$\frac{P}{E_0}$

This classic valuation ratio assumes constant annual earnings and no discounting, implying a direct recovery of the stock price over time through static earnings.

Here, E₀ is the most recent annual earnings per share.

This simplification provides an intuitive but incomplete measure of a stock's value.

$\text{PEG}=\frac{P/E}{g}$

where:

The PEG ratio introduces growth into valuation by dividing the P/E by g. However, this is a linear adjustment that neither reflects compounding effects nor the time value of money, and it makes no provision for discounting or risk. It thus provides a crude and often misleading representation of valuation under non-static conditions.

$\text{PPP}=\frac{\log \left(\frac{P/E\cdot \left(g-r\right)}{1+r}+1\right)}{\log \left(\frac{1+g}{1+r}\right)}$

where:

This formula calculates the number of years theoretically required for the present value of growing earnings to equal the current stock price, assuming full earnings distribution. Unlike the P/E and PEG ratios, the PPP captures nonlinear growth, incorporates the cost of capital, and yields a time-based valuation metric that reflects both earning power and risk-adjusted return expectations.

To make the PPP model’s foundation transparent, we derive it from fundamental financial valuation principles. Let a stock’s earnings grow at a rate g and be discounted at a rate r. Assuming full earnings distribution and starting earnings of E₀, the present value of future earnings, represented as an infinite series, is:

$P=\underset{t=1}{\overset{\infty }{{\displaystyle \sum }}}\frac{E_0{\left(1+g\right)}^t}{{\left(1+r\right)}^t}$

This infinite geometric series becomes:

$P=E_0\underset{t=1}{\overset{\infty }{{\displaystyle \sum }}}{\left(\frac{1+g}{1+r}\right)}^t=E_0\cdot \frac{\left(\frac{1+g}{1+r}\right)}{1-\left(\frac{1+g}{1+r}\right)}=E_0\cdot \frac{1+g}{r-g}$

This result applies only when g < r. When g ≈ r, we assume that g declines linearly toward r over time to maintain convergence.

The PPP formula generalizes this relationship by transforming the infinite series into a finite one. It removes the limiting condition g < r and defines the number of years N required for the present value of growing earnings to equal the price P, using a finite horizon:

$P=\underset{t=1}{\overset{N}{{\displaystyle \sum }}}\frac{E_t}{{\left(1+r\right)}^t}$

Solving this sum algebraically to isolate N is complex and does not lead to a simple formula. To overcome this difficulty, we use a logarithmic approximation method. This approach assumes that earnings grow geometrically and are discounted geometrically over time, meaning each year’s earnings are multiplied by a growth factor (1 + g) and discounted by a factor (1 + r). This leads to a clean formula for PPP, expressed in logarithmic terms.

Assuming $E_t=E_0{\left(1+g\right)}^t$, and simplifying algebraically, the PPP becomes:

$\text{PPP}=\frac{\log \left(\frac{P/E\cdot \left(g-r\right)}{1+r}+1\right)}{\log \left(\frac{1+g}{1+r}\right)}$

The assumption that the earnings growth rate g gradually declines toward the discount rate r reflects the real-life cycle of companies. While young firms may experience rapid growth, this growth typically slows as they mature, eventually converging toward the cost of capital. This assumption promotes conservative valuation practices and helps prevent the overestimation of long-term growth by avoiding unrealistic extrapolations.

Below is a sensitivity table showing how PPP varies as the relationship between g and r changes, with g linearly declining toward r:

Assuming full earnings distribution (i.e., a 100% payout ratio), we substitute:

$P=\frac{E_1}{r-g}\Rightarrow \frac{P}{E_1}=\frac{1}{r-g}$

Since $E_1=E_0\left(1+g\right)$, we express the price in terms of current earnings E₀, yielding the growth-adjusted P/E ratio:

$\frac{P}{E_0}=\frac{1+g}{r-g}$

This formulation introduces growth and discounting, improving upon the static P/E model. However, it remains dependent on idealized assumptions, including:

Moreover, the Gordon Growth Model suffers from a serious mathematical limitation: the requirement that g < r in order for the formula to be valid and convergent. This constraint severely restricts its applicability in high-growth scenarios, precisely when a more robust valuation tool is needed.

In contrast, the Potential Payback Period (PPP) provides a more flexible and realistic structure. It explicitly connects valuation to time-based recovery, risk, and nonlinear earnings dynamics, and does not suffer from the same restrictive condition. Thanks to its finite-horizon framework and logarithmic formulation, the PPP remains valid even when g ≥ r, accommodating real-world cases where short- to medium-term growth can exceed the discount rate.

The PPP formula relies on logarithmic expressions that can be expanded via the Taylor series:

$\log \left(1+x\right)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$

While the PEG ratio uses only the first-order term, the PPP retains the full structure, incorporating higher-order compounding effects. This makes the PPP more accurate and better aligned with how earnings accumulate over time under growth and risk.

The denominator of the PPP is:

$\log \left(\frac{1+g}{1+r}\right)=\log \left(1+g\right)-\log \left(1+r\right)$

Applying Taylor series expansions:

$\log \left(1+g\right)\approx g-\frac{g^2}{2},\log \left(1+r\right)\approx r-\frac{r^2}{2}$

Thus:

$\log \left(\frac{1+g}{1+r}\right)\approx \left(g-r\right)-\frac{g^2-r^2}{2}$

This shows that the PPP implicitly includes nonlinear terms that reflect compound effects, whereas the PEG ratio oversimplifies the growth-discounting relationship by omitting these corrections.

As g → r, the PPP becomes an indeterminate form:

$\text{PPP}=\frac{0}{0}$

Applying L’Hospital’s Rule:

$\underset{g\to r}{\lim }\text{PPP}=\frac{P/E}{1+r}\cdot \left(1+r\right)=\frac{P}{E}$

Thus, the PPP smoothly converges to the P/E ratio when growth and discount rates are equal, confirming that P/E is a special (or degenerate) case of PPP. In contrast, the PEG ratio diverges as g → 0, producing meaningless results in low-growth contexts and demonstrating its instability.

The PPP captures the core elements of modern financial theory:

By embedding these components directly into its structure, the PPP offers a more realistic and theory-consistent valuation model.

The PPP provides a clear time-based interpretation: how many years of discounted earnings are required to recover the current stock price. This interpretation is absent in both P/E and PEG, making PPP uniquely valuable for investors focused on long-term earning power and capital recovery.

This hierarchy reveals how PPP subsumes and improves upon both traditional metrics, offering a unified conceptual and mathematical framework for equity valuation.

Example – NVIDIA (as of October 2024):

$g_{\text{avg}}=\frac{g+r}{2}=\frac{0.5738+0.0403}{2}=0.30705$

g_avg is the earnings average growth rate over the PPP period.

$\text{PPP}=\frac{\log \left(\frac{69.31\cdot \left(0.30705-0.0403\right)}{1+0.0403}+1\right)}{\log \left(\frac{1+0.30705}{1+0.0403}\right)}\approx 12.85\text{years}$

PPP is the number of years required for future earnings—growing at an average rate of 30.705% and discounted at a rate of 4.03%—to equal the stock’s current price corresponding to a P/E ratio of 69.31 (concept of “potential payback period”).

$\text{PEG}=\frac{P/E}{g}=\frac{69.31}{57.38}=1.21$

This shows how the Price-to-Earnings to Growth (PEG) ratio—here at a level above the overvaluation threshold of 1—may falsely signal overvaluation, while the PPP indicates a fast payback period and robust earnings power.