4.1. Synthetic Values
To see how this new formula is similar numerically to existing formulas, we can start with a simple example. Suppose Company A has a price of
$10 and an EPS of
$1. The P/E ratio is 10. For a fair valuation, the growth rate expectation must be 10%. If the PEG ratio is to remain at 1 over the next 5 years, the price by the end of the fifth year will be
$16.1 and the EPS will be
$1.61. The P/E ratio will be 10 and the PEG ratio will be 1 by the end of the fifth year. You can use any earning growth rate assumptions, so long as the current PEG is 1, and draw the same conclusion.
Table 1 displays the details.
Thus, for a fairly priced stock with a constant growth rate, PEGF, PEGL and PEGC are the same.
What if the price over a given period remains the same while earning growths at 14.87% a year? If the stock starts the period fairly-valued, then the P/E had to be 14.87. We do not really need to know what the starting EPS is. We can use
$1 for simplicity. If the growth rate is 14.87%, then by the fifth year, EPS would have doubled and the price would remain at
$14.87. The P/E ratio by the end of fifth year would be 7.435, and with an earnings growth rate of rate of 14.87%, the PEG value will be 0.5.
Table 2 provides the numerical results.
Using the new equation, we will arrive with the same value:
This the value of PEGF.
What insights can we gain from this new equation? Using the value in Equation (4), if we assume that the earning growth rate remains at the same rate over the next 5 years, the PEG ratio for the next 5 years will have to be 2 for the entire 10-year period to have a PEG ratio of 1 (stock will be fairly valued over a 10-year period). In other words,
where
PEGC1 is the
PEG ratio over the last 5 years and
PEGC2 is the
PEG ratio for the next 5 years. For
PEGC2 to be at the value of 2, the price over the next 5 years will have to increase by 300%. The usefulness of the new equation is that it allows for any combination of earning and price growth rates, so long as it satisfies the condition that the PEG ratio for the next 5 years is 2. In this example, for a PEG value of 2, the price will have to increase from
$14.87 to
$59.48, with EPS increasing to
$4 if the growth rate assumption remains the same. This gives a P/E ratio of 14.87 and thus a PEG ratio of 1 over a 10-year period. The following table (
Table 3) shows the numerical outcomes.
What if the price over the last 5 years has been growing at a rate far higher than the earnings growth? If we assume that the price has been growing twice (in geometric relative) as much as the earnings over the last 5 years, the end of the 5-year period price will be
$44.61, with an EPS of
$2. This gives a P/E of 22.3 and a PEG ratio of 1.5. For the PEG for the entire period to be 1, the next 5-year PEG has to be 0.667. Assuming that the earnings increased to
$4 (100%) by the end of the next 5 years, the price has to increase only by 33.33% over the next 5-year period. The end price will, of course, be
$59.48. Thus, a P/E of 14.87 and a PEG of 1 is restored. The results are shown in the
Table 4.
The proposed formula also allows for negative price growth and any given time length. Consider a stock with PEG of 1 and P/E of 10 at the beginning of the period. By the end of the period, the PEG is 0.5, with EPS growth at 60% over the same period. Using the new formula, price growth over the same period had to be −20%. Numerically, if we start with price of
$10 and an EPS of
$1, the EPS by the end of the period will be
$1.6. If the price declined by 20%, the price will be
$8 at the end of the period, and the P/E will be 5 at the end of the period. At what annual growth rate would you obtain an EPS from
$1 to
$1.6? The answer is about 10% over 5 years. Therefore, the PEG will be 0.5 using the existing formula, which agrees with the new formula.
Table 5 shows the numerical results.
From the numerical examples shown above, we can see that the conclusions we can draw from the new formula are the same as the existing formulas. However, the new formula improved upon the existing formula in that it shows what the earnings growth and price growth must be within the period that an analyst is considering. It also allows for any growth rate assumptions (including negative growth) so long as they produce the value needed to bring PEGC to the desired level.
Another important question is does the ending or beginning value for PEG have to be 1 for the new equation to work? The answer is that it does not. We can generalize Equation (5) as:
where
PEGm is the PEG ratio in period m,
PEGn is the PEG ratio in period n, and
PEGm+n is the PEG ratio for the combined (end) period. For example, if the ending PEG ratio is 0.9 and
PEGn is 1.5, then
PEGm has to be 0.6, and, so, an analyst who spotted a stock with a PEG ratio of 1 currently, looking back at the PEG ratio for the last 3- or 5-year period as having PEG ratio of 1.5, can deduce that the stock must have had a PEG ratio of 0.67 over the same time distance in the past (3 or 5 years ago). Note that using Equations (5) and (6), an analyst can feel comfort in recommending a stock with short term PEG near or even above 2, as long as the previous period PEG is near or below 0.5.
Trombley (
2008) and
Schnabel (
2009) show that firms with a low cost of equity and a high growth rate can have a PEG ratio above 1. A low cost of equity implies a higher forward growth rate potential. With the stock price growth lagging in previous years, it is only reasonable for the stock price growth to catch up in following period.
The proposed new equation also has another unique property: it connects the forward PEG and the backward PEG. Let us say that period
m is time zero and the stock starts with a known forward PEG (P/E ratio at time 0 and forward growth rate). If the growth rate in
PEGL is the unbiased forecast of future growth rate, by the time we reached time
n, we would have the backward PEG, which is
PEGF. Thus, we have:
Equation (7) connects the Farina PEG ratio (backward PEG) and the Lynch PEG ratio (forward PEG).
Equation (7) can be further generalized. Since Equation (2) is a ratio of geometric return relatives, the generalized PEG is:
Whether you believe in Lynch’s argument that P/E ratio and growth rate have to be the same, or that the market is efficient in the long run, the value of Equation (8) has to be 1 given long enough T.
What is more striking with Equation (8) is that it gives us a new way to calculate the PEG ratio without growth rate assumptions. Since Equation (8) is a product of geometric relatives in both the numerator and denominator, the equation can be simplified to:
Equation (9) shows that the
PEGC ratio from time 0 to time
T is simply the
P/E ratio at time T divided by the
P/E ratio in time zero. This can be generalized into any specific time length T. Consider a stock with a
P/E ratio of 30, with a stock price of
$30 and an EPS of
$1. Assuming that the forward EPS growth rate is 20% for the next period (5 years), it would give us a PEG ratio of 1.5. Further, we can assume that the stock price growth rate is lower than the earnings growth at 15%. The following table (
Table 6) shows the numerical results:
The EPS will increase to $2.49 and stock price will increases to $60.34. Thus, it gives us a P/E ratio of 24.25 in year 5. The PEG ratio calculated with the P/E ratios (PEG-PE) from the period is 0.808, the same as the result with the value obtained by applying Equations (3) and (9).
It is important to note that the PEG ratio in the fifth year in this example is 1.215, which is a little more than 1. For the 10-year period to have a PEG of 1, earnings will have to keep growing at a rate above stock price increases over the next 5 years. This is reasonable considering that the stock had a PEG of 1.5 in time 0. The ending PEG is 1.215, which is the product of 1.5 and 0.808 (the PEG for the 5-year period we considered), and is the outcome we can expect by applying Equation (6). If we assume that earnings can keep growing at 20% for the next 5 years, the EPS will be
$6.19 by year 10. To have a P/E of 20 (thus giving us a forward PEG of 1), the stock price has to be
$123.8 by year 10. The PEG ratio over the next 5 years will be 0.824. The product of the first 5-year PEG and next 5-year PEG gives us a 10-year PEG of 0.667, which, when timed with the initial PEG of 1.5, gives us an overall PEG of 1––exactly the same as the outcome of applying Equation (5).
Dukes et al. (
2006) found that one of the most commonly used methods for valuating common stocks among practitioners is the current P/E ratio multiplied by the forward EPS. Equation (9) shows how such a valuation method can produce the PEG ratio.
What insight could an analyst take from this? We often think that a stock with high P/E might be a poor choice for long-term investment. However, the higher P/E is justified if the earnings growth rate is high enough and remains high for some time into the future. For example, if an analyst forecasts that a company currently has a P/E of 10 and has a forward EPS growth rate of 20%, the PEG ratio is 0.5. This would mean that the P/E ratio in 5 years would be 5. Let us assume that the company has an EPS of $1, thereby giving us a current price of $10. The growth rate forecast would imply an EPS of $2.49 in year 5, which would imply the stock price to be $12.45 in 5 years. That is highly unlikely given that the company’s growth rate is at 20%. Therefore, the stock price would have to increase to $24.49 a share for the P/E ratio to be 10 by year 5. However, at a P/E ratio of 10 and a growth rate of 20%, the PEG ratio will remain at 0.5. Therefore, the stock price growth figure can be adjusted upward. If we take the assumption that a PEG ratio between 1 and 2 is a fair value, the stock price could have a range of $50 to $100. This would mean that the stock price could increase anywhere between 400% and 900%, and the stock can still be fairly valued. It could also mean that the 5-year PEG ratio could be between 2 and 4, if the previous PEG ratio is 0.5. Conversely, if a stock that has a current P/E of 40 and a PEG ratio of 2, it would imply that the future P/E has to be 80 at the end of the timeframe. This P/E ratio would put the stock in a highly speculative range.
4.2. Industry Examples
4.2.1. Unusually Large Value for Earnings Growth
The last example mentioned above also shows another advantage of the proposed new formula: it can accommodate unusually high growth rates. With a price growth of 400% to 900% in a 5-year period, it would require a P/E ratio of a similar value for the old PEG formulas to work. For mature companies that experience periods of a high growth rate, they will not have the high P/E ratio to match the high growth rate. Thus, the PEG ratio will be completely off. However, the formula proposed in this paper can handle any growth rates.
Let us consider a real-world example. LRCX had an EPS of $6.3 in the fiscal year 2014 and an average stock price of $72 near the end of the fiscal year. It had a P/E ratio of 11.43. By the fiscal year of 2019, the company had an EPS of $14.6, a 5-year geometric growth rate of 18%. The PEGF ratio value is 0.63 during that period. The stock price in October of 2019 was $280 a share, implying a P/E ratio of 19.18. The PEGC ratio in October of 2019 was 1.07. The cumulative EPS growth during that 5-year period was 132%, while the stock price cumulative growth was 289% (LRCX started to pay a dividend in 2017, so this figure will be higher when adjusted for dividends). The stock price was increasing at a rate more than twice the stock price, yet the PEGC ratio still implies LRCX a value stock (below 1). The main question is whether investors/analyst foresaw the rapid growth rate of LRCX earning? In the fiscal year of 2010, LRCX had an EPS of $2.73. So, the geometric growth rate from the 2010–2015 period was 18%. The company’s stock price was $46 at the end of the fiscal year of 2010, which implies a price growth of 9.4%. The P/E ratio in 2010 was 16.85. With a forward growth rate of 18%, LRCX had a PEG ratio below 1 in 2010. However, by 2014, the PEG ratio dropped to 0.63 due to rapid a EPS growth relative to stock price growth (PEGC ratio between 2010 and 2014 was 0.65, and the price increased by 150% and EPS by 230%). So, the price increase between 2014 and 2019 was simply the company’s stock price catching up with the EPS growth. From the fiscal year of 2019 to 2021, EPS growth was at about 100%, which was about the same as the price growth ($560 per share), giving us a PEG value of 1. This would imply that the price of LRCX is at most fairly valued. It could go to as high as $1120 a share and still have a PEG below 2.
How does the performance of the proposed formula compare to the exiting formulas? Let us assume that the 18% EPS growth rate can continue for another 5 years, which would mean that by 2024, the company would have an EPS of $33.4 a share. Using the proposed formula, we can form a better idea of what the potential price range could be for LRCX, since the geometric growth relative to the earnings will be 12.23 between 2010 and 2024. To achieve a PEGC ratio of 1, the price has to increase at the same multiple, from $46 a share to $562.78 a share, about double the current price of $280 a share (the first draft was written in November 2019, when LRCX was trading for $280 a share). Even at $562.78 a share, the company’s stock would still be considered a value stock since the PEGC ratio would still be at 0.94, according to Equation (6). If we take the average of 1 and 2 and a PEG ratio of 1.5, the price could be as high as $844 a share, and the company’s stock would still be considered fairly valued (LRCX has an average stock price of $600 over the 2021–2022 period). This would imply a price growth of about 18 times over 15 years. According to Farina’s formula, the P/E ratio for LRCX has to be in the 100+ range. LRCX never achieved an average P/E above 50 during the last 12 years. According to Lynch’s formula, the P/E ratio has to be below 20 (since the forward growth rate is 18%) for the stock to be considered fair value. Again, LRCX never achieved an average P/E of below 20 over the last 12 years. Therefore, one would conclude that LRCX is an expensive stock according to Lynch’s formula, and that it is an ultra-cheap stock according to Farina’s formula. However, according to the proposed formula, the price of LRCX is considered a fair value between $560 and $1120. The price of LRCX reached above $800 a share by late 2021.
4.2.2. Negative Earnings and Price Growth
Perhaps the most useful application of the proposed formula is it can accommodate negative growth rates. In 2014, Exxon Mobile (XOM) stock was trading for $100 a share. Five years later, it was trading for $76 (−24% return). During the same period, the EPS went from $7.37 to $4.88 (−33.79% EPS growth). By plugging these numbers into Equation (3), we obtain a PEGC value of 1.15 for XOM. For a company with a negative EPS growth, a PEG ratio above 1 seems expensive. The price for XOM continued to decline to as low as $33 a share by late 2020. With an estimated EPS of $3.36 for the fiscal year of 2019 (reported in the middle of 2020), the resulting PEG for XOM was 0.72, making XOM a value stock by definition. By the beginning of 2022 (before the oil price shock caused by Russia’s invasion of Ukraine), XOM stock price recovered to $80 a share.