Stock Market Ambiguity and the Equity Premium

Transcription

1 Stock Market Ambiguity and the Equity Premium Panayiotis C. Andreou Anastasios Kagkadis Paulo Maio Dennis Philip January 2014 Abstract We create a measure of stock market ambiguity based on investors trading activity in the S&P 500 index options market. We show that, for the 1996: :12 period, ambiguity consistently forecasts negative market excess returns for horizons up to two years ahead, exhibiting a predictive ability comparable to that of the variance risk premium (VRP) and outperforming all other variables examined. Furthermore, its forecasting power remains intact at all horizons when alternative predictors are added into the predictive model. An out-of-sample predictability analysis shows that only ambiguity and VRP outperform the simple historical average. This result is economically significant as revealed by the performance of a trading strategy based on the out-ofsample predictability. The information embedded in stock market ambiguity cannot be subsumed by other option-implied variables that proxy for variance and jump risk or reflect investors hedging demand. Our results are in line with equilibrium models that incorporate Bayesian learning and an elasticity of intertemporal substitution lower than one. Department of Commerce, Finance and Shipping, Cyprus University of Technology, 140, Ayiou Andreou Street, 3603 Lemesos, Cyprus; [email protected] Durham Business School, Durham University, Mill Hill Lane, Durham DH1 3LB, UK; s: [email protected], [email protected] Hanken School of Economics, P.O. Box 479, Helsinki, Finland; [email protected] 1

2 1 Introduction The Ellsberg paradox reveals that decision makers exhibit ambiguity (or Knightian uncertainty ) aversion in the sense that they prefer lotteries with known probability measures from lotteries with unknown probability measures. Since the probability measure governing future stock market prices is clearly unknown, it is apparent that market discount rates reflect investors aversion to both risk and ambiguity. Unlike conditional risk which can be quantified using GARCH and MIDAS models or implied volatility estimates, conditional ambiguity is much more difficult to be estimated. Therefore, the relationship between ambiguity about expected stock market returns and the equity premium remains hardly investigated. One exception is the study of Anderson, Ghysels and Juergens (2009) who use analysts forecasts regarding aggregate corporate profits in order to construct a measure of ambiguity about expected market returns and find that it is significantly positively related to future market returns. In this study, we propose a novel measure of ambiguity about expected market returns that is based on the trading activity of participants in the S&P 500 index options market and investigate its relationship with future market returns. In particular, we capture stock market ambiguity by the dispersion in the volume-weighted strike prices of S&P 500 index options. A high dispersion of trading volume across different strike prices implies that many investors have different subjective beliefs about future market returns, i.e. ambiguity is high. In contrast, a low dispersion across strike prices suggests that investors subjective beliefs tend to be similar, i.e. ambiguity is low. In the limiting case where trading volume exists for only one strike price, ambiguity is equal to zero, since all investors share the same expectations about future returns. Compared to other proxies of ambiguity that rely on analysts forecasts (e.g. Diether, Malloy and Scherbina, 2002 and Anderson, Ghysels and Juergens, 2009), stock turnover (Connolly, Stivers and Sun, 2005), or the press (Baker, Bloom and Davis, 2013), our stock market ambiguity proxy has several advantages. First, it is purely forward-looking as it is based on investors trading activity in the options market. 2

3 Second, it is related directly to expected stock returns and not to aggregate profits, GDP, or any alternative economic activity indicator. Third, it provides a better representation of the true ambiguity as it reflects the subjective beliefs of all the investors who trade in the highly liquid S&P 500 index options market and not the beliefs of only a small group of professional forecasters. Fourth, it can be estimated on a daily basis. We document a significant and robust negative relationship between stock market ambiguity and future market returns. While this result seemingly contradicts the conventional wisdom that higher ambiguity should be associated with a higher equity premium, it is in line with equilibrium models such as Veronesi (2000) that incorporate Bayesian learning and an elasticity of intertemporal substitution (EIS) lower than one. In this setting, investors prefer consumption smoothing and therefore when they are ambiguous averse and there is high ambiguity in the market they will increase their hedging demand for the aggregate risky asset as they are willing to substitute current consumption with increased future consumption. This increased hedging demand will increase the price of the risky asset and will lower the equity premium. In the opposite case of an EIS higher than one, investors do not prefer consumption smoothing and hence their hedging demand will decrease if ambiguity in the market is high, leading to a lower price of the risky asset and a higher equity premium. In that respect, the empirical evidence in this study has important implications about the level of EIS of the representative investor and contributes to the respective long lasting debate in the macro-finance literature (see Beeler and Campbell, 2012 and Bansal et al., 2012). The empirical results show that at the 1-month horizon, stock market ambiguity is a strong predictor of future returns and outperforms in terms of predictive power all other predictors examined in prior literature apart from the variance risk premium (VRP), which explains a higher proportion of the variation in future returns. The significance of stock market ambiguity remains intact when considering bivariate models with alternative predictors as control variables and when considering trivariate models with stock market ambiguity and VRP as the common variables. Therefore, our analysis reveals that stock market ambiguity 3

4 and VRP have different information content and can be used complementarily for forecasting the equity premium. The results from long-horizon regression analysis show that stock market ambiguity remains significant at all horizons and for horizons of 12 and 24 months ahead exhibits an adjusted R 2 higher than 9% which is remarkable given the low persistence of this predictor (about 0.50). Trivariate models with stock market ambiguity and VRP as the main variables confirm that the two measures continue to be jointly successful in predicting long-horizon future market returns. The results of out-of-sample predictive analysis reveal that stock market ambiguity has significantly higher forecasting power than the historical mean and it outperforms all other predictive variables apart from VRP. Following Campbell and Thompson (2008), by imposing a constraint of positive forecasted equity premia improves further the out-of-sample predictability of ambiguity. The results are also economically significant since an active trading strategy based on the out-of-sample predictive power of stock market ambiguity offers increased utility to a meanvariance investor that would otherwise follow a passive buy-hold strategy. Despite the fact that in terms of economic significance ambiguity is outperformed by the VRP, the dividendpayout ratio (d-e), and the default spread (DEF), it is the only variable that when combined with VRP improves the performance of the trading strategy. Therefore, it is confirmed again that stock market ambiguity and VRP act as complementary variables and their joint use for investment decisions can prove very beneficial to an active investor. Finally, we compare stock market ambiguity with other popular option-implied variables in order to alleviate potential concerns about the information embedded in our measure. More specifically, ambiguity is compared to the slope of the implied volatility smirk, the risk-neutral variance, skewness and kurtosis,and the out-of-the-money (OTM) puts to the atthe-money (ATM) calls open interest ratio proxying for investors hedging pressure. Higher ambiguity is associated with higher variance, more negative skewness, higher kurtosis, more negatively sloped volatility smirk, and less hedging pressure. However, the highest correla- 4

5 tion coefficient, which is the one between ambiguity and risk-neutral variance, is only 0.30 revealing that stock market ambiguity does not proxy for any type of variance or tail risk and is not driven by the well-known hedging demand for OTM puts. Bivariate and multivariate regression analysis confirms that in the presence of the alternative option-implied measures, stock market ambiguity remains highly significant in forecasting the equity premium at all horizons. The remainder of the paper is structured as follows. Section 2 presents the theoretical motivation and the description of our stock market ambiguity measure. Section 3 describes the data and the variables used in the study. Section 4 provides the empirical evidence from in-sample regression analysis. Section 5 discusses the results from out-of-sample regression analysis. Section 6 presents the economic significance of the out-of-sample empirical evidence. Section 7 presents the comparison between stock market ambiguity and other option-related variables. Finally, Section 8 concludes. 2 Ambiguity Measurement Since the seminal works of Knight (1921) and Keynes (1936), the term risky has been used to describe a lottery with unknown outcome but known probability distribution of all possible outcomes, while ambiguous has been used to describe a lottery with an unknown probability distribution of possible outcomes. The experimental evidence in Ellsberg (1961) showed that the subjective expected utility theory of Savage (1954) can be violated in reality and underlined the importance of taking into consideration ambiguity aversion in addition to risk aversion when modelling investors decision making. The literature of ambiguity averse behavior can be split into two main categories (see Epstein and Schneider, 2010 for a review). The first category comprises the maxmin expected utility theory of Gilboa and Schmeidler (1989), Epstein and Wang (1994) and Epstein and Schneider (2003). In this setting the decision maker is ambiguous about the probability mea- 5

6 sure governing future consumption and evaluates her expected utility under the worst-case scenario. The second category consists of studies that capture ambiguity averse behavior using lack of reduction between compound probabilities. In particular, an ambiguity averse decision maker evaluates her expected utility in two stages, a first stage that assumes certainty about the probability measure and a second stage that considers all possible probability measures, weighting each of them nonlinearly. Such papers that differ in terms of the axiomatic foundation and generality include Segal (1987, 1990), Klibanoff, Marinacci and Mukerji (2005, 2009), Nau (2006), Seo (2009), Ergin and Gul (2009), and Neilson (2010). 1 In their seminal paper, Klibanoff, Marinacci and Mukerji (2005) introduce the smooth ambiguity model and conceptualize ambiguity as mean-preserving spreads in the distribution of expected utility values induced by all possible probability measures. In particular, the smooth ambiguity model consists of an inner integral that captures expected utility under an (objectively known) first-order probability measure in addition to an outer integral that captures utility under subjective second-order probabilities about the set of all possible probability measures. The shape of the function in the inner integral captures attitudes towards risk and as in standard expected utility theory, it is increasing and concave if the decision maker is risk averse. The shape of the function in the outer integral reflects attitudes towards ambiguity and is increasing and concave if the decision maker is ambiguity averse. An important feature of the smooth ambiguity model in relation to the maxmin expected utility model of Gilboa and Schmeidler (1989) is that it can separate ambiguity itself captured by the second-order probabilities and sensitivity to ambiguity captured by the shape of the function in the outer integral. Hayashi and Miao (2011) extend the smooth ambiguity model into a dynamic setting and Ju and Miao (2012) examine the asset pricing implications of their model by considering Epstein and Zin (1989) and Weil (1989) recursive preferences. This way, they achieve a three- 1 A closely related strand of the literature includes the multiplier preferences (or model uncertainty) theory of Hansen (2007), Hansen and Sargent (2001, 2007, 2008), and Strzalecki (2011) where the decision maker evaluates the relative entropy between her best guess probability measure and all other plausible probability measures, thus capturing possible model misspecification. 6

7 way separation between risk aversion, ambiguity aversion, and the elasticity of intertemporal substitution (EIS). Unlike this framework, prior studies that examine asset pricing implications of ambiguity aversion using the maxmin expected utility framework (e.g. Epstein and Wang, 1994, Chen and Epstein, 2002, Epstein and Miao, 2003, Epstein and Schneider, 2008, Leippold et al., 2008, and Drechsler, 2013) cannot separate risk aversion from intertemporal substitution or ambiguity from ambiguity aversion. Ju and Miao (2012) consider a representative investor that is averse to both risk and ambiguity with an EIS higher than one who updates her beliefs in a Bayesian way. In such a case, periods of low consumption are associated with a decreased demand and a lower price for the risky asset, which creates a highly negative covariance between the pricing kernel and realized market returns and leads to a large equity premium. When the representative investor is ambiguity averse and ambiguity is high in the stock market, 2 the pricing kernel increases even more and the phenomenon is even more pronounced. In the same vein, periods of high consumption are associated with an increase in the price of the risky asset and a low equity premium, and this effect is reinforced when ambiguity is high. It is important to note, however, that as shown in Veronesi (2000) and Ai (2010), in the case of an EIS that is lower than one, then the exactly opposite argument holds and hence higher ambiguity leads to a lower equity premium. Specifically, low current consumption decreases the price of the risky asset but also increases investors hedging demand since their expectations about future consumption are now lower. The latter effect tends to raise the price, cancelling out the initial effect, and therefore the covariance between current consumption and asset returns becomes low leading to a low risk premium. When ambiguity is high the effect is even more pronounced since investors are willing to pay more in order to hedge against the possibility of low future consumption. Therefore, any empirical study that examines the effect of ambiguity on expected stock returns implicitly 2 Ju and Miao (2012) assume that higher ambiguity is mostly associated with bad news about future consumption (i.e. low current consumption) since the posterior belief about the economy being in the high-growth state is almost always close to unity. In this paper, we assume that the level of ambiguity is time-varying but does not necessarily increase with bad news. It can change for example due to an imprecise signal of time-varying information quality. 7

8 provides evidence about the level of EIS as well. Our interest is in quantifying ambiguity about future stock market returns by assuming a Lucas-type economy with aggregate dividends being equal to aggregate consumption. Following the relevant literature we assume that the distribution of expected return values of the representive investor can be inferred from the different subjective expected market returns of a group of investors. Previous studies have tried to quantify ambiguity about individual or aggregate stock returns using dispersion in analysts forecasts (e.g. Diether, Malloy and Scherbina, 2002 and Anderson, Ghysels and Juergens, 2009), stock turnover (Connolly, Stivers and Sun, 2005) and the content of newspapers (Baker, Bloom and Davis, 2013). The aforementioned methods, however, exhibit important disadvantages that make the usage of those measures as proxies of expected stock returns ambiguity questionable. For example, dispersion in analysts forecasts is based on the views of only a few professional forecasters 3 who express their opinions about expected corporate earnings but not specifically about expected stock returns. Moreover, analysts forecasts data are updated on either a monthly or a quarterly basis thus implying that ambiguity remains unchanged within the respective period. Stock turnover or trading volume is clearly affected by the different subjective expected returns among investors but it cannot quantify how much different are those subjective expectations. Finally, it is very difficult to distinguish risk from ambiguity based on the content of newspapers, a fact which is apparent from the high correlation (73.3%) of Baker, Bloom and Davis s (2013) equity market uncertainty index with VIX, which is a measure of risk and not ambiguity. In contrast to the previous studies, our stock market ambiguity measure refers directly to expected market returns and stems from the subjective forward-looking beliefs of all the investors that trade in the highly liquid Standard and Poors (S&P) 500 index options market. Moreover, this proxy increases when the range of pessimistic-optimistic views become more extreme and can be estimated on a daily basis. In particular, our stock market ambiguity 3 The average number of forecasters in Anderson, Ghysels and Juergens (2009) is 36. 8

9 index is based on the simple fact that investors with different subjective beliefs about future expected returns will trade options of different strike prices. For example, more optimistic investors will choose to trade out-of-the-money calls or in-the-money puts while more pessimistic investors will choose to trade out-of-the-money puts or in-the-money calls. Following Garleanu et al. (2009), we assume that the type of option (whether put or call) is irrelevant for the investor and her expectations are reflected only on the strike price. Therefore, using S&P 500 index options data we construct the following two measures that proxy for the dispersion of the expected returns distribution: AMB t = K K w j X j w j X j, (1) j=1 j=1 ) AMBt = K 2 K w j (X j w j X j, (2) j=1 j=1 where w j is the proportion of the total trading volume attributed to the jth strike price X j. AMB corresponds to the mean absolute deviation of the strike prices, while AMB* corresponds to its standard deviation. The two measures have the same information content but AMB* will always be higher or equal to AMB due to Jensen s inequality. The intuition of these measures is straightforward: a higher dispersion of trading volume across strike prices increases ambiguity about future expected returns since investors hold different expectations about future returns. In the limiting case where trading volume is concentrated only at one strike price then ambiguity is equal to zero, since all investors have the same expectations about future returns. 3 Data and Variables In order to create our aggregate stock market ambiguity measures we use call and put options data on the S&P 500 index. Our sample period is 1996:01 to 2012:12 and for each month 9

10 we estimate AMB and AMB* using options on the last trading day of the month with maturities between 10 and 360 calendar days. We consider options with maturities up to one year ahead since we want to capture investors ambiguity regarding both short-term and long-term expected returns. Consistent with this intuition, unreported results show that using an AMB (or AMB*) measure created solely by short-maturity options exhibits similar predictability for short horizons but has limited power for long horizons. We compare the predictive ability of our measures of stock market ambiguity with a set of variables that have been found in the literature to predict stock market returns. The main alternative predictor is the variance risk premium (VRP) introduced by Bollerslev, Tauchen and Zhou (2009) and also examined by Drechsler and Yaron (2011), Bollerslev et al. (2012) and Zhou (2012). VRP is the difference between the expected 1-month ahead stock return variance under the risk neutral measure and the expected 1-month ahead variance under the physical measure. When investors are more averse to future variance risk, they are willing to pay more in order to hedge against variance and therefore increase the VRP. Monthly VRP data are obtained from Hao Zhou s website. 4 The rest of the predictor variables include the tail risk (TAIL, Kelly and Jiang, 2013), the aggregate dividend-price ratio (d-p, Fama and French, 1988 and Cambell and Shiller, 1988a,b), the market dividend-payout ratio (d-e, Cambell and Shiller, 1988a and Lamont, 1998), the yield gap (YG, Maio, 2013a), the yield term spread (TERM, Campbell, 1987 and Fama and French, 1989), the default spread (DEF, Keim and Stambaugh, 1986 and Fama and French, 1989), the relative short-term risk free rate (RREL, Campbell, 1991) and the realized stock market variance (SVAR, Guo, 2006). TAIL captures the probability of extreme negative market returns and is constructed by applying the Hill s (1975) estimator to the whole NYSE/AMEX/NASDAQ cross-section (share codes 10 and 11) of daily returns within a given month. d-p is the difference between the log aggregate annual dividends and 4 Following Bollerslev, Tauchen and Zhou (2009), we use the past 1-month realized variance as the expected 1-month ahead variance under the physical measure. Our results remain robust to the choice of the VRP proxy. 10

11 the log level of the S&P 500 index, while d-e is the difference between the log aggregate annual dividends and the log aggregate annual earnings. YG is the difference between the aggregate earnings-price ratio and the 10-year bond yield, both in levels. TERM is the difference between the 10-year bond yield and the 1-year bond yield, while DEF is the difference between BAA and AAA corporate bonds yields from Moody s. Finally, RREL is the difference between the 3-month t-bill rate and its moving average over the preceding twelve months and SVAR is the monthly variance of the S&P 500 index. Data on monthly prices, dividends, and earnings are obtained from Robert Shiller s website. All interest rate data are obtained from the FRED database of the Federal Reserve Bank of St. Louis. SVAR is downloaded from Amit Goyal s website. As a proxy for stock market returns we use the value-weighted index from the Chicago Center for Research in Security Prices (CRSP). In order to create a series of monthly excess stock market returns we subtract from the monthly log-return the (log of) the 1-month Treasury bill rate obtained from Kenneth French s website. Longer horizons continuously compounded excess market returns are created by taking cumulative sums of monthly excess market returns. Figure 1 plots AMB along with VIX, a popular investor fear indicator capturing market forward-looking variance risk. 5 Both series are standardized for easier comparison. While the two series exhibit some common variation (the correlation coefficient is 30%) they tend to peak at different times. For example, unlike VIX, AMB is increasing but not very high during the 1997 Asian crisis and the 1998 Russian crisis, showing that there was not high ambiguity about the state of the economy during those periods. On the contrary, it exhibits several spikes during the period of the dot-com bubble showing that there were concerns about the very high stock market prices driven by the technology sector. In particular, AMB peaks in 2000:03 when NASDAQ reaches its all-time record high and the U.S. Federal Reserve decreases the fed funds rate for a second time within two months, in 2000:09 when 5 The respective graph for AMB* is very similar and thus omitted. 11

12 NASDAQ slightly recovers before it finally bursts, and finally in 2001:01 when the fed funds rate is decreased twice within one month just before the recession period begins. AMB also peaks in 2001:09 due to the 9/11 terrorist attack and remains relatively low until its peak in 2007:11 just before the beginning of the recent recession period. After the collapse of Lehman Brothers in 2008:09 it increases but not as extremely as VIX showing that ambiguity about the state of the economy unlike risk was not that elevated. Finally, AMB substantially increases during the latest period of the European sovereign debt crisis and takes its all-time high value in 2012:03 after the Eurogroup agreement regarding the second bailout package for Greece, following the concerns about the success of the Private Sector Involvement (PSI) program. Table 1 Panel A reports descriptive statistics about the stock market ambiguity measures and the alternative predictive variables, while Panel B presents the respective correlation coefficients. Both ambiguity measures exhibit very similar statistics with slightly positive skewness and excess kurtosis. Unlike the majority of the alternative predictors, they are only moderately persistent with autocorrelation coefficient of and for AMB and AMB* accordingly. This mitigates the problem of potentially spurious regression results caused by highly persistent regressors (see Valkanov, 2003, Torous, Valkanov, and Yan, 2004, and Boudoukh, Richardson, and Whitelaw, 2008). VRP has also a low autocorrelation coefficient of and exhibits negative skewness and very large kurtosis. The two ambiguity measures are very highly correlated (97%) and close to uncorrelated with VRP (-0.03 and for AMB and AMB* respectively) showing that the stock market ambiguity contains different information than VRP. Finally, ambiguity is negatively correlated with TAIL and to a lesser extent with RREL, while being weakly positively correlated with YG, TERM, DEF, and SVAR. On the other hand, the correlations between ambiguity and both d-p and d-e are very close to zero. Overall, both ambiguity measures are not significantly correlated with any of the alternative predictors, the biggest correlation occurring with TAIL. 12

13 4 In-Sample Predictability In order to gauge the predictive power of our proposed stock market ambiguity measures, we run multiple-horizon regressions of excess stock market returns of the following form: re t+h,h = α h + β hz t + ε t+h,h, (3) where re t+h,h = ( ) 1200 [ret+1 + re h t re t+h ] is the annualized h-month excess return of the CRSP value-weighted index and z t is the vector of predictors. The regression analysis covers the period 1996:1-2012:12 and for each forecasting horizon we lose h observations. Under the null of no predictability the overlapping nature of the data imposes an MA (h 1) structure to the error term ε t+h,h process. To overcome this problem we base our statistical inference on both Newey and West (1987) and Hodrick (1992) standard errors with lag length equal to the forecasting horizon. In general, the Hodrick (1992) standard errors tend to be more conservative, especially in long horizons when the null of no predictability is true (Ang and Bekaert, 2007) but have lower statistical power when the null is false (Bollerslev et al., 2012). The beta coefficients reported in the subsequent tables have been scaled and can be interpreted as the percentage annualized excess market returns caused by a one standard deviation increase in each regressor. 4.1 One-month ahead predictability Table 2, Panel A provides the results for 1-month ahead univariate predictive regressions. The results show that the two ambiguity measures are strong predictors of stock market excess returns as the null hypothesis of no predictability is rejected at either 5% or 1% level based on Newey-West or Hodrick standard errors. The slope estimates are negative and economically significant in both cases: a one standard deviation increase in AMB predicts a negative annualized market excess return of 10.17%, while a one standard deviation increase in AMB* leads to a negative annualized market excess return of 9.37%. The adjusted R 2, 13

14 denoted by R 2, is 2.52% and 2.06% for AMB and AMB* respectively. Turning to the rest of the predictor variables, VRP has a positive slope (16.09), which is significant at the 1% and 5% levels based on Newey-West and Hodrick standard errors, respectively. The corresponding forecasting ratio is relatively large (7.04%). None of the other variables is statistically significant at the 5% level (there is only marginal significance for both RREL and SVAR), a finding which is in line with Goyal and Welch s (2008) conclusion that most of the traditional predictors have performed poorly over the last decades. Moreover, the R 2 s of most of the alternative predictors are either negative or below 1% similar to Goyal and Welch (2008) and Campbell and Thompson (2008). Again, the exceptions are RREL and SVAR, which still deliver lower explanatory ratios than AMB. Next, we assess the robustness of the significant results for AMB and AMB* to the presence of other predictive variables by conducting bivariate regressions. Panel B of Table 2 reports the results. The significance of both AMB and AMB* remains intact in all cases, showing that the information content of stock market ambiguity is distinct from that of other variables that have been used in the literature. It is also interesting to note, that the combination of AMB (AMB*) with VRP renders both variables strongly significant and increases R 2 to 9.32% (8.74%), showing that the stock market ambiguity and the variance risk premium are complementary measures and capture different features of investors expectations. Since stock market ambiguity and VRP appear to be the only successful predictors during our sample period, as a final robustness exercise we run trivariate predictive regressions considering combinations of AMB (or AMB*), VRP, and each of the other variables. Results in Panel C of Table 2 show that stock market ambiguity and VRP continue to be significant at either 1% or 5% level in all cases. Moreover, now YG becomes also strongly significant with a positive predictive slope, in line with Maio (2013a). Overall, the results in Table 2 suggest that in our sample period only stock market ambiguity and VRP are consistently successful in predicting the equity premium, and this predictive power is enhanced when they are combined in the same model. 14

15 The negative sign of the predictive slopes for AMB and AMB* shows that higher stock market ambiguity lowers the expected equity premium. This finding is ostensibly in contrast to conventional wisdom that dictates investors requiring a premium for bearing ambiguity. However, it can be explained in the context of equilibrium models that incorporate Bayesian learning and a risk and ambiguity averse representative investor with EIS less than one, which leads to a scenario in which the hedging demand dominates the static demand for the risky asset (see for example, Veronesi, 2000, Ai, 2010, and Ju and Miao, 2012). It is apparent that the documented negative relationship between stock market ambiguity and expected excess market returns implies an EIS lower than one, meaning that investors prefer consumption smoothing. In that respect, the empirical evidence in this study indirectly contributes to the long existing debate in the literature about the level of EIS for the representative investor. For example, Hall (1988), Campbell and Mankiw (1989), Campbell (1996, 1999, 2003), Yogo (2004, 2006), and Lioui and Maio (2012) estimate the EIS to be lower than one, while Attanasio and Weber (1989), Vissing-Jorgensen (2002), Attanasio and Vissing-Jorgensen (2003), and Guvenen (2006) show that the EIS is higher than one (see also Beeler and Campbell, 2012 and Bansal et al., 2012, for recent discussions on the topic). 4.2 Long-horizon predictability Table 3 provides the results for 3-, 6-, 12- and 24-month ahead univariate predictive regressions. Both AMB and AMB* consistently forecast negative excess market returns and are significant for all horizons except for the case of the 6-month horizon when their significance is limited only to the 10% level based on Newey-West standard errors. The slope estimates continue to be economically significant as a one standard deviation increase in AMB (AMB*) predicts a negative annualized market excess return in the range of 4.54%-6.23% (4.51%-6.33%). In terms of fit, R 2 stays between 2% and 3% for 3- and 6- month horizons but increases substantially and exceeds 9% for 12- and 24-month horizons. This last result is of particular importance given the relatively low persistence of our stock market ambiguity 15

16 variables. The only variables that exhibit higher R 2 at the 24- month horizon are d-p, d-e, and TERM all of which have an autocorrelation coefficient higher than Therefore, we conclude that AMB and AMB* can successfully capture investors ambiguity about both short and long horizon expected market returns. Turning to the alternative predictors, VRP remains strongly significant for 3- and 6- month horizons with large R 2 s, yet its predictive power becomes less significant for longer horizons as in Bollerslev, Tauchen and Zhou (2009), Drechsler and Yaron (2011) and Bollerslev et al. (2012). From the rest of the variables d-p, d-e, TERM, DEF, and SVAR become significant as the horizon increases with almost monotonically increasing R 2 s. Moreover, their Newey-West t-statistics are always considerably higher than the Hodrick t-statistics implying that in many of these cases the significance may have arisen spuriously due to the high persistence of the predictive variables (Ang and Bekaert, 2007). Since the results in Table 3 suggest that only AMB, AMB*, and VRP exhibit a strong and consistent predictive pattern across all horizons, we proceed by examining trivariate 3-, 6-, 12- and 24-month ahead regressions considering combinations of AMB (or AMB*), VRP, and each of the other variables. The results reported in Table 4 suggest that the significance of AMB and AMB* follows the same pattern as in the univariate regressions. In particular, Panels A and B show that AMB and AMB* are significant in almost all the cases for the 3- month horizon, while they become less significant for the 6-month horizon. The predictive slopes, however, remain economically significant ranging from to In all the models considered, VRP continues to be strongly significant. Panels C and D show that for 12- and 24- month horizons both AMB and AMB* become again strongly significant in almost all the cases with economically significant slopes ranging from to As in the univariate analysis, the significance of VRP for 12- and 24- month horizons is weaker. Summarizing, the empirical evidence regarding long-horizon predictability confirms that stock market ambiguity embeds important information about future excess market returns that is not included in any of the other variables considered. Moreover, a combination of 16

17 stock market ambiguity and VRP can provide significant long-horizon predictive power for the equity premium. 5 Out-of-Sample Predictability The results of the previous Section provide convincing evidence that stock market ambiguity can significantly predict future excess stock market returns in-sample (IS). In this Section, we evaluate the out-of-sample (OS) performance of our ambiguity measures following Lettau and Ludvigson (2001), Goyal and Welch (2003, 2008), Guo (2006), and Campbell and Thompson (2008) among others. The purpose of this exercise is to assess the usefulness of stock market ambiguity for an investor who has access only to real time data when making her forecasts and also to gauge regression parameter instability over time. Following the literature we mainly rely on OS regressions of 1-month horizon but for robustness purposes we also report results for 3- and 6-month horizons, keeping in mind the relatively low statistical power of OS regression analysis compared to IS analysis (Inoue and Kilian, 2004). As in Goyal and Welch (2008), Campbell and Thompson (2008), Rapach et al. (2010), and Ferreira and Santa-Clara (2011) we use an expanding window of s h observations to estimate the following predictive model: re t+h,h = α h + β hz t + ε t+h,h, and based on the estimated parameters we form our OS forecasts for the expected excess market return using the concurrent values of the predictor variables: re s+h,h = α s;h + β s;hz s. The initial estimation period is from 1996: :12. and the first prediction is made for 2000:1. This way we create a series of T OS = T s h + 1 OS forecasts that is compared 17

18 to a series of recursively estimated historical averages, which correspond to OS forecasts of a restricted model with only a constant as a regressor. We employ four measures to assess the OS predictability performance of our ambiguity measures. The first measure is the OS R 2 denoted by R 2 OS which takes the form: R 2 OS = 1 MSE U MSE R, where MSE U = 1 T OS T t=s (re t+h,h re t+h,h ) 2 is the mean square error of the unrestricted model and MSE R = 1 T OS T t=s (re t+h,h re t+h,h ) 2 is the mean square error of the restricted model with re t+h,h being the recursively estimated historical average. ROS 2 takes positive values whenever the unrestricted model outperforms the restricted model in terms of predictive power (i.e. MSE U < MSE R ). The second measure of OS performance is the F-test from McCracken (2007): MSE F = (T OS h + 1) MSE R MSE U MSE U, which tests whether MSE U is statistically significantly lower than MSE R. The third OS performance test is the encompassing test of Clark and McCracken (2001): ENC NEW = (T T [ OS h + 1) t=s (ret+h,h re t+h,h ) 2 (re t+h,h re t+h,h ) (re t+h,h re t+h,h ) ], T OS MSE U which examines whether the restricted model encompasses the unrestricted model, meaning that the unrestricted model does not improve the forecasting ability of the restricted model. Statistical inference for the MSE F and the ENC NEW tests relies on the critical values derived by McCracken (2007) and Clark and McCracken (2001) using Monte Carlo simulations. The final measure of OS forecasting performance is the constrained OS R 2 denoted by RC OS 2 suggested by Campbell and Thompson (2008). This measure is the same with R2 OS 18

19 apart from the fact that it sets the OS forecasts of the unrestricted model equal to zero whenever they take negative values. Therefore, an investor s real time equity premium prediction becomes in accordance with standard asset pricing theory. Table 5 presents the results for 1-, 3- and 6-month horizon OS predictability. In the case of 1-month horizon, AMB and AMB* exhibit positive ROS 2 s of 2.04% and 1.53% respectively. For both measures, the MSE F test rejects at 5% level the null hypothesis that the mean square error of the unrestricted model is equal to the mean square error of the restricted model while the ENC NEW test rejects at 5% level the null hypothesis that the restricted model encompasses the unrestricted model. When we impose the restriction of positive expected equity premium the results are improved for both ambiguity measures, with R 2 C OS becoming 3.26% for AMB and 2.74% for AMB*. Turning to the rest of the predictors, only VRP provides a positive R 2 OS of 7.96%. Moreover, the MSE F and ENC NEW tests strongly reject the respective null hypotheses at 5% level. Since univariate analysis suggests that only stock market ambiguity and VRP have significant OS forecasting performance, we proceed by combining the two ambiguity measures with VRP. The results show that the bivariate models increase the ROS 2, which becomes 9.20% in the regression including AMB and VRP and 8.52% in the case of AMB* and VRP confirming that the information content of stock market ambiguity is different from that of VRP. Moreover, the MSE F and ENC NEW tests reject the respective null hypotheses even more decisively. The results for the 3-month horizon are very similar to those for the 1-month horizon albeit slightly less significant for AMB and AMB* and a bit stronger for VRP. This is in line with the IS regression results presented in the previous Section. As in the 1-month horizon analysis, apart from AMB, AMB* and VRP, none of the other predictors exhibit positive ROS 2 s. Furthermore, the bivariate model of stock market ambiguity with VRP is even more successful in OS equity premium predictability. The results for the 6-month horizon are also in the same vein with the evidence from the IS analysis. In particular, the ROS 2 s of AMB and AMB* become slightly negative and the MSE F test cannot reject the null hypothesis 19

20 of equality between the mean square error of the restricted and the unrestricted models. However, the ENC NEW test still rejects the null hypothesis that the restricted model encompasses the unrestricted model for both ambiguity measures. Moreover, the negative ROS 2 s of AMB and AMB* still outperform the R2 OSs of all other predictors except for VRP. If we impose the constraint of positive expected equity premium, R 2 C OS turns positive taking a value of 5.31% and 5.18% for AMB and AMB* accordingly, while all the other predictors (except DEF) have negative values. Finally, despite the negative ambiguity ROS 2 s, combining AMB or AMB* with VRP increases the R 2 OS in comparison to a model that includes only VRP. Overall, the empirical evidence regarding OS equity premium predictability suggests that only stock market ambiguity and VRP are successful predictors at short horizons and that their predictive power is enhanced when they are combined in one bivariate model. 6 Economic Significance In this Section, we evaluate the economic significance of the documented OS predictive power of stock market ambiguity for 1-month ahead stock market excess returns. Following, Goyal and Santa-Clara (2003), Campbell and Thompson (2008), Ferreira and Santa-Clara (2011), and Maio (2013a,b) we create an active trading strategy that relies on the OS forecasting power of our stock market ambiguity measures. In particular, we follow the procedure described in the previous Section and estimate a series of OS excess market return forecasts. 6 Then we consider two scenarios: one where short-sales are not allowed and one where shortsales are allowed. More specifically, in the first scenario we have: a = 1 if re t+1 0 a = 0 if re t+1 < 0, 6 In this Section, the term return refers to simple arithmetic return and not to logarithmic return. 20

21 where a represents the portfolio weight attributed to the stock market index. In the second scenario we have: a = 1.5 if re t+h 0 a = 0.5 if re t+h < 0. The realized returns from the active trading strategy can be represented by: R p,t+1 = ar m,t+1 + (1 a) R f,t+1, where R m,t+1 denotes the simple market return and R f,t+1 denotes the return of the riskless asset. Therefore, following this procedure we create a series of realized portfolio returns based on the OS forecasting power of each forecasting variable and we compare the results with those from a buy-hold strategy. This strategy invests only in the market in case of the first scenario and allocates 150% to the market and -50% to the risk-free asset in case of the second scenario. For each trading strategy, we estimate the mean portfolio return, the standard deviation, and its Sharpe ratio. Moreover, since the Sharpe ratio weights equally the mean and volatility of the portfolio returns, we follow Campbell and Thompson (2008), Ferreira and Santa- Clara (2011), and Maio (2013b) and additionally create a certainty equivalent return (CER) in excess of the buy-hold strategy, assuming a mean-variance investor with risk aversion coefficient equal to three. CER represents the change in investor s utility resulting from her choice to follow the active instead of the passive trading strategy. As an additional performance measure we also estimate the maximum drawdown (MDD), which represents the maximum loss than an investor can incur if she enters the strategy at any-time during its implementation period. All measures apart from the MDD are in annualized terms. Finally, we also report the percentage of months that each active strategy goes long the stock market index. 21

22 The performance results from the strategies are presented in Table 6. When short sales are not allowed, the relatively low means for the AMB and AMB strategies (2.93% and 2.57 % respectively) are accompanied by remarkably low volatilities (9.38% and 9.42% respectively). This leads to annualized Sharpe ratios of 0.31 and 0.27 accordingly, both of which outperform the Sharpe ratio of the buy-hold strategy (0.23). Furthermore, the CER associated with the AMB strategy is 1.94% per year and that of the AMB* strategy is 1.57%, thus showing that the utility associated with the active stock market ambiguity strategies is higher than the utility of the buy-hold strategy. These CER values are outperformed by the CER of the strategy related to VRP, but also by those related to d-e and DEF. This is because stock market ambiguity goes long the risky asset in about only half of the periods thereby avoiding a lot of negative market return realizations, but also ignoring a few large positive spikes. In contrast, both d-e and DEF, despite their poor OS performance at the 1-month horizon, tend to invest much more in the market (in 82.69% and 77.56% of the months, respectively), but also go long the riskless rate during the turbulent periods after the dot-com bubble and the Lehman Brothers collapse. Not surprisingly, the strategies based on AMB, AMB*, VRP, d-e and, DEF strategies also exhibit very low MDDs, with the one related to AMB* having the lowest cumulative loss (-26.59%). The most successful variable in terms of both Sharpe ratio (0.54) and CER (4.80%) is VRP. However, when we combine d-e and DEF in bivariate models with VRP the performance of the respective trading strategies deteriorates in comparison to the trading strategy based solely on VRP. The CER in the bivariate model with d-e becomes 4.61%, while the CER when we include DEF is only 1.87%. In contrast, when we combine either AMB or AMB* in bivariate models with VRP the performance of the respective trading strategies improves substantially in comparison to the strategy based solely on VRP. In particular, the CER of the strategy related to the combination of AMB and VRP is 7.20%, while in the case of AMB* and VRP we obtain 6.42%. These results show that unlike d-e and DEF, the information content of stock market ambiguity is significantly beneficial in economic terms 22

23 for an investor who already uses the VRP in her investment decisions. The pattern in the performance of the different trading strategies is very similar when short-sales are allowed, with the main difference being that extreme realizations (highly positive and highly negative market returns) have now a larger impact on the portfolio returns. Therefore, it turns out that stock market ambiguity underperforms the buy-hold strategy in terms of the Sharpe ratio (marginally so in the case of AMB), but clearly outperforms the passive strategy in terms of the CER measure. As in the first scenario, the CER of the AMB and AMB* strategies (3.89% and 3.14% respectively) is only outperformed by the strategies associated with VRP, d-e, and DEF (9.63%, 8.83% and, 4.66% respectively). However, when the trading strategies rely on bivariate models with VRP as the common variable, the strategies based on combinations of VRP and either d-e or DEF offer a lower CER than those based only on VRP (9.24% and 3.74% accordingly), while the opposite is true for combinations of VRP with either AMB or AMB* (14.48% and 12.91% accordingly). In summary, the empirical evidence in this Section shows that the OS forecasting ability of stock market ambiguity is economically significant, especially for an investor who already considers the information from VRP for her investment decisions. 7 Comparison with Option-Implied Measures The empirical evidence presented in the previous Sections suggests that stock market ambiguity has significant IS and OS predictive power for future market excess returns and its information content is distinct from and complementary to that of VRP. However, one might still argue that our ambiguity measures are driven by the well documented volatility smirk anomaly (Rubinstein, 1994, Jackwerth and Rubinstein, 1996), the hedging demand for OTM puts (Bollen and Whaley, 2004, Garleanu et al., 2009), or that they just proxy for common variance risk captured by VIX. To alleviate such concerns, in this Section we compare stock market ambiguity and its predictive power with a set of popular option-implied variables. 23

24 The first variable is the slope of the implied volatility curve measured as the difference between the (volume-weighted) implied volatility of OTM puts and that of ATM calls (Slope, Xing et al., 2010, Bali et al., 2011). The second variable is the ratio of open interest of OTM puts to the open interest of ATM calls which proxies for hedging pressure in the S&P 500 index options market (HP, Han, 2008). The last three variables are the second, third, and fourth risk-neutral moments (VIX, Skewness and Kurtosis, Ang et al., 2006, Chang et al., 2013). 7 Panel A of Table 7 reports the correlation coefficients between the two ambiguity measures and the other option-implied measures. While AMB (AMB*) displays some common variation with all the other variables, the maximum correlation is 0.30 (0.27) showing that the information embedded in stock market ambiguity is unique and is not subsumed by any other option-implied measure studied in the literature. In general, higher AMB and AMB* values are related to higher implied volatility, more negative skewness, higher kurtosis, and a more negatively sloped implied volatility curve. Moreover, AMB and AMB* are negatively correlated with HP showing that in periods of high demand for portfolio insurance there is less ambiguity about expected returns since the majority of investors anticipate negative jumps. Panel B of Table 7 shows the results of 1-month ahead bivariate predictive regressions with AMB or AMB* as the main variables. In all the models considered, both AMB and AMB* remain significant at either 5% or 1% level based on both Newey-West and Hodrick standard errors. Moreover, none of the other option-related measures exhibits significant predictive ability at the 1-month horizon. Panel C of Table 7 considers the case of multivariate regressions with all the option-implied variables being included into the predictive regression. The results show that AMB and AMB* are strongly significant while again all the other variables remain highly insignificant. 7 Risk-neutral moments are calculated using the model-free method of Bakshi, Kapadia and Madan (2003). The estimated implied volatility has a correlation of 99.7% with VIX and thus for better comparability with other studies we proceed by keeping VIX as our proxy for risk-neutral volatility. 24

25 The multivariate analysis is extended in Table 8 for horizons of 3, 6, 12, and 24 months ahead. The results reveal that both AMB and AMB* exhibit significant forecasting power for all horizons examined at either 5% or 1% level irrespectively of which standard errors are considered. Turning to the rest of the predictors, only VIX appears to be consistently and strongly significant, predicting positive excess market returns for all horizons longer than a quarter ahead. In the case of the 24-month horizon, Slope becomes significant at the 5% level, while Skewness and Kurtosis appear significant at the 5% level only when statistical inference is based on Newey-West standard errors. Overall, the results of this Section suggest that stock market ambiguity is not highly related to other well-established option-implied variables that proxy for hedging demand, crash risk, or variance risk and its predictive power for excess market returns remains intact in the presence of such measures. 8 Conclusion In this article, we create a novel measure for capturing ambiguity about expected market returns that is based on the trades taking place in the S&P 500 index options market. In particular, stock market ambiguity is proxied by the dispersion in trading volume across strike prices. A high dispersion implies that there is little consensus in the market regarding the expected market return, while a low dispersion suggests that investors expectations are similar. Our stock market ambiguity measure is by definition forward-looking, can be easily implemented on a daily basis, is associated directly with market returns (and not with a related indicator such as corporate earnings), and reflects the expectations of all participants in a highly liquid options market instead of those of a small number of professional forecasters. We provide empirical evidence for a strong and robust negative relation between stock market ambiguity and subsequent market returns. Moreover, the low autocorrelation coefficient of our measure alleviates the common concern of potentially spurious regression 25

26 results stemming from a highly persistent predictive variable. In-sample analysis shows that at the 1-month horizon, stock market ambiguity compares favorably to the well-established variance risk premium (VRP) and clearly outperforms all other alternative predictors examined. At longer horizons, ambiguity remains significant and exhibits a high adjusted R 2, outperformed only by highly persistent variables. Most importantly, the significance of stock market ambiguity remains intact at all horizons when VRP and other predictors are added into the predictive model. It is therefore evident that the information content of stock market ambiguity is different from that of VRP and the two variables can be used complementarily for forecasting purposes. The results of out-of-sample analysis reveal that stock market ambiguity has significantly higher predictive power than the simple historical average and its forecasting ability can be enhanced by imposing a restriction of positive forecasted equity premia. Apart from VRP, none of the other alternative predictors examined can improve the simple historical average model. The out-of-sample forecasting power of stock market ambiguity is also economically significant, as indicated by the additional utility offered to an investor who follows an active trading strategy associated with its predictive ability. Moreover, unlike other variables, ambiguity improves the performance of a trading strategy based on VRP, when it is added into the predictive model. We also investigate the relationship of stock market ambiguity with other popular optionimplied variables and show that it does not proxy for any of them. More specifically, ambiguity is associated with higher implied volatility, more negative skewness, higher kurtosis, a more negatively sloped implied volatility curve, and less hedging pressure. However, its correlation with these variables lies between 10% and 30% showing that the information content of stock market ambiguity is largely distinct. Most importantly, a regression analysis confirms that ambiguity remains highly significant at all horizons when combined with the other option-related variables. The documented significant and negative relationship between stock market ambiguity 26

27 and subsequent market returns while paradoxical at a first glance, is in line with equilibrium models that incorporate Bayesian learning and an elasticity of intertemporal substitution (EIS) less than one. Therefore, our study has important implications about two strands of the literature. First, it contributes to the return predictability literature by showing that ambiguity about expected market returns, as captured by our measure, is a strong determinant of market discount rates. Second, it contributes to the long lasting debate about the value of the representative investor s EIS by suggesting that it is below one, i.e. that investors prefer consumption smoothing. 27

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34 Figure 1: Stock Market Ambiguity vs VIX This figure plots the monthly time series of AMB versus VIX for the period 1996:1-2012:12. Both variables have been standardized to have zero mean and variance one. 34

35 Table 1: Descriptive statistics and correlation coefficients of forecasting variables Panel A: Descriptive Statistics AMB AMB* VRP TAIL d-p d-e YG TERM DEF RREL SVAR Mean Median Maximum Minimum Std. Dev Skewness Kurtosis AR(1) AMB 1.00 Panel B: Correlation Coefficients AMB AMB* VRP TAIL d-p d-e YG TERM DEF RREL SVAR AMB* VRP TAIL d-p d-e YG TERM DEF RREL SVAR This table reports descriptive statistics (Panel A) and correlation coefficients (Panel B) of the forecasting variables used in the study. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). The sample period is 1996:1-2012:12. 35

36 Table 2: One-month horizon predictability Panel A: Univariate Panel B: Bivariate Panel C: Trivariate R2 (%) AMB Z R2 (%) AMB* Z R2 (%) AMB VRP Z R2 (%) AMB* VRP Z R2 (%) AMB (-2.51)** [-2.62]*** AMB* (-2.36)** [-2.43]** VRP (4.66)*** (-2.56)** (4.37)*** (-2.37)** (4.33)*** [2.49]** [-2.51]** [2.45]** [-2.25]** [2.43]** TAIL (0.93) (-2.52)** (-0.08) (-2.31)** (0.05) (-2.62)*** (4.36)*** (-0.46) (-2.34)** (4.28)*** (-0.27) [0.85] [-2.67]*** [-0.07] [-2.47]** [0.05] [-2.71]*** [2.47]** [-0.36] [-2.43]** [2.44]** [-0.22] d-p (1.04) (-2.56)** (1.08) (-2.27)** (0.98) (-2.65)*** (4.58)*** (1.38) (-2.31)** (4.52)*** (1.27) [1.12] [-2.62]*** [1.13] [-2.32]** [1.03] [-2.51]** [2.52]** [1.33] [-2.13]** [2.50]** [1.24] d-e (0.31) (-2.60)*** (0.47) (-2.39)** (0.38) (-2.58)** (4.58)*** (-0.10) (-2.37)** (4.53)*** (-0.17) [0.34] [-2.70]*** [0.51] [-2.46]** [0.42] [-2.52]** [2.38]** [-0.10] [-2.25]** [2.38]** [-0.18] YG (1.11) (-2.89)*** (1.67)* (-2.74)*** (1.66)* (-3.22)*** (4.54)*** (2.69)*** (-3.03)*** (4.49)*** (2.65)*** [1.05] [-2.91]*** [1.65] [-2.74]*** [1.62] [-2.95]*** [2.64]*** [2.28]** [-2.72]*** [2.61]*** [2.24]** TERM (0.21) (-2.59)** (0.64) (-2.43)** (0.60) (-2.57)** (4.39)*** (0.43) (-2.37)** (4.34)*** (0.39) [0.21] [-2.64]*** [0.64] [-2.45]** [0.60] [-2.50]** [2.43]** [0.42] [-2.25]** [2.41]** [0.37] DEF (-0.33) (-2.60)** (0.15) (-2.34)** (0.04) (-2.57)** (4.40)*** (0.19) (-2.28)** (4.32)*** (0.08) [-0.37] [-2.60]** [0.16] [-2.36]** [0.05] [-2.51]** [2.45]** [0.20] [-2.20]** [2.44]** [0.08] RREL (1.57) (-2.38)** (1.30) (-2.22)** (1.37) (-2.31)** (4.21)*** (1.71)* (-2.11)** (4.17)*** (1.78)* [1.77]* [-2.41]** [1.42] [-2.23]** [1.49] [-2.24]** [2.51]** [1.70]* [-2.00]** [2.50]** [1.77]* SVAR (-1.66)* (-2.27)** (-1.26) (-2.09)** (-1.32) (-2.60)** (3.27)*** (-0.25) (-2.32)** (3.17)*** (-0.34) [-1.23] [-2.26]** [-0.94] [-2.08]** [-0.98] [-2.55]** [2.45]** [-0.19] [-2.26]** [2.40]** [-0.26] This table reports the results of 1-month ahead predictive regressions for the excess return on the CRSP value-weighted index. The sample period is 1996:1-2012:12. Panel A reports the results of univariate regressions, Panel B the results of bivariate regressions and Panel C the results of trivariate regressions. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). Reported coefficients indicate the percentage annualized excess return resulting from a one standard deviation increase in each predictor variable. Newey and West (1987) and Hodrick (1992) t-statistics with lag length equal to the forecasting horizon are reported in parentheses and square brackets respectively. ***, ** and * denote significance in 1%, 5% and 10% level. 36

37 Table 3: Univariate long-horizon predictability h=3 h=6 h=12 h=24 R 2 (%) R2 (%) R2 (%) R2 (%) AMB (-1.98)** (-1.75)* (-2.43)** (-2.01)** [-1.83]* [-1.53] [-2.12]** [-1.77]* AMB* (-2.19)** (-1.85)* (-2.74)*** (-2.14)** [-1.90]* [-1.55] [-2.21]** [-1.91]* VRP (5.25)*** (3.96)*** (2.40)** (2.29)** [3.62]*** [3.33]*** [2.06]** [1.77]* TAIL (-0.38) (-0.49) (0.74) (1.25) [-0.32] [-0.43] [0.77] [1.32] d-p (1.28) (1.84)* (2.72)*** (4.46)*** [1.25] [1.50] [1.76]* [1.96]* d-e (0.73) (1.23) (2.26)** (3.25)*** [0.67] [0.99] [1.17] [2.18]** YG (1.13) (0.96) (0.77) (0.22) [0.95] [0.76] [0.61] [0.23] TERM (0.28) (0.54) (1.38) (3.27)*** [0.26] [0.46] [1.05] [2.35]** DEF (-0.04) (0.45) (1.23) (2.00)** [-0.05] [0.38] [0.78] [1.27] RREL (1.95)* (1.94)* (1.59) (-0.33) [1.82]* [1.71]* [1.42] [-0.18] SVAR (-1.21) (0.20) (2.05)** (2.64)*** [-0.93] [0.10] [0.77] [1.19] This table reports the results of 3-, 6-, 12- and 24-month ahead univariate predictive regressions for the excess return on the CRSP value-weighted index. The sample period is 1996:1-2012:12. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). Reported coefficients indicate the percentage annualized excess return resulting from a one standard deviation increase in each predictor variable. Newey and West (1987) and Hodrick (1992) t-statistics with lag length equal to the forecasting horizon are reported in parentheses and square brackets respectively. ***, ** and * denote significance in 1%, 5% and 10% level. 37

38 Table 4: Trivariate long-horizon predictability Panel A: 3-month horizon Panel B: 6-month horizon AMB VRP Z R2 (%) AMB* VRP Z R2 (%) AMB VRP Z R2 (%) AMB* VRP Z R2 (%) TAIL (-2.97)*** (5.82)*** (-2.51)** (-3.14)*** (5.68)*** (-2.34)** (-2.58)** (4.52)*** (-1.86)* (-2.62)*** (4.42)*** (-1.81)* [-2.65]*** [3.69]*** [-1.64] [-2.63]*** [3.66]*** [-1.52] [-2.63]*** [3.56]*** [-1.75]* [-2.58]** [3.52]*** [-1.62] d-p (-2.17)** (6.33)*** (1.95)* (-2.15)** (6.21)*** (1.84)* (-2.19)** (5.09)*** (2.46)** (-2.05)** (5.03)*** (2.37)** [-1.72]* [3.71]*** [1.41] [-1.57] [3.69]*** [1.35] [-1.45] [3.62]*** [1.61] [-1.25] [3.62]*** [1.57] d-e (-2.21)** (5.50)*** (0.33) (-2.35)** (5.40)*** (0.26) (-1.86)* (3.92)*** (0.90) (-1.83)* (3.87)*** (0.85) [-1.80]* [3.44]*** [0.26] [-1.76]* [3.43]*** [0.21] [-1.59] [2.70]*** [0.70] [-1.51] [2.69]*** [0.66] YG (-2.97)*** (4.38)*** (2.60)** (-3.23)*** (4.33)*** (2.62)*** (-2.55)** (3.48)*** (2.10)** (-2.64)*** (3.45)*** (2.10)** [-2.16]** [3.88]*** [1.93]* [-2.19]** [3.86]*** [1.91]* [-1.81]* [3.62]*** [1.44] [-1.79]* [3.61]*** [1.42] TERM (-2.21)** (4.87)*** (0.42) (-2.38)** (4.83)*** (0.40) (-1.87)* (3.76)*** (0.69) (-1.88)* (3.73)*** (0.68) [-1.75]* [3.55]*** [0.34] [-1.74]* [3.53]*** [0.32] [-1.54] [3.23]*** [0.54] [-1.51] [3.21]*** [0.52] DEF (-2.65)*** (5.34)*** (0.52) (-2.68)*** (5.18)*** (0.41) (-2.24)** (4.59)*** (1.16) (-2.13)** (4.45)*** (1.04) [-1.92]* [3.60]*** [0.34] [-1.82]* [3.57]*** [0.27] [-1.91]* [3.37]*** [0.73] [-1.77]* [3.34]*** [0.66] RREL (-1.76)* (4.01)*** (2.60)*** (-1.89)* (3.99)*** (2.63)*** (-1.20) (3.08)*** (2.21)** (-1.19) (3.07)*** (2.22)** [-1.34] [3.71]*** [1.86]* [-1.36] [3.69]*** [1.88]* [-1.02] [3.40]*** [1.69]* [-1.02] [3.39]*** [1.71]* SVAR (-2.40)** (4.37)*** (0.53) (-2.52)** (4.21)*** (0.43) (-2.22)** (5.70)*** (2.90)*** (-2.21)** (5.69)*** (2.79)*** [-1.90]* [3.38]*** [0.29] [-1.84]* [3.35]*** [0.24] [-2.00]** [3.97]*** [1.12] [-1.91]* [3.93]*** [1.06] (to be continued) 38

39 (continued) Panel C: 12-month horizon Panel D: 24-month horizon AMB VRP Z R2 (%) AMB* VRP Z R2 (%) AMB VRP Z R2 (%) AMB* VRP Z R2 (%) TAIL (-3.67)*** (2.43)** (-0.65) (-4.16)*** (2.29)** (-0.61) (-2.42)** (1.89)* (0.67) (-2.39)** (1.79)* (0.65) [-2.82]*** [2.15]** [-0.69] [-2.90]*** [2.09]** [-0.64] [-2.03]** [1.69]* [0.75] [-2.19]** [1.64] [0.71] d-p (-3.12)*** (2.89)*** (3.42)*** (-3.21)*** (2.76)*** (3.20)*** (-4.62)*** (2.17)** (7.72)*** (-3.85)*** (2.08)** (7.03)*** [-1.98]** [2.36]** [1.77]* [-1.85]* [2.33]** [1.69]* [-1.46] [2.15]** [1.93]* [-1.35] [2.12]** [1.88]* d-e (-2.93)*** (1.69)* (1.93)* (-3.08)*** (1.60) (1.86)* (-4.20)*** (0.97) (3.90)*** (-3.67)*** (0.88) (3.82)*** [-2.37]** [1.36] [1.23] [-2.36]** [1.33] [1.14] [-2.38]** [0.74] [2.43]** [-2.36]** [0.69] [2.32]** YG (-2.72)*** (2.19)** (1.50) (-3.06)*** (2.11)** (1.50) (-1.99)** (1.89)* (0.45) (-2.09)** (1.80)* (0.42) [-2.22]** [2.45]** [1.05] [-2.29]** [2.41]** [1.04] [-1.74]* [2.49]** [0.42] [-1.86]* [2.43]** [0.39] TERM (-3.04)*** (2.15)** (1.93)* (-3.39)*** (2.01)** (1.93)* (-3.68)*** (1.64) (4.26)*** (-3.67)*** (1.48) (4.23)*** [-2.31]** [1.87]* [1.25] [-2.38]** [1.80]* [1.25] [-2.10]** [1.36] [2.47]** [-2.18]** [1.28] [2.45]** DEF (-3.71)*** (2.69)*** (2.98)*** (-3.78)*** (2.47)** (2.75)*** (-4.21)*** (2.14)** (4.57)*** (-3.57)*** (1.96)* (4.36)*** [-2.83]*** [2.06]** [1.39] [-2.72]*** [1.99]** [1.27] [-2.73]*** [1.81]* [1.95]* [-2.62]*** [1.72]* [1.81]* RREL (-1.78)* (1.96)* (1.50) (-1.93)* (1.89)* (1.51) (-2.20)** (2.00)** (-0.81) (-2.22)** (1.86)* (-0.77) [-1.77]* [2.10]** [1.25] [-1.84]* [2.05]** [1.27] [-1.97]* [1.55] [-0.45] [-2.04]** [1.48] [-0.42] SVAR (-3.62)*** (3.77)*** (4.37)*** (-3.93)*** (3.94)*** (4.60)*** (-3.88)*** (3.23)*** (4.10)*** (-3.67)*** (3.28)*** (4.26)*** [-2.82]*** [2.98]*** [2.20]** [-2.82]*** [2.93]*** [2.10]** [-2.55]** [2.54]** [2.81]*** [-2.62]*** [2.49]** [2.71]*** This table reports the results of 3- (Panel A), 6- (Panel B), 12- (Panel C) and 24-month (Panel D) ahead trivariate predictive regressions for the excess return on the CRSP value-weighted index. The sample period is 1996:1-2012:12. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). Reported coefficients indicate the percentage annualized excess return resulting from a one standard deviation increase in each predictor variable. Newey and West (1987) and Hodrick (1992) t-statistics with lag length equal to the forecasting horizon are reported in parentheses and square brackets respectively. ***, ** and * denote significance in 1%, 5% and 10% level. 39

40 Table 5: Out-of-sample predictability 1-month horizon 3-month horizon 6-month horizon R 2 OS (%) MSE-F ENC-NEW R2 C OS (%) R2 OS (%) MSE-F ENC-NEW R2 C OS (%) R2 OS (%) MSE-F ENC-NEW R2 C OS (%) AMB ** 6.15** ** 3.98** ** 5.31 AMB* ** 5.43** ** 4.11** ** 5.18 VRP ** 11.27** ** 22.44** ** 14.59** TAIL d-p ** d-e YG ** ** ** TERM DEF ** RREL SVAR * AMB & VRP ** 18.41** ** 28.16** ** 19.97** AMB* & VRP ** 17.50** ** 28.36** ** 20.29** This table reports the results of 1-, 3- and 6-month ahead out-of-sample predictability for the excess return on the CRSP value-weighted index. The total sample period is 1996:1-2012:12 and the forecasting period begins in 2000:1. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). R OS 2 is the out-of-sample coefficient of determination, MSE-F is the McCracken (2007) F-statistic, ENC-NEW is the encompassing test of Clark and McCracken (2001) and R C OS 2 is the out-of-sample coefficient of determination when the out-of-sample prediction is restricted to be positive. ** and * denote significance in 5% and 10% level. 40

41 Table 6: Economic significance Mean (%) St. Dev. (%) Sharpe CER (%) MDD (%) Long (%) Panel A: No Short Sales Buy & Hold AMB AMB* VRP TAIL d-p d-e YG TERM DEF RREL SVAR AMB & VRP AMB* & VRP VRP & d-e VRP & DEF Panel B: Short Sales Buy & Hold AMB AMB* VRP TAIL d-p d-e YG TERM DEF RREL SVAR AMB & VRP AMB* & VRP VRP & d-e VRP & DEF This table reports the results of trading strategies based on the 1-month ahead out-of-sample predictability for the excess return on the CRSP value-weighted index. The total sample period is 1996:1-2012:12 and the forecasting period begins in 2000:1. Panel A shows the results when short sales are not allowed and Panel B when short sales are allowed. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), variance risk premium (VRP), tail risk (TAIL), dividend-price ratio (d-p), dividend payout ratio (d-e), yield gap (YG), yield term spread (TERM), default spread (DEF), relative short-term risk-free rate (RREL) and realized stock market variance (SVAR). Mean denotes the average return, St. Dev. denotes the standard deviation of returns, Sharpe stands for the Sharpe ratio, CER is the certainty equivalent return, MDD stands for the maximum drawdown and Long is the percentage of months that the strategy goes long the market index. All measures of performance apart from MDD are in annualized terms. 41

42 Table 7: Comparison with other option-implied measures - One-month horizon Panel A: Correlation Coefficients Slope HP VIX Skewness Kurtosis AMB AMB* Panel B: Bivariate Regressions Slope HP VIX Skewness Kurtosis AMB (-2.51)** (-2.41)** (-2.99)*** (-2.40)** (-2.38)** [-2.57]** [-2.52]** [-3.03]*** [-2.51]** [-2.47]** Z (0.42) (-0.05) (0.69) (0.15) (-0.09) [0.43] [-0.05] [0.74] [0.16] [-0.10] R 2 (%) AMB* (-2.36)** (-2.24)** (-2.75)*** (-2.24)** (-2.21)** [-2.41]** [-2.33]** [-2.79]*** [-2.31]** [-2.27]** Z (0.36) (-0.07) (0.60) (0.14) (-0.07) [0.37] [-0.07] [0.65] [0.15] [-0.08] R 2 (%) Panel C: Multivariate Regressions AMB Slope HP VIX Skewness Kurtosis R2 (%) (-2.77)*** (0.50) (0.08) (0.56) (0.70) (0.73) [-2.79]*** [0.51] [0.08] [0.60] [0.69] [0.73] AMB* Slope HP VIX Skewness Kurtosis R2 (%) (-2.51)** (0.44) (0.04) (0.49) (0.65) (0.68) [-2.55]** [0.45] [0.04] [0.52] [0.63] [0.68] This table reports the results of 1-month ahead predictive regressions for the excess return on the CRSP value-weighted index. The sample period is 1996:1-2012:12. Panel A reports the correlation coefficients, Panel B the results of bivariate regressions and Panel C the results of multivariate regressions. The forecasting variables are the two stock market ambiguity proxies (AMB, AMB*), slope of the implied volatility curve (Slope), hedging pressure (HP), implied volatility (VIX), risk-neutral skewness (Skewness) and risk-neutral kurtosis (Kurtosis). Reported coefficients indicate the percentage annualized excess return resulting from a one standard deviation increase in each predictor variable. Newey and West (1987) and Hodrick (1992) t-statistics with lag length equal to the forecasting horizon are reported in parentheses and square brackets respectively. ***, ** and * denote significance in 1%, 5% and 10% level. 42

43 Table 8: Comparison with other option-implied measures - Long horizons Panel A: h=3 AMB Slope HP VIX Skewness Kurtosis R2 (%) (-2.37)** (0.63) (1.59) (1.01) (-0.72) (-0.52) [-2.02]** [0.56] [1.36] [1.11] [-0.76] [-0.62] AMB* Slope HP VIX Skewness Kurtosis R2 (%) (-2.44)** (0.59) (1.51) (0.95) (-0.76) (-0.55) [-2.00]** [0.51] [1.31] [1.05] [-0.79] [-0.65] Panel B: h=6 AMB Slope HP VIX Skewness Kurtosis R2 (%) (-2.55)** (0.50) (1.46) (3.14)*** (-0.09) (0.62) [-2.27]** [0.37] [1.21] [1.86]* [-0.09] [0.75] AMB* Slope HP VIX Skewness Kurtosis R2 (%) (-2.58)** (0.41) (1.38) (3.02)*** (-0.13) (0.60) [-2.29]** [0.31] [1.16] [1.79]* [-0.14] [0.72] Panel C: h=12 AMB Slope HP VIX Skewness Kurtosis R2 (%) (-3.90)*** (0.72) (1.67)* (3.46)*** (0.45) (0.94) [-2.80]*** [0.45] [1.18] [2.51]** [0.51] [1.26] AMB* Slope HP VIX Skewness Kurtosis R2 (%) (-4.18)*** (0.56) (1.47) (3.54)*** (0.40) (0.93) [-2.89]*** [0.35] [1.07] [2.45]** [0.46] [1.25] Panel D: h=24 AMB Slope HP VIX Skewness Kurtosis R2 (%) (-5.38)*** (2.25)** (0.58) (3.17)*** (2.30)** (2.57)** [-2.69]*** [2.14]** [0.67] [2.53]** [1.51] [1.44] AMB* Slope HP VIX Skewness Kurtosis R2 (%) (-5.20)*** (2.25)** (0.47) (3.17)*** (2.27)** (2.49)** [-2.90]*** [2.17]** [0.53] [2.48]** [1.51] [1.44] This table reports the results of 3- (Panel A), 6- (Panel B), 12- (Panel C) and 24-month (Panel D) ahead multivariate predictive regressions for the excess return on the CRSP value-weighted index. The sample period is 1996:1-2012:12. The forecasting variables are stock market ambiguity (AMB), slope of the implied volatility curve (Slope), hedging pressure (HP), implied volatility (VIX), risk-neutral skewness (Skewness) and risk-neutral kurtosis (Kurtosis). Reported coefficients indicate the percentage annualized excess return resulting from a one standard deviation increase in each predictor variable. Newey and West (1987) and Hodrick (1992) t-statistics with lag length equal to the forecasting horizon are reported in parentheses and square brackets respectively. ***, ** and * denote significance in 1%, 5% and 10% level. 43