The Returns on Human Capital
H. Lustig and S. Van Nieuwerburgh
Sept 18, 2007
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Introduction
Many asset pricing models try to relate consumption growth
to asset returns
Lustig and Van Nieuwerburgh ask the question: What
restrictions does the single agent framework impose on the
joint distribution of aggregate consumption growth and
market returns?
Market returns are a weighted average of the returns on
financial and human wealth
Instead of making assumptions directly on the unobserved
human wealth return process, Lustig and Van Nieuwerburgh
impute consumption innovations not attributed to news about
current or future financial returns to the returns on human
wealth
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Results
Other models cannot match certain moments of their implied
consumption growth
Innovations in financial asset returns are negatively correlated
with innovations in human capital returns, for any EIS
Implied total market return is negatively correlated with
returns on financial wealth if EIS < 1
Hedging component of the risk premium is positive, unlike
most models
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Consumption Volatility and Correlation Puzzle
In the data, the volatility of financial asset returns is much
higher than that of aggregate consumption and the series are
only weakly correlated
If an agents portfolio contains only financial wealth, the model
implied volatility of consumption is 5 times too high, and the
correlation of innovations with financial assets is four times
too high, regardless of the EIS
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Epstein-Zin Preferences
Agents maximize:
(1−γ)/θ 1−γ
Ut = ((1 − β)Ct + β(Et Ut+1 )1/θ )θ/(1−γ)
Subject to
m
Wt = Rt+1 (Wt − Ct )
Where
Ct is consumption
m
Rt+1 is return on market portfolio
1−γ
θ = 1−(1/σ)
γ is relative risk aversion
σ is EIS
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Distribution of Consumption and Asset returns
Following Campbell (1993), linearize the budget constraint
and Euler Equation and assume that consumption and returns
are conditionally homoskedastic and jointly log normal to get:
∞
m m
ct+1 − Et ct+1 = rt+1 − Et rt+1 + (1 − σ)(Et+1 − Et ) m
ρj rt+1+j
j=1
Rest of the paper: study the properties of aggregate implied
by this relationship between aggregate consumption and the
market return process
How to measure market returns?
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Measuring the Market Return
Return on the Market Portfolio:
rtm = (1 − νt−1 )rta + νt−1 rty
rtm is the log return on the market portfolio
rta is the return on financial wealth
rty is the return on human wealth
νt is the ratio of human wealth to total wealth
Only rta is observed, need to model rty and νt
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Measuring Financial Asset Returns: 2 ways
1 CRSP Value Weighted Returns
2 ”Firm Value”
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Cointegration
Following Lettau and Ludvigson (2001), a cointegrating
relationship exists between consumption and aggregate
wealth, proxied by cayt = λct − (1 − ν)at − νyt
ˆ
λ = 1.0395, ν = 0.7761
Imposes restrictions on the transitions of ∆ct , ∆at , ∆yt
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Computing Innovations: VAR
a
State Vector zt = (∆at , ∆yt , dpt , reltbt , yspt , st , ∆ct )
Estimate VECM:
zt+1 = Azt + Γcayt + t+1
Re-write this as a VAR:
ˆ z
zt+1 = Aˆt + ˆt
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Notation
(c)t = ct − Et−1 [ct ] = ∆ct − Et−1 [∆ct ] = e7 t
DRta = rta − Et−1 [rta ]
a ∞ j a
DR∞ = (Et − Et−1 ) j=1 ρ rt+j
CFty = ∆yt − Et−1 [∆yt ]
y ∞ j
CFt,∞ = (Et − Et−1 ) j=0 ρ ∆yt+j
CFta = ∆dt − Et−1 [∆dt ]
a ∞ j
CFt,∞ = (Et − Et−1 ) j=0 ρ ∆dt+j
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Stylized Facts
Return innovations are much more volatile than consumption
deviations (13.5% vs 0.8%)
Consumption innovations are only weakly correlated with
return innovations Corr ((c)t , DRta ) = 0.21
News about future financial returns is volatile, St.dev = 14.3%
Current return innovations are negatively correlated with news
a
about future expected returns, Corr (DRta , DR∞ ) = −0.86
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Stylized Facts, Cont
Current and future dividend growth and labor income growth
a y
are negatively correlated, Corr (CFt,∞ , CFt,∞ ) = −.423
Periods with good news about current financial asset returns
tend to be periods with good news about current and future
y
labor income growth, Corr (CFt,∞ , DRta ) = .493
Periods with good news about future financial asset returns
tend to be periods with bad news about current and future
y a
labor income growth, Corr (CFt,∞ , DRt,∞ ) = −.633
Current and future labor income growth is not very volatile,
y
St.Dev (CFt,∞ ) = .030
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Modeling Human Capital Returns
Campbell (1991) framework gives;
∞ ∞
rty −Et−1 [rty ] = (Et −Et−1 ) ρj ∆yt+j −(Et −Et−1 ) y
ρj rt+j
j=0 j=1
y y
Equivalently write: DRty = CFt,∞ − DR∞
y
Only CFt,∞ is observed
y
How to model Et [rt+1 ]?
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Modeling Human Capital Returns
Model 1: Only financial wealth, νt = 0∀t
Model 2 (Campbell 1996): Et−1 [rty ] = Et−1 [rta ]
Model 3 (Shiller 1995): Et−1 [rty ] = 0
Model 4 (Jagannathan and Wang 1996):
rty − Et−1 [rty ] = ∆yt − Et−1 ∆yt
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Modeling Human Capital Returns
y
All of these can be written as Et [rt+1 ] = C zt for the
appropriate choice of C.
Once we have C, can compute:
y
DR∞ = C ρ(I − ρA)−1 t
y y
DRty = CFt,∞ − DR∞ = (e2 − C ρ)(I − ρA)−1 t
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Modeling Human Capital Returns
Lustig and Van Nieuwerburgh: All of the above models imply
aggregate consumption is too volatile and too highly
correlated with financial returns
Model 5: Choose the vector C which minimizes the distance
between the model-implied consumption volatility and
correlation and the same moments in the data.
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Constant Wealth Shares
¯
Suppose we had an estimate of ν , then:
¯ y y a
(c)t = (1 − ν )DRta + ν CFt,∞ − σ¯DR∞ + (1 − σ)(1 − ν )DR∞
¯ ν ¯
Lustig and Van Nieuwerburgh choose C so that the moments
of this equation match those in the data.
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Time Varying Wealth Shares (Sketch)
Under some conditions, we can write
1
νt = y
1+ exp(dpt − dpt + log ( 1−st ))
a
st
The consumption innovations become:
y y
(c)t = (1 − νt−1 )DRta + νt−1 CFt,∞ − (νt−1 + (σ − 1)¯)DR∞
ν
+ (1 − σ)(1 − ν )DR∞ − (1 − σ)(DRtw ,a − DRtw ,y ) (1)
¯ a
∞
DRtw ,a = (Et − Et−1 ) j
j=1 ρ (νt−1+j ¯ a
− ν )rt+j
∞
DRtw ,y = (Et − Et−1 ) j
j=1 ρ (νt−1+j ¯ y
− ν )rt+j
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Results
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Why does Model 5 work?
Large negative correlation between human and financial
wealth returns, Corr (DRty , DRta ) < 0
y y
Corr (DRty , DRta ) = Cov [CFt,∞ , CFt,∞ ] − Cov [CFt,∞ , DR∞ ]
a a
y a y a
− Cov (DR∞ , CFt,∞ ) + Cov (DR∞ , DR∞ ) (2)
Good news about current and future cash flows on human
wealth coincides with bad news about current and future cash
flows for financial assets as well as lower future risk premia on
financial assets
Discount rates on human wealth are high when expected
future dividend growth is high and future risk premia on
financial assets are low.
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Implications for Market Returns
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Sensitivity to EIS
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Implications for asset pricing
From Campbell (1996)
a m
Et rt+1 −r f +1/2Vaa,t = γCov (DRta , DRtm )+(γ−1)Cov (DRta , DR∞ )
Model 5 delivers positive hedging risk premia for all EIS,
negative myopic risk premia for low EIS, else positive
Models 1-4 exactly the opposite
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Title
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Other Explanations
Innovations in aggregate consumption growth not accounted
for by innovations in financial returns was attributed to
human wealth returns
Other models explored include habit formation, adding
housing wealth, heteroskedastic market returns, and
heterogeneity
None can explain the volatility and correlation puzzles
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital
Conclusion
Consumption volatility and correlation puzzles are hard to
reconcile in other asset pricing models
The returns to human wealth needs to be negatively
correlated with returns on financial assets in order to generate
a consumption process that is consistent with the data,
contrary to standard theoretical models.
H. Lustig and S. Van Nieuwerburgh The Returns on Human Capital