The asset pricing methodology we employ follows the stochastic discount factor (SDF) approach, similar to that of Cochrane (2005), and utilized by modern asset pricing studies. We specialize our analysis to a linear setting, in the tradition of Fama and French (1993).25 We employ three different tests: the J-test, the HJ distance, the delta-J test and the supLM test. The basic setup may be expressed in the following manner.
Consider an n× 1 vector of gross returns R and a vector of asset prices p. Under conditions of no arbitrage there can be shown to exist a stochastic discount factor m, such that the following pricing relation holds:
E(Rm) = p (7)
If we introduce a k-vector of risk factors f , and specialize to linear factor models, the relevant SDF is of the form
m = b0+ fb1 (8)
where b0 is a constant, and b1 is a k-vector of coefficients.26 It can be demonstrated (Cochrane (1996);
Ferson (2003)) that there is an equivalence between the linear discount factor of equation (8) and a factor pricing model expressed using factor risk premiums and betas. The equivalence can be expressed in the following manner, E(R) = R0p + βλ. In this notation, the unconditional riskless rate is R0= b 1
0+E(f)b1, the vector of projections of asset returns on factors is denoted β = cov(r, f)var(f)−1, and the risk premia may be calculated as
λ =−R0cov(f, f)b1 (9)
Now that we have displayed the basic framework we shall discuss the three tests in turn. First is the J test.
We estimate the parameters b = {b0 b1} by optimal GMM of Hansen (1982). From the data b is chosen to minimize the following objective function: b = arg min Jt = gT(b)W gT(b). Here g is defined as gT(b) = T1 Tt=1Rtyt− p, the vector of sample pricing errors, y is the candidate SDF, and W is the optimal weighting matrix. Hansen (1982) derives the distribution of the associated J-test statistic as
T∗ JT ∼ χ2(n − k) (10)
where n is the number of orthogonality conditions and k is the number of parameters estimated.
One shortcoming of the J test is that it is model-specific; one might improve Jt = gT(b)S−1gT(b) by inflating estimates of S rather than by lowering pricing errors gT. Therefore we also consider a second test,the HJ distance of Hansen and Jagannathan (1997) To understand the HJ distance one can proceed in
25For further exposition of the SDF and linear factor model approaches, see Campbell (2003) and Ferson (2003), respectively. For asset pricing studies that apply these frameworks, see Ang et al. (2006b); and Vassalou and Xing (2004).
26The setup here closely follows that of Cochrane (2005), which derives the results in more detail.
the following fashion. Consider a proxy SDF y and the set of correct SDFs, M . The HJ distance δ is the minimum distance to the nearest correct SDF, and may be defined as
δ = min
m∈L2 y − m (11)
subject to E(mR) = p or, equivalently, δ2 = min
m∈L2 sup
λ∈Rn
E(y− m)2+ 2λ[E(mR) − p]. (12)
Hansen and Jagannathan (1997) show that the solution to this program can be expressed as δ = [E(yR− p)E(RR)−1E(yR − p)]1/2, and that the estimation of the model’s parameters can be cast in a GMM framework such that HJ distance is minimized. Empirically, this amounts to choosing b as
b = arg min δ2 = arg min gT(b)WTgT(b) (13) where WT = T1 Tt=1(RtRt)−1. This is the approach we use for constructing the HJ distance metric.
Hansen and Jagannathan (1997) also note that the HJ distance can be interpreted as the maximum pricing error for the test portfolios, with (portfolio) return having a norm of unity.
The third test is the delta-J test. The delta-J test examines whether other risk factors (in this case, HML and SMB) have any additional explanatory power in the presence of the proposed models. Suppose we have two sets of factors, f1 and f2, and wish to determine whether the set f2 is irrelevant in the presence of f1. One method, akin to the classical Likelihood Ratio test, is to estimate both the unrestricted and restricted models, respectively, m = b1f1+ b2f2and m = b1f1, then compare the J test statistic defined in (10) above.
The J statistic should be larger for the restricted case since there are fewer parameters to estimate. To assess whether the increase in the J statistic is significant, we utilize the delta-J statistic, which is distributed as
ΔJ = T Jrestricted − T Junrestricted ∼ χ2(q) (14) where q is the number of restrictions. For example, in the context of our framework, f1 can correspond to CAPM augmented with liquidity and the tail risk factor, and f2corresponds to HML and SMB.
Figure 1: Average Monthly Tail Risk in Asset Returns
The figure shows the tail index estimator of Hill (1975), applied to stock returns. The tail index is estimated from equation (2) in the cross-section of stocks every day to obtain a market tail risk index. The threshold level is the lowest 5% of the data. In this figure the market tail index is the averaged over all days in each month to obtain a measure of monthly market tail risk. The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. Shaded areas denote NBER recessions. The sample period is 1964 to 2010.
Figure 2: Daily Return Tail Index using Alternative Thresholds
The figure shows the return tail index estimated using the method of Hill (1975), from equation (2). We use both 10% and 5% thresholds, that is, we estimate the index using the lowest 10 and 5 percent of returns in the cross section of stocks each day, respectively. The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. The sample period is 1964 to 2010.
Figure 3: Tail Risk and Volatility: Daily Data
The figure shows the tail index estimator of Hill (1975), applied to stock returns. The tail index is estimated from equation (2) in the cross-section of stocks every day to obtain a daily tail risk index for the market. The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. Also shown is the volatility measure VXO, which measures the implied volatility of an-at-the- money option. The tail index is in green, while the volatility VXO is blue. The sample period is 1986 to 2010.
Figure 4: Tail Risk and Volatility: Monthly Data
The figure shows the monthly average of the tail risk estimator of Hill (1975), applied to stock returns.
The tail index is estimated from equation (2) in the cross-section of stocks every day to obtain a daily tail risk index for the market. This daily index is then averaged each month to obtain a monthly tail index.
The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. Also shown is the monthly average of the volatility measure VXO, which measures implied volatility of an at-the-money option. The tail index is in green, while the volatility VXO is blue. The sample period is 1986 to 2010.
Table 1: Performance of Portfolios based on Return Tail Risk and Liquidity
The table presents the average returns of portfolios sorted on tail risk in returns and liquidity. St. Dev.
denotes the standard deviation. The letters ’RTI’ and ’LIQ’ denote portfolios sorted on sensitivity to the tail index for returns and the liquidity measure of Amihud (2002), respectively, as in equation (5). For example, the portfolio ’LIQ2’ corresponds to the returns on firms that are in the second (2) most sensitive quintile to liquidity risk. Portfolio returns are annualized and in percentages, so that 1 represents 1%). The data comprise firms with prices between $5 and $1000, and include common stocks listed on NYSE, AMEX and NASDAQ during the sample period. The time period is 1964 through 2010.
Panel A: Portfolios Sorted onβR, Sensitivity to Return Tail Index
Panel B: Portfolios Sorted onβliq, Sensitivity to Liquidity
Table 2: Properties of Return Tail Risk Factor, ’TR’
The table presents summary statistics of the tail risk factor, in terms of average returns and correlation with other factors. t-statistics are in square brackets. The letters ’TR’ (’LIQ’) denote the return tail risk (liquidity) factor, which is computed as the difference in returns between stocks with highest and lowest sensitivity to the tail index (liquidity). Our measure of liquidity is that of Amihud (2002). We sort all firms in June each year according to their respective sensitivity to tail risk, as estimated in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year. The return difference between the highest (5) and lowest (1) sensitivity portfolios represents the tail risk factor TR. All factors are annualized and in percentage points, so that 1 represents 1%). Data comprise firms with prices between $5 and $1000, and include firms listed on NYSE, AMEX and NASDAQ during the sample period. The time period is 1964 through 2010.
Panel A: Average Returns
MKT SMB HML LIQ TR
Mean 5.15 3.49 4.61 1.52 4.87
[2.21] [2.14] [3.05] [0.95] [3.25]
Panel B: Correlations
MKT SMB HML LIQ TR
MKT 1 0.3088 -0.3066 -0.0501 -0.0322
SMB 1 -0.2348 -0.0559 0.1309
HML 1 0.1765 0.0023
LIQ 1 0.0671
TR 1
Figure 5: The Return Tail Risk Factor
The figure shows the tail risk factor TR, which is computed as the difference in returns between stocks with highest and lowest sensitivity to the daily tail index. We sort all firms in June each year according to their respective sensitivity to tail risk as described in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year. The return difference between the highest (5) and lowest (1) sensitivity portfolios represents the tail risk factor. The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. Shaded areas denote NBER recessions. The sample period is 1964 to 2009.
Figure 6: The Exposure of Financial Firms to Tail Risk
The figure presents the proportion of financial firms and other characteristics of the stocks in our Tail Risk portfolios. We sort all firms in June each year according to their respective sensitivity to tail risk as in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year.
The letters ’TI’ in the bottom bar denote portfolios sorted on sensitivity to the tail index. For example, the portfolio ’TI2’ corresponds to the returns on firms that are in the second (2) most sensitive quintile to tail risk. All portfolios are value weighted. Financial firms are those with SIC code 6000. The proportion of financial firms is in percentage points. Leverage is calculated as the ratio of total debt to total assets. Book-to-Market and leverage are computed using the value in Compustat, as of December 31 of year t-1, for the portfolios that are formed in June of year t. Book-to-market and leverage are winsorized at 1% and 99%.
The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000. The sample period is 1964 to 2009.
19600 1970 1980 1990 2000 2010
10
19600 1970 1980 1990 2000 2010
1 2 3
C. Book−to−market Ratio
TI1 TI2 TI3 TI4 TI5
Table 3: Characteristics of Portfolios for Return Tail Risk and Liquidity
The table presents characteristics of the firms in our Return Tail Risk and Liquidity portfolios. We sort all firms in June each year according to their respective sensitivity to return tail risk as in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year. The letters ’TI’ and LIQ denote portfolios sorted on sensitivity to the return tail index and liquidity, respectively. For example, the portfolio ’TI2’ corresponds to the returns on firms that are in the second (2) most sensitive quintile to tail risk. Financial firms are those with SIC code 6000. The proportion of financial firms is in percentage points.
Leverage is calculated as the ratio of total debt to total assets. Book-to-Market and leverage are computed using the value in Compustat, as of December 31 of year t-1, for the portfolios that are formed in June of year t. Book-to-market and leverage are winsorized at 1% and 99%. Standard deviations are in parentheses.
All portfolios are value weighted. The time period comprises 1964 through 2009.
Panel A: Tail Index Portfolios
TI1 TI2 TI3 TI4 TI5
% Financial Firms 10.06 15.28 17.93 16.32 11.47 (5.09) (7.50) (8.90) (8.45) (5.21)
% Financial Firms 11.01 15.74 17.48 15.84 11.02 (6.10) (8.14) (9.09) (8.37) (5.27) Leverage 0.5148 0.5223 0.5334 0.5277 0.5135 (0.0384) (0.0453) (0.0526) (0.0442) (0.0355)
Book-to-Market 0.81 0.82 0.82 0.79 0.78
(0.37) ( 0.31) (0.30) (0.29) ( 0.36)
Figure 7: Tail Risk for Financial Firms
The figure’s upper panel displays the return tail index, computed for all firms, and for only financial firms.
The tail index is estimated for daily stock data using the method of Hill (1975), from equation (2), using the cross section of returns each day. Then, we average it across all stocks to obtain a market tail index, reported in the figure’s upper panel. The lower panel presents the Dow Jones Industrial Average (DJIA), both levels and returns. The data comprise NYSE, AMEX and NASDAQ stocks with prices between $5 and $1000.
The sample period is 1964 to 2010.
Table 4: Properties of Return Tail Index for Financial Firms
The table presents summary statistics of the return tail index, computed both for all firms and for financial firms only. The tail index is estimated from the cross section of daily returns using the method of Hill (1975), as in equation (2). All tail indices refer to the left tail, unless otherwise specified. The market tail index is computed in two steps. First we compute the tail index for each stock from the cross section of returns.
Then we average the tail index across all stocks to obtain a market return tail index, reported in the Table.
P-values are presented in parentheses. Min, Max and St. Dev. denote minimum, maximum and standard deviation, respectively. RTI and RTI(Fin) denote the return tail index for all companies and for financial companies only, respectively. DJIA is the level of the Dow Jones Industrial Average, and DJIA(Ret) is the daily return on the Dow Jones Industrial Average. Data comprise firms with prices between $5 and $1000, and include firms listed on NYSE, AMEX and NASDAQ during the sample period, January 1964 through December 2010.
Table 5: Stationarity and Unit Root Tests
The table presents the unit root test of Phillips and Perron (1988), and the KPSS stationarity test of Kwiatkowski et al. (1992). St. Dev is the standard deviation. Month-end TI denotes the tail index cal-culated at the end of the month. Avg. TI denotes the average tail index each month. TR denotes the tail risk factor. Yield denotes the spread between BAA and AAA bonds, available from the Federal Reserve Bank of St. Louis. Variables are evaluated at the one month frequency. P-values are in parentheses. The time period comprises 1964 through 2009.
MKT Yield TR Month-end TI Avg. TI Mean 0.0900 0.0105 0.0047 -2.5913 -2.6919 St. Dev. 0.1588 0.0014 0.0276 0.4735 0.2420 H0: unit root -20.50 -1.07 -20.21 -2.84 -0.86
( 0.001) (0.259) (0.001) (0.005) (0.338) H0: stationarity 0.12 3.50 0.15 0.27 1.64
(0.100) (0.010) (0.048) (0.010) (0.010)
Table 6: Causality Tests
The table presents causality tests based on the approach of Granger (1969). The framework is a 2-variable vector autoregression (VAR). The null hypothesis is that all coefficients are zero, which is assessed by an F-test. The symbols TR and MKT denote the tail risk factor and market return, respectively. Yield denotes the spread between BAA and AAA bonds, available from the Federal Reserve Bank of St. Louis. The tail index is evaluated at the month’s end. AIC and BIC are the Akaike and Bayesian information criteria. All dependent variables are annualized. The symbols ***, **, and * denote p-values smaller than 0.01, 0.05, and 0.10, respectively. The time period comprises 1964 through 2009.
Panel A: Choice of Optimal Lag in Bivariate Vector Autoregression
Model 1 Model 2 Model 3 Model 4
TR vs Yield TR vs MKT Tail Index vs Yield Tail Index vs MKT
Lags AIC BIC AIC BIC AIC BIC AIC BIC
1 -6075.59 -6049.90 -7129.25 -7103.57 -3051.74 -3026.06 -4101.41 -4075.73 2 -6126.04 -6083.23 -7123.85 -7081.05 -3111.37 -3068.56 -4111.34 -4068.54 3 -6142.19 -6082.27 -7120.08 -7060.15 -3117.88 -3057.95 -4108.20 -4048.28 4 -6137.17 -6060.12 -7112.71 -7035.66 -3117.13 -3040.08 -4106.25 -4029.20 5 -6142.05 -6047.88 -7114.19 -7020.02 -3123.22 -3029.05 -4116.80 -4022.63 6 -6157.90 -6046.61 -7110.22 -6998.92 -3134.50 -3023.21 -4116.80 -4005.51 7 -6153.01 -6024.59 -7103.13 -6974.72 -3130.73 -3002.32 -4110.59 -3982.18 8 -6154.61 -6009.08 -7098.34 -6952.81 -3135.31 -2989.78 -4106.94 -3961.41 9 -6150.43 -5987.78 -7092.98 -6930.33 -3134.19 -2971.54 -4101.88 -3939.22 10 -6146.08 -5966.30 -7089.22 -6909.44 -3135.35 -2955.58 -4102.89 -3923.11 11 -6149.27 -5952.38 -7088.50 -6891.61 -3133.68 -2936.78 -4102.03 -3905.13 12 -6145.03 -5931.01 -7082.72 -6868.70 -3137.82 -2923.80 -4103.86 -3889.84
Panel B: F-statistic from Granger Causality test (lag determined by BIC)
Model 1: TR vs Yield Model 2: TR vs MKT Model 3: Tail Index vs Yield Model 4: Tail Index vs MKT Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2
TR Yield TR MKT Tail Index Yield Tail Index MKT
Variable 1 4.40** 5.73*** 6.34** 0.20 7.66*** 1.95 3.01* 3.35*
Variable 2 1.65 4110.61*** 4.20** 4.61** 1.46 3973.72*** 3.34* 5.18**
Panel C: F-statistic from Granger Causality test (lag determined by AIC)
Model 1: TR vs Yield Model 2: TR vs MKT Model 3: Tail Index vs Yield Model 4: Tail Index vs MKT Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2 Variable 1 Variable 2
TR Yield TR MKT Tail Index Yield Tail Index MKT
Variable 1 2.16** 6.15*** 6.34** 0.20 4.86*** 0.72 6.47*** 1.35
Variable 2 2.36** 1339.12*** 4.20** 4.61** 1.50 692.76*** 2.25** 1.90*
Table 7: Predictability Tests
The table presents tests of predictability for the market return (MKT) and yield spread (Yield), using the tail index and tail risk factor TR. In Panel A, the regression tests are of the form M KTt= α+βT R·T Rt−1+εt and Y ieldt= α +βT R·T Rt−1+εt. Panel B presents the same estimation except that the tail index replaces TR on the right hand side, and the coefficient is now βT I. All dependent variables are annualized so that coefficients can be compared directly. Yield denotes the spread between BAA and AAA bonds, available from the Federal Reserve Bank of St. Louis. The tail index is evaluated at the month’s end. We consider horizons of 1 month, 1 year, 3 years, and 5 years. Standard errors are in square brackets, and computed using the method of Hodrick (1992). The time period comprises 1964 through 2009.
Panel A: Predictability using Tail Risk Factor TR
Dependent Variable:MKTt Dependent Variable:Y ieldt
1 Month 1 Year 3 Year 5 Year 1 Month 1 Year 3 Year 5 Year α 0.0904 0.0900 0.0937 0.0954 0.0104 0.0104 0.0104 0.0105 [0.0236] [0.0238] [0.0242] [0.0244] [0.0002] [0.0002] [0.0002] [0.0002]
βT R -0.0893 -0.7188 0.0486 0.1002 0.0113 0.0271 0.0079 0.0064 [1.0322] [0.2974] [0.1109] [0.0892] [0.0085] [0.0025] [0.0009] [0.0007]
Panel B: Predictability Using Market Tail Index
Dependent Variable:MKTt Dependent Variable:Y ieldt
1 Month 1 Year 3 Years 5 Years 1 Month 1 Year 3 Years 5 Years α -0.1286 0.0943 0.1267 0.1292 0.0142 0.0123 0.0109 0.0105 [0.1395] [0.0558] [0.0543] [0.0501] [0.0013] [0.0004] [0.0005] [0.0005]
βT I -0.0820 0.0029 0.0123 0.0125 0.0014 0.0007 0.0002 0.0000 [0.0528] [0.0200] [0.0189] [0.0181] [0.0005] [0.0001] [0.0002] [0.0002]
Table 8: The Price of Exposure to Tail Risk
The table presents the estimated risk premia, which measure the return per unit of exposure to each risk factor. More details are in the Appendix. The letters ’TR’ (’LIQ’) denote the return tail risk (liquidity) factor, which is computed as the difference in returns between stocks with highest and lowest sensitivity to the tail index (liquidity). We sort all firms in June each year according to their respective sensitivity to tail risk as in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year. The return difference between the highest (RTI5) and lowest (RTI1) sensitivity portfolios represents the tail risk factor. FF3 denotes the Fama-French 3-factor model. The test statistics are the 5x5 book to market portfolios, available from the website of Kenneth French. Estimation is performed by GMM of Hansen (1982). Robust t-statistics are in square brackets. The data comprise common stocks on NASDAQ, NYSE and AMEX with at least 120 trading days in the relevant year. The time period is years 1964 through 2009.
Model: CAPM CAPM CAPM CAPM FF3 FF3
& LIQ & TR & LIQ, TR & LIQ, TR Estimated Risk Premia
MKT 0.0037 0.0030 0.0058 0.0063 0.0046 0.0060 [1.89] [1.57] [2.38] [2.46] [2.14] [2.10]
Table 9: Asset Pricing Tests
The table presents the results of asset pricing tests on our sample. Estimation is performed using GMM.
Robust t-statistics are in square brackets, and p-values are in parentheses. The J-test is the over-identifying restriction test of Hansen (1982). HJ-distance refers to the distance metric of Hansen and Jagannathan (1997). Large p-values for the J-statistic and HJ distance indicate that the particular model fits well. The delta-J test of Newey and West (1987) assesses whether the inclusion of HML and SMB improves model fit. A small p-value for the delta-J test indicates that additional factors improve model fit. The letters ’TR’
(’LIQ’) denote the return tail risk (liquidity) factor, which is computed as the difference in returns between stocks with highest and lowest sensitivity to the tail index (liquidity). We sort all firms in June each year according to their respective sensitivity to tail risk as in equation (5). We then form 5 quintile portfolios and compute monthly returns over the subsequent year. The return difference between the highest (RTI5) and lowest (RTI1) sensitivity portfolios represents the tail risk factor. All portfolios are value weighted.
FF3 denotes the Fama-French 3-factor model. The data comprise common stocks on NASDAQ, NYSE and AMEX with at least 120 trading days in the relevant year. The time period is 1964 through 2009.
Model: CAPM CAPM CAPM CAPM FF3 FF3
& LIQ & TR & LIQ, TR & LIQ, TR J-Statistic 46.45 46.89 31.78 29.53 36.82 20.13
(0.00) (0.00) ( 0.08) ( 0.10) (0.02) (0.39)
HJ Distance 0.36 0.36 0.31 0.31 0.31 0.25
(0.00) ( 0.00) (0.07) (0.05) (0.00) (0.41)
Delta-J 9.63 13.27 9.58 9.39
(0.01) (0.00) (0.01) (0.01)
Table 10: Robustness to Value-Weighted Liquidity
The table presents estimated premia and results of asset pricing tests on our sample, where we calculate liquidity portfolios that are value-weighted. Estimation is performed using GMM. Robust t-statistics are in
The table presents estimated premia and results of asset pricing tests on our sample, where we calculate liquidity portfolios that are value-weighted. Estimation is performed using GMM. Robust t-statistics are in