Chapter 5 Lecture (Part II)
As we have seen, investments have two types of risk: systematic and unsystematic.
Unsystematic risk can be diversified away making it irrelevant for diversified investors.
You might be asking yourself the following question: “How much diversification is
enough to diversify away unsystematic risk?” To answer this question, consider the
following graph
Unsystematic Risk
Total
Risk
Systematic Risk
30
Number of
Investments
The graph shows risk on the vertical axis and the number of investments on the
horizontal axis. As the number of investments in a portfolio rises, the unsystematic (or
diversifiable) risk falls. Notice the curved (unsystematic risk) line gets closer and closer
to the dark (systematic risk) line but never actually touches it. The curved line is
therefore said to be an asymptote to the dark line, meaning we can diversify away nearly
all of the unsystematic risk, but we can never get rid of all of it. However, what remains
is insignificant. Notice that virtually all of the unsystematic risk is gone after investing in
as few as 30 different investments. This does not mean you are diversified if you invest
in 30 oil companies because the returns of all the oil companies in your portfolio would
be highly correlated with the price of oil. The idea is to invest in securities whose returns
are imperfectly correlated. If you invest in 30 diverse investments in different industries
whose returns are imperfectly correlated with one another, it is possible to eliminate
nearly all unsystematic risk from your portfolio. All that remains is the systematic risk
and this is the risk for which investors demand compensation.
Since this idea of correlation of returns is so important to reducing unsystematic risk,
let’s spend some time studying it. Investment correlation is a statistical measure of the
tendency of the returns of two investments to move together. Correlation is denoted by
the Greek letter Rho ( The value of ranges from -1, when the returns of two
investments always move in the opposite direction (not likely in the real world) to +1
when the returns of two investments always move in the same direction (also not likely in
the real world). When is -1 we say the two investments are perfectly negatively
correlated. When  is +1 we say the two investments are perfectly positively correlated.
The correlation of most investments is positive, but not perfectly so. Most pairs of
investments have a  value ranging from .3 to .7. The returns of oil company stocks
would obviously be positively correlated with one another and with the returns of oil
service stocks (Transocean and Schlumberger to name two). Can you think of any other
industries that might be positively correlated with oil company returns?1
Investment pairs with low positive or negative correlation are more interesting since
investing in these reduces risk. Suppose you have some airline stocks in your portfolio
(This would be so sad because it would mean your portfolio returns have really taken a
beating with the rising oil prices – so many of the airlines have declared bankruptcy or
are close to bankruptcy) and you would like to make investments having low correlations
with airline stocks to reduce the risk of your portfolio (a better idea would be to sell your
airline stocks: their returns won’t be increasing anytime soon, but bear with me). Which
sectors have low or perhaps even a negative correlation with airline stocks? If you
guessed oil companies (or any of the industries in footnote 1) you would be correct.
When people complain about the huge profits being earned by oil companies, I wonder
why they don’t simply invest in their stocks.
Review figures 5.3 and 5.4 on pages 213 and 214 to get a better idea of how correlation
works to reduce risk.
Since investors can eliminate unsystematic risk by investing in a diverse portfolio of
investments with imperfectly correlated returns, the only risk remaining is systematic risk
and this is the risk that investors must be compensated for bearing. The amount of the
compensation investors will require will depend on the amount of the systematic risk in
the investment. For this, we will need a measure of the systematic risk an investment
contains. This measure of systematic risk is called beta and is denoted by the Greek letter

BETA
Beta is an index of systematic risk. Beta measures the tendency of an investment’s
returns to move with the universe of possible investments which is known as the
market portfolio. Theoretically, the market portfolio contains every asset on the planet.
Since some assets’ returns are difficult to measure (e.g., the return on people or the return
on Merrill Hall), investors settle on a proxy for the market portfolio such as the Standard
1
How about coal companies, solar energy firms, natural gas companies, geothermal companies, wind
turbine companies (rather difficult to find a pure play here since conglomerates like GE and Kyocera are so
involved in this industry), and fuel cell companies.
and Poor’s 500 index (composed of 500 large U.S. firms) or the Wilshire 5000 (5000
firms).
To see how beta works, consider E-Solar (the firm makes solar panels) which has a beta
of 2 (actually its 1.9, but 2 is easier to work with). This tells us that, on average, the
return on E-Solar changes by 2% for every 1% change in the market portfolio’s return.
For example, if the return on the market portfolio were expected to change by +5%
tomorrow we might expect the return of E-Solar to change by 2 X 5% = 10%.
General Electric has a beta of .7. If we expected the return on the market portfolio to fall
by -10% tomorrow (this would be a huge decline), we might expect GE to change by
.7 X -10% = -7%.
In general:
%Ki = i X %KM
Where:
%Ki = the percentage change in investment i
i = the beta of investment i
%KM = the percentage change in the market portfolio
How do we find the beta of an investment? The answer is by using a statistical tool
called linear regression. Linear regression shows the relationship between two variables.
To see how it works, consider the following
graph:
20%
.
%Ki
.
15%
10%
. . E-Solar
.
. .
.
. .
run .
.
rise
.
-20%
-15%
-10%
-5%
-5%
.
.
.
-10%
.
.
.
.
.
.
5%
. . .
. ..
.
.
-15%
. .
.
-20%
10%
15%
20%
%KM
The graph show the change in the return in E-Solar versus the change in the return on the
market portfolio on the same day. Consider the dot in front of the arrow. On the day the
market return fell by 4%, E-Solar’s return fell by 19%. Each dot shows how E-Solar’s
return changed compared to the change in the market’s return on that same day. In
reality, several years of returns would be plotted on the graph. Now examine the straight
line running through the middle of the dots. This line is drawn such that the mean
squared distance between the line and the dots is minimized. A simpler way to say this is
that the line is drawn through the middle of the dots. What this line shows us is the
historical relationship between the change in the return on the market portfolio and the
change in the return for E-Solar. This line is called a characteristic line. The slope (rise
over the run) of the characteristic line is the beta for E-Solar. Notice the rise is about
twice the size of the run. This tells us that, on average, E-Solar’s return changed by twice
that of the market return. The steeper the characteristic line, the higher the beta.
We can compute an investment’s beta using our calculators. All we need to do is to enter
the returns of the market portfolio and the returns for some investment. Refer to problem
P5-16 on page 236 and the table at the top of page 237. You will see market returns for
the market portfolio and for assets A and B from 2000 to 2009. We are going to find the
Beta for asset A.
Step 1: Enter the returns for the Market and for Asset A into the calculator.
Hit 2nd Data (Data is found above the #7 button on the keypad). You will see X01 = 0 (if
you see X01 = something else other than zero, you need to clear out your data worksheet:
to do this hit 2nd CLR Work). You will notice on the graph of the characteristic line that
the market returns appear on the X axis. The X01 is the calculator’s way of asking you
for the first market return. The first market return (for 2000) is 6%.
Hit 6 Enter then hit the down arrow
You will see Y01 = 0. You will notice on the graph of the characteristic line that the
returns for E-Solar were on the Y-axis. The Y01 is the calculator’s way of asking you for
the first return for investment A.
Hit 11 Enter, then hit the down arrow
Hit 2 Enter, then hit the down arrow
Hit 8 Enter, then hit the down arrow
Hit 13 +/- Enter, then hit the down arrow
Hit 4 +/- Enter, then hit the down arrow.
Continue in this fashion until all the returns for the market and for investment A have
been entered.
After you have finished entering the returns,
Hit 2nd STAT (the STAT button appears above the #8 button on the keypad)
Continually hit the down arrow until you see b = .7907. This tells you the beta for
investment a is .7907. If you are not getting this answer, you probably miss-entered one
or more observations).
Now try to find the beta for Asset B. The Market returns (X’s) won’t change. All you
have to do is change the Y’s to reflect Asset B’s returns. You should find that the Beta
for Asset B is 1.3787.