[ ,t,

Zi = i + bi,1Y1 + ... + bi,k-1Yk-1 + i, i = 1,..., n. (41) Here, n k 2, the vector = ( 1, 2, ..., n)' is independent of Y1,..., Yk-1 and follows a joint sub-Gaussian symmetric k-stable distribution with 1 <

k < 2, zero mean and characteristic function (t) = exp -

the random variables Yj d= SYj ( y j, y j,0), j = 1 , . . . , k - 1 are mutually independent j-stable distributed with 1 < j < 2 and zero mean. In order to estimate the parameters, we need to know the joint law of the vector (Y1,...,Yk-1). Therefore, we assume independent random variables Yj, j = 1 , . . . ,k — 1. Then the characteristic function of the vector of returns

z = [z1,...,zn] ' is given by

z(t) = (t) k-1 j=1 Yj (t'b •,J)e

Under this additional assumption, we can approximate all parameters of any optimal portfolio using a similar procedure of the previous three fund separation model. However, if we assume a given joint ( 1, . . . , k - 1) stable law for the vector (Y1, . . . ,Yk - 1), we can generally determine estimators of the parameters studying the characteristics of the multivariate stable law.