The Shape Of The Portfolio Possibilities Curve
Reexamine the earlier figures in this chapter and note that the portion of the portfolio possibility curve that lies above the minimum variance portfolio is concave while that which lies below the minimum variance portfolio is convex.8 This is not due to the peculiarities of the examples we have chosen but rather is a general characteristic of all portfolio problems.
This can easily be demonstrated. Remember that the equations and diagrams we have developed are appropriate for all combinations of securities and portfolios. We now examine combinations of the minimum variance portfolio and an asset that has a higher return and risk.
8A concave curve is one where a straight line connecting any two points on the curve lies entirely under the curve. M a curve is convex, a straight line connecting any two points lies totally above the curve. The only exception to this is that a straight line is both convex and concave and so can be referred to as either.
Figures 5.6a, 5.6b, and 5.6c represent three hypothesized shapes for combinations of Colonel Motors and the minimum variance portfolio. The shape depicted in 5.6b cannot be possible since we have demonstrated that combinations of assets cannot have more risk than that found on a straight line connecting two assets (and that only in the case of perfect positive correlation). But what about the shape presented in Figure 5.6c? Here all portfolios have less risk than the straight line connecting Colonel Motors and the minimum variance portfolio. However, this is impossible. Examine the portfolios labeled u and v. These are simply combinations of the minimum variance portfolio and Colonel Motors. Since u and V are portfolios, all combinations of v and Vmust lie either on a straight line connecting u and v or above such a straight line.9 Hence 5.6c is impossible and the only legitimate shape is that shown in 5.6a, which is a concave curve. Analogous reasoning can be used to show that if we consider combinations of the minimum variance portfolio and a security or portfolio with higher variance and lower return, the curve must be convex, that is, it must look like Figure 5.7a rather than 5.7b or 5.7c.
Now that we understand the riskreturn properties of combinations of two assets, we are in a position to study the attributes of combinations of all risky assets.
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