1. Asset Pricing and Skewness
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

2. 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.
(1)
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. .

3. A quick and eloquent example: The two lotteries example
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 $ or an unexpected loss of $998001?

4. Skewness and Portfolio Choice Theory
Skewness is the third centered moment of a random variable:
(2)
with the corresponding co-moments:
(3)
(4)

5. (5)
A Convenient Notation
(6)

6. Mean-Variance Pricing Theory
mean-variance efficient portfolio weights
w. r. t.
analytically tractable solution
two fund separation theorem and the properties of conjugated portfolios
(assuming there is a risk free asset)
the CAPM emerges

8. Mean variance efficient portfolios
are not necessarily
mean-variance-skewness efficient

9. ? Mean-Variance-Skewness Pricing w. r. t.
mean-variance-skewness efficient portfolio weights
specialized non-linear optimization software
high non-linearity
analytically intractable solution ?
unrevealed pricing formula

11. 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).
(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.

12. s.t.
First order condition:
three moment CAPM
(8)

13. Model 1.1 (classical CAPM)
, for i = 1,2,…,N assets
, for i = 1,2,…,N assets
equations
identification condition p > q is satisfied
parameters

15. Model 1.2 , for i = 1,2,…,N assets , for i = 1,2,…,N assets equations
identification condition p > q is satisfied
parameters

16. Model 2 (quadratic market model incorporating skewness)

17. for i = 1,2,…,N assets
for i = 1,2,…,N assets
for i = 1,2,…,N assets
equations
identification condition p > q is satisfied
parameters

18. Coping with non-linearity: a Taylor series expansion about a set of consistent estimators
(9)
Then, for f(x,y) = xy,
(10)

19. GMM Estimation and Testing
with j = 1,2,…,p and t = 1,2,…,T
(11)
the moment function
(12)
sample mean of is
(13)
, j = 1,2,…,p and

20. Having p (moment conditions) > q (parameters), then
According to Hansen [1982], the GMM estimator minimizes
(14)
the covariance matrix estimator for the model’s parameters
(15)
estimating the parameter vector
(16)
The J-test.
Having p (moment conditions) > q (parameters), then
(17)

21. 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.

22. Descriptive statistics for asset returns and market index

23. The empirical cumulative distribution
function versus the normal
cumulative distribution function for
the market index
The empirical density vs. the normal
density for the market index returns

24. GMM test statistics

25. Convergence

27. Model 1.1
Model 1.2
Model 2: betas
Model 2: gammas

28. Conclusions
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.