TakeawaySmall and value stocks require a smaller hedging premium than large and growth stocks, and that difference, rather than cash-flow beta, is what the modern data supports.

I replicate the return decompositions of Campbell and Vuolteenaho (2004) and Campbell, Giglio, Polk, and Turley (2018), which explained the size and value “anomalies” in stock returns. Because cash-flow and discount-rate news co-move in the modern period, I find little evidence that value stocks have significantly higher cash-flow betas than growth stocks; news about expected future market volatility, however, effectively disentangles the size-and-value puzzle. Small and value stocks require a smaller hedging premium than large and growth stocks. Their methodology also shows some ability to price the abnormal returns of popular equity strategies and currency carry trade premiums.

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Introduction

It is well known that the CAPM, with the value-weighted S&P 500 return as the market proxy, fails to explain average stock returns in the modern period (after 1963). In Sharpe (1964) and Lintner (1965), a stock’s beta with a single total-wealth portfolio summarizes its return characteristics. However, Fama and French (1995) documented the size and value anomalies and showed that portfolios tilted toward small and value stocks tend to earn excess returns. Campbell and Vuolteenaho (2004) solved Merton’s (1973) intertemporal capital asset pricing model (ICAPM) for a representative agent with Epstein and Zin (1989) utility and offered an elegant explanation: they split beta into a “bad” beta on cash-flow news and a “good” beta on discount-rate news. They found evidence that value and small stocks have considerably higher cash-flow betas than growth and large stocks. In other words, for a conservative long-term investor under the ICAPM, small and value stocks are intertemporal hedges that perform well when investment opportunities deteriorate; however, in equilibrium, these assets should have delivered lower average returns, so rational long-term investors should not tilt their portfolios toward small and value stocks. In a subsequent paper, Campbell et al. (2018) argued that either rising stock-return volatility or falling expected stock returns can worsen investment opportunities in the stock market. Adding low-frequency movements in equity volatility, they proposed a three-beta ICAPM with cash-flow, discount-rate, and risk news to explain the cross-section of asset returns, with results consistent with “Bad Beta, Good Beta.”

I examine whether the two-beta ICAPM of Campbell and Vuolteenaho (2004) and the three-beta ICAPM of Campbell et al. (2018) still explain the size and value puzzle in the latest modern data (1963:07-2018:12). For both vector-autoregressive (VAR) economies, I estimated the parameters on the full sample (1929:06-2018:12). I first describe both models and the data, then compare my estimates for the 25 ME- and BE/ME-sorted portfolios from Ken French’s website with both Campbell papers, and finally test both models out of sample on popular equity and currency strategies. I find that, because cash-flow and discount-rate news co-move in the modern data, cash-flow betas no longer separate value from growth stocks, but betas on news about market volatility do: small and value stocks require a smaller hedging premium than large and growth stocks.

Model

Following Campbell and Shiller (1988), Campbell and Vuolteenaho (2004) used the log-linear approximate return decomposition of Campbell (1993):

\[r_{t+1} - E_tr_{t+1} = (E_{t+1} - E_t)\sum_{j=0}^{\infty}\rho^j \Delta d_{t+1+j} - (E_{t+1} - E_t)\sum_{j=0}^{\infty}\rho^j \Delta r_{t+1+j} = N_{CF, t+1} - N_{DR, t+1}\]

where \(r_{t+1}\) is a log stock return, \(d_{t+1}\) is the log dividend paid by the stock, \(\Delta\) denotes a one-period change, \(E_t\) denotes a rational expectation at time \(t\), and \(\rho\) is a discount coefficient. I use \(\rho = 0.95^{1/12}\) throughout. \(N_{CF}\) denotes news about future cash flows and \(N_{DR}\) denotes news about future discount rates. They assume the data are generated by a first-order VAR model

\[z_{t+1} = a + \Gamma z_t + u_{t+1}\]

where \(r_{t+1}\) is the first element of \(z_{t+1}\). Given this process, cash-flow and discount-rate news are

\[N_{CF, t+1} = (e_1'+e_1'\lambda)\mu_{t+1}\] \[N_{DR, t+1} = e_1'\lambda \mu_{t+1}\]

where \(\lambda = \rho \Gamma(I-\rho\Gamma)^{-1}\) and \(e_1\) is a vector whose first element is 1 and whose other elements are 0. The two-beta ICAPM is then

\[E_t[R_{i, t+1} - R_{j, t+1}] = \gamma Cov_t[r_{i, t+1} - r_{j, t+1}, N_{CF, t +1}] + Cov_t[r_{i, t+1} - r_{j, t+1}, -N_{DR, t +1}]\]

Campbell et al. (2018) argued that when the intertemporal CAPM allows for stochastic volatility, its log stochastic discount factor has three factors:

\[m_{t+1} - E_tm_{t+1} = -\gamma N_{CF, t+1} - [-N_{DR, t+1}] + \frac{1}{2} N_{RISK, t+1}\]

where \(N_{RISK, t+1}\) is news about volatility. They therefore assumed the economy follows a first-order VAR

\[z_{t+1} = a + \Gamma z_t + \sigma_tu_{t+1}\]

where the first two elements of \(z_{t+1}\) are \(r_{t +1}\) and \(\sigma_{t+1}^2\). They also assume that \(u_{t+1}\) has a constant variance-covariance matrix \(\Sigma\), with element \(\Sigma_{11}=1\), and \(\sigma_t^2\) is equal to the conditional variance of market returns. Given this structure, news about discount rates can be written as

\[N_{DR, t+1} = e_1'\lambda\sigma_t\mu_{t+1}\]

while implied cash-flow news is

\[N_{CF, t+1} = (e_1'+e_1'\lambda)\sigma_t\mu_{t+1}\]

In their specification, news about risk is proportional to news about market return variance, \(N_V\):

\[N_{RISK, t+1} = \omega\rho e_2'(I-\rho\Gamma)^{-1}\sigma_tu_{t+1} = \omega N_{V, t+1}\]

They then derived the three-beta ICAPM:

\[E_t[R_{i, t+1} - R_{j, t+1}] = \gamma Cov_t[r_{i, t+1} - r_{j, t+1}, N_{CF, t +1}] + Cov_t[r_{i, t+1} - r_{j, t+1}, -N_{DR, t +1}]-\frac{1}{2}\omega Cov_t[r_{i, t+1} - r_{j, t+1}, N_{V, t +1}]\]

Beta Measurement

For consistency with Campbell (1993), I define the betas as

\[\beta_{i, CF} \equiv \frac{Cov(r_{i, t}, N_{CF, t})}{Var(r_{M, t} - E_{t-1} r_{M, t})}\] \[\beta_{i, DR} \equiv \frac{Cov(r_{i, t}, -N_{DR, t})}{Var(r_{M, t} - E_{t-1} r_{M, t})}\] \[\beta_{i, V} \equiv \frac{Cov(r_{i, t}, N_{V, t})}{Var(r_{M, t} - E_{t-1} r_{M, t})}\]

and estimate them with one lag:

\[\widehat{\beta}_{i, CF} = \frac{Cov(r_{i, t}, N_{CF, t})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})}+\frac{Cov(r_{i, t}, N_{CF, t-1})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})}\] \[\widehat{\beta}_{i, DR} = \frac{Cov(r_{i, t}, -N_{DR, t})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})} +\frac{Cov(r_{i, t}, -N_{DR, t-1})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})}\] \[\widehat{\beta}_{i, V} = \frac{Cov(r_{i, t}, N_{V, t})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})} + \frac{Cov(r_{i, t}, N_{V, t-1})}{Var(N_{CF, t} - N_{DR, t}+ I_{V}N_{V, t})}\]

where \(I_V = 1\) in the three-beta model and \(I_V = 0\) otherwise. Following Campbell, I include one lag of the market’s news terms in the numerator because, early in the sample, not all stocks in the test-asset portfolios traded frequently and synchronously.

Data Description

My VAR economy has six state variables: the excess market return (the log excess return on the Center for Research in Security Prices (CRSP) S&P value-weighted index over 30-day Treasury bills); the expected market volatility, estimated with a lagged regression model; the log yield of the 30-day Treasury bill; the market’s smoothed price-earnings ratio (the log ratio of the S&P 500 price index to a ten-year moving average of S&P 500 earnings, from Shiller’s website); the default credit spread (the log yield spread between Moody’s BAA and AAA bonds, from the Federal Reserve Bank of St. Louis); and the small-stock value spread (the difference between the log book-to-market ratios of small value and small growth stocks).

Campbell et al. (2018) argued that stochastic market volatility contributes significantly to explaining the cross-sectional risk premia of asset prices. They introduced expected market volatility (EVOL) to capture the variance of market returns, \(\sigma_t^2\), conditional on information available at time \(t\). To construct \(EVOL_t\), I first computed \(RVOL_t\), the within-month realized variance of daily returns at each time \(t\). I then regressed \(RVOL_{t+1}\) on lagged realized variance and the other state variables at time \(t\). The predicted value from this regression is the expected market variance (\(EVOL_t\equiv \widehat{RVOL}_{t+1}\)). Campbell et al. (2018) also included the default spread (DEF) because it tracks time-series variation in expected real returns on the market portfolio, and shocks to DEF should reflect news about aggregate default probabilities. Brennan, Wang, and Xia (2004) showed that the value spread predicts future market returns. Table I reports descriptive statistics for the monthly VAR state variables between 1929:06 and 2018:12.

Results

VAR Estimates

Tables II and IV report parameter estimates for the two-beta and three-beta VAR models, respectively, over my full sample period, June 1929 to June 2018. I excluded expected market volatility from the two-beta model, consistent with the Campbell and Vuolteenaho (2004) formulation, and used all state variables in the three-beta model. Ordinary least squares standard errors appear in parentheses below the coefficients, along with the \(R^2\) for each regression. Both tables show that the lagged market excess return, together with the other state variables, has some ability to predict future market returns; the coefficients on the term yield spread and the small value spread are consistent with Campbell and Vuolteenaho (2004). Campbell et al. (2018) found that expected volatility and the default credit spread positively predict excess returns. In my three-beta VAR economy, by contrast, the default credit spread has a negative but insignificant coefficient. One explanation is that market volatility tends to widen the credit spread, so EVOL absorbs most of the default spread’s information and DEF’s negative coefficient acts as an offset. The smoothed price-earnings ratio predicts returns negatively. Overall, an \(R^2\) of about 2 percent for both forecasting models is reasonable for monthly returns.

Tables III and V summarize the implied news terms: cash-flow and discount-rate news from the two-beta model, and cash-flow, discount-rate, and volatility news from the three-beta model, respectively. My cash-flow news has a lower variance than my discount-rate news, which agrees with the finding of Campbell (1991) that discount-rate news is the dominant component of the market return.

Figures 1 and 2 show the VAR model’s view of stock market history around NBER recessions; the shaded area indicates the amber recession alert period. My results are similar to Campbell’s in the common period (1963-2001). During the 2008 financial crisis, both discount-rate and cash-flow news drop significantly, a pattern Campbell and Vuolteenaho (2004) call a “mixed recession” (as in the Great Depression). News about volatility also peaks sharply in 2008.

Betas

Table VI reports beta estimates for the 25 size- and book-to-market portfolios over the 1963-2018 period under both the two-beta (Panel A) and three-beta (Panel B) ICAPMs. Each matrix places growth stocks at the left, value stocks at the right, small stocks at the top, and large stocks at the bottom. In each panel, the first matrix shows the cash-flow betas and the second shows the discount-rate betas; Panel B adds a third matrix of volatility-news betas. Unsurprisingly, cash-flow betas are significantly lower than discount-rate betas, because cash-flow news comes from VAR economies with limited forecastability. In Panel B, because the excess return should include a volatility (Jensen’s inequality) term reflecting a precautionary saving motive, the negative volatility betas are consistent with investors paying a hedging premium against volatility news. In both ICAPMs, small stocks still have higher discount-rate and cash-flow betas, and value stocks have lower discount-rate betas. However, value stocks also have lower cash-flow betas than growth stocks. This suggests that the two types of news co-move in the modern period, so the two-beta model may struggle to disentangle the size-and-value puzzle. In the three-beta ICAPM, by contrast, my volatility betas are consistent with the earlier-sample evidence in Campbell et al. (2018): small and value stocks require a lower hedging premium, which may explain the size-and-value anomalies in the modern period.

Empirical Estimates of Risk Premia

With the estimated betas, I run the cross-sectional regressions

\[\bar{R}_i^e = g_1\widehat{\beta}_{i, CF} + g_2\widehat{\beta}_{i, DR} + e_i\] \[\bar{R}_i^e = g_1\widehat{\beta}_{i, CF} + g_2\widehat{\beta}_{i, DR} + g_3\widehat{\beta}_{i, V} + e_i\]

for the two-beta and three-beta models, respectively, where \(\bar{R}_{i}^e\equiv \bar{R}_i -\bar{R}_{rf}\) denotes the sample average simple excess return on asset \(i\). The implied risk-aversion coefficient is \(g_1/g_2\). Table VII shows that the traditional CAPM and the two-beta ICAPM explain the cross-sectional variation only fairly. This is expected: as the previous subsection shows, cash-flow and discount-rate news co-move for most of the updated period (2001-2018), especially during the financial crisis, which weakens the two-beta ICAPM. The three-beta ICAPM, by contrast, explains more than 30% of the cross-sectional variation in average excess returns. The two-beta ICAPM has an implied risk aversion of 12.31, while the three-beta ICAPM implies 5.25. Figure 3 summarizes these results visually; I multiply the estimates by 1,200 to express them as annualized percentage returns.

Pricing Popular Equity and Currency Strategies

I next use my ICAPMs to assess common anomalies from the asset pricing literature. Table VIII analyzes several well-known equity anomalies using data from Ken French’s website. The sample period is 1963:07-2018:12. The anomaly portfolios are the market (RMRF), size (SMB), and value (HML) factors discussed in this paper, the profitability (RMW) and investment (CMA) factors added in Fama and French (2015), the momentum (UMD) factor of Carhart (1997), and the short-term (STR) and long-term (LTR) reversal factors. I also include the long-short currency carry trade portfolios of Lustig, Roussanov, and Verdelhan (2011), for all countries and for developed countries only. For each portfolio, Table VIII reports the mean excess return in the first column and the return standard deviation in the second. Columns 3-5 report the portfolios’ betas on my estimates of discount-rate, cash-flow, and volatility news. Columns 6-9 use these betas to construct the components of fitted excess returns based on discount-rate news (\(\lambda_{DR}\)), cash-flow news in the two-beta ICAPM (\(\lambda^{2−BETA}_{CF}\)), cash-flow news in the three-beta ICAPM (\(\lambda^{3−BETA}_{CF}\)), and variance news in the three-beta ICAPM (\(\lambda_V\)). These fitted excess returns use the two-beta and three-beta parameter estimates reported in Table VII. Because I did not re-estimate any parameters, this analysis is an out-of-sample test. Columns 10-12 report the alphas of the anomalies (their sample average excess returns less their predicted excess returns) under the CAPM, the two-beta ICAPM, and the three-beta ICAPM. All portfolios except RMRF were chosen to have positive CAPM alphas.

Table VIII shows that volatility risk exposure explains most of the abnormal returns. Most anomaly portfolios have negative betas on volatility news, which makes them riskier and helps explain their positive excess returns; the exceptions are SMB, RMW, and CMA. The two-beta model is not clearly better than the CAPM, but it does explain the developed-country carry trade return well. The three-beta ICAPM explains the value and size portfolio returns much better than the two-beta ICAPM. It also better explains the all-country carry trade return, which suggests that the carry premium arises from volatility news in US equity markets. Overall, the three-beta ICAPM predicts anomaly returns more accurately than both the CAPM and the two-beta ICAPM.

Conclusions

I find that value and small stocks carry considerably lower premiums for hedging future volatility shocks, which can explain their higher average returns. However, I find no clear evidence of higher cash-flow betas for these stocks in either the two-beta or the three-beta ICAPM. Chen and Zhao (2009) argued that because cash-flow news is computed as a residual from discount-rate news, it captures all the noise in the VAR model: any variable omitted from the VAR but in rational investors’ information set ends up in cash-flow news. Cash-flow news is therefore highly sensitive to the VAR specification. Under alternative VAR specifications, they showed that value companies do not have larger cash-flow betas, which is consistent with my estimates. The disagreement over cash-flow betas may therefore stem from misspecified VAR economies.

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