Conclusions - 高階系統性風險之研究

Market volatility-of-volatility appears to be a state variable that is important for asset pricing. We develop a market-based three-factor model, in which market risk, market volatility risk, and market volatility-of-volatility risk determine the cross-sectional asset prices. We find that market volatility-of-volatility risk is priced in the cross-sectional stock returns. Stocks with negative larger return exposure to market volatility-of-volatility have substantially higher future stock returns, even after we account for exposures to the

Fama and French four factors, market skewness factor, firm characteristics and volatility characteristics. We also find that market volatility-of-volatility risk is priced in the cross-sectional variance risk premium.

Our measure of market volatility-of-volatility generates leverage effect and feedback effect. Stocks with negative larger return exposure to market volatility-of-volatility have substantially lower contemporaneous stock returns, which suggests that market volatility-of-volatility is priced such that an anticipated increase in market volatility-volatility-of-volatility risk raises the required rate of return, leading to an immediate stock price decline and higher future returns. Our evidence on return predictability for the aggregate market portfolio supports feedback effect implied by our model. The predictability evidence afforded by the market volatility also suggests that economic volatility-of-volatility is an important state variable.

Our study shows that market volatility-of-volatility risk affects the cross-sectional expected variance risk premium. One direction for future research is to explore whether market volatility-of-volatility risk plays a role in tradable volatility-related assets such as equity option returns or index option returns. Future research could also investigate whether our measure of market volatility-of-volatility affects the VIX option returns.

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Table 1. 1: Properties of the daily factors

We report summary statistics and Spearman correlations for the daily factors, including Fama and French (1993) four factors (MKT, SMB, HML, and UMD), the market variance risk premium (VRP), the VIX index, our measure of variance of market variance (VoV), and Chang, Christoffersen, and Jacobs (2013) market skewness factor (SKEW) and market kurtosis factor (KURT). ΔVIX is the first difference of VIX.

ΔVoV, ΔSKEW, and ΔKURT are the residuals from fitting an ARMA(1,1) regression using VoV, SKEW, and KURT, respectively. The sample period is from January 1996 to December 2010.

Panel A: Summary statistics

MKT(%) SMB(%) HML(%) UMD(%) VRP(%) VIX(%) VoV(%) SKEW KURT Mean 0.023 0.010 0.016 0.024 17.260 23.098 0.054 -1.663 9.313 Median 0.070 0.030 0.020 0.070 15.082 22.150 0.003 -1.637 8.672 Std.Dev. 1.300 0.629 0.682 1.035 21.182 9.509 0.563 0.485 3.466

Panel B: Spearman correlation

MKT SMB HML UMD VRP ΔVIX ΔVoV ΔSKEW ΔKURT

MKT 1.000

SMB 0.038 1.000

HML -0.279 -0.082 1.000

UMD -0.047 0.053 -0.078 1.000

VRP -0.222 -0.050 0.006 0.062 1.000

ΔVIX -0.779 0.031 0.210 0.020 0.180 1.000

ΔVoV -0.044 -0.003 -0.038 0.036 0.145 0.049 1.000

ΔSKEW -0.237 -0.022 0.026 0.034 0.076 0.248 -0.017 1.000

ΔKURT 0.311 0.014 -0.057 -0.017 -0.106 -0.307 -0.010 -0.863 1.000

Table 1. 2: Portfolios sorted on

At the end of each month, we run the following regression for each stock using daily returns:

, , , , , , .

We sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5). After the portfolio formation, we calculate the value-weighted daily and monthly stock returns for each portfolio. The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. For each portfolio, we estimate the same time-series regression as above using the post-formation daily portfolio returns to obtain the post-formation factor loadings. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD) from the intercept estimate of a time-series regression of the monthly portfolio returns on the four factors. Numbers in parentheses are t-statistics. Size reports the average market capitalization (in billion) for firms within the portfolio; B/M reports the average book-to-market ratios; RET_2_12 reports the average of past 11-month returns prior to last month; ILLIQ reports the average of Amihud’s (2002) illiquidity measure. The sample period is from January 1996 to December 2010.

Portfolios ranking

1 2 3 4 5 5-1 Risk-adjusted performance of _, sorted portfolios (monthly return)

Excess return 0.90 0.64 0.40 0.34 0.02 -0.88

Table 1. 3: Portfolios sorted on

, controlling for Size, B/M, momentum, reversal, and illiquidity

This table shows performance of portfolios sorted on _, , controlling for market capitalization (Size), book-to-market ratio (B/M), past 11-month returns (RET_2_12), past 1-month return (RET_1), and Amihud’s illiquidity (ILLIQ), respectively. We first sort stocks into five quintiles based on their market capitalization (Size). Then, within each quintile, we sort stocks based on their _, loadings into five portfolios. All portfolios are rebalanced monthly and are value weighted. The five portfolios sorted on

, are then averaged over each of the five Size portfolios, resulting , quintile portfolios controlling for Size. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD). B/M, RET_2_12, RET_1, and ILLIQ are analyzed with the same procedure as described above. Numbers in parentheses are t-statistics.

Portfolios ranking on _, α-FF4

1 2 3 4 5 5-1 5-1

Controlling for Size 0.81 0.94 0.76 0.75 0.39 -0.42 -0.45

( 1.30) ( 2.12) ( 2.00) ( 1.80) ( 0.70) (-1.93) (-2.14)

Controlling for B/M 1.10 0.71 0.49 0.47 0.29 -0.81 -0.88

( 2.19) ( 1.95) ( 1.48) ( 1.31) ( 0.59) (-2.81) (-3.15)

Controlling for RET_2_12 0.53 0.52 0.41 0.29 0.04 -0.49 -0.52

( 1.05) ( 1.21) ( 0.99) ( 0.69) ( 0.07) (-2.01) (-2.12)

Controlling for RET_1 0.69 0.64 0.46 0.43 0.11 -0.59 -0.61

( 1.29) ( 1.54) ( 1.20) ( 1.08) ( 0.20) (-1.95) (-2.06)

Controlling for ILLIQ 0.79 0.81 0.75 0.67 0.28 -0.51 -0.53

( 1.38) ( 2.09) ( 2.24) ( 1.84) ( 0.56) (-2.18) (-2.30)

Table 1. 4: Portfolios sorted on

At the end of each month, we run the following regression for each stock using daily returns:

, , , , , , .

We sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5). After the portfolio formation, we calculate the value-weighted daily and monthly stock returns for each portfolio. The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. For each portfolio, we estimate the same time-series regression as above using the post-formation daily portfolio returns to obtain the post-formation factor loadings. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD) from the intercept estimate of a time-series regression of the monthly portfolio returns on the four factors. Numbers in parentheses are t-statistics. Size reports the average market capitalization (in billion) for firms within the portfolio; B/M reports the average book-to-market ratios; RET_2_12 reports the average of past 11 month returns prior to last month; ILLIQ reports the average of Amihud’s (2002) illiquidity measure. The sample period is from January 1996 to December 2010.

Portfolios ranking

1 2 3 4 5 5-1 Risk-adjusted performance of _, sorted portfolios (monthly return)

Excess return 0.76 0.58 0.37 0.38 -0.11 -0.87

Table 1. 5: Two-way sorted portfolios on

and

At the end of each month, we run the following regression for each stock using daily returns:

, , , , , , .

We sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5), and independently sort stocks into quintile portfolios based on _, . All portfolios are rebalanced monthly and are value weighted. The five portfolios sorted on , are then averaged over each of the five _, portfolios, resulting _, quintile portfolios controlling for _, . Similar approach gives

, quintile portfolios controlling for , . The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. For each portfolio, we estimate the same time-series regression as above using the post-formation daily portfolio returns to obtain the post-formation factor loadings. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD). Numbers in parentheses are t-statistics. Panel A and Panel B present the results for _, quintile portfolios and , quintile portfolios, respectively. The sample period is from January 1996 to

Table 1. 6: Two-way sorted portfolios on

and

At the end of each month, we separately run the following regressions for each stock using daily returns:

, , , , ,

, ,

, , , , , , .

, quintile portfolios controlling for , . The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. For each portfolio, we estimate the same time-series regression as above using the post-formation daily portfolio returns to obtain the post-formation factor loadings. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD). Numbers in parentheses are t-statistics. Panel A presents the results for the , quintile portfolios. Panel B shows the results for the _, quintile portfolios controlling for _, quintile portfolios while Panel C shows the results for the , quintile portfolios controlling for ,

quintile portfolios. The sample period is from January 1996 to December 2010.

Portfolios ranking

Table 1. 7: The price of volatility-of-volatility risk

Panel A reports the Fama–MacBeth (1973) factor premiums on 25 portfolios sorted on intersection of ,

quintile portfolios and _, quintile portfolios, using our market-based three factors (MKT, ΔVIX, and ΔVoV), Chang, Christoffersen, and Jacobs (2013) market skewness factor (ΔSKEW), and Fama-French four factors (MKT, SMB, HML, and UMD). We estimate the first stage return betas using the daily full-sample post-formation value-weighted returns. Then, we regress the cross-sectional monthly portfolio returns on daily return betas from the first stage, using Fama–MacBeth (1973) cross-sectional regression. Panel B reports the Fama–MacBeth (1973) factor premiums on 25 portfolios sorted on intersection of ,

quintile portfolios and , quintile portfolios. Robust Newey and West (1987) t-statistics with six lags that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2010.

Table 1.7 (continued.)

Fama-MacBeth cross-sectional regressions

[1] [2] [3] [4] [5] [6]

Panel B: 25 portfolios sorted on _, × _, (5×5)

MKT 0.48 0.45 0.61 0.43 0.64 0.64

( 1.16) ( 1.12) ( 1.57) ( 1.06) ( 1.64) ( 1.63)

ΔVIX -0.31 4.51 -0.79 8.47 8.46

(-0.17) ( 1.27) (-0.45) ( 1.73) ( 1.62)

ΔVoV -2.00 -2.07 -1.76 -1.87 -1.86

(-1.84) (-1.77) (-1.77) (-1.86) (-1.75)

SMB -0.31 -0.51 -0.50

(-0.37) (-0.60) (-0.58)

HML -1.40 -0.87 -0.87

(-2.30) (-1.32) (-1.24)

UMD 1.70 2.33 2.33

( 1.09) ( 1.44) ( 1.43)

ΔSKEW 2.73 2.77 2.76

( 2.53) ( 1.45) ( 1.36)

ΔKURT 5.69 0.58

( 0.59) ( 0.05)

Adj. R² 0.10 0.15 0.26 0.13 0.26 0.26

Table 1. 8: Two-way portfolios sorted on volatility spreads and

This table shows performance of portfolios sorted on implied-realized volatility spread (IVOL-TVOL), the call-put implied volatility spread (CIVOL-PIVOL), the expected individual variance risk premium (EVRP), and _, using stocks with available equity options. We independently sort stocks into quintile portfolios based on each of the four variables, from the lowest (quintile 1) to the highest (quintile 5). All portfolios are rebalanced monthly and are value weighted. We compute the risk-adjusted return of each portfolio with respect to Fama-French four factors (MKT, SMB, HML, and UMD). Panel A reports the results for the one-way sorted portfolios. We construct two-one-way sorted portfolios formed on intersection of each of the volatility spread quintile portfolios and _, quintile portfolios. Panel B shows the results for the _, quintile portfolios controlling for volatility spread quintile portfolios. Numbers in parentheses are t-statistics. The sample period is from January 1996 to December 2010.

Portfolios ranking α-FF4 Panel B: Two-way sorted portfolios, ranking on _,

Controlling for IVOL-TVOL 0.97 0.77 0.45 0.32 0.28 -0.69 -0.76

Table 1. 9: Firm-level Fama-MacBeth regressions

This table reports the results for the firm-level Fama-MacBeth regressions. We run the following cross-sectional regression:

, , , , , , , ,

, , , ,

where the dependent variable is the monthly individual stock returns; _, _, , _, _,, and _, _, are post-ranking betas estimated from the 25 portfolios formed on intersection of , quintile portfolios and

, quintile portfolios; _, consists of Size, B/M, RET_2_12, RET_1, and ILLIQ;

, includes IVOL-TVOL, CIVOL-PIVOL, and EVRP. Following the methodology of Fama and French (1992), we assign each of the 25 portfolio-level post-ranking beta estimates to the individual stocks within the portfolio at that time. Robust Newey and West (1987) t-statistics with six lags that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2010.

Table 1. 10: Portfolios sorted on

We sort stocks into quintile portfolios based on _, , from the lowest (quintile 1) to the highest (quintile 5). After the portfolio formation, we calculate the value-weighted daily 30-day model-free implied variance and monthly 30-day variance risk premium for each portfolio. The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. For each portfolio, we estimate the post-ranking variance betas by running the following regression using daily portfolio implied variance:

, , , , .

Numbers in parentheses are t-statistics. The sample period is from January 1996 to December 2010.

Portfolios ranking

Table 1. 11: The price of volatility-of-volatility risk in cross-sectional

EVRP

This table reports the Fama–MacBeth (1973) factor premiums on 25 portfolios sorted on _, , using our market-based three factors (MKT, ΔVIX, and ΔVoV), Chang, Christoffersen, and Jacobs (2013) market skewness factor (ΔSKEW), and Fama-French four factors (MKT, SMB, HML, and UMD). For each portfolio, we estimate the post-ranking variance betas by running the following regression using daily portfolio implied variance:

, , , , .

Then, we regress the cross-sectional monthly portfolio expected variance risk premium on the post-ranking variance betas using Fama–MacBeth (1973) cross-sectional regression:

, , .

In column from 4 to 6, we include the post-ranking return betas obtained from running regression using daily portfolio returns on the risk factors:

, , , , , ,

, , .

Robust Newey and West (1987) t-statistics with six lags that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2010.

Fama-MacBeth cross-sectional regressions

Table 1. 12: Return predictability regressions

Panel A reports the estimates of the one-period return predictability regression using daily market return on the lagged variance risk premium (VRP), variance of market variance (VoV), market skewness (SKEW), and market kurtosis (KURT). In panel B, we use the monthly market return as the dependent variable, and the independent variables are sampled at the end of the previous month. We multiply Daily market return in Panel A is multiplied by 22. Robust Newey and West (1987) t-statistics with sixteen lags in Panel A and with six lags in Panel B that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2010.

Figure 1. 1: Daily market volatility-of-volatility (VoV)

We plot daily market volatility-of-volatility over the time period January 1996 through December 2010.

We partition the tick-by-tick S&P500 index options data into five-minute intervals, and then we estimate the model-free implied variance for each interval. For each day, we use the bipower variation formula on the five-minute based annualized 30-day model-free implied variance estimates within the day, resulting in our daily measure of market volatility-of-volatility (VoV).

Figure 1. 2: Performance of portfolios sorted on

in event time

At the end of each day, we estimate the regression of equation (1.33) using daily stock returns over the past 22 days. We then sort stocks into quintile portfolios on the estimated , for each day and calculate the event-time daily value-weighted portfolio returns ranging from -11 to 11 in days.

Figure 1. 3: Cross-correlations

The plots are based on the pre-formation and post-formation of quintile portfolio return differentials (low-minus-high; long the lowest quintile and short the highest quintile) formed on _, . The top panel shows the sample cross-correlation between the VIX and portfolio formation time leads and lags of the low-minus-high ranging from -11 to 11 days. The bottom panel shows the sample cross-autocorrelations between the market volatility-of-volatility (VoV) and the returns.

Chapter 2 A Model-Free CAPM with High Order Risks

2.1 Introduction

The concept of linear risk-return trade-off has been the keystone in finance theory. For example, in addition to the market risk of the classical capital asset-pricing model (e.g.

Sharpe (1964) and Lintner (1965)), prior literature has illustrated the important role of the stock return exposures to multiple factors (e.g. Fama and French (1992; 1993; 1995; 1996) and Carhart (1997)) and to the high order of market moments (see, for example, Kraus

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