the academy of economic studies bucharest doctoral school of finance and banking
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THE ACADEMY OF ECONOMIC STUDIES BUCHAREST DOCTORAL SCHOOL OF FINANCE AND BANKING. DISSERTATION PAPER Asset Pricing and Skewness. Student: Penciu Alexandru Supervisor: Professor Moisǎ Altǎr. BUCHAREST, JUNE 2004. - PowerPoint PPT PresentationTRANSCRIPT
THE ACADEMY OF ECONOMIC STUDIES BUCHARESTDOCTORAL SCHOOL OF FINANCE AND BANKING
DISSERTATION PAPERAsset Pricing and Skewness
Student: Penciu AlexandruSupervisor: Professor Moisǎ Altǎr
BUCHAREST, JUNE 2004
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In asset pricing theory there is very often assumed that variance or the squared root of variance, standard deviation, is the appropriate measure of risk to the investor.
But, while returns follow non-normal, non-symmetrical distributions it is possible that investors care not only for variance but perhaps they develop some form of preferences for higher moments.
(1)
A quick and eloquent example: The two lotteries example
001.0999.0
9999991:1LR
001.0999.0
9980011999:2LR
Having equal values for expected return and variance, based on which question you choose between these two lotteries:
1. do you like an almost certain loss of $1 or an almost certain win of $1999?
2. do you like an unexpected win of $999999 or an unexpected loss of $998001?
Skewness and Portfolio Choice Theory
Skewness is the third centered moment of a random variable:
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with the corresponding co-moments:
2))())(((),( YEYXEXEYXcoasim
))())(())(((),,( ZEZYEYXEXEZYXcoasim
(2)
(3)
(4)
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(5)
(6)A Convenient Notation
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1min)var(
2
1min
w. r. t.ex
xRE P
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analytically tractable solution
two fund separation theorem and the properties of conjugated portfolios (assuming there is a risk free asset)
the CAPM emerges fMifi RRERRE )()(
mean-variance efficient portfolio weights
Mean-Variance Pricing Theory
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.002 -0.001 0 0.001 0.002 0.003 0.004
x1
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
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x2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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x3
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
std. dev.
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n re
turn
Mean-Variance-Skewness Pricing
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w. r. t.
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high non-linearity
analytically intractable solution
unrevealed pricing formula
specialized non-linear optimization software
mean-variance-skewness efficient portfolio weights
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-0.4
-0.2
0
0.2
0.4
0.6
0.8
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x2
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x3
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mean return
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w.
An Utility Approach
Assume the investor has a NIARA-class utility function with positive marginal utility and risk aversion for wealth and income (as a stochastic variable).
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(7)
By expanding the utility function in a Taylor series about final (expected)wealth (initial wealth plus period’s income) and taking expectations we get expected utility as a function of the stochastic variable’s moments.
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First order condition: 0x
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three moment CAPM
(8)
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Model 1.1 (classical CAPM)
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22 Np equations
2Nq parameters identification condition p > q is satisfied
Model 1.2
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22 Np equations
2Nq parameters identification condition p > q is satisfied
Model 2 (quadratic market model incorporating skewness)
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),cov(
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0)( qMMMiqMRiMRi qqqE for i = 1,2,…,N assets
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23 Np equations
22 Nq parameters identification condition p > q is satisfied
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Coping with non-linearity: a Taylor series expansion about a set of consistent estimators
(9)
(10)
GMM Estimation and Testing
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sample mean of is )(tg TT
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,)1(, qTjx
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According to Hansen [1982], the GMM estimator minimizes
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the covariance matrix estimator for the model’s parameters
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0)]([ tgE:0HThe J-test.
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Having p (moment conditions) > q (parameters), then
(17)
Data
The data used for this analysis consists of series of daily returns of nineteen liquid stocks traded at the Bucharest Stock Exchange, on both the first and second tiers, starting from April 17th 1998 to December 19th 2002, giving a total of 1136 observations. These stocks are: Alro, Arctic, Antibiotice, Azomures, Oltchim, Rulmentul, Terapia, Banca Transilvania, Amonil, Compa, Carbid-Fox, Electroaparataj, Mefin, OilTerminal, Policolor, Mopan, Sinteza, Sofert and Silcotub. The BSE is a young market so these stocks were chosen in order to provide a large sample of observations for estimation. The BET-C index was used as a proxy for the market portfolio. For the risk free rate there was used the medium interest rate for deposits on the inter-banking market, BUBID.The whole period was divided in three equal sub-periods in order to check if there are major discrepancies in test statistics across time.
The empirical density vs. the normal density for the market index returns
The empirical cumulative distribution function versus the normal cumulative distribution function for the market index
betas
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sample Equivalents
GMM estimates
gammas
-6
-4
-2
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sample Equivalents
GMM estimates
betas
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sample Equivalents
GMM estimates
gammas
-6
-4
-2
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sample Equivalents
GMM estimates
Model 1.1 Model 1.2
Model 2: betas Model 2: gammas
We have briefly illustrated how to deal with multi-moment portfolio analysis by using appropriate conventional notation, specifically designed procedures programmed in Gauss and nlp solvers like MINOS or CONOPT provided by optimization software like GAMS in order to compensate for the intractability of analytical solutions.
In the absence of an analytical solution provided by a non-linear optimization problem and believing that an utility based pricing model is to restrictive in it’s assumptions to be supported by actual data, we test if a quadratic model as the ones suggested by Barone-Adesi, Urga and Gagliardini [2003] or Harvey and Siddique [2000b] is indeed incorporating a measure of systematic skewness. Using the Generalized Method of Moments, and specifically designed procedures implemented with Gauss we find that all the models tested perform well, the test statistic confirming the null hypothesis of orthogonality.
Conclusions