
Timing: the first payment is assumed to occur one full period from today. If your first payment arrives immediately (an annuity-due), multiply the result by $(1 + r)$.
r must exceed g: if the growth rate equals or exceeds the discount rate, the stream has no finite present value and the calculation fails.
Sensitivity: because $PV = C / (r - g)$, the output is highly sensitive to the difference $(r - g)$. Test a range of realistic rates before relying on a single figure.
Use in DCF models: when valuing a business, the growing perpetuity typically applies only to cash flows after an explicit forecast period, with a conservative long-term growth rate (often close to inflation or GDP growth).

If you need to value a payment stream that ends after a fixed number of years, use the Present Value for Annuity Calculator instead. To discount a single future amount to today, see the Present Value for Future Value Calculator, and to evaluate the overall return of an investment, try the Return on Investment Calculator. For longer planning horizons, the Retirement Planner can help you model growing savings over time.

Frequently Asked Questions

How do I calculate the present value of a growing perpetuity by hand?

Subtract the growth rate from the discount rate, then divide the first payment by that difference: $PV = \frac{C}{r - g}$. For example, with a first payment of \$2,000, a discount rate of 10%, and growth of 4%, you get $2000 / (0.10 - 0.04) = 33,333.33$. Remember to express both rates as decimals, not percentages, before subtracting.

What is a good or typical discount rate to use?

There is no universal 'good' rate — it should reflect the risk of the cash flows and your opportunity cost of capital. Common choices range from government bond yields for very safe streams to 8–12% for risky business cash flows. Because the result depends heavily on the small difference $r - g$, always test a range of rates rather than trusting one number. For financial decisions, consider consulting a professional.

Can the growth rate ever be higher than the discount rate?

No, not in this formula. If $g \geq r$, the payments grow at least as fast as they are discounted, so the sum of the infinite series diverges and no finite present value exists. If you encounter this situation, the growth assumption is unrealistic over the long run — cap the horizon at a fixed number of years and use a growing annuity calculation instead.

What happens if the growth rate is zero?

With $g = 0$, the formula simplifies to $PV = C / r$, which is the present value of an ordinary (non-growing) perpetuity. A payment of \$1,000 discounted at 5% is worth $1000 / 0.05 = \$20,000$. So the growing perpetuity formula includes the simple perpetuity as a special case.

How do I handle payments that arrive at the start of each period?

The standard formula assumes the first payment arrives one full period from now (an ordinary perpetuity). If your first payment is due immediately, you have a perpetuity-due: multiply the result by $(1 + r)$. For example, if the calculator shows \$20,000 with $r = 8\%$, the due version is $20000 \times 1.08 = \$21,600$.

How is this formula used in the Gordon Growth Model for stocks?

The Gordon Growth Model applies the same formula to dividends: a stock's value equals next year's dividend divided by the required return minus the dividend growth rate. If a share will pay a \$3 dividend next year, growing at 4%, and you require a 9% return, the stock is worth $3 / (0.09 - 0.04) = \$60$. Note that $C$ must be the next-period dividend, not the one just paid.

What are the most common mistakes when using this formula?

The most frequent errors are using percentages instead of decimals, plugging in today's payment instead of the payment one year from now, forgetting that $r$ must exceed $g$, and applying an unrealistically high long-term growth rate in DCF terminal values. Also watch timing: quarterly or monthly payments require rates converted to the matching period before applying the formula.