Variance Risk Premia, Asset Predictability Puzzles, and Macroeconomic Uncertainty

Transcription

1 Variance Risk Premia, Asset Predictability Puzzles, and Macroeconomic Uncertainty Hao Zhou Federal Reserve Board First Draft: May 2009 This Version: November 2010 Abstract This paper presents predictability evidence from the difference between implied and expected variances or variance risk premium that: (1) variance risk premium predicts a significant positive risk premium across equity, bond, and credit markets; (2) the predictability is short-run, in that it peaks around one to four months and dies out as the horizon increases; and (3) such a short-run predictability is complementary to that of the standard predictor variables P/E ratio, forward spread, and short rate. Using a general equilibrium model of economic uncertainty under recursive preference, I provide calibration evidence that stochastic volatility-of-volatility can potentially explain both the variance risk premium dynamics and its short-run predictability across financial markets. However, it remains a challenge to simultaneously match both the level and the predictability in bond returns and credit spreads. JEL classification: G12, G13, G14. Keywords: Short-run predictability, variance premium dynamics, equity premium puzzle, bond risk premia, credit spread puzzle, macroeconomic uncertainty, recursive preference. I benefited from helpful discussions with Gurdip Bakshi, Ravi Bansal, Tim Bollerslev, Andrea Buraschi, John Campbell, Peter Christoffersen, Alex David, Bjørn Eraker, Jeremy Graveline, Pete Kyle, Mark Ready, Philippe Mueller, Anna Cieslak, George Tauchen, Andrea Vedolin, Jonathan Wright, Hong Yan, Moto Yogo, Harold Zhang, and Guofu Zhou. I am also grateful for the comments received from seminar participants at Bank for International Settlements and Hong Kong Monetary Authority, Tsinghua University, Federal Reserve Bank of San Francisco, Renmin University, University of Texas at Dallas, Duke University, Purdue University, University of Wisconsin-Madison, UBC Winter Finance Conference, Mont Tremblant Risk Management Conference, University of Calgary, and Empirical Asset Pricing Retreat in Amsterdam. I would like to acknowledge the recognitions from Chicago Quantitative Alliance (CQA) Academic Competition (3rd place) and Crowell Memorial Prize from PanAgora Asset Management (3rd place). James Marrone and Paul Reverdy provided excellent research assistance. The views presented here are solely those of the author and do not necessarily represent those of the Federal Reserve Board or its staff. Risk Analysis Section, Federal Reserve Board, Mail Stop 91, Washington DC 20551, USA; hao.zhou@frb.gov, Phone , Fax

2 Variance Risk Premia, Asset Predictability Puzzles, and Macroeconomic Uncertainty Abstract This paper presents predictability evidence from the difference between implied and expected variances or variance risk premium that: (1) variance risk premium predicts a significant positive risk premium across equity, bond, and credit markets; (2) the predictability is short-run, in that it peaks around one to four months and dies out as the horizon increases; and (3) such a short-run predictability is complementary to that of the standard predictor variables P/E ratio, forward spread, and short rate. Using a general equilibrium model of economic uncertainty under recursive preference, I provide calibration evidence that stochastic volatility-of-volatility can potentially explain both the variance risk premium dynamics and its short-run predictability across financial markets. However, it remains a challenge to simultaneously match both the level and the predictability in bond returns and credit spreads. JEL classification: G12, G13, G14. Keywords: Short-run predictability, variance premium dynamics, equity premium puzzle, bond risk premia, credit spread puzzle, macroeconomic uncertainty, recursive preference.

3 1 Introduction Option implied volatility, such as the Chicago Board Option Exchange s VIX index, is widely viewed by investors as the market gauge of fear (Whaley, 2000). 1 In recent research, the difference between the implied and expected volatilities has been interpreted as an indicator of the representative agent s risk aversion (Rosenberg and Engle, 2002; Bakshi and Madan, 2006; Bollerslev, Gibson, and Zhou, 2010a). An alternative interpretation is that the impliedexpected variance difference, as a proxy for variance risk premium, is due to the macroeconomic uncertainty (Bollerslev, Tauchen, and Zhou, 2009; Drechsler and Yaron, 2010). Such an approach relies on the non-standard recursive utility framework of Epstein and Zin (1991) and Weil (1989), such that the consumption uncertainty commands a time-varying risk premium. This method follows the spirit of the long-run risks models as pioneered by Bansal and Yaron (2004), but focuses on the stochastic consumption volatility as a primary source of the time variation in market risk premia by shutting down the long-run risk channel completely. This paper demonstrates that the difference between implied and expected variances, as a measure for variance risk premium, provides a significant predictability for short-run equity returns, bond returns, and credit spreads. The documented return predictability peaks around one-to-four month horizons across these markets, and then dies out as the forecasting horizon increases. More importantly, such a short-term forecastability of market risk premia is complementary to the standard predictors P/E ratio, forward spread, and short rate that are established by asset pricing theories or vast empirical evidence. When combined together with these standard predictors, the statistical significance of the variance risk premium proxy is rather increased or at least stable, instead of being crowded out by the standard predictor variables. This constitutes an important evidence that risk premia across major financial markets comove in the short-run, and such a common component 1 For example, in the final quarter of 2008, the VIX index has closed above 50 percent for almost twelve weeks and peaked around 90 percent. As reported by the Wall Street Journal on November 12, 2008, if market volatility continues to remain above 50 percent for just over five weeks, it would have surpassed the Great Depression in the 1930s; and such a high volatility signifies all those unknowns that are a greater cloud of what we call Uncertainty. 1

4 seems to be intimately related to the variance risk premium, which is constructed from the high quality derivatives market data and high-frequency underlying market prices. This type of common short-run risk factor may be a proxy for stochastic economic uncertainty or consumption volatility risk that varies independently with the consumption growth. To interpret the empirical findings in this paper, I adopt a self-contained general equilibrium framework incorporating the effect of time-varying economic uncertainty, where the uncertainty risk is priced only under the recursive utility function. Calibration evidence shows that such a framework can replicate to a certain degree the observed skewness and kurtosis in variance premium dynamics, without introducing jumps into the endowment process and/or the volatility process. The stylized model and its extended versions can potentially explain the unconditional risk premia on equity, bond, and credit markets. More importantly, under calibrated or simulated regressions, such a framework can partially replicate the short-run predictability pattern of variance risk premium across these asset markets. However, there seems to be a conflict for the proposed stochastic volatility model in simultaneously matching both the level and the predictability for bond and credit markets without introducing richer fundamental dynamics. Focusing exclusively on the volatility channel for generating time-varying risk premia (Tauchen, 2005; Eraker, 2008) represents a departure from the long-run risk models (Bansal and Yaron, 2004), where consumption growth and consumption volatility are both priced risk factors under recursive preference. 2 Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2010) present the predictability evidence of variance risk premium for stock market returns and single out volatility or jump risk as the primary driver of economic uncertainty. Eraker and Shaliastovich (2008) and Bansal and Shaliastovich (2008b) design long-run risk models with jump and volatility risk factors for explaining the predictability puzzles in bond and currency. Buraschi, Trojani, and Vedolin (2010a) and Wang, Zhou, and Zhou (2010) interpret the close linkage between variance risk premia and credit spreads as 2 There is a fundamental link between the notion of option-implied volatility risk premium (Heston, 1993; Bates, 1996; Bakshi and Kapadia, 2003) and the notion of variance risk premium embedded in underlying assets as examined here. However, as a unique risk factor in a consumption-based asset pricing framework, variance risk premium must be positive as more risk requires more return compensation. 2

5 endogenously determined by economic uncertainty. Benzoni, Collin-Dufresne, and Goldstein (2008), Drechsler (2008), and Shaliastovich (2009) try to jointly explain the option pricing puzzles and observed variance risk premium dynamics under recursive preference. In this paper, I try to explain the common short-run predictability of variance risk premium for equity, bond, and credit risk premia, by focusing on the stochastic volatility channel only without relying on jump risk or long-run risk or the correlation between volatility and growth components. 3 Asset pricing with economic uncertainty can be studied with other techniques under the recursive preference setup, for example, by information learning (Bansal and Shaliastovich, 2008a; Shaliastovich, 2009), Markov switch (Lettau, Ludvigson, and Wachter, 2008; Chen and Pakos, 2008), Knightian uncertainty (Drechsler, 2008), among others. Conceptually, the stochastic volatility approach taken here can be justified by a real business cycle model with shocks in total factor productivity and firm-specific technology (Bloom, 2009; Bloom, Floetotto, and Jaimovich, 2009). Such an approach treats the short-run economic uncertainty as a fundamental priced risk factor, without relying on informational learning, heterogeneous beliefs, or behavioral assumptions. The emphasis here is also complementary to the long-run risk perspective (Bansal and Yaron, 2004) or rare disaster angle (Barro, 2006; Gabaix, 2009), and is orthogonal to the channel of time-varying risk aversion (Campbell and Cochrane, 1999). Lastly, but not the least, this framework has sharp empirical testable hypotheses, since the economic uncertainty factor or stochastic volatility-of-volatility is uniquely loaded on the variance risk premium proxy. Therestofthepaperwillbeorganizedasthefollows, thenextsectiondefinesthevariance risk premium and describes its empirical measurement; Section 3 presents the main empirical evidence of the short-run predictability from the implied-expected variance difference for risk 3 Beeler and Campbell (2009) argue that the calibration exercise in Bansal, Kiku, and Yaron (2007) puts more emphasis on the persistent volatility channel than the long-run risk channel. Bekaert, Hoerova, and Scheicher (2009) rely on VIX to assess the relative importance between time-varying risk aversion and economic uncertainty. Pástor and Stambaugh (2009) use the uncertainties of expected future and current returns to argue that long-run stock returns are indeed more volatile. Buraschi, Trojani, and Vedolin (2010b) examine volatility risk premium under heterogeneous beliefs with independent Lucas trees. Zhou and Zhu (2009) incorporate two volatility factors into the long-run risks model. Bollerslev, Sizova, and Tauchen (2010b) apply a similar framework to account for volatility asymmetry and dynamic dependency. 3

6 premia across asset markets; the following section discusses a general equilibrium model with stochastic economic uncertainty and provides some calibration implications for explaining the short-run risk premia dynamics; and Section 5 concludes. 2 Variance Risk Premium and Empirical Measurement The central empirical finding of this paper is that market risk premia have a common shortrun component variance risk premium that is not directly observable. However, an empirical proxy can be constructed from the difference between model-free option-implied variance and the conditional expectation of realized variance. 2.1 Variance Risk Premium: Definition and Measurement To define the procedure in quantifying the model-free implied variance, let C t (T,K) denote the price of a European call option maturing at time T with strike price K, and B(t,T) denote the price of a time t zero-coupon bond maturing at time T. As shown by Carr and Madan (1998) and Britten-Jones and Neuberger (2000), among others, the market s riskneutral expectation of the return variance between time t and t + 1 conditional on time t information, or the implied variance IV t,t+1, can be expressed in a model-free fashion as a portfolio of European calls, ) K C t (t+1, C IV t,t+1 E Q B(t,t+1) t (t,k) t (Var t,t+1 ) = 2 dk, (1) 0 K 2 which relies on an ever increasing number of calls with strikes spanning from zero to infinity. 4 This equation follows directly from the classical result in Breeden and Litzenberger (1978), that the second derivative of the option call price with respect to strike equals the risk-neutral density, such that all risk neutral moments payoff can be replicated by the basic option prices (Bakshi and Madan, 2000). In practice, IV t,t+1 must be constructed on the basis of a finite number of strikes; which turns out to be a fairly accurate approximation 4 Such a characterization abstracts from the realistic economic environment that allows for (1) lumpy dividend payment, (2) stochastic interest rate, (3) underlying asset jumps, and (4) limited number and range of option strikes resulting in discretization and truncation errors. See Jiang and Tian (2005) for detailed discussions. 4

7 to the true (unobserved) risk-neutral expectation of return variance, under reasonable assumptions about the underlying asset dynamics (Jiang and Tian, 2005; Carr and Wu, 2009; Bollerslev, Gibson, and Zhou, 2010a). In order to define the measure in quantifying the actual return variance, let p t denote the logarithmic price of the asset. The realized variance over the discrete t to t+1 time interval can be measured in a model-free fashion by RV t,t+1 n j=1 [ p t+ j n p t+ j 1 n ] 2 Vart,t+1, (2) where the convergence relies on n ; i.e., an increasing number of within period price observations. As demonstrated in the literature (see, e.g., Andersen, Bollerslev, Diebold, and Ebens, 2001; Barndorff-Nielsen and Shephard, 2002), this model-free realized variance measure based on high-frequency intraday data offers a much more accurate ex-post observation of the true (unobserved) return variance than the traditional ones based on daily or coarser frequency returns. In practice, various market microstructure frictions invariably limit the highest sampling frequency that may be used in reliably estimating RV t,t+1. Variance risk premium (VRP) at time t is defined as the difference between the ex-ante risk-neutral expectation and the objective or statistical expectation of the return variance over the [t,t+1] time interval, VRP t E Q t (Var t,t+1 ) E P t (Var t,t+1 ), (3) which is not directly observable in practice. To construct an empirical proxy for such a VRP concept, one needs to estimate various reduced-form counterparts of the risk neutral and physical expectations, i.e., VRPt ÊQ t (Var t,t+1 ) ÊP t (Var t,t+1 ). In practice, the risk-neutral expectation ÊQ t (Var t,t+1 ) is typically replaced by the CBOE implied variance (VIX 2 /12) and the true variance Var t,t+1 is replaced by realized variance RV t,t+1. To estimate the objective expectation, ÊP t (Var t,t+1 ), I use a linear forecast of realized variance as RV t,t+1 = α+βiv t 1,t +γrv t 1,t +ǫ t,t+1, with one lag of implied variance and one lag of realized variance. The model-free implied variance from options market is an 5

8 informationally more efficient forecast for future realized variance than the past realized variance (see, e.g., Jiang and Tian, 2005, among others), while realized variance based on high-frequency data also provides additional power in forecasting future realized variance (Andersen, Bollerslev, Diebold, and Labys, 2003). Therefore, a joint forecast model with one lag of implied variance and one lag of realized variance seems to capture the most forecasting power from the time-t available information (Drechsler and Yaron, 2010) Data Description and Summary Statistics For the option-implied variance of the S&P500 market return, I rely on the end-of-month Chicago Board of Options Exchange (CBOE) volatility index on a monthly basis (VIX 2 /12). Following literature, the monthly realized variance for the S&P500 index is the summation of the 78 intra-day five-minute squared log returns from 9:30am to 4:00pm including the close-to-open interval. Three market risk premiums are considered with their traditional predictor variables. Specifically, monthly P/E ratios and index returns for the S&P500 are obtained from Standard & Poor s, bond returns and forward rates are from the monthly CRSP Fama t-bill data set with 1 to 6 month maturities, and AAA and BAA corporate bond spread indices are from Moody s with Fama-Bliss risk-free interest rates from CRSP. The empirical analysis here is for the sample period from January 1990 to December 2008, when the new model-free VIX measure based on S&P500 index becomes available. To give a visual illustration, Figure 1 plots the monthly time series of variance risk premium (VRP), implied variance, and realized variance. The VRP proxy is moderately high around the 1990 and 2001 economic recessions but much higher around the financial crisis and Asia-Russia-LTCM crisis. The variances spike during October 2008 already surpasses the initial shock of the Great Depression in October The huge run-up of VRP in the fourth quarter of 2008 leads the equity market bottom reached in March 2009, consistent with the short-run predictability pattern discovered in Section 3. 5 Alternative approaches for estimating expected variance include: lag realized variance (Bollerslev and Zhou, 2007), jump-diffusion model-implied expectation (Todorov, 2010), ex-post realized variance (Carr and Wu, 2009), and AR(12) forecast of realized variance (previous version of this paper). 6

9 Basic summary statistics for the monthly risk premia and their predictor variables are given in Table 1 Panels A to C. The mean excess return of S&P500 equals 3.58 percent annually, reflecting the significantly lowered market returns during the financial crisis and economic downturn. The one month holding period returns for 2-6 month t-bills are ranging from 0.44 to 0.86 percent annually, and the credit spread for Moody s AAA rating is 1.25 percent and BAA 2.14 percent. The sample mean of VRP is (in percentages squared), with a standard deviation of Notice that the extraordinary skewness (4.51) and kurtosis (34.61) signal a highly non-gaussian process for VRP. For standard predictor variables, P/E ratio and short rate are very persistent with first order autocorrelations being 0.98 and Forward spread are rather stationary with a serial correlation between to Note that VRP has a low autocorrelation of Of particular interest, is that the VRP variable has very small correlations with standard longrun predictor variables 0.11 with P/E ratio, around to 0.01 with forward spreads, and effectively zero with short rate which may partially explain why the short-run predictability from VRP is complementary to those of the established predictor variables. 3 Short-Run Predictability Puzzles of Financial Assets This section presents new predictability evidence of the variance risk premium proxy for equity returns, bond return, and credit spreads, with and without the standard predictor variables P/E ratio, forward spread, and short rate. Data are monthly observations with horizons up to one year. All of the reported t-statistics are based on heteroskedasticity and serial correlation consistent standard errors (Newey and West, 1987). The discussion focuses on the estimated slope coefficients and their statistical significance as determined by the robust t-statistics. The forecasting accuracy of the regressions are also measured by the corresponding adjusted R 2 s. 6 6 For the highly persistent predictors like P/E ratio and short rate, the conventional t-statistics and R 2 s for the overlapping multi-period return regressions need to be interpreted with caution (Stambaugh, 1999; Campbell and Yogo, 2006; Goyal and Welch, 2008; Boudoukh, Richardson, and Whitelaw, 2008). 7

10 3.1 Equity Returns For equity returns, I focus on the regression of S&P500 returns on a long-run predictor P/E ratio and a short-run predictor variance risk premium (VRP), xr t+h = b 0 (h) + b 1 (h) VRP t + b 2 (h) log(p t /E t ) + u t+h,t, (4) where xr t+h is the horizon-scaled market excess return and the horizon h goes out to 12 months. Table 2 top row shows that the predictability or R 2 of VRP, starts out at 0.02 percent at monthly, peaks around 6 percent at four month, and then gradually decreases toward zero with longer forecasting horizons. The robust t-statistic is the highest at four month at 5.07 and remains marginally significant at the nine month horizon. On the other hand, as shown in the middle row in Table 2, the usual long-run predictor, log P/E ratio, starts out as marginally significant at 10 percent level at one month and then becomes gradually insignificant. However, the adjusted R 2 of P/E ratio monotonically increases from 0.92 percent at one month to 6.34 percent at twelve month. Turning to the joint regressions reported in the bottom row of Table 2, it is clear that combining VRP with the P/E ratio results in an even higher R 2 of percent at the four month horizon, which is higher than the sum of two R 2 s in the respective univariate regressions. The t-statistics for VRP and P/E ratio are also somewhat higher at four month, 6.11 and -1.70, than their univariate counterparts respectively. Figure 2 visualizes the short-run predictability pattern VRP in R 2 and t-statistics. Such a predictability has a tent shape pattern maximizes at four month horizon. While P/E ratio has no statistical significance at one-to-twelve month horizons. The discussion is brief here as detailed results on equity return predictability in comparison with other established predictors are discussed in Bollerslev and Zhou (2007). 3.2 Bond Returns The failure of the Expectations Hypothesis (EH) of interest rates can be best characterized as that bond excess returns, projected onto forward rates, are largely predictable and time- 8

11 varying (Fama, 1984, 1986; Stambaugh, 1988). 7 Here I adopt the conventional forward rate setup as in Fama (1984) and augment it with variance risk premium (VRP), xhpr n t+h = b n 0(h) + b n 1(h) VRP t + b n 2(h) [f t (n h,h) y t (h)] + u n t+h,t, (5) where xhpr n t+h is the excess holding period return of zero coupon bonds with hold period h = 1,2,3,4,5 month and maturity n = 2,3,4,5,6 month (in excess of the yield on a h- month zero coupon bond); f t (n h,h) is the forward rate for a contract h-month ahead with n h-month length; and y t (h) is the h-month zero coupon bond yield. As shown in Table 3, VRP can significantly forecast the one month holding period excess returns of the two-six month t-bills, with a positive slope coefficient around to Considering the average level of VRP, these magnitudes translate to average bond risk premia induced by the variance risk around 7 to 23 basis points. More importantly, longer maturity bond seems to load more the variance risk: as the Newey-West t-statistics are from 1.68 to 3.90 and R 2 from 1.13 to 3.46 percent. Moving to the two month holding period, the t-ratios reduce to 1.22 to 3.05, and the R 2 decreases to 0.22 to 3.92 percent. The predictability of VRP converges to zero as the holding period increase to five month, so longer holding period reduces the forecasting power of VRP. As Table 4 indicates, the forward rate (spread) is indeed a powerful predictor for excess bond returns for two-to-six month bonds with one-to-five month holding periods t-statistics all above the 1 percent level and R 2 between 6.03 and percent. Another pattern is that the magnitudes of t-statistics and R 2 are generally higher at the one-month horizon and lower toward the five-month horizon. Overall, the predictability of forward spreads for short-term bills reported here are similar to those in Fama (1984). More importantly, when VRP is combined with forward rates, as shown in Table 5, the predictability of the VRP remains intact. For example, at one month horizon, the t- statistics for VRP in joint regressions are around 1.85 to 4.05, all higher than their univariate 7 The forward rate regression is recently extended by Cochrane and Piazzesi (2005) to multiple forward rates, by Ludvigson and Ng (2009) to incorporate extracted macroeconomic factors, and by Wright and Zhou (2009) to include a realized jump risk measure. However, these studies use 2-5 year zero coupon bonds with one year holding period, where variance risk premium has virtually zero forecasting power for bond returns. 9

12 counterparts (Table 3 top row). Note that the adjusted R 2 s for the one month horizon with both VRP and forward spread are all higher than the ones with forward spreads alone (Table 4 top row), suggesting that the VRP variable indeed contributes to the short-run bond return predictability, independent of what has already been captured by the forward spread. These results can be visualized in Figure 3, where for 6 month t-bill returns, the predictability of the VRP variable is significant at one and two month but is monotonically decreasing with the holding period. While for the forward spread variable, the predictability is significant at all horizons. This result suggests that VRP and forward spread are proxies for different components in bond risk premia that are largely orthogonal to each other. 3.3 Credit Spreads The relatively large and time-varying credit spread on corporate bond has long been viewed as an anomaly in the literature (see, e.g., Huang and Huang, 2003). Here I provide some new evidence that, in additional to the standard predictor, namely the short term interest rate (Longstaff and Schwartz, 1995), variance risk premium (VRP) also helps to explain the short-run movement in credit spreads. The following standard forecasting regression is adopted, CS t+h = b 0 (h) + b 1 (h) VRP t + b 2 (h) r f,t + u t+h,t, (6) where the credit spread CS t+h of h month ahead is being forecasted by the short rate and VRP. As shown in Table 6, short term interest rate is indeed a predominant predictor of the future credit spread levels, with t-statistics of for high investment grade (Moody s AAA rating)and-3.14forlowinvestmentgrade(moody sbaarating). TheadjustedR 2 isaround 32 percent, and the negative sign of the slope coefficient is consistent with the structural model interpretation in Longstaff and Schwartz (1995). 8 Although the significance of the short rate level extends to the six-month horizon, it is a very persistent variable with an autoregressive coefficient of 0.99 (Panel E in Table 1). As shown in the lower two panels 8 If one includes term spread alone, it is marginally significant; but when combined with short rate, term spread is driven out as insignificant. This result in tabular form is available upon request. 10

13 of Figure 4, the predictability of short rate for credit spread shows a monotonic pattern of decreasing, and the slope coefficient becomes indifferent to zero for horizons longer than six months. Table 6 indicates that if we include VRP alone in the forecasting regressions, its statistical level is higher than 1 percent at one month horizon, with t-statistic being 3.95 for AAA grade and 3.54 for BAA grade. Given the average level of VRP, it translates into an average effect on credit spread in the order of 12 to 21 basis points. Once the forecasting horizon increases to 3, 6, 9, and 12 months, the t-statistics for the VRP variable decreases and becomes marginally significant. As shown in the upper two panels of Figures 4, the predictability of VRP for BAA credit spread shows a hump-shape pattern, which peaks at two month and then becomes insignificant beyond five month. When VRP is combined with the short rate, as seen in Table 6, both become more significant at short horizons. For example, at one month, the t-statistics for short rate is for AAA and for BAA, while for VRP is 4.26 and 4.89; all much higher than their univariate counterparts. In fact, the VRP variable maintains its significance in the joint regressions even at the twelve month horizon, while the short rate becomes insignificant beyond the six month horizon. This is an important finding, in that VRP or the impliedexpected variance difference captures an important component in credit spread, which is independent with and complementary to the systematic risk factor(s) that has already been captured by the short-term interest rate. 4 A Model of Macroeconomic Uncertainty It is challenging to provide a conceptual framework to jointly explain the short-run risk premium dynamics in equity, bond, and credit markets. Here I draw from a self-contained general equilibrium model with stochastic consumption volatility-of-volatility (Bollerslev, Tauchen, and Zhou, 2009), and try to give a unified qualitative interpretation of these shortterm risk premia predictability puzzles. One should admit upfront that such a stylized model cannot provide a satisfactory quantitative explanation for various aspects of asset 11

14 pricing puzzles, constrained by the same parameter setting and by matching the consumption dynamics. Tobemorespecific,Iwilltrytocalibrateamodelwitheconomicuncertainty,bymatching the equity risk premium as in Bansal and Yaron (2004) and equity return predictability as in Bollerslev, Tauchen, and Zhou (2009), constrained by reasonable preference structure and consumption dynamics. And then I explore how far such a modeling framework can go for jointly explaining some salient features in variance, bond, and credit risk premiums. 4.1 Model Assumptions and Wealth-Consumption Ratio The representative agent in the economy is equipped with Epstein-Zin-Weil recursive preferences, and has the value function V t of her life-time utility as V t = [(1 δ)c 1 γ θ t +δ ( E t [ V 1 γ t+1 ] ])1 θ 1 γ θ, (7) where C t is consumption at time t, δ denotes the subjective discount factor, γ refers to the coefficient of risk aversion, θ = 1 γ, and ψ equals the intertemporal elasticity of substitution 1 1 ψ (IES). The agent maximizes her utility subject to the budget constraint, W t+1 = (W t C t ) r t+1, where W t is the wealth of the agent and r t is the return on the consumption asset. The key assumptions are that γ > 1, implying that the agents are more risk averse than the log utility investors; and ψ > 1 hence θ < 0, implying that agents prefer an earlier resolution of economic uncertainty. These restrictions ensure that the uncertainty or volatility risk in asset markets carries a positive risk premium. Suppose that log consumption growth and its volatility follow the joint dynamics g t+1 = µ g +σ g,t z g,t+1, (8) σ 2 g,t+1 = a σ +ρ σ σ 2 g,t + q t z σ,t+1, (9) q t+1 = a q +ρ q q t +ϕ q qt z q,t+1, (10) whereµ g > 0denotestheconstantmeangrowthrate, σg,t+1 2 representstime-varyingvolatility in consumption growth, and q t introduces the volatility uncertainty process in the consumption growth process. The parameters satisfy a σ > 0,a q > 0, ρ σ < 1, ρ q < 1, ϕ q > 0; 12

15 and {z g,t }, {z σ,t } and {z q,t } are iid Normal(0,1) processes jointly independent with each other. The time-variation in σg,t+1 2 is one of the two components that drives the equity risk premium, or the consumption risk ; while the time-variation in q t is not only responsible for the uncertainty risk component in equity risk premium, but also constitutes the main driver of variance and bond risk premia as explained bellow. Let w t denote the logarithm of the price-dividend or wealth-consumption ratio, of the asset that pays the consumption endowment, {C t+i } i=1; and conjecture a solution for w t as an affine function of the state variables, σg,t 2 and q t, w t = A 0 +A σ σ 2 g,t +A q q t. (11) One can solve for the coefficients A 0, A σ and A q (as in Bollerslev, Tauchen, and Zhou, 2009), using the standard Campbell and Shiller (1988) approximation r t+1 = κ 0 +κ 1 w t+1 w t +g t+1. The aforementioned restrictions that γ > 1 and ψ > 1, hence θ < 0, imply that the impact coefficient associated with both consumption and volatility state variables are negative; i.e., A σ < 0 and A q < 0. So if consumption and uncertainty risks are high, the price-dividend ratio is low, hence risk premia are high. In response to high economic uncertainty risks, agents sell risky assets, and consequently the wealth-consumption ratio falls; so that risk premiums rise. 4.2 Model-Implied Equity, Variance, and Bond Risk Premia Given the solution of the price-dividend ratio, one can easily solve for the variables of interest. The model-implied equity risk premium can be shown as E t (r t+1 ) r f,t = γσ 2 g,t + (1 θ)κ 2 1(A 2 qϕ 2 q +A 2 σ)q t > 0. (12) The premium is composed of two separate terms. The first term, γσg,t, 2 is compensating for the classic consumption risk as in a standard consumption-based CAPM model. The second term, (1 θ)κ 2 1(A 2 qϕ 2 q + A 2 σ)q t, represents a true premium for variance risk. The existence of the variance or uncertainty risk premium depends crucially on the dual assumptions of recursive utility, or θ 1, as uncertainty would otherwise not be a priced factor; and time 13

16 varying volatility-of-volatility, in the form of the q t process. The restrictions that γ > 1 and ψ > 1 implies that the variance risk premium embedded in the equity risk premium is always positive by construction. And since the variance risk premium embedded in equity returns loads on the same uncertainty risk factor, q t, as in the variance risk premium (demonstrated bellow), the latter becomes a perfect predictor for equity premium variation that is induced by the stochastic economic uncertainty. The conditional variance of the time t to t+1 return, σ 2 r,t Var t (r t+1 ), can be shown as σ 2 r,t = σ 2 g,t +κ 2 1( A 2 σ +A 2 qϕ 2 q) qt. The variance risk premium can be defined as the difference between risk-neutral and objective expectations of the return variance, 9 VRP t E Q t ( σ 2 r,t+1 ) E P t ( σ 2 r,t+1 ) (θ 1)κ 1 [ Aσ +A q κ 2 1( A 2 σ +A 2 qϕ 2 q) ϕ 2 q ] qt > 0, (13) One key observation here is that any temporal variation in the endogenously generated variance risk premium, is due solely to the volatility-of-volatility or economic uncertainty risk, q t, but not the consumption growth risk, σ 2 g,t+1. Moreover, provided that θ < 0, A σ < 0, and A q < 0, as would be implied by the agents preference of an earlier resolution of economic uncertainty, this difference between the risk-neutral and objective expectations of return variances is guaranteed to be positive. If volatility-of-volatility is constant, q t = q, the variance premium reduces to a constant (θ 1)κ 1 A σ q, and one cannot replicate the large skewness and kurtosis found in the observed variance risk premium series. The real bond yield in this economy can be shown as an affine function of the state [ ][ ]. variables, y t (n) = 1 n A(n) B(n) C(n) 1 σg,t 2 q t Let rxt+1 (n 1) be the bond excess return from t to t+1 for an n-period bond holding one period, then its expected value brp n t or real bond risk premium is given by, brp n t = [ B(n 1)(θ 1)κ 1 A σ +C(n 1)(θ 1)κ 1 A q ϕ 2 q] qt > 0, (14) where the risk premia has two time-varying components consumption risk and uncertainty risk, but they are co-linear in the same state variable q t. This is driven by the fact that 9 The approximation comes from the fact that the model-implied risk-neutral conditional expectation cannot be computed in closed form, and a log-linear approximation is applied. 14

17 the variances of both the consumption volatility and the volatility-of-volatility processes are loading on the same state variable, q t. Therefore in a real economy, variance risk premia can perfectly predict the bond risk premia, and one cannot replicate the empirical predictability pattern discussed earlier without introducing exogenous inflation. 4.3 Calibrating Equity and Variance Risk Premia To more directly gauge how the model adopted here can explain the documented risk premium dynamics, I perform a limited calibration exercise. The basic strategy is to find the preference and distribution parameters that are constrained by reasonable consumption dynamics and can simultaneously match the observed equity and variance premium dynamics as much as possible. The following subsections examine whether such a modeling framework can be extended to explain the short-run predictability in bond return and credit spread Calibration Design and Parameter Setting As shown in Table 7, the two parameter settings are adapted from Bansal and Yaron (2004) such that the consumption growth rate (µ g = ) is 2.4 percent annually. I also use the same intertemporal elasticity of substitution (IES) parameter (ψ = 1.5) and time preference δ = across two scenarios. The Campbell-Shiller approximation constants are chosen as κ 1 = 0.9, hence κ 0 = Note that κ 1 = exp( z)/(1+exp( z)) and κ 0 = log(1 + exp( z)) κ 1 z, where z is the mean log price-consumption ratio. However, these two parameter settings differ dramatically in terms of risk aversion and volatility risk. In Bollerslev, Tauchen, and Zhou (2009) or BTZ2009 for short, the risk aversion coefficient is γ = 10, same as in Bansal and Yaron (2004); but this paper chooses γ = 2. Ontheotherhand,inBTZ2009,theconsumptionvolatilityisE(σ g ) = ,similar as in Bansal and Yaron (2004), while it increases to E(σ g ) = ( ) 2 in this paper. 10 In addition, the persistence parameter of consumption volatility ρ σ is in BTZ2009, while here it equals 0.1 here. Furthermore, BTZ2009 chooses the economic uncertainty process, 10 A material implication of these parameter values is that the consumption volatility is increased to 6 times of the 2.7 percent annually, which may be justifiable if one leverages up the dividend shocks several times larger than the consumption shocks (Abel, 1999). For example, the dividend volatility is levered up to 5.96 times of the consumption volatility in Bansal, Kiku, and Yaron (2007). 15

18 q t, to have monthly persistence level of ρ q = 0.80, long-run mean of E(q) = , and volatility-of-volatility parameter as ϕ q = 0.001; but this paper uses ρ q = 0.98, E(q) = , and ϕ q = 0.008, implying that the q t process here is not only magnified but also more persistent and volatile Calibrated Equity and Variance Risk Premia The calibration results in Table 8 indicate that one can achieve a reasonable compromise to simultaneously match the equity premium and variance risk premium (VRP), without relying on introducing jumps into the endowment and volatility processes (as in Eraker and Shaliastovich, 2008; Drechsler and Yaron, 2010). Since the calibration result on predictability pattern of variance risk premia for equity excess returns has been reported in Bollerslev, Tauchen, and Zhou (2009), it is omitted here. 11 The resulting equity risk premium is 5.34 percent and real interest rate 1.25 percent, which are different than BTZ2009 (7.79 and 0.69 percent) but similar to the observed sample of 3.58 and 1.13 percent for the sample period of The parameter choice here has a advantage of getting a better fit of equity volatility at percent, as opposed to 4 percent in BTZ2009; comparing to percent observed in the recent period and 20 percent in the longer histories. On the other hand, this paper overfits the volatility of risk-free rate at percent, versus 3.37 percent in recent period. In short, the current model calibration achieves reasonable values for equity premium and risk-free rate, improved the matching of equity volatility, but worsening the fit of risk-free rate volatility. The model-implied VRP has a mean of 4.62 and a standard deviation of 7.23, which are close to the result in BTZ2009 but fall short of the observed values of and More importantly, the model produces reasonable values in skewness (2.73) and kurtosis (14.23), which are less than the observed values of 4.51 and but improve a lot upon previous result in BTZ2009(1.70 and 11.42). These results are non-trivial in that a stochastic volatility-of-volatility model can generate reasonable skewness and kurtosis of VRP, similar 11 Admittedly such a short-run risk model lacks the long-run component to match the consumption growth predictability, since the long-run component in consumption growth is totally shut down here. 16

19 as in Drechsler and Yaron (2010) but without introducing jumps. Of course, the level of VRP is still too low, reflecting the constraint of simultaneously matching equity and variance premiums, without producing outrageous bond risk premium as discussed bellow. 4.4 Calibrating Bond Risk Premia To price the nominal bond, one need to introduce an exogenous inflation process (see, e.g., Wachter, 2006; Bansal and Shaliastovich, 2008b). Here one can expand the state space to specify an autonomous inflation dynamics π t+1 = a π +ρ π π t +ϕ π σ g,t z π,t+1, (15) where {z π,t } is an iid Normal(0,1) and independent of all other shocks in the model. The three parameters, a π, ρ π, and ϕ π, control the level, persistence, and variance of the inflation dynamics. The non-standard part of the inflation process is the volatility channel linkage to the consumption dynamics, ϕ π σ g,t z π,t+1, which ensures the existence of a time-varying inflation risk premium, without violating the money-neutrality assumption. The calibrated inflation parameters are taken down directly from Gallmeyer, Hollifield, Palomino, and Zin (2008), as shown in Table 7, which implies annualized inflation rate of 4.46 percent and inflation volatility of 2.67 percent. The bond yield in the nominal economy can be shown as an affine function of the state [ ][ ], variables, y t (n) = 1 n A(n) B(n) C(n) D(n) 1 σg,t 2 q t π t where the coefficients A(n), B(n), C(n), and D(n) are functions of underlying dynamics and preference parameters. The assumed inflation dynamics implies a nominal bond risk premium (brp) of the following form brp n t = [ B(n 1)(θ 1)κ 1 A σ +C(n 1)(θ 1)κ 1 A q ϕ 2 q] qt D(n 1)ϕ 2 πσ 2 g,t. (16) The first two items in brackets reflect consumption and uncertainty risks, while the third item captures inflation risk. In a regression of bond risk premium on variance risk premium brp n t = â+ˆbvrp t, the model-implied slope coefficient is B(n 1)A σ +C(n 1)A q ϕ ˆb 2 q = ), A σ +A q κ1( 2 A 2 σ +A 2 qϕ 2 q ϕ 2 q 17

20 and the model implied R 2 is R 2 = [ ] B(n 1)(θ 1)κ1 A σ +C(n 1)(θ 1)κ 1 A q ϕ 2 2Var(qt q ) [ ] B(n 1)(θ 1)κ1 A σ +C(n 1)(θ 1)κ 1 A q ϕ 2 2Var(qt q )+[D(n 1)ϕ 2 π] 2 Var (. σg,t) 2 Using these two metrics, one can assess how the proposed simple model is able to reproduce the empirical bond return predictability from VRP. As shown in Table 8, the observed bond risk premium for 2-6 month t-bills holding one month are positive and mildly upward sloping at 0.44 to 0.86 percent. If one chooses the specification in Bollerslev, Tauchen, and Zhou (2009), the real bond risk premia would be extremely hump-shaped with about 3 percent at 2 month, 7 percent at 4 month, and percent at 6 month. In contrast, the specification chosen in this paper would produce a term structure of bond risk premia from 0.73 to 0.94 percent for 2-6 month t-bills, which are slightly highly than the observed ones but quite closer. This is a non-trivial result because the inflation dynamics introduced here is autonomous. Figure 5 reports calibrated slope coefficients and R 2 s for regressing one month excess returns of 2-6 month t-bills on VRP, in comparison with the empirical estimated slopes and R 2 s. Model 1 is the benchmark scenario in Table 7; and three alternative calibrations as detailed in Table 9 are Model 2 with lower risk aversion γ = 1.1 (versus 2), Model 3 with higher intertemporal elasticity of substitution (IES) ψ = 10 (versus 1.5), and Model 4 with higher inflation volatility ϕ π = 0.23 (versus 0.115). The model-implied slope coefficients from the calibrated benchmark model, as shown in the top panel of Figure 5, are positive and upward sloping, like the empirical estimates, but about 15 to 20 times larger in magnitudes. Altering the model calibration by reducing risk aversion (Model 2) and increasing IES (Model 3) will make the calibrated projection slopes closer to the empirical ones. On the other hand, increasing inflation volatility (Model 4) will increase the predictability of VRP for bond risk premia. Note that none of the four models have slope coefficients within the 95 percent confidence band of the empirical estimates. Turning to the model-implied R 2 s as depicted in the bottom panel of Figure 5, the benchmark model are close to 100 percent for 2-month t-bill and down to about 60 percent for 6-month t-bill, while the empirical counterparts are merely 1-4 percent. reducing risk 18

21 aversion to γ = 1.1 (Model 2) will make the R 2 s almost match the empirical ones, while increasing IES to ψ = 10 (Model 3) will make things even worse. However, increasing inflation volatility to ϕ π = 0.23 (Model 4) will improve the fitting of R 2 s in the long end of maturity like 4-6 months. Although the calibration result indicates that altering certain preference and dynamics parameter may improve the matching between model-implied predictability of VRP for bond return with its empirical counterpart, the fitting of unconditional bond risk premia may become worse. As shown in Table 9, reducing risk aversion will underfit the bond risk premia, while increasing IES or inflation volatility will overfit. Therefore, it remains a challenge to simultaneously reproduce both the unconditional level of bond risk premia and the predictability pattern of VRP for bond returns, which may calls for a departure from the underlying assumption of money neutrality (see, e.g., Pennacchi, 1991). I will leave this for future research. 4.5 Calibrating Credit Spread Predictability In order to assess the implication of variance risk premium (VRP) for credit spread, one can examine a discrete-time version Merton (1974) model with stochastic variance (similar to Zhang, Zhou, and Zhu, 2009). Such a model is partial equilibrium in nature, without establishing the market clearing conditions between macroeconomic fundamentals and firm value dynamics, as in Chen (e.g., 2010); Bhamra, Kuehn, and Strebulaev (e.g., 2009); Chen, Collin-Dufresne, and Goldstein (e.g., 2009). Instead, the simple objective here is to show that a firm value dynamics with stochastic variance, which may be consistent with the macroeconomic fundamentals set out earlier, is able to generate the predictability pattern of VRP for credit spread. Suppose that a representative firm s asset dynamics and its variance process can be described by the following stochastic difference equations: A t+1 A t A t = µ a δ a + V t z a,t+1, (17) V t+1 V t = κ v (θ v V t )+σ v Vt z v,t+1, (18) 19

22 where A t is the firm value, µ a is the expected asset return, and δ a is the asset payout ratio. The asset return variance, V t, follows a discrete-time square-root process, with long-run mean θ v, mean reversion κ v, and volatility-of-volatility parameter σ v. Finally, the correlation between asset return and return volatility is corr(z a,t+1,z v,t+1 ) = ρ. With proper bankruptcy assumptions, one can solve for the equity price, S t, as a European call option on firm asset A t with maturity T. The debt value can be expressed as D t = A t S t, anditspriceisp t = D t /B, whereb isthefacevalueofdebt. Thecreditspread, CS t, is given by CS t = 1 T t log(p t) r. Inside simulation, one can examine the predictability for credit spread CS t from variance risk premium, or VRP t = E Q t (RV t+1 ) E P t (RV t+1 ), where RV t+1 is the realized variance from 5-minute equity returns. The risk-neutral and physical expectations, E Q t ( ) and E P t ( ), are approximated with the asset volatility dynamics (18). The calibrated firm value dynamics, as shown in Table 7, are taken down directly from Zhang, Zhou, and Zhu (2009) for a BAA representative firm. 12 The model is simulated for 240 months or 7200 days at every 5-minute interval and is solved for equity price and credit spread at each 5-minute interval. Then monthly realized variance is calculated as the sum of log 5-minute squared returns, and VRP is calculated using the asset volatility dynamics (18). Table 8 shows the simulated credit spread of 2.17 percent, which is quite close to the 2.14 observed value, similar to the sample average reported in Zhang, Zhou, and Zhu (2009). As suggested by Huang and Huang (2003), standard structural models without stochastic variance can only capture percent of investment-grade credit spread. For higher order moments, the model overfits the standard deviation but fall short of skewness and kurtosis. Figure 6 reports simulated slope coefficients and R 2 s for regressing a BAA representative s credit spread on VRP for 1 to 12 month horizons, in comparison with their empirical counterparts. Model 1 is the benchmark scenario in Table 9; and three alternative simulations are also shown in Table 9: Model 2 with asset-volatility correlation increased to three 12 Other parameters in calibrated model are chosen as risk-free rate 5 percent, initial leverage percent, recovery rate percent, and volatility risk premium , same as those in Zhang, Zhou, and Zhu (2009). These parameters are calibrated to match both the historical default probability and the observed credit spread for the BAA rated firms. 20