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Abstract Value premiums, which we define as value portfolio returns in excess of market portfolio returns, are on average much lower in the second half of the July 1963–June 2019 period. But the high volatility of monthly premiums prevents us from rejecting the hypothesis that expected premiums are the same in both halves of the sample. Regressions that forecast value premiums with book-to-market ratios in excess of market (BM–BMM) produce more reliable evidence of second-half declines in expected value premiums, but only if we assume the regression coefficients are constant during the sample period. (JEL G11, G12, G14) Received: January 21, 2020; editorial decision: July 21, 2020; Editor: Jeffrey Pontiff. Researchers and investors often use the ratio of book value to market value of equity (BM) to sort stocks into value and growth portfolios. Using data for the United States, Fama and French (FF 1992, 1993) document that high BM value stocks produce higher average returns than low BM growth stocks during the 28 years of July 1963–June 1991 (see also Stattman 1980; Rosenberg, Reid, and Lanstein 1985; and Chan, Hamao, and Lakonishok 1991). Acknowledging the importance of out-of-sample robustness, Davis, Fama, and French (2000) document strong value premiums in U.S. average stock returns for the July 1926–June 1963 period preceding the FF (1992, 1993) sample. Likewise, Fama and French (FF 2017) find that there are large average value premiums in Europe, Japan, and the Asia Pacific region during the mostly out-of-sample July 1990–December 2015 period. Foreshadowing the results presented here, however, they find that U.S. value premiums for that period are rather weak. Fama and French (1993) argue that the value premium is compensation for risk in a multifactor asset-pricing model. If investors do not judge that value stocks are, on some multifactor dimension, riskier than either growth stocks or the market as a whole, or that value stocks have other drawbacks, such as higher tax burdens or transaction costs, discovery of the value premium and its exploitation by investors should lead to its demise (see, for example, Pontiff 1996 and Dimson, Nagel, and Quigley 2003). Armed with 28 years (July 1991–June 2019) of out-of-sample United States returns, we ask whether we can confidently conclude that the expected value premium in the United States declines or perhaps disappears after FF (1992, 1993). Researchers often focus on the spreads between average returns on value and growth portfolios. We focus instead on value and growth returns in excess of the value-weight (VW) market return. From an investment perspective, the market portfolio is the base from which tilts toward value or growth are typically judged. The market portfolio is also the centerpiece in most asset-pricing models. Finally, using the market portfolio is the natural choice as the base asset if we are to compare value and growth premiums. Value stocks in the United States produce higher average returns for the full July 1963–June 2019 period than the VW market portfolio of all listed U.S. stocks (Market). Average value premiums are larger in the 28-year Fama-French (1992) period, July 1963–June 1991, than in the 28-year out-of-sample period, July 1991–June 2019, and we do not reject the hypothesis that out-of-sample expected monthly premiums are zero. But inferences from average premiums are clouded by the high volatility of monthly premiums, and we also cannot reject the hypothesis that out-of-sample expected premiums are the same as in-sample expected premiums. In this situation, the full sample arguably provides the best evidence on long-term expected premiums. The lower average premiums of the second half lean against the strong average premiums of the first, but full-period average value premiums provide statistically reliable evidence of positive expected premiums. Tests on average premiums target inferences about unconditional expected premiums. Given the imprecision of average premiums, estimates of conditional expected premiums from forecasting regressions can perhaps better identify variation in expected premiums. For value portfolios, regressions of returns in excess of Market, R–RM, on lagged excess book-to-market ratios, BM–BMM, uncover reliable evidence of variation in expected premiums. Like average R–RM, average BM–BMM for value portfolios is lower in the second half of the sample. Thus, if the regression coefficients are constant, the forecasting regressions are evidence that expected value premiums are lower in the second half of the July 1963–June 2019 period. But if we allow the regression coefficients to change from the in-sample to the out-of-sample period, the noise that prevents average returns from identifying reliable in- and out-of-sample differences in unconditional expected premiums also prevents the regressions from identifying reliable in- and out-of-sample differences in regression estimates of conditional expected premiums. From Schwert (2003) to Linnainmaa and Roberts (2018), many papers study post-1991 value premiums. The common finding is that average out-of-sample value premiums are low and statistically indistinguishable from zero. Focusing only on this result, the inference is that expected value premiums have disappeared. We emphasize that the high volatility of month-to-month value premiums in fact rules out inferences about whether expected value premiums change post-1991. McLean and Pontiff (2016) and Linnainmaa and Roberts (2018) pool results for many average return anomalies, and in this way produce reliable evidence that expected returns are lower outside the periods used to identify the anomalies. The cost of improved power is that they can draw inferences only about average expected returns for the population of anomalies they study, not about expected returns for individual anomalies. Our goal is specific inferences about the value premium, which is important both in asset-pricing research after Fama and French (1993) and in investor portfolio decisions. Going back at least to Cohen, Polk and Vuolteenaho (2003), many papers use regressions of monthly value premiums on lagged book-to-market spreads to identify variation in conditional expected value premiums. Our insight, to our knowledge absent from the previous literature, is that reliable inferences from this regression about conditional expected value premiums depend critically on the assumption that the coefficients in the forecasting regression are constant. This assumption, implicit but often unspoken in inferences from forecasting regressions, seems tenuous in tests intended to determine whether investment opportunities have changed. If we allow the regression coefficients to change from the in-sample to the out-of-sample period, reliable inferences about conditional expected premiums from lower out-of-sample BM spreads disappear. But we also find that allowing regression coefficients to change produces no improvement in explanatory power and no evidence against the assumption that the coefficients in the forecasting regression are constant during the 1963–2019 sample period. In the end, the specification checks provided by split-sample regressions lean toward the conclusion that the constant coefficient regressions are well specified, so we can perhaps accept their evidence that conditional expected value premiums are lower in the second half of 1963–2019. The split-sample regressions are, however, plagued by estimation error, so this inference is on shaky ground. 1. The Portfolios The tests focus on seven portfolios. The baseline is Market (M), the value-weight market portfolio of NYSE, AMEX, and (after 1972) NASDAQ stocks. As in Fama and French (1993) and many of our papers thereafter, we form VW value and growth portfolios of small and big stocks at the end of each month from June 1963 to May 2019. Small stocks are NYSE, AMEX, and NASDAQ stocks with end-of-month market cap below the NYSE median, and big stocks are those above the NYSE median. Value stocks are NYSE, AMEX, and NASDAQ stocks with BM at or above the 70th percentile of BM for NYSE stocks, and growth stocks are those below the 30th percentile. A stock’s BM at the end of month τ from June of year t to May of t+1 is book equity for the fiscal year ending in t–1 divided by market cap at the end of month τ, with market cap adjusted for shares issued or repurchased from the t–1 fiscal year-end to month τ. We exclude stocks with negative BM from the portfolios and when forming BM breaks. There are three VW value portfolios: Small Value (SV), Big Value (BV), and their VW combination, Market Value (MV). The corresponding VW portfolios of growth stocks are Small Growth (SG), Big Growth (BG), and Market Growth (MG). 2. Average Returns Table 1 shows summary statistics for returns in excess of Market, R–RM, for the portfolios described in the previous section. The table shows results for the 56 years (672 months) from July 1963 to June 2019, which we denote 1963–2019, and for the July 1963–June 1991 and July 1991–June 2019 28-year half-periods, henceforth 1963–1991 and 1991–2019. Table 1 Summary statistics for R–RM, average percent of aggregate market cap, correlations of portfolio returns with RM, and correlations of return premiums, July 1963–June 2019 Panel A. Summary statistics for monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . 1963–2019 Average 0.26 0.21 0.45 –0.03 –0.02 0.06 Standard deviation 2.88 2.94 3.66 1.10 1.16 3.64 t-statistic 2.37 1.81 3.21 –0.70 –0.54 0.45 1963–1991 Average 0.42 0.36 0.58 –0.07 –0.06 0.09 Standard deviation 2.36 2.30 3.32 1.23 1.28 3.48 t-statistic 3.25 2.91 3.19 –0.98 –0.89 0.48 1991–2019 Average 0.11 0.05 0.33 0.01 0.01 0.03 Standard deviation 3.31 3.46 3.98 0.96 1.03 3.80 t-statistic 0.60 0.24 1.52 0.11 0.25 0.17 Difference between 1963–1991 and 1991–2019 average monthly premiums Average 0.31 0.32 0.25 –0.07 –0.08 0.06 Standard deviation 4.07 4.15 5.18 1.56 1.64 5.16 t-statistic 1.39 1.41 0.87 –0.84 –0.85 0.20 F statistic for T2 test that 1963–1991 and 1991–2019 expectations are equal = 0.410, p-value = 87.2% Panel A. Summary statistics for monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . 1963–2019 Average 0.26 0.21 0.45 –0.03 –0.02 0.06 Standard deviation 2.88 2.94 3.66 1.10 1.16 3.64 t-statistic 2.37 1.81 3.21 –0.70 –0.54 0.45 1963–1991 Average 0.42 0.36 0.58 –0.07 –0.06 0.09 Standard deviation 2.36 2.30 3.32 1.23 1.28 3.48 t-statistic 3.25 2.91 3.19 –0.98 –0.89 0.48 1991–2019 Average 0.11 0.05 0.33 0.01 0.01 0.03 Standard deviation 3.31 3.46 3.98 0.96 1.03 3.80 t-statistic 0.60 0.24 1.52 0.11 0.25 0.17 Difference between 1963–1991 and 1991–2019 average monthly premiums Average 0.31 0.32 0.25 –0.07 –0.08 0.06 Standard deviation 4.07 4.15 5.18 1.56 1.64 5.16 t-statistic 1.39 1.41 0.87 –0.84 –0.85 0.20 F statistic for T2 test that 1963–1991 and 1991–2019 expectations are equal = 0.410, p-value = 87.2% Panel B. Average percent of aggregate market cap . . MV . BV . SV . MG . BG . SG . Average percent of MC 13.9 11.6 2.3 51.9 48.4 3.5 Panel B. Average percent of aggregate market cap . . MV . BV . SV . MG . BG . SG . Average percent of MC 13.9 11.6 2.3 51.9 48.4 3.5 Panel C. Correlations of monthly portfolio and market returns, and of monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . Cor(R, RM) 0.85 0.84 0.82 0.97 0.97 0.87 MV 1.00 0.97 0.76 –0.72 –0.72 –0.03 BV 0.97 1.00 0.60 –0.70 –0.67 –0.19 SV 0.76 0.60 1.00 –0.54 –0.64 0.45 MG –0.72 –0.70 –0.54 1.00 0.97 0.14 BG –0.72 –0.67 –0.64 0.97 1.00 –0.06 SG –0.03 –0.19 0.45 0.14 –0.06 1.00 Panel C. Correlations of monthly portfolio and market returns, and of monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . Cor(R, RM) 0.85 0.84 0.82 0.97 0.97 0.87 MV 1.00 0.97 0.76 –0.72 –0.72 –0.03 BV 0.97 1.00 0.60 –0.70 –0.67 –0.19 SV 0.76 0.60 1.00 –0.54 –0.64 0.45 MG –0.72 –0.70 –0.54 1.00 0.97 0.14 BG –0.72 –0.67 –0.64 0.97 1.00 –0.06 SG –0.03 –0.19 0.45 0.14 –0.06 1.00 We use independent sorts on market cap (MC) and book-to-market equity (BM) to form four VW portfolios at the end of each month. The breakpoint for Small and Big (S and B) is the median NYSE MC. Growth and Value are stocks with BM below the 30th or at or above the 70th NYSE percentile. Market Value and Growth (MV and MG) are VW portfolios of all Value or all Growth stocks. A portfolio’s return premium is its monthly return R minus the return on the VW market portfolio RM. The Hotelling T2 statistic tests the hypothesis that expected premiums for the six portfolios do not change from 1963–1991 to 1991–2019. Open in new tab Table 1 Summary statistics for R–RM, average percent of aggregate market cap, correlations of portfolio returns with RM, and correlations of return premiums, July 1963–June 2019 Panel A. Summary statistics for monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . 1963–2019 Average 0.26 0.21 0.45 –0.03 –0.02 0.06 Standard deviation 2.88 2.94 3.66 1.10 1.16 3.64 t-statistic 2.37 1.81 3.21 –0.70 –0.54 0.45 1963–1991 Average 0.42 0.36 0.58 –0.07 –0.06 0.09 Standard deviation 2.36 2.30 3.32 1.23 1.28 3.48 t-statistic 3.25 2.91 3.19 –0.98 –0.89 0.48 1991–2019 Average 0.11 0.05 0.33 0.01 0.01 0.03 Standard deviation 3.31 3.46 3.98 0.96 1.03 3.80 t-statistic 0.60 0.24 1.52 0.11 0.25 0.17 Difference between 1963–1991 and 1991–2019 average monthly premiums Average 0.31 0.32 0.25 –0.07 –0.08 0.06 Standard deviation 4.07 4.15 5.18 1.56 1.64 5.16 t-statistic 1.39 1.41 0.87 –0.84 –0.85 0.20 F statistic for T2 test that 1963–1991 and 1991–2019 expectations are equal = 0.410, p-value = 87.2% Panel A. Summary statistics for monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . 1963–2019 Average 0.26 0.21 0.45 –0.03 –0.02 0.06 Standard deviation 2.88 2.94 3.66 1.10 1.16 3.64 t-statistic 2.37 1.81 3.21 –0.70 –0.54 0.45 1963–1991 Average 0.42 0.36 0.58 –0.07 –0.06 0.09 Standard deviation 2.36 2.30 3.32 1.23 1.28 3.48 t-statistic 3.25 2.91 3.19 –0.98 –0.89 0.48 1991–2019 Average 0.11 0.05 0.33 0.01 0.01 0.03 Standard deviation 3.31 3.46 3.98 0.96 1.03 3.80 t-statistic 0.60 0.24 1.52 0.11 0.25 0.17 Difference between 1963–1991 and 1991–2019 average monthly premiums Average 0.31 0.32 0.25 –0.07 –0.08 0.06 Standard deviation 4.07 4.15 5.18 1.56 1.64 5.16 t-statistic 1.39 1.41 0.87 –0.84 –0.85 0.20 F statistic for T2 test that 1963–1991 and 1991–2019 expectations are equal = 0.410, p-value = 87.2% Panel B. Average percent of aggregate market cap . . MV . BV . SV . MG . BG . SG . Average percent of MC 13.9 11.6 2.3 51.9 48.4 3.5 Panel B. Average percent of aggregate market cap . . MV . BV . SV . MG . BG . SG . Average percent of MC 13.9 11.6 2.3 51.9 48.4 3.5 Panel C. Correlations of monthly portfolio and market returns, and of monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . Cor(R, RM) 0.85 0.84 0.82 0.97 0.97 0.87 MV 1.00 0.97 0.76 –0.72 –0.72 –0.03 BV 0.97 1.00 0.60 –0.70 –0.67 –0.19 SV 0.76 0.60 1.00 –0.54 –0.64 0.45 MG –0.72 –0.70 –0.54 1.00 0.97 0.14 BG –0.72 –0.67 –0.64 0.97 1.00 –0.06 SG –0.03 –0.19 0.45 0.14 –0.06 1.00 Panel C. Correlations of monthly portfolio and market returns, and of monthly return premiums, R–RM . . MV . BV . SV . MG . BG . SG . Cor(R, RM) 0.85 0.84 0.82 0.97 0.97 0.87 MV 1.00 0.97 0.76 –0.72 –0.72 –0.03 BV 0.97 1.00 0.60 –0.70 –0.67 –0.19 SV 0.76 0.60 1.00 –0.54 –0.64 0.45 MG –0.72 –0.70 –0.54 1.00 0.97 0.14 BG –0.72 –0.67 –0.64 0.97 1.00 –0.06 SG –0.03 –0.19 0.45 0.14 –0.06 1.00 We use independent sorts on market cap (MC) and book-to-market equity (BM) to form four VW portfolios at the end of each month. The breakpoint for Small and Big (S and B) is the median NYSE MC. Growth and Value are stocks with BM below the 30th or at or above the 70th NYSE percentile. Market Value and Growth (MV and MG) are VW portfolios of all Value or all Growth stocks. A portfolio’s return premium is its monthly return R minus the return on the VW market portfolio RM. The Hotelling T2 statistic tests the hypothesis that expected premiums for the six portfolios do not change from 1963–1991 to 1991–2019. Open in new tab Average premiums in excess of Market for the three growth portfolios are small and indistinguishable from zero for 1963–2019 and for the 1963–1991 and 1991–2019 half-periods. For example, the average growth premiums for 1963–2019 are –0.03% per month (t = –0.70) for MG, –0.02% (t = –0.54) for BG, and 0.06% (t = 0.45) for SG. Average value premiums over Market are strong for the first half of the sample, 1963–1991. BV beats M by 0.36% per month (t = 2.91), and SV wins by an impressive 0.58% per month (t = 3.19). Since it is a value-weight portfolio, Market Value is mostly Big Value and its average monthly premium for the first half of the sample, 0.42% (t = 3.25), is close to that of Big Value. Despite the low 1963–1991 average premiums for the three growth portfolios, the Hotelling T2 test rejects the hypothesis that the 1963–1991 expected premiums for the six value and growth portfolios are all zero (F = 3.35, p-value = 0.3%). Average value premiums are much lower in the second half of the sample, 1991–2019. The BV average is 0.05% per month (t = 0.24). The 1991–2019 average premium for SV is larger, 0.33% per month (t = 1.52), but well below the 1963–1991 average, 0.58%. Average R–RM for the portfolio of all value stocks (MV) falls from 0.42% per month in the first half to 0.11% (t = 0.60) in the second. Despite the low 1991–2019 average premiums, the T2 test is only weakly consistent with the hypothesis that expected 1991–2019 premiums for the six value and growth are all zero (F = 1.80, p-value = 9.8%). Is this marginal result from the T2 test for 1991–2019 average premiums sufficient to conclude that expected premiums disappear during the second half of the sample? A different and perhaps more direct test is provided by the Table 1 summary statistics for differences between first-half and second-half averages of R–RM. The largest declines in average premiums are 0.31% for MV and 0.32% for BV. These are economically large but only 1.39 and 1.41 standard errors from zero. For the other portfolios, including SV, average differences between premiums for the two half-periods are within one standard error of zero. The F-statistic for the T2 test of the six changes in average premiums against zero is 0.41 and its p-value is 87.2%, so the six differences between first- and second-half average premiums are far from unusual if expected premiums do not change from the first to the second half of the sample. The declines from 1963–1991 to 1991–2019 in average premiums for the value portfolios seem large, but statistically they are indistinguishable from zero. The culprit is volatility. For example, Table 1 shows that, on average, Big Value accounts for only 11.6% of the market capitalization (cap) of the overall Market portfolio, and the correlation between BV and M returns, though a substantial 0.84, leaves room for lots of independent variation. The standard deviation of Big Value’s monthly premium is indeed large, 2.94% for the full sample (Table 1). Moreover, monthly premiums for the two half-periods are close to uncorrelated, so variances of differences between half-period average monthly premiums are close to the sums of the variances of the subperiod averages. As a result, inferences about differences between expected premiums for the two subperiods are clouded by imprecision. In contrast, on average Big Growth is about 48% of aggregate market cap, the correlation between BG and Market returns is high (0.97), and the full-sample standard deviation of monthly BG premiums is only 1.16%. The average BG premium is close to zero in both half-periods, however, so despite its relatively low volatility, the difference between the average premiums of the first and second half-periods, –0.08%, is economically and statistically trivial. Since we cannot reject the hypothesis that expected premiums over RM are the same for the two halves of the sample, average premiums for the full sample are arguably the best evidence on long-run expected premiums. Full-period average premiums are halfway between average premiums for 1963–1991 and 1991–2019, but doubling the sample size increases precision. The full-period average premiums for MV and SV, 0.26% and 0.45% per month, are 2.37 and 3.21 standard errors from zero. The full-period average premium for BV, 0.21% per month, is 1.81 standard errors from zero. Thus, for MV (the market portfolio of value stocks) and especially for SV, full-period expected premiums over RM are reliably positive. The inference leans in that direction for BV, but it is on shakier ground. The meager full-period average premiums for the three growth portfolios are within a standard error of zero. The T2 test cleanly rejects the hypothesis that 1963–2019 expected premiums for the six value and growth portfolios are all zero (F = 3.37, p-value = 0.3%). A referee suggests that the power of the tests for the existence of 1991–2019 out-of-sample expected value premiums can perhaps be enhanced by extending the in-sample 1963–2019 period back to 1926. The logic is that Davis, Fama, and French (1999) confirm that the strong value premiums of 1963–1991 extend back to 1926. It is thus legitimate to treat the entire 1926–1991 period as in-sample relative to 1991–2019. Skipping the details, we can report that the potential power gained from a longer in-sample period is largely offset by the high volatility of the additional 1926–1963 monthly returns, and the conclusions from the tests that use the shorter 1963–1991 in-sample period stand. Specifically, the tests lack power to determine whether expected value premiums for the out-of-sample 1991–2019 period differ from zero or from the in-sample expected premiums of either 1963–1991 or 1926–1991. 3. Forecasting Regressions A portfolio’s expected return almost certainly varies through time. The challenge is to measure the variation. In the literature, the lagged dividend/price ratio, DP, is often used to forecast stock returns. The logic is that the price of a stock is the present value of expected dividends, where the discount rate is (roughly speaking) the expected stock return. Variation in the expected return thus has an inverse effect on price, which means variation in DP is in part due to variation in the expected stock return. The “in part” is important. Price is the present value of future expected dividends, and lagged dividends, the numerator in DP, may be a poor proxy for expected dividends, both because of differences in expected earnings growth across firms and because dividends can be affected by financing decisions that have little or no effect on expected earnings (Miller and Modigliani 1961). Here we use lagged book-to-market ratios in excess of Market, BM–BMM, to forecast returns in excess of Market, R–RM. Like the price in DP, the market cap in BM responds to variation in expected dividends as well as to variation in expected returns. This makes BM a noisy measure of expected returns. But B, the book value of equity-financed assets, is less subject to arbitrary financing decisions than D. This (perhaps arguably) makes B a less noisy scaling variable for extracting variation in the expected stock return from the stock’s price. We regress returns in excess of RM on lagged BM in excess of BMM for evidence on whether value premiums decline and perhaps disappear in the second half of 1963–2019, Rit–RMt=ai+bi(BMit–1–BMMt–1)+eit.(1) In this regression, Rit and RMt are returns on portfolio i and Market in month t, BMit–1 and BMMt–1 are the book-to-market-equity ratios of portfolio i and Market at the end of month t–1, ai and bi are the intercept and slope for portfolio i, and eit is a residual. (We use subscripts i and t only when needed for clarity.) Table 2 shows summary statistics for BM–BMM. Time-series plots of BM for M, MV, and MG are in Figure 1, and plots of BM–BMM for MV and MG are in Figure 2. The BM ratios for the six value and growth portfolios share a common market component. The BM ratios for MV and MG in Figure 1 move with BMM, and the correlation of BMM with BM for each of the six portfolios (panel B of Table 2), ranges from 0.69 (SV) to 0.96 (MG and BG). After subtracting the common market component, book-to-market ratios for value and growth portfolios tend to be negatively correlated. The correlation between the BM–BMM of MV and MG, for example, is –0.30 (panel B of Table 2), and the negative relation is apparent in Figure 2. Thus, if the regressions say BM–BMM tracks expected premiums over Market, expected premiums for value and growth portfolios tend to move in opposite directions. Figure 1 Open in new tabDownload slide BM for Market, Market Value, and Market Growth Figure 1 Open in new tabDownload slide BM for Market, Market Value, and Market Growth Figure 2 Open in new tabDownload slide Excess BM for Market Value and Market Growth Figure 2 Open in new tabDownload slide Excess BM for Market Value and Market Growth Table 2 Summary statistics for BM–BMM, correlations of portfolio and market BM and of portfolio BM–BMM, and autocorrelations of BM–BMM, July 1963–June 2019 Panel A. Summary statistics for monthly excess book-to-market equity ratios, BM–BMM . . MV . BV . SV . MG . BG . SG . July 1963–June 2019 Average 0.78 0.74 0.97 –0.28 –0.28 –0.28 Standard deviation 0.33 0.30 0.39 0.10 0.10 0.12 July 1963–June 1991 Average 0.85 0.80 1.05 –0.35 –0.35 –0.35 Standard deviation 0.30 0.27 0.35 0.09 0.09 0.12 July 1991–June 2019 Average 0.71 0.67 0.89 –0.21 –0.20 –0.20 Standard deviation 0.34 0.31 0.42 0.06 0.06 0.07 Difference between July 1963–June 1991 and July 1991–June 2019 averages Average 0.13 0.13 0.16 –0.15 –0.15 –0.14 Standard Deviation 0.45 0.41 0.55 0.11 0.10 0.14 Panel A. Summary statistics for monthly excess book-to-market equity ratios, BM–BMM . . MV . BV . SV . MG . BG . SG . July 1963–June 2019 Average 0.78 0.74 0.97 –0.28 –0.28 –0.28 Standard deviation 0.33 0.30 0.39 0.10 0.10 0.12 July 1963–June 1991 Average 0.85 0.80 1.05 –0.35 –0.35 –0.35 Standard deviation 0.30 0.27 0.35 0.09 0.09 0.12 July 1991–June 2019 Average 0.71 0.67 0.89 –0.21 –0.20 –0.20 Standard deviation 0.34 0.31 0.42 0.06 0.06 0.07 Difference between July 1963–June 1991 and July 1991–June 2019 averages Average 0.13 0.13 0.16 –0.15 –0.15 –0.14 Standard Deviation 0.45 0.41 0.55 0.11 0.10 0.14 Panel B. Correlations of monthly portfolio and market BM, and correlations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . Cor(BM, BMM) 0.74 0.76 0.69 0.96 0.96 0.85 MV 1.00 1.00 0.96 –0.30 –0.32 –0.07 MG –0.30 –0.32 –0.30 1.00 1.00 0.94 SV 0.96 0.94 1.00 –0.30 –0.32 –0.07 SG –0.07 –0.10 –0.07 0.94 0.93 1.00 BV 1.00 1.00 0.94 –0.32 –0.34 –0.10 BG –0.32 –0.34 –0.32 1.00 1.00 0.93 Panel B. Correlations of monthly portfolio and market BM, and correlations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . Cor(BM, BMM) 0.74 0.76 0.69 0.96 0.96 0.85 MV 1.00 1.00 0.96 –0.30 –0.32 –0.07 MG –0.30 –0.32 –0.30 1.00 1.00 0.94 SV 0.96 0.94 1.00 –0.30 –0.32 –0.07 SG –0.07 –0.10 –0.07 0.94 0.93 1.00 BV 1.00 1.00 0.94 –0.32 –0.34 –0.10 BG –0.32 –0.34 –0.32 1.00 1.00 0.93 Panel C. Autocorrelations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . 1 0.95 0.94 0.95 0.99 0.99 0.99 2 0.88 0.87 0.88 0.98 0.98 0.99 3 0.82 0.81 0.82 0.97 0.97 0.98 4 0.75 0.74 0.75 0.96 0.96 0.97 5 0.68 0.67 0.68 0.95 0.95 0.97 6 0.63 0.62 0.63 0.94 0.94 0.96 7 0.59 0.57 0.58 0.93 0.92 0.95 8 0.56 0.54 0.54 0.92 0.91 0.95 9 0.53 0.51 0.52 0.91 0.90 0.94 10 0.50 0.48 0.50 0.90 0.90 0.93 11 0.47 0.45 0.48 0.89 0.89 0.93 12 0.44 0.42 0.45 0.88 0.88 0.92 Panel C. Autocorrelations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . 1 0.95 0.94 0.95 0.99 0.99 0.99 2 0.88 0.87 0.88 0.98 0.98 0.99 3 0.82 0.81 0.82 0.97 0.97 0.98 4 0.75 0.74 0.75 0.96 0.96 0.97 5 0.68 0.67 0.68 0.95 0.95 0.97 6 0.63 0.62 0.63 0.94 0.94 0.96 7 0.59 0.57 0.58 0.93 0.92 0.95 8 0.56 0.54 0.54 0.92 0.91 0.95 9 0.53 0.51 0.52 0.91 0.90 0.94 10 0.50 0.48 0.50 0.90 0.90 0.93 11 0.47 0.45 0.48 0.89 0.89 0.93 12 0.44 0.42 0.45 0.88 0.88 0.92 MV, BV, and SV are the value portfolios, and MG, BG, and SG are the growth portfolios of Table 1. Open in new tab Table 2 Summary statistics for BM–BMM, correlations of portfolio and market BM and of portfolio BM–BMM, and autocorrelations of BM–BMM, July 1963–June 2019 Panel A. Summary statistics for monthly excess book-to-market equity ratios, BM–BMM . . MV . BV . SV . MG . BG . SG . July 1963–June 2019 Average 0.78 0.74 0.97 –0.28 –0.28 –0.28 Standard deviation 0.33 0.30 0.39 0.10 0.10 0.12 July 1963–June 1991 Average 0.85 0.80 1.05 –0.35 –0.35 –0.35 Standard deviation 0.30 0.27 0.35 0.09 0.09 0.12 July 1991–June 2019 Average 0.71 0.67 0.89 –0.21 –0.20 –0.20 Standard deviation 0.34 0.31 0.42 0.06 0.06 0.07 Difference between July 1963–June 1991 and July 1991–June 2019 averages Average 0.13 0.13 0.16 –0.15 –0.15 –0.14 Standard Deviation 0.45 0.41 0.55 0.11 0.10 0.14 Panel A. Summary statistics for monthly excess book-to-market equity ratios, BM–BMM . . MV . BV . SV . MG . BG . SG . July 1963–June 2019 Average 0.78 0.74 0.97 –0.28 –0.28 –0.28 Standard deviation 0.33 0.30 0.39 0.10 0.10 0.12 July 1963–June 1991 Average 0.85 0.80 1.05 –0.35 –0.35 –0.35 Standard deviation 0.30 0.27 0.35 0.09 0.09 0.12 July 1991–June 2019 Average 0.71 0.67 0.89 –0.21 –0.20 –0.20 Standard deviation 0.34 0.31 0.42 0.06 0.06 0.07 Difference between July 1963–June 1991 and July 1991–June 2019 averages Average 0.13 0.13 0.16 –0.15 –0.15 –0.14 Standard Deviation 0.45 0.41 0.55 0.11 0.10 0.14 Panel B. Correlations of monthly portfolio and market BM, and correlations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . Cor(BM, BMM) 0.74 0.76 0.69 0.96 0.96 0.85 MV 1.00 1.00 0.96 –0.30 –0.32 –0.07 MG –0.30 –0.32 –0.30 1.00 1.00 0.94 SV 0.96 0.94 1.00 –0.30 –0.32 –0.07 SG –0.07 –0.10 –0.07 0.94 0.93 1.00 BV 1.00 1.00 0.94 –0.32 –0.34 –0.10 BG –0.32 –0.34 –0.32 1.00 1.00 0.93 Panel B. Correlations of monthly portfolio and market BM, and correlations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . Cor(BM, BMM) 0.74 0.76 0.69 0.96 0.96 0.85 MV 1.00 1.00 0.96 –0.30 –0.32 –0.07 MG –0.30 –0.32 –0.30 1.00 1.00 0.94 SV 0.96 0.94 1.00 –0.30 –0.32 –0.07 SG –0.07 –0.10 –0.07 0.94 0.93 1.00 BV 1.00 1.00 0.94 –0.32 –0.34 –0.10 BG –0.32 –0.34 –0.32 1.00 1.00 0.93 Panel C. Autocorrelations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . 1 0.95 0.94 0.95 0.99 0.99 0.99 2 0.88 0.87 0.88 0.98 0.98 0.99 3 0.82 0.81 0.82 0.97 0.97 0.98 4 0.75 0.74 0.75 0.96 0.96 0.97 5 0.68 0.67 0.68 0.95 0.95 0.97 6 0.63 0.62 0.63 0.94 0.94 0.96 7 0.59 0.57 0.58 0.93 0.92 0.95 8 0.56 0.54 0.54 0.92 0.91 0.95 9 0.53 0.51 0.52 0.91 0.90 0.94 10 0.50 0.48 0.50 0.90 0.90 0.93 11 0.47 0.45 0.48 0.89 0.89 0.93 12 0.44 0.42 0.45 0.88 0.88 0.92 Panel C. Autocorrelations of monthly BM–BMM . . MV . BV . SV . MG . BG . SG . 1 0.95 0.94 0.95 0.99 0.99 0.99 2 0.88 0.87 0.88 0.98 0.98 0.99 3 0.82 0.81 0.82 0.97 0.97 0.98 4 0.75 0.74 0.75 0.96 0.96 0.97 5 0.68 0.67 0.68 0.95 0.95 0.97 6 0.63 0.62 0.63 0.94 0.94 0.96 7 0.59 0.57 0.58 0.93 0.92 0.95 8 0.56 0.54 0.54 0.92 0.91 0.95 9 0.53 0.51 0.52 0.91 0.90 0.94 10 0.50 0.48 0.50 0.90 0.90 0.93 11 0.47 0.45 0.48 0.89 0.89 0.93 12 0.44 0.42 0.45 0.88 0.88 0.92 MV, BV, and SV are the value portfolios, and MG, BG, and SG are the growth portfolios of Table 1. Open in new tab Figure 2 and panel A of Table 2 also say that BM in excess of Market BM is more variable for value than for growth portfolios. The full-sample standard deviation of BM–BMM for MV, for example, is 0.33, versus 0.10 for MG. Figure 2 suggests and panel C of Table 2 confirms that BM–BMM is autocorrelated, and more so for growth than for value portfolios. For example, the autocorrelations of BM–BMM for MG decline from 0.99 at lag 1 to 0.88 at lag 12, versus 0.95 to 0.44 for MV. Thus, if BM–BMM tracks variation in expected return premiums, the conditional expected premiums are persistent. At least for the value portfolios, the decay of the autocorrelations suggests that BM–BMM reverts to a long-run mean. Panel A of Table 3 summarizes the regressions of monthly return premiums on lagged BM–BMM. For the three value portfolios, we can conclude that BM–BMM captures variation in expected premiums. The BM–BMM slopes for MV, BV, and SV, 2.20, 2.26, and 2.26, are more than six standard errors from zero. The slopes for MG and BG, 0.66 (t = 1.63) and 0.76 (t = 1.75), are about one third those for the value portfolios and the SG slope is a trivial 0.02 (t = 0.02). Even for the value portfolios, the estimated variation in expected premiums tracked by BM–BMM is small relative to the variation in realized premiums. The regression R2 for the value portfolios are 0.05 and 0.06. For all portfolios, residual standard errors (RSE in Table 3) are close to the standard deviations of return premiums (Table 1). Table 3 Full-period and half-period regressions of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–2019 Panel A. Full-period regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963-2019 . . Coefficient . Standard Error . t-statistic . . a . b . a . b . a . b . R2 . RSE . MV –1.46 2.20 0.28 0.33 –5.24 6.71 0.06 2.79 BV –1.46 2.26 0.29 0.37 –4.97 6.12 0.05 2.86 SV –1.74 2.26 0.37 0.35 –4.76 6.48 0.06 3.56 MG 0.15 0.66 0.12 0.41 1.28 1.63 0.00 1.10 BG 0.19 0.76 0.13 0.43 1.46 1.75 0.00 1.16 SG 0.07 0.02 0.35 1.16 0.20 0.02 –0.00 3.65 Panel A. Full-period regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963-2019 . . Coefficient . Standard Error . t-statistic . . a . b . a . b . a . b . R2 . RSE . MV –1.46 2.20 0.28 0.33 –5.24 6.71 0.06 2.79 BV –1.46 2.26 0.29 0.37 –4.97 6.12 0.05 2.86 SV –1.74 2.26 0.37 0.35 –4.76 6.48 0.06 3.56 MG 0.15 0.66 0.12 0.41 1.28 1.63 0.00 1.10 BG 0.19 0.76 0.13 0.43 1.46 1.75 0.00 1.16 SG 0.07 0.02 0.35 1.16 0.20 0.02 –0.00 3.65 Panel B. Split-sample regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–1991 and 1991–2019 . . Coefficient . Standard Error . t-statistic . . a . da . b . db . a . da . b . db . a . da . b . db . R2 . RSE . MV –1.21 –0.40 1.93 0.49 0.45 0.58 0.50 0.68 –2.67 –0.69 3.82 0.72 0.06 2.79 BV –1.27 –0.30 2.04 0.37 0.48 0.61 0.57 0.76 –2.62 –0.49 3.57 0.48 0.05 2.87 SV –1.11 –1.01 1.62 1.13 0.62 0.77 0.56 0.72 –1.81 –1.31 2.89 1.56 0.06 3.55 MG 0.21 –0.01 0.78 0.16 0.24 0.33 0.67 1.26 0.86 –0.03 1.16 0.13 –0.00 1.10 BG 0.28 –0.05 0.97 0.08 0.26 0.36 0.73 1.34 1.05 –0.13 1.33 0.06 0.00 1.16 SG –0.06 0.62 –0.44 3.02 0.61 0.89 1.64 3.44 –0.10 0.70 –0.27 0.88 –0.00 3.65 Wald test that da and db are zero in all 6 regressions = 0.647, DOF = (12, 648), = 80.3% Panel B. Split-sample regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–1991 and 1991–2019 . . Coefficient . Standard Error . t-statistic . . a . da . b . db . a . da . b . db . a . da . b . db . R2 . RSE . MV –1.21 –0.40 1.93 0.49 0.45 0.58 0.50 0.68 –2.67 –0.69 3.82 0.72 0.06 2.79 BV –1.27 –0.30 2.04 0.37 0.48 0.61 0.57 0.76 –2.62 –0.49 3.57 0.48 0.05 2.87 SV –1.11 –1.01 1.62 1.13 0.62 0.77 0.56 0.72 –1.81 –1.31 2.89 1.56 0.06 3.55 MG 0.21 –0.01 0.78 0.16 0.24 0.33 0.67 1.26 0.86 –0.03 1.16 0.13 –0.00 1.10 BG 0.28 –0.05 0.97 0.08 0.26 0.36 0.73 1.34 1.05 –0.13 1.33 0.06 0.00 1.16 SG –0.06 0.62 –0.44 3.02 0.61 0.89 1.64 3.44 –0.10 0.70 –0.27 0.88 –0.00 3.65 Wald test that da and db are zero in all 6 regressions = 0.647, DOF = (12, 648), = 80.3% Panel C. Subperiod averages of conditional expected premiums from full-period and split-sample regressions . . Average BM–BMM . Full-Period Regressions . Split-sample Regressions . X1 . X2 . Y^11 . Y^12 . Y^11–Y^12 . Std Err . t-stat . Y^21 . Y^22 . Y^21–Y^22 . Std Err . t-stat . MV 0.85 0.71 0.41 0.12 0.30 0.04 6.71 0.42 0.11 0.31 0.22 1.44 BV 0.80 0.67 0.35 0.06 0.29 0.05 6.12 0.36 0.05 0.32 0.22 1.44 SV 1.05 0.89 0.63 0.28 0.35 0.05 6.48 0.58 0.33 0.25 0.27 0.90 MG –0.35 –0.21 –0.08 0.02 –0.10 0.06 –1.63 –0.07 0.01 –0.07 0.09 –0.84 BG –0.35 –0.20 –0.08 0.03 –0.11 0.06 –1.75 –0.06 0.01 –0.08 0.09 –0.85 SG –0.35 –0.20 0.06 0.06 –0.00 0.17 –0.02 0.09 0.03 0.06 0.28 0.20 Panel C. Subperiod averages of conditional expected premiums from full-period and split-sample regressions . . Average BM–BMM . Full-Period Regressions . Split-sample Regressions . X1 . X2 . Y^11 . Y^12 . Y^11–Y^12 . Std Err . t-stat . Y^21 . Y^22 . Y^21–Y^22 . Std Err . t-stat . MV 0.85 0.71 0.41 0.12 0.30 0.04 6.71 0.42 0.11 0.31 0.22 1.44 BV 0.80 0.67 0.35 0.06 0.29 0.05 6.12 0.36 0.05 0.32 0.22 1.44 SV 1.05 0.89 0.63 0.28 0.35 0.05 6.48 0.58 0.33 0.25 0.27 0.90 MG –0.35 –0.21 –0.08 0.02 –0.10 0.06 –1.63 –0.07 0.01 –0.07 0.09 –0.84 BG –0.35 –0.20 –0.08 0.03 –0.11 0.06 –1.75 –0.06 0.01 –0.08 0.09 –0.85 SG –0.35 –0.20 0.06 0.06 –0.00 0.17 –0.02 0.09 0.03 0.06 0.28 0.20 See Table 1 for description of portfolios. Panels A and B report coefficients, standard errors, and t-statistics for the coefficients, R2, and residual standard errors (RSE) for regressions (1) and (2), Rit–RMt=ai+bi(BMit–1–BMMt–1)+eit(1) Rit–RMt=ai+daiD+bi(BMit–1–BMMt–1)+dbiD(BMit–1–BMMt–1)+eit.(2) Panel C reports average values of the independent variable, BM–BMM, and average forecasts from regression (1), Y^11and Y^12 , and regression (2), Y^21 and Y^22 , for the first and second half-period, differences between the average forecasts, and standard errors and t-statistics for the differences. Open in new tab Table 3 Full-period and half-period regressions of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–2019 Panel A. Full-period regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963-2019 . . Coefficient . Standard Error . t-statistic . . a . b . a . b . a . b . R2 . RSE . MV –1.46 2.20 0.28 0.33 –5.24 6.71 0.06 2.79 BV –1.46 2.26 0.29 0.37 –4.97 6.12 0.05 2.86 SV –1.74 2.26 0.37 0.35 –4.76 6.48 0.06 3.56 MG 0.15 0.66 0.12 0.41 1.28 1.63 0.00 1.10 BG 0.19 0.76 0.13 0.43 1.46 1.75 0.00 1.16 SG 0.07 0.02 0.35 1.16 0.20 0.02 –0.00 3.65 Panel A. Full-period regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963-2019 . . Coefficient . Standard Error . t-statistic . . a . b . a . b . a . b . R2 . RSE . MV –1.46 2.20 0.28 0.33 –5.24 6.71 0.06 2.79 BV –1.46 2.26 0.29 0.37 –4.97 6.12 0.05 2.86 SV –1.74 2.26 0.37 0.35 –4.76 6.48 0.06 3.56 MG 0.15 0.66 0.12 0.41 1.28 1.63 0.00 1.10 BG 0.19 0.76 0.13 0.43 1.46 1.75 0.00 1.16 SG 0.07 0.02 0.35 1.16 0.20 0.02 –0.00 3.65 Panel B. Split-sample regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–1991 and 1991–2019 . . Coefficient . Standard Error . t-statistic . . a . da . b . db . a . da . b . db . a . da . b . db . R2 . RSE . MV –1.21 –0.40 1.93 0.49 0.45 0.58 0.50 0.68 –2.67 –0.69 3.82 0.72 0.06 2.79 BV –1.27 –0.30 2.04 0.37 0.48 0.61 0.57 0.76 –2.62 –0.49 3.57 0.48 0.05 2.87 SV –1.11 –1.01 1.62 1.13 0.62 0.77 0.56 0.72 –1.81 –1.31 2.89 1.56 0.06 3.55 MG 0.21 –0.01 0.78 0.16 0.24 0.33 0.67 1.26 0.86 –0.03 1.16 0.13 –0.00 1.10 BG 0.28 –0.05 0.97 0.08 0.26 0.36 0.73 1.34 1.05 –0.13 1.33 0.06 0.00 1.16 SG –0.06 0.62 –0.44 3.02 0.61 0.89 1.64 3.44 –0.10 0.70 –0.27 0.88 –0.00 3.65 Wald test that da and db are zero in all 6 regressions = 0.647, DOF = (12, 648), = 80.3% Panel B. Split-sample regressions of of return premiums, R–RM, on excess book-to-market equity, BM–BMM, 1963–1991 and 1991–2019 . . Coefficient . Standard Error . t-statistic . . a . da . b . db . a . da . b . db . a . da . b . db . R2 . RSE . MV –1.21 –0.40 1.93 0.49 0.45 0.58 0.50 0.68 –2.67 –0.69 3.82 0.72 0.06 2.79 BV –1.27 –0.30 2.04 0.37 0.48 0.61 0.57 0.76 –2.62 –0.49 3.57 0.48 0.05 2.87 SV –1.11 –1.01 1.62 1.13 0.62 0.77 0.56 0.72 –1.81 –1.31 2.89 1.56 0.06 3.55 MG 0.21 –0.01 0.78 0.16 0.24 0.33 0.67 1.26 0.86 –0.03 1.16 0.13 –0.00 1.10 BG 0.28 –0.05 0.97 0.08 0.26 0.36 0.73 1.34 1.05 –0.13 1.33 0.06 0.00 1.16 SG –0.06 0.62 –0.44 3.02 0.61 0.89 1.64 3.44 –0.10 0.70 –0.27 0.88 –0.00 3.65 Wald test that da and db are zero in all 6 regressions = 0.647, DOF = (12, 648), = 80.3% Panel C. Subperiod averages of conditional expected premiums from full-period and split-sample regressions . . Average BM–BMM . Full-Period Regressions . Split-sample Regressions . X1 . X2 . Y^11 . Y^12 . Y^11–Y^12 . Std Err . t-stat . Y^21 . Y^22 . Y^21–Y^22 . Std Err . t-stat . MV 0.85 0.71 0.41 0.12 0.30 0.04 6.71 0.42 0.11 0.31 0.22 1.44 BV 0.80 0.67 0.35 0.06 0.29 0.05 6.12 0.36 0.05 0.32 0.22 1.44 SV 1.05 0.89 0.63 0.28 0.35 0.05 6.48 0.58 0.33 0.25 0.27 0.90 MG –0.35 –0.21 –0.08 0.02 –0.10 0.06 –1.63 –0.07 0.01 –0.07 0.09 –0.84 BG –0.35 –0.20 –0.08 0.03 –0.11 0.06 –1.75 –0.06 0.01 –0.08 0.09 –0.85 SG –0.35 –0.20 0.06 0.06 –0.00 0.17 –0.02 0.09 0.03 0.06 0.28 0.20 Panel C. Subperiod averages of conditional expected premiums from full-period and split-sample regressions . . Average BM–BMM . Full-Period Regressions . Split-sample Regressions . X1 . X2 . Y^11 . Y^12 . Y^11–Y^12 . Std Err . t-stat . Y^21 . Y^22 . Y^21–Y^22 . Std Err . t-stat . MV 0.85 0.71 0.41 0.12 0.30 0.04 6.71 0.42 0.11 0.31 0.22 1.44 BV 0.80 0.67 0.35 0.06 0.29 0.05 6.12 0.36 0.05 0.32 0.22 1.44 SV 1.05 0.89 0.63 0.28 0.35 0.05 6.48 0.58 0.33 0.25 0.27 0.90 MG –0.35 –0.21 –0.08 0.02 –0.10 0.06 –1.63 –0.07 0.01 –0.07 0.09 –0.84 BG –0.35 –0.20 –0.08 0.03 –0.11 0.06 –1.75 –0.06 0.01 –0.08 0.09 –0.85 SG –0.35 –0.20 0.06 0.06 –0.00 0.17 –0.02 0.09 0.03 0.06 0.28 0.20 See Table 1 for description of portfolios. Panels A and B report coefficients, standard errors, and t-statistics for the coefficients, R2, and residual standard errors (RSE) for regressions (1) and (2), Rit–RMt=ai+bi(BMit–1–BMMt–1)+eit(1) Rit–RMt=ai+daiD+bi(BMit–1–BMMt–1)+dbiD(BMit–1–BMMt–1)+eit.(2) Panel C reports average values of the independent variable, BM–BMM, and average forecasts from regression (1), Y^11and Y^12 , and regression (2), Y^21 and Y^22 , for the first and second half-period, differences between the average forecasts, and standard errors and t-statistics for the differences. Open in new tab Skipping the details, we can report that the autocorrelations of the residuals from the estimates of (1) are close to zero. Thus, the high autocorrelations of BM–BMM say BM spreads capture highly autocorrelated variation in conditional expected returns in excess of Market, but residuals close to white noise suggest that the autocorrelation of BM–BMM is not a problem for interpreting the regressions. A referee suggests that the high autocorrelations of BM–BMM and the relation between the unexpected excess return in (1) and the contemporaneous change in excess BM may cause important Stambaugh (1999) bias in the estimates of (1). We can report that adjustments for the bias have no effect on our inferences from the regression. Bias-adjusted slopes remain more than six standard errors from zero for value portfolios and less than two standard errors from zero for growth portfolios. Stambaugh (1999) bias is probably less relevant for inter-period comparisons of slopes in regression (2) below because the expected effects should be similar in the two halves of the sample. Panel A of Table 2 shows that from the first to the second half of the sample, the positive average values of BM–BMM for value portfolios and the negative averages for growth portfolios shrink toward zero. The shrinkage is similar in magnitude for all six portfolios. Average BM–BMM falls 0.14 (from 0.85 to 0.71) for MV, for example, and rises 0.14 (from –0.35 to –0.21) for MG. Is this compression of BM–BMM for value and growth portfolios reliable evidence that conditional expected return premiums shrink toward zero in the second half of the sample? The answer is yes if we assume the regression coefficients in (1) are the same for the first and second halves of the sample, but the answer is no if we allow the coefficients to change. In tests to determine whether changes in market conditions, such as increasing demand for value stocks, cause expected value and growth premiums to shrink toward zero from the first to the second half of 1963–2019, the assumption in regression (1) that the intercept and slope are constant is open to challenge. Regression (2) allows the intercept and slope in (1) to change from 1963–1991 to 1991–2019, Rit–RMt=ai+daiD+bi(BMit–1–BMMt–1)+dbiD(BMit–1–BMMt–1)+eit.(2) In (2), D is a dummy variable, zero in 1963–1991 and one in 1991–2019. The new variables in regression (2) are zero for 1963–1991, so ai and bi are the regression (1) coefficients for this period, and dai and dbi are the changes in the regression (1) coefficients from 1963–1991 to 1991–2019. Table 3 shows that measured to two decimal places, the regression R2 from (2) are the same as those from (1), and residual standard errors (RSE) from (1) and (2) differ by at most 0.01 (one higher and one lower by 0.01). We can infer that the split-sample regression (2) adds nothing to the 1963–2019 explanatory power of regression (1). The estimates of regression (2) in panel B of Table 3 also offer little evidence that the coefficients in regression (1) change from the first to the second half of 1963–2019. All estimates of dai and dbi in regression (2) are within 1.56 standard errors of zero. The Wald test that da and db are jointly zero in all six estimates of (2) fails to reject with a p-value greater than 80%. The standard errors of dai and dbi from (2) are large, however, so there is also little evidence against a wide range of non-zero true values of dai and dbi. Table 3 shows that the standard errors of dai and dbi from (2) are larger than the standard errors of ai and bi from (2), which in turn are much larger than the standard errors of ai and bi from the univariate regression (1). We show next that the relative precision of the slope estimates from the constant-slope regression (1) leads to strong inferences about declines in value premiums from the first to the second half of 1963–2019. But if we use (2) to allow changes in regression coefficients, imprecise estimates rule out reliable inferences about changes in expected premiums. An error in the intercept estimate from regression (1) affects first- and second-half forecasts equally, so it has no effect on the difference between the regression’s average forecasts of R–RM for the two periods. Conditional on the sample values of BM–BMM, the only source of noise in the difference between the first- and second-half average forecasts from regression (1), Y^11–Y^12 , is sampling error in b, the full-period slope estimate. Thus, for a given portfolio, the standard error of the difference between the first- and second-half average forecasts of R–RM from (1) is SE(Y^11–Y^12)=|A(BM–BMM)1–A(BM–BMM)2|SE(b).(3) In this equation, |A(BM–BMM)1 – A(BM–BMM)2| is the magnitude of the difference between a portfolio’s average excess book-to-market ratios for the two periods, and SE(b) is the standard error of the estimate of b for the portfolio. Panel C of Table 3 reports summary statistics for the conditional forecasts of R–RM. The bottom line is that, up to a possible sign change, the t-statistic for testing whether the change from 1963–1991 to 1991–2019 in average expected R–RM from regression (1) is reliably different from zero is just the t-statistic for the slope in (1), t(Y^11–Y^12)=(Y^11–Y^12) /SE(Y^11–Y^12)=[A(BM–BMM)1–A(BM–BMM)2]b/ [|A(BM–BMM)1–A(BM–BMM)2|SE(b)]=sign[A(BM–BMM)1–A(BM–BMM)2]t(b).(4) The intuition is straightforward. The book-to-market spread BMit–1–BMMt–1 in regression (1) is a predetermined explanatory variable whose sample values are known when we estimate the standard error in (3). If the intercept and slope in (1) are constant, the estimated slope is the only source of sampling error in the difference between the regression’s first- and second-half average forecasts of R–RM for a portfolio. Thus, if the absolute value of the sample difference between average BM spreads for the two half-periods, |A(BM–BMM)1 – A(BM–BMM)2|, is nonzero, our confidence that the true difference between average premium forecasts is also nonzero matches our confidence that the true slope in (1) is not zero. If regression (1) is well specified, slope estimates more than six standard errors from zero and excess book-to-market ratios that shrink toward zero are strong evidence that expected premiums for the value portfolios are smaller in the second half of 1963–2019. Smaller t-statistics for the growth portfolio slopes from (1), 1.63, 1.75, and 0.02, imply less confidence that the shrinkage of the excess book-to-market ratios for growth portfolios implies shrinkage toward zero of their negative expected return premiums. The analysis that produces (4) implies that the t-statistic for the test that any two different BM–BMM spreads (no matter how close in value) deliver different conditional expected premiums is just the t-statistic for the slope in regression (1). The power of regression (1) estimates of conditional expected premiums is then due to the fact that its univariate slope, b, is estimated precisely. Regression (2) estimates different intercepts and slopes for the two halves of 1963–2019. If we switch from regression (1) to (2), we can no longer infer that any of the six expected premiums change from the first to the second half of the sample. The culprit is the low precision of the estimates of the regression coefficients in (2), documented in panel B of Table 3. Reversing the subtraction to simplify the notation, the difference between average premium forecasts for 1991–2019 and 1963–1991 from regression (2), (Y^22–Y^21) , is Y^22–Y^21 = da+dbiA(BM–BMM)2+b[A(BM–BMM)2–A(BM–BMM)1].(5) Define V = [1, A(BM–BMM)2, A(BM–BMM)2 – A(BM–BMM)1]ʹ as the vector of loadings on da, db, and b in (5). If Σ is the covariance matrix of the regression (2) coefficients da, db, and b, the standard error of Y^22–Y^21(and of Y^21–Y^22 ) is SE(Y^22–Y^21)=Sqrt(V′ΣV).(6) Like the message from (3), the message from (6) is that uncertainty about the expected value of the regression forecasts of changes in average return premia from the first to the second half of the sample centers on the precision of the estimated regression coefficients. In the constant-slope regression (1) the uncertainty is about the univariate regression slope b. In regression (2), which allows regression coefficients to change from the first to the second half of 1963–2019, the uncertainty is about b (the first-half regression slope), and da and db (the changes in the intercept and slope from the first to the second half of the sample). The uncertainty about the true values of these three coefficients is captured by Σ, their covariance matrix. Panel C of Table 3 shows that the standard error of the Y^21–Y^22 estimates from (6) for the three value portfolios, 0.22 (MV), 0.22 (BV), and 0.27 (SV), are roughly five times those from (3), 0.04, 0.05, and 0.05, which assume constant regression coefficients. Differences between SE( Y^11–Y^12 ) from (3) and SE( Y^21–Y^22 ) from (6) are smaller for growth portfolios, but even for the growth portfolios, the standard errors from regression (2) are roughly 50% larger than those from regression (1). The low precision of the coefficient estimates from (2) rules out reliable inferences about changes in average expected value and growth premiums from the first to the second half of the sample. The t-statistics for differences between average premium forecasts for 1963–1991 and 1991–2019 for the three value portfolios (panel C of Table 3) are 1.44 (MV), 1.44 (BV), and 0.90 (SV). For the three growth portfolios, the differences are less than a standard error from zero. There is a different perspective on regression (2). The regression in effect estimates separate coefficients for 1963–1991 and 1991–2019, so the average residual for each of the two half-periods is zero, and the average conditional forecast of the premium for each half-period matches the average realized premium. Regression (2) thus provides a more powerful test for changes in expected premiums than the tests on unconditional average premiums in Table 1 only if regression (2) absorbs some of the variance of monthly premiums. With R2 of 0.00, regression (2) explains virtually none of the monthly variation in R–RM for the three growth portfolios. As a result, the standard errors of the differences in average half-period regression (2) forecasts for the growth portfolios (panel C of Table 3) match the standard errors for differences in unconditional average premiums in Table 1. Matching standard errors and matching differences in average premiums produce matching t-statistics for the average differences. Similarly, regression (2) explains only 5% to 6% of the variance of R–RM for the value portfolios. As a result, the standard errors of differences in half-period average premium forecasts from (2) in panel C of Table 3 are only a bit smaller than the standard errors of differences in average premiums in panel A of Table 1. The premium regression (2) thus produces only slightly improved t-statistics for differences in half-period average premiums. The t-statistics testing whether the average unconditional expected return premiums for the value portfolios fall from 1963–1991 to 1991–2019 are 1.39 for MV, 1.41 for BV, and 0.87 for SV (panel A of Table 1). The t-statistics testing whether, conditional on the observed values of BM–BMM, the average expected premiums produced by regression (2) fall (panel C of Table 3), are only slightly higher, 1.44 for MV, 1.44 for BV, and 0.90 for SV (panel C of Table 3). In short, just as the volatility of average return premiums in panel A of Table 1 prevents us from making reliable inferences about changes in unconditional expected value and growth premiums from the first to the second half of 1963–2019, imprecision in the regression (2) coefficients prevents us from making reliable inferences about changes in conditional expected premiums. 4. Conclusions Our goal is to determine whether expected value premiums—the expected differences between value portfolio returns and the VW market return—decline or perhaps disappear after Fama and French (1992, 1993). The 1963–2019 period used here doubles the 1963–1991 sample of the earlier papers, so we compare results for the first (in-sample) and second (out-of-sample) halves of 1963–2019. The initial tests confirm that realized value premiums fall from the first half of the sample to the second. The average premium for Big Value drops from 0.36% per month (t = 2.91) to a puny 0.05% (t = 0.24). The Small Value average premium is a hefty 0.58% (t = 3.19) for 1963–1991, versus 0.33% (t = 1.52) for 1991–2019. Market Value, which is mostly Big Value, produces a first-half average premium of 0.42% (t = 3.25), declining to 0.11% (t = 0.60) for the second half. The high volatility of monthly value premiums clouds inferences about whether the declines in average premiums reflect changes in expected premiums. Comparing the first and second half-period averages, we do not come close to rejecting the hypothesis that out-of-sample expected premiums are the same as in-sample expected premiums. But the imprecision of the estimates implies that we also can’t reject a wide range of lower values for second half expected premiums. Given its assumption that the intercept and slope are constant, regression (1) says conditional expected value premiums are lower in the second half of 1963–2019. For the three value portfolios, regressions of premiums in excess of Market, R–RM, on book-to-market ratios in excess of the Market, BM–BMM, produce positive slopes more than six standard errors above zero. Since the average BM–BMM for value portfolios are lower in the second half of the sample, the estimates of the constant-slope regression (1) imply lower conditional expected premiums for 1991–2019—if the regressions are well-specified. When we use the split-sample regression (2) to accommodate changes in the regression coefficients in (1), the estimates of (2) say it adds nothing to the explanatory power of (1), and it provides no evidence against the constant coefficient assumption of (1). It seems that (2) sacrifices precision of coefficient estimates with no apparent benefits over the simple alternative provided by regression (1). 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The Review of Asset Pricing Studies – Oxford University Press
Published: Jun 11, 20
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