**6.1 OVERVIEW**

In Chapter 3, Naive Diversification you saw that you can reduce risk simply by dividing your wealth among a large number of securities. In Chapter 4, Markowitz Diversification, you saw that dividing your wealth equally is not the best you could do, and you learned how to choose portfolio weights optimally. You also saw that the "best" or undominated portfolios form the efficient frontier.

This chapter has two aims.

The first is to derive the efficient frontier under two natural extensions : without short sales and with a risk-free asset. These are both natural extensions. In practice, there are many restrictions on short sales, including rules that prohibit short selling when prices fall (i.e., on a "down tick") and impose margin requirements. CAPM Tutor allows you to see what happen to the minimum-variance frontier when short sales are prohibited. In this case the minimum-variance frontier has to be computed numerically.

A risk-free asset is an asset whose return has zero variance, such as a default-free government bond. The existence of such an asset leads to a simple characterization of the risk/return trade-off in the market, called the capital market line.

The second aim of this chapter is to derive an important relationship, which shows that the expected return on any individual security can **always** be written as a linear combination of two portfolios on the minimum-variance frontier. This is the** capital asset pricing model **which we** **interpret further when we study market equilibrium. In topic 6.8, the Security Market Line, we derive this when there is a risk-free asset. The general case is covered in topic 6.11, Zero-Beta Portfolios.

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