Determinants of Bond Risk Premia Jing-zhi Huang and Zhan Shi Smeal - - PowerPoint PPT Presentation

Introduction Data and Empirical Method Empirical Results Conclusion

Determinants of Bond Risk Premia

Jing-zhi Huang and Zhan Shi Smeal College of Business Penn State University

Waterloo Research Institute in Insurance, Securities and Quantitative Finance University of Waterloo May 15, 2012

Huang and Shi (Penn State Smeal) Determinants of Bond Risk Premia 1 / 31

Introduction Data and Empirical Method Empirical Results Conclusion

Outline

1

Introduction Motivation What we do

2

Data and Empirical Method Data Used Methods for Model Selection The Proposed Two-Step Model Selection

3

Empirical Results In-sample Analysis Out of Sample Analysis

4

Conclusion

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Focus of the Study

Prediction of (default-free) bond excess returns: r(n)

t+1 − y(1) t

= rx(n)

t+1 = γ′Zt + ǫt+1

(1) where r(n)

t+1: an n-yr zero-cpn Treasury bond’s 1-yr log holding period

return over [t, t + 1] y(1)

t

: the log yield of a 1-yr zero-cpn at time t rx(n)

t+1: excess return of an n-yr zero over [t, t + 1]

Zt: forecasting variables

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Examples of Predictors Identified Empirically:

Yield curve factors

Forward Spreads: Fama and Bliss (1987) Yield Spreads: Campbell and Shiller (1991) Forward Rates: Stambaugh (1988), Cochrane and Piazzesi (2005) Factors estimated from a DTSM: Duffee (2011) Decomposition of yields: Cieslak-Povala (2010)

Macroeconomic factors

Predetermined Economic Measures: Duffee (2007) (using inflation, GDP growth, 3m T-bill) Factors extracted from a set of macros: Ludvigson and Ng (2009)

Interest rate derivatives factors

Realized jump measures constructed from Treasury futures prices: Wright and Zhou (2009)

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Recent Empirical Evidence

Cochrane-Piazzesi (2005; CP): Zt=5 fwd rates

R2 = 30 ∼ 35% by a linear comb of the fwd rates

Ludvigson-Ng (2009; LN): Zt=Factors estimated from a set of 131 macro

R2 = 26% by the LN factor alone Containing substantial info beyond CP: raising the R2 from 31% to 44%

Duffee (2011): Zt=5 factors estimated from a DTSM

R2 = 37% by the factors

Wright and Zhou (2009): Zt= Realized jump measures

R2 ∼ 15% by the Jump mean R2 = 60 − 62% when augmented w/ the CP factor

Cieslak-Povala (2010): cycle factor cf : R2 = 41 − 46%

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Recent Empirical Evidence (cont’d)

The notion of the “hidden” forecasting factor (CP, LN, Duffee)

Has substantial forecasting power for bond risk premia But has little impact on yields and “hidden” from the term structure E.g., Duffee’s 5th factor “has an imperceptible affect on yields but has substantial forecast power for future bond excess returns”

Implications for term structure modeling:

Affine TSMs: the important determinants of expected future yields are the important determinants of current yields Duffee (2011) and Barillas (2010) build DTSM where a factor affects dynamics but not current bond prices

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Objectives

Explore the predictability in bond markets beyond what have been documented from macroeconomic aggregates by Ludvigson-Ng (2008) Extract new macro factors w/ higher forecast power for risk premia using a shrinkage-type method Say more about the source of additional predictability Examine the link between the new predictors and existing ones (Cochrane-Piazzesi, Ludvigson-Ng, Duffee, Wright-Zhou, and Cieslak-Povala) Investigate the robustness of the explanatory power of macro variables for forecasting bond risk premia Provide new implications for term structure modeling

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Introduction Data and Empirical Method Empirical Results Conclusion Motivation What we do

Preview of the Main Results

Identifying a single macro factor with an R2 up to 43%:

Parsimonious and easy to interpret Related to economic measures in employment, housing and inflation Like other documented factors w/ forecast power, this factor

Unspanned (affecting dynamics but not current bond prices)

Subsuming both the Ludvigson-Ng (2009) single macro and Cooper-Priestley (2009) output gap factors Different from Duffee’s hidden factor, Wright-Zhou’s jump factor, and Cieslak-Povala’s cycle factor The above 3 factors + our housing fac: ¯ R2 ∼ 54 − 58%

Provide evidence on the robustness of this predictability (out

• f sample and small-sample analysis)

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Introduction Data and Empirical Method Empirical Results Conclusion Data Used Methods for Model Selection The Proposed Two-Step Model Selection

Data Used

131 monthly macroeconomic time series used by Ludvigson-Ng

From Global Insight Basic Economic database or the Fed Sample period: Jan. 1964 - Dec. 2007 8 categories: (1) output; (2) labor market; (3) housing sector; (4) orders and inventories; (5) money and credit; (6) bond and FX markets; (7) prices; and (8) stock market. Transformed data to ensure stationarity

Treasury bond yields

the Fama-Bliss dataset of CRSP monthly yields with maturities of 1 to 5 years

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Introduction Data and Empirical Method Empirical Results Conclusion Data Used Methods for Model Selection The Proposed Two-Step Model Selection

Lasso

Run the OLS using all macro variables available

Over-fitting (poor out-of-sample performance) The resulting OLS predictor not intuitive

Consider the following penalized least squares (PLS) func ||y − Xβ||2 + λ

N

• i=1

|βi|, (2) where λ ≥ 0 is a tuning parameter.

The resulting estimator called the “least absolute shrinkage and selection operator” (lasso) estimator (Tibshirani, 1996) Many “useless” X variables dropped out Interpretability: The resultant model is much less complex than the OLS Forecasting accuracy: Improving the forecast power

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Introduction Data and Empirical Method Empirical Results Conclusion Data Used Methods for Model Selection The Proposed Two-Step Model Selection

Two Variations of Lasso

The Adaptive Lasso (Zou, 2006): min

β∈RN ||y − Xβ||2 + λ

• i

wi|βi|. (3) where wi allows for different tuning parameters used for different predictors. The Group Lasso (Yuan and Lin, 2006): min

β∈RN ||y − Xβ||2 + λ

• h

||βh||. (4) where h is the group index.

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Introduction Data and Empirical Method Empirical Results Conclusion Data Used Methods for Model Selection The Proposed Two-Step Model Selection

Motivation for Using the Group Lasso

A large set of macro variables Macroeconomic data often have cluster structure, where the clusters consist of “similar” time series

Selected factors more intuitive

Potential forecast power of lagged variables Limitations of factor analysis

The analysis focusing on predictors themselves (unsupervised)

(In statistical learning, a problem is supervised if the goal is to predict the value of an outcome based on numerous input measures)

Factors involving linear combinations of “too many” macros Factors’ economic interpretation not always clear

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Introduction Data and Empirical Method Empirical Results Conclusion Data Used Methods for Model Selection The Proposed Two-Step Model Selection

Supervised Adaptive Group Lasso (SAGLasso)

We propose a two-step procedure (SAGLasso):

1 In each group/cluster, apply the adaptive lasso

Y variable used: arx (a vector of average excess bond returns across maturity)

2 Apply the group lasso at the cluster level 3 Construct the single predictor using those selected groups Huang and Shi (Penn State Smeal) Determinants of Bond Risk Premia 13 / 31

Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Group Factors Selected after the 2nd Step

Four out of eight groups selected: ˆ gh

t = X h t ˆ

βh, h = 1, . . . , 4. (5) (corresponding to the original groups #2, 3, 6, and 7)

g 1

t (Employment and Income): Wachter (2006)

g 2

t (Housing): Piazzesi et al. (2007)

g 3

t (Bond Market)

g 4

t (Prices): Brandt and Wang (2003)

Single factor:

• G ≡

4

• h=1

˜ Xh ˜ βg

h .

(6) consisting of only 21 (out of 131) series and 38 variables (including lags)

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Predictive Power of the Group Factors (Table 2)

Regressions of Annual Excess Bond Returns on Group Factors:

maturity ˆ g1t ˆ g2t ˆ g3t ˆ g4t 2-yr 4.353 2.909 1.772 6.965 HH (4.496) (6.115) (5.537) (2.736) NW (5.040) (6.873) (5.916) (3.025) ¯ R2 0.245 0.303 0.208 0.129 mat. ˆ g1t ˆ g2t ˆ g3t ˆ g4t

• CPt

¯ R2 JointTest P-val 2 1.952 1.560 0.700

• 0.581

0.242 0.48 HH (1.992) (2.639) (2.716) (-0.263) (3.531) 196.3 [0.0] NW (2.171) (2.832) (2.928) (-0.287) (3.896) 196.3 [0.0] “HH:” Hansen-Hodrick GMM correction for overlap. “NW–18:” Newey-West lags to correct serial corr

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Predictive Power of the Single Factor G (Table 3)

rx(2)

t+1 = β

Gt + et+1 maturity (yr)

• Gt

¯ R2 Joint Test P-val 2 1.064 0.437 HH (10.797) 116.57 [0.00] NW (11.519) 132.68 [0.00] rx(2)

t+1 = β

Gt + γ LNt + et+1 maturity (yr)

• Gt
• LNt

¯ R2 Joint Test P-val 2 0.903 0.132 0.448 HH (4.544) (1.230) 153.30 [0.00] NW (5.111) (1.340) 154.80 [0.00]

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

The SAGLasso Factor vs. the CP Factor (Table 3)

Table: Comparison to the CP Forward-Rate Factor maturity

• CPt
• Gt

¯ R2 Joint Test P-val 2 0.183 0.883 0.466 HH (2.170) (7.317) 121.20 [0.000] NW (2.445) (8.110) 141.50 [0.000]

Unlike the Ludvigson-Ng macro factor, the Cochrane-Piazzesi (2005) factor is not subsumed by our Gt

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

SAGLasso vs. the Output Gap Factor (Table 4)

gap: the gap factor used in Cooper-Priestley (2009):

maturity gapt−1

• CP

⊥ t

• G ⊥

t

ˆ g1t ¯ R2 2

• 8.798

0.094 NW (-2.480) 2

• 8.798

0.433 0.254 NW (-2.812) (5.241) 2

• 8.798

1.213 0.451 NW (-3.681) (11.356) 2

• 4.623

3.887 0.268 NW (-1.309) (3.741)

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Countercyclical Pattern of the SAGLasso Factor (Fig. 1)

Time Variations of the SAGLasso Factor and the IP Growth:

Normalized series of the SAGLasso Factor and the IP Growth are reported here. Shaded bars denote months designated as recessions by the NBER. Huang and Shi (Penn State Smeal) Determinants of Bond Risk Premia 19 / 31

Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

On the Hidden Factor

In a standard affine TSM, determinants of current yields also determine the expected future yields: In Duffee’s (2011) DTSM:

By construction, one factor affects both terms on the RHS (dynamics) but not prices Here, xm is a hidden factor

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Is the SAGLasso factor G a hidden factor?

• G explains only 0.09% of variations of yield changes

Projecting G onto the first three PCs of yield changes:

• Gt = θ0 + θ1 ˜

H1t + θ2 ˜ H2t + θ3 ˜ H3t + εt. (7)

R2 = 32% = ⇒ A large % of the variation in ˆ Gt orthogonal to the (contemporaneous) info in the bond market As such, ˆ Gt is unspanned (CP (2008), Duffee (2011), Joslin-Priebsch-Singleton (2009))

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Why G is hidden/unspanned?

Rewriting p(n)

t

’s decomposition as follows: y(n)

t

= 1 nEt n−1

• i=0

y(1)

t+i

• + TP(n)

t

. (8) Like Duffee’s xm, G has opposite effects on expected future short rates and the term premium TP(n)

t

Namely, a negative shock to consumption growth (or consumer prices, housing market) = ⇒

Risk Aversion ↑ = ⇒ Term Premium ↑ (on one hand) Belief in Monetary Policy = ⇒ Exp. Future Short Rate ↓ (on the other hand)

= ⇒ Little impact on the current yield

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Impulse Response Analysis (Fig. 2)

Impulse Response Generated from a factor-augmented VAR (FAVAR) model:

24 48

• 2
• 1.5
• 1
• 0.5

0.5 1 FFR Months Ahead 24 48

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 3m TREASURY BILLS Months Ahead 24 48

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 5y TREASURYBONDS Months Ahead 24 48

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 CONSUMER EXPECTATIONS Months Ahead

Estimated impulse responses, with 90 percent confidence intervals, of key macroeconomic indicators to Huang and Shi (Penn State Smeal) Determinants of Bond Risk Premia 23 / 31

Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

The SAGLasso Factor Gt versus Yield Factors (Table 5)

Table: Predictive regressions of excess returns on the

SAGLasso factor Gt and Duffee’s five yield factors

maturity

• Gt
• H1t
• H2t
• H3t
• H4t
• H5t

¯ R2 2-yr 0.888 0.619 1.726

• 3.146

27.656 134.905 0.501 HH (6.999) (1.606) (0.677) (-0.755) (1.494) (2.480) NW (7.633) (1.785) (0.747) (-0.717) (1.567) (2.608)

Both G and Duffee’s hidden factor H5 survive But risk premia contain info not captured by the two factors How to explain the remaining 50% of the variations?

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Evidence on the Cycle Factor (Table 6)

• cf : the cycle factor of Cieslak-Povala (2010):

mat.

• cf t

R2

• Gt
• cf t

¯ R2 ˆ g1t ˆ g2t ˆ g3t ˆ g4t

• cf t

¯ R2 2 0.454 0.28 0.850 0.244 0.50 1.215 2.031 0.352 0.309 0.335 0.54 HH (5.013) (7.686) (3.241) (1.187) (3.279) (1.035) (0.157) (4.353) NW (5.634) (8.209) (3.497) (1.309) (3.541) (1.184) (0.168) (4.776) The Cycle Factor vs. Yield Factors maturity

• H1t
• H2t
• H3t
• H4t
• H5t
• cf t

¯ R2 2 70.300 254.283 934.318 2227.017 16323.859 0.325 0.376 HH (1.725) (0.902) (1.040) (0.971) (3.203) (3.327) NW (1.928) (0.966) (1.167) (1.018) (3.083) (3.565)

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Wright and Zhou’s (2009) Jump Factor

Regressing excess returns on the Jump Mean and Fwd Rates

maturity JMt F (1)

t

F (3)

t

F (5)

t

¯ R2 2

• 9.870
• 0.288

0.974

• 0.199

0.587 HH ( -5.172) ( -1.047) ( 1.916) ( -0.484) NW ( -5.420) ( -1.184) ( 2.120) ( -0.538) 3

• 18.807
• 0.750

2.191

• 0.576

0.591 HH ( -5.560) ( -1.532) ( 2.377) ( -0.771) NW ( -5.745) ( -1.723) ( 2.611) ( -0.853) 4

• 25.017
• 1.289

3.229

• 0.713

0.588 HH ( -5.308) ( -1.987) ( 2.612) ( -0.705) NW ( -5.508) ( -2.216) ( 2.830) ( -0.772) 5

• 30.206
• 1.572

3.476

• 0.395

0.567 HH ( -5.078) ( -2.032) ( 2.327) ( -0.325) NW ( -5.280) ( -2.253) ( 2.511) ( -0.354)

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

• G, JM, Duffee’s

Hi, and cf 1984-2007 (Table 7)

maturity JMt R2

• Gt

R2 JMt

• Gt

R2 2

• 6.794 0.143

1.109 0.308 -5.324 1.014 0.389 HH (-4.282) ( 5.438) (-2.784) (4.353) NW (-4.359) ( 5.791) (-2.945) (4.737)

maturity JMt

• Gt
• H1t
• H2t
• H3t
• H4t
• H5t

¯ R2 2

• 8.373

0.472 -5744.607 3564.557 1250.825 423.294 213.014 0.656 HH (-3.949) (2.024) (-1.095) (1.510) (1.379) (1.659) (3.601) NW (-4.203) (2.194) (-1.078) (1.355) (1.382) (1.779) (4.015)

maturity

• H5t

ˆ g2t

• cf t

JMt ¯ R2 2 17519.0 0.388 0.450

• 4.286 0.537

HH (2.143) (4.824) (3.520) (-2.153) NW (2.255) (5.163) (3.626) (-2.277)

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Out-of-Sample Evidence (Table 8)

Panel A: SAGLasso Factor Gt v.s. constant maturity (yr) Ericsson MSEu/MSEr Clark-McCracken 2 3.364 0.7217 129.6905

Panel B: SAGLasso Factor Gt + AR(6) v.s. AR(6)

maturity (yr) Ericsson MSEu/MSEr Clark-McCracken 2 6.728 0.6446 177.076

Panel C: SAGLasso Factor Gt + LNt v.s. LNt + constant

maturity (yr) Ericsson MSEu/MSEr Clark-McCracken 2 2.730 0.9384 42.988

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Introduction Data and Empirical Method Empirical Results Conclusion In-sample Analysis Out of Sample Analysis

Out-of-Sample Evidence (Table 8): Cont’d

Panel D: CPt + Gt v.s. CPt + constant maturity (yr) Ericsson MSEu/MSEr Clark-McCracken 2 4.893 0.666 120.587

Panel E: CPt + Gt + cf t v.s. CPt + Gt

maturity (yr) Ericsson MSEu/MSEr Clark-McCracken 2 0.340 1.216 4.205

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Introduction Data and Empirical Method Empirical Results Conclusion

Conclusion

Reexamine the predictive power of macroeconomic indicators by developing a SAGLasso approach Provide new and robust evidence on the power of macroeconomic variables for forecasting excess bond returns

Stronger evidence than documented in the literature

Identify a single macro factor

Parsimonious (including only 21 out of 131 series) Easy to interpret (a combination of 4 group factors) Including a housing (group) factor (a new factor) Subsuming the LN factor A hidden factor: Affecting expectations of future yields, but having little impact on current yields Out of sample predictability

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Introduction Data and Empirical Method Empirical Results Conclusion

Conclusion (cont’d)

Complement Ang-Piazzesi (2003), Ludvigson-Ng (2008), and Duffee (2011) Implications for DTSMs

Reinforce the implication of existing studies for incorporating macroeconomic factors that are not spanned by bond yields, yet that have predictive power for excess returns on these bonds May want to incorporate jumps

Candidates for the additional factors (aside from the macro, jump, and Duffee’s hidden factors)?

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