Intertemporal CAPM, Refreshed

Key figures

In plain English

The idea. Stocks fall for two reasons: investors expect lower profits, or investors get more fearful and demand a higher return. Profit losses are permanent; fear tends to reverse, because a higher required return means higher returns later. A long-term investor should therefore demand much more pay for exposure to profit news (the "bad" beta) than to fear news (the "good" beta). The newer version adds a third risk: news that the market is about to get more turbulent.

How it becomes a risk premium. Each stock gets one beta for each kind of news, and each beta has a price. In the data, a unit of profit beta is paid % a month and a unit of fear beta %, a ratio of about seven to one. For the market as a whole, × % + × % gives an equity risk premium of about % a month (~% a year), against % actually earned.

What I find. The model gets the market right but not the 25 size and value portfolios: value stocks no longer carry more of the bad beta, and small growth stocks earn far less than their betas imply. Volatility news helps explain value, momentum and carry, but its price is not statistically reliable.

Introduction

It is well known that the CAPM fails to explain the average stock return in the modern period (after 1963) using the value-weighted market return as a proxy for the market portfolio. According to Sharpe (1964) and Lintner (1965), a single total wealth portfolio can summarize a stock's return characteristics with its beta. However, Fama and French (1992) documented the size and value anomalies, and indicated that portfolios tilting towards small and value stocks tend to generate excess returns. By solving Merton's (1973) intertemporal capital asset pricing model (ICAPM) within the Epstein and Zin (1989) utility framework of a representative agent, Campbell and Vuolteenaho (2004) provided an elegant explanation by splitting the beta risk into "bad" and "good" varieties: the beta associated with news about cash flows and the beta associated with news about discount rates. They showed some evidence that value stocks and small stocks have considerably higher cash-flow betas than growth stocks and large stocks. In other words, for a conservative long-term investor in the ICAPM framework, small and value stocks are intertemporal hedges and perform well when investment opportunities deteriorate; however, in equilibrium, these assets should have delivered a lower average return, so the rational long-term investor should not tilt their portfolio towards small and value stocks. Furthermore, in a subsequent paper, Campbell, Giglio, Polk and Turley (2018) claimed that either increasing volatility of stock returns or decreasing expected stock returns can send negative shocks to the investment opportunities in the stock market. In addition to the low-frequency movements in equity volatility, they proposed a three-beta ICAPM with cash-flow, discount-rate and risk news to explain the cross-sectional asset return puzzle, and provided results consistent with their "Bad Beta, Good Beta" paper.

In this paper, I examine whether the two-beta ICAPM in Campbell and Vuolteenaho (2004) and the three-beta ICAPM in Campbell et al. (2018) can still explain the size and value puzzle in the latest modern data (). Under both vector-autoregressive (VAR) economies, I estimate the parameters using the full sample data (). In the first two sections (tabs 01 and 02), I describe both ICAPM models and the data used to estimate the parameters of the VAR economies. In the third section (tabs 03 to 05), I walk through my estimated results on the 25 ME- and BE/ME-sorted portfolios from Ken French's website in the modern data, and compare them against the results in both of Campbell's papers. In the fourth section (tab 06), I conduct an out-of-sample test on some popular equity and currency trading strategies using the estimated two-beta and three-beta models. In the final section (tab 07), I summarize the findings.

Model

Campbell (1991); Campbell and Vuolteenaho (2004); Campbell et al. (2018)

Bottom lineA first-order VAR turns each month's unexpected market return into news about cash flows and news about discount rates; the three-beta model adds news about variance, and each stock's exposure to each news term is a beta.

Following Campbell and Shiller (1988), Campbell and Vuolteenaho (2004) used a log-linear approximate decomposition of returns from Campbell (1991):

where is a log stock return, is the log dividend paid by the stock, denotes a one-period change, denotes a rational expectation at time , and is a discount coefficient. I use throughout the paper. denotes news about future cash flows and denotes news about future discount rates. They assume the data are generated by a first-order VAR model

where is the first element of . Given this process, I can obtain the news about cash flows and discount rates as follows:

where and is a vector whose first element is 1 and every other element is 0. As a result, the two-beta ICAPM is

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

where is news about volatility. Following this, they assumed the economy is described by a first-order VAR

where the first two elements of are and . They also assume that has a constant variance-covariance matrix , with element , and that is equal to the conditional variance of market returns. Given this structure, news about discount rates can be written as

while implied cash-flow news is

Their specification also implies that news about risk is proportional to news about market return variance, :

Hence, they derived the three-beta ICAPM:

Beta measurement

To be consistent with Campbell and Vuolteenaho (2004), I define the betas as follows:

and estimate the betas with a lag:

Following Campbell, I include one lag of the market's news terms in the numerator because, during the early sample period, not all stocks in the test-asset portfolios were traded frequently and synchronously. The denominator is the variance of the unexpected market return, .

Six state variables from public sources, June 1929 to

Monthly · observations

Bottom lineEvery input is public and runs to the latest month all sources share; the valuation, rate and spread variables are highly persistent, and the market return is close to unpredictable.

In my VAR economy, I have six state variables: the excess market return (measured as the log excess return on the Center for Research in Security Prices (CRSP) value-weighted market over one-month Treasury bills, from Ken French's website); the expected market volatility, estimated using a lagged regression model; the log yield of the one-month Treasury bill; the market's smoothed price-earnings ratio (measured as 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 bonds and AAA bonds, obtained from the Federal Reserve Bank of St. Louis); and the small-stock value spread (measured as the difference between the log book-to-market ratios of small value and small growth stocks).

Campbell et al. (2018) argued that stochastic volatility in the market makes a significant contribution to explaining the cross-sectional risk premia of asset prices. They introduced the expected market volatility term (EVOL), which is meant to capture the variance of market returns, , conditional on information available at time . To construct , I first create a series of the within-month realized variance of daily returns for each time , . I run a regression of on a constant, lagged realized variance and the rest of the state variables at time . The predicted value from this regression is defined as the expected market variance (). Campbell et al. (2018) also included the default spread (DEF) because it is known to track time-series variation in expected real returns on the market portfolio, and shocks to DEF should reflect news about aggregate default probabilities. Lastly, the value spread is shown by Campbell and Vuolteenaho (2004) to predict future market returns. Table I reports the descriptive sample statistics of the monthly VAR state variables between .

Table I — Descriptive statistics of the VAR state variables

ReadBILL, PE, DEF and VS have first-order autocorrelations of 0.98 or more, so the VAR forecasts them well; the market return, at 0.09, is close to unpredictable. EVOL moves with DEF (correlation 0.67), as expected for two measures of market stress.

Variable	Mean	Median	Stdev.	Min	Max	Autocorr.
r_M^e	0.005	0.010	0.053	(0.344)	0.328	0.092
EVOL	0.002	0.002	0.003	(0.001)	0.031	0.711
BILL	0.003	0.002	0.003	(0.001)	0.013	0.978
PE	2.870	2.887	0.430	1.717	3.789	0.995
DEF	0.010	0.008	0.006	0.003	0.052	0.976
VS	1.737	1.632	0.343	1.283	2.949	0.989

Correlations	r_M^et+1	EVOLt+1	BILLt+1	PEt+1	DEFt+1	VSt+1
r_M^e, t+1	1.000	(0.460)	(0.060)	0.009	(0.037)	(0.012)
EVOL, t+1	(0.460)	1.000	(0.144)	0.002	0.671	0.414
BILL, t+1	(0.060)	(0.144)	1.000	(0.200)	(0.107)	(0.453)
PE, t+1	0.009	0.002	(0.200)	1.000	(0.417)	0.014
DEF, t+1	(0.037)	0.671	(0.107)	(0.417)	1.000	0.543
VS, t+1	(0.012)	0.414	(0.453)	0.014	0.543	1.000
r_M^e, t	0.092	(0.239)	(0.063)	0.073	(0.127)	0.005
EVOL, t	(0.030)	0.711	(0.162)	(0.021)	0.702	0.417
BILL, t	(0.047)	(0.144)	0.978	(0.202)	(0.094)	(0.450)
PE, t	(0.061)	0.047	(0.195)	0.995	(0.410)	0.015
DEF, t	0.022	0.621	(0.117)	(0.412)	0.976	0.552
VS, t	(0.024)	0.415	(0.455)	0.011	0.538	0.989

Notes: The table shows the descriptive statistics of the VAR state variables estimated from the full sample period , monthly data points. r_M^e is the excess log return on the CRSP value-weighted market. BILL is the log of the one-month Treasury bill yield. EVOL is the expected market variance. PE is the log ratio of the S&P 500's price to the S&P 500's ten-year moving average earnings. DEF is the default credit spread, the difference in the log Moody's BAA bond yield and log Moody's AAA bond yield. VS is the small-stock value spread, the difference in the log book-to-market ratios of small value and small growth stocks. The small-value and small-growth portfolios are two of the six elementary portfolios constructed by Davis et al. (2000). "Stdev." denotes standard deviation and "Autocorr." the first-order autocorrelation of the series. The lower panel reports correlations of each variable at t+1 and at t with each variable at t+1. Negatives in parentheses.

Each input comes from a public source

Variable	Construction	Source
r_M^e	Log excess return on the value-weighted market over the one-month T-bill	Ken French, Mkt-RF and RF
EVOL	Fitted value from regressing next month's realized variance of daily returns on a constant, this month's state variables and realized variance	Ken French daily factors
BILL	Log yield of the one-month T-bill	Ken French RF
PE	Log S&P 500 price over ten-year average earnings	Robert Shiller, ie_data
DEF	Log yield spread of Moody's BAA over AAA bonds	FRED
VS	Log book-to-market of small value less small growth	Ken French 6 portfolios
Test assets	25 size and book-to-market portfolios, value-weighted, from July 1963	Ken French
Strategies	RMRF, SMB, HML, RMW, CMA, UMD, STR, LTR; currency carry from November 1983	Ken French; Lustig, Roussanov and Verdelhan (2011)

Downloaded October 2026; Ken French files run to September 2026, carry to .

Discount-rate news drives most market moves

VAR(1), · OLS standard errors

Bottom line

Tables II and IV report parameter estimates for the two-beta and three-beta VAR models respectively over the full sample period, June 1929 to . In the two-beta model, I exclude the expected market volatility to keep it consistent with the Campbell and Vuolteenaho (2004) formulation; in the three-beta model, all state variables are used. Ordinary least squares (OLS) standard errors are reported in brackets below the coefficients. Finally, I report the R² for each regression. Both tables show that the lagged market excess return, together with the other state variables, provides some ability to predict the future market return; the T-bill yield and the small-stock value spread have coefficients consistent with the results in Campbell and Vuolteenaho (2004). Campbell et al. (2018) show that expected volatility and the default credit spread positively predict the excess return. The expected volatility has a positive coefficient () but is not significant, and the default credit spread has a negative sign in the three-beta VAR economy, also insignificant. One can argue that market volatility tends to widen the credit spread, so EVOL explains the majority of the default spread, and the negative contribution from DEF provides some offsetting effect. The smoothed price-earnings ratio shows negative predictive power. Overall, an R² of about 2% for both forecasting models ( and ) is reasonable for a monthly return model.

Tables III and V summarize the behavior of the implied news about cash flows and discount rates from the two-beta model, and the implied news about cash flows, discount rates and volatility from the three-beta model, respectively. The estimated news about cash flows has a lower variance than the news about discount rates, which agrees with the finding of Campbell (1991) that discount-rate news is the dominant component of the market return.

Figures 1 and 2 illustrate the VAR model's view of stock market history in relation to NBER recessions. The shaded areas indicate NBER recession periods. My results are similar to Campbell's in the common period (1963–2001); during the 2008 financial crisis, I see both the news about discount rates and the news about cash flows drop. Campbell and Vuolteenaho (2004) call this a "mixed recession" (as in the Great Depression). In particular, the news about volatility peaked substantially in 2008, second only to the early 1930s.

Table II — VAR parameter estimates for the two-beta model

ReadThe market equation explains of monthly returns; the T-bill yield and the price-earnings ratio predict lower returns, and the other state variables are close to random walks.

	Constant	r_M^et	BILLt	PEt	DEFt	VSt	R²%
r_M^e, t+1	0.051[0.016]	0.088[0.029]	(1.863)[0.716]	(0.008)[0.004]	0.263[0.341]	(0.012)[0.006]	1.88
BILL, t+1	–[0.000]	–[0.000]	0.970[0.007]	–[0.000]	(0.005)[0.003]	–[0.000]	95.68
PE, t+1	0.013[0.011]	0.503[0.020]	(0.338)[0.481]	0.998[0.003]	0.604[0.229]	(0.009)[0.004]	99.32
DEF, t+1	–[0.000]	(0.011)[0.001]	0.026[0.017]	–[0.000]	0.960[0.008]	–[0.000]	96.04
VS, t+1	0.002[0.015]	0.110[0.027]	(0.523)[0.665]	0.011[0.004]	1.603[0.317]	0.972[0.006]	97.95

Notes: The table shows the OLS parameter estimates for a first-order VAR model including a constant, the log excess market return (r_M^e), log Treasury bill yield (BILL), price-earnings ratio (PE), default credit spread (DEF) and small-stock value spread (VS). Each row corresponds to a different dependent variable. The first six columns report coefficients on the six explanatory variables, and the remaining column shows R². OLS standard errors are in brackets. Sample period for the dependent variables is , monthly data points. Negatives in parentheses.

Table III — Cash-flow and discount-rate news for the market portfolio

ReadDiscount-rate news has a monthly standard deviation of against for cash-flow news; a shock to the price-earnings ratio is almost entirely discount-rate news (correlation −0.87).

News covariance	N_CF	N_DR	News corr/std	N_CF	N_DR
N_CF	0.00095	(0.00010)	N_CF	0.031	(0.078)
N_DR	(0.00010)	0.00165	N_DR	(0.078)	0.041
Shock correlations and news functions
r_M^e shock	0.642	(0.814)	r_M^e shock	0.727	(0.273)
BILL shock	(0.598)	(0.366)	BILL shock	(34.198)	(34.198)
PE shock	0.183	(0.873)	PE shock	(0.744)	(0.744)
DEF shock	(0.107)	0.242	DEF shock	(2.494)	(2.494)
VS shock	0.046	(0.049)	VS shock	(0.016)	(0.016)

Notes: The table shows the properties of cash-flow news (N_CF) and discount-rate news (N_DR) implied by the VAR model of Table II. The upper-left section of the table shows the covariance matrix of the news terms. The upper-right section shows the correlation matrix of the news terms with standard deviation on the diagonal. The lower-left section shows the correlation of shocks to individual state variables with the news terms. The lower-right section shows the functions (e₁′ + e₁′λ, e₁′λ) that map the state-variable shocks to cash-flow and discount-rate news. I define λ ≡ ρΓ(I − ρΓ)⁻¹, where Γ is the estimated VAR transition matrix from Table II and ρ is set to 0.95 per annum. r_M^e is the excess log return on the CRSP value-weighted market. BILL is the log Treasury bill yield. PE is the log ratio of the S&P 500's price to the S&P 500's ten-year moving average of earnings. DEF is the default credit spread. VS is the small-stock value spread, the difference in log book-to-markets of small value and small growth stocks. Negatives in parentheses.

Table IV — VAR parameter estimates for the three-beta model

ReadEVOL's own equation explains of next month's expected variance. In the market equation it enters with a positive coefficient, but at it is not significant.

	Constant	r_M^et	EVOLt	BILLt	PEt	DEFt	VSt	R²%
r_M^e, t+1	0.056[0.017]	0.105[0.038]	0.694[1.029]	(1.828)[0.718]	(0.010)[0.005]	(0.006)[0.525]	(0.012)[0.006]	1.92
EVOL, t+1	(0.006)[0.001]	(0.003)[0.001]	0.370[0.037]	0.009[0.026]	0.002[0.000]	0.218[0.019]	–[0.000]	57.98
BILL, t+1	–[0.000]	(0.001)[0.000]	(0.041)[0.010]	0.968[0.007]	–[0.000]	0.011[0.005]	–[0.000]	95.74
PE, t+1	0.025[0.012]	0.541[0.026]	1.599[0.690]	(0.257)[0.481]	0.995[0.003]	(0.016)[0.352]	(0.009)[0.004]	99.32
DEF, t+1	0.001[0.000]	(0.009)[0.001]	0.076[0.024]	0.029[0.017]	–[0.000]	0.930[0.012]	–[0.000]	96.07
VS, t+1	0.009[0.016]	0.132[0.036]	0.903[0.956]	(0.477)[0.667]	0.009[0.005]	1.253[0.487]	0.972[0.006]	97.95

Notes: The table shows the OLS parameter estimates for a first-order VAR model including a constant, the log excess market return (r_M^e), expected market variance (EVOL), log Treasury bill yield (BILL), price-earnings ratio (PE), default credit spread (DEF) and small-stock value spread (VS). Each row corresponds to a different dependent variable. The first seven columns report coefficients on the seven explanatory variables, and the remaining column shows R². OLS standard errors are in brackets. Sample period for the dependent variables is , monthly data points. Negatives in parentheses.

Table V — Cash-flow, discount-rate and risk news for the market portfolio

ReadVolatility news is driven mainly by default-spread shocks (correlation 0.82) and is negatively correlated with both cash-flow and discount-rate news.

News covariance	N_CF	N_DR	N_V	News corr/std	N_CF	N_DR	N_V
N_CF	0.00096	(0.00009)	(0.00009)	N_CF	0.031	(0.075)	(0.176)
N_DR	(0.00009)	0.00164	(0.00015)	N_DR	(0.075)	0.041	(0.236)
N_V	(0.00009)	(0.00015)	0.00025	N_V	(0.176)	(0.236)	0.016
Shock correlations and news functions
r_M^e shock	0.643	(0.812)	0.078	r_M^e shock	0.744	(0.256)	0.036
EVOL shock	(0.366)	0.612	0.266	EVOL shock	0.802	0.802	3.390
BILL shock	(0.598)	(0.372)	0.144	BILL shock	(34.452)	(34.452)	6.016
PE shock	0.183	(0.873)	0.130	PE shock	(0.739)	(0.739)	0.272
DEF shock	(0.113)	0.241	0.821	DEF shock	(2.797)	(2.797)	11.732
VS shock	0.043	(0.050)	0.122	VS shock	(0.017)	(0.017)	0.052

Notes: The table shows the properties of cash-flow news (N_CF), discount-rate news (N_DR) and volatility news (N_V) implied by the VAR model of Table IV. The upper-left section of the table shows the covariance matrix of the news terms. The upper-right section shows the correlation matrix of the news terms with standard deviation on the diagonal. The lower-left section shows the correlation of shocks to individual state variables with the news terms. The lower-right section shows the functions (e₁′ + e₁′λ, e₁′λ, ρe₂′(I − ρΓ)⁻¹) that map the state-variable shocks to cash-flow, discount-rate and volatility news. I define λ ≡ ρΓ(I − ρΓ)⁻¹, where Γ is the estimated VAR transition matrix from Table IV and ρ is set to 0.95 per annum. r_M^e is the excess log return on the CRSP value-weighted market. EVOL is the expected market variance. BILL is the log Treasury bill yield. PE is the log ratio of the S&P 500's price to the S&P 500's ten-year moving average of earnings. DEF is the default credit spread. VS is the small-stock value spread. Negatives in parentheses.

Figure notes: Figure 1 plots the cash-flow news and negative discount-rate news from the two-beta VAR; Figure 2 plots the cash-flow news, negative discount-rate news and volatility news from the three-beta VAR. Each series is smoothed with a trailing exponentially weighted moving average with a decay parameter of 0.08 per month, MA_t(N) = 0.08N_t + (1 − 0.08)MA_t−1(N). Gray shaded areas denote NBER recessions. The sample is ; the charts start in June 1931, after a 24-month burn-in of the moving average.

Value stocks carry lower cash-flow betas than growth, not higher

25 size and book-to-market portfolios,

Bottom line

Table VI reports beta estimates for the 25 size- and book-to-market-sorted portfolios over the period under both the two-beta (Panel A) and three-beta (Panel B) ICAPMs. The portfolios are organized in a square matrix with growth stocks at the left, value stocks at the right, small stocks at the top and large stocks at the bottom. For each panel, the first matrix displays the cash-flow betas, and the second matrix displays the discount-rate betas. In Panel B, the last matrix shows the estimated betas for the news about volatility. First of all, it is no surprise to see that the cash-flow betas are significantly lower than the discount-rate betas, because cash-flow news comes from VAR economies with limited forecastability. In Panel B, the volatility betas are positive for all 25 portfolios in the updated sample, so each portfolio partly hedges news of higher volatility; with a negative price of volatility risk, the portfolios with the lowest volatility betas are the riskiest. For both ICAPMs, the small stocks still show higher betas in both discount rates and cash flows. Moreover, the value stocks show lower discount-rate betas. However, the value stocks have lower cash-flow betas than the growth stocks. This suggests that the two innovations co-move in the modern period and that the two-beta model may have trouble disentangling the size-value puzzle. Based on the three-beta ICAPM, the volatility betas are lower for value stocks than for growth stocks, consistent with the story from the earlier sample in Campbell et al. (2018), but higher for small stocks than for large stocks. The lower hedging premium required for value stocks may explain the value anomaly in the modern period; it does not explain the size anomaly.

Table VI, Panel A — Two-beta estimates,

	Growth	2	3	4	Value
Cash-flow betas, β̂_CF
Small	0.622[0.059]	0.528[0.051]	0.469[0.046]	0.452[0.045]	0.491[0.048]
2	0.570[0.053]	0.482[0.046]	0.434[0.043]	0.391[0.041]	0.464[0.048]
3	0.510[0.049]	0.446[0.042]	0.394[0.040]	0.374[0.039]	0.418[0.046]
4	0.468[0.045]	0.415[0.041]	0.390[0.039]	0.354[0.037]	0.403[0.044]
Large	0.364[0.039]	0.336[0.036]	0.297[0.035]	0.315[0.036]	0.358[0.042]
Discount-rate betas, β̂_DR
Small	0.994[0.074]	0.849[0.063]	0.792[0.058]	0.737[0.057]	0.791[0.062]
2	0.903[0.067]	0.793[0.058]	0.737[0.054]	0.694[0.053]	0.793[0.063]
3	0.848[0.063]	0.726[0.051]	0.666[0.048]	0.678[0.052]	0.729[0.060]
4	0.746[0.055]	0.704[0.050]	0.661[0.050]	0.666[0.055]	0.727[0.060]
Large	0.630[0.044]	0.595[0.041]	0.557[0.041]	0.581[0.051]	0.687[0.057]

Table VI, Panel B — Three-beta estimates,

	Growth	2	3	4	Value
Cash-flow betas, β̂_CF
Small	0.619[0.060]	0.526[0.051]	0.466[0.047]	0.449[0.045]	0.488[0.049]
2	0.567[0.054]	0.478[0.047]	0.431[0.043]	0.388[0.041]	0.460[0.049]
3	0.507[0.050]	0.443[0.043]	0.391[0.040]	0.371[0.040]	0.415[0.047]
4	0.466[0.046]	0.411[0.041]	0.387[0.040]	0.351[0.039]	0.401[0.046]
Large	0.363[0.039]	0.335[0.036]	0.295[0.036]	0.314[0.037]	0.355[0.043]
Discount-rate betas, β̂_DR
Small	0.991[0.074]	0.847[0.063]	0.793[0.058]	0.739[0.058]	0.792[0.064]
2	0.902[0.067]	0.794[0.058]	0.738[0.055]	0.695[0.054]	0.795[0.065]
3	0.848[0.062]	0.727[0.052]	0.667[0.049]	0.680[0.053]	0.730[0.061]
4	0.745[0.055]	0.705[0.050]	0.663[0.051]	0.668[0.057]	0.729[0.061]
Large	0.629[0.043]	0.595[0.041]	0.559[0.041]	0.584[0.052]	0.689[0.058]
Volatility betas, β̂_V
Small	0.133[0.040]	0.109[0.034]	0.095[0.033]	0.086[0.032]	0.081[0.035]
2	0.121[0.037]	0.099[0.033]	0.085[0.031]	0.086[0.030]	0.083[0.036]
3	0.111[0.035]	0.094[0.031]	0.080[0.028]	0.083[0.030]	0.079[0.034]
4	0.100[0.030]	0.094[0.030]	0.074[0.030]	0.077[0.032]	0.068[0.033]
Large	0.087[0.024]	0.092[0.024]	0.068[0.025]	0.045[0.030]	0.070[0.031]

Notes: Panel A shows the estimated cash-flow (β̂_CF) and discount-rate betas (β̂_DR) for the 25 ME- and BE/ME-sorted portfolios. Panel B shows the estimated cash-flow (β̂_CF), discount-rate (β̂_DR) and volatility betas (β̂_V) for the 25 ME- and BE/ME-sorted portfolios. "Growth" denotes the lowest BE/ME, "value" the highest BE/ME, "small" the lowest ME and "large" the highest ME stocks. Standard errors (in brackets) are bootstrapped using 2,000 samples. Negatives in parentheses.

The same betas as heat maps

Cash-flow risk is priced; volatility risk is not, at conventional levels

Cross-section of 25 portfolios,

Bottom line

Lastly, with the estimated betas, I can run the cross-sectional regressions

respectively, where denotes the sample average simple excess return on asset . The implied risk-aversion coefficient can be recovered as . Table VII shows that neither the traditional CAPM nor the two-beta ICAPM explains the cross-sectional variation in the updated sample. As the earlier subsections suggest, the innovations to cash flows and discount rates co-move in parts of the updated period, especially during the financial crisis, so one can expect the two-beta ICAPM's performance to fall off. The three-beta ICAPM does better: allowing an intercept and free premia, it explains of the variation in the cross-section of average excess returns, against less than 5% for the CAPM and the two-beta ICAPM. Under the models' own restrictions (no intercept, with fixed at the market variance), all three R² values are negative. The two-beta ICAPM has implied risk aversion of , while the three-beta ICAPM shows . A visual summary of these results is provided in Figure 3, where I multiply the estimates by 1,200 to show annualized percentage points.

Table VII — Asset pricing tests for the modern sample

Parameter	CAPM	Two-beta ICAPM	Three-beta ICAPM
β̂_CF premium (g₁)	0.0063	0.0136	0.0309
Std. err.	0.0004	0.0011	0.0096
β̂_DR premium (g₂)	0.0063	0.0020	0.0020
Std. err.	0.0004	fixed	fixed
β̂_V premium (g₃)	n.a.	n.a.	(0.084)
Std. err.	n.a.	n.a.	0.046
Implied risk aversion, γ	n.a.	6.75x	15.41x
R²	(58.1%)	(72.5%)	(51.0%)
R², with an intercept and free premia	1.6%	3.1%	29.0%

Notes: The table shows premia estimated from the sample for the CAPM, the two-beta ICAPM and the three-beta ICAPM. The test assets are the 25 ME- and BE/ME-sorted portfolios. Estimates are from a cross-sectional regression without an intercept of average simple excess test-asset returns (monthly, in fractions) on estimated cash-flow (β̂_CF), discount-rate (β̂_DR) and volatility betas (β̂_V). In the ICAPMs, g₂ is fixed at the market variance, . R² is 1 − SSE/SST around the cross-sectional mean; the last row refits each model with an intercept and every premium estimated freely. Negatives in parentheses.

Figure 3 notes: The three diagrams correspond to the CAPM, the two-beta ICAPM and the three-beta ICAPM. The predicted values are from the regressions in Table VII. Dots denote the 25 ME and BE/ME portfolios; small growth is marked in red.

Volatility news cuts the alphas of value, momentum and carry

Out of sample: premia from Table VII, not re-estimated

Bottom line

In this section, I use my ICAPM models to assess some common anomalies that have been discussed in the asset pricing literature. Table VIII analyzes a number of well-known equity anomalies using data taken from Kenneth French's website. The sample period is . 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 reversal (STR) and long-term reversal (LTR) factors. I also include the currency carry trade long-short portfolio strategies proposed in Lustig, Roussanov and Verdelhan (2011), with all countries and with developed countries only. For each of these portfolios, Table VIII reports the mean excess return in the first column and the standard deviation of return in the second column. Columns 3–5 report the portfolios' betas with my estimates of cash-flow news, discount-rate news and volatility news. These are used in Columns 6–9 to construct the components of fitted excess returns based on cash-flow news in the two-beta ICAPM (), cash-flow news in the three-beta ICAPM (), discount-rate news () and variance news in the three-beta ICAPM (). These fitted excess returns use the parameter estimates of the two-beta and three-beta models reported in Table VII. I do not re-estimate any parameters, so this can be read as this analysis as an out-of-sample test. Columns 10–12 report the alphas of the anomalies (their sample average excess returns less their predicted excess returns) calculated using the CAPM, the two-beta ICAPM and the three-beta ICAPM. All the portfolios, with the obvious exception of RMRF, have been chosen to have positive CAPM alphas.

Table VIII shows that the volatility risk exposure explains a large part of the abnormal returns. Most of the anomaly portfolios have negative betas on volatility news, which make them riskier and help to explain their positive excess returns. The exceptions are the SMB, RMW, CMA and STR portfolios, whose volatility betas are close to zero. Here, the two-beta model is not necessarily better than the CAPM. Against the two-beta ICAPM, the three-beta ICAPM gives a much more promising interpretation of the value and momentum portfolio returns: the HML alpha falls from % to % a month and the UMD alpha from % to %. Interestingly, the three-beta model also gives the better result in explaining the carry trade return with all countries, with an alpha of % against % under the CAPM. This indicates that part of the carry premium arises from volatility news in the US equity markets. Overall, the three-beta model gives more accurate predictions than the CAPM and the two-beta ICAPM.

Table VIII — Pricing popular equity and currency strategies

Strategy	μ(1) %	σ(2) %	β̂_CF(3)	β̂_DR(4)	β̂_V(5)	λ_CF^2-beta(6) %	λ_CF^3-beta(7) %	λ_DR(8) %	λ_V(9) %	α_CAPM(10) %	α_ICAPM^2-beta(11) %	α_ICAPM^3-beta(12) %
RMRF	0.60	4.46	0.37	0.64	0.08	0.50	1.14	0.13	(0.70)	(0.04)	(0.03)	0.03
SMB	0.18	3.02	0.11	0.17	0.02	0.15	0.33	0.03	(0.14)	0.01	–	(0.05)
HML	0.29	2.97	(0.07)	(0.05)	(0.03)	(0.10)	(0.22)	(0.01)	0.27	0.37	0.40	0.25
RMW	0.25	2.29	(0.04)	(0.07)	–	(0.06)	(0.13)	(0.01)	0.01	0.32	0.32	0.39
CMA	0.24	2.07	(0.06)	(0.09)	(0.01)	(0.09)	(0.20)	(0.02)	0.10	0.35	0.35	0.36
UMD	0.60	4.21	(0.04)	(0.16)	(0.04)	(0.06)	(0.13)	(0.03)	0.37	0.73	0.70	0.40
STR	0.40	3.18	0.06	0.12	–	0.08	0.18	0.02	(0.04)	0.28	0.29	0.23
LTR	0.22	2.64	(0.01)	0.01	(0.02)	(0.01)	(0.03)	–	0.17	0.22	0.23	0.08
CarryALL	0.57	2.48	0.07	0.12	(0.02)	0.09	0.21	0.02	0.18	0.45	0.45	0.15
CarryDEV	0.39	2.78	0.05	0.16	(0.04)	0.07	0.16	0.03	0.37	0.26	0.29	(0.18)

Notes: The table decomposes the average monthly returns on well-known equity strategies and currency carry trade strategies using the capital asset pricing model (CAPM), the two-beta intertemporal CAPM and the three-beta ICAPM. I estimate α_CAPM from the CAPM premium in Table VII. I estimate α_ICAPM^2-beta and α_ICAPM^3-beta using the corresponding premia from Table VII. The sample of equity strategies covers and that of the currency carry trade strategies . The market variance over is . The strategies are the market (RMRF), size (SMB), value (HML), profitability (RMW), investment (CMA), momentum (UMD), short-term reversal (STR), long-term reversal (LTR), carry trade with all countries (CarryALL) and carry trade with developed countries only (CarryDEV). All returns except the currency carry trade strategies are from Kenneth French's website; the currency carry trade returns are from Hanno Lustig's website. Negatives in parentheses.

The results hold on the sample to 2018, except the volatility premium

Method, robustness and next steps

Bottom lineEnding the sample in December 2018 changes the cash-flow results little but removes the volatility premium entirely, so the evidence for priced volatility risk rests on the last nearly eight years.

Robustness: the sample to 2018 against the full sample

	Sample to 2018VAR 1929–2018 · test 1963–2018	Full sampleVAR 1929–2026 · test 1963–2026
Two-beta ICAPM
Implied risk aversion, γ	8.27x	6.75x
Cash-flow premium, t-statistic	10.6	12.7
R²	(57.7%)	(72.5%)
Three-beta ICAPM
Implied risk aversion, γ	8.01x	15.41x
Variance premium, g_V	0.001	(0.084)
Variance premium, t-statistic	0.0	(1.8)
R²	(57.8%)	(51.0%)
Betas, value less growth
Cash-flow beta	(0.067)	(0.080)
Volatility beta	(0.030)	(0.034)
Other models
CAPM R²	(43.2%)	(58.1%)
R², with an intercept and free premia
CAPM	0.8%	1.6%
Two-beta ICAPM	4.3%	3.1%
Three-beta ICAPM	18.7%	29.0%

VAR from June 1929; test from July 1963. t-statistics from OLS standard errors. The last block refits each model with an intercept and every premium estimated freely. Negatives in parentheses.

Method

EVOL. Next month's realized variance of daily returns, regressed with a constant on this month's R_M^e, PE, DEF, VS and realized variance; the fitted value is dated at the month whose data it uses.
VAR. OLS on the full sample; ρ = 0.95^1/12; the two-beta VAR omits EVOL.
Betas. One lag of news in the numerator; the variance of N_CF − N_DR in the denominator; bootstrap standard errors from 2,000 draws.
Cross-section. OLS without an intercept on average monthly excess returns; g_DR fixed at the market variance.
Strategies. Excess returns as published; carry is already a long-short excess return.

Conclusions

Empirically, I find that value stocks have a considerably lower insurance premium to hedge future volatility shocks, and this can explain their higher average returns; the price of volatility risk, however, is not statistically significant (t-statistic of ), and small stocks do not show the same pattern. I cannot find clearly higher cash-flow betas for value stocks in either the two-beta or the three-beta ICAPM. Chen and Zhao (2009) argued that news about cash flows is computed residually relative to news about discount rates, so the cash-flow news captures all the noise in the VAR model. If a variable is omitted from but belongs to rational investors' information set, it ends up in the innovation of the cash-flow news. Therefore, the cash-flow news is very sensitive to the VAR specification. They showed that value companies do not have larger cash-flow betas under an alternative VAR specification. This is consistent with my beta estimates. Hence, the disagreement on cash-flow betas may stem from the misspecification of the VAR economies.

Limits and next steps

Negative variance forecasts. The linear EVOL regression gives a negative forecast in months; a floored or log-variance specification is the next test.
Cross-section design. A regression with an intercept and GLS weights, and rolling out-of-sample premia from 1990, would test the fit more fairly.
VAR specification. Because cash-flow news is a residual, an alternative state vector (for example with the term spread) could change the cash-flow betas.

References

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Important information

Personal research. Prepared by Johnew Zhang in a personal capacity. It was not prepared for, reviewed or endorsed by any current or former employer and does not reflect their views or activities.

Public data only. Inputs come from the Kenneth French data library, Robert Shiller's online data, FRED and a published currency-carry series. They may be revised and have not been independently verified.

Model outputs, not forecasts. Estimates depend on the specification described. Nothing here is investment advice or a recommendation to buy or sell any security.