Western European Stock Market
Vincent Hoitink
1University of Groningen (Netherlands)
June 21, 2009
Abstract
This paper introduces a technique to perform nonparametric statistical tests on the three factor model of Fama and French (1993). Because of the non-normal distribution of returns we can thus draw more valid conclusions regarding the P/E effect on the bases of the nonparametric tests performed on the three factor model alpha. This paper does not find any evidence that the nonparametric statistical tests performed on the three factor model alpha yield any other conclusions than the parametric statistical tests performed in the literature and thus, according to the three factor model, the P/E effect in the Western European market does not constitute any real investment opportunities. This paper also introduces a technique to perform statistical tests on the Sharpe (1966) ratio. The statistical findings of the new technique show, that on the basis of the Sharpe (1966) ratio, the P/E effect does yield investment opportunities in the Western European market. The debate whether the P/E effect is caused by naïve investors or by risk factors associated with firm size and book-to-market equity remains present.
Keywords: return anomalies, price-earning effect, value, growth JEL Classification Number: G11
1 MScBa Finance Thesis; student number 1644394
1st Supervisor: Brunia, N.
1. Introduction
The effect, that on average stocks with low price-earnings (P/E) ratios tend to outperform stocks with high P/E ratios, is called the P/E effect. Value investment is the investment strategy that involves buying stocks with low P/E ratios. In this paper stocks or portfolios with low P/E ratios are called “value stocks” or “value portfolios”, and stocks or portfolios with high P/E ratios are called “growth stocks” or “growth portfolios”.2 Value
investment styles are employed by fund managers all around the world. Academic papers report a P/E effect of 7 to 8% a year on average (see for example Basu (1977), Fama and French (1998)). The P/E effect of 7 to 8% a year is reported for different time periods and for different markets (see for example Basu (1977), Fama and French (1998)).
Whether the P/E effect is a real investment opportunity or not, is a problem that is faced by many fund managers and private investors. An investor has to judge what the cause of the reported P/E effect is. If the market is efficient the implied risk of the value portfolio is higher than the implied risk of the growth portfolio. The investor has to decide whether he wants to take the extra risk of the value portfolio in order to yield a higher average return. But if the market is inefficient and the P/E effect is caused by naïve investors, the P/E effect leads to arbitrage possibilities. The investor could yield a return of 7 to 8% with only a minimum of systematic risk by holding a long position in the value portfolio and a short position in the growth portfolio. Also an investor has to make a judgment about the validity of the tests performed in the papers that report the P/E effect. If the reported P/E effect is caused by data limitations, the P/E effect is not a real investment opportunity.
The main purpose of this study is to examine if there exists a significant P/E effect in Western Europe in the period 1981 till 2008. To test the implications of the P/E effect for an arbitrage investor, seeking for arbitrage opportunities on the bases of the P/E effect, I test whether a portfolio that is long in the value stocks and short in growth stocks yields a return that is significantly larger than zero. To test implications of the P/E effect for a value investor, that wants to compare the value portfolio against the growth portfolio, I test whether a value portfolio yields a significantly larger return than the return of a growth portfolio.
I also use two empirical models to correct the returns for the risk they bear according
2 In other papers, stocks with low valuation multiples other than the P/E ratio, like the market-to-book (M/B)
to the two models. First the monthly returns of both portfolios are adjusted with the Capital Asset Pricing Model for the market risk that both portfolios bear. The second risk adjustment is based on the three factor model of Fama and French (1993). By this model the average monthly returns are adjusted for market risk, and the common risk factors associated with firms size and book-to-market equity (Fama and French (1993)). To enhance the power of the significance test I want the sample period to cover as many years as possible. Unfortunately the data source used in this study starts recording the data needed to measure the P/E effect in the year 19803.
The literature regarding the P/E effect, can be divided into three broad “camps” that try to explain what the causes of the reported return difference between value portfolios and growth portfolios are. The first camp is the proponents of an efficient market. They argue that the value premium is caused by differences in risk between value portfolios and growth portfolios. In an efficient market taking higher risks should be compensated on average with higher returns and so the higher returns of the value portfolio implies that the value portfolio is riskier in comparison to the growth portfolio.
The second camp argues that the inferior returns of growth portfolios in empirical studies are the result of data and methodological problems. Banz and Breen (1986) put forward that the reported P/E effect before 1986 is caused by the problem that firms that have merged, gone bankrupt or are not traded any more for any other reason, are not included in the samples used to measure the P/E effect. According to Banz and Breen (1986) firms with low valuation multiples have a higher chance of going bankrupt than firms with high valuation multiples, and thus the stock performance of firms with low valuation multiples are seriously overestimated in papers that do not include the firms that have gone bankrupt.
The last explanation for the P/E effect comes from the proponents of behavioural finance. These proponents (like for example Lakonishok et al. (1994)) argue that the P/E effect is caused by naïve investors. In forming expectations about the future performance of a company, investors give too less weight to the fact that the performance of a firm is mean reverting, and too much weight to the past performance of the firm. Naïve investors think that the good past performance is representative for the future performance of the company. The psychological error of giving too much weight to recent events, when forming expectations about the future, causes investors to extrapolate past performance too far into the future. Lakonishok et al. (1994) show that value stocks are characterized by low past performance and growth stocks are characterized by high past performance. They conclude that the P/E
effect is caused by naïve investors who have overvalued growth stocks and undervalued value stocks.
In this study I introduce a technique to measure the risk adjusted P/E effect with a non-parametric test. Most financial studies use some form of the t-test to draw conclusions upon. The problem of all t-tests is that the returns should be normal distributed for the t-test to be valid. If the returns are non-normal distributed the results of the matched pairs t-test are invalid. In that case a nonparametric technique is appropriate, because nonparametric techniques do not have the assumption of normality. Thus the inferences, decisions or conclusions on the bases of the nonparametric technique are more appropriate and meaningful in comparison to the inferences, decisions or conclusions on the bases of some sort of t-test, if the returns are non-normal distributed.
To further deepen the research about the significance of the P/E effect I also introduce a technique to test whether the Sharp (1966) ratio of the value portfolio is significantly higher than the Sharpe (1966) ratio of the growth portfolio. The Sharpe (1966) ratio is a ratio that measures the reward-to-risk efficiency of a portfolio. In an efficient market taking higher risk should on average be compensated with higher returns, and thus the Sharpe (1966) ratio should on average be the same for all portfolios. Risk is defined in the Sharpe (1966) ratio as the standard deviation of return. A portfolio with a higher Sharpe (1966) ratio, if compared to another portfolio, means that the portfolio with the higher ratio has a higher return per unit of standard deviation. Thus if the Sharpe (1966) ratio of the value portfolio is significantly larger than the Sharpe (1966) ratio of the growth portfolio, an investor could yield a higher return than the growth portfolio with the same amount of risk as the growth portfolio and thus the P/E effect constitutes a real investment opportunity.
The paper starts with an overview of the literature in section 2. It continues in the sections 3 and 4 with a discussion of the methodology and the data. Section 5 discusses the empirical results and section 6 concludes the paper.
2. Literature Review
In this section I only discus what the causes of the reported P/E effect might be, because it is well documented that the P/E effect of 7 to 8% is reported in many markets and in many time periods (see for example Basu (1977) and Fama and French (1998)).
biases. The first bias is the ex-post-selection bias, or survivorship bias, meaning that companies which had gone bankrupt or merged were not recorded in the samples used by the studies before 1986. The second bias is the look-ahead bias. By this Banz and Breen (1986) mean that portfolios are formed at the beginning of the year with accounting data which was not available at that time in the past. Banz and Breen (1986) collected the Compustat data in the period 1974 till 1981 on a monthly basis. It contains firm data which was available for investors at the end of each month and it also includes firms that have gone bankrupt, merged, or otherwise disappeared from the normal Compustat data. Banz and Breen (1986) find that there is a significant size effect (small stocks outperforming large stocks) in their sample but that the P/E effect is not significant. They conclude (page 793): “the ex-post-selection bias and the look-ahead bias appear to create the “low P/E” effect”. With the “low P/E” effect Banz and Breen (1986) mean the effect that low P/E portfolios outperform high P/E portfolios.
Many studies show that a single index model in the form of the CAPM is not able to describe the cross-section of average stock returns. Ball (1992) his theoretical paper reviews the literature regarding the P/E anomaly and argues that publicly available P/E ratios, at little cost, cannot be the source of returns that are abnormal in comparison to the CAPM. His argument is that, if the market is efficient and becomes aware of the P/E anomaly the P/E effect should disappear. Based on the fact that the P/E effect does not disappear, Ball (1992) hypothesizes that the reported P/E effect could be the result of systematic experimental errors. By experimental errors Ball (1992) means that researchers are not able to construct an adequate theory or adequate empirical model that is able to capture the cross-section of average returns.
returns, these two fundamental variables are priced by the market. According to Fama and French (1993) stocks with high book-to-market ratios are financial distressed and thus face a larger risk and small stocks are more risky because these stocks are illiquid. The three factor model explains the cross-section of average returns for many markets (see for example Bundoo (2008).
Danielson and Dowdell (2001) construct a model, named the return-stages valuation model, to show what expectations of the future return on equity are embedded in the P/E ratio and the price-to-book ratio (P/B ratio). Their model writes a firm’s stock price as a function of three future returns on equity levels: the future return on equity from past investments, the return on equity from new investments during the firm’s growth or restructuring phase, and the return on equity from the firm’s total investment after its growth or restructuring phase ends.
According to the return-stages model a firm has a P/B ratio of 1 if the return on existing and future (incremental) equity equals the cost of capital k. For a P/B ratio to be greater than one, the firm must be able to earn excess returns on existing equity, on new equity, or both. According to the model a firm’s P/E ratio equals 1/k, if a firm operates in a perfectly competitive industry. In that case the maximum sustainable value of the return on past investments, the return form new investments, and the return of the total assets in the future all equal the cost of capital k. Assuming that the return on past investments, the return from new investments, and the return of the total assets in the future all equal the cost of capital k the P/E ratio is equal to 1/k. If, for example, it is expected by investors that the firm is able to invest in new positive net present value projects, the P/E ratio will be higher than 1/k, because the expected value of the future investments is higher than the return on capital k. This firm can still have a return on total assets that is less than the cost of capital k. Thus for a firm to have a P/E ratio that is higher than 1/k, the firm must be able to increase the dollar amount of excess returns (return above the cost of capital). The dollar amount of the excess returns does not have to be positive.
Thus a firm with a P/B ratio higher (lower) than 1 is expected to earn a positive (negative) excess return in the future. A firm that has a P/E ratio that is higher (lower) than 1/k is expected to increase (decrease) the dollar amount of excess returns (return on equity minus the cost of capital) over time until it converges to zero due to competition. According to the return-stages valuation model a firm that meets or exceeds the expectation embedded in the firm’s P/E and P/B ratio has a stock return that is equal or higher as the costs of capital.
sample into four broad groups shown in figure 1.
Figure 1
Classification of firms by their P/E and P/B ratio
P/B>1 P/B<1
P/E>1/k Growth firms Turnaround firms
P/E<1/k Mature firms Declining firms
According to the model the expectations for a “growth firm” are that the firm is able to earn excess returns in the future, and the dollar amount of the excess returns is expected to increase over time until they converge to zero. A “mature firm” is expected to earn excess returns, but the dollar amount of the excess returns is expected to decrease due to for example competition. The “turnaround” firms are expected to earn a negative excess return, but the dollar amount of the excess return is expected to increase in the future. The high P/E ratios of these firms are justified by expectation about for example restructuring activities of these firms. Due to the restructuring activities of these firms the dollar amount of the excess return is expected to increase, but these firms are still expected to earn a negative excess return. For “declining firms” the expectations are that the firms will earn a negative excess return, and the dollar amount of the negative excess return is expected to decline.
Danielson and Dowdell (2001) find that the P/B and P/E ratios can be used to understand the expectations of investors with different P/B and P/E ratios. The future excess returns of the sample of Danielson and Dowdell (2001) follow the patterns predicted by the returns-stages valuation model. The sample of Danielson and Dowdell (2001) includes all NYSE, Amex and Nasdaq firms and the sample period ranges from 1981 until 1996. They conclude that the stock returns of their sample depend on how the operating performance of the firms in their sample compares to the expectations implied by their model. The empirical results of this paper shows that a firm’s performance relative to the expectations implied by the model helps to explain the stock return of that firm.
weight to the prior stock return of a company. Kahneman and Tversky (1974) name the psychological error of overweighting recent events and information and underweighting prior events or base rate data the “respresentativeness heuristic”.
Lakonishok et al. (1994) relate the performance, as measured by earnings growth over the last five years, to the anomaly that past losers outperform past winners. Lakonishok et al. (1994) find that in their sample, including NYSE and Amex stocks in the period April 1968 until April 1990, past winners (losers) have high (low) valuation multiples and have high (low) past earnings growth rates. Lakonishok et al. (1994) mention as possible explanation for the value premium the representativeness heuristic. Investors extrapolate the past performance too far into the future, resulting in too optimistic expectations for growth stocks (winners) and too pessimistic expectations for value stocks (losers). They further hypothesis that if the earnings expectations are not met, stocks with high valuation multiples decline in value and stocks with low valuation multiples rise in value, causing the value premium and thus also P/E effect. The sample Lakonishok et al. (1994) uses is free of the look-ahead bias. Although the sample of Lakonishok et al. (1994) suffers from the survivorship bias, they also show that investing in firms with low valuation multiples also yields a higher return for the largest 50% of the NYSE and Amex firms. The largest 50% of the NYSE and Amex firms where largely free of bankruptcies and mergers.
3. Methodology
3.1 Portfolio formation
is 1023, portfolios 4, 5, and 6, contain an additional stock. The same amount of money is invested in each stock in each portfolio, at the end of June in each formation year t. The end of June is chosen to prevent the results from the look-ahead bias. The total returns to shareholder of each portfolio are monthly calculated from July of year t to June of year t + 1. In the total sample period, there are 27 years in which portfolios are formed, based on the P/E decile breakpoints, in each of the 27 years.
3.2 Significance of the Price Earnings Effect
To see if the value portfolio yields significantly higher returns than the growth portfolio does, I form a portfolio that is long in the value portfolio (portfolio 10) and short in the growth portfolio (portfolio 1) and name it the “value minus growth” portfolio. If this portfolio yields a return that is significantly higher than zero, the P/E effect is present in the Western European market in the period 1981 until 2008. The one-sample t-test (Keller (2005) is performed to test if the average monthly return of the value minus growth portfolio is significantly larger than zero. The test statistic t of the one-sample t-test, to test whether the average return of a portfolio is significantly different from zero, is calculated with the following equation: n x S x t p p / ) ( ˆ = (1)
where x is the average return of portfolio p, Sˆ(xp)is the standard deviation of the monthly returns of portfolio p, and n the number of observations (324 months). The degrees of freedom is n − 1.
whether the average difference in return between the value portfolio and the growth portfolio is significantly larger than zero.
The matched pairs t-statistic is calculated as follows:
D D D n x S x t / ) ( ˆ = (2)
where x is the average difference between the monthly returns of the value portfolio and the D
monthly difference of the growth portfolio, Sˆ(xD)is the standard deviation of the average difference between the monthly returns of the value portfolio and the monthly returns of the growth portfolio, and n the number of observations (324 months). The degrees of freedom is D
n − 1.
The average difference in return, x , between the value portfolio and the growth D
portfolio, is exactly the same as the average return, x ,of the value minus growth portfolio. p
Also the standard deviation of the average difference in return, Sˆ(xD), between the value
portfolio and the growth portfolio, and the standard deviation of the returns, Sˆ(xp),of the value minus growth portfolio are the same. From the above we thus can conclude that, both the one-sample t-test and the matched pairs t-test, lead to the same test statistics. Thus we only have to perform the one-sample t-test on the value minus growth portfolio to conclude what the implications of the P/E effect are for both the arbitrage investor and the value investor.
The Wilcoxon signed ranks-test (Keller (2005)) is performed as follows. First the absolute differences between the monthly returns of the value minus growth portfolio and the monthly returns of the zero portfolio are calculated. Next the absolute values of the differences are ranked. The sum of the positive differences (denotedT ) and the sum of the +
ranks of the negative differences (denotedT ) are then calculated. − T Is used for the +
calculation of the test statistic, because I hypothesis from the literature that the value minus growth portfolio yields a median return that is higher than the median return of the zero portfolio. According to Keller (2005) for samples larger than 30, like in this study, T is +
approximately normally distributed with mean:
4 ) 1 ( ) (T+ = n n+ E (3)
and standard deviation
24 ) 1 2 )( 1 ( ) ( ˆ T+ = n n+ n+ S (4)
where n is the number of absolute differences (324 months).
Thus, the standardized test statistic z is
) ( ˆ ) ( + + + − = T S T E T z (5)
growth strategy. Although the median return of the value minus growth portfolio may be different than the difference between the median return of the value portfolio and the median return of the growth portfolio, the 324 monthly absolute differences important for the arbitrage investor are the same as the 324 monthly absolute differences important for the value investor. Because the 324 monthly absolute differences important for the arbitrage investor are the same as the 324 monthly absolute differences important for the value investor, the ranks of the two absolute differences are also the same. Thus the sum of the positive differences (denoted T ) and the sum of the ranks of the negative differences (denoted + T ) −
that are important for the arbitrage investor are both the same as the sum of the positive differences (denoted T ) and the sum of the ranks of the negative differences (denoted + T ) −
that are important for the value investor, respectively. Thus the Wilcoxon signed ranks test z-statistic important for the arbitrage investor is also the same as the Wilcoxon signed ranks test z-statistic important for the value investor. From the above we can conclude that we only have to perform the Wilcoxon signed ranks test on the value minus growth portfolio to see what the implications of the P/E effect are for both the arbitrage investor and the value investor.
3.3 Risk Adjusted Price Earnings Effect
The returns of each decile are also calculated on a risk adjusted basis with Jensen’s (1968) alpha, and the Fama and French (1993) three factor model.
3.3.1 Market Adjusted P/E Effect
The capital asset pricing model (CAPM) defines the relation between risk and the expected return on risky assets. According to the CAPM a risky asset carries two types of risk; the market risk (systematic risk), which is measured by beta, and unsystematic risk. Sharpe (1964) argues that a diversified portfolio of risky assets does not carry unsystematic risk. According to the empirical work of Campbell et al. (2001) investors need to own at least 20 stocks to replicate the risk of the market as a whole. The smallest number of stocks in the portfolios in this study is 26 and thus according to Campbell et al. (2001) the portfolios are diversified enough.
expected returns is as follows: ) ) ( ( ) (Ri rf i E Rm Rf E = + β − , (6)
where E(Ri) is the expected return on security i, E(Rm) is the expected return on the market
portfolio, r is the risk-free rate and f βi is a measure of systematic risk for security i.
Jensen (1968) uses the CAPM to measure the return of a portfolio in excess of the CAPM’s theoretical expected return. According to Jensen (1968) the excess return above the CAPM’s theoretical expected return (named Jensen’s alpha, are market adjusted return) is the return which is not explained by the systematic risk of a portfolio. This means that if the value minus growth portfolio has a Jensen’s alpha that is significantly larger than zero, the price earnings effect is also present after correcting for market risk.
Jensen’s Alpha is measured with the time-series regression below:
t p t f t m p p t f t p R R R R , − , = α + β ( , − , )+ ε , , (7)
where Rp,tis the return of portfolio p in month t, Rf,tis the one month government
zero-coupon rate for the country in which the stock is traded at the beginning of month t, α pis
Jensen's alpha (a constant) of portfolio p, β pis the slope of the regression equation (the index
of systematic risk), Rm,t is the simple average return of all firms in the sample in month t, and
t p,
ε is a error term.
For simplicity I choose for an equally weighted return of all stocks (equally weighted index). According to Ross (1978) both an equally-weighted index and a value-weighted index may be used and both indices lead to the same conclusions in empirical tests.
where α p,tis the monthly Jensen’s alpha of portfolio p, βˆ is the estimated regression p
coefficient of regression (8).
Now we can test whether the average monthly Jensen’s alpha (the average monthly market adjusted return) of the value minus growth portfolio is significantly larger than zero and whether the median monthly Jensen’s alpha of the value minus growth portfolio is significantly larger than zero as in section 3.2 with the one-sample t-test and the Wilcoxon signed ranks test, respectively. As explained in section 3.2 the test results of both the one-sample t-test performed on the value minus growth portfolio and the Wilcoxon signed ranks test performed on the value minus growth portfolio have implications for both the arbitrage investor and the value investor. The t-statistic of the one-sample t-test, to test whether the average monthly Jensen’s Alpha of the value minus growth portfolio is significantly larger than zero is exactly the same as the regression t-statistic to test whether the regression coefficient α p of the value minus growth portfolio in (7) is statistical different from zero.
Thus we can conclude on the basis of the regression t-statistic of α p in (7) of the value minus
growth portfolio whether the P/E effect constitutes a real investment opportunity for both the arbitrage investor and the value investor.
To the best knowledge of the author the above technique to calculate monthly alphas as in (8) is never used before in the literature. The advantage of this new technique is that with this new technique it is also possible to perform the nonparametric Wilcoxon signed ranks test on the median Jensen’s alpha of the value minus growth portfolio. Thus if the monthly Jensen’s alphas of the value minus growth portfolio calculated as in (8) is non-normal distributed we can draw more valid conclusions on the bases of the Wilcoxon signed ranks test.
3.3.2 Three Factor Model Adjusted P/E effect
The Fama and French three factor model for expected returns is: ), ( ) ( ) ) ( ( ) (R R b E R R sE SMB hE HML E i − f = i m − f + i + i (9)
where E(Ri)− Rf is the expected excess return of portfolio p,E(Rm)− Rf is the expected
market premium, E(SMB)is the expected difference between the equally weighted returns on portfolios of small and big stocks (below or above the median, respectively), and E(HML)is the expected difference between the equally weighted returns on portfolios of high- and low-book-to-market equity stocks (above and below the 0,7 and 0,3 fractiles of low-book-to-market equity, respectively), and the factor sensitivities or loadings, b , i s , and i h are the slopes in i
the time-series regression (10) shown below. The times series regression for the factor sensitivities is: t p p p t f t m p p t f t p R b R R s SMB h HML R , − , = α + ( , − , )+ + + ε , . (10) where α pis a constant.
The three factor model alpha of portfolio p, α p, is the excess return above the Fama
and French three-factor model theoretical return, the return that is not explained by the three factor model.
In this study the exogenous variables SMB and HML are formed following almost the same procedure as in the study of Fama and French (1993). At the end of June of each year t, all stocks are sorted on market value. The median market value of equity (ME) of the ranked stocks is then used to split the sample into two groups, small (S) and big (B).
Book-to-market equity is the book-to-market ratio for the end of December year t – 1 for all firms. Based on the breakpoints of the bottom 30% (Low (L)), middle 40 % (Medium (M)), and top 30% (High (H)) of the ranked values of market equity, three book-to-market equity groups are formed. Negative book-equity firms are excluded from the sample.
total returns to shareholders for each portfolio are monthly calculated from July of year t to June of year t +1.
According to Fama and French (1993), the exogenous variable SMB in (9) is meant to mimic the risk factor in returns related to size. SMB is the difference, each month, between the simple average of the returns on the three small-stock portfolios (S/L, S/M, and S/L) and the simple average of the returns on the three big-stock portfolios (B/L, B/M, and B/H). Thus SMB is the difference of returns between the returns on the three small-stock portfolios and the three big-stock porfolios with about the same average book-to-market ratio.
According to Fama and French (1993) the exogenous variable HML in (9) is meant to mimic risk factors in returns related to book-to-market equity. HML is the difference, each month, between the simple average of the returns on the two high-book-to-market portfolios (S/H and B/H) and the simple average of the two low-book-to-market equity portfolios (S/L and B/L).
The market factor in (9), Rm,t − Rf,t, is the excess market return. Rm,t is the simple
average return of all stocks in the sample in month t . Rf,tis the one month government
zero-coupon rate for the country in which the stock is tradedat the beginning of month t.
If the value portfolio has a higher three factor model alpha then the growth portfolio has, the price earnings effect is also present after correcting for market risk, and the common risk factors associated with firm size and book-to-market-equity (Fama and French (1993)).
To test whether the median three factor model alpha, α p, of the value minus growth
portfolio in (10) is significantly higher than the zero I rearrange regression (10) so that the Wilcoxon signed ranks test can be performed. The rearranged version of regression (10) to calculate the monthly three factor model alphas looks like follows:
HML h SMB s R R b R Rpt f t p mt f t p p t p, = , − , − ˆ ( , − , )− ˆ − ˆ α (11)
where α p,tis the three factor model monthly alpha of portfolio p, and bˆ , p sˆ , andp hˆ are the p
estimated regression coefficient of regression (11).
respectively. As explained in section 3.2 the test results of both the one-sample t-test and the Wilcoxon signed ranks test have implications for both the arbitrage investor and the value investor. The t-statistic of the one-sample t-test, to test whether the average monthly three factor model alpha of the value minus growth portfolio is significantly larger than zero is exactly the same as the regression t-statistic to test whether the regression coefficient α p of
the value minus growth portfolio in (10) is statistical different from zero.
3.4 The Sharpe (1966) ratio
To relate the risk of a portfolio to the return of a portfolio I calculate the Sharpe (1966) ratio. The Sharpe (1966) ratio is the average return above or below the risk free rate of a portfolio divided by the standard deviation of that particular portfolio. Stated otherwise, the Sharp ratio is the return in excess of the risk free rate per unit of risk (risk as measured by the standard deviation of the returns). In an efficient market bearing higher risk should on average be compensated with higher returns and vice versa. And thus in an efficient market the Sharpe (1966) ratios of all portfolios should be the same.
The Sharpe (1966) ratio, Sp , is defined as:
p t f t p p R R S σ ) ( , − , = (12)
where (Rp,t − Rf,t) is the average return of portfolio p above or below the risk free rate over the entire sample period of 324 month, and σ pis the standard deviation of all 324 monthly
returns of portfolio p.
I also calculate monthly Sharpe (1966) ratios per portfolio to measure whether the Sharpe (1966) ratio of the value portfolio is significantly larger than the Sharpe (1966) ratio of the growth portfolio. To calculate the monthly Sharpe (1966) ratio I use the following equation: p t f t p t p R R S σ ) ( , , , − = (13)
Calculating monthly Sharpe (1966) ratios, calculated with an equation that looks like equation (13), is to the best knowledge of the author, never used in the literature. The advantage of this technique is that it is now possible to test for example whether the Sharpe (1966) ratio of a portfolio is significantly larger than zero. For an arbitrage investor it is important whether the average Sharpe (1966) ratio of the value minus growth portfolio is significantly larger than zero. To test the arbitrage strategy I use the one-sample t-test described in section 3.2. For a value investor it is important whether the average Sharpe (1966) ratio of the value portfolio is significantly larger than the average Sharpe (1966) ratio of the growth portfolio. To test whether the average Sharpe (1966) ratio of the value portfolio is significantly larger than the average Sharpe (1966) ratio of the growth portfolio, we need to use the matched pairs t-test, described in section 3.2. The standard deviation of the value minus growth portfolio is not the same as the difference in standard deviation between the monthly Sharpe (1966) ratios of the value portfolio and the monthly Sharpe (1966) ratios of the growth portfolio4, and thus the one-sample t-test to test the arbitrage strategy does not
yield the same results as the matched pairs t-test to compare the value strategy against the growth strategy. We thus have to perform both, the one-sample t-test on the average Sharpe (1966) ratio of the value minus growth portfolio to test the arbitrage strategy, and the matched pairs t-test on the average difference between the monthly Sharpe (1966) ratios of the value portfolio and the monthly Sharpe (1966) ratios of the growth portfolio, because the two t-tests lead to different t-statistics.
If the monthly Sharpe (1966) ratio is non-normal distributed the one-sample t-test and the matched pairs t-test are not the appropriate tests, and we thus have to perform the Wilcoxon signed ranks test. For the implications of the monthly Sharpe (1966) ratios for the arbitrage investor, we have to test whether the median Sharpe (1966) ratio of the value minus growth portfolio is significantly larger than zero. For the implications of the monthly Sharpe (1966) ratios for the value investor we have to test whether the median Sharpe (1966) ratio of the value portfolio is significantly larger than the median Sharpe (1966) ratio of the growth portfolio.
4 The difference comes from the fact that the Sharpe (1966) ratio divides the average return by its standard
4. Data
The sample period in this study covers the years 1981 through 2008. Datastream is used as the data source in this study. The accounting data available in Datastream is supplied by Worldscope. Firms without accounting data available in Datastream are excluded from the sample, because Datastream does not have P/E ratios for these firms. Datastream has returns data available from 1973 and thereafter. Worldscope started supplying Datastream with accounting data in 1980 for the countries that are studied in this paper. Because accounting data of one year prior to portfolio formation is needed, the first year in which portfolios are formed is 1981. The last formation year is 2007.
The following countries are included in the sample: United Kingdom, Ireland, the Netherlands, Belgium, Luxembourg, and France. These countries together are defined by the 2008 CIA World Factbook (2008) as Western Europe. Local currencies prior to the euro from Ireland, the Netherland, Belgium, Luxembourg, or France are calculated to the euro with the exact exchange rate set in 1999, and still relevant for old contracts stated in the old currency. For data stated in the United Kingdom pound sterling prior to the euro, the pound sterling-French franc currency rate and then the sterling-French franc exact exchange rate set in 1999 is used to calculate the values in the euro currency. For data stated in the United Kingdom pound sterling after 1999 the euro-pound sterling exchange rate is used.
does not report negative earnings. In a pure value/contrarian investment strategy, investors buy stocks that are out of favor. So a pure contrarian investor would buy stocks with negative earnings and consequentially negative P/E ratios. But if the decision making process in selecting stocks is at least a little affect heuristic driven5, even the most purist contrarian
investor has strong bad emotional feelings (with or without consciousness) with negative earnings, and neglects the firms with negative earnings. Thus it can be assumed, that the strategy of a pure value/contrarian investor does not significantly differ form the strategy used in this study; buying stock with low positive P/E ratios.
The return of a stock is measured with the “Total Return Index” (Datastream datatype “RI”). The Total Return index is adjusted for subsequent stock splits, and includes all types of dividends. If a firm is traded at more then one exchange, the data of the exchange with the highest trade volume are used; the data of the other exchange(s) are deleted from the sample.
Market value (Datastream datatype “MV”) is the share price at the required date multiplied by the number of ordinary shares outstanding at the required date. The amount in issue is updated whenever new tranches of stock are issued or after a capital change.
Market-to-book ratio (Datastream datatype “MTBV”) is defined as the market value of the common equity at the required date divided by the balance sheet value of the common equity in the company of fiscal year t – 1. For the Fama and French (1993) three factor model the inverse of the market-to-book ratio is used (the book-to-market ratio).
As a proxy for the risk free rate of return I use the one month government zero-coupon rate for the country in which the stock is tradedat the beginning of month t.
From 1980 until 1985, the database of Datastream has accounting records for about 30% of the firms in the total database in these years. From 1986 until 1995 this number has changed to about 70%. From 1996 until 2008, about 95% of the total firms in the database have balance sheet records available. From the above, we can conclude that Worldscope has expanded her database rapidly during the late nineteen eighties. It is unknown which process Worldscope follows when they add firms to their database. The process Worldscope follows could be a potential source of bias. If Worldscope only adds firms that have performed well during the last fiscal year before adding the firm to the database, the Worldscope database is skewed towards good performing firms especially during the late nineteen eighties until the mid-nineteen nineties.
5 Simple stated the affect heuristic is a rule of thumb in decision making. Affect, in the context of the affect
To prevent the sample from the ex-post-selection bias, stocks no longer traded, due to bankruptcies, mergers, takeovers, etc., are also included in the sample. The date and reason why the companies stock is no longer traded are provided by Datastream. In the case of bankruptcy the return index is set manually to zero in the particular month of bankruptcy, assuming a one hundred percent capital loss. It is assumed, that a stock that is merged, delisted, or acquired yields the average return, of the portfolio it was part of, in the months after the event took place.
The dataset contains 3421 stocks in total. The average number of stocks in the sample in a formation year is 1177. The minimum number of stocks in the sample is 263 in formation year 1981; the maximum is 1996 in formation year 1998.
Table 1 shows the descriptive statistics of the sample. The table shows that on average the firms in the value portfolio have low market-to-book ratios. We can observe a negative relation between the P/E ratio and the book-to-market ratio. The same negative relation is also found by Fama and French (1992). From table 1 we can observe that the relation between the P/E ratio and firm size is inverted u-shaped.
Table 2 shows the descriptive statistics of the yearly portfolio returns in excess of the risk free rate calculated from July of year t to June of year t + 1. Table 2 also shows the average and median return in excess of the risk free rate over the entire 27 year sample period for all portfolios. As a risk measure, the standard deviations of the yearly returns in excess of the risk free rate for all portfolios are also included in the table. As a risk-return measure (risk as measured by the standard deviation of the yearly returns) the Sharpe (1966) ratio is shown.
Table 1
Descriptive Statistics for the Ten Stock Portfolios Formed on P/E Ratios
From the growth portfolio down to the value portfolio, we move from the portfolio containing the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, to the portfolio containing the bottom ten percent of stocks as ranked by the P/E ratio at the end of June in each year t. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t.
The average or median P/E ratio of a portfolio is the average or median P/E ratio of all firms in the portfolio at the end of June year t requested in Datastream. The average or median market value of a portfolio is the average or median market value of all firms at the end of June year t requested in Datastream. The average or median market-to-book ratio of a portfolio is the average or median market-to-book ratio of all the firms in a portfolio at the end of December year t – 1 requested in Datastream. All averages and medians are than averaged across the 27 years sample period. The row “minimum and “maximum” of the P/E ratio, the firm size, or the market-to-book ratio show the minimum and maximum value of the average P/E ratio, firms size, or market-to-book ratio of the year in which the average P/E ratio, firm size, or market-to-book ratio has the lowest or highest value, respectively.
Growth 2 3 4 5 6 7 8 9 Value
Average of average P/E ratio 44.6 24.9 19.9 17.1 15.0 13.4 11.8 10.3 8.5 5.4 Average of median P/E ratios 43.6 25.3 20.6 17.4 15.3 13.2 11.6 10.1 8.8 5.5 Minimum P/E ratios 29.7 19.3 15.6 13.3 11.8 10.5 9.2 7.6 5.7 3.5 Maximum P/E ratios 60.4 33.3 24.8 20.4 18.8 17.3 16.0 14.5 12.5 8.6 Average of average firm size a 124 175 195 199 200 150 132 119 83 52 Average of median firm size a 108 177 201 187 173 132 127 104 70 42
Minimum firm size a 29 82 82 61 71 63 46 48 25 21
Maximum firm size a 319 378 451 543 631 355 248 244 158 139
Average of average market-to-book ratio 3.71 3.54 2.89 2.86 2.79 1.77 1.99 1.72 1.85 1.51 Average of median market-to-book ratio 3.39 2.85 2.57 2.52 2.35 2.24 1.95 1.87 1.48 1.34 Minimum market-to-book ratio 1.69 1.32 0.30 0.84 1.29 -12.55 0.47 0.77 0.61 0.31 Maximum market-to-book ratio 8.26 9.10 8.65 11.31 7.81 6.81 5.33 2.66 6.60 5.08 a In millions of euro
From table 2 we can also observe that the relation between the P/E ratio and the Sharpe (1966) ratio is negative. If the CAPM holds the Sharpe (1966) ratio of each portfolio should on average be the same, because taking higher risk should be compensated with a higher return on average. The negative relation between the P/E ratio and the Sharpe ratio (1966) is thus another indication for the anomaly that portfolios with low P/E ratios tend to outperform the portfolios with high P/E ratios without being riskier (risk as measured by the standard deviation of the return).
Table 2
Yearly Euro Currency Returns in Excess of the Risk Free Rate per Portfolio
From the growth portfolio down to the value portfolio, we move from the portfolio containing the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, to the portfolio containing the bottom ten percent of stocks. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t. The 1 year buy-and-hold percent returns in excess of the risk free rate of each portfolio are calculated per month from July of year t to June of year t + 1. The percent returns are calculated for a total period of 27 years, starting in July 1981 and ending June 2008. The value minus growth portfolio is a portfolio that is long in the value portfolio and short in the growth portfolio. The portfolio “Western European market” contains all stocks in the sample.
The Sharp ratio is the average yearly return in excess of the risk free rate of a portfolio divided by the yearly standard deviation of that particular portfolio. The column “Minimum” shows the minimum yearly return in excess of the risk free rate for each portfolio: the column “Maximum” shows the maximum yearly return in excess of the risk free rate for each portfolio.
Portfolio p yearly return^ Average Standard de-viation of yearly return^ Yearly Sharpe ratio Minimum yearly return^ Maximum yearly return^ Median yearly return^
Western European market 10.49 19.63 0.53 -30.82 39.18 14.28
Growth (high P/E) 9.38 24.81 0.38 -32.80 67.39 12.86
2 7.63 20.94 0.36 -29.80 35.76 14.00 3 8.24 21.29 0.39 -34.22 38.29 9.16 4 9.25 18.12 0.51 -28.05 34.24 15.91 5 7.10 16.21 0.44 -31.41 29.25 9.89 6 10.34 18.92 0.55 -28.73 43.37 9.58 7 10.10 19.34 0.52 -33.68 43.74 12.62 8 10.81 19.60 0.55 -33.76 44.36 12.62 9 13.66 21.87 0.62 -35.05 57.24 17.46
Value (low P/E) 18.24 26.00 0.70 -26.39 63.93 21.91
Value minus Growth 8.86 16.81 0.53 -40.28 33.71 7.27
^ In excess of the risk free rate
Figure 2
Yearly Euro Currency Returns in Excess of the Risk Free Rate of the Value minus Growth Portfolio from July 1981 to June 2008
For the formation-details of the value minus growth portfolio see the caption of table 1.
5. Results
5.1 Unadjusted P/E effect
Table 3 shows the average and median of the 324 monthly returns in excess of the risk free rate for the growth portfolio, the value portfolio, and the value minus growth portfolio. To compare the average and median monthly returns in excess of the risk free rate of the value and growth portfolio to the average return of the sample, the portfolio “Western European market”, containing all the stocks in the sample, is also included in the table. This table also shows the results of the t-test and the Wilcoxon signed ranks test, to test whether the average monthly returns and the median monthly returns are significantly larger than zero.
Table 3
Monthly Euro Currency Returns in Excess of the Risk Free Rate
The growth portfolio contains the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, and the value portfolio contains the bottom ten percent. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t. The 1 year buy-and-hold returns in excess of the risk free rate of each portfolio are calculated per month from July of year t to June of year t + 1. The percent returns are calculated for a total period of 324 months, starting in July 1981 and ending June 2008. The “Western European market” portfolio contains al stocks in the sample. The portfolio “Value minus Growth” is a portfolio that is long in the value portfolio and short in the growth portfolio and measures the P/E effect.
The one-sample t-test is performed to measure whether the average monthly return in excess of the risk free rate per portfolio is significantly larger than zero. The Wilcoxon signed ranks test is performed to measure whether the median monthly return in excess of the risk free rate per portfolio is significantly larger than zero. The Jarque-Bera test measures whether the monthly returns in excess of the risk free rate are normal distributed.
Portfolio p excess of the risk free rateAverage monthly return in ^
Median monthly return in excess of the risk free rate#
Jarque-Bera of monthly re-turns in excess of the risk
free rate
Western European market (3.11)0.81* (4.43)1.39* 266.68*
Growth (high P/E) 0.70**(2.25) (3.19)1.33* 256.55*
Value (low P/E) (4.45)1.32* (5.42)1.70* 170.41*
Value minus Growth (2.86)0.62* (3.51)0.66* 443.42*
^ One-sample t-statistics in parentheses
# Wilcoxon signed ranks test z-statistic in parentheses *** Significant at 10%
** Significant at 5% * Significant at 1%
signed ranks z-statistic shows that the median return in excess of the risk free rate of the value minus growth portfolio is statistically larger than zero (z-statistic of 3.51). The z-statistic of 3.51 is thus also the z-statistic for the Wilcoxon signed ranks test that tests whether the value portfolio yields a significantly larger return than the growth portfolio yields (see section 3.2). Thus the unadjusted P/E effect constitutes a real investment opportunity for both arbitrage investors and value investors. Arbitrage investors yield a significant median unadjusted return of 0.66% per month without investing any money. Value investor can conclude that the value strategy yields a return that is 0.37% (median return of the value portfolio minus the median return of the growth portfolio) significantly larger than the median return of the growth strategy.
5.2 Market adjusted P/E effect
Table 4 shows the estimation results of regression (7) per portfolio, and the median of Jensen’s alpha (the market adjusted return) calculated with equation (8) per porfolio. The estimation results for the value minus growth portfolio shows that the value portfolio has a significant smaller CAPM beta than the growth portfolio, because the beta estimate of the value minus growth portfolio is statistical smaller than zero at the 1% significance level (regression t-statistic of -3.07). Although the value portfolio has a smaller beta estimate than the growth portfolio has, the average monthly return in excess of the risk free rate of the value portfolio is higher than the average monthly return in excess of the risk free rate of the growth portfolio (see table 3). If the CAPM holds, the growth portfolio should have a higher average return than the value portfolio, because the growth portfolio has a higher beta. In this paper the growth portfolio has a higher beta and lower return if compared to the beta and return of the value portfolio. Thus the P/E anomaly is also present in the Western European stock market, because, without being fundamentally riskier than the growth portfolio, the value portfolio yields an average return that is significantly higher than the average return of the growth portfolio.
Table 4
Monthly Jensen’s Alphas Calculated with the Estimated Regression Coefficients of Rp.t – Rf,t = αp + βp (Rm.t – Rf,t) + εp,t
The growth portfolio contains the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, and the value portfolio contains the bottom ten percent. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t. The 1 year buy-and-hold returns in excess of the risk free rate of each portfolio are calculated per month from July of year t to June of year t + 1. The percent returns are calculated for a total period of 324 months, starting in July 1981 and ending June 2008. The portfolio “Value minus Growth” is a portfolio that is long in the value portfolio and short in the growth portfolio and measures the P/E effect.
For the regression (regression 7) shown in the table Rp.t – Rf,t is the monthly return of portfolio p in excess of the risk
free rate, αp is Jensen’s alpha of portfolio p, βp is a factor sensitivity, Rm.t – Rf,t is the monthly return of the Western European
market in excess of the risk free rate, and εp,t is an error term.
The monthly Jensen’s alphas are calculated with the following equation: ap,t= Rp.t – Rf,t - βp (Rm.t – Rf,t), where ap,t is the
Jensen’s alpha of portfolio p in month t, and βp is the estimated regression coefficient of regression (7) shown in the table.
The Wilcoxon signed ranks test is performed to measure whether the median monthly Jensen’s alpha per portfolio is significantly larger than zero. The Jarque-Bera test measures whether the monthly alphas are normal distributed.
Portfolio p
Estimation results of Rp.t – Rf,t = αp + βp (Rm.t – Rf,t) + εp,t
αp (Average monthly
Jensen’s alpha)^ βp^ Adjusted R2
Median of monthly Jensen’s alpha# Jarque-Bera of monthly Jensen’s alpha Growth (-1.65)-0.20 (43.94)1.12* 0.86 (-2.51)-0.17* 1205.00* Value (3.65)0.53* (31.53)0.98* 0.76 (3.74)0.34* 771.35*
Value minus Growth (3.38)0.73* (-3.07)-0.14* 0.03 (4.00)0.73* 341.16*
^ Regression t-statistics in parentheses
# Wilcoxon signed ranks test z-statistic in parentheses *** Significant at 10%
** Significant at 5% * Significant at 1%
the growth portfolio, the Jensen’s alpha of regression (6) is significantly higher than zero for the value minus growth portfolio. This means that the P/E effect is also present after correcting for market risk. As explained in section 3.3.1, if we test whether the average monthly Jensen’s alpha of the value minus growth portfolio (calculated with equation 8) is significantly larger than zero, we get a one-sample t-statistic of 3.38, that is the same as the regression t-statistic of the regression coefficient ap of the value minus growth portfolio. From
the significantly higher than zero average market adjusted return of the value minus growth portfolio we can conclude that, on the bases of the CAPM, the P/E effect constitutes real investment opportunities for both the arbitrage investor and the value investor.
From the Jarque-Bera results we can conclude that the returns of all three portfolios are non-normal distributed and we thus have to base our conclusions on the nonparametric test.
explained in section 3.2, the z-statistic of the Wilcoxon signed ranks test, that tests whether the median Jensen’s alpha of the value portfolio is significantly larger than the median Jensen’s alpha of the growth portfolio, is thus also 4.00. Thus also on the bases of a nonparametric technique the market-adjusted P/E effect is present in the Western European market, between 1981 and 2008. This result indicates that also on the bases of the more valid nonparametric tests the market adjusted P/E effect constitutes a real investment opportunity for both value investors and arbitrage investors.
5.3 Three factor model adjusted P/E effect
Table 5 shows the estimation results of regression (10) per portfolio, and the median of the monthly three factor model alpha calculated with equation (11) per portfolio. The regression coefficient, bp, for all 25 portfolios in the paper of Fama and French (1993) are
close to 1 (ranging from 0.91 to 1.18). Table 5 shows the factor loading, bp, for the value
portfolio and the growth portfolio are also close to 1.
Table 5 shows us that the average three factor model alpha of the growth portfolio is significantly larger than zero (regression t-statistic 2.16) at the 5% significance level. This means that the growth portfolio has an average monthly return of 0.21% that is unexplained by the three factor model. From the insignificance of the average three factor model alpha of the value portfolio (regression t-statistic of -0.06) we can conclude that the returns of value portfolio are entirely explained by the three factor model.
Although the growth portfolio has an average monthly three factor model alpha that is significantly higher than zero and the average monthly three factor model alpha of the value portfolio is insignificantly different from zero, the average monthly alpha of the value minus growth portfolio is insignificantly different from zero (regression t-statistic of -1.27). From the insignificant average three factor model alpha of the value minus growth portfolio we can conclude that the three factor model adjusted P/E effect is also insignificant.
Table 5
Monthly Three Factor Model Alphas Calculated with the Estimated Regression Coefficients of
Rp.t – Rf,t = αp + bp (Rm.t – Rf,t) + spSMB + hpHML + εp,t
The growth portfolio contains the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, and the value portfolio contains the bottom ten percent. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t. The 1 year buy-and-hold returns in excess of the risk free rate of each portfolio are calculated per month from July of year t to June of year t + 1. The percent returns are calculated for a total period of 324 months, starting in July 1981 and ending June 2008. The portfolio “Value minus Growth” is a portfolio that is long in the value portfolio and short in the growth portfolio and measures the P/E effect.
For the regression (regression 10) shown in the table Rp.t – Rf,t is the monthly return of portfolio p in excess of the risk
free rate, αp is the three factor model alpha of portfolio p, Rm.t – Rf,t is the monthly return of the Western European market in
excess of the risk free rate, SMB is the difference between the equally weighted returns on portfolios of small and big stocks (below or above the median of the sample market value, respectively), and HML is the difference between the equally weighted returns on portfolios of high- and low-book-to-market equity stocks (above and below the 0,7 and 0,3 fractiles of book-to-market equity, respectively), bp , sp , hp are factor sensitivities and εp,t is an error term.
The monthly three factor model alphas are calculated with the following equation: ap,t= Rp.t – Rf,t - bp (Rm.t – Rf,t) - spSMB - hpHML,
where ap,t is the three factor model alpha of portfolio p in month t, bp , sp , hs are the estimated regression coefficient of the
regression shown in the table (regression (10)).
The Wilcoxon signed ranks test is performed to measure whether the median monthly three factor model alpha per portfolio is significantly larger than zero. The Jarque-Bera test measures whether the monthly three factor model alphas are normal distributed. Portfolio p Estimation results of Rp.t – Rf,t = αp + bp (Rm.t – Rf,t) + spSMB + hpHML + εp,t αp (Average monthly three factor alpha)^ bp^ sp^ hp^ Adjusted R2 Median of monthly three factor model
al-pha3
Jarque-Bera of monthly three factor model
al-pha Growth 0.21**(2.16) (51.86)1.04* (4.03)0.19* (-13.58)-0.53* 0.92 0.12**(2.21) 27.18* Value (-0.06)-0.01 (39.84)1.05* (7.45)0.47* (11.37)0.58* 0.83 (-0.55)0.05 380.93* Value minus Growth -0.22 (-1.27) 0.01 (0.27) 0.28* (3.27) 1.11* (16.20) 0.46 -0.21 (-1.64) 75.28*
^ Regressions t-statistic in parentheses
# Wilcoxon signed ranks test z-statistic in parentheses *** Significant at 10%
** Significant at 5% * Significant at 1%
Table 6
The Monthly Sharpe (1966) ratio
The growth portfolio contains the top ten percent of all stocks in the sample as ranked by the P/E ratio at the end of June in each year t, and the value portfolio contains the bottom ten percent. In a portfolio, the same amount of money is invested in each stock at the end of June, in each year t. The 1 year buy-and-hold returns in excess of the risk free rate of each portfolio are calculated per month from July of year t to June of year t + 1. The percent returns are calculated for a total period of 324 months, starting in July 1981 and ending June 2008. The portfolio “Value minus Growth” is a portfolio that is long in the value portfolio and short in the growth portfolio and measures the P/E effect. The portfolio “Western European market” is a portfolio containing all the stocks in the sample. Row 5 shows the differences between the average and median monthly Sharpe (1966) ratio of the value portfolio and the growth portfolio, respectively, and is important for a value investor comparing his value investment strategy against the growth investment strategy (investing in high P/E portfolio).
The monthly Sharpe (1966) ratios are calculated with the following equation:
p t f t p t p R R S σ ) ( , , , − =
where Sp,t is the Sharpe (1966) ratio of portfolio p in month t, Rp.t – Rf,t is the monthly return of portfolio p in excess of the
risk free rate, and σp is the standard deviation of all 324 monthly returns of portfolio p.
The one-sample t-test is performed to measure whether the average monthly Sharpe (1966) ratio per portfolio is significantly larger than zero. The matched pairs t-test tests whether the average monthly Sharpe (1966) ratio of the value portfolio is significantly larger than the average monthly Sharpe (1966) ratio of the growth portfolio. The Wilcoxon signed ranks test is performed to measure whether the median monthly Sharpe (1966) ratio per portfolio is significantly larger than zero (row 1 to 4) or whether the median monthly Sharpe (1966) ratio of the value portfolio is significantly larger than the median monthly Sharpe (1966) ratio of the growth portfolio (row 5). The Jarque-Bera test measures whether the monthly Sharpe (1966) ratios are normal distributed.
Row number Portfolio p Average monthly Sharpe ratio Median Monthly Sharpe ratio
Jarque-Bera of monthly Sharpe
ra-tio
1 Western European market 0.17*^(3.12) 0.30*(4.41)# 266.68*
2 Growth 0.13**^(2.25) 0.24*(3.17)# 256.56*
3 Value 0.25*^(4.54) 0.32*(5.39)# 170.41*
4 Value minus Growth 0.16*^(2.86) 0.17*(3.50)# 443.42*
5 Difference between row 2 and row 1 0.12*(3.21)~ 0.08*(3.81)#
-^ One-sample t-statistics in parentheses ~ Matched pairs t-statistic in parentheses
# Wilcoxon signed ranks test z-statistic in parentheses *** Significant at 10%
** Significant at 5% * Significant at 1%
5.4 The Sharpe (1966) ratio
than zero at the 1% level (z-statistic of 3.50). Thus for an arbitrage investor the P/E effect leads, on the bases of the monthly Sharpe (1966) ratio, to significant arbitrage opportunities.
The median monthly Sharpe (1966) ratio of the value portfolio is significantly larger than the median monthly Sharpe (1966) ratio of the growth portfolio (Wilcoxon signed ranks test z-statistic 0f 3.81). This means that also for the value investor the P/E effect, on the basis of the monthly Sharpe (1966) ratio leads to investment opportunities. The value investor could yield a median return that is 0.08% larger per unit of standard deviation.
6. Conclusion
The results of the statistical tests performed on the unadjusted monthly P/E effect show that the P/E effect in the Western European stock market is significant for both the arbitrage investor and the value investor.
The tests performed on the market adjusted monthly P/E effect also indicate that the P/E effect is a real investment opportunity. As a result of the significant lower CAPM beta for the value portfolio, if compared to the CAPM beta of the growth portfolio, the market adjusted monthly P/E effect is even larger than the unadjusted P/E effect. According to the CAPM an arbitrage investor can yield a significant positive return without bearing any market risk and the value investor can yield a significant higher market adjusted return than a growth investor.
The question remains whether the CAPM is the correct model to describe the cross- sectional variation in average stock returns. If the CAPM holds, the returns differential between the value portfolio and the growth portfolio should be explained by sample biases. This study is free of the survivorship bias and the look-ahead bias, so these biases cannot be the source of the P/E effect. The 27 yearly returns of the value minus growth portfolio do not indicate that the process Worldscope follows in adding firms to their database influences the results of this study. So the only source of bias could be that the sample used in this study does not include stocks with negative P/E ratios.
If we compared the adjusted R2 of three factor model with the adjusted R2 of the
better than the CAPM, we can conclude that the three factor model is preferred above the CAPM.
On the basis of parametric tests, the three factor model has proven to explain the cross section of returns in markets all over the world (see for example Bundoo (2008). This paper does not find any evidence that the nonparametric statistical tests performed on the three factor model alpha yield any other conclusions than the parametric tests performed on the three factor model alpha. The regression coefficients on the exogenous variable SMB and HML of the three factor model for the value minus growth portfolio are significant different from zero. Thus we can conclude that the P/E effect is significantly caused by the size effect as well as the book-to-market effect. This means that an arbitrage investor is not able to yield any arbitrage possibilities from the P/E effect. Also the value investor is also not able to profit from the P/E effect. The value investor does not yield a risk adjusted return that is higher than the risk adjusted return of the growth portfolio. Thus on the bases of the three factor model we have we cannot accept the hypotheses that the P/E effect in the Western European market constitutes a real investment opportunity.
