A growth adjusted price-earnings ratio
aa r X i v : . [ q -f i n . GN ] J a n A growth adjusted price-earnings ratio
Graham Baird ∗ , James Dodd † , Lawrence Middleton ‡ January 24, 2020
Abstract
The purpose of this paper is to introduce a new growth adjusted price-earnings measure(GA-
P/E ) and assess its efficacy as measure of value and predictor of future stock returns.Taking inspiration from the interpretation of the traditional price-earnings ratio as a period oftime, the new measure computes the requisite payback period whilst accounting for earningsgrowth. Having derived the measure, we outline a number of its properties before conductingan extensive empirical study utilising a sorted portfolio methodology. We find that the returnsof the low GA-
P/E stocks exceed those of the high GA-
P/E stocks, both in an absolute senseand also on a risk-adjusted basis. Furthermore, the returns from the low GA-
P/E porfoliowas found to exceed those of the value portfolio arising from a
P/E sort on the same poolof stocks. Finally, the returns of our GA-
P/E sorted porfolios were subjected to analysis byconducting regressions against the standard Fama and French risk factors.
The classical strategy of value investing involves purchasing stocks which are deemed to be under-valued relative to their intrinsic value. In practice, when it comes to determining whether a givenstock is undervalued or not, investors typically rely on a number of standard value metrics, forexample, a stock possessing a high book-to-market (
B/M ), or alternatively a low price-earningsratio (
P/E ) would generally be seen as a ‘value stock’. At the other end of the spectrum are‘growth stocks’ which typically trade at high multiples of both book value (low
B/M ) and earnings(high
P/E ). Generally investors are willing to pay this higher price, relative to the current intrinsicvalue, as it is anticipated that the company will experience significant growth in their earnings andfuture book value. Often growth stocks are associated with newly listed companies operating inrapidly expanding industries, such as technology and healthcare.The value approach has long been employed by investors, with Benjamin Graham and DavidDodd being amongst the earliest proponents of the style, espousing its merits at length in theseminal text Security Analyis [27]. This led to Graham being given the title “the father of valueinvesting”, and the pair have attracted many notable followers within the investment community.There has also been a great deal of debate within the academic literature regarding both the generalefficacy of value investing and also in explaining the strategy’s historical performance. Focussingon the price-earnings ratio in particular, significant works include Nicholson [40], who presented thefirst study to systematically demonstrate a ‘
P/E effect’, whereby stocks with lower price-earningsratios provide greater subsequent returns than those with higher price-earnings ratios. The laterwork by Basu [6], offered more comprehensive evidence for the existence of such a
P/E effect, withthe author further demonstrating that the outperformance persists even when returns are adjustedfor risk. In the intervening period since these early articles there has been much discussion as tothe veracity of the price-earnings effect. Some have suggested that the price-earnings ratio simplyacts as a proxy for other factors [3], for example firm size [43], or that the observed effect occurs asa consequence of biases in the datasets utilised [5], whilst other studies have continued to supportthe existence of the price-earnings effect [31]. ∗ Mathematical Institute, University of Oxford † St. Cross College, University of Oxford ‡
1t is now generally accepted that stocks with low price-earnings ratios offer higher returnsthan those with a high price-earnings ratio, a finding consistent across markets and time periods.Furthermore, this represents just one manifestation of a wider value effect, with similar resultsobtained across the class of value metrics [21]. This being the case, the discussion has largely turnedto providing an explanation for the additional returns offered by value stocks. One explanation,advanced most notably by Eugene Fama and Kenneth French, is that value stocks tend to beassociated with firms which have experienced a period of distress, and as such provide inherentlyriskier investment opportunities [19]. In light of the higher risk, the additional return observedwith value stocks simply represents the necessary compensation required by the market. Thishas led to the inclusion of a ‘value risk factor’ in a number of multi-factor extensions of thetraditional capital asset pricing model (CAPM), for example Fama and French’s own three [20]and five [23] factor models, and Carhart’s four factor model [11]. Alternatively, some authors, forexample De Bondt and Thaler [15, 16], Chopra, Lakonishok and Ritter [39], Lakonishok, Shleifer,and Vishny [32] and Haugen [28] argue that this additional return for value stocks arises becausethe market has a tendency to overreact to temporary short-term circumstances. In particular,undervaluing currently distressed stocks whilst overvaluing the stocks of recently successful firms(growth or glamour stocks). When the circumstances are resolved or the exceptional success cannotbe sustained, these pricing errors are corrected and the distressed (value) stocks experience higherreturns in comparison to growth stocks.As discussed, traditional value stocks are those which trade at a low price relative to theirintrinsic value, a value which is determined largely with respect to the current circumstances ofthe firm. However, one might argue that in order to determine the true intrinsic value of a stock itis necessary to take account of the growth prospects of the underlying firm. If such prospects aretaken into account within our calculations, then we might expect to find true value prospects acrossthe traditional value/growth spectrum. It is the purpose of this paper to propose a new ‘value’measure which takes account of both present value and earnings growth. Having introduced themeasure and examined its properties, we examine its efficacy as a measure and predictor of futurereturns through an extensive empirical study, covering stocks listed on the the NYSE, Nasdaqand AMEX over the period 1990 to 2015. However, we begin by reviewing the presently availableprice-earnings based measures and outlining some of their features.
Of the range of value measures, the price-earnings ratio or
P/E -ratio is perhaps the most widelydiscussed metric when it comes to stock valuation. Defined as the ratio of a company’s stock priceto (some measure of) its annual earnings per share (EPS), the price-earnings ratio may be viewedas the price investors must pay per unit of current earnings, or alternatively as the period requiredin order for a stock to accrue its current value in earnings, assuming that earnings remain constant.Whilst an apparently simple measure, multiple variants of the
P/E -ratio exist, with these mostlyvarying in the earnings figure used within the calculation.When the earnings figure is derived from the previous fiscal year, or alternatively the previouslyreported four quarters, then it is common to refer to the ratio as the ‘trailing’
P/E -ratio. Theadvantage of this variant is that it provides an objective measure of value. However, by nature itis a backward looking measure, which may make it less relevant in predicting future stock returns.Alternatively, it is common to use estimates of the next set of annual earnings, or else predictionsof earnings over the following four quarters. Calculating the ratio using such figures provides uswith the ‘forward’
P/E -ratio. The requisite forward earnings figures are typically estimated byaveraging the predictions of a group of market analysts. By using the forward earnings the objectiveis to provide a measure that is more representative of the future prospects of the firm, however thedownside is that we introduce a subjective element to our measure.A further variant on the
P/E -ratio is the cyclically adjusted price-earnings ratio (CAPE).Instead of using a single year’s earnings in the calculation, the CAPE utilises the average of theprevious ten years earnings, after adjusting for inflation. In averaging the earnings, the objectiveis to reduce the effect of earnings volatility over the course of the business cycle, which mayprove beneficial especially when making longer term forecasts of future returns. The CAPE waspopularised by Robert Shiller and John Campbell [9, 10], who applied the CAPE on a stock index2evel, however the idea of averaging earnings over a number of years in order to stabilise the
P/E value goes back at least to Graham and Dodds [27].In theory the price-earnings ratio allows an investor to make value judgments between stocks,however caution must be exercised. Intuitively we might expect that investors should be willing topay a higher multiple of current earnings only if they expect the stock to reliably produce higherearnings going forward. Such a hypothesis is supported by those studies which set out to derivea theoretically ‘fair’
P/E value, for example [26, 37] and [18]. However, the empirical evidence[8, 42, 47] is mixed as to whether the observed discrepancies in
P/E -ratios may be justified bygrowth. In any event, naïvely comparing vastly differing stocks based solely on price-earnings ratiois likely to provide the investor with an incomplete picture of their comparative value.Having outlined a number of the variants of the standard price-earnings ratio, and its unsuit-ability for making value judgments between stocks with differing growth expectations, we nowspend some time examining a number of models and metrics which attempt to incorporate growthexpectations with a measure of value.
As discussed, investors are generally willing to accept a higher
P/E -ratio when they believe a firmhas credible prospects for strong earnings growth. Therefore, when assessing a stock, investorsmust weigh up the value it currently offers with their expectations for earnings growth. The price-earnings to growth ratio, or PEG ratio, attempts to quantify this trade-off between growth andcurrent value, enabling a more ready comparison to be made between stocks. The PEG ratio isderived from the
P/E -ratio in the following manner:PEG Ratio = P/E
RatioAnnual EPS growth rate in % . (1)As with the P/E -ratio, there are a number of variants of the PEG ratio depending on the
P/E -ratioused in the calculation and how the earnings growth rate is determined. If the trailing
P/E -ratiois combined with a growth rate estimated from historical earnings, then it is common to refer tothe resulting PEG as the trailing
PEG ratio. However, if the forward
P/E -ratio is used alongside aforecast of future earnings growth, then we obtain the forward
PEG ratio. Once again the forwardearnings and growth rate are obtained from surveys of professional analysts’ forecasts.Originally developed by Farina in [24], the PEG ratio was most notably lauded by Peter Lynchin his book
One Up On Wall Street [36]. In this book, Lynch states that “The
P/E -ratio of anycompany that’s fairly priced will equal its growth rate", implying that a fairly priced stock shouldhave a PEG ratio of . With this rule of thumb, stocks with a PEG ratio below are deemedto provide good value, whilst those with a PEG above are seen as overpriced. However, asopposed to the P/E -ratio which we may interpret as the price of future earnings or alternativelyas a payback period, the PEG ratio is simply a heuristic measure and lacks any real intrinsicsignificance in terms of valuation [17]. In the upcoming example featured in Table 1, we shalldemonstrate the shortcomings of the PEG ratio. The two stocks considered both have a PEG of , however they are shown to provide unequal investment opportunities.An alternative measure which attempts to account for earnings growth within stock valuation isthe PEG payback period [25, 38]. The PEG payback period follows similar lines to the P/E -ratioin its interpretation as a time period, however, the PEG payback period accounts for earningsgrowth. Simply put, the PEG payback period is the minimum number of whole years a stock mustbe held in order for its accrued earnings to equal or exceed its initial price, assuming that theannual earnings grow at a constant rate g . However, it is somewhat naive in its approach, typicallyproceeding via a simplistic calculation.For example, let us consider a stock with price P = 10 and earnings per share of E = 1 overthe past four quarters, hence the stock has a trailing P/E -ratio of . If we assume that earningsper share grow at a constant per annum, then the table below (Table 1) sets out the futureannual earnings and associated cumulative earnings for the stock. From this table we can see thatafter years the cumulative earnings of the stock surpass the initial stock price, hence the stockhas a PEG payback period of . In comparison, if we consider a stock with P = 20 and E = 1 with earnings growth then we can see that the initial price is only recouped after years, bywhich point the P/E = 10 stock has earned of the initial price, in comparison to forthe
P/E = 20 stock. As such, these two stocks with the same PEG ratio clearly do not offer the3ame value to the investor. However, if earnings continued to grow at the given rates then the
P/E = 20 stock would soon surpass the
P/E = 10 stock in terms of its return. This example alsoclearly demonstrates the imprecision of the PEG payback period as a metric of value. Examiningthe earnings schedule for the stock with P = 20 , after years we can see cumulative earningshave almost reached . However, we require an additional year to surpass P in full, by whichpoint cumulative earnings exceed P by almost . As with the PEG ratio, the PEG paybackperiod originates more from the popular investment literature, and to date has received little orno coverage in the academic literature. P/E = 10 , with 10% growth
P/E = 20 , with 20% growthYear Earnings Cumulative Earnings Cumulative0 1.00 1.001 1.10 1.10 1.20 1.202 1.21 2.31 1.44 2.643 1.33 3.64 1.73 4.374 1.46 5.11 2.07 6.445 1.61 6.72 2.49 8.936 1.77 8.49 2.99 11.927 1.95
Table 1: Future earnings for stocks with
P/E ’s of 10 and 20 and growth rates of 10% and 20%respectively.
The previous measures take a stocks price, the earnings per share and earnings growth rate as givenand attempt to ascertain whether it represents a favourable investment opportunity. However, thereexists another group of models which take the earnings per share and growth rate, amongst othervariables, and look to derive a fair price for the underlying stock, having done so it is then possibleto obtain a theoretically ‘correct’
P/E -ratio for the stock. The first such model derives from theGordon growth model [26], which equates a stocks worth to the sum of all of its future dividendpayments, discounted back to their present value. Letting D denote the most recent dividend paidby the stock, the model assumes that the stock will pay a dividend in all future years, with aconstant annual dividend growth rate of g D . If r ( > g D ) represents the cost of equity capital forthe firm, that being the rate at which we will discount future payments, then the model gives thefollowing value for the stock: P = ∞ X n =1 (cid:18) g D r (cid:19) n D = (1 + g D ) Dr − g D . Assuming that the firm pays a constant proportion γ of earnings as dividends each year, so that D = γE , where E represents the most recent EPS figure, and noting that we must therefore alsohave earnings growth of g = g D , it is straightforward to obtain PE = (1 + g ) γr − g , as given in [37], as the correct P/E -ratio for the stock. Although simple, the above relationexhibits a number of features that would be expected given the underlying fundamentals. Firstand foremost, as we let g ր r , the fair P/E -ratio grows without bound. This is in-line with theassertion that investors should be willing to accept higher
P/E multiples for higher anticipatedearnings growth. In contrast, the larger the value of r , corresponding to a higher required rateof return, the lower the fair P/E should be. Higher required rates of return typically indicate ahigher level of uncertainty for future cash-flows, as such investors should expect to pay a lowermultiple of current earnings. Finally, as the payout ratio γ is increased, so does the P/E value,4ue to the greater cash-flows to the investor. However, we should note that the ability to maintaina constant growth rate of g requires a certain level of re-investment, restricting the payout ratio[29]. The Gordon growth model may be generalised to account for a non-constant growth rateover the lifetime of the stock. For example, it is common for immature growth companies toexperience rapid earnings growth during the early stages of their development, before settling toa more conservative growth rate as they mature. Such a two stage growth profile can be simplybuilt into the model by suitably adjusting the future cash-flows [14].Whilst the basic Gordon growth model assumes a constant and equal growth rate for bothearnings and dividends, a more general framework is provided by the abnormal earnings growth(AEG) model of Ohlson and Juettner-Nauroth [41]. The model relates the stock price to thefollowing year’s expected earnings per share (12 months forward EPS), the short-term growthin EPS (FY-2 versus FY- l), the long-term (asymptotic) growth in EPS and the cost-of-equitycapital. Following the derivations of [41, Corollary 1], it is possible to obtain an expression forthe forward price-earnings ratio which is dependent on the aforementioned factors [41, Equation7]. Furthermore, if we impose the assumption of constant and equal growth rates for earnings anddividends, then we recapture the basic Gordon growth model from the AEG. P /E -ratio
Having outlined a number of existing value metrics, we now derive our own growth adjusted price-earnings ratio or ‘GA-
P/E ’. The motivation for the form of the GA-
P/E comes from the interpre-tation of the
P/E -ratio as a period of time, that being the number of years required for the stocksaccrued earnings per share to equal its current price. As such, the GA-
P/E bears somewhat of asimilarity to the PEG payback period of the previous section. However, through its construction itoffers an increased level of precision and a greater ability to differentiate the value offered by stocks.Let us consider a stock with annual (previous 4 quarters) earnings per share of
E > and anearnings growth rate of g > − , which we assume for now to be known. The total earningsattributable to each stock over the next N ≥ years are then given by E N = N X n =1 E (1 + g ) n = E (1 + g ) (1 + g ) N − g , (2)where the last equality comes as a consequence of the standard result for the first N terms ofa geometric series. Relaxing the assumption on N being a positive integer and recalling theinterpretation of the standard P/E -ratio as the earnings payback period, we equate the abovequantity to the stock price P to obtain P = E (1 + g ) (1 + g ) N − g . (3)The N which solves (3), assuming a solution exists, then gives us the required payback period. Asimple rearrangement of (3) produces (1 + g ) N = 1 + g g PE . Taking the logarithm of both sides allows us to extract the exponent N , and dividing both sidesby log(1 + g ) then gives N = log (cid:16) g g PE (cid:17) log(1 + g ) , (4)as the number of years required to repay the stock price P through earnings, i.e. the growthadjusted P/E -ratio (GA-
P/E ). Having derived our growth adjusted price earnings ratio, we nowspend time examining its properties and its dependence on the constituent factors PE and g .In our construction of the GA- P/E we assumed the existence of a solution to (3). Implicitlythis amounts to an assumption that at some point accrued future earnings will surpass the currentstock price P . However, it is possible given a sufficiently negative − < g < , that annual earningscontract sufficiently rapidly that total future earnings amount to less than P . Therefore we now5erive a lower bound on the growth (contraction) rate g such that we can ensure a solution to (3)exists. Let us assume that − < g < , then for a solution to (3) to exist we require that E ∞ = ∞ X n =1 E (1 + g ) n > P, which, after evaluating the summation, is equivalent to − E (1 + g ) g > P. A simple re-arrangement of the previous inequality provides the following bound on the growthrate g : g > − EP + E . (5)If we consider the situation as g decreases towards this bound, then from a financial perspective wewould expect it to take an increasingly long period for accrued earnings to repay the initial stockprice. Examining the behaviour of (4) as we let g ց − E/ ( P + E ) , we see that N grows withoutbound as we approach the limit, confirming our intuitive expectations. In Figure 1, we can seethis graphically illustrated for the case of P = 15 and E = 1 . For the given choice of P and E ,the bound (5) computes to approximately -0.0625 and as the growth rate g is reduced towards thislevel, we observe the GA- P/E N as given in (4), increase sharply. -0.25 0 0.25 0.50102030405060708090100
Figure 1: GA-
P/E
Payback period N as growth rate g approaches − E/ ( P + E ) .Another special case we might wish to consider is that of constant earnings, i.e. g = 0 . If we recall,under such an assumption the original price-earnings ratio gives us the payback period for a stock.Therefore, if we substitute g = 0 in to (4) then we would hope to recover PE . However, if we naivelyattempt to evaluate (4) at g = 0 then we quickly run in to difficulties as both the numerator anddenominator return a value of for g = 0 . Instead, an application of L’Hôpital’s rule yields thefollowing: lim g → log (cid:16) g g PE (cid:17) log(1 + g ) = lim g → ( g + 1) P ( g + 1)( gE + gP + E ) = PE .
Hence, in the limit as we let g → , we recover the original price-earnings ratio as we would hope.Having examined some specific cases of g , we now consider the behaviour of (4) as g is varied morewidely. The charts in Figure 2 show the GA- P/E N plotted against the underlying price-earningsratio
P/E for a selection of growth rates g . Observing the case g = 0 (blue line), as discussedabove, we see behaviour consistent with the standard P/E -ratio, whereby the payback period N equals P/E . However, as the growth rate g increases, we see that larger initial price-earnings ratioscan be supported within a given payback period. In Figure 3 we see the GA- P/E plotted againstthe earnings growth rate for a selection of initial price-earnings ratios. As you would expect, givenan initial price-earnings ratio
P/E , as the growth rate g increases the GA- P/E N decreases.6
Figure 2: GA-
P/E payback period N against underlying P/E for various growth rates
Figure 3: GA-
P/E payback period N against growth rate g for various underlying P/E -ratios
P /E
Having outlined the derivation of our growth adjusted price-earnings ratio and examined someof its properties, we now seek to determine its efficacy as a value metric and predictor of futurestock returns. In order to assess this we conducted an empirical study utilising a sorted portfoliomethodology. Such studies have found widespread application across empirical finance and inparticular have been applied in many studies examining the effectiveness of the traditional price-earnings ratio (or alternatively its inverse, the earnings yield
E/P ), for example [6, 7, 5, 31, 2].For an extensive list of such studies the reader may consult the introductory section of [12]. Thisstudy proceeds by taking a sample pool of stocks and calculating the GA-
P/E for each of thesecurities within the set. The stocks are then ranked according to their GA-
P/E and placedwith equal weighting in to quantile portfolios based on this ranking. In this way we end up with acollection of portfolios, ranging from a portfolio containing the lowest GA-
P/E stocks to a portfolioconsisting of the highest GA-
P/E stocks. For each portfolio we compute the monthly returns overthe subsequent year. This process of portfolio formation and return tracking is repeated each yearover the duration of the study period, although the stocks within the selection pool might varyfrom year to year as the result of new listings and de-listings. Through this process we obtaina track-record of the historical returns for each of the portfolios. If the GA-
P/E is indeed aneffective measure of value and predictor of future returns then we would expect to observe aninverse relationship between the ranking of portfolio returns and their GA-
P/E ordering, that islower GA-
P/E portfolios providing higher returns and vice versa.
The calculation of the GA-
P/E and of the subsequent portfolio returns requires the use of bothearnings data as well as a record of historical stock prices. For earnings data we utilised the7ompustat North American fundamentals dataset, whilst for historical prices we made use ofthe Center for Research in Security Prices (CRSP) monthly file dataset. The data within theCompustat dataset is provided on a firm by firm basis and is predominantly obtained from SECfilings, whilst the data from CRSP is security specific, with data originating from a number ofexchanges. The datasets were then combined using the CRSP/Compustat Merged (CCM) linktable. The CCM link table details the linkage histories between firms and securities, allowing usto associate company earnings records with the appropriate securities. For the purposes of ourstudy we restricted our attention to primary listings of common shares on the NYSE, Nasdaq andAMEX over the 25 year period from 1990 to 2015.Considering specifically the earnings data form Compustat, if we attempt to use reported annualearnings then the fact that companies follow different reporting calendars raises an issue for theannual portfolio construction exercise. For example, if we consider the fiscal year 1996 then wemay find this ending as early as June 1996, or as late as May 1997. This misalignment meansthat whichever date we select to carry out the annual portfolio construction, there will always bea group of firms for which the most recent annual earnings figures are stale. Previous studies, forexample [6, 31], get around this issue by considering only those firms whose fiscal year ends inDecember. However, to avoid further reducing our stock universe, beyond the restrictions made inthe upcoming Section 4.2, we instead make use of reported quarterly earnings. Each firm’s fiscalyear is then redefined to consist of four successive quarters, with the final quarter being the oneending between October and December. In the event that earnings data was unavailable for anyof the quarters, then the firm was excluded from the subsequent year’s portfolios.Another issue that we must be aware of is the potential introduction of look-ahead bias in ourresults. In practice, earnings figures relating to a period are not publicly available immediatelyfollowing the end of that period. If we were to naively construct our portfolios following the endof our annual fiscal period each December, then we would be utilising data not available to theinvestor at that time, biasing our results [5]. To counteract this we introduced a lag to our study.Since the vast majority of firms report within three months of the earnings period end, we follow[6, 13, 31] and form our portfolios at the end of March each year, using the month end price andthe previous year’s earnings to form the GA-
P/E .A further source of potential error in our study is that of survivorship bias, occurring as a resultof the way in which firms are added to the Compustat database [46]. Typically firms are added toCompustat only after they have been operating viably for a number of years, at which point theyare introduced with multiple years of historical accounting data, in a process known as ‘backfilling’.Firms which disappear before reaching such a stage fail to make it onto the database and thereforewould be missing from our study. However, there are a number of factors particular to our studywhich reduce the potential significance of this effect. Firstly, the construction of our GA-
P/E measure requires us to attribute an earnings growth rate to each stock under consideration, inour case this is estimated from historical earnings data. The calculation of this historical growthrate is outlined in the following section, however, for now we note that the firm is required tohave positive earnings for at least two of its most recent fiscal years. Any firms not satisfyingthis requirement would automatically be discounted from consideration, hence we might expecta sizeable proportion of the firms missing from Compustat to be ineligible anyway. Furthermore,following 1978, Compustat initiated a major database expansion project, greatly improving theircoverage and reducing the significance of the selection biases [46]. The period of our study occursafter this expansion and therefore the impact of this survivorship bias should be further reduced.Having covered the addition of new listings to our stock universe, we must also consider howwe handle the removal of stocks due to delisting. When a stock within our universe is delisted fromits exchange and hence disappears from the CRSP monthly pricing dataset we require a return beattributed to its final month. To compute said return, we ustilised the CRSP monthly stock eventdelisting dataset, and in particular the delisting return (excluding dividends) data item. Althoughsuch delistings may occur at any point throughout the month, the realisation of any return isattributed to the end of the month in which it occurs. In the case of a stock disappearing fromthe pricing dataset without a delisting return being available, we assume a complete loss for theinvestor and attribute a return of -1. 8 .2 Estimating growth
The calculation of the GA-
P/E measure (4) requires us to attribute an earnings growth rate g to each stock in any given year. Within our study, these rates were approximated from historicalearnings data in the following manner. Let us suppose we are calculating the GA- P/E for a givenstock as part of the portfolio formation process during the calendar year n . As such, the mostrecent earnings figure will be derived from the four quarter period ending in the final quarter ofthe calendar year n − , which we denote by E n − . Now assuming we wish to estimate the growthover a historical period of k years, i.e. the growth between E n − k − and E n − , then we estimatethe growth rate g for the year n as follows: g = (cid:18) E n − − E n − k − E n − k − (cid:19) k − . (6)In our empirical study we utilised a range of period lengths k (= 1 , , when computing our growthestimates. We also note that the requirement for historical earnings from which to estimate thegrowth rate necessitates a longer history for our earnings data than for our price data. Given thatthe period of our study commences with the first portfolio formation in March 1990, and that weuse up to a 3 year window to estimate the growth rate g , we therefore require 4Q earnings figuresdating as far back as October 1986. Figure 4: Total number of matched securities and the number meeting the earnings conditions for k = 1 , , over the period 1990-2014The calculation (6) requires that both the earnings figures E n − k − and E n − are positive. There-fore, when forming our portfolios in year n we must exclude any stock which does not meet thisrequirement. The charts in Figure 4 depict the scale of this reduction throughout the period ofthe study. The uppermost blue line plots the number of securities resulting from the matchingprocedure and represents the maximum extent of our stock universe. The remaining lines showthe number of securities left after we have imposed the restriction that both E n − k − and E n − bepositive, where we have considered the values k = 1 , , .We would expect company earnings to display a degree of autocorrelation, with firms makinga profit (loss) one year being more likely to make a profit (loss) in subsequent years. The effectof this serial correlation is likely to be strongest in consecutive years and diminish as the yearsin question become further apart. The impact of this may be observed in Figure (4), where thenumber of securities meeting the earnings condition is generally highest for k = 1 and the lowestfor k = 3 . P/E
As determined earlier, when the growth rate g approaches and drops below the value (5) the GA- P/E (4) grows without bound, as the totality of projected future earnings falls below the currentstock price. This raises an issue regarding how to handle these ‘infinite GA-
P/E ’ stocks, should9hey be excluded from the study, or else if we choose to include them, which portfolio should theybe allocated to? The following figure (Figure (5)) depicts the proportion of those stocks satisfyingthe positive earnings requirements, which produce a finite and GA-
P/E s. Over the duration ofthe study period, we observe that a non-negligible proportion of stocks (generally around 25-30%per year) have an infinite GA-
P/E . Therefore, if we were to remove them from the study thenwe would be significantly reducing our sample size, beyond the cuts made on the basis of positiveearnings. Preferring to avoid the further reduction of our stock universe, we proceed to outline ameans by which to rank the infinite GA-
P/E stocks within our framework.
Figure 5: Proportions of stocks with finite and infinite GA-
P/E s when growth is calculated over1,2 and 3 yearsClearly we should consider stocks with an infinite GA-
P/E to lie at the costly end of our valuespectrum, and as such they should be allocated to the portfolio consisting of the highest GA-
P/E stocks. However, it is not necessarily the case that the quantile portfolios will always be sufficientlylarge to allow the inclusion of all infinite GA-
P/E stocks within a single portfolio. In fact, withtypically around 25% of stocks having an infinite GA-
P/E , there is likely to be some overspill fromthe highest GA-
P/E portfolio for at least some of the years. One possible solution might see usallocate all infinite GA-
P/E stocks to a standalone highest GA-
P/E portfolio, before separatingthe remaining finite GA-
P/E stocks into a number of other portfolios. A similar approach wasadopted by [5], whilst studying the
E/P ratio. The authors placed all stocks with negative earnings,hence a negative
E/P , in a single portfolio before allocating the remaining stocks into equally sizedportfolios based on their
E/P . However, in our case, given that the proportion of stocks with aninfinite GA-
P/E is not constant, and can be seen from Figure (5) to exceed 40% in some years, suchan approach would lead to significant discrepancies in portfolio size for some of years. Therefore,we require a means of ranking infinite GA-
P/E stocks amongst themselves in order to determinewhich of them to attribute to the highest GA-
P/E portfolio, and which if any to allocate to lowerGA-
P/E portfolios. In order to do so, we consider the proportion of the stock price P that isprojected to be paid back. In the case that (5) is not satisfied, this payback proportion is given by E ∞ P = P ∞ n =1 E (1 + g ) n P = E (1 + g ) − gP . (7)In contrast to the GA- P/E measure where a lower figure signifies greater value, the higher (7) iscalculated to be, the greater the value offered by the the underlying stock. In order to remainconsistent with the GA-
P/E we therefore take the reciprocal of the above ratio to get
P/E ∞ .Furthermore, if N max signifies the highest finite GA- P/E observed in a given year, then for all theinfinite GA-
P/E stocks that year, we add N max to the inverted ratio P/E ∞ to give the followingmeasure: N ∗ = N max + P/E ∞ = N max − gPE (1 + g ) . By using the standard GA-
P/E for those stocks which return a finite value of (4), and N ∗ forthose stocks registering an infinite GA- P/E , we obtain a value measure which facilitates a coherent10anking across both finite and infinite GA-
P/E stocks. However, we note that the use of N ∗ is onlyto allow a ranking to be taken, the measure N ∗ has no intrinsic significance and the underlyingstocks all possess an infinite GA- P/E . Following the approach outlined, we conducted our empirical study over the period from 1990 to2015, with portfolio construction occurring at the end of each March from 1990 to 2014. Duringeach construction event we formed five quintile portfolios, labeled P to P , corresponding witha low to high GA- P/E ordering. The choice of five portfolios is in itself arbitrary, however itmatches the number selected by Basu [6]. From Figure 4, we can see that the number of stockswithin our pool typically varies between 2000 and 4000 depending upon the year and the periodover which growth is calculated. Therefore, upon construction we would expect our portfolios tocontain between 400 and 800 individual stocks. Furthermore, if we consider the proportion ofstocks displaying an infinite GA-
P/E (Figure 5), then it allows us to apportion the majority ofsuch stocks to the highest GA-
P/E portfolio, whilst retaining a reasonable number of portfoliosand hence a range of typical GA-
P/E values. This process was followed using a one, two andthree year window for estimating our growth rates, with the corresponding restrictions of our stockuniverse as per Figure 4. The following table (Table 2) provides a summary of the typical value ofsome key variables for the resulting GA-
P/E sorted portfolios. The values given in the table areobtained by averaging the median values (for individual years) over the duration of the study.Portfolio P P P P P P/E * NAwindow
P/E
P/E ** NAwindow
P/E
P/E ***
NAwindow
P/E * Excluding ten years in which the median GA-
P/E was infinite. ∗∗ Excluding seven yearsin which the median GA-
P/E was infinite. ∗∗∗
Excluding four years in which the medianGA-
P/E was infinite. NA signifies that the median GA-
P/E was infinite in all years.
Table 2: Summary of statistics for the GA-
P/E portfolios when growth was estimated using a one,two and three year window.Examining the statistics given in Table 2, we can see that in each case, the lowest GA-
P/E stockstend to be those with the lowest
P/E -ratios and the highest earnings growth, with the oppositebeing true for the highest GA-
P/E stocks. Although this might be expected from the form ofthe GA-
P/E , it does somewhat run contrary to the distinction between value and growth stocksand the idea that the cost of high earnings growth is inevitably a high
P/E -ratio. Consideringthe combination of extremely high/low growth rates coupled with low/high
P/E -ratios, it is likelythat many of the extreme GA-
P/E stocks are those that have experienced a significant changein their earnings over the short-term which has not been matched by a comparable change inshare price. Comparing the statistics across the different growth windows, it is noticeable thatthe growth rates associated with our end portfolios P and P are significantly moderated as thewindow length increases. This might have been expected, since the likelihood of a firm recordingsignificant changes in earnings over a second or third consecutive year diminishes, as compared tothe likelihood of observing a single year of high earnings growth/contraction.Having formed the portfolios P to P using one, two and three year growth windows, weaverage the holdings across the three realisations of each portfolio, to create a single set of fiveportfolios. This is done in order to mitigate the noise within the estimated earnings growth rates,which will often display high levels of variability from year to year [33]. By averaging across the11hree instances, we place a greater weight upon those stocks which have consistently displayed agrowth rate g , which when combined with the associated P and E , produce a GA- P/E that wouldplace the stock in the given portfolio.As explained in Section 4.2, the requirement for positive earnings for the end years of ourgrowth windows restricts the pool of stocks from which we may select the portfolios. Giventhe pools associated with one, two and three year windows, we also form three sets of quintileportfolios based upon a ranking of
P/E -ratios. As with the GA-
P/E sorted portfolios, we averagethe holdings across the three instances of each portfolio, in order to obtain one set of five
P/E sorted portfolios.The average annual returns of our five GA-
P/E portfolios can be found in the first row ofTable 3. Listed alongside these returns, we have included the those of the analogous
P/E sortedportfolios. Average annual rate of return Portfolio P P P P P GA-
P/E sort 0.1982 0.1660 0.1367 0.1336 0.1580
P/E sort 0.1889 0.1569 0.1377 0.1424 0.1598Difference 0.0093 0.0091 -0.0010 -0.0087 -0.0017 signifies that the return from the GA-
P/E portfolio exceeds that of thecorresponding
P/E portfolio, whilst signifies the return from the
P/E port-folio exceeds that of the corresponding GA-
P/E portfolio.
Table 3: Average annual returns for the averaged GA-
P/E and
P/E portfolios. Figure 6: Cumulative performance over the course of the study (1990-2015) of portfolios P and P based upon GA- P/E and
P/E sorts, compared with the S&P 500.From the briefest examination of Table 3 it is clear that in general the lower GA-
P/E portfoliosprovide the highest average returns, with P and P providing average annual returns of 19.82%and 16.60% respectively. The only outlier to the trend being P , whose return of 15.80% exceedsthat of both P and P . However, it is not unexpected that the ordering of returns is not perfectlymonotonic, with similar deviations having been observed in many studies of the P/E -ratio. Forexample in [6], the highest
P/E (lowest
E/P ) portfolios were found to outperform a number ofthe middle ranking portfolios, whilst both [31, 19] comment on a U-shaped returns profile. Indeed,examining the returns from our
P/E ranked portfolios we observe a more pronounced U-shaped The period of the study covers 300 months. If r , . . . , r denote the associated monthly returns, then theaverage annual rate of return is given by Q j =1 (1 + r j ) / − . P/E portfolio P delivering the second highest return, whilst the lowestreturns are generated by the middle ranked portfolio P . However, once again the highest returnsare provided by the value portfolio P . Comparing the level of returns between the GA- P/E and
P/E portfolios, for the value portfolios ( P and P ), the GA- P/E noticeably outperforms the
P/E ,providing an additional 0.9% annually in both cases. Whereas at the other end, for the portfolios P and P , the returns for the GA- P/E can be seen to be below those for the
P/E . The cumulativeeffect of the return differential between the low and high GA-
P/E portfolios P and P can be seenin Figure 6. The chart also details the cumulative performance of the low and high P/E portfolios,allowing us not only to observe the difference between these two portfolios, but also to comparebetween analogous GA-
P/E and
P/E portfolios.At first glance, the returns in Table 3 and Figure 6 may appear rather high especially incomparison to the S&P 500 which gave an average annual return of 7.49% over the same period.Considering this further, there are a number of reasons which might explain the strong performance.The first point to note is that in order to estimate the earnings growth rate, we were required todrop those firms which recorded a loss in either the year starting or ending our growth window.The firms that are left are those with a more robust profitability record. Secondly, by includingall primary listings on the NYSE, Nasdaq and AMEX we would expect the average size of a firmwithin our stock universe to be somewhat smaller than the typical S&P 500 firm. Furthermore,before averaging holdings over the different growth windows, each firm within our base portfoliosinitially receives an equal weighting. As such, following the averaging procedure, the maximumdifference in weighting that we might observe between two stocks within the same portfolio isapproximately 3 to 1. In contrast, the S&P 500 employs a weighting system based upon marketcapitalisation, and therefore we observe significantly larger differences in weightings (greater than3 to 1), in favour of larger firms over smaller firms. As a result, the significance of smaller firmswithin our portfolios is increased in comparison to an equivalent market capitalisation weightedportfolio, or indeed the S&P 500. Since both high profitability and smaller firm size are factorswhich correlate with higher stock returns [23], it is unsurprising that our portfolios outperform theS&P 500 by a margin.Having observed that the returns of the lower GA-
P/E portfolios were greater than those ofthe high GA-
P/E portfolios over our test period, we now investigate whether these differencesin returns are statistically significant. We also tested the differences in returns between the
P/E portfolios so as to allow comparison. We conducted paired sample, two-tailed t -tests betweenthe monthly returns { R i } and { R j } of portfolios P i and P j , for both the GA- P/E and the
P/E portfolios. The resulting p -values may be found in Table 4. p -values for GA- P/E monthly returns p -values for P/E monthly returns P P P P P P P P P × − *** × − *** × − *** ** P ** × − *** ** P × − *** × − *** P * P P P * P ∗ , ∗∗ , ∗∗∗ denote significance at the 5%, 1% and 0.1% levels respectively.signifies that the observed return differential was negative, counter to the proposed monotonic ordering. Table 4: p -values for the differential ¯ R i − ¯ R j in average monthly returns between portfolio P i ( i th row) and portfolio P j ( j − th column) using a two-tailed t -test. Portfolios formed on the basis ofGA- P/E (left) and
P/E -ratio (right).Examining the results of Table 4, we see a greater number of statistically significant results whencomparing the monthly returns between the GA-
P/E sorted portfolios than we do for the
P/E sorted portfolios. Most notably, when we test the returns of the lowest GA-
P/E portfolio P against those of the other portfolios, we obtain a significant result in each case. However, this isnot the case with the P/E sorted portfolio, where the difference in returns between P and P isnot significant. Furthermore, in those cases where the returns differential for both the GA- P/E and the
P/E portfolios are significant, we invariably find that the result for the GA-
P/E showsa higher level of significance. This evidence further supports the hypothesis that the GA-
P/E is13redictive of future stock returns, and when selection is restricted to stocks with positive earnings,the predictive strength of the GA-
P/E exceeds that of the
P/E -ratio.We now examine the returns of the portfolios in more depth, adjusting the returns to account fortheir observed volatility, and also determine the portfolios’ exposure to some common risk factors.In order to assess the risk-adjusted returns of our portfolios we utilise the Sharpe ratio. Developedby William Sharpe in [44] and later revised in [45], the Sharpe ratio measures the expected excessreturn earned by an asset per unit of volatility, where the excess return is measured against the risk-free rate. In our analysis, when computing the Sharpe ratio we utilise monthly returns data andtake the 3-month U.S. Treasury as a proxy for the risk-free asset. We also measure the exposure ofthe portfolios’ returns to the standard risk factors identified in Fama and French’s [20] three factormodel (market risk factor, HML value risk factor and SMB size risk factor). In our assessment ofthese exposures we utilise the monthly returns of the S&P 500 as a proxy for the market returnsand once again take the 3-month U.S. Treasury as a proxy for the risk-free asset. The data forboth the monthly S&P and 3-month U.S. Treasury returns were obtained from WRDS, as werethe monthly values of the HML and SMB Fama-French factors.Performance measures and factor coefficients for GA- P/E sorted portfoliosPortfolio P P P P P Sharpe ratio α β b HML b SMB R P/E sorted portfolios. The values in parentheses below the regression coefficients are the relevant t -statistics. Performance measures and factor coefficients for P/E sorted portfoliosPortfolio P P P P P Sharpe ratio α β b HML b SMB R P/E sorted portfolios. The values in parentheses below the regression coefficients are the relevant t -statistics. Whilst the computed Sharpe ratios are given on a monthly basis, we can approximate their annual equivalentsby multiplying the resulting figure by √ , for further details see [35]. For reference, following the same calculation,the S&P 500 gives a Sharpe ratio of . over the period of our study. P/E sorted portfolios, we see a similar U-shaped profile across the portfolios (as observed in Table 3) once the returns are adjusted for risk.The value portfolio P gives the highest risk adjusted return, with the Sharpe ratio decreasingas we move down the value range through to portfolio P . However, once again P proves anexception to the monotonic ordering, although after adjusting for risk, the returns from P nowrank th as opposed to rd previously. In comparison, if we examine the risk adjusted returnsof the P/E sorted portfolios from Table 6, then we can see a strict monotonic ranking as we gofrom P through to P , matching the results of [6]. Nonetheless, the risk adjusted return for theGA- P/E sorted P exceeds that of the P/E sorted P .Examining the GA- P/E sorted portfolios’ exposure to the risk factors (Table 5), we see thatat least some of P ’s additional return may be explained by higher exposures to the market riskfactor ( β = 0 . , st overall), value risk factor ( b HML = 0 . , nd overall) and the size riskfactor ( b SMB = 0 . , nd overall). However, these exposures do not explain all of the additionalreturns as evidenced by P having the highest (monthly) alpha value ( α = 0 . ), which measuresthe average abnormal return not explained by the three standard risk factors. Contrasting theexposures for the two P ’s, we see that the GA- P/E derived P has a higher exposure to thegeneral market risk factor (higher β ) and also the size risk factor (higher b SMB ), than does the
P/E sorted P . On the other hand, as we might expect, the portfolio P originating from the P/E sort displays a significantly higher exposure to the value risk factor (higher b HML ), than theGA-
P/E sorted P . Overall, exposure to these standard risk factors is insufficient to explain theadditional returns from the GA- P/E portfolio P , as evidenced by its higher alpha intercept.More widely it is interesting to compare the pattern of b HML coefficients from the two sets ofportfolios. Looking at the
P/E sorted portfolios (Table 6), we see a very clear decrease in theexposure to the value factor as we go from P to P , with P having a negative b HML coefficient.The
P/E -ratio is a traditional value metric much like the
B/M -ratio which is used to construct thevalue factor risk premium, as such it is unsurprising that the portfolios exhibit increasing b HML exposure as we go from the high
P/E non-value portfolio P through to the low P/E value portfolio P . In contrast, the GA- P/E portfolios produce a more uniform pattern of b HML coefficients,indicating a more even exposure to the traditional value risk factor across the portfolios. Thissuggests, that by adjusting our value measure for growth, the GA-
P/E finds ‘true value’ acrossthe traditional value spectrum.We noted that at least some of the additional returns for the GA-
P/E portfolio P might beattributed to its higher exposure to the size risk factor. The size-effect was first noted by Banz[4], and has now been accepted as a standard risk factor in the asset pricing literature [20, 23, 11].Whilst it would seem plausible that smaller firms experience greater risk in general than their largercounterparts, specific explanations include the lower liquidity of smaller stocks [1, 34], and alsoincreased uncertainty owing to a reduced availability of information about small firms [48]. In anyevent, evidence suggests that these effects are largely restricted to the smallest of publicly tradedfirms [30, 22], and that by removing these microcap stocks we can mitigate the effect. Therefore,in order to test the GA- P/E further, we restrict our stock universe to the largest x % of firms bymarket capitalisation on NYSE, Nasdaq and AMEX, prior to carrying out the same procedure asbefore. We then reduce the percentage x % of firms retained, and record the average annual return,the Sharpe ratio and α of the P portfolios resulting from both a GA- P/E and a
P/E sort. Thecharts below (Figure 7) detail the results of this process.15 -3 Figure 7: Average annual return (top left), Sharpe ratio (top right) and monthly α (bottom) forthe GA- P/E (blue) and
P/E (orange) sorted P portfolios as the stock pool is restricted to thelargest x % of market capitalisations.From the above charts it is apparent that the outperformance of the low GA- P/E stocks overthe low
P/E stocks cannot be attributed to a bias for smaller cap stocks. As we increasinglyrestrict the pool of stocks from which the portfolios are selected, the margin of outperformance, asmeasured by our three metrics remains fairly consistent. Indeed, if anything the gap between thetwo portfolios increases as we eliminate the very smallest microcap stocks from our selection pool.
This paper has proposed a new growth-adjusted
P/E ratio to improve on the noted shortcomings ofthe existing price-earnings based value measures. To start, a simple dimensional analysis revealedthat the standard
P/E ratio is not a simple dimensionless ratio, but is a period of time, that beingthe number of years required for a company’s accumulated earnings per share to equal the currentshare price, assuming earnings remain constant.Whilst the PEG ratio provides a simple attempt to judge an appropriate
P/E given a certainlevel of growth, it is revealed as a meaningless quantity having the dimensions of time/percentagegrowth rate. What is more the PEG is unable to distinguish the relative value of paying, forinstance, a
P/E of 10 for a growth rate of 10% or a
P/E of 20 for a growth rate of 20%, in eachcase the PEG being 1. The GA-
P/E shown in (4) has the merit of remaining a period of time, asis the case with the
P/E ratio, however the GA-
P/E adjusts for a series of growing earnings.The paper proceeds with an extensive anaylsis of stock market returns of the US markets overthe period from 1990 to 2015. It shows that, divided into quintile portfolios, the lowest quintileGA-
P/E stocks generate superior returns to the higher quintiles. This is consistent with the resultsfor the conventional
P/E , however the differential in returns obtained with the GA-
P/E is shown16o be greater than that obtained using the
P/E . Furthermore, the return provided by the lowestquintile GA-
P/E portfolio exceeds that for the lowest quintile
P/E portfolio.We further regress the returns of each set of portfolios against the standard Fama-French riskfactors. Noting that each of the GA-
P/E portfolios may contain stocks of arbitrary
P/E , we findthese portfolios display a uniform exposure to the traditional HML value risk factor . This contrastswith the results for the
P/E sorted portfolios which exhibit increasing exposure to HML as wemove from high through to low
P/E . The GA-
P/E portfolios as a whole exhibited greater variancein their exposures to the SMB size factor as compared to those for the set of
P/E portfolios. Inparticular, the lowest GA-
P/E portfolio produced a higher loading to the size factor than wasobserved for the comparable
P/E portfolio. However, this was not found sufficient to fully explainthe outperformance of the low GA-
P/E portfolio, as evidenced by its higher alpha value. Toinvesitgate this size dependence further, we concluded with a comparative study of lowest GA-
P/E and
P/E quintiles, whilst sequentially restricting our stock universe to include those stockswith the largest market capitalisations. As the market capitalisation threshold was increased,we found that the lowest GA-
P/E quintile continued to outperform the lowest
P/E quintile, asmeasured by absolute return, risk adjusted return (Sharpe ratio) and excess return (Fama-Frenchalpha).Whilst we believe that this market wide analysis indicates the superiority of the GA-
P/E toconventional
P/E in forecasting returns, we believe also that the utility of the GA-
P/E will beparticularly obvious in comparing similar stocks with differing
P/E s and growth rates, e.g. similarstocks in the same sector (oil majors etc) and specialised subgroups (e.g. the FAANGS etc).
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