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

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. .

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 $999999 or an unexpected loss of $998001?

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

(5) A Convenient Notation (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

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

? 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

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.

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

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

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

Model 2 (quadratic market model incorporating skewness)

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

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

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

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)

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.

Descriptive statistics for asset returns and market index

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

GMM test statistics

Convergence

Model 1.1 Model 1.2 Model 2: betas Model 2: gammas

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.