Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Asset Prices and the Return to Normalcy
Ole Wilms
(University of Zurich)
joint work with Walter Pohl and Karl Schmedders
(University of Zurich)
Economic Applications of Modern Numerical Methods
Becker Friedman Institute, University of Chicago
Rosenwald Hall, Room 301 November 22, 2013
Professor Kenneth L. Judd
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Motivation
I
I
Financial market characteristics:
I
high stock returns (high volatility)
I
low risk free rate (low volatility)
I
stock return predictability
I
...
Common assumption: random walk component in consumption
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Motivation (Continued)
I
Empirical evidence: trend-stationarity vs random walk
(see Nelson and Plosser (1982), Dejong and Whiteman (1991), Perron
(1989), Andreou and Spanos (2003) or Christiano and Eichenbaum
(1990))
I
Research focus: impact of trend-stationarity in time-series on asset
prices and returns (as in DeJong and Ripoll (2007), Tallarini (2000) or
Rodriguez (2006))
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
The Economy
I
Discrete, infinite time, complete markets
I
Representative investor with CRRA-utility
I
Consumption process:
ln ct
gt
I
=
(1 − ρc )gt + ρc ln ct−1 + c,t ,
c,t ∼ N(0, σ2c ),
= ḡ + gt−1
Consumption modelled as in Tallarini (2000) but the paper only
concentrates on special case with IES = 1 (EZ-Utility)
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
The Economy (Continued)
Asset pricing equations:
I
Risk free rate:
ptf
I
= βEt
ct+1 −γ
ct
Consumption claim:
pt
= βEt
ct
ct+1 1−γ
ct
pt+1
+1
ct+1
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Theoretical Results
We have have closed form solutions for the model with a permanent and a
temporary shock when ρc = 0:
I
I
ln ct
=
gt + νt
gt
=
ḡ + gt−1 + t
t
∼
N(0, σ2 ), νt ∼ N(0, σ2ν )
σ = 0 ⇒ trend-stationarity
σν = 0 ⇒ random walk
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Theoretical Results (Continued)
Table: Analytical Solutions
σν
σ
E (rts )
E (rtf )
EP
γ=2
0.01
0.005
0
0
0.005
0.01
0.0936
0.0617
0.0513
0.0513
0.0461
0.0305
0.0423
0.0157
0.0208
γ=6
0.01
0.005
0
0
0.005
0.01
0.5883
0.2027
0.0101
0.1389
0.0888
-0.0486
0.4494
0.1139
0.0588
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Detrending
I
Remove linear trend: ct∗ = ct /(1 + g)t
I
Detrended consumption:
∗
log(ct∗ ) = (1 − ρc )µc + ρc log(ct−1
) + c,t ,
I
c,t ∼ N(0, σ2c )
Pricing of the consumption claim:
∗ ct+1 1−γ pt+1
pt
1−γ
= β(1 + g)
Et
+1
ct
ct∗
ct+1
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Projection Method
pt
ct
I
Define xt =
I
We solve for x which is a function of c ∗ :
∗ ∗
c+ 1−γ
∗
x (c ∗ ) = β(1 + g)1−γ E
x
(c
)
+
1
c
+
c∗
I
Solution functions are approximated by n-degree chebychev
polynomials:
x (c ∗ ) =
n
X
αi Φi (c ∗ )
i=1
∗
Here c+
denotes the next periods state of c ∗ , Φi are the basis functions and
αi the unknown solution coefficients of the chebychev polynomials.
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Projection Method
Residual function:
R(c ∗ ; α) ≡ x (c ∗ ) − β(1 + g)1−γ E
I
∗ 1−γ
∗
c+
∗
x
(c
)
+
1
c
+
c∗
Collocation projection:
R(ci∗ ; α) = 0, i = 1, . . . , n
I
Galerkin projection:
Z
R(c ∗ ; α)Φi (c ∗ ) = 0, i = 1, . . . , n
c∗
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Projection Method
I
We solve for x (c ∗ ) on a range of ±6σc ∗ around the unconditional
mean of c ∗
I
We take chebychev nodes for the state-space
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Integral in pricing equation:
Z +∞ −∞
∗
c+
c∗
1−γ
∗
∗ ∗
∗
x (c+
) + 1 f (c+
|c )dc+
⇒ Gauss-Hermite quadrature for expectation
Integral in galerkin projection:
Z +6σc ∗
R(ci∗ ; α)Φi (c ∗ ) = 0, i = 1, . . . , n
−6σc ∗
⇒ Gauss-chebychev quadrature
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Data
We use three datasets to check robustness;
I
First dataset (Mehra & Prescott 1985): 1889-1978
I
Second dataset (Mehra & Prescott 1985): 1889-2004
I
Third dataset (Robert Shiller): 1889-2009
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Estimation Procedure
1. Estimate linear trend g and detrend consumption: ct∗ = ct /(1 + g)t
2. Estimate AR(1) process for detrended consumption by OLS:
∗
log(ct∗ ) = (1 − ρc )µc + ρc log(ct−1
) + c,t ,
c,t ∼ N(0, σ2c )
Table: Parameter Estimates
MP1889−1978
MP1889−2004
SH1889−2009
µc
σc
g
ρc
95% conf. interv. (ρc )
1.12
1.11
2.64
0.035
0.031
0.034
0.018
0.018
0.021
0.92
0.92
0.90
(0.84, 1.00)
(0.84, 0.99)
(0.82, 0.98)
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Moments of the Consumption Process
Table: Comparison of Empirical and Model Moments
E (ln ct+1
)
ct
σ(ln ct+1
)
ct
ρ(ln ct+1
)
ct
∗
ρ(ln ct+1
,
lnc
t )
ct
89 − 78
Model
89 − 04
Model
89 − 09
Model
0.0175
0.0357
-0.1362
-0.1980
0.0181
0.0357
-0.0501
-0.1946
0.0173
0.0319
-0.1203
-0.2071
0.0178
0.0319
-0.0426
-0.2059
0.0200
0.0352
-0.0640
-0.2274
0.0206
0.0352
-0.0498
-0.2237
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Empirical Moments of Returns
Table: Empirical Moments of Realized Returns for Different Periods
E (rts )
Vol(rts )
E (rtf )
Vol(rtf )
EP
MP1889−1978
0.0698
0.1654
0.008
0.0567
0.0618
MP1889−2004
0.0776
0.1660
0.0134
0.0520
0.0642
SH1889−2009
0.0760
0.1873
0.0197
0.0580
0.0563
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Structure of the Following Tables
I
Data shows empirical moments found in the data (as given on the last
slide)
I
Bench refers to the benchmark model. We take the basic model by
Mehra and Prescott (1985) where consumption growth follows an
AR(1) process.
I
CC is our trend-stationary model of the pricing of the consumption
claim
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Results
Table: Results for the dataset from 1889-2009
Data
ρc
E (rts )
Vol(rts )
E (rtf )
Vol(rtf )
EP
0.9
0.0760
0.1873
0.0197
0.0580
0.0563
γ=2
Bench
CC
0.95
0.9
0.85
0.0514
0.0395
0.0487
0.0055
0.0027
0.0533
0.0545
0.0558
0.0423
0.0657
0.0841
0.0511
0.0497
0.0490
0.0082
0.0163
0.0246
0.0022
0.0047
0.0068
γ=8
Bench
CC
0.95
0.9
0.85
0.1557
0.0632
0.1395
0.0243
0.0162
0.1837
0.1894
0.2026
0.0855
0.1681
0.2480
0.1687
0.1487
0.1324
0.0364
0.0717
0.1071
0.0150
0.0407
0.0701
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Risk Free Rate Puzzle
I
Kocherlakota (1990) shows that in growth economies well defined
equilibria can exist even though the discount factor is larger than one
and the agent might still prefer consumption today over future
consumption.
I
β > 1 is used e.g. in Piazzesi, Schneider and Tuzel (2007).
I
We show that an equilibrium exists if β <
1
(1+g)1−γ
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Results
Table: Results with adjusted β for the dataset from 1889-2009
β
Data
E (rts )
Vol(rts )
E (rtf )
Vol(rtf )
EP
0.0760
0.1873
0.0197
0.0580
0.0563
0.0197
0.0197
0.0054
0.0160
0.0027
0.0048
0.0197
0.0197
0.0217
0.0640
0.0150
0.0612
γ=2
Bench
CC
1.019
1.02
0.0224
0.0244
0.0385
0.0717
γ=8
Bench
CC
1.106
1.116
0.0347
0.0809
0.0580
0.2428
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Robustness Across Datasets
MP−Data 1889−1978
MP−Data 1889−2004
0.1
Shiller−Data 1889−2009
0.1
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.07
0.06
0.06
0.06
ρec + 0.05
0.09
ρec
ρec − 0.05
0.08
0.05
0.04
Equity Premium
Equity Premium
Equity Premium
Bench
0.05
0.04
0.05
0.04
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
2
4
6
γ
8
10
0
2
4
6
γ
8
10
0
2
4
6
γ
8
10
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Model Two: Pricing of the Dividend Claim
I
Data on dividends
I
Consumption = Divided Income + Labor Income
I
Vector autoregressive process for consumption and labor income to
dividend ratio
I
We assume a common linear trend g in consumption, prices, dividends
and labor income
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Pricing of Dividend Claim
We have now two states: detrended consumption c ∗ and the ratio of labor
income to dividend income δ:
ct
δt
xt
= dt + et (⇒ ct∗ = dt∗ + et∗ )
e∗
et
= t∗
=
dt
dt
= (I − Φ)µ̄ + Φx t−1 + t
with
xt =
log ct∗
log δt
,Φ =
ρc
ρδc
ρcδ
µc
c,t
, µ̄ =
, t =
∼ N(0, Σ)
ρδ
µδ
δ,t
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
Pricing of Dividend Claim
Pricing equation for the dividend claim:
pt
= β(1 + ḡ)1−γ Et
dt
Define xt =
pt
dt .
∗ 1−γ ct+1
1
pt+1
+
.
ct∗
dt+1 1 + δt+1
Now xt is a function of ct∗ and δt .
⇒ Apply solution method as described before but with the two-dimensional
state space (c ∗ , δ).
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Robustness - Pricing of Dividend Claim
Table: Results for the dataset from 1889-2009
Data
E (rts )
Vol(rts )
E (rtf )
Vol(rtf )
EP
0.0760
0.1873
0.0197
0.0580
0.0563
0.0187
0.0182
0.0059
0.0048
0.0700
0.0628
0.0423
0.0563
γ=2
β = 0.99
β = 1.02
0.0561
0.0244
0.0990
0.0731
0.0502
0.0197
γ=7
β = 0.99
β = 1.10
0.1784
0.0760
0.2313
0.2618
0.1361
0.0197
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Return Predictability
Regression: Cumulative Returns on log Price Dividend Ratio
Table: Predictability of Stock Returns
h=1
R
Data
Bench
CC
DC
2
0.0317
0.1207
0.0673
0.0204
h=3
2
β
R
-0.0880
-1.2735
-0.1493
-0.0732
0.0644
0.0617
0.1769
0.0565
h=5
2
β
R
-0.1962
-1.2576
-0.4634
-0.2387
0.1048
0.0431
0.2587
0.0888
β
-0.3268
-1.3274
-0.7927
-0.4328
Conclusion
Motivation
The Model
Solution Method
Results
Model Two
Return Predictability
Conclusion
I
Large impact of underlying process
I
Model is able to match financial market characteristics like
I
High equity premium
I
Low risk free rate
I
High stock and low bond volatilities
I
Return predictability
with standard preferences and risk aversion below 10
Conclusion
Appendix
Empirical evidence: trend-stationarity vs. random walk
Table: Test Statistics and Critical Values of Unit Root Tests
MP1889−1978
MP1889−2004
SH1889−2009
ct
ADF-Test
dt
pt
ct
PP-Test
dt
pt
ct
KPSS-Test
dt
pt
-2.0325
-2.1832
-2.3218
-3.6731
-4.2298
-4.1468
-2.6208
-2.3758
-2.7335
-2.3314
-2.5224
-2.5763
-3.2025
-3.6364
-3.5601
-2.4199
-2.2484
-2.5933
0.1016
0.1576
0.2088
0.0772
0.0535
0.0909
0.1219
0.1327
0.1496
1%
5%
10%
1%
-3.99
-3.43
-3.13
-4.04
Critical Values
5%
-3.45
10%
1%
5%
10%
-3.15
0.216
0.146
0.119
Appendix
Hansen-Jagannathan Bounds
1
0.9
0.8
0.7
σ(m)
0.6
0.5
0.4
0.3
0.2
γ=1
0.1
0
0.94
0.96
0.98
1
1.02
E(m)
1.04
1.06
1.08
1.1