Asset Pricing with Real Estate and Skewed Returns: Work

Transcription

1
Asset Pricing with Real Estate and Skewed
Returns:
Work in Progress
Benoît Carmichael : Université Laval
Alain Coën : Université du Québec à Montréal (UQAM)
© 2012, Carmichael, Coën
ACFAS_ Real_Estate_Finance
Montreal May 2012
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Stylized Facts
² Since Samuelson (1970) and Kraus and
Litzenberger (1976), skewness and coskewness
are seen as potentially important determinants
of equilibrium asset returns.
² Problem: It is difficult to build a model linking
analytically skewness premiums to deep
structural parameters governing preference and
the distributions of stocks returns.
² Wealth should be decomposed into: financial
and real estate assets.
² A CAPM with real estate including
idiosyncratic coskewness.
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Stylized Facts
² Solution: Harvey and Siddique (2000, among others) :
explicit expressions for the risk premiums in skewed
environment.
² Advantage: their premiums are distribution free (no specific
distributional assumption).
² Drawbacks: difficult to isolate the influence of skewness on asset
returns.
² Second and third moments are tied by a common set of structural
parameters.
² Risk premiums are reduced form expressions of deeper structural
parameters determining returns.
² No specific distribution: empirical analysis cannot be made with fully
efficient estimation methods.
A need for a more powerful method: Azzalini
(1985)’s Skew-Normal distribution
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Our contribution
² A natural departure point: Azzalini (1985)’s SkewNormal (SN) distribution in a financial economics
framework.
² Main motivations:
-1- The standard normal distribution is a special case of the
SN distribution.
-2- The skewness and coskewness of joint stochastic skewnormal variates have explicit expressions.
² Analytic expressions for the skewness premiums:
² Coskewness and idiosyncratic coskewness.
-3- The financial returns and real estate returns, as joint skewnormal variates have explicit expressions for their
skewness and coskewness.
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Theoretical Framework: The Skew Normal
Distribution
² The Skew-Normal distribution: Azzalini(1985)
The density function for a standardized univariate skewed z
variate.
The skew-normal (SN()) distribution shares many formal
properties with the normal distribution.
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Distribution
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Distribution
² skew-normal variable y
location
Scale parameter
The three moments of y
Reliance on the
deeper structural
parameters, 
and 
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Distribution
² The skew-normal distribution in multivariate
environments: Azzalini and Dalla Valla (1996)
The density
The first three moments
Reliance on the
deeper structural
parameters,
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Distribution
² Explicitit expression of the coskewness
Analytic solutions should help to elucidate intricate role of
skewness and coskewness in asset pricing.
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Theoretical Framework: A Model of Asset
Returns
² Here we develop the theoretical restrictions imposed by the skewnormal distribution on asset returns.
² The starting point is the Euler equation of optimal portfolio
diversification.
² Harvey and Siddique assume that the marginal rate of substitution is a
quadratic function of the market return.
² We adopt Brown and Gibbons (1985)’s assumption: the marginal rate
of substitution is a power function of the gross return on wealth.
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Returns
² Brown and Gibbons (1985)’s assumption is that optimal
consumption is proportional to wealth.
² Decomposition of wealth between financial wealth and
real estate wealth
Mt +1 = b
1- q - d
q
RF ,t +1 RE ,t +1
² Euler equation:
1 = b.Et
1- q - d
q
RF ,t +1 RE ,t +1
×Ri ,t +1
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Returns
1 = b .E t
R qF ,t + 1
- d
1- q
R E ,t + 1
×R i ,t + 1
ijkHL
B HL F
Euler equation:
xi- d
1= b e
.2 F
2 2
2
1- q xE- d q xF+xi+ d wE - d2 q wE2 + 1 d2 q2 wE2 +d2 q r FE wE wF- d2 q2 r FE wE wF+ 1 d2 q2 wF2 - d r iE wE wi+d q r iE wE wi- d q r iF wF wi+ wi
2
2
2
2
mzi wi - dqmzF wF - d 1 - q mzEwE
y
N.B.: for aggregated wealth:
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y
z
{
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Returns
² We obtain the main equation , the key expression
determining expected log returns in a skew-normal
environment:
@
DJ
HHLH
L
L
BBHL F
N
E ri = mzi wi + 1 2u- d2 wE2 + 2d2 qwE2 - d2 q2 wE2 + 2d ErE - mzEwE - 2dq ErE- mzEwE
2
- 2d2 qr FEwEwF + 2d2 q2 r FEwEwF - d2 q2 wF2 + 2dq ErF - mzF wF + 2dr iEwEwi
- 2dqr iEwEwi + 2dqr iF wF wi - wi2 - 2Log 2F
d - 1+q mzE wE- d q mzF wF+mzi wi
y
² It holds for all returns, including the safe return:
HL
rf = 1 2u- d2wE2+2d2qwE2- d2q2wE2+2d ErE- mzEwE - 2dq ErE- mzEwE - 2d2qr FEwEwF
2
+2d2q2r FEwEwF- d2q2wF2+ 2dq ErF- mzFwF - 2Log 2F
d -1+q mzEwE-dqmzFwF
y
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Returns
² To highlight the contribution of skewness we use a third order
Taylor expansion
Approximate real
interest rate: for aggregated
wealth
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Returns
² Equations of the expected log return for the risky asset
and the risk-free asset together with (15) ( lead the
following expression for the aggregated wealth
coskewness
Idiosync.coskewness
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Returns
The idiosyncratic coskewness is a
«volatility» risk.
Systemic variance
risk
skewness risk
The idiosyncratic coskewness of asset i is one of the
Elements determining the systemic variance of asset i
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Empirical Framework: The Log likelihood
function
² We report maximum likelihood estimates of constrained
and unconstrained version of the model.
² The empirical specification is based on the asset pricing
equation.
² Expected and actual skew-normal log returns are linked
by the following relation:
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function
² We complete the empirical specification with the
assumption that the expected market return is linear
function of forecasting variables known at time t. We use
the same assumption for the FTSE NAREIT Real Estate
index
² With skew-normal error, the empirical market log return
equation becomes (we obtain the same decompositon for
NAREIT US index) :
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function
² The log likelihood function of a sample of T observations
is:
² (25) differs from the log likelihood of the normal
distribution because of the third term.
² If asset returns are symmetric, ’is a zero vector and (25)
reduces to the Gaussian log likelihood.
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Empirical Framework: The Data
² Estimates are based on monthly portfolio returns from
Center for Research in Security Prices (CRSP) dataset.
² The returns are the monthly returns of NYSE/AMEX
decile portfolios ranked by market equity: CRSP Stock
File Indices.
² The market return is proxied by the CRSP ValueWeighted Index of the S&P 500 Universe.
² The data cover the period from1971:12 to 2007:12.
² Real returns are obtained by deflating nominal returns
with the rate of inflation, measured by the CRSP CPI
index.
² For real estate, we use as a proxy FTSE NAREIT US
Real Estate index.
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² The forecasting variables are:
The inflation rate;
The log market return;
The log return on human capital;
The log price/earnings ratio
The spread between the log gross returns on long (10
years) and short (30 days) bonds.
² The log return of the FTSE NAREIT US index.
² These variables are all lagged one period in the
forecasting equations.
²
²
²
²
²
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² Sample summary statistics
r1
r2
r3
r4
r5
r6
r7
r8
r9
r10
r. market
r. real est.
Mean
S.E.
Min
Max
Skewness Kurtosis
0,01535186 0,07615109 ‐0,306655 0,534829 1,51198746 9,38540382
0,01264907 0,06385139 ‐0,300949 0,461993 0,70894223 9,07202395
0,01211059 0,05696148 ‐0,297883 0,397882 0,2833992 8,35731014
0,01181936 0,05533863 ‐0,293397 0,371884 0,13740836 7,60921824
0,01158394 0,05409892 ‐0,290756 0,335461 ‐0,11845072 6,35869416
0,01251645 0,0527319 ‐0,280157 0,317161 ‐0,32913639 5,67816433
0,01220951 0,05084083 ‐0,272037 0,250385 ‐0,54233391 4,23193045
0,01140755 0,04802515 ‐0,262605 0,247008 ‐0,39286354 4,04437753
0,01146519 0,04752104 ‐0,248313 0,216687 ‐0,36420239 3,15606022
0,00970279 0,04180037 ‐0,199221 0,176525 ‐0,28313561 2,26606031
0,01011776 0,04247369 ‐0,218138 0,165042 ‐0,40349802 2,66626066
1,10158608 4,03515098 ‐15,2393736 14,0847253 ‐0,40257948 1,78906511
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Empirical Framework: The Method and
Expected Results
² We report unconstrained equation maximum likelihood
estimates.
² These estimates are called unconstrained because the
theoretical restrictions on equilibrium expected log returns
imposed by skew-normal distribution are not taken into
account.
² This procedure allows to focus on the coskewness of joint
log returns in an environment that is not contamined by
potentially false elements of the asset pricing model.
² Equations (22) and (24) are jointly estimated taking into
account all cross-equation restrictions. We report
constrained maximum likelihood estimates.
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Empirical Framework: The Results
² We report implied unconstrained estimates of factor
loadings.
² We expect coefficients of skewness (for financial and real
estate returns) to be highly significant in all cases.
² Coskewness and idiosyncratic coskewness should have an
important role in the explanation of the cross-sectional
variations of returns.
² To recover information about the premiums related to
volatily and skewness, we integrate the theoretical
restrictions imposed by (22) in the estimation procedure.
² We can easily compute implied risk premiums and
associated factor loadings of the constrained and
unconstrained estimates (Covariance and Skewness risks).
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Conclusions
Main results:
- Modelling with Azzalini (1985)’s skew-normal
distribution taking into account real estate in the
wealth function.
- Explicit and analytic expressions of the skewness
premiums: the roles of Coskewness and Idiosyncratic
coskewness in an asset pricing with real estate.
- Maximum likelihood estimates reveal very small
statistically significant skewness premiums.
- With Azzalini (1985)s’ distribution, the skewness
market premium and the skewness real estate premium
extend the CAPM focusing on idiosyncratic risk.
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Conclusions
² Following Poterba (1984, 1991), Mankiw
and Weil (1989), Case and Shiller (1988),
Ayuso and Restoy (2006), extension of an
equilibrium asset pricing approach in a
real estate context where returns are
skewed.
² A CAPM with real estate including
idiosyncratic coskewness.
Montreal May 2012

Asset Pricing with Real Estate and Skewed Returns: Work

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