The Onefund Theorem

When risk-free borrowing and lending are available, the efficient set consists of a single straight line, which is the top of the triangular feasible region. This line is tangent to the original feasible set of risky assets (See Figure 6.14.) There will be a point F in the original feasible set that is on the Sine segment defining the overall efficient set. It is clear that any efficient point (any point on the line) can be expressed as a combination of this asset and the risk-free asset. We obtain different efficient points by changing the weighting between these two (including negative weights of the risk-free asset to borrow money in order to leverage the buying of the risky asset) The portfolio

FIGURE 6.14 One-fund theorem. When both borrowing and lending at the risk-free rale are allowed, there is a unique fund F of risky assets that is efficient All points on the efficient frontier are combinations of F and the risk-free asset

represented by the tangent point can be thought of as a fund made up of assets and sold as a unit. The role of this fund is summarized by the following statement:

The one-fund theorem There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk-free asset

This is a final conclusion of mean-variance portfolio theory, and this conclusion is the launch point for the next chapter It is fine to stop reading here, and (after doing some exercises) to go on to the next chapter But if you want to see how to calculate the special efficient point F, read the specialized subsection that follows.

Solution Method*

How can we find the tangent point that represents the efficient fund? We just characterize that point in terms of an optimization problem. Given a point in the feasible region, we draw a line between the risk-free asset and that point We denote the angle between that line and the horizontal axis by 0 For any feasible (risky) portfolio p, we have tan 0 =

The tangent portfolio is the feasible point that maximizes 6 or, equivalently, maximizes tan0. It turns out that this problem can be reduced to the solution of a system of linear equations

To develop the solution, suppose, as usual, that there are n risky assets We assign weights w\, u>2, w„ to the risky assets such that Yl'Ui w< = ^ There is zero weight on the risk-free asset in the tangent fund. (Note that we are allowing short selling among the risky assets ) For rp — Yl'l-i w>'»1 we liave 7,, — i and if = EL I w>'f Thus>

anWiWt

It should be clear that multiplication of all uV s by a constant will not change the expression, since the constant will cancel Hence it is not necessary to impose the constraint Wj = 1 here

We then set the derivative of tan9 with respect to each wk equal to zero. This leads (see Exercise 10) to the following equations:

where X is an (unknown) constant. Making the substitution = Xw-, for each /, (6.9) becomes n y crki Vj ~rk-rf, k ~ 1,2,...,«. (6.10)

We solve these linear'equations for the u/s and then normalize to determine the wf s; that is,

Example 6.12 (Three uncorrelated assets) We consider again Example 6 9, where the three risky assets were uncorrelated and each had variance equal to 1. The three mean rates of return were n = 1, Ti = 2, and ô = 3 We assume in addition that there is a risk-free asset with rate ty = 5.

We apply (6,9), which is very simple in this case because the covariances are all zero, to find us = 1 - 5 = .5 V2 - 2 - .5 = 1.5 U3 = .3 — .5 = .2.5 We then normalize these values by dividing by their sum, 4 5, and find

Example 6.13 (A larger portfolio) Consider the five risky assets of Example 6.11 Assume also that there is a risk-free asset with /y = 10%. We can easily find the special fund F

We note that the system of equations (6.10) is identical to those used to find v1 and v2 in Example 6.11, but with a different right-hand side. Actually the right-hand side is a linear combination of those used for v! and v2; namely, Tk—> y = 1 xrk—ij-x 1. Therefore the solution to (6.10) is v = v2 — /yv1 Thus (using iy = lOto be consistent with the units used in the earlier example), v = (2.242, -.,427, 2.728, — 786, .3..306) We normalize this to obtain the final result w = (..317, - 060, .386, -.111, 468)

Basically, we have used the fact that portfolio F is a combination of two known efficient points.

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