Again, just as in the Markowitz framework, we define the efficient frontier of the feasible region to be the upper left-hand portion of the boundary. This frontier is efficient in the sense of growth as spelled out by the growth efficiency proposition of Section 15 4. In this case we can be quite specific and state that the efficient frontier is the portion of the boundary curve lying between the minimum-variance point and the log-optimal point.

f>Of course we must temper our enthusiasm with an accounting of the commissions associated with frequent trading

Minimum log variance

Maximum expected log

Maximum expected log

Minimum log variance

In fact, we obtain a strong version of the two-fund theorem. Any point on the efficient frontier can be achieved by a portfolio consisting of a mixture of the minimum-variance portfolio and the log-optimal portfolio We now state this formally as a theorem. We also give a proof using vector-matrix notation, (The reader may safely skip the proof)

"V The Wo-fund theorem Any point on the efficient frontier can be achieved as a mixture ~y of any two points on that frontier, In particular the minimum-log-variance portfolio and the log-optimal portfolio can be used

Proof: Assume there are n securities. Let u = p.2, ■ - , p„), and let w — (ufj, W2, . . ., w„) be portfolio weights, If vv is efficient, it must solve the following problem for some a:

maximize w7u~ ~w7Sw subject to w7 X = 1

By introducing Lagrange multipliers X and y /2, we form the Lagrangian L = wTu - ~w7Sw — A(w71 - 1) - \y(w7Sw — s) The first-order conditions are u — Sw — A.1 -ySw = 0. Hence the solution has the form w = —i—S~"l(u — XI) . 1 + y

The constants X and y are determined so that the solution w satisfies the two constraints of the original problem.

Setting y = 0 means that the second constraint is not active, and hence this solution corresponds to the log-optimal portfolio.

All solutions are linear combinations of the two vectors S~!u and S~'l, so any two such solutions can be used to generate all others. In particular, the log-optimal and the minimum-variance solutions can be used. 1

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