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