A. Firm-Level Implied Cost of Capital

Following Pastor, Sinha, and Swaminathan (Reference Pastor, Sinha and Swaminathan2008), Lee, Ng, and Swaminathan (Reference Lee, Ng and Swaminathan2009), and Li, Ng, and Swaminathan (Reference Li, Ng and Swaminathan2013), we define firm-level ICC as the internal rate of return that equates the present value of expected future dividends or free cash flows to the current stock price:Footnote ⁶

(1) $$ {P}_t=\sum \limits_{k=1}^{\infty}\frac{E_t\left({D}_{t+k}\right)}{{\left(1+{r}_e\right)}^k}. $$

We make two key assumptions in implementing the free cash flow model: i) short-run earnings growth rates converge in the long run to the growth rate of the overall economy, and ii) competition will drive economic profits on new investments to zero in the long run, that is, the marginal rate of return on investment (the ROI on the next dollar invested) converges to the cost of capital. As discussed later, these assumptions are used to forecast earnings growth rates and free cash flows during the transition from the short-run to the long-run steady state. We implement equation (1) in two parts: i) the present value of free cash flows up to a terminal period $ t+T $ , and ii) a continuing value that captures free cash flows beyond the terminal period. Free cash flows through year $ t+T $ are estimated as the product of annual earnings forecasts and one minus the plowback rate:

$$ {E}_t\left({FCFE}_{t+k}\right)={FE}_{t+k}\times \left(1-{b}_{t+k}\right), $$

where $ {FE}_{t+k} $ and $ {b}_{t+k} $ are the earnings forecasts and the plowback rate forecasts for year $ t+k $ , respectively.

We forecast earnings up to year $ t+T $ in three stages:

1. i) We explicitly forecast earnings (in dollars) for year $ t+1 $ using analyst forecasts. IBES analysts provide earnings per share (EPS) forecasts for the next two fiscal years, $ {FY}_1 $ and $ {FY}_2 $ , for each firm in the IBES database. We construct a 12-month ahead earnings forecast $ {FE}_1 $ as a weighted average of the median $ {FY}_1 $ and $ {FY}_2 $ forecasts: $ {FE}_1=w\times {FY}_1+\left(1-w\right)\times {FY}_2 $ , where $ w $ is the number of months remaining until the next fiscal year-end divided by 12. We use median forecasts instead of means in order to alleviate the effects of extreme forecasts.

2. ii) We use the growth rate implied by $ {FY}_1 $ and $ {FY}_2 $ forecasts to forecast earnings for year $ t+2 $ . The implied growth is $ {g}_2={FY}_2/{FY}_1-1 $ , and the 2-year-ahead earnings forecast is given by $ {FE}_2={FE}_1\left(1+{g}_2\right) $ . Constructing $ {FE}_1 $ and $ {FE}_2 $ in this way ensures a smooth transition from $ {FY}_1 $ to $ {FY}_2 $ throughout the fiscal year and guarantees that our forecasts are always 12 and 24 months ahead from the current month. To avoid extreme values, we limit growth rates to between 1% and 75%.

3. iii) We forecast earnings from year $ t+3 $ through year $ t+T+1 $ by assuming that the year $ t+2 $ earnings growth rate, $ {g}_2 $ , mean-reverts exponentially to a long-run steady-state growth by year $ t+T+2 $ . Starting in year $ t+T+2 $ , we assume earnings grow at the long-run nominal GDP growth rate, $ g $ , computed as a rolling average of annual nominal GDP growth rates. The EPS growth rates and the resulting EPS forecasts are computed for years $ t+3 $ to $ t+T+1 $ ( $ k=3,\dots, T+1 $ ) using an exponential rate of mean reversion:

(2) $$ {g}_{t+k}={g}_{t+k-1}\times \exp \left[\log \left(g/{g}_2\right)/T\right]\;\mathrm{and} $$

(3) $$ {FE}_{t+k}={FE}_{t+k-1}\times \left(1+{g}_{t+k}\right). $$

The exponential rate of mean reversion corresponds to linear interpolation in logs and results in a faster decay for higher growth rates, consistent with the empirical evidence in Nissim and Penman (Reference Nissim and Penman2001).

We forecast plowback rates using a two-stage approach.

1. i) We explicitly forecast the plowback rate for year $ t+1 $ as one minus the most recent year’s dividend payout ratio. The payout ratio is estimated as actual dividends paid during the most recent fiscal year divided by earnings over the same period.Footnote ⁷ We exclude share repurchases and new equity issues due to the practical challenges associated with assessing their recurrence in future periods.Footnote ⁸ Payout ratios below zero (above one) are capped at zero (one).

2. ii) We assume that the plowback rate in year $ t+1 $ , denoted $ {b}_1 $ , reverts linearly to a steady-state value by year $ t+T+1 $ , computed using the sustainable growth rate formula. In the steady state, the product of the return on new investments and the plowback rate, $ ROE\times b $ , equals the sustainable earnings growth rate, $ g $ . We further impose the condition that $ ROE $ equals the cost of equity, $ {r}_e $ , in the steady state, reflecting the idea that competition will drive return on new investment down to the cost of equity. Substituting $ {r}_e $ for $ ROE $ in the sustainable growth rate formula and solving for the steady-state plowback rate gives $ b=g/{r}_e $ . Intermediate plowback rates from $ t+2 $ to $ t+T $ ( $ k=2,\dots, T $ ) are computed as follows:

(4) $$ {b}_{t+k}={b}_{t+k-1}-\frac{b_1-b}{T}. $$

The terminal value $ TV $ is computed as the present value of a perpetuity equal to the ratio of the year $ t+T+1 $ earnings forecast divided by the cost of equity:

(5) $$ {TV}_{t+T}=\frac{FE_{t+T+1}}{r_e}, $$

where $ {FE}_{t+T+1} $ is the earnings forecast for year $ t+T+1 $ .Footnote ⁹ It is straightforward to show that the constant growth model simplifies to equation (5) when $ ROE $ equals $ {r}_e $ .

Substituting equations (2) to (5) into the infinite horizon free cash flow valuation model in equation (1) provides the following empirically tractable finite horizon model:

(6) $$ {P}_t=\sum \limits_{k=1}^T\frac{FE_{t+k}\times \left(1-{b}_{t+k}\right)}{{\left(1+{r}_e\right)}^k}+\frac{FE_{t+T+1}}{r_e{\left(1+{r}_e\right)}^T}. $$

Following Li et al. (Reference Li, Ng and Swaminathan2013), we use a $ 15 $ -year horizon $ \left(T=15\right) $ to implement the model in (6) and compute $ {r}_e $ as the rate of return that equates the present value of free cash flows to the current stock price. The resulting $ {r}_e $ is the firm-level ICC measure used in our empirical analysis. To reduce the influence of outliers, we trim the firm-level ICC by removing the top and bottom $ 0.5\% $ in each month.

B. Value and Growth Portfolios and Realized Value Premium

For each month $ t $ from January 1977 to December 2023, all NYSE stocks in CRSP are ranked by size (market capitalization). The median NYSE market cap is used to split NYSE, Amex, and NASDAQ stocks into two size portfolios: small (S) and big (B). Stocks are also sorted into three book-to-market (B/M) portfolios using NYSE breakpoints: the bottom 30% form the low B/M group (L), the middle 40% the medium group (M), and the top 30% the high B/M group (H). Book equity is calculated as stockholders’ equity plus balance sheet-deferred taxes and investment tax credits, minus the book value of preferred stock. Depending on data availability, we use redemption value, liquidation value, or par value, in that order, to represent preferred stock. Book-to-market equity (B/M) is then computed as the most recent Compustat book equity, lagged by 3 months to ensure public availability, divided by market equity at the end of month $ t $ . Following Fama and French (Reference Fama and French1993), we exclude firms with negative book equity when calculating breakpoints and when forming portfolios. Intersecting the two size groups and three B/M groups yields six size–B/M portfolios: S/L, S/M, S/H, B/L, B/M, and B/H. We define the value portfolio (H) as the equal-weighted average of S/H and B/H, $ \left(\mathrm{S}/\mathrm{H}+\mathrm{B}/\mathrm{H}\right)/2 $ , and the growth portfolio (L) as the equal-weighted average of S/L and B/L, $ \left(\mathrm{S}/\mathrm{L}+\mathrm{B}/\mathrm{L}\right)/2 $ . The value premium, HML, is the difference in returns between $ \left(\mathrm{S}/\mathrm{H}+\mathrm{B}/\mathrm{H}\right)/2 $ and $ \left(\mathrm{S}/\mathrm{L}+\mathrm{B}/\mathrm{L}\right)/2 $ .

Although B/M is the most commonly used measure to define value and growth stocks in the academic literature, it has several well-documented limitations. Since at least 2015, the literature has raised concerns about the traditional book-to-market ratio (see Asness et al. (Reference Asness, Frazzini, Israel and Moskowitz2015), Fama and French (Reference Fama and French2015), Eisfeldt, Kim, and Papanikolaou (Reference Eisfeldt, Kim and Papanikolaou2022), Park (Reference Park2022), and Gonçalves and Leonard (Reference Gonçalves and Leonard2023)), citing factors such as rising amounts of intangible assets and changes in accounting standards. Gonçalves and Leonard (Reference Gonçalves and Leonard2023) propose a fundamental equity-to-price ratio as a potentially better proxy than B/M. Motivated by these limitations, we introduce two alternative measures of value in addition to B/M.

The first measure is a composite value rank combining B/M, the cash flow-to-price ratio (C/P), and two earnings-to-price ratios: FE $ {}_1 $ /P and FE $ {}_2 $ /P based on 1- and 2-year-ahead earnings forecasts, respectively. Following Lakonishok et al. (Reference Lakonishok, Shleifer and Vishny1994), C/P is defined as net income before extraordinary items plus depreciation and amortization from the most recent Compustat data (lagged by 3 months), divided by market equity at month t. For each measure, firms are ranked monthly from 0 (most expensive) to 1 (most value-like). The composite rank is calculated as a weighted average:

$$ \frac{1}{3} RnkB/M+\frac{1}{3}\left(\frac{1}{2}{RnkFE}_1/P+\frac{1}{2}{RnkFE}_2/P\right)+\frac{1}{3} RnkC/P, $$

where $ RnkB/M $ , $ {RnkFE}_1/P $ , $ {RnkFE}_2/P $ , and $ RnkC/P $ are the individual ranks of corresponding value measures.Footnote ¹⁰ Each month from January 1977 to December 2023, we form value and growth portfolios based on a two-way sort using size and the composite value rank. The sorting procedure follows the same portfolio construction steps as with the B/M-based portfolios: (S/H + B/H)/2 is the value portfolio (H), (S/L + B/L)/2 is the growth portfolio (L), and HML = (S/H + B/H)/2 –(S/L + B/L)/2.

Finally, we use a value measure based on operating cash flows, motivated by Ball et al. (Reference Ball, Gerakos, Linnainmaa and Nikolaev2015), (Reference Ball, Gerakos, Linnainmaa and Nikolaev2016) and Hou et al. (Reference Hou, Mo, Xue and Zhang2024), who show that the operating cash flow-to-market ratio (OCF/M) is a superior measure of value compared to B/M. For nonfinancial firms, OCF is defined as EBITDA plus R&D expenses minus the change in working capital. We further require that (EBITDA + R&D) be positive for the operating cash flow-to-market measure to prevent loss firms from distorting the portfolio rankings. Similar to B/M, OCF/M is calculated using the most recent Compustat data (lagged by 3 months), divided by market equity at the end of month $ t $ . High OCF/M stocks are classified as value stocks, while low OCF/M stocks are considered growth stocks. Each month from January 1977 to December 2023, we form value and growth portfolios based on a two-way sort using size and OCF/M.

We construct three HML factors within our sample, in addition to the Fama and French HML factor, yielding four HML factors in total for our empirical tests:

1. i) HML(FF), the Fama and French HML based on B/M ratios from Ken French’s data library (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html).

2. ii) HML(B/M), also based on B/M ratios but constructed using only the firms in our sample.

3. iii) HML(Comp), based on the composite value measure, again using only the firms in our sample.

4. iv) HML(OCF/M), based on the OCF/M measure, again using only the firms in our sample.

C. Implied Value Premium and Value Spread

We construct the implied value premium (IVP) as follows: Each month, we form value and growth portfolios using the latest B/M ratio, the composite value rank, or the OCF/M ratio. We compute the ICCs of S/L, B/L, S/H, and B/H by value weighting the ICCs of constituent firms using month-end market capitalizations. The ICC for the H portfolio is the simple average of S/H and B/H, and the ICC for the L portfolio is the average of S/L and B/L. The implied value premiums are then:

$$ {\displaystyle \begin{array}{c} IVP{\left(B/M\right)}_t= ICCH{\left(B/M\right)}_t- ICCL{\left(B/M\right)}_t,\\ {} IVP{(Comp)}_t= ICCH{(Comp)}_t- ICCL{(Comp)}_t,\\ {} IVP{\left( OCF/M\right)}_t= ICCH{\left( OCF/M\right)}_t- ICCL{\left( OCF/M\right)}_t,\end{array}} $$

where $ ICCH $ is the ICC for the value portfolio (H) and $ ICCL $ is the ICC for the growth portfolio (L).

An important control variable in our regression analysis is the value spread (VS) (e.g., Asness et al. (Reference Asness, Friedman, Krail and Liew2000), Cohen et al. (Reference Cohen, Polk and Vuolteenaho2003)). For value and growth portfolios based on B/M and composite value rank, VS is defined as the difference in the log book-to-market ratios of value and growth portfolios. The book-to-market ratio of the value portfolio is computed as the simple average of the value-weighted ratios of S/H and B/H, while the growth portfolio ratio is computed analogously using S/L and B/L. The value spreads based on B/M and the composite value rank are denoted as VS(B/M) and VS(Comp), respectively:

$$ {VS}_t= logB/M{(H)}_t- logB/M{(L)}_t. $$

For OCF/M, since operating cash flows can be negative, we construct a scaled value spread by dividing the raw value spread by the cross-sectional median absolute deviation (MAD). Thus, the OCF/M-based value spread is

$$ {VS}_t=\left( OCF/M{(H)}_t- OCF/M{(L)}_t\right)/ MAD. $$

A. Data

We obtain market capitalization and return data from CRSP; accounting data, including common dividends, net income, EBITDA, book value of common equity, depreciation and amortization, working capital items, and fiscal year-end date, from COMPUSTAT; and analyst earnings forecasts and share prices from IBES. To ensure that we use only publicly available information, we obtain accounting data items for the most recent fiscal year ending at least 3 months prior to the month in which the ICC is computed. Data on nominal GDP growth rates are obtained from the Bureau of Economic Analysis. Our GDP data begin in 1930. Each year, we compute the steady-state GDP growth rate as the historical average of GDP growth rates using annual data up to that year.

The control variables used in the forecasting regressions include the business cycle variables term spread (Term) and default spread (Default). The term spread is defined as the difference between Moody’s AAA bond yield and the 3-month T-bill rate, and represents the slope of the Treasury yield curve. The default spread is the difference in yields between BAA- and AAA-rated corporate bonds. All data are obtained from the economic research database at the Federal Reserve Bank of St. Louis (FRED) (https://fred.stlouisfed.org/).

B. Summary Statistics

Table 1 presents summary statistics for the variables used in this article. Panel A reports the implied value premiums based on B/M (IVP(B/M)), the composite measure (IVP(Comp)), and OCF/M (IVP(OCF/M)) for the full sample period (January 1977–December 2023) and for two subperiods: January 1977–December 2007 and January 2008–December 2023. Over the full sample, the means of IVP(B/M), IVP(Comp), and IVP(OCF/M) are 3.44%, 4.28%, and 2.99%, respectively. Across the two subperiods, the means of all three measures declined. For example, the mean of IVP(B/M) fell from 3.81% to 2.70%, IVP(Comp) decreased from 4.49% to 3.87%, and IVP(OCF/M) decreased from 3.07% to 2.85%.

TABLE 1 Summary Statistics

Panel B reports summary statistics for the four HML measures. Our in-sample HML(B/M) closely tracks the standard HML(FF), with mean annualized returns of 2.78% and 2.62%, respectively. By contrast, HML(Comp) delivers a stronger premium of 4.39%, while HML(OCF/M) is highest at 5.43%, consistent with Hou et al. (Reference Hou, Mo, Xue and Zhang2024). The four measures are highly correlated (0.73–0.93 in Panel D). Except for HML(OCF/M), all exhibit negative long-horizon autocorrelations, most pronounced at the 3-year horizon, motivating our use of the smoothed HML as a control.

The ex post value premium declined sharply across all four measures when comparing the pre-2007 and post-2007 periods. From 1977 to 2007, the average annualized returns for HML(B/M), HML(Comp), HML(OCF/M), and HML(FF) were 3.79%, 5.92%, 7.09%, and 4.54%, respectively. In contrast, over 2008–2023 the corresponding values dropped to 0.82%, 1.43%, 2.21%, and −1.09%. A nonparametric perspective confirms this pattern: the proportion of months in which HML was positive fell from 55.38% to 47.92% for HML(B/M), from 57.80% to 45.31% for HML(Comp), from 58.60% to 51.56% for HML(OCF/M), and from 55.65% to 43.75% for HML(FF). While HML(OCF/M) continues to perform best post-2007, its average return of 2.21% is far below its pre-2007 level of 7.09%, underscoring that the weak post-2007 value premium is not simply a B/M phenomenon.

The average implied value premium in Panel A is of similar magnitude to the ex post value premium in Panel B over the full sample. The means of IVP(B/M), IVP(Comp), and IVP(OCF/M) are 3.44%, 4.28%, and 2.99%, respectively, comparable to the means of the four HML factors, which range from 2.62% to 5.43%. The implied value premium is also highly persistent, with first-order autocorrelations of 0.90 for IVP(B/M), 0.89 for IVP(Comp), and 0.91 for IVP(OCF/M). Unit root tests strongly reject the null of non-stationarity for all IVP measures. Panel C reports summary statistics for the value spread and the business cycle variables. With the exception of VS(OCF/M), all variables exhibit first-order autocorrelations between 0.95 and 0.98, indicating much higher persistence than IVP.

Panel D shows that all three measures of IVP exhibit modest positive correlations with the business cycle variables, suggesting that some of the time variation in the implied value premium may be related to the business cycle. This supports the view that the underperformance of value strategies, and the time variation in IVP, may be linked to changing macroeconomic conditions. Figures 2–4 plot the time series of IVP(B/M), IVP(Comp), and IVP(OCF/M), with NBER recessions shaded. All three exhibit similar time variation and mean reversion: the implied value premium was high in January 2000, low in June 2007, and high again in March 2009 and March 2020.

FIGURE 2 Implied Value Premium IVP(B/M) (January 1977–December 2023)

Figure 2 plots the implied value premium based on B/M ratios, IVP(B/M), expressed in annualized percentages. The three lines surrounding the time-series correspond to the rolling median and the + or – 2-standard deviation bounds calculated using a rolling average up to that month, starting from January 1987. The shaded areas indicate the NBER recession periods.

FIGURE 3 Implied Value Premium IVP(Comp) (January 1977–December 2023)

Figure 3 plots the implied value premium based on the composite value rank, IVP(Comp), expressed in annualized percentages. The three lines surrounding the time-series correspond to the rolling median and the + or – 2-standard deviation bounds calculated using a rolling average up to that month, starting from January 1987. The shaded areas indicate the NBER recession periods.

FIGURE 4 Implied Value Premium IVP(OCF/M) (January 1977–December 2023)

Figure 4 plots the implied value premium based on OCF/M ratios, IVP(OCF/M), expressed in annualized percentages. The three lines surrounding the time-series correspond to the rolling median and the + or – 2-standard deviation bounds calculated using a rolling average up to that month, starting from January 1987. The shaded areas indicate the NBER recession periods.

A. Univariate Regressions

We examine the univariate predictive power of IVP for HML based on the following multiperiod forecasting regression:

(7) $$ \frac{Y_{t,t+K}}{K}=a+b\times {X}_t+{u}_{t,t+K}, $$

where $ b $ is the slope coefficient, $ K $ is the forecasting horizon in months or quarters, and $ {u}_{t,t+K} $ is the regression residual. $ {Y}_{t,t+K} $ is the cumulative return from $ t $ to $ t+K $ , which is computed based on either the Fama–French HML factor, HML(FF), or our constructed HML factors: HML(B/M), HML(Comp), or HML(OCF/M). If a stock is delisted, its cumulative return is computed using the corresponding value-weighted market return. $ {X}_t $ represents the implied value premium, IVP(B/M), IVP(Comp), or IVP(OCF/M), the value spread, the business cycle variables, or the smoothed HML.Footnote ¹¹

We estimate the forecasting regression for multiple horizons: $ K=6 $ , $ 12 $ , $ 24 $ , and $ 36 $ months for monthly regressions, and $ K=2 $ , $ 4 $ , $ 8 $ , and $ 12 $ quarters for quarterly regressions. We use the generalized method of moments (GMM) standard errors with the $ K-1 $ Newey–West lag correction (Newey and West (Reference Newey and West1987)) to correct for both autocorrelation and heteroscedasticity. We refer to the resulting statistic as the $ Z $ statistic. While GMM standard errors consistently estimate the asymptotic variance–covariance matrix, Richardson and Smith (Reference Richardson and Smith1991) show these standard errors are biased in small samples due to the sampling variation in estimating the autocovariances. To address this issue, we generate small sample distributions of the test statistics using Monte Carlo simulations (see Hodrick (Reference Hodrick1992), Nelson and Kim (Reference Nelson and Kim1993), Swaminathan (Reference Swaminathan1996) and Lee, Myers, and Swaminathan (Reference Lee, Myers and Swaminathan1999)). The Appendix describes the Monte Carlo simulation methodology.

Since the forecasting regressions use overlapping data across horizons, the regression slopes are correlated. Richardson and Stock (Reference Richardson and Stock1989) propose a joint test based on the average slope coefficient to test the null hypothesis that the slopes across horizons are jointly zero. Following their methodology, we compute the average slope statistic as the arithmetic average of regression slope coefficients at different horizons, and conduct Monte Carlo simulations to assess its statistical significance.

If the implied value premium is a good proxy for the expected value premium, it should positively predict HML. Accordingly, the slope coefficients associated with IVP(B/M), IVP(Comp), and IVP(OCF/M) in (7) should be positive. We also expect a positive sign for the value spread, as Asness et al. (Reference Asness, Friedman, Krail and Liew2000) and Cohen et al. (Reference Cohen, Polk and Vuolteenaho2003) find that it positively predicts the ex post value premium. If the value premium is countercyclical, as argued in risk-based theories, then business cycle variables might also positively predict future realized value premium. Due to mean reversion in HML, we expect the smoothed HML to negatively predict future value premium. Therefore, a one-sided test of the null hypothesis is appropriate for all forecasting variables.

Table 2 presents univariate regression results for the implied value premium, IVP(B/M), IVP(Comp), and IVP(OCF/M), the value spread, VS(B/M), VS(Comp), and VS(OCF/M), the business cycle variables, and the smoothed HML. Panel A presents results for predicting HML(FF), while Panels B, C, and D present results for predicting HML(B/M), HML(Comp), and HML(OCF/M), respectively. We include Panels B–D to show that our results are robust to using value factors constructed from a smaller sample. In Panels B–D, we omit the business cycle variables and the smoothed HML results to save space and avoid repetition.

TABLE 2 Univariate Regressions Predicting Future Realized Value Premium

The regression results provide strong evidence that the implied value premium predicts future realized value premium. The slope coefficients of IVP(B/M), IVP(Comp), and IVP(OCF/M) are uniformly positive and statistically significant at the 1% level (based on simulated $ p $ values) at every horizon. Not surprisingly, the average slope statistics are also strongly significant at the 1% level or better. The adjusted $ R $ ²s are also high: for example, in Panel A, the adjusted $ R $ ² of IVP(B/M) is 14% at the 12-month horizon and 36% at the 36-month horizon. Results in Panels B–D yield adjusted $ R $ ²s of similar magnitude.

The results are also economically significant. At the 12-month horizon, a 1-standard-deviation increase in IVP(B/M) ( $ 1.97\% $ ) translates into an annualized increase of about $ 5.36\% $ ( $ 1.97\%\times 2.72 $ ) for HML(FF) (Panel A) and $ 5.87\% $ ( $ 1.97\%\times 2.98 $ ) for HML(B/M) (Panel B). Similarly, a 1-standard-deviation increase in IVP(Comp) ( $ 2.10\% $ ) leads to an annualized increase of $ 5.71\% $ ( $ 2.10\%\times 2.72 $ ) for HML(Comp) (Panel C), while a 1-standard-deviation increase in IVP(OCF/M) ( $ 2.07\% $ ) results in an annualized increase of $ 5.24\% $ ( $ 2.07\%\times 2.53 $ ) for HML(OCF/M) (Panel D).

Neither VS nor the business cycle variables reliably predict HML(FF), and the estimated slope coefficients on the business cycle variables display inconsistent signs. The smoothed HML exhibits the expected negative signs but is not statistically significant. Overall, the implied value premium remains the strongest predictor of the ex post value premium in univariate regressions.

B. IVP Predictability Controlling for Other Predictors

In this section, we examine whether the implied value premium continues to predict ex post value premium in regressions that include the value spread, the business cycle variables, and the smoothed HML. Table 3 presents the regression results. The dependent variable is HML(FF) in Panel A, HML(B/M) in Panel B, HML(Comp) in Panel C, and HML(OCF/M) in Panel D.

TABLE 3 Regressions of Realized Value Premium on IVP and Other Predictors

The results show that the implied value premium strongly predicts future realized value premium, even after controlling for the value spread, business cycle variables, and the smoothed HML. In all four panels, IVP has positive slope coefficients that are statistically significant at every horizon. The average slope statistics are all significant at the 1% level or better. By contrast, none of the other variables exhibit any consistent predictive power. The unavoidable conclusion is that the implied value premium is the most robust predictor of the ex post value premium.

The underperformance of value stocks since 2007 coincides with a period of historically low global interest rates, raising the possibility that value’s relative performance may depend on the interest rate environment. To test this, we examine the predictive power of IVP controlling for long-term and short-term real interest rates. Monthly inflation is computed as the rolling 12-month average of annualized changes in the Consumer Price Index for All Urban Consumers. The short-term real interest rate (Real Tbill) is calculated as the difference between the 3-month Treasury bill rate and inflation, and the long-term real interest rate (Real Yield) is the difference between the 10-year Treasury constant maturity rate and inflation. All data are obtained from FRED. Panel E of Table 3 shows that IVP remains a strong predictor of realized value premium, whereas neither interest rate is statistically significant.

In Table A1 in the Supplementary Material, we conduct a variety of robustness checks on our implied value premium measures and confirm that the predictive power of IVP for future realized value premium remains strong and significant. For instance, we demonstrate that our results are robust to using alternative steady-state earnings growth rates and plowback rates in constructing firm-level ICCs. Our IVP measures are based on value and growth portfolios formed monthly using the most recent data (e.g., B/M, C/P). Additionally, we show that our findings hold when using IVP constructed from value and growth portfolios following Fama and French (Reference Fama and French1993), where portfolios are formed annually instead of monthly. Furthermore, we confirm that IVP derived from small-cap firms predicts the small-cap value premium, while IVP from large-cap firms predicts the large-cap value premium. Finally, we demonstrate that time variations in relative analyst forecast bias between value and growth stocks do not account for our findings.

C. Sources of Predictability

In this section, we examine the sources of the strong predictive power of the implied value premium (IVP). Recent studies by Hou et al. (Reference Hou, Xue and Zhang2015) and Zhang (Reference Zhang2017) emphasize the close link between the investment premium and the value premium. We first analyze this relationship through the lens of the investment CAPM. Second, because earnings announcements convey important information about firms’ fundamental values, we investigate whether IVP predicts the cumulative return differences between value and growth stocks around these announcements.

1. IVP and the Investment Premium

Zhang (Reference Zhang2017) shows that within the Investment CAPM framework, the value premium can be explained by the investment premium, defined as the return spread between low- and high-investment firms. The investment premium arises because, holding profitability constant, firms with lower costs of capital face higher marginal q, invest more, and subsequently earn lower average returns. In the same framework, the value premium arises because value firms have lower marginal q, fewer growth opportunities, and invest less, which leads them to earn higher expected returns than growth firms. Thus, the value premium originates from firm investment behavior rather than mispricing. In this subsection, we examine whether the implied value premium (IVP) predicts the investment premium, proxied by the investment factor from the q-factor model.

We obtain the investment factor data from the q-factor website (https://global-q.org/factors.html). Table 4 reports summary statistics and correlations for the investment premium. Overall, the investment premium is strongly positive, with an average annualized return of 3.68% over the full sample period from January 1977 to December 2023. Similar to HML, the investment premium has declined in the post-2007 period, falling from 5.06% in the early sample (1977–2007) to just 0.99% in the later period (2008–2023). The fraction of months with positive investment premium also declined, from 59.14% to 45.31%. Panel B of Table 4 further shows that the investment factor is highly correlated with realized value premium, with the correlations between various HML measures and the investment factor ranging from 0.43 to 0.64. The strong positive correlation suggests that value and investment premia may be driven by similar sources of return variation. To investigate this link more directly, we test whether the implied value premium predicts the investment factor, with results reported in Table 5.

TABLE 4 Summary Statistics of Investment Premium

TABLE 5 Regressions of Investment Premium on IVP and Other Predictors

Panel A of Table 5 shows that all measures of IVP strongly predict the investment premium in univariate regressions. The slope coefficients are statistically significant at the 1% or 5% level across forecasting horizons, with adjusted R $ {}^2 $ values ranging from 9% to 16% at the 12-month horizon. The economic significance is also substantial: at the 12-month horizon, a 1-standard-deviation increase in IVP(OCF/M) corresponds to a 3.44% ( $ 2.07\%\times 1.66 $ ) increase in the investment premium. Panel B further shows that all three IVP measures remain strong predictors after controlling for value spreads, business cycle variables, and a smoothed investment factor measured as the past 3-year average of the monthly investment premium. These results suggest that the value and investment premia share common sources of time variation, consistent with the Investment CAPM interpretation of the value premium.

2. Predicting Price Reactions Around Quarterly Earnings Announcements

La Porta et al. (Reference La Porta, Lakonishok, Shleifer and Vishny1997) find that value stocks earn positive abnormal returns, while growth stocks earn negative abnormal returns in the days surrounding their future quarterly earnings announcements. We extend this test to a time-series context. For each stock in each quarter, we compute cumulative (market-adjusted) abnormal returns (CAR) from day −1 to +1 around its quarterly earnings announcement. We then average the CAR over the next $ K $ quarters (where $ K=\mathrm{2,4,8,12} $ ). CAR(HML) is calculated by subtracting the average CAR of the growth portfolio, CAR(L), from that of the value portfolio, CAR(H). This difference captures the relative price reaction to earnings surprises between value and growth stocks.

We consider four measures of CAR(HML) corresponding to the four measures of HML discussed earlier: i) CAR(HML(FF)) for the Fama and French value and growth portfolios, ii) CAR(HML(B/M)) for value and growth portfolios formed using B/M ratios within our sample, iii) CAR(HML(Comp)) for portfolios formed using the composite value rank, and iv) CAR(HML(OCF/M)) for portfolios formed using OCF/M. Quarterly earnings announcement dates are obtained from the quarterly COMPUSTAT file, while daily stock and market returns are obtained from the daily CRSP files. We use the WRDS value-weighted market return with dividends as the market return. Quarterly values of the implied value premium and other forecasting variables are as of the end of each quarter. Unreported results are similar when quarterly values are computed as the average of monthly observations within the quarter. The sample period spans 1977:Q1 to 2023:Q4.

To save space, the univariate regression results are reported in Table A2 in the Supplementary Material. Table 6 presents regression results controlling for other forecasting variables. The results show that the implied value premium strongly predicts CAR(HML). For all measures of IVP, the coefficients are statistically significant at every horizon, with the average slope significant at the 1% level.

TABLE 6 Predicting Cumulative Abnormal Returns Around Earnings Announcements with IVP and Other Predictors

What explains the CAR results? Under the mispricing scenario, a high IVP, indicating that value stocks are more undervalued relative to growth stocks due to investors’ pessimistic expectations about future profitability, should predict a high CAR(HML), reflecting larger positive earnings surprises for value stocks than for growth stocks. Thus, the finding that IVP positively predicts future CAR(HML) is consistent with this mispricing interpretation. Because CAR measures returns over only a few days, neither risk nor the price of risk is likely to change meaningfully over this period. Consequently, these results are difficult to reconcile with rational asset pricing models, though we cannot entirely rule it out.

The investment CAPM predicts firm characteristics should explain the cross section of average stock returns and is able to explain a number of anomalies including accruals (Wu, Zhang, and Zhang (Reference Wu, Zhang and Zhang2010)), price and earnings momentum (Liu and Zhang (Reference Liu and Zhang2014)), and others (Lin and Zhang (Reference Lin and Zhang2013), Zhang (Reference Zhang2017)). In this framework, owning a stock is economically equivalent to owning the firm’s investment, so stock returns theoretically equal fundamental returns. A positive (negative) earnings surprise implies a higher (lower) fundamental return and, hence, a higher (lower) stock return, providing a direct profitability channel from earnings surprises to stock returns. Our results show that a widening IVP is associated with higher CAR(HML), which is consistent with value stocks experiencing more positive earnings surprises than growth stocks. Why this pattern arises, and whether rational asset pricing, including both the Investment CAPM and the Consumption CAPM, can account for it, remains an open question for future research.

D. Out-of-Sample Analysis

So far, we have shown strong in-sample evidence that the implied value premium is a powerful predictor of the ex post value premium. In this section, we evaluate its out-of-sample predictive power, an issue that has received considerable attention in the literature (e.g., Spiegel (Reference Spiegel2008), Welch and Goyal (Reference Welch and Goyal2008)).

1. Econometric Specification and Forecast Evaluation

Consider the following predictive regression:

(8) $$ {r}_{t+1}={\alpha}_i+{\beta}_i{x}_{i,t}+{\varepsilon}_{i,t+1}, $$

where $ {r}_{t+1} $ is HML(FF), HML(Comp), HML(OCF/M), or the investment premium (InvPrem) at month $ t+1 $ , $ {x}_{i,t} $ is the $ i $ th monthly predictive variable, which includes the implied value premium IVP(B/M), IVP(Comp), or IVP(OCF/M), as well as other variables: the value spread VS(B/M), VS(Comp), or VS(OCF/M); the term spread Term; the default spread Default; and the smoothed factor (smoothed HML or smoothed InvPrem). $ {\varepsilon}_{i,t+1} $ is the error term.

Following Welch and Goyal (Reference Welch and Goyal2008), we use a recursive method to estimate (8) and generate out-of-sample forecasts of the value premium. A similar procedure is applied to forecasting the investment premium. Specifically, we divide the full sample $ T $ into two periods: an estimation period with the first $ m $ observations and an out-of-sample forecast period with the remaining $ q=T-m $ observations.

We begin by using the first $ m $ observations to estimate (8) and obtain the OLS estimators $ {\hat{\alpha}}_{i,m} $ and $ {\hat{\beta}}_{i,m} $ , which gives us the first out-of-sample forecast:

$$ {\hat{r}}_{i,m+1}={\hat{\alpha}}_{i,m}+{\hat{\beta}}_{i,m}{x}_{i,m}. $$

We then re-estimate the model using $ m+1 $ observations to obtain $ {\hat{\alpha}}_{i,m+1} $ and $ {\hat{\beta}}_{i,m+1} $ to generate the second out-of-sample forecast:

$$ {\hat{r}}_{i,m+2}={\hat{\alpha}}_{i,m+1}+{\hat{\beta}}_{i,m+1}{x}_{i,m+1}. $$

Proceeding in this manner through the end of the sample, for each predictive variable $ {x}_i $ , we obtain a time series of predicted value premium $ {\left\{{\hat{r}}_{i,t+1}\right\}}_{t=m}^{T-1} $ . We use the historical average realized value premium returns $ {\overline{r}}_{t+1}={\sum}_{j=1}^t{r}_j $ as a benchmark forecasting model (e.g., Campbell and Thompson (Reference Campbell and Thompson2008), Welch and Goyal (Reference Welch and Goyal2008), and Rapach, Strauss, and Zhou (Reference Rapach, Strauss and Zhou2010)). To compare the performance of different predictors, we use the out-of-sample $ {R}^2 $ statistic, $ {R}_{os}^2 $ , which measures the reduction in mean squared error (MSPE) achieved by using the predictive regression (8) with a given predictor, relative to the historical average forecast:

$$ {R}_{os}^2=1-\frac{\sum_{k=1}^q{\left({r}_{m+k}-{\hat{r}}_{i,m+k}\right)}^2}{\sum_{k=1}^q{\left({r}_{m+k}-{\overline{r}}_{m+k}\right)}^2}. $$

A positive $ {R}_{os}^2 $ indicates that the predictive variable improves forecast accuracy relative to the historical average. A predictive variable that has a higher $ {R}_{os}^2 $ performs better in the out-of-sample forecasting test. We formally test the null of $ {R}_{os}^2\le 0 $ against the alternative of $ {R}_{os}^2>0 $ by using the adjusted MSPE statistic of Clark and West (Reference Clark and West2007).Footnote ¹²

Finally, we explore the information content of IVP relative to other forecasting variables by conducting a forecast encompassing test developed by Harvey, Leybourne, and Newbold (Reference Harvey, Leybourne and Newbold1998) (see also Rapach et al. (Reference Rapach, Strauss and Zhou2010)). The null hypothesis is that the model $ i $ forecast encompasses the model $ j $ forecast against the one-sided alternative that the model $ i $ forecast does not encompass the model $ j $ forecast.Footnote ¹³

2. Out-of-Sample Forecasting Results

We focus on the forecast period from December 2000 to December 2023, which represents roughly half of the full sample. In Table A3 in the Supplementary Material, we also consider an alternative forecast period from December 1993 to December 2023, and the results remain robust under this specification. The 12-month moving averages of IVP(B/M), IVP(Comp), and IVP(OCF/M) are used as out-of-sample predictors. We examine four forecasting scenarios: i) $ Y= HML(FF) $ , $ IVP= IVP\left(B/M\right) $ , $ VS= VS\left(B/M\right) $ ; ii) $ Y= HML(Comp) $ , $ IVP= IVP(Comp) $ , $ VS= VS(Comp) $ ; iii) $ Y= HML\left( OCF/M\right) $ , $ IVP= IVP\left( OCF/M\right) $ , $ VS= VS\left( OCF/M\right) $ ; and iv) $ Y= InvPrem $ , $ IVP= IVP\left( OCF/M\right) $ , $ VS= VS\left( OCF/M\right) $ .

Panel A of Table 7 reports the $ {R}_{os}^2 $ test results. We find that all three implied value premium measures are the strongest out-of-sample predictors of HML. Specifically, when predicting HML(FF), IVP(B/M) yields an $ {R}_{os}^2 $ of 2.11%; when predicting HML(Comp), IVP(Comp) yields 2.85%; and when predicting HML(OCF/M), IVP(OCF/M) yields 2.27%. In addition, IVP(OCF/M) achieves an $ {R}_{os}^2 $ of 2.58% when predicting the investment premium. All estimates are statistically significant at the 1% level. Campbell and Thompson (Reference Campbell and Thompson2008) argue that for monthly data, even $ {R}_{os}^2 $ values as low as 0.5% can indicate an economically meaningful degree of return predictability for a mean–variance investor, which provides a simple assessment of predictability in practice. Against this benchmark, the out-of-sample forecasting performance of the implied value premium is quite impressive. Among the other predictors, the term spread and default spread produce positive $ {R}_{os}^2 $ values, but neither is statistically significant. All remaining variables generate negative out-of-sample $ {R}^2 $ values, indicating that they fail to outperform the naïve historical average benchmark.

TABLE 7 Out-of-Sample Analysis

We further examine whether IVP(B/M), IVP(Comp) and IVP(OCF/M) contain distinct information relative to existing variables such as the value spread. The forecast encompassing test results of Harvey et al. (Reference Harvey, Leybourne and Newbold1998), presented in Panel B of Table 7, provide strong support for this view. We strongly reject the null hypothesis that IVP(B/M), IVP(Comp) or IVP(OCF/M) is encompassed by any other predictor at the 1% (or 5%) level. In contrast, we cannot reject the null that IVP(B/M), IVP(Comp), or IVP(OCF/M) encompasses the other variables, indicating that IVP contains incremental information not captured by existing predictors.