Chapter 4

ARCH (autoregressive conditional heteroscedasticity) is a special case of GARCH that allows future variances to rely only on past disturbances, whereas GARCH allows future variances to depend on past variances as well.

The autocorrelation of a time series of returns from an investment refers to the possible correlation of the returns with one another through time.

Autoregressive refers to when subsequent values to a variable are to explained by past values of the same variable.

The beta of an asset is defined as the covariance between the asset's returns and a return such as the market index, divided by the variance of the index's return, or, equivalently, as the correlation coefficient multiplied by the ratio of the asset volatility to market volatility: ßi = Coc(Rm,Ri) / Var(Rm) = oim/o2 where ßi is the beta of the returns of asset i (Ri) with respect to a market index of returns, Rm.

Conditionally heteroskedastic financial market prices have different levels of return variation even when specified conditions are similar (e.g., when they are viewed ad similar price levels).

The correlation coefficient (also called the Pearson correlation coefficient) measures the degree of association between two variables, but unlike the covariance, the correlation coefficient can be easily interpreted.

The covariance of the return of two assets is a measure of the degree or tendency of two variables to move in relationship with each other.

Future possible returns and their probabilities are referred to as expectational or ex ante returns.

Ex post returns are realized outcomes rather than anticipated outcomes.

Excess kurtosis provides a more intuitive measure of kurtosis relative to the normal distribution because it has a value of zero in the case of the normal distribution: Excess Kurtosis = {E[(R-u)4]/o4}-3

First-order autocorrelation refers to the correlation between the return in time period t and the return in the immediately previous time period, t-1.

GARCH (generalized autoregressive conditional heteroskedasticity) is an example of a time-series method that adjusts for varying volatility.

Heteroskedasticity is when the variance of a variable changes with respect to a variable, such as itself or time.

Homoskedasticity is when the variance of a variable is constant.

The Jarque-Bera test involves a statistic that is a function of the skewness and excess kurtosis of the sample: JB = (n/6)[S2+(K2/4)] where JB is the Jarque-Bera test statistic, n is the number of observations, S is the skewness of the sample, and K ist the excess kurtosis of the sample.

Kurtosis serves as an indicator of the peaks and tails of a distribution. Kurtosis = E[(R-u)4]/o4

If a return distribution has positive excess kurtosis, meaning it has more kurtosis than the normal distribution, it is said to be leptokurtotic, or fat tailed, and to exhibit leptokurtosis.

A variable has a lognormal distribution if the distribution of the logarithm of the variable is normally distributed.

The most common raw moment is the first raw moment and is known as the mean, or expected value, and is an indication of the central tendency of the variable.

If a return distribution has no excess kurtosis, meaning it has the same kurtosis as the normal distribution, it is said to be mesokurtic, mesokurtotic, or normal tailed, and to exhibit mesokurtosis.

The normal distribution is the familiar bell-shaped distribution, also known as the Gaussian distribution.

A correlation coefficient of -1 indicates that the two assets move in the exact opposite direction and in the same proportion, a result known as perfect linear negative correlation.

A correlation coefficient of +1 indicates that the two assets move in the exact same direction and in the same proportion, a result known as perfect linear positive correlation.

If a return distribution has negative excess kurtosis, meaning less kurtosis than the normal distribution, it is said to be platykurtic, platykurtotic, or thin tailed, and to exhibit platykurtosis.

The skewness is equal to the third central moment divided by the standard deviation of the variable cubed and serves as a measure of asymmetry: Skewness = E[(R-u)3]/o3

The Spearman rank correlation is a correlation designed to adjust for outlieres by measuring the relationship between variable ranks rather than variable values.

The square root of the variance is an extremely popular and useful measure of dispersion known as the standard deviation: Standard Deviation = SRo2 = o

The variance is the second central moment and is the expected value of the deviations squared.

In investment terminology, volatility is a popular term that is used synonymously with the standard deviation of returns.