"Frontmatter". In: Analysis of Financial Time Series

DISTRIBUTIONAL PROPERTIES OF RETURNS 11program with six stocks and two options classes. The NYSE added 57 stocks and94 stocks to the program on September 25 and December 4, 2000, respectively. AllNYSE and AMEX stocks started trading in decimals on January 29, 2001.Equation (1.16) suggests that conditional distributions are more relevant thanmarginal distributions in studying asset returns. However, the marginal distributionsmay still be of some interest. In particular, it is easier to estimate marginal distributionsthan conditional distributions using past returns. In addition, in some cases,asset returns have weak empirical serial correlations, and, hence, their marginal distributionsare close to their conditional distributions.Several statistical distributions have been proposed in the literature for themarginal distributions of asset returns, including normal distribution, lognormal distribution,stable distribution, and scale-mixture of normal distributions. We brieflydiscuss these distributions.Normal DistributionA traditional assumption made in financial study is that the simple returns {R it | t =1,...,T } are independently and identically distributed as normal with fixed meanand variance. This assumption makes statistical properties of asset returns tractable.But it encounters several difficulties. First, the lower bound of a simple return is−1. Yet normal distribution may assume any value in the real line and, hence, hasno lower bound. Second, if R it is normally distributed, then the multiperiod simplereturn R it [k] is not normally distributed because it is a product of one-period returns.Third, the normality assumption is not supported by many empirical asset returns,which tend to have a positive excess kurtosis.Lognormal DistributionAnother commonly used assumption is that the log returns r t of an asset is independentand identically distributed (iid) as normal with mean µ and variance σ 2 .The simple returns are then iid lognormal random variables with mean and variancegiven by( )E(R t ) = exp µ + σ 2− 1, Var(R t ) = exp(2µ + σ 2 )[exp(σ 2 ) − 1]. (1.17)2These two equations are useful in studying asset returns (e.g., in forecasting usingmodels built for log returns). Alternatively, let m 1 and m 2 be the mean and varianceof the simple return R t , which is lognormally distributed. Then the mean andvariance of the corresponding log return r t are⎡E(r t ) = ln ⎣m 1 + 1√1 + m 2⎦ ,(1+m 1 ) 2 ⎤[]m 2Var(r t ) = ln 1 +(1 + m 1 ) 2 .Because the sum of a finite number of iid normal random variables is normal,r t [k] is also normally distributed under the normal assumption for {r t }. In addition,