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Required Growth Probability: An Objective Approach to Fundamental Security Analysis Dan W. French And David Javakhadze Contact: Dan W. French Department of Finance Robert J. Trulaske, Sr. College of Business University of Missouri 403 Cornell Hall Columbia, MO 65211 573-882-3739 [email protected] ABSTRACT We argue that it is possible to construct an objective set of decision criteria for security selection that can enable an active fund manager to employ fundamental analysis and outperform. We consider an approach that uses probability-based security analysis and find considerable evidence that the approach produces abnormal returns. By studying a probability-based approach to security selection we are able explore an area of asset pricing that has received little attention. JEL Classification: G10, G11, G14 Draft Date: February 28, 2013 Required Growth Probability: An Objective Approach to Fundamental Security Analysis 1. Introduction Every day there are thousands of security analysts and portfolio managers who use publicly available financial data in an effort to identify overvalued and undervalued securities and even to rank such securities by their extent of their over or undervaluation. It is analysts’ belief in the superiority of their judgment (relative to the other analysts) that motivates their work. In order to justify the efforts they spend on valuation, they all must accept that pricing inefficiencies exist even though observed prices reflect the collective expectations of the market (i.e. other analysts). They therefore believe that their valuations are more precise as a reflection of “true” values than are valuations of the “average” analyst. In other words, they believe that they can correctly identify over and undervalued securities more often than would occur by chance alone and construct portfolios based on these valuations that would yield superior risk-adjusted performance. This paper analyzes the efficacy of a different approach to stock selection. In this method, they analyst uses the market price to infer the market’s collective expectation for a firm’s future revenue growth. The analyst then uses financial data to estimate probability that the firm can actually meet this expectation. The process involves reversing the traditional discounted cash flow (DCF) approach to back out the firm’s implied revenue growth and fitting the firm’s historical growth distribution. Then, firms more likely to achieve their required revenue growth (and portfolios constructed using this criterion) should exhibit superior performance. 2 This alternative approach relies on fewer subjective and potentially biased judgments on the part of analysts because it avoids making assumptions and forecasts about future growth as does the valuation approach. It simply solves for the growth rate implied by the current price. The subjective nature of this alternative approach could come into play when the analyst estimates the probability of the firm actually achieving the required growth rate. However, an objective estimate of the probability can result from fitting a probability distribution using observed patterns of historical firm revenue growth. We call this measure of the probability that a firm will achieve the growth rate needed to justify its current market price the Required Growth Probability (RGP). It is the thesis of this paper that implementing a portfolio construction strategy using RGPs will be more effective at achieving superior performance than traditional valuation measures because RGP relies less on subjective estimates and therefore less influenced by analyst bias. Our findings suggest the RGP approach can in fact be fruitful to an active fund manager and more generally suggest that disciplined, objective stock selection is a superior approach to traditional fundamental analysis and the subjectivity it typically entails. That is, an alternative strategy that is based on conventional fundamental analysis yet applied with less subjectivity can result in superior performance. The abnormal performance we observe appears to be strongest among growth stocks but exists throughout the growth-value spectrum. It is also robust to size and momentum effects. The organization of the paper is as follows. First we review the theory that motivates our interest in this approach to investing. Second we describe our proprietary dataset and the methodology used to generate the probability measure that will be the focus of the remainder of the paper. 3 Section 4 contains our empirical work which begins with preliminary tests of outperformance and concludes with tests that explain the outperformance we observe. Section 5 discusses the findings and concludes. 2. Theoretical Motivation Financial literature contains tests of a variety of fundamental investment analysis techniques beginning with studies based on price/earnings (P/E) multiples. For example, Whitbeck and Kisor (1963) devise a model for estimating fundamental P/E multiple for a firm’s stock and demonstrate a strategy that would have yielded high investment returns. Basu (1977) finds that strategies using merely price multiples to select stocks appeared to capture abnormal returns. Similar strategies such as the use of book-to-market ratio, first identified by Rosenberg, Reid and Lanstein (1984) and later codified by Fama and French (1992) would ultimately prove to be of seminal importance to the asset pricing theory. In practice, however, most fundamental analysis is done quite subjectively. In most cases this is necessarily so. Financial statements contain overwhelming amounts of information that make it difficult for an analyst or fund manager to distill into a single decision criterion. Yet a single decision criterion is often just what an active fund manager needs in order to select within a large universe of stocks. One study that summarizes multiple fundamental data items is that of Piotroski (2000), who argues using that many fundamental indicators used in conjunction may result in abnormal returns while using any one in isolation may not. The approach we study also summarizes much fundamental data, but additionally results in a single decision criterion. In this way, the approach 4 is in the same vein as Ou and Penman (1989) who summarize a variety of financial statement data items into a single probability measure that predicts an increase or decrease in earnings. Yet in general the use of probabilities is conspicuously absent in the field of asset pricing, despite the fact that probability as a means to evaluate risk is pervasive in the financial services industry, from measuring credit default risk to insurance underwriting risk. Moreover, probability is simple metric that could give a clear signal to a fund manager making stock selection decisions. By studying a probability-based approach to investing we are able explore an area of asset pricing that has received little attention. The approach we study makes important use of the discounted cash flow analysis (DCF). As an expansion on the dividend discount model introduced by Gordon (1959), DCF analysis holds that the value of an asset is equal to the present value of future cash flows. While widely used by practitioners, the model also has support in the literature (see Myers (1974), Kaplan and Ruback (1995, 1996), Gilson, Hotchkiss and Ruback (2000), Inselbag and Kaufold (1997) among others). Typically such models make projections about a firm’s future free cash flow and then discount these cash flows at the firm’s cost of capital. Common DCF models have two stages, often ten years of cash flows in the first stage and a second stage representing the terminal growth rate in years 11 and thereafter. 3. Data and Empirical Approach The method we use to generate the probabilities is based loosely on ideas formalized by Rappaport and Mauboussin (2001) who argue that measuring price-implied market expectations is key to generating abnormal returns because the stock price itself is the most unambiguous 5 source of firm-specific information. Accordingly, the method uses the stock price as the input into a reversed discounted cash flow valuation. By reversing this process the analyst must solve for implied revenue growth rather than intrinsic value. To perform the reversed DCF, the analyst first calculates enterprise value, a metric often used by practitioners in place of market cap due to its ability to capture all sources of financing. The method is not assumption-free however, but the assumptions required to complete the process are made in accordance with a set of predetermined guidelines and applied uniformly. For instance, in order to solve for the first stage free cash flow growth rate, the analyst must make assumptions about the long-term terminal growth rate and the allocation of free cash flow within the first stage. The discount rate he uses is the cost of capital estimated under a general version of the CAPM (Sharpe (1964), Lintner (1965) and Black (1972)). With this information, he can calculate the first stage FCF growth rate and, by deduction, the first stage revenue growth rate. The next step in the process is to determine the probability that the required revenue growth will be met. To calculate this, the firm’s history of revenue growth over the previous eight quarters is fit to a log-normal distribution, overweighting the most recent quarters under the theory that the most recent observations should have the most predictive content. To eliminate the effect of seasonality, growth rates are calculated on a quarter-on-quarter basis as (1) where and is the growth rate in quarter t, is the revenue in the current quarter t is the revenue in the same quarter of the previous year. 6 The use of the log-normal distribution is this case is logical for several reasons. First, growth defined this way is a non-negative variable and heavily right-skewed. Also, the assumption of log normality has been used in the accounting literature for a variety of fundamental variables (see Hilliard and Leitch (1975)). After transforming the data, the weighted mean and variance can be used to fully describe the cumulative distribution function of the normal distribution. The mean is weighted under the assumption that more recent observations have more predictive content. We define as follows: we simply integrate the probability density function of the normal distribution at the log-transformed required revenue growth. More formally, √ where ∫ ( ) ) (2) equals the natural log of the required revenue growth rate, mean and is the sample weighted is the sample standard deviation. The integral is subtracted from one because we are concerned with the probability that the realized growth rate is greater than or equal to . Our proprietary dataset is provided by Transparent Value, LLC, an asset manager utilizing such a probability-based strategy with more than $500 million under management. That data includes the required revenue growth calculated by the firm using its set of guidelines as well as the probabilities that result. We have independently replicated the probabilities using the required revenue growth numbers and found them to be nearly identical. The sample we analyze includes the required revenue growth and probabilities of stocks over the period 1999 to 2011. This sample includes both real-time calculated probabilities beginning in 2008 as well as back-calculated probabilities over the period 1999 through 2007. Descriptive 7 statistics of the sample can be found in Table 1. Due to the nature of our data source, the firms in our sample are generally large and thus tend to have lower betas. 4. Empirical Analysis A. Fama-Macbeth Regressions Our initial examination of the abnormal return generating potential of the required growth probability is a stepwise Fama Macbeth (1973) approach to return analysis using the three factors of Fama French (1992) plus the Carhart (1997) momentum factor. We perform 156 crosssectional regressions of return in month t on size (market capitalization), book-to-market, prior return and probability. Prior return is defined as one plus the six month return in months j-6 through j-2. The regressions are performed over the period June 1999 through December 2011. The average coefficients of the times series are shown in Table 2. In the first column we add ( ) to the market model and find that it is statistically significant in terms of ( explaining returns. Columns (2) and (3) show ) added to the three-factor and ( four-factor models. The coefficient estimates of )in these regressions are all highly significant. The results of Fama Macbeth analysis indicate that the probability has explanatory power. The coefficient on probability is positive, as expected, and statistically significant under all model specifications. This provides compelling preliminary evidence that the probability measure we test may in fact have the potential to contribute to abnormal returns. 8 B. Performance of Probability-based Portfolios Having established preliminary evidence that probability has explanatory power, it is important that we begin to assess the actual potential profitability of strategy that uses . To do this we sort all stocks in to quintiles and test the absolute performance. We form the quintile portfolios monthly, and calculate the mean raw return to each quintile over our entire sample period. The results, shown in Panel A of Table 3, reveal that performance increases monotonically by quintile, from 0.63% in the (equally-weighted) low probability quintile to 0.99% in the (equally-weighted) high probability quintile, suggesting a long/short strategy could generate as much as 0.36% per month. Results using value-weighted quintiles are similar. After this analysis, however, it remains possible that the high versus low quintile results are due to exposures to known risk factors, suggesting that further testing is needed to determine how robust they are to multifactor asset pricing models. To test the robustness of the results, we regress the returns of each quintile on the Fama French 3 factors and Carhart factor. 1 Panel B of Table 3 displays these results. It does not appear that the results shown in Panel A are due to exposure to known risk factors. Specifically, the alphas (using both equal-weighted and value-weighted portfolios) are positive and significantly different from zero for the highest probability portfolio while not significantly different from zero for the lowest probability portfolios. The result is a high-low difference in alpha of 0.608% per month using equallyweighted portfolios and 0.682% per month using value-weighted portfolios. Does the unusual nature of our sample bias our results due to exposure to small or large stocks? To examine this we sort all stocks in to quintiles based on both probability and size. If the size 1 Data used in this analysis are taken from Ken French’s website. 9 effect is affecting our results, we would expect to see higher high-probability-minus-lowprobability differences among small stocks. The size effect does not appear pronounced. Returns consistently increase with probability quintile. Further, as shown in Table 4, there appears to be little disparity in results among the various size quintiles. Surprisingly, the largest difference actually occurs among large stocks, contrary to what we would expect to see due to the size effect. The value-weighted quintile results shown in Panel B of Table 4 reinforce this finding. The large value-weighted size quintile, for which the largest firms in the universe tend to dominate results, has the highest highlow difference in performance. From this analysis we conclude that the size effect is not driving the results. Yet we must still establish that size quintiles are not exposed to known risk factors that might impact our findings. Table 5 shows the results of regressions under the three and four factor models. We perform five regressions, one for each size quintile, where the dependent variable is the return of the high probability quintile minus the return of the low probability quintile. The alphas in all regressions are positive and significant, indicating that, regardless of size, the high probability minus low probability approach has alpha-producing potential. C. Performance of Probability Factor How does our measure perform when treated as a factor itself? To answer this, we form a high-minus-low probability factor in the same way as the HML factor returns of the Fama French 3-factor model. Specifically, we form six value-weighted portfolios by sorting size into two portfolios and in to three portfolios. The 10 factor return equals the average return of the two high portfolios minus the average return of the two low portfolios. We can then regress our factor, High-Minus-Low-Probability (HMLP), on the Fama French three factors and Carhart factor. First we examine the nature of HMLP by calculating summary statistics. Table 6, Panel A, displays the mean and variance of HMLP as compared to the four other factors. While the mean return to HMLP over our sample period is not as high as SMB, its Sharpe Ratio is higher than all other factors. For a factor to capture the fundamental riskiness of a stock, it must sometimes have negative returns despite its positive mean return. The Sharpe Ratio of HMLP, being higher than that of other factors due to the overall lower volatility of HMLP, suggests that HMLP is capturing abnormal returns that are less fully explained by risk and instead may be a real source of abnormal returns. In Panel B of Table 6 we show the correlations of HMLP with the four factors. HMLP is correlated with SMB, HML and MOM, yet the correlation with HML is negative. The correlation of HMLP with SMB is understandable, as our sample consists of mainly large stocks. As Table 9, Panel A shows, around 95% for all the stocks classified in Fama-French 25 book-tomarket portfolios as a small, we are unable to estimate large stocks have greater . Also, as Panel B shows, then then small stocks. Similarly, the correlation with MOM is also expected to the extent that HMLP has a positive mean return and probabilities in consecutive quarters are correlated. However the strong negative correlation with HML is noteworthy. It may mean that HMLP is positive when HML is negative, suggesting that our probability measure is best at identifying outperforming stocks with growth outperforms value. 11 Alternatively, it may simply indicate that probability works best among growth stocks regardless of the direction of HML. The results of the factor regressions, shown in Table 7, indicate that neither the three-factor nor four-factor model can explain the returns to a high minus low portfolio. Consistent with the correlation analysis, HMLP loads very heavily negatively on HML. The HMLP/HML relation warrants further investigation. In Table 8 we examine the performance of quintiles based on probability and book-to-market. For robustness, we also use alternative measures of value, market leverage and prior 36 month return. From this analysis we find that the high-minus-low difference is not larger among growth stocks as we would expect if probability works when growth outperforms value. Rather, we see no discernible trend in the high-minus-low difference across value/growth quintiles, confirming that the HMLP effect is robust to the value/growth effect. Fama and French (1996) divide all stocks in to three portfolios based on book-to-market ratio: value, neutral and growth. In Figure 1 we chart monthly proportion (percent) of the value, growth and neutral stocks in the top 30% and bottom 30% high (low) buckets. That is, among stocks we identify percent of the value stocks, percent of growth stocks and percent of neutral stocks in each month and plot it for the period 1999-2011.Growth stocks tend to occur more frequently among stocks in the top HMLP portfolio, indicating that the highest probabilities occur among growth stocks. If high probability stocks outperform low probability stocks, this provides evidence that the negative relation of HMLP with HML is due to 12 the probability measure’s ability to identify mispriced growth stocks with slightly greater frequency than value stocks. As we know that value, as defined by book-to-market ratio, tends to outperform growth, and that high probability stocks tend to outperform low probability stocks, the negative relation suggests that much of the return to HMLP comes from growth stocks rather than value stocks. If a growth stock outperforms it is likely that there is a divergence from the standard efficient pricing of stocks because ordinarily we expect growth stocks to underperform. Thus, the negative relation we see is evidence that the abnormal returns our probability measure appears to generate may be the result of behavioral biases. That is to say, the ability of our measure to identify outperforming growth stocks (and as Table 9 Panel B shows growth stocks on average have greater than value stocks) is evidence that prices are more likely to be irrational due to behavioral biases. The coincidence of this phenomenon with the ability of our probability measure to produce abnormal returns indicates HMLP, or probability more generally, may work due to its ability to identify when prices are more likely to be inefficient due to behavioral biases. D. Performance over Different Periods A potential concern with our data is the relatively short time series that we have available. While the data cover only the period 1999 through 2011, several distinct market and economic cycles occurred during this period. We therefore test how robust the probability measure is as a stock selection tool to different environments by calculating the alphas generated by regressing the returns of each probability quintile on the four-factor model. The difference in alphas between the high quintile and low quintile tells us how a theoretical long/short trading strategy might perform in each period. 13 Table 10 presents these results. In general we conclude the stock selection system works well in both bull and bear markets, recession and expansion. That is to say, our more general results reported in previous sections are not driven by any one period in our time series. 5. Discussion What explains the surprisingly robust ability of the required revenue growth probability measure that we study to generate abnormal returns? Our findings are interesting primarily because only public information is used in the calculation of the probability. However, in contrast to more typical fundamental analysis, the probability measure is far more objective and applied in a uniform fashion without human judgment or subjectivity. This suggests that the abnormal returns are at least in part the result of biases by the balance of fundamental analysts and investors – those who ultimately set market prices. In general, our results suggest the probability approach picks up on systematic behavioral biases among market participants. Lakonishok, Shliefer and Vishny (1994) as well as La Porta, Lakonishok, Shleifer and Vishny (1997) questioned the nature of value strategies in a rational pricing framework. They concluded that value stocks outperform growth not because they are truly riskier, but because they exploit the behavioral shortcomings of other investors. What the authors term “glamour” stocks tend to have prices that imply growth that the firm ultimately cannot deliver because investors have extrapolated past growth rates too far in to the future. Another way of saying this is that market expectations for glamour stocks tend to be too optimistic. That prices could be set by systematically biased market participants became very clear just a few years later. Indeed, 14 subsequent work such as that of Chan, Karceski and Lakonishok (2000) found enormous divergences between “value” metrics and operating performance. They concluded that the latenineties outperformance of growth over value was unsubstantiated by performance and instead the result of irrational pricing. Chan, Karceski and Lakonishok (2000) refer to the divergence between value and performance as a disruption, and clearly in the late nineties it was. However, smaller and less systematic disruptions can occur among subsets of stocks. The evidence in this paper suggests that the probability of meeting required revenue growth can help identify and avoid “glamour” stocks by measuring the implied market expectations and objectively balancing these expectations against performance. While value stocks may still outperform growth in general, this approach appears to be able to identify which value stocks have overly optimistic implied expectations and thus can lead to outperformance among such growth stocks. The objectivity of the approach is crucial. For a behavioral bias to be exploited it is very important that the analyst not commit the same behavioral bias himself. The fashion in which our probability measure is calculated suggests this does not happen. The stock price is the most unambiguous piece of firm-specific information, so using this as the input rather than subjective or even arbitrary projections about future performance mitigates analyst input. In summary, the probability measure we test in this paper appears to have alpha-producing potential that arises from the ability for the measure to identify and exploit the irrational pricing of securities by market participants. The probability seems to work best among growth stocks for which investors have driven prices to irrationally high levels, yet we find outperformance among value stocks as well. We interpret this as evidence that removing the subjectivity from the 15 analysis process and distilling the investment decision to a single criterion may enable active managers to identify mispricings. 16 References Basu, S., 1977, “Investment Performance of Common Stocks in Relation to Their Price-Earnings Ratios: A Test of the Efficient Market Hypothesis, Journal of Finance 32, 663-682. Black, Fischer, 1972, Capital market equilibrium with restricted borrowing,” Journal of Business 45, 444-455. Carhart, Mark 1997, “On persistence in mutual fund performance,” Journal of Finance 52, 57-82 Chan, Louis K. C., Jason Karceski, and Josef Lakonishok, 2000, “New paradigm or same old hype in equity investing?,” Financial Analysts Journal 56, 23–36. Fama, Eugene and Kenneth French, 1992, “The cross-section of expected stock returns,” Journal of Finance 47, 427-465. Gilson, Stuart, Edith Hotchkiss and Richard Ruback, 2000, “Valuation of Bankrupt Firms,” Review of Financial Studies 13, 43-74. Gordon, Myron, 1959, “Dividends, Earnings and Stock Prices,” Review of Economics and Statistics 41, 99-105. Hilliard, Jimmy E. and Robert A. Leitch, 1975, “Cost-volume-profit analysis under uncertainty: a log-normal approach,” The Accounting Review 50, 69-80. Holthausen, RW, DF Larcker, 1992, “The prediction of stock returns using financial statement information,” Journal of Accounting and Economics, 373-411. Inselbag, Isik and Howard Kaufold, 1997, “Two DCF approaches for valuing companies under alternative financing strategies and how to choose between them,” Journal of Applied Corporate Finance, 10, 1 (Spring), 114-122 Kaplan, Steven and Richard Ruback, 1995, “The Valuation of Cash Flow Forecasts: An Empirical Analysis,” Journal of Finance 50, 1059-1093. Kaplan, Steven and Richard Ruback, 1996, “The market pricing of cash flow forecasts: discounted cash flow vs. the method of comparables,” Journal of Applied Corporate Finance 8, 4, 45-60. Lakonishok, Josef, Andrei Shleifer and Robert Vishny, 1994, “Contrarian Investment, Extrapolation, and Risk,” The Journal of Finance 49, 1541-1578. Laporta, Rafael, Lakonishok, Josef, Andrei Shleifer and Robert Vishny, 1997, “Good News for Value Stocks: Further Evidence on Market Efficiency,” The Journal of Finance 52, 859-874. 17 Lintner, John, 1965, “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets,” Review of Economics and Statistics 47, 13-37. Myers, Stewart C, 1974, .Interactions of corporate financing and investment decisions. implications for capital budgeting,. The Journal of Finance, 29, 1-25. Ou, JA and SH Penman, 1989, “Financial Statement Analysis and the prediction of stock returns,” Journal of Accounting and Economics 11, 295-330. Ou, JA and SH Penman, 1989, “Accounting Measurement, Price-Earnings Ratio, and the Information Content of Security Prices,” Journal of Accounting Research 27, 111-144. Piotroski, J., 2000, “Value investing: The use of historical financial statement information to separate winners from losers,” Journal of Accounting Research 38 (Supplement), 1-41. Piotroski, Joseph and Eric So, 2012, “Identifying Expectation Errors in Value/Glamour Strategies: A Fundamental Analysis Approach,” Review of Financial Studies 25, 2841-2875. Rappaort, A, and M. J. Mauboussin, 2001, Expectations Investing: Reading Stock Prices for Better Returns, Harvard Business School Press. Rosenberg, Barr, Kenneth Reid, and Ronald Lanstein, 1984, “Persuasive evidence of market inefficiency,” Journal of Portfolio Management 12, 9-17. Ruback, Richard S., 1986, "Calculating the market value of riskless cash flows," Journal of Financial Economics, Volume 15, No. 3 (March 1986), pp. 323-339 Sharpe, William F., 1964, “Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442. Sharpe, William F., 1966, "Mutual Fund Performance," Journal of Business 39, 119–138. Whitbeck, Volkert S. and Manown Kisor , 1963, “A New Tool in Investment Decision Making,” Financial Analysts Journal, 55-62. 18 Table 1: Sample Description This table shows descriptive statistics. Panel A shows statistics for our sample. Panel B shows the same statistics for the universe of stocks in the Compustat database. Variable definition is as follows: CHat – cash scaled by total assets, CAPXat – capital expenditures scaled by total assets, PPESALES – property, plant and equipment scaled by total assets, EBITSALE – earnings before interest and taxes scaled by total assets, DEBTAT – debt scaled by total assets, Q – Tobin’s Q defined as total assets plus market value of equity minus book value of equity, scaled by total assets, ROE – returns on equity, ROA – return on assets, BM – book-to-market ratio, Mkval – market value of equity, BETA – ninety-day beta. Panel A: Sample Descriptive Statistics Variable Mkval BETA PPESALES EBITSALE DEBTAT Q ROE ROA CHat CAPXat Mean 8572.17 0.87 1.73 -0.78 0.43 2.15 -0.07 0.01 0.11 0.06 Median 2166.87 1.07 0.21 0.10 0.42 1.59 0.11 0.05 0.06 0.04 Std Dev 25906.38 23.55 90.48 39.87 0.22 2.00 9.39 0.32 0.12 0.06 Maximum 508329.45 275.61 10189.91 4.34 3.56 78.56 103.97 2.17 0.95 0.90 Minimum 0.94 -2376.78 -0.26 -4347.58 0.00 0.31 -790.61 -19.45 0.00 -0.03 Std Dev 11334.770 290.359 124.537 136.820 103.634 1071.360 70.386 353.405 0.192 1.078 Maximum 508329.45 50761.88 19269.94 4037.11 25968.97 222021.00 7770.330 2369.430 1.000 341.000 Minimum 0.000 -35936.12 -646.44 -30175.70 0.00 0.03 -23155.88 -130077.00 -0.53 -2.09 Panel B: Compustat Descriptive Statistics Variable Mkval BETA PPESALES EBITSALE DEBTAT Q ROE ROA Chat CAPXat Mean 1867.200 2.830 3.935 5.405 2.867 0.755 0.086 2.508 0.133 0.073 Median 106.068 0.791 0.208 0.053 0.429 1.381 0.073 0.015 0.052 0.037 19 Table 2: Fama-MacBeth Regressions This table presents the time-series average coefficients for 156 cross-sectional regressions for each month from June 1999 to December 2011. The dependent variable is the stock return for month t. A firm’s size is its market capitalization (price times shares outstanding). The book-tomarket ratio (BE/ME) is the prior year’s book value of equity divided by the firm’s market value. Prior return is the raw buy-and-hold return from month j−6 to month j−2. ( ) is natural log estimated by equation (2). T-statistics are in parenthesis. *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. ( ) ( ) ( ) ( Model (1) (2) (3) Intercept 0.973* (1.98) 0.091 (0.30) 3.851* (1.98) 0.078 (0.26) -0.132 (-1.17) 0.952*** (4.82) 0.124* (1.73) 0.169*** (2.58) 3.653** (2.14) 0.003 (0.01) -0.131 (-1.28) 0.923*** (5.66) 0.513 (0.89) 0.187*** (3.06) Beta ln(Size) Ln(BE/ME) LN(1+prior) ln(Probability) 20 ) Table 3: Performance of High minus Low Probability Portfolios Panel A shows Equal-weighted and value-weighted returns for each quintile. Panel B shows four-factor regressions of monthly excess equally-weighted and value-weighted portfolio returns. T-statistics are presented in brackets. *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. Panel A: Monthly Returns for Probability Quintile Quintile 1 (Low) 2 3 4 5 (High) High-Low Monthly EW Returns 0.63% 0.80% 0.83% 0.84% 0.99% 0.37% Monthly VW Returns 0.16% 0.32% 0.42% 0.43% 0.66% 0.50% Panel B: Monthly High minus Low Probability Returns as Dependent Variable Equal-Weighted Alpha RMRF SMB HML UMD Adjusted R2 Low 2 Probability Quintiles 3 4 -0.020 (-0.18) 1.169 (44.95)*** 0.402 (12.43)*** 0.366 (11.33)*** -0.093 (-4.83)*** 95.5% 0.224 (1.96)** 1.104 (41.49)*** 0.427 (12.92)*** 0.191 (5.77)**** -0.143 (-7.21)*** 95.1% 0.316 (2.92)*** 1.072 (42.53)*** 0.410 (13.07)*** 0.052 (1.67)* -0.178 (-9.48)*** 95.6% Low 2 3 -0.277 (-1.83)* 1.108 (31.51)*** 0.049 (1.12) 0.223 (5.11)*** -0.044 (-1.69) 89.5% -0.024 (-0.23) 0.958 (38.00)*** -0.038 ( -1.20) 0.074 (2.37)** 0.026 (1.39) 89.5% 0.083 (0.73) 0.936 ( 35.08)*** -0.117 ( -3.53)*** 0.152 ( 4.58)*** 0.037 (1.88)* 90.3% Value-Weighted Alpha RMRF SMB HML UMD Adjusted R2 0.309 (2.47)*** 1.076 (36.91)*** 0.475 (13.10)*** -0.002 ( -0.07) -0.175 (-8.08)*** 64.5% Probability Quintiles 4 21 0.125 ( 1.08) 0.992 (36.68)*** -0.068 (-2.03)** -0.029 ( -0.88) 0.013 (0.65) 91.6% High High-Low 0.588 (3.31)*** 1.148 ( 27.80)*** 0.423 ( 8.24)*** -0.308 ( -6.01)*** -0.221 ( -7.21) 91.2% 0.608 (2.91)*** -0.021 ( -0.44) 0.021 ( 0.35) -0.675 (-11.16)*** -0.128 ( -3.54) 48.3% High High-Low 0.405 ( 3.61)*** 0.993 ( 37.95)*** -0.034 ( -1.05) -0.192 ( -5.91)*** -0.035 ( -1.82)* 92.9% 0.682 (3.05)*** -0.115 ( -2.22)** -0.083 ( 1.28) -0.416 (-6.43)*** 0.009 ( 0.23) 21.8% Table 4: Monthly Returns by Size and Probability Quintiles This table shows monthly returns by size and probability quintiles for the period 1999-2011. Portfolios are determined by using only NYSE cutoffs and are rebalanced monthly2. A firm’s size is its market capitalization (stock price multiplied by shares outstanding). is defined by equation (2). *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. Panel A: Equal-Weighted Monthly Returns Probability Quintiles Size Quintile Low 2 3 4 High Diff High-Low Small 0.87% 1.57% 1.53% 1.39% 1.29% 0.42% 2 3 4 Large 0.75% 0.66% 0.61% 0.25% 0.86% 0.66% 0.69% 0.31% 0.86% 0.47% 0.77% 0.47% 0.91% 0.96% 0.61% 0.36% 1.10% 1.04% 0.92% 0.93% 0.35% 0.38% 0.31% 0.69% Average 0.63% 0.82% 0.82% 0.85% 1.06% 0.43% Panel B: Value-Weighted Monthly Returns Probability Quintiles Size Quintile Low 2 3 4 High High-Low Small 2 3 4 0.93% 0.72% 0.65% 0.63% 1.40% 0.82% 0.66% 0.71% 1.32% 0.84% 0.48% 0.77% 1.25% 0.90% 0.98% 0.58% 1.16% 1.09% 1.04% 0.88% 0.22% 0.37% 0.39% 0.26% Large 0.15% 0.20% 0.46% 0.30% 0.60% 0.44% Average 0.62% 0.76% 0.77% 0.80% 0.95% 0.34% 2 Qualitatively similar results are obtained when portfolios are rebalanced quarterly. 22 Table 5: Four-factor Regressions on Monthly Size and Probability Quintiles (High minus Low) This table shows regression results of High minus Low portfolio monthly returns for each size quintile on Fama-French-Carhart factors for the period 1999-2011. Portfolios are determined by using only NYSE cutoffs and are rebalanced monthly3. is defined by equation (2). *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. Panel A: Monthly Probability Quintile (High-Low) Equal-Weighted Returns as the Dependent Variable Small 2 3 4 Large 0.652 0.645 0.512 0.484 0.865 Alpha (3.32)** (2.00)** (1.97**) (1.98)** (1.91)* * -0.030 -0.015 0.054 0.032 -0.020 MKT (-0.4) (-0.2) (0.89) (0.54) (-0.33) 0.002 -0.193 0.051 -0.013 0.021 SMB (0.02) (-2.04)** (0.69) (-0.17) (0.27) -0.589 -0.474 -0.463 -0.491 -0.595 HML (-6.25)*** (-5.00)*** (-6.21)*** (-6.71)*** (-7.9)*** -0.152 -0.154 -0.058 -0.070 -0.018 MOM (-2.68)*** (-2.71)*** (-1.29) (-1.6) (-0.4) 21.81% 14.85% 24.32% 24.40% 31.10% Adjusted R2 Panel B: Monthly Probability Quintile (High-Low) Value-Weighted Returns as the Dependent Variable Small 2 3 4 Large 0.396 0.641 0.505 0.418 0.636 Alpha (1.27) (1.99)** (1.99)** (1.63)* (2.23)** 0.054 -0.026 0.051 0.025 -0.083 MKT (0.75) (-0.34) (0.87) (0.42) (-1.25) 0.012 -0.134 0.092 0.028 -0.095 SMB (0.14) (-1.44) (1.25) (0.38) (-1.15) -0.507 -0.474 -0.474 -0.523 -0.473 HML (-5.64)*** (-5.08)*** (-6.47)*** (-7.02)*** (-5.73)*** -0.099 -0.165 -0.058 -0.063 0.034 MOM (-1.84)* (-2.95)** (-1.32) (-1.41) (0.7) 19.9% 15.35% 27.39% 27.14% 17.78% Adjusted R2 3 Qualitatively similar results are obtained when portfolios are rebalanced quarterly. 23 Table 6: Factor Summary Statistics and Correlations This table shows monthly summary statistics and correlations of HMLP and Fama-FrenchCarhart four factors. HMLP factor is created to mimic return difference between high and low probability portfolios. HMLP factor is created similar to Fama-French HML factor. HMLP is the sample average of the value-weighted returns on the two high probability portfolios minus the average value-weighted return of the two low probability portfolios. Returns are monthly (%). Pvalues in the Panel 2 are italicized. Panel A: Descriptive Statistics Quarterly Rebalancing Annual Rebalancing Variable Mean Std Dev Sharpe Ratio Mean Std Dev Mkt Smb Hml Mom HMLP 0.099 0.497 0.302 0.307 0.335 4.865 3.873 3.742 6.349 2.388 0.02 0.128 0.081 0.048 0.14 0.078 0.507 0.356 0.401 0.35 4.865 3.854 3.714 6.339 2.625 Panel B: Factor Pearson Correlations Quarterly Rebalancing Mkt Smb 1 0.29484 Mkt 0.0002 0.29484 1 Smb 0.0002 -0.17011 -0.36651 Hml 0.0355 <.0001 -0.32736 0.13415 Mom <.0001 0.0983 -0.16643 0.02174 HMLP 0.0398 0.7897 Annual Rebalancing Mkt Smb 1 0.29156 Mkt 0.0003 0.29156 1 Smb 0.0003 -0.15274 -0.38185 Hml 0.062 <.0001 -0.33344 0.14549 Mom <.0001 0.0757 0.02431 0.22402 HMLP 0.7678 0.0059 Hml -0.17011 0.0355 -0.36651 <.0001 1 -0.1579 0.0513 -0.13911 0.0863 Hml -0.15274 0.062 -0.38185 <.0001 1 -0.14488 0.0769 -0.36922 <.0001 24 Mom -0.32736 <.0001 0.13415 0.0983 -0.1579 0.0513 1 0.21513 0.076 Mom -0.33344 <.0001 0.14549 0.0757 -0.14488 0.0769 1 0.21425 0.0085 Sharpe Ratio 0.016 0.132 0.096 0.063 0.133 HMLP -0.16643 0.0398 0.02174 0.7897 -0.13911 0.0863 0.21513 0.0076 1 HMLP 0.02431 0.7678 0.22402 0.0059 -0.36922 <.0001 0.21425 0.0085 1 Table 7: Fama-French-Carhart factors and HMLP This table shows regression results of the HMLP factor on both Fama-French three and FamaFrench-Carhart four factors. HMLP factor is created to mimic return difference between high and low probability portfolios. HMLP factor is created similarly to Fama-French HML factor. HMLP is the sample average of the value-weighted returns on the two high probability portfolios minus the average value-weighted return of the two low probability portfolios. In (1)-(2) probability portfolios are rebalanced quarterly. In (3)-(4) probability portfolios are rebalanced monthly. is defined by equation (2). *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. Alpha MKT SMB HML (1) 0.371 (1.93)** -0.098 (-2.4)** 0.012 (0.23) -0.106 (-1.94)** MOM Adjusted R2 3.8% (2) 0.355 (1.85)* -0.068 (-1.54) -0.006 (-0.1) -0.091 (-1.66)* 0.056 (1.69)* 4.9% 25 (3) 0.397 (1.95)** -0.032 (0.75) 0.077 (1.33) -0.237 (-4.05)*** 13.0% (4) 0.375 (1.85)* 0.003 (0.07) 0.054 (0.91) -0.223 (-3.81)*** 0.066 (1.89)* 14.53% Table 8: Equal and Value-Weighted Monthly Returns across Measures of Value This table shows returns by and value/growth measures quintiles. Each year the sample is sorted into quintiles on the basis of 3 alternative definitions of value (book-to-market, market leverage and prior 36 month returns). is defined by equation (2). Panel A: Equal-Weighted Returns Book-to-market Quintile Growth 2 3 4 Value Average Low -0.20% 0.36% 0.72% 1.13% 2.09% 0.82% 2 -0.02% 0.55% 0.69% 1.14% 1.86% 0.84% Market Leverage Quintile Growth 2 3 4 Value Average Low -0.10% 0.13% 0.95% 1.16% 1.01% 0.63% 2 0.17% 0.59% 0.98% 1.04% 0.97% 0.75% Prior 36 Month Return Quintile Value 1 2 3 Growth Average Low 0.82% 0.41% 0.70% 0.50% 0.51% 0.59% 2 0.87% 0.76% 0.62% 0.40% 0.48% 0.63% 26 Probability Quintiles 3 4 High 0.00% -0.08% -0.02% 0.44% 0.43% 0.98% 0.92% 0.87% 1.21% 1.43% 1.38% 1.46% 2.32% 1.98% 2.61% 1.02% 0.91% 1.25% Probability Quintiles 3 4 High -0.35% -0.26% -0.26% 0.69% 0.48% 1.15% 0.91% 1.00% 1.42% 1.29% 1.00% 1.41% 1.12% 1.61% 1.36% 0.73% 0.76% 1.02% Probability Quintiles 3 4 High 1.34% 1.54% 1.21% 0.93% 0.94% 1.07% 1.04% 0.81% 0.98% 0.73% 0.66% 0.96% 0.67% 0.71% 1.01% 0.94% 0.93% 1.05% High-Low 0.18% 0.62% 0.49% 0.34% 0.52% 0.43% High-Low -0.16% 1.03% 0.47% 0.25% 0.34% 0.39% High-Low 0.19% 0.65% 0.28% 0.46% 0.70% 0.46% Table 8 (Continued) Panel B: Value-Weighted Returns Book-to-market Quintile Growth 2 3 4 Value Average Low -0.40% 0.28% 0.48% 0.54% 1.10% 0.40% 2 0.03% 0.66% 0.62% 0.49% 1.23% 0.61% Market Leverage Quintile Growth 2 3 4 Value Average Low -0.29% -0.29% 0.82% 0.42% 0.51% 0.23% 2 0.21% 0.40% 0.30% 0.76% 0.40% 0.41% Prior 36 Month Return Quintile Value 1 2 3 Growth Average Low 0.33% 0.11% 0.40% 0.10% 0.12% 0.21% 2 0.34% 0.48% 0.35% 0.55% -0.12% 0.32% 27 Probability Quintiles 3 4 High 0.24% 0.09% 0.25% 0.29% 0.58% 1.07% 0.36% 0.63% 0.90% 0.93% 0.95% 0.91% 1.36% 1.04% 1.79% 0.64% 0.66% 0.99% Probability Quintiles 3 4 High 0.22% -0.19% 0.17% 0.37% 0.34% 0.84% 0.59% 0.72% 0.75% 0.56% 0.71% 1.38% 0.78% 0.89% 0.93% 0.50% 0.49% 0.81% Probability Quintiles 3 4 High 1.02% 0.91% 0.97% 0.62% 0.61% 0.76% 0.57% 0.45% 0.87% 0.46% 0.47% 0.63% 0.36% 0.44% 0.66% 0.60% 0.58% 0.78% High-Low 0.65% 0.80% 0.42% 0.37% 0.69% 0.59% High-Low 0.45% 1.13% -0.07% 0.96% 0.42% 0.58% High-Low 0.64% 0.65% 0.48% 0.53% 0.53% 0.57% Table 9: Probabilities Across Fama-French 25 Size and Book-to-Market Potfolios Panel A shows percentage of the firms in Fama-French 25 book-to-market portfolio for which probabilities are NOT available. Panel B shows time-series mean value of probabilities for each of the Fama-French 25 book-to-market portfolios. Panel A: % of the firms for which probabilities are NOT available Small 2 3 4 Big Growth 96.07% 63.82% 26.44% 4.75% 0.68% 2 95.92% 65.89% 28.38% 2.55% 0.22% 3 97.57% 69.74% 31.78% 2.76% 0.53% 4 97.87% 71.88% 32.61% 2.64% 0.20% Value 98.38% 66.06% 21.81% 3.14% 0.64% Panel B: Probabilities across Size and Market-to-book Quintiles Small 2 3 4 Big Small-Big Growth 0.515 0.537 0.558 0.547 0.523 -0.008 2 0.457 0.510 0.518 0.476 0.475 -0.018 3 0.451 0.461 0.482 0.476 0.481 -0.031 4 0.387 0.435 0.488 0.471 0.459 -0.072 28 Value 0.365 0.418 0.432 0.445 0.447 -0.081 Value-Growth -0.149 -0.119 -0.126 -0.102 -0.076 Table 10: Performance in Market and Economic Cycles This table presents the alphas from four-factor regressions of monthly excess equally weighted and value-weighted portfolio returns. The column Diff High-Low refers to the difference in four-factor alphas between high and low portfolios. Portfolios are created based on probabilities and are rebalanced monthly. T-statistics are presented in brackets. *** represents a significant difference at the 1% level, ** at the 5% level, and * at the 10% level. NBER Expansion Alpha, Equally-weighted Alpha, Value-weighted Probability Quintile 3 4 0.306 0.354 (2.16)** (1.95)* -0.025 0.354 (-0.17) (2.35)** Low -0.076 (-0.53) -0.311 (-1.72)* 2 0.081 (0.57) -0.077 (-0.51) Low -0.022 (-0.07) 0.227 ( 0.46) Probability Quintile 2 3 4 0.272 0.649 0.347 (1.18) (2.58)*** (1.42) -0.177 0.983 -0.268 ( -0.63) ( 1.23) ( -0.83) High 0.768 (2.96)*** 0.418 (2.56)*** Diff High-Low 0.844 (2.81)*** 0.728 (2.47)*** High 0.732 (1.96)* 0.273 ( 1.08) Diff High-Low 0.754 (1.46) 0.045 ( 0.07) High 0.489 (3.15)*** 0.395 (3.17)*** Diff High-Low 0.543 (2.82)*** 0.448 (1.80)*** High 1.716 (4.33)*** 0.675 (2.47)** Diff High-Low 1.671 (3.67)*** 1.371 (2.9)*** NBER Recession Alpha, Equally-weighted Alpha, Value-weighted Bull Market Alpha, Equally-weighted Alpha, Value-weighted Probability Quintile 3 4 0.139 0.155 (1.16) (1.12) -0.089 0.073 (-0.7) (0.6) Low -0.054 (-0.38) -0.053 (-0.33) 2 -0.022 (-0.21) -0.088 (-0.79) Low 0.045 (0.21) -0.696 (-2.12)** Probability Quintile 2 3 4 0.843 0.963 0.877 (2.93)*** ( 4.28)*** ( 3.18)*** -0.397 0.714 0.308 (-1.69)* (2.8)*** 1.15 Bear Market Alpha, Equally-weighted Alpha, Value-weighted 29 Figure 1: This figure shows % of the value, growth and neutral firms in the top and bottom 30% of the HMLP quintile. Time-series mean values are also reported. Stocks in the top 30% of the HMLP Jun-11 Dec-11 Dec-11 Dec-10 Jun-10 Dec-09 Jun-09 Dec-08 Jun-08 Dec-07 Jun-07 Jun-11 Value (top 30% in the FF HML) Dec-06 Jun-06 Dec-05 Jun-05 Dec-04 Jun-04 Dec-03 Jun-03 Dec-02 Jun-02 Dec-01 Jun-01 Dec-00 Jun-00 Dec-99 Jun-99 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Neutral (middle 40% in the FF HML) Growth (bottom 30% in the FF HML) Stocks in the bottom 30% of the HMLP 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% Value (top 30% in the FF HML) Dec-10 Jun-10 Dec-09 Jun-09 Dec-08 Jun-08 Dec-07 Jun-07 Dec-06 Jun-06 Dec-05 Jun-05 Dec-04 Jun-04 Dec-03 Jun-03 Dec-02 Jun-02 Dec-01 Jun-01 Dec-00 Jun-00 Dec-99 Jun-99 0.00% Neutral (middle 40% in the FF HML) Growth (bottom 30% in the FF HML) Time-series Average Value 11.22% 21.93% Stocks in the top 30% of the HMLP Stocks in the bottom 30% of the HMLP 30 Neutral 32.01% 38.56% Growth 56.77% 39.51%