Any portfolio for which each wi becomes consistently smaller as n gets large (specifically, where each w 2 ap- proaches zero as n gets large) will satisfy the condition that the portfolio nonsystematic risk will approach zero as n gets large.

In fact, this property motivates us to define a well-diversified portfolio as one that is di- versified over a large enough number of securities with proportions wi, each small enough that for practical purposes the nonsystematic variance, 2(eP), is negligible. Because the ex- pected value of eP is zero, if its variance also is zero, we can conclude that any realized value of eP will be virtually zero. Rewriting equation 11.1, we conclude that for a well- diversified portfolio, for all practical purposes

rP E(rP) PF and 2 2 2 P P F; P P F

Large (mostly institutional) investors can hold portfolios of hundreds and even thou- sands of securities; thus the concept of well-diversified portfolios clearly is operational in contemporary financial markets. Well-diversified portfolios, however, are not necessarily equally weighted.

As an illustration, consider a portfolio of 1,000 stocks. Let our position in the first stock be w%. Let the position in the second stock be 2w%, the position in the third 3w%, and so on. In this way our largest position (in the thousandth stock) is 1,000w%. Can this portfo- lio possibly be well diversified, considering the fact that the largest position is 1,000 times the smallest position? Surprisingly, the answer is yes.

III. Equilibrium In Capital Markets

11. Arbitrage Pricing Theory The McGraw−Hill Companies, 2001

326 PART III Equilibrium in Capital Markets

Figure 11.1 Returns as a function of the systematic factor. A, Well-diversified Portfolio A. B, Single stock (S).

A Return (%)

A

B Return (%)

S