Managing Credit Risk in Corporate Bond Portfolios : A Practitioner's ...

Mathematical Preliminaries 7the distribution from the random variable, then they are known as centralmoments. The second central moment represents the variance of the distributionand is given bys 2 qqs 2 ani1(x ) 2 f(x)dx (continuous distribution)(x i ) 2 p(x i ) (discrete distribution)Following the definition of the expected value of a random variable, thevariance of the distribution can be represented in the expected value notationas E[(X ) 2 ]. The square root of the variance is referred to as thestandard deviation of the distribution. The variance or standard deviationof a distribution gives an indication of the dispersion of the distributionabout the mean.More insight into the shape of the distribution function can be gainedby specifying two other parameters of the distribution. These parametersare the skewness and the kurtosis of the distribution. For a continuous distribution,the skewness and the kurtosis are defined as follows:skewness qqkurtosis qq(x ) 3 f(x)dx(x ) 4 f(x)dxIf the distribution is symmetric around the mean, then the skewness is zero.Kurtosis describes the “peakedness” or “flatness” of a distribution. A leptokurticdistribution is one in which more observations are clusteredaround the mean of the distribution and in the tail region. This is the case,for instance, when one observes the returns on stock prices.In connection with value at risk calculations, one requires the definitionof the quantile of a distribution. The pth quantile of a distribution, denotedX p , is defined as the value such that there is a probability p that the actualvalue of the random variable is less than this value:X pp P(X X p ) f(x)dxq