WFU Law School
Law & Valuation
2.3.2 Covariance Defined

2.3.3 Diversification

Combining negatively correlated assets can reduce the overall variability of returns -- or risk as measured by (lower-case sigma). This is known as diversification. Consider an example. Suppose three businesses are available for you to acquire, each with different prospects depending on the economy:

Company
Economy booms
Economy busts
X sells silicon chips.
High-tech explodes, and X's returns are phenomenal.
High-tech sinks, and X's returns are dismal.
Y sells automobiles.
Many people buy cars, and Y's returns are strong.
Fewer people buy cars, and Y's returns are weak.
Z sells cigarettes.
Smokers try to kick the habit, and Z's returns are poor.
Everyone despondently lights up, and Z's returns are great.

Suppose the chances of the economy booming, staying normal or busting are the same -- a 33% probability of each happening. What is the effect of buying a portfolio of X and Y, or a portfolio of X and Z, or a portfolio of Y and Z?

Effect of diversification
Assets
Portfolios
X
Y
Z
XY (50/50)
XZ (50/50)
Bust (33% chance)
4%
8%
16%
6%
10%
Normal (33% chance)
12%
12%
12%
12%
12%
Boom (33% chance)
20%
16%
8%
18%
14%
Statistics:
Expected return
12%
12%
12%
12%
12%
Std deviation
6.53%
3.27%
3.27%
4.90%
1.63%

Notice some amazing things. The expected returns of each portfolio are the same -- 12%. But notice how combining financial assets has affected variability of returns (risk). The XY portfolio, which combines positively correlated assets, has less risk than X alone. And the XZ portfolio, which combines negatively correlated assets, has less risk than X or Z alone! In fact, the XZ portfolio has 25% less risk than investing in X alone, with the same expected return!

What if you had invested in a YZ portfolio? Combining Y and Z would have created a risk-free portfolio. A guaranteed 12% return -- no downside, but then again no upside either.

Note on Stock Indexing

Diversification is widely touted as a way for investors to reduce investment risk. In fact, some investors have sought to diversify their portfolios so as to eliminate individual-company risk by investing in the market as a whole.

They do this by investing in index funds that purport to track the market - the most widely-used index is the S&P 500. But even then, the portfolio is not completely diversified. Not only are smaller companies excluded, but the S&P 500 is constantly choosing new stocks and booting old ones. And in the past few years, the S&P 500's picking method has created a portfolio of "recently arrived" companies added at or near their tops.

In fact, funds that seek to track the S&P 500 index overpay for their portfolio since each time the S&P gurus annonce a new company has been added to the index, its price immediately rises - causing S&P 500 fund to overpay for it. And when a company gets booted from the index, its price immediately falls - again leading to losses when S&P funds then sell the stock.

See an interesting story in Slate.


2.3.2 Covariance Defined

©2003 Professor Alan R. Palmiter

This page was last updated on: March 16, 2004