Equity Risk Premium Predictability from Cross-Sectoral Downturns Brazilian Finance Society (SBFin) Annual Meeting, Jul 2018

Transcription

1 Equity Risk Premium Predictability from Cross-Sectoral Downturns Brazilian Finance Society (SBFin) Annual Meeting, Jul 2018 Juan Arismendi Zambrano (University of Reading) with José Afonso Faias (Católica Lisbon School of Business & Economics) March 26, 2018 ICMA Centre, Henley Business School at Reading, UK Contact: Juan Arismendi Zambrano March 26, /35

2 Outline 1 Introduction Motivation Contribution Literature Review 2 Theoretical Motivation 3 Empirical Results Data Description Tail Estimation Predictability 4 Conclusions Juan Arismendi Zambrano March 26, /35

3 Introduction Motivation Motivation Juan Arismendi Zambrano March 26, /35

4 Introduction Motivation Motivation Rare events, such as the global financial crisis, are crucial in asset pricing (Rietz, 1988; Barro, 2006; Gourio, 2012; Wachter, 2013). Juan Arismendi Zambrano March 26, /35

5 Introduction Motivation Motivation Rare events, such as the global financial crisis, are crucial in asset pricing (Rietz, 1988; Barro, 2006; Gourio, 2012; Wachter, 2013). One typically uses aggregate-level variables, but if endogenous sectorial shocks hold specific information, then they should be used to reflect uncertainty in asset prices Juan Arismendi Zambrano March 26, /35

6 Introduction Motivation Motivation Rare events, such as the global financial crisis, are crucial in asset pricing (Rietz, 1988; Barro, 2006; Gourio, 2012; Wachter, 2013). One typically uses aggregate-level variables, but if endogenous sectorial shocks hold specific information, then they should be used to reflect uncertainty in asset prices This is even more important when addressing tail comovements since at the aggregate level tail risk is partially diversified away Juan Arismendi Zambrano March 26, /35

7 Introduction Motivation Motivation Rare events, such as the global financial crisis, are crucial in asset pricing (Rietz, 1988; Barro, 2006; Gourio, 2012; Wachter, 2013). One typically uses aggregate-level variables, but if endogenous sectorial shocks hold specific information, then they should be used to reflect uncertainty in asset prices This is even more important when addressing tail comovements since at the aggregate level tail risk is partially diversified away Is there a benefit of incorporating sectorial tail shocks in predicting asset returns? Juan Arismendi Zambrano March 26, /35

8 Introduction Contribution Contribution We use a time-varying endogenous sectoral disaster-risk consumption-based asset pricing model to explain a substantial portion of the equity-risk premium. Juan Arismendi Zambrano March 26, /35

9 Introduction Contribution Contribution We use a time-varying endogenous sectoral disaster-risk consumption-based asset pricing model to explain a substantial portion of the equity-risk premium. This model illustrates the crucial role of left cross-sectoral bivariate tail dependence (LTM) which endogenously incorporates shocks imperceptible at the aggregate level. Juan Arismendi Zambrano March 26, /35

10 Introduction Contribution Contribution We use a time-varying endogenous sectoral disaster-risk consumption-based asset pricing model to explain a substantial portion of the equity-risk premium. This model illustrates the crucial role of left cross-sectoral bivariate tail dependence (LTM) which endogenously incorporates shocks imperceptible at the aggregate level. We proxy left tail dependence by the average of pairwise left tail dependency among major equity sectors, and we confirm that it significantly predicts the equity risk premium in- and out-of-sample and its information is also crucial to improve other predictors forecast. Juan Arismendi Zambrano March 26, /35

11 Introduction Literature Review Literature Review On the one hand, one way to incorporate rare events in finance is to use extreme value theory (e.g., Longin & Solnik, 2001; Bae et al., 2003; Hartmann et al., 2004). 1 On the other hand, researchers have shown that a powerful solution when examining aggregate-level variables is the use of sectoral information since different shocks can be recognized at the sectoral level but are invisible at the aggregate level (e.g., Horvath, 2000; Veldkamp & Wolfers, 2007; Comin & Mulani, 2009; Holly & Petrella, 2012). This paper provides a positive answer to the earlier question. 1 For example, Poon et al. (2004) advocate the use of risk measures based on extreme value theory rather than traditional risk measures, such as volatility or value-at-risk. They demonstrate that the latter are unsuitable for measuring tail risk, which may lead to inaccurate portfolio risk assessment. Juan Arismendi Zambrano March 26, /35

12 Theoretical Motivation Theoretical Motivation Juan Arismendi Zambrano March 26, /35

13 Theoretical Motivation Theoretical Motivation We consider a standard economy with a single household that represents all consumers. This household maximizes the utility of consumption U (C t ) in an infinite horizon: [ ] E β t U(C t ), (1) t=0 where β is the time discount factor. We introduce a multi-asset endowment economy to model the n sectors of the economy with consumption C t =(C 1,t,...,C n,t ) in which the sectors have a weighting ω t =(ω 1,t,...,ω n,t ). Hence, the aggregated consumption is given by C M,t = ω t C t. Juan Arismendi Zambrano March 26, /35

14 Theoretical Motivation Theoretical Motivation The representative agent must decide between consuming at time t or investing and waiting to consume at time t +1. Hence, in equilibrium, the fundamental relation holds: P e M,tU (C M,t )=βe [ P e M,t+1 + C M,t+1 U (C M,t+1 ) ]. (2) where the equity market price PM,t e = ω t Pe t is the aggregated price process of the sectors and P e t =(P 1,t e,...,pe n,t ) is the joint price process of the sectors. The dividend process is defined by D t =(D 1,t,...,D n,t ), where the aggregated dividend is D M,t = ω td t. RM,t e = ω tr e t is the corresponding aggregated equity return where R e t =(Re 1,t,...,Re n,t ), and the equity risk premium is given by ERP t = ω t Re t Rb t with Rb t being the treasury bill rate. Juan Arismendi Zambrano March 26, /35

15 Theoretical Motivation Theoretical Motivation In the rare disaster consumption models (Rietz, 1988; Barro, 2006; Wachter, 2013), an additional factor that represents the consumption collapse is added to the classic equity risk premium formulation: ERP t = ERP standard t + ERP disaster t (3) where ERPt standard premium. is the Mehra & Prescott (1985) equity risk Barro (2006) models ERPt disaster as a time-static factor that prices a low probability consumption collapse while Wachter (2013) includes a time-varying factor in ERPt disaster that models the high volatility of stock market returns. Juan Arismendi Zambrano March 26, / 35

16 Theoretical Motivation Endogenous Sectors in the Mehra & Prescott (1985) Model In an explanatory paper, Mehra (2003) demonstrates that the source of the equity risk premium comes from the covariance between the derivative of the utility of future consumption and equity returns: ERP standard t = 1 E [U (C M,t+1 )] cov ( U (C M,t+1 ),RM,t+1 e ) where U(C M,t+1 )=(C 1 θ M,t )/(1 θ) is a power utility function with θ the risk aversion parameter and the joint distribution of consumption and equity prices is modelled by a bivariate lognormal distribution. This assumption of a lognormal distribution is not present in the original formulation of the ERP puzzle. Hence, we can use a different distribution to explain sectoral predictability (bivariate jump-diffusion process or a bivariate Student s t). Juan Arismendi Zambrano March 26, / 35 (4)

17 Theoretical Motivation Endogenous Sectors in the Mehra & Prescott (1985) Model 5 std U ' (C t ) std LTM std LTM, U ' (C t ) Year Figure 1: The LTM of U.S. economic sectors and the marginal utility of U.S. personal consumption core price index This figure presents the standardized LTM of U.S. economic sectors and the standardized marginal utility of personal consumption core price index between January 1993 and December A CRRA utility function is used with a risk aversion parameter θ =4. Juan Arismendi Zambrano March 26, / 35

18 Theoretical Motivation However, a bivariate heavy-tailed distribution of the marginal utility and the equity returns can explain a high level of the ERP in the classic framework. Juan Arismendi Zambrano March 26, / 35 Endogenous Sectors in the Mehra & Prescott (1985) Model From Figure 2, we can observe that there is a strong correlation between the LTM and the marginal utility U (C M,t+1 ).Both standardized marginal utility and standardized LTM series are detrended. We regress the standardized marginal utility on the standardized LTM and we obtained an in-sample (IS) R-squared of 19.77%. This high correlation between the two variables is not sufficient to explain the full ERP; a high volatility of the marginal utility U (C M,t+1 ) and/or a high volatility of the equity returns are still required to explain the ERP puzzle in the classic framework.

19 Theoretical Motivation Endogenous Sectors in the Barro (2006) Model Barro (2006) consumption model incorporates the disaster factor that was originally proposed in Rietz (1988) but he uses worldwide data to calibrate the parameters. In the Barro framework, the consumption change is split into three components: ( ) Ci,t+1 log = γ + xi standard + xi disaster, (5) C i,t where γ is a constant. Notice that all the production is fully consumed (C M,t = D M,t ) as in the Lucas (1978) tree. In this model, it is assumed a power utility function. Juan Arismendi Zambrano March 26, / 35

20 Theoretical Motivation Endogenous Sectors in the Barro (2006) Model Barro (2006) assumes no parametric form of the normal times consumption and the rare disaster times consumption factors; nevertheless, for the purpose of comparing with the time-varying model of Wachter (2013) and being able to calculate the LTM we assume a jump-diffusion form for the consumption change: ( ) Ci,t+1 log = μ i + σ i W i,t + ( e K i,t 1 ) N i (λ t ), (6) C i,t where μ i and σ i are the standard (normal) periods sector i consumption growth mean and volatility. The normal (standard) and the disaster consumption components are defined by: log (C i,t+1 /C i,t ) standard = μ i + σ i W i,t, (7) log (C i,t+1 /C i,t ) disaster = ( e K i,t 1 ) N i (λ t ), (8) where W i,t is a discrete Brownian motion stochastic process, and N i (λ t ) is a discrete Poisson jump process. Juan Arismendi Zambrano March 26, / 35

21 Theoretical Motivation Endogenous Sectors in the Barro (2006) Model We use a multivariate jump-diffusion to model sectoral consumption rare disasters for ease of presentation and computation of the method of moments so that we can disentangle the consumption process in normal periods from the rare disaster consumption process. Previous modelling of rare disasters with jump-diffusion include, as examples, Liu et al. (2003) and Das & Uppal (2004). In the case in which only the sectoral effects in the static rare disaster premium are considered, the ERP is given by: [ ] ERP t = θσ 2 + λ t E e θω K t 1. (9) where λ t is the probability of a rare disaster, and K t =(K 1,t,...,K n,t ) is the joint distribution of the disaster contraction for each sector, with μ J i and ν J i the mean size and the volatility of the jumps. Juan Arismendi Zambrano March 26, / 35

22 Theoretical Motivation Endogenous Sectors in the Wachter (2013) Model A main problem with the static rare disaster models is that they are not able to explain ERP out-of-sample predictability, but only ERP in-sample predictability. One of the main findings in this paper is the significant out-of-sample predictability of LTM for ERP over the traditional ERP predictors. For this reason, we develop an endogenous sector time-varying rare disaster consumption model having as a baseline Wachter (2013). There is an endowment economy where the sector i-th consumption evolves according to: dc i,t+1 = μ i + σ i dw i,t + ( e K M,t 1 ) dn i,t (λ t ), (10) C i,t where dw i,t is a Brownian motion, μ i and σ i are the standard (normal) periods sector i consumption growth mean and volatility, K i,t is the sector disaster decline return, and dn i,t (λ t ) is a Poisson jump process. Juan Arismendi Zambrano March 26, / 35

23 Theoretical Motivation Endogenous Sectors in the Wachter (2013) Model The objective of the disaster period dependence factor is to price extreme value tail-dependent events. The model in Equation (9) can be expanded for the time-varying case as: ERP t =Lθσ 2 λ t G t G t bσ 2 λ + λ t E[ ( e ( θω K t) 1) ( (1 q) ( 1 e (Lω K t) ) + ( ) )] q e (ω K t) e (Lω K t), (11) where L is the leverage of the consumption (D t = Ct L ), G t is a function of the price-dividend ratio (see Wachter, 2013), σλ 2 the total volatility of a rare disaster and q the probability of a default by a disaster. Juan Arismendi Zambrano March 26, / 35

24 Theoretical Motivation LTM Impact in Static and Time-varying Rare Disaster Models To compute the LTM, weusearesultbypoonet al. (2004): If the disaster percentage decline e Kt is multivariate lognormal distributed, then (i) the multivariate distribution of the sectoral consumption growth is multivariate lognormal, and the bivariate tail dependence of the sectors are equal to the Pearson correlation: ρ ( dci,t C i,t, dc j,t C j,t ) = ( σi σ j + λ t (μ J i μj j + νj i νj j )) ( σ 2 i + λ t ( μ J i ) 2 +(ν i ) 2)) 1/2 ( σ 2 j + λ t ( μ J j ) 2 +(ν j ) 2)) 1/2 (12) (ii) the bivariate tail dependence of the jumps is equal to 1, i.e., Hence, the resulting LTM is an average of the tail dependence of all sectors in the Equation (12). Juan Arismendi Zambrano March 26, / 35

25 Theoretical Motivation LTM Impact in Static and Time-varying Rare Disaster Models 0.2 Risk premium time-varying disaster risk (Wachter 2013) static disaster risk (Wachter 2013) time-varying disaster risk MVLN (LTM = 0.77 at λ = ) static disaster risk MVLN (LTM = 0.77 at λ = ) time-varying disaster risk MVLN (LTM = 0.57 at λ = ) static disaster risk MVLN (LTM = 0.57 at λ = ) Disaster probability Figure 2: The impact of LTM changes in the ERP in a endogenous sectoral rare disaster model This figure presents the instantaneous equity risk premium from two models: (i) the time-varying rare disaster model of Wachter (2013) and (ii) a sectoral static rare disaster model as in Barro (2006). The multi-asset time-varying rare disaster model uses a multivariate lognormal distribution for modeling joint disaster events, and it is calibrated with the multinomial distribution of the disaster contractions, as in Barro and Ursua (2008). Panel A presents the static and dynamic disaster risk for different levels of tail dependence (LTM) for the multi-asset model. Panel B presents the monotone increasing relation of the LTM with the disaster probability in the sectoral time-varying rare disaster model. Juan Arismendi Zambrano March 26, / 35

26 Empirical Results Data Description Data Description Juan Arismendi Zambrano March 26, / 35

27 Empirical Results Data Description Data Predictability Monthly observations from January 1993 to December 2013 Datastream: MSCI US index Ibbotson: US long term Bonds FRED: 1-month T-Bill Goyal and Welch (2008): e.g. D/P, TMS, DFS Tail Dependence Weekly observations from January 1973 to December 2013 Datastream: 10 MSCI US sector-indices Juan Arismendi Zambrano March 26, / 35

28 Empirical Results Tail Estimation Tail Estimation Juan Arismendi Zambrano March 26, / 35

29 Empirical Results Tail Estimation Measuring Extreme Dependence Extreme dependence is the comovement of two assets in the tails distribution Can be analyzed in two perspectives: (i) Left tail dependence - Negative joint extreme events (ii) Right tail dependence - Positive joint extreme events Juan Arismendi Zambrano March 26, / 35

30 Empirical Results Tail Estimation Measuring Extreme Dependence Poon et al. (2004) base their analysis on EVT, using log Pr(S >x) χ = 2 lim s log Pr(S >x,t>x) 1 (13) where S and T are Fréchet marginals obtained respectively from the bivariate returns X and Y (i) Measures asymptotic dependence (ii) Equal to Pearson correlation coefficient for normal bivariate distribution (iii)takes values between 1 and 1 Juan Arismendi Zambrano March 26, / 35

31 Empirical Results Tail Estimation Measuring Extreme Dependence 10 US sector indices 45 pairs of dependence measures (i) 1,040-week rolling windows In each month, aggregate dependence measures through an arithmetic average of the 45 pairs (i) Similar to Kelly & Pruitt (2015) Why should one use bivariate distributions? (i) Captures the intra-country relations between all sectors (ii) Reduces excessive impact of one specific sector (iii)more complete picture of the tail dependence structure Juan Arismendi Zambrano March 26, / 35

32 Empirical Results Tail Estimation Evolution of the dependence measures Figure 3: Levels of the measures This figure presents the evolution of the dependence measures RTM (bivariate right tail sectors mean), LTM (bivariate left tail sectors mean), CORR (sectors correlation mean), ALTM (univariate left tail of the market), and SLTM (univariate left tail sectors mean). The figure presents presents the levels. Juan Arismendi Zambrano March 26, / 35

33 Empirical Results Tail Estimation Evolution of the dependence measures Figure 4: Standardized measures This figure presents the evolution of the dependence measures RTM (bivariate right tail sectors mean), LTM (bivariate left tail sectors mean), CORR (sectors correlation mean), ALTM (univariate left tail of the market), and SLTM (univariate left tail sectors mean). The figure presents the standardized variables. The standardization is performed using the unconditional moments. The gray vertical bands indicate the NBER-defined recessionary periods. Juan Arismendi Zambrano March 26, / 35

34 Empirical Results Predictability Predictability Juan Arismendi Zambrano March 26, / 35

35 Empirical Results Predictability Predictability Based on the framework of Goyal and Welch (2008) Predictive regression ERP t = α + βx t 1 ε t (14) Performance against the historical risk premium ) 2 T RIS 2 =1 t=2 (ERP t ÊRP t T ( ) 2 (15) t=2 ERPt ERP t ) 2 T ROOS 2 t=m+1 (ERP t ÊRP t =1 T ( ) 2 (16) t=m+1 ERPt ERP t Clark and West (2007) test of equal forecast ability Juan Arismendi Zambrano March 26, / 35

36 Empirical Results Predictability Predictability DFS TMS DP TBILL BM DY DE EP SV RIS 2 (%) ROOS 2 (%) NTIS INFL LTY VRP CSTR CORR RTM LTM RIS 2 (%) ROOS 2 (%) Table 1: Predictability by traditional and dependence variables This table presents the R-squared of in- and out-of-sample predictive regressions of one single variable of the risk premium for the next month using several common predictors. Panel A presents the results for common variables and dependence variables. The non-dependence variables are DFS (default spread), TMS (term spread), DP (dividend-price ratio), TBILL (detrended T- bill rate), BM (book-to-market ratio), DY (dividend yield), DE (dividend payout ratio), EP (earnings-price ratio), SV (realized stock variance), NTIS (net equity expansion), INFL (inflation), LTY (long-term yield), VRP (variance risk premium), and CSTR (cross-sectional tail risk). The VRP is from Hao Zhou s website. CSTR is computed by the authors. The variables are defined in Section 3.2. The dependence variables are CORR (correlation sectors mean), RTM (bivariate right tail sectors mean), and LTM (bivariate left tail sectors mean). All these variables are defined in Section 3.1. Panel B presents the results for different versions of LTM, including ALTM (univariate left tail of the market) and SLTM (univariate left tail sectors mean). These variables are defined in Section 2.1. The time span is January 1993 to December The stars represent the statistically significant predictors at a 5% significance level. Juan Arismendi Zambrano March 26, / 35

37 Empirical Results Predictability Predictability Left tail of aggregate market Average of univariate sector left tails Most important bivariate sector VW LTM R 2 IS (%) R 2 OOS (%) Table 2: Predictability by different versions of the LTM Juan Arismendi Zambrano March 26, / 35

38 Conclusions Conclusions I An explicit and economically significant relation between the increase of the probability of a rare consumption disaster and the increase in the tail dependence of the sectors of the economy is derived with implications for the asset pricing literature. Anewmeasureofcountry left tail dependence is proposed, which is based on the cross-sectional left tail behavior of its pairs of sectors. We offer evidence of the predictability of stock market premium using joint sectoral shocks in exercises in-sample and out-of-sample. Juan Arismendi Zambrano March 26, / 35

39 Conclusions Conclusions II No other variable, including the variance risk premium, can predict the stock market risk premium better than the historical average. We show that the information at the industry level and their dependences in the tails are crucial for this outcome. Moreover, the new tail measure that is based on more granular information about dependency of sectors is superior to aggregate univariate tail measures. These results provide evidence that joint negative tail sector relations play an important role in stock market predictability. Juan Arismendi Zambrano March 26, / 35

40 References Juan Arismendi Zambrano March 26, / 35

41 References Bae, K-H, Karolyi, G. A., & Stulz, R. M A New Approach to Measuring Financial Contagion. The Review of Financial Studies, 16(3), Barro, Robert J Rare Disasters and Asset Markets in the Twentieth Century. Quarterly Journal of Economics, 121(3), Comin, Diego, & Mulani, Sunil A theory of growth and volatility at the aggregate and firm level. Journal of Monetary Economics, 56(8), Das, S. R., & Uppal, R Systemic Risk and International Portfolio Choice. The Journal of Finance, 59(6), Gourio, François Disaster Risk and Business Cycles. American Economic Review, 102(6), Hartmann, P., Straetmans, S., & de Vries, C. G Asset Market Linkages in Crisis Periods. Review of Economics and Statistics, 86(1), Holly, Sean, & Petrella, Ivan Factor Demand Linkages, Technology Shocks, and the Business Cycle. Review of Economics and Statistics, 94(4), Juan Arismendi Zambrano March 26, / 35

42 References Horvath, Michael Sectoral shocks and aggregate fluctuations. Journal of Monetary Economics, 45(1), Kelly, Bryan, & Pruitt, Seth The three-pass regression filter: A new approach to forecasting using many predictors. Journal of Econometrics, 186(2), Liu, Jun, Longstaff, Francis A., & Pan, Jun Dynamic Asset Allocation with Event Risk. The Journal of Finance, 58(1), Longin, F., & Solnik, B Extreme Correlation of International Equity Markets. The Journal of Finance, 56(2), Lucas, Robert E Asset Prices in an Exchange Economy. Econometrica, 46(6), Mehra, Rajnish, & Prescott, Edward C The equity premium: A puzzle. Journal of Monetary Economics, 15(2), Poon, S.-H., Rockinger, M., & Tawn, J Extreme Value Dependence in Financial Markets: Diagnostics, Models, and Financial Implications. The Review of Financial Studies, 17(2), Rietz, Thomas A The equity risk premium a solution. Journal of Monetary Economics, 22(1), Juan Arismendi Zambrano March 26, / 35

43 Veldkamp, Laura, & Wolfers, Justin Aggregate shocks or aggregate information? Costly information and business cycle comovement. Journal of Monetary Economics, 54(sep), Wachter, J. A Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility? The Journal of Finance, 68(3), Juan Arismendi Zambrano March 26, / 35