The “Fed-model” and the changing
correlation of stock and bond returns:
An equilibrium approach
Henrik Hasseltoft
Stockholm School of Economics &
The Institute for Financial Research-SIFR
June 6, 2009
Introduction
• Objective: Provide an economic understanding of
two “puzzling” features of data:
1. The positive correlation between dividend yields
and nominal yields - “Fed-model”
2. The time-varying correlation between stock and
bond returns
• Several statistical papers but few economic models
• I take a consumption-based approach
1. The “Fed-model”
16
US 5y Treasury rate
US dividend yields
14
12
10
8
6
4
2
0
1950
1960
1970
1980
1990
2000
2010
1. Why is it puzzling ?
• Historically, changes in inflation and bond risk premiums have
been main drivers of nominal yields
• Drivers of dividend yields, Gordon growth formula:
D
 R  G  rf  rp  G
P
• Inflation illusion (e.g., Modigliani and Cohn, 1979, Campbell and
Vuolteenaho, 2004)
• Bekaert and Engstrom (2008): Rational mechanisms are at work.
High correlation between inflation and the equity risk premium
2. The time-varying correlation of
stock and bond returns
1
0.5
0
-0.5
-1
1950
1960
1970
1980
1990
2000
2010
2. The time-varying correlation of
stock and bond returns
• Possible channels:
Changes in real rates – Positive correlation
Dividend growth – Wedge between stocks and bonds
Changes in risk premiums – Positive correlation if both assets
are risky
• Empirically, higher short rates, steeper yield curve,
higher inflation, higher inflation uncertainty &
volatility predict correlations positively
E.g., Li (2002), Viceira (2007), David and Veronesi (2008), Yang
et. al (2009)
Main Results
• Model provides rational explanation of the Fed-model and the
time-varying correlation of returns
Key mechanism
• US data - inflation has signalled low future consumption growth
• Investors dislike positive inflation shocks
• Equity and nominal bonds are poor inflation hedges
Implications
• Equity/bond risk premiums are positively correlated
• Corr(Dividend yields, nominal yields) > 0
• Corrt(stock ret, bond ret) move positively with macro volatility
Model
• Builds on Bansal and Yaron (2004), Piazzesi and
Schneider (2006)
• Recursive preferences of Epstein-Zin (1989) and Weil
(1989)
• Dynamics: Specification I and II
zt 1  ( ct 1 ,  t 1 , d t 1 ) '
zt 1    xt   t 1
xt 1   xt  t 1
 t 1 ~ N .i.i.d . (0, )
• Interaction between real variables and inflation
Asset prices
• Log asset prices linear functions of state variables
(homoscedastic case):
• Dividend growth drives a wedge between stocks and
bonds
• Compare to Bansal and Yaron 2004 and Piazzesi and
Schneider 2006
Model Implications
• Quarterly US data 1952 – 2007: Expected and
unexpected inflation signal lower future consumption
growth
• Innovation to pricing kernel (homoscedastic case)
• Given elasticity of intertemporal substitution > 1:
Investors’ dislike positive inflation shocks
Investors’ dislike positive shocks to macro volatility
• PD-ratios negatively related to inflation, macro volatility
as in data. Power utility gives opposite implications
The equity risk premium
• Positive inflation shocks lower both investors’ well-being and real
stock returns → positive risk premium
 Covt (mt 1 , rm,t 1 ) 
 ( AC c2,t  BD  2,t  ( AD  BC ) c ,t  AE cd ,t  BE  d ,t  F )
• Higher macroeconomic volatility raises risk premiums, in
particular inflation volatility
• Positive covariance between dividend growth and inflation
lowers risk premiums. Important in late 1990s
The inflation risk premium on bonds
• Decompose nominal short rate
• The inflation risk premium:
• Bonds have low payoffs in bad times → positive risk premium
• Inflation volatility plays key role (again)
Conditional volatilities
• Consumption growth
• Inflation
Conditional covariances
• Dividend growth and inflation
• Consumption growth and inflation
Explaining the Fed-model
• Positive unconditional correlation between equity and bond risk
premiums due to common exposure to macroeconomic volatility
• Turning off risk premium channel yields a correlation of 0.17
Explaining the time-varying correlation
of stock and bond returns
• Recall: Higher macroeconomic volatility increases both equity
and bond risk premiums
positive covariance of returns
• Covariance is increasing in the volatility of consumption growth
and inflation
• Covariance is decreasing in covariance between inflation and
dividend growth
• Recall: Dividend growth drives a wedge between stocks and
bonds
Predicting realized
correlations
Realized correlations
1
Model
Data
0.5
0
1970s-early 1980s: High macro volatility
Early 1980s-2000: “The Great Moderation”
-0.5
Late 1990s: Low volatility + positive
covariance of dividend growth and inflation
-1
1950
1960
1970
1980
1990
2000
2010
Conclusion
1. Risk premiums on equity and nominal bonds comove positively
through changes in macroeconomic volatility.
2. Positive correlation of risk premiums explain the Fed-model,
which stands in contrast to the inflation illusion argument
3. Conditional correlation of stock and bond returns loads
positively on macroeconomic volatility
4. Low macro volatility + pos correlation between dividend growth
and inflation = negative stock-bond correlation
5. Model suggests that inflation volatility is key driver of both
equity and bond risk premia
•
Key: inflation has real effects + recursive preferences
Extra Slides
Estimating homoscedastic case
• Maximum likelihood
zt 1    xt  t 1
 xc ,t 1   0.533  0.104
 xc ,t   0.245  0.107
  c ,t 1 

 
  



  (0.157) (0.052)
   (0.068) (0.092)


 x   0.281 1.019
 x   0.076
 
0.495
 ,t 1
 ,t


   
  ,t 1 

  (0.122) (0.038)
   (0.050) (0.064)


 x   0.564
 x    0.203
 
0
.
799
0
.
295
 d ,t 1  
 d ,t  
 d ,t 1 

  (0.413)

(0.071)    (0.235)
(0.055) 

 

• Both expected and unexpected inflation signal low future
consumption growth
Asset prices
• Risk aversion = 10, EIS = 1.5, discount factor = 0.997
Asset prices
• Valuation ratios are negatively related to expected
inflation in data