CHAPTER 8

The Geometry of

Mean Variance

Unconstrained Portfolios

W

e have now covered how to find the optimalfs for a given market
system from a number of different standpoints. Also, we have
seen how to derive the efficient frontier. In this chapter we show
how to combine the two notions of optimalfand classical portfolio theory.
Furthermore, we will delve into an analytical study of the geometry of
portfolio construction.

The Capital Market Lines (CMLs)

We can improve upon the performance of any given portfolio by combin-
ing a certain percentage of the portfolio with cash. Figure 8.1 shows this
relationship graphically.
In Figure 8.1, point A represents the return on the risk-free asset. This
would usually be the return on 91-day Treasury bills. Since the risk, the
standard deviation in returns, is regarded as nonexistent, point A is at zero
on the horizontal axis.
Point B represents the tangent portfolio. It is the only portfolio lying
upon the efficient frontier that would be touched by a line drawn from the
risk-free rate of return on the vertical axis and zero on the horizontal axis.
Any point along line segment AB will be composed of the portfolio at Point B
and the risk-free asset. At point B, all of the assets would be in the portfolio,
and at point A all of the assets would be in the risk-free asset. Anywhere in
between points A and B represents having a portion of the assets in both the
portfolio and the risk-free asset. Notice that any portfolio along line segment

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