# THE POTENTIAL PAYBACK PERIOD (PPP): A NEW METRIC FOR STOCK EVALUATION

TO ADDRESS THE SHORTCOMINGS OF THE PRICE-EARNINGS (PE) RATIO

### ABSTRACT

The Potential Payback Period (PPP) is proposed as a new metric for stock evaluation, aiming to address the limitations of the Price-Earnings (PE) Ratio. While the PE Ratio offers simplicity in assessing stock attractiveness, it fails to account for fundamental variables like earnings growth rate and interest rate, particularly affecting evaluations of high-growth stocks. Attempts like the Price-Earnings to Growth (PEG) Ratio offer improvements but remain empirical (being just a rule of thumb). The PPP, however, integrates earnings growth rigorously through a formula defining the time needed to equalize the stock price with future earnings, also incorporating interest rates for discounting future earnings. This synthetic forward-looking metric, which can be called "Dynamic PE Ratio", allows for meaningful stock comparisons, especially in cases where the traditional PE Ratio falls short, such as startups or companies with erratic earnings. The PPP's logical coherence translated into meaningfull figures, makes it a valuable tool, demonstrating the financial market's rationality and providing a dynamic approach for stock evaluation.

### KEY WORDS

Stock evaluation, Potential Payback Period (PPP), Price-Earnings (PE) Ratio, earnings growth rate, interest rate, Price-Earnings to Growth (PEG) Ratio, financial market, synthetic metric, dynamic perspective, high-growth stocks, startups.

#
INTRODUCING THE POTENTIAL PAYBACK PERIOD (PPP): A NEW METRIC FOR STOCK EVALUATION

TO ADDRESS THE SHORTCOMINGS OF THE PRICE-EARNINGS (PE) RATIO

Initially popularized by Benjamin Graham widely known as the "father of value
investing" who wrote the
discipline’s founding text “Security Analysis” in 1934, the Price-Earnings (PE) Ratio remains the cornerstone
metric for stock investors.

This measure has the merit of simplicity as it is obtained through a simple division of the stock price by the
corresponding earnings per share.

But although it's only meant to offer a quick glance at the relative “attractiveness” or “expensiveness” of a
stock, the PE Ratio soon reveals its shortcomings or operational limitations as a management tool.

In particular, there are two fundamental variables in stock evaluation that the PE Ratio does not explicitly take
into account – earnings growth rate and interest rate – whose variations strongly and daily influence stock
prices.

The questionable applicability of the PE Ratio to high-growth stocks

The first shortcoming of the PE Ratio is its inability to properly assess the value of high-growth stocks.

Why do some high-growth stocks with "prohibitive" PE Ratios continue to outperform the market?

For instance, a group of US stocks known as the “Seven Magnificent” (Alphabet, Amazon, Apple, Meta, Microsoft,
Nvidia and Tesla) have in common their relatively recent creation and exponential growth in sales and profit,
coupled with exponential increases in their share prices. Such valuations have led to PE Ratios which, by
traditional criteria, appear extremely high. Nevertheless, those high-growth stocks have been outperforming the
market over the last ten years or more.

Intuitively, we accept the idea of relatively high PE Ratios for companies with fast growth. But how high is
justified, and where is the limit? Can we still buy these shares and at what price should they be sold?

The PE Ratio seems ill-suited to measure the "attractiveness” or "expensiveness" of a stock because it does not
explicitly take earnings growth into account. We empirically notice a correlation between the PE Ratio and the
expected earnings growth rate but there is no proportionality between the two. Hence, the difficulty to make
well-founded evaluations and comparisons.

The Price-Earnings to Growth (PEG) Ratio is the first metric to adjust the PE Ratio
based on the earnings growth rate, but it’s just rule of thumb

The PEG Ratio is considered to be a convenient approximation, being simply the result of the division of the PE
Ratio by the earnings growth rate. If the PE Ratio is symbolized by “PER” and the earnings growth rate by “g” then

According to Wikipedia the PEG concept was originally developed by Mario Farina who wrote about it in his 1969
Book “A Beginner's Guide to Successful Investing in The Stock Market.” It was later popularized by Peter Lynch,
who wrote in his 1989 book “One Up on Wall Street” that "The PE Ratio of any company that's fairly priced will
equal its growth rate," meaning a fairly valued company will have its PEG Ratio equal to 1. This implies that any
stock with a PEG Ratio above 1 would be considered “overvaluated.”

As an equities valuation tool the PEG Ratio is thought to enhance the PE Ratio because it provides a more complete
picture by introducing expected earnings growth into the calculation. However, the way earnings growth rate “g” is
factored in to obtain the PEG Ratio (just by dividing the PE Ratio by “g”) and the way the result of the division
is interpreted (with the figure “1” being the threshold above which a stock would be considered relatively
“overvalued”) show that the PEG Ratio essentially rests on a rule of thumb. Such an empirical and approximative
method may be useful occasionally but it has clear limitations because it lacks rigorous theoretical foundation
and therefore cannot be applied in all circumstances.

The Potential Payback Period (PPP) is a new synthetic metric that incorporates earnings
growth in a most rigorous manner

The stock’s Potential Payback Period (PPP) is defined as the precise amount of time necessary to equalize the
current stock price with the sum of future earnings per share after taking into account their growth rate “g”.
Through a slightly more complex formula that that of the PEG Ratio, the PPP moves from an empirical and
approximative approach to mathematical logic and precision with the following formula

In the above PPP formula, "g" is the same earnings growth rate as the "g" in the PEG formula

The formula of the Potential Payback Period (PPP) for stock valuation is based on that of the sum of the first n
terms of a geometric sequence.

Consider the geometric progression: E_{0} , E_{0}q , E_{0}q^{2},
E_{0}q^{3}, ………… E_{0}q^{n-1}

Let’s the sum of n terms be P, the first term be E0 and the common ratio of geometric progression be q.

Then P = E_{0} + E_{0}q + E_{0}q^{2} + E_{0}q^{3} + ……. +
E_{0}q^{n-1} [A]

Multiply both sides of P by q, we get,

Pq = E_{0}q + E_{0}q^{2} + E_{0}q^{3} + ……. + E_{0}q [B]

Subtracting [B] from [A], we get,

Pq - P = (E_{0}q + E_{0}q^{2} + E_{0}q^{3} + … +
E_{0}q^{n}) – (E_{0} + E_{0}q + E_{0}q^{2} +
E_{0}q^{3} + ……. + E_{0}q^{n-1})

Pq - P = E_{0}q^{n} – E_{0}

P (q – 1) = E_{0} (q^{n} – 1)

P = E_{0}> (q^{n} – 1) / (q – 1)

Hence the sum of n terms of the geometric sequence is

P = E_{0} (q^{n} – 1) / (q – 1)

P/E_{0} = (q^{n} – 1) / (q – 1)

Now let’s calculate n, which is the Potential Payback Period (PPP), while P is the current share price.

E0 is the earnings per share for the current year.

P/E0 is the Price Earnings Ratio (PER) , which we name X for now.

X = (q^{n} – 1) / (q – 1)

q^{n} – 1 = X (q – 1)

q^{n} = X (q – 1) + 1

Log q^{n} = Log [X (q – 1) + 1]

n Log q = Log [X (q – 1) + 1]

n = Log [X (q – 1) + 1] / Log q [C]

In our sequence of future earnings per share, if g is the annual earnings growth rate, then the common ratio q of
geometric progression is q = 1 + g based on the compound interest formula.

On the right side of equation [C]: (q – 1) = 1 + g – 1 = g

Noting that n is the Potential Payback Period (PPP), X is the PE Ratio or PER, q – 1 = g
and q = g + 1, we have from [C]

By adjusting the PE Ratio to rigorously incorporate earnings growth the PPP is meant to allow more meaningful
stock comparisons.

The PPP is viewed in the same way as the P/E Ratio, meaning the smaller the amount of the metric the more
attractive the stock, all else being the same.

The higher the earnings growth rate “g” the shorter the PPP, which varies with respect to the PE Ratio in a
pattern shown in the following graph.

As seen from formula [1], when g = r =
0%, PPP = PE Ratio. Then the PPP and PE
Ratio curves correspond to the
bisector.

The PPP curve deviates below the bisector (PPP < PE Ratio) when the
earnings growth rate is greater than the discount rate applied to these earnings (g> r). The PPP decreases
more rapidly relative to the PE Ratio with the
acceleration of the earnings growth rate (here from 0% to 30% per year).

After all, the PE ratio is only a special case of the PPP where there is no earnings growth. As explained in
Investopedia article “What Is the Price-to-Earnings (P/E) Ratio?”, “Simply put, a PE ratio of 15
would mean that
the current market value of the company is equal to 15 times its annual earnings. Put literally, if you were to
hypothetically buy 100% of the company’s shares, it would take 15 years for you to earn back your initial
investment through the company’s ongoing profits assuming the company never grew in the future.” In this specific
case where g = 0%, PPP = P/E Ratio, as represented by the bisector in the above graph.

Because we must reject the unrealistic “0 growth” assumption, we have to factor in the earnings growth rate “g” as
in the above PPP formula [1].

For example, if g = 8 (percent per year), the PPP would amount to 10.24 (down from 15
if g = 0).

Instant calculations of the PPP can be performed at at
https://stockinternalrateofreturn.com/instant_calculations.html

PPP’s sensitivity to the earnings growth rate "g"

The stock market is highly sensitive to any change in the expected earnings growth rate. It is not the growth rate
itself or the speed (concept of "first derivative" in physics) that matters most, but its acceleration or
deceleration (concept of "second derivative").

This explains why stock prices can be so sensitive to quarterly earnings revisions, which can lead – through
extrapolations – to revisions of earnings growth rates over a period well beyond the quarters in question.

Any new earnings growth rate immediately modifies the level of the PPP, automatically resulting in an adjustment
of the stock price.

In any case we must remember that the reliability and the precision of any evaluation model, no matter how
relevant it may be, depend on the reliability and the precision of the data put into it. For the PPP the most
sensitive data is "g", the estimated earnings growth rate for the next two to three years (possibly and
temporarily extrapolated beyond), which has to be updated according to the most relevant and most recent
information. The same remark applies to the "g" used to calculate the PEG Ratio.

Incorporation of the interest rate "r" in the PPP

Furthermore, in order to take into account inflation and/or an opportunity cost for an investor in shares, future
earnings must be discounted to their present values using a long-term risk-free interest rate as the discount
rate, which we call "r".

With the introduction of the interest rate "r" and based on the discount rate formula, the above-mentioned
sequence of future earnings per share becomes a geometric progression with (1 +
g) / (1 + r) as common ratio.

In the previous demonstration of formula [1], we came to the point where

n = Log [X (q – 1) + 1] / Log q [C]

where n is the Potential Payback Period (PPP), X is the PE Ratio or PER, and q is the common ratio of the
geometric sequence of future earnings per share.

Now we have q = (1 + g) / (1 + r)

On the right side of equation [C], (q – 1) can be developed as follows:

(q – 1) = [(1 + g) / (1 + r)] – [(1 + r) / (1 + r)] = (g – r) / (1 + r)

Then

n = Log [PER x [(g – r) / (1 + r )] + 1] / Log [(1 + g) / (1 + r)]

Finally here is the formula of the PPP after introduction of interest rate "r" :

Therefore, with the introduction of an interest rate to discount future earnings, the formula of the Potential
Payback Period (PPP), transforms

from

to

Where

PPP = Potential Payback Period

PER = PE Ratio

g = annual earnings per share growth rate

r = long-term interest rate used as discount rate

Continuing with the above example (with P/E ratio = 15 and g = 8), if r = 3 (percent), then PPP = 11.54 (up from
10.24 when r = 0).

The inverse relationship between interest rates and stock prices is known intuitively and empirically, but the PPP
allows for the measurement of the impact of a change in interest rate on the price of a stock.

The PPP is a “Dynamic PE Ratio”

As seen from formula [2] the PPP only coincides with the PE Ratio
(PPP = PE Ratio) in the very theoretical case where g = r = 0% (assuming a stagnant world) or in very exceptional
cases where g = r = any value. But normally, g is different from r, and the PPP differs from the PE Ratio.

Therefore the PPP appears as a generalization of the PE Ratio with possibilities to take into account various and
varying earnings growth rates and interest rates in the stock evaluation process. Professor Emeritus of Finance at
ESSEC (a top French business school) Patrice Poncet considers the PPP to be a "nice generalization of the P/E
Ratio" and that it represents an “undeniable improvement” in financial analysis.

Because it starts from the PE Ratio and incorporates two additional variables that are fundamental in determining
the value of a share – namely the earnings growth rate and the interest rate – the PPP can be considered a
synthetic metric from a dynamic perspective.

If the PE ratio is a snapshot based on earnings for a single year, the PPP is a video that captures the evolution
of these earnings in real terms over a more extended period, defined as the time required ("period") for the
discounted future earnings flow to equal the current stock price. To underline its difference from the traditional
PE Ratio, which is a static metric based on earnings from a single year, the PPP can be called “Dynamic PE Ratio.”

Other attempts to incorporate earnings from several consecutive years in the PE
Ratio

A metric such as the traditional PE Ratio that is based on earnings for a single year inevitably leads to an
incomplete or biased view of the financial reality.

Using the Trailing-Twelve-Month PE Ratio or the Forward P/E Ratio enables investors to more closely track a
dynamic reality and even anticipate it by a year. However, the calculations are still based on earnings from a
single year.

The task of smoothing the PE Ratio based on earnings from several consecutive years was methodically undertaken
for the first time by Robert Shiller, who created the Cyclically Adjusted PE Ratio, or CAPE Ratio, or Shiller PE,
or PE 10. The CAPE Ratio is based on the average inflation-adjusted earnings from the previous 10 years.

The CAPE Ratio helps assess whether the stock market – as represented by the S&P 500 – is overvalued or
undervalued. The higher the ratio, the more overvalued a market. Over more than 100 years, the average and median
CAPE Ratio has been around 16 or 17, spiking up significantly higher often before market crashes.

It reached an all-time high in December 1999 at 44.19. Based in part on that record high ratio, Shiller correctly
predicted the market crash that occurred at the beginning of year 2000, as he explained in his book “Irrational
Exuberance” published the same year. Shiller received the Economics Nobel Prize in 2013.

But like the traditional PE Ratio, the CAPE Ratio is based on historical performance and therefore has the
limitations of a backward-looking metric when it comes to estimating individual companies’ profit-making capacity
for the future, knowing that it is this profit-making capacity that essentially determines the value of their
shares.

Contrary to the CAPE Ratio, the PPP or “Dynamic PE Ratio” is a forward-looking metric through the expected rate of
earning growth “g”.

The PPP as an alternative metric when the traditional PE Ratio is not applicable in the
cases of startups, temporarily loss-making companies or those in a turnaround situation

Investors often find themselves needing to evaluate companies for which the PE Ratio cannot be calculated, at
least temporarily. This is particularly true for startups or companies in a turnaround situation, or those
undergoing restructuring.

In a common scenario, these companies often incur losses in the current year (referred to as "year 0" in this
article), followed by results close to zero for the year after ("year 1"). Finally, they show a more "normal"
profit in the following year ("year 2"). It is from that "year 2" onward that the company enters a steady growth
phase characterized by more or less regular profit growth.

In such a scenario, calculating the PE Ratio for the company is impossible or meaningless for years 0 and 1.
Comparing the company with others in the same sector that have well-defined PE Ratios is also not feasible.

In such a situation the PPP can be an alternative metric for stock evaluation and comparison

Unlike the traditional PE Ratio, which measures the "attractiveness" or "expensiveness" of the stock based on the
earnings of a single year, the PPP does so on the basis of earnings generated over a much longer period of time,
in fact over as many years as it takes to equalize those future earnings with the current share price. By doing
so, it reduces the impact on the evaluation of any "exceptional" earnings or losses for one or two particular
years.

In the case of special situations like those described earlier, the above PPP basic formula [2] can be adapted by
directly introducing the loss for year 0, the near-zero profit for year 1, and the "normal" profit for year 2,
finally giving the following adapted formula:

First, let’s take the example of a company A for which the PE Ratio cannot be calculated based on the following
data, which we apply to the adapted formula [3]:

P = 100

E0 = −10

E1 = 0

E2 = +10

g = + 8%,

r = 3%

Instant calculations of the PPP with all possible simulations can be performed at
https://www.stockinternalrateofreturn.com/instant_calculations.html

This example gives a PPP = 11.47. This PPP is truly significant as a "period" expressed in years and can be
meaningfully compared with the PPP of any other company, regardless of its PE Ratio and whether it is significant
or not.

Now, consider the case of a company B that is somewhat similar to company A but with more regular results. Its
earnings per share start at 10 in year 0 and steadily increase by 8% per year. With a PE Ratio of 10 from the
start (100/10) and an interest rate held at 3%, applying the PPP basic
formula [2] gives a PPP = 8.35.

Company A's PPP (11.47) can be directly compared with company B's PPP (8.35), while no comparison is possible
based on the PE Ratio. Based on the PPP we can say that B is more attractive than A, all else being equal.

Relying solely on the PE Ratio to assess the value of a stock can lead to absurd conclusions. Indeed, the PE Ratio
can reach astronomical levels when the earnings per share for the selected year are close to 0, and it loses all
meaning in the case of losses for the considered year. On the other hand, the PPP can be calculated meaningfully
for any stock at any time, even for startups or companies in a turnaround situation or undergoing restructuring
involving temporary losses.

Homogeneity and rationality of the financial market

Whereas, in practice, the PE Ratio can take any value (up to infinity), the PPP varies within a relatively narrow
range of about 5 to 15 (years). Such figures can be considered significant, realistic, and credible due to their
reasonable order of magnitude and relative stability, demonstrating the homogeneity and rationality of the
financial market.

Unlike the PE Ratio, which can lose all meaning for certain stocks during certain periods, the PPP is a more
stable because more logical measure that remains always meaningful in space and time and can always be used for
comparisons between companies.

Because the question "how many years does it take to potentially recover the current share price through future
earnings?" makes sense, the answer provided by the PPP as a dynamic metric, also makes sense.

The financial market confirms its rationality when we use appropriate metrics.

Origin of the PPP

The concept of PPP for stock evaluation is derived from the well-known "payback period" in corporate finance. I
first exposed it in a series of articles in the French-language review Analyse Financière in the 1980s. The title
of the first article (second quarter 1984) was "Le PER, un instrument mal adapté à la gestion mondiale des
portefeuilles. Comment remédier à ses lacunes (The PE Ratio, an instrument ill-suited for global portfolio
management. How to remedy its shortcomings). The approach was subsequently explained in the reference book
"Finance d'Entreprise" (Corporate Finance)
by Pierre Vernimmen, a professor at France’s HEC, the country’s most
prestigious business school.

I have applied the PPP concept to evaluate and compare financial markets and company stocks over the past 12
months on my Website and my Linkedin page.

Rainsy Sam

https://www.stockinternalrateofreturn.com/

https://www.linkedin.com/in/rainsy-sam-2891a347/