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

Mathematical Preliminaries 21It immediately follows from this relation that the proportion of variance ofthe original random variables explained by the ith principal component isgiven byl il 1 l 2 l nThe principal components derived by performing an eigenvector decompositionof the covariance matrix are optimal in explaining the variance structureover some historical time period. Outside this sample period overwhich the covariance matrix is estimated, the eigenvectors may not be optimaldirection vectors in the sense of maximizing the observed varianceusing a few principal components. Moreover, the principal componentdirection vectors keep changing as new data come in, and giving a riskinterpretation to these vectors becomes difficult. Given these difficulties,one might like to know whether one could choose some other direction vectorsthat lend themselves to easy interpretation, but nonetheless explain asignificant amount of variance in the original data using only a few components.The answer is yes, with the only requirement that the directionvectors be chosen to be linearly independent.If, for instance, one chooses two direction vectors s and t, denotedshift and twist vectors, respectively, then the variance of the new randomvariables iss 2 s / Ts ©/ ss 2 t / Tt ©/ tThe proportion of variance in the original data explained by the twodepends on how much correlation there is between the two random variablesconstructed. The correlation between the random variables is given byr cov(/ s, / t)s s s t / Ts ©/ ts s s tThe proportion of total variance explained by the two random variables iss 2 s (1 r)s 2 tl 1 l 2 l nQUESTIONS1. A die is rolled 10 times. Find the probability that the face 6 will show(a) at least two times and (b) exactly two times.