ABSTRACT
We begin by considering the class of proportionate investment vectors α˜ that have the same average return as the orthogonal portfolio considered earlier, or μα ≡ α˜T · e˜ = Q˜ T · e˜ = μQ. Here, μα ≡ α˜T · e˜ is the average return on the portfolio whose proportionateinvestment weights are the elements of α˜ and, as previously, e˜ is the vector whose elements are the average returns on theN assets comprising the portfolio. The proportionate investment vectors α˜ for these inefficient portfolios are related to the proportionate investment vector for the orthogonal portfolio, Q˜ , through the formula
α˜ = Q˜ + N−2∑ j=1
where the k˜ j are self-financing ‘kernel’ or ‘arbitrage’ portfolios and the ψj are parametersthat can take on any numerical value. The arbitrage portfolios k˜ j are determined by solvingthe following system of equations:( e1 e2 e3 . . . en 1 1 1 . . . 1
) k˜ j = 0˜
where the e1, e2, e3, . . . , en are the average returns on the individual risky assets (i.e. i the
components of the vector e˜) and 0˜ = (
0 0
) is the null vector (i.e. the vector both of whose
elements are zero). We can solve the above system of equations and thereby show that the arbitrage portfolios take the form
k˜1 =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
e3 − e2 −(e3 − e1) e2 − e1
0 0 ...
