1078 The Journal of finance has no systematic risk; i.e., for eachy ∑x Any such arbitrage portfolio, x, will have returns of x=(xE)+(xb1)61+…+(xbk)8k+(x∈) E+(xb1)61+ (xbk)8K xe The term(xe)is(approximately) eliminated by applying the law of large numbers For example, if a denotes the average variance of the E, terms, and if, for implicity, each x, exactly equals +l/n, then var(xe=var(1/n∑e) =[var()]/n = here we have assumed that the E are mutually independent It follows that for large numbers of assets, the variance of xe will be negligible and we can diversify the Recapitulating, we have shown that it is possible to choose arbitrage portfolios with neither systematic nor unsystematic risk terms! If the individual is in equilibrium and is content with his current portfolio, we must also have XE No portfolio is an equilibrium(held) portfolio if it can be improved upon without incurring additional risk or committing additional resources To put the matter somewhat differently, in equilibrium all portfolios of these n assets which satisfy the conditions of using no wealth and having no risk must also earn no return on average The above conditions are really statements in linear algebra. Any vector, x which is orthogonal to the constant vector and to each of the coefficient vectors, b, (j=1,., k), must also be orthogonal to the vector of expected returns. An algebraic consequence of this statement is that the expected return vector, E must be a linear combination of the constant vector and the b, vectors, In algebraic terms, there exist k+1 weights,λ,A1,……,λ k such that E2=Ao+λ1b2 λkb If there is a riskless asset with return, Eo, then bo, =0 and E=入o E2-E0=λ1b1+…+Akbk, with the understanding that Eo is the riskless rate of return if such an asset exists