Time-varying Risk of Nominal Bonds: How Important Are Macroeconomic Shocks?

Transcription

1 Time-varying Risk of Nominal Bonds: How Important Are Macroeconomic Shocks? Andrey Ermolov Columbia Business School February 7, / 45

2 Motivation: Time-varying stock and bond return correlation Unconditional correlation is 0.02 Computed quarterly from daily data Expectations=dynamic conditional correlation of Colacito et.al. (2009) 2 / 45

3 Stock and bond return correlation - important but difficult to explain moment Important: First order effect on portfolio variance Stocks and bonds large and closely integrated markets: should be modeled jointly Difficult to explain: Theoretically: starting from Shiller and Beltratti (1992) Empirically: e.g., even in dynamic factor models (Baele et.al., 2010) 3 / 45

4 Question Are macroeconomic shocks (consumption growth and inflation) related to time-varying stock and bond return correlation? Can they generate correlations of observed magnitudes? Historically, how much do they matter at different points in time? 4 / 45

5 Contribution: Methodology Tractable stuctural model for analyzing macroeconomic risk of nominal assets Campbell et.al., 2014 Burkhardt and Hasseltoft, Me 2012; Song, 2014 Type Habit Long-run risk Habit Non-Gaussian macro dynamics No Yes Yes Exact closed form solutions No No Yes Realistic term structure Yes No Yes Macroeconomic shocks from consumption and inflation data Do macroeconomic shocks matter for the risk of nominal assets? No No Yes Not much A lot Half of the sample 5 / 45

6 Contribution: Empirical results Economically intuitive characterization of macroeconomic shocks Implications for stock and bond return correlation: macroeconomic shocks generate sizeable positive and negative correlations, although negative correlations smaller and less frequent than in data historically, macroeconomic shocks are important in explaining high correlations from late 70 s to early 90 s and low correlations pre- and during the Great Recession 6 / 45

7 Overview of the model External habit utility: realistic asset pricing moments: in particular, realistic term structure Macroeconomic dynamics from Bekaert, Engstrom, and Ermolov (2014c): convenient for modeling time-varying bond risk: drives time-varying stock and bond return correlations 7 / 45

8 Consumption growth and inflation Consumption growth: g t+1 = ḡ + ɛ g t+1 Constant mean ḡ Heteroskedastic 0-mean shock ɛ g t+1 Inflation: π t+1 = π + x π t + ɛ π t+1 Unconditional mean π Persistent 0-mean inflation expectations x π t Heteroskedastic 0-mean shock ɛ π t+1 8 / 45

9 Macroeconomic shocks ɛ g t+1 = σd g ut+1 d + σg s u }{{}}{{} t+1, s >0 >0 ɛ π t+1 = σ }{{} π d ut+1 d σπ s u }{{} t+1, s >0 >0 Cov(ut+1, d ut+1) s = 0, Var(ut+1) d = Var(ut+1) s = 1. u d t+1 - demand shock : moves g t+1 and π t+1 in the same direction nominal bonds hedge well u s t+1 - supply shock : moves g t+1 and π t+1 in opposite directions nominal bonds hedge poorly 9 / 45

10 Macroeconomic environments If supply and demand shocks are heteroskedastic, Cov t (ɛ g t+1, ɛπ t+1) will vary over time: Cov t (ɛ g t+1, ɛπ t+1) = σ d g σ d πvar t (u d t+1) σ s gσ s πvar t (u s t+1) Demand shock environment: Cov t (ɛ g t+1, ɛπ t+1) > 0 stock and bond correlations relatively low Supply shock environment: Cov t (ɛ g t+1, ɛπ t+1) < 0 stock and bond correlations relatively high 10 / 45

11 Modeling demand and supply shocks Demand and supply shocks modeled using Bad Environment-Good Environment (BEGE) structure (Bekaert and Engstrom, 2014): component models of two 0-mean shocks } ut+1 d = σp d ωp,t+1 d σn d ωn,t+1, d ω p,t+1 - good shock ut+1 s = σpω s p,t+1 s σnω s n,t+1, s ω n,t+1 - bad shock Shocks follow demeaned gamma distributions: ωp,t+1 d Γ(pt d, 1) pt d, ωn,t+1 d Γ(nt d, 1) nt d, gamma distribution with Γ(x, y) shape parameter x and ωp,t+1 s Γ(pt s, 1) pt s, scale parameter y ωn,t+1 s Γ(nt s, 1) nt s. 11 / 45

12 Bad Environment-Good Environment structure: Probability density function 12 / 45

13 Time-varying variances p t can be interpreted as good variance and n t as bad variance Variances are persistent and driven by the realization shocks, capturing volatility clustering (Gourieroux and Jasiak, 2006): p d t+1 = pd + ρ d p(p d t p d ) + σ d ppω d p,t+1, Similar processes for n d t+1, ps t+1, ns t+1 13 / 45

14 Time-varying variances: Probability density functions Intuitive expressions for the moments 14 / 45

15 Model: Why gamma distributed shocks? Empirically supported to capture non-gaussian features prevalent in consumption and inflation data (Bekaert and Engstrom, 2009; Bekaert, Engstrom, and Ermolov, 2014a,b) Non-Gaussian features facilitate theoretically matching risk-premia Intuitive closed form solutions Efficient estimation 15 / 45

16 Data US quarterly observations: 1969Q4-2012Q4 Working (1960) adjusted consumption of non-durables and services Inflation: St.Louis Fed Inflation expectations: Survey of Professional Forecasters 16 / 45

17 Estimation Maximum likelihood estimation using only macroeconomic data (no financial data) Input: consumption growth and inflation time series Output 1: macroeconomic dynamics parameters estimates Output 2: expected p d t, n d t, p s t, n s t time series Methodology: sequentially computing likelihood over observations - in characteristic function domain formulas for computing likelihood available in closed form (Bates, 2006) Detailed estimation overview Maximum likelihood estimation overview 17 / 45

18 Consumption growth and inflation shocks ɛ g t+1 = (0.0003) ud t (0.0003) us t+1 ɛ π t+1 = (0.0010) ud t (0.0006) us t+1 Consumption growth shocks: supply driven Inflation shocks: demand driven 18 / 45

19 Demand and supply variances 19 / 45

20 Supply shocks Supply shock parameter estimates 20 / 45

21 Correlation between industry portfolio returns and bad supply shocks (ω s n,t+1 ) More correlations 21 / 45

22 Demand shocks Demand shock parameter estimates 22 / 45

23 Correlation between industry portfolio returns and bad demand shocks (ω d n,t+1 ) More correlations 23 / 45

24 Conditional correlation between consumption growth and inflation 24 / 45

25 Utility Representative agent Habit utility: E 0 t=0 βt (C t H t) 1 γ 1 γ Discount factor β Risk-aversion coefficient γ (always assumed >1) C t - consumption H t - external habit: e.g., exogeneous standard of living 25 / 45

26 Habit Inverse surplus ratio: q t = ln q t+1 = q + ρ q (q t q) Ct C t H t γ q }{{} const>0 ɛ g t+1 Habit = weighted average of past consumption shocks Here Campbell and Cochrane (1999) Price of risk Constant Time-varying Amount of risk Time-varying Constant Ermolov (2014a) shows that the time-varying amount of risk specification has advantages in term structure modeling (+asset prices in closed-form!) 26 / 45

27 Financial Assets Risk-free 0-coupon nominal bonds Aggregate equity = claim to the aggregate dividends 27 / 45

28 Dividends and expected inflation Real dividend growth: d t+1 = ḡ + ɛ d t+1 ɛ d t+1 heteroskedastic 0-mean shock, 0 < Corr(ɛ d t+1, ɛg t+1 ) < 1 Persistent inflation expectations x π t, 0 < Corr(x π t, ɛ π t ) < 1 ḡ - consumption growth mean, ɛ g t+1 - consumption growth shock, ɛ π t+1 - inflation shock More details 28 / 45

29 Pricing Stochastic discount factor (SDF): M t+1 = βe γg t+1+γ(q t+1 q t) Innovations to SDF: m t+1 E t (m t+1 ) = a }{{} p ωp,t+1+ d a }{{} n const<0 const>0 ω d n,t+1+ a p }{{} const<0 ω s p,t+1+ a n }{{} const>0 Positive consumption shocks decrease marginal utility Negative consumption shocks increase marginal utility ω s n,t+1 Nominal SDF: m $ t+1 = m t+1 π t+1 29 / 45

30 Asset prices Time t n-period nominal bond prices: P $ n,t = exp(c $ n +Q $ nq t +X π n x π t +P d$ n p d t +N d$ n n d t +P s$ n p s t +N s$ n n s t ) Time t aggregate equity P D -ratio: P t n=1 exp(c e n + Q e nq t + P de n p d t + N de n n d t + P se n p s t + N se D t = Coefficients recursively defined n n s t ) 30 / 45

31 Price impact of demand shocks Suppose a positive demand shock occurs 31 / 45

32 Price impact of demand shocks Suppose a positive demand shock occurs 32 / 45

33 Price impact of supply shocks Suppose a positive supply shock occurs 33 / 45

34 Price impact of supply shocks Suppose a positive supply shock occurs 34 / 45

35 Conditional return comovements In the model: Cov t (r e t+1, r b t+1) adpa e dp b pt d + a }{{} dna e dn b nt }{{} d + aspa e sp b pt s + a }{{} sna e sn b }{{} <0 <0 >0 >0 Demand shock environment: Cov t (r e t+1, r b t+1) < 0 - nominal bonds hedge well n s t Supply shock environment: Cov t (r e t+1, r b t+1) > 0 - nominal bonds hedge poorly 35 / 45

36 Data US quarterly observations: 1969Q4-2012Q4 Corporate earnings payout (Longstaff and Piazzesi, 2004): NIPA Aggregate stock returns: CRSP Treasury yields: Gürkaynak et.al. (2006) 36 / 45

37 Estimation Macroeconomic dynamics already estimated from consumption and inflation data Generalized method of moments (GMM) estimation 5 preference parameters to estimate: β, γ, q, ρ q, γ q 9 unconditional moments to match: 1 quarter nominal interest rate and its variance 5 year bond excess return and its variance price-dividend ratio and its variance equity premium and its variance unconditional 5 year bond and stock return covariance 37 / 45

38 Estimated preference parameters β 0.99 fixed γ 4.12 (0.51) q 1.00 fixed ρ q 0.99 (0.02) γ q (0.84) 38 / 45

39 GMM moments match Moment Data Model E(y 1q $ ) 1.33% 1.53% (0.18%) Var(y 1q $ ) 6.48E E-05 (2.00E-05) E(r5y bx ) 0.49% 0.62% (0.24%) Var(r5y bx ) (0.0003) E(pd) (0.10) Var(pd) (0.04) E(r ex ) 1.08% 0.90% (0.58%) Var(r ex ) (0.0013) Cov(r ex, r bx ) (0.0005) Overidentification test p-value Implied macro moments Implied local risk-aversion Implied financial moments 39 / 45

40 Implied stock and bond return correlations Unconditional correlation Data Model (0.13) Conditional correlations Data (expectations) Model Min Max st percentile (0.05) 2.5 th percentile (0.04) 97.5 th percentile (0.02) 99 th percentile (0.03) Macroeconomic shocks generate sizeable positive and negative stock and bond return correlations Negative correlations less extreme and frequent than in data 40 / 45

41 Historical stock and bond return correlations Macroeconomic shocks important from late 70 s until early 90 s and pre- and during Great Recession Excluding and : Corr(Model, Data)=0.58, Corr(r ex,r bx )=0.27 Additional results 41 / 45

42 Defining flights to safety episodes High-frequency episodes of simultaneous extreme positive bond and negative stock returns unlikely to be related to macroeconomic factors (Baele et.al. 2014) 42 / 45

43 Explaining residual stock and bond return correlations with flights to safety episodes 43 / 45

44 Comparision to the literature Studies finding weak links between risk of nominal assets and macroeconomy: restrictive macroeconomic dynamics (difficult to incorporate realistic dynamics into asset pricing frameworks in a tractable manner) Studies finding strong links between risk of nominal assets and macroeconomy: rely on financial data to estimate macroeconomic shocks 44 / 45

45 Conclusions Tractable structural framework for understanding macroeconomic risk of nominal assets: tons of applications! Economically characterizing macroeconomic shocks Macroeconomic shocks: produce sizeable positive and negative stock and bond return correlations, although negative correlations smaller and less frequent than in data historically most important for correlations from late 70 s to early 90 s and pre- and during the Great Recession 45 / 45

46 Appendix 1: BEGE conditional moments Intuitive theoretical expressions for (unscaled) moments: Var t (u t+1 ) = σ 2 pp t + σ 2 nn t Skw t (u t+1 ) = 2(σ 3 pp t σ 3 nn t ) Ex.Kur t (u t+1 ) = 6(σ 4 pp t + σ 4 nn t ) Back 46 / 45

47 Appendix 2: Macroeconomic dynamics estimation procedure Stage 1: Filter ɛ g t+1 and ɛπ t+1 using OLS Stage 2: Estimate σ d g, σ s g, σ d π, σ s π to invert ɛ g t+1 and ɛπ t+1 to ud t+1 and us t+1 using GMM (based on unconditional second and third moments, including cross-moments) Stage 3: From ut+1 d and us t+1, estimate macroeconomic volatility parameters ( p d, n d, p s, n s, ρ d p, ρ d n, ρ s p, ρ s n, σpp, d σnn, d σpp, s σnn) s using the characteristic function domain approximate maximum likelihood (Bates, 2006) Stage 4: Estimate inflation expectations and dividend dynamics by regressing them on u d t+1 and us t+1 Back 47 / 45

48 Appendix 3: Maximum likelihood estimation procedure Below is the algorithm for u d t, algorithm for u s t is identical Sequentially computing likelihood over {u d t = σ d p ω d p,t σ d n ω d n,t} T t=1 Step 1: Computing likelihood of ut+1 d given pd t and nt d distributions Step 2: Updating pt d and nt d distributions given ut+1 d Step 3: Computing conditional distribution of pt+1 d and nt+1 d given ud t+1 In characteristic function domain (approximate) Steps 1-3 formulas available in closed form (Bates, 2006) Back 48 / 45

49 Appendix 4: Supply shocks parameters Good variance Bad variance σp s 0.15 σs n 0.26 (0.03) (0.07) p s 7.69 n s (0.71) (1.12) ρ s p 0.92 ρ s n 0.99 (0.09) (0.14) σpp s 0.92 σnn s 0.40 (0.30) (0.21) Back 49 / 45

50 Appendix 5: Correlation between industry portfolio returns and good supply shocks (ω s p,t+1 ) Back 50 / 45

51 Appendix 6: Demand shocks parameters Good variance Bad variance σp d 0.07 σd n 5.39 (0.03) (1.32) p d n d 0.01 (7.17) (0.01) ρ d p 0.96 ρ d n 0.75 (0.03) (0.20) σpp d 0.96 σnn d 0.08 (0.14) (0.04) Gaussian good component Rare-disaster type bad component Back 51 / 45

52 Appendix 7: Correlation between industry portfolio returns and good demand shocks (ω d p,t+1 ) Back 52 / 45

53 Appendix 8: Dividends and expected inflation specifications Real dividend growth: d t+1 = ḡ + γ d ɛ g t+1 + γ dd u d t+1 + ɛdiv t+1, ɛdiv t+1 N (0, σ d ) Inflation expectations: xt+1 π = ρ x π xπ t ɛ xπ t+1 N (0, σπ x ) + γ x π ɛ π t+1 + γ x π d u d t+1 + ɛxπ t+1, Parameter Estimate Standard error ḡ 0.42% 0.04% π 1.06% 0.07% γ d γ d d σ d ρ x π γ x π γ x π d σ x π Back 53 / 45

54 Appendix 9: Implied local risk-aversion Percentile 1% 5% 25% 50% 75% 95% 99% Value Back 54 / 45

55 Appendix 10: Unconditional consumption growth and inflation dynamics Consumption growth Inflation Data Model Data Model Mean 0.42% 0.42% 1.06% 1.06% (0.04%) (0.07%) Standard deviation 0.41% 0.44% 0.86% 0.86% (0.03%) (0.08%) Skewness (0.26) (0.78) Excess kurtosis (0.56) (2.53) Pr(<mean-2 Standard deviation) 2.91% 3.11% 0.58% 1.62% (0.97%) (0.60%) Pr(<mean-4 Standard deviation) 0.00% 0.00% 0.58% 0.19% (0.12%) (0.60%) Pr(>mean+2 Standard deviation) 2.91% 2.05% 5.54% 2.71% (1.04%) (1.64%) Pr(>mean+4 Standard deviation) (0.00%) (0.03%) 0.00% 0.03% (0.00%) (0.14%) Corr(g t, π t ) (0.11) (0.18) Back 55 / 45

56 Appendix 11: Implied financial moments Data Model y 5y $ y 1y $ 0.18% 0.12% (0.04%) y 5y y 1y 0.11% 0.09% (0.02%) Fama-Bliss (1987) slope: 5 years vs 1 year (0.36) AC 1 (pd) (0.03) Slope rt+1 ex wrt pdt (0.0171) Back 56 / 45

57 Appendix 12: Time pattern in stock and bond return correlations Difference Data: expectations *** (0.17) (0.22) Model *** (0.09) (0.15) Back 57 / 45