REVIEW OF RISK & UNCERTAINTY
When we were pricing the share of stock, we estimated that next year’s dividend
would be $1.15. In actuality, the dividend that will be paid next year is a range of
possible outcomes:
Distribution
A
Probability
Outcome
0.25 *
$1.00 =
0.50 *
$1.15 =
0.25 *
$1.30 =
1.00 E(V)= mean =
$0.25
$0.58
$0.33
$1.15
The estimated dividend of $1.15 represents a weighted average of the possible
outcomes, where each possible outcome is weighted by its probability of occurrence.
The weighted average is known as the mean of the distribution, or the expected value in
the parlance of finance. Why must the probabilities sum to 1.00?
If the possible outcomes are plotted versus their respective probabilities of
occurrence, the resulting graph appears as
DISTRIBUTION A
60%
50%
40%
30%
20%
10%
$2
.0
0
$1
.8
0
$1
.6
0
$1
.4
0
$1
.2
0
$1
.0
0
$0
.8
0
$0
.6
0
$0
.4
0
$0
.2
0
$0
.0
0
0%
This is known as a point distribution since it represents three points of possible
outcomes. In reality, there are a large number of possible outcomes, each with a
specific probability of occurrence. Rather than try to enumerate every possible outcome
and its probability of occurrence, each point in this distribution really represents a range
of possible outcomes. Thus, a more appropriate representation of the distribution might
look as follows:
POINT DISTRIBUTION A
60%
50%
40%
30%
20%
10%
$2
.0
0
$1
.8
0
$1
.6
0
$1
.4
0
$1
.2
0
$1
.0
0
$0
.8
0
$0
.6
0
$0
.4
0
$0
.2
0
$0
.0
0
0%
Even this is a simplification. The true distribution of possible outcomes is
continuous, although it would be virtually impossible to specify every possible dividend
that might be paid and the probability of each. Nonetheless, the true distribution
CONTINUOUS DISTRIBUTION A
60%
50%
40%
30%
20%
10%
$2
.0
0
$1
.8
0
$1
.6
0
$1
.4
0
$1
.2
0
$1
.0
0
$0
.8
0
$0
.6
0
$0
.4
0
$0
.2
0
$0
.0
0
0%
appears as on the previous page.
Now consider a second distribution of possible outcomes for next year’s
dividend:
Distribution
Probability
0.10 *
0.20 *
0.40 *
0.20 *
0.10 *
B
Outcome
$0.80 =
$1.00 =
$1.15 =
$1.30 =
$1.50 =
E(V)= mean =
$0.08
$0.20
$0.46
$0.26
$0.15
$1.15
The expected value of this distribution is also $1.15 for next year’s dividend. If
we plot Distribution B along with Distribution A, we obtain the following graph:
CONTINUOUS DISTRIBUTIONS A & B
60%
50%
40%
30%
20%
10%
$2
.0
0
$1
.8
0
$1
.6
0
$1
.4
0
$1
.2
0
$1
.0
0
$0
.8
0
$0
.6
0
$0
.4
0
$0
.2
0
$0
.0
0
0%
Your “best guess” as to the actual dividend that will be paid is $1.15 for both
distributions. If you could choose between the two distributions, which one would you
prefer?
Most individuals would prefer Distribution A over Distribution B since there is less
variability about the expected value with Distribution A. The variability about the
expected value is how we define “risk” in finance.
How do is the risk of a distribution of possible outcomes measure? Some would
be inclined to define risk as the range of possible outcomes, where Distribution A ranges
from $1.00 to $1.30 while Distribution B ranges from $0.80 to $1.50 in possible
outcomes. However, if we give Distribution A a one-millionth probability of the dividend
only being $0.80 and a one-in-a-million chance of the dividend being $1.50 then the
range of the two distributions is identical.
A better measure of the variability of a distribution is the standard deviation:

 ( x  x)
i
2
* pi
i
The standard deviation of Distribution A would be calculated as
 A  (1. 00  1.15) 2 *. 25  (1.15  1.15) 2 *. 5  (1. 30  1.15) 2 *. 25
 ( 0.15) 2 *. 25  ( 0) 2 *. 5  ( 0.15) 2 *. 25
 . 005625  0. 005625
 . 01125
 $0.1061
or a little over ten and one-half cents. The standard deviation not only reflects the range
of possible outcomes, but also there probabilities.
What is the interpretation of the standard deviation? Recall from statistics that,
for a normal distribution, the mean plus/minus one standard deviation represents
approximately 68% of the area of the entire distribution. In the case of our expected
dividend of $1.15 for next year, the mean ($1.15) minus one standard deviation ($0.106)
yields $1.044 while the mean plus one standard deviation yields a value of $1.256.
Thus, there is a 68% probability that the actual dividend will fall between $1.04 and
$1.26 (rounding to the nearest penny). The mean plus/minus two standard deviations
represents approximately 95% of the area under the curve of the distribution. There is a
96% probability that the actual dividend next year will be between $0.94 and $1.36 in
value. What is the probability that the dividend will be above $1.36?
68%
95%
Mean
The standard deviation of Distribution B is $0.183 (check it at home). As we could see
from both the numbers in the distribution as well as from the graphs, Distribution B has
more variability than Distribution A as confirmed by the standard deviations.
Consider a third distribution of possible outcomes, Distribution C:
Distribution
Probability
0.25 *
0.50 *
0.25 *
C
Outcome
$1.10 =
$1.25 =
$1.40 =
E(V)= mean =
$0.275
$0.625
$0.350
$1.25
Distribution C is identical in shape as Distribution A but has a higher expected value.
CONTINUOUS DISTRIBUTIONS A & C
60%
50%
40%
30%
20%
10%
$2
.0
0
$1
.8
0
$1
.6
0
$1
.4
0
$1
.2
0
$1
.0
0
$0
.8
0
$0
.6
0
$0
.4
0
$0
.2
0
$0
.0
0
0%
The standard deviation of Distribution C is $0.106 like that of Distribution A. But if you
had to choose between Distribution A and Distribution C, which would you prefer?
Distribution C has a higher expected value and is therefore preferred to Distribution A.
Does that mean that Distribution C will turn out to be better than Distribution A? Only if
you could do it over and over again, then on average Distribution C would be better than
Distribution A.
A better measure of risk is the coefficient of variation which is defined as
cv 

E (V )
The coefficient of variation measures the variability relative to the expected value and
thus expresses the standard deviation as a percentage of the expected value. For our
three distributions, the coefficients of variation are
cv A 
$0.106
 0. 0922
$1.15
cv B 
$0.183
 0.159
$1.15
cvC 
$0.106
 0. 085
$1. 25
According to the coefficients of variation, Distribution B is the most risky with a variability
of 15.9 cents per dollar of expected return. Distribution A is the second most risky with a
variability of 9.2 cents per dollar of expected return. Distribution C is the least risky with
a variability of only 8.5 cents per dollar of expected return. The coefficient of variation
has ranked the three distributions in the same manner that risk-averse individuals would.
These three distributions were designed in such a manner that it is clear which
distribution is preferred to another. In reality, it is generally the case that the distribution
with the higher expected value also has the higher variability. This is why the coefficient
of variation is a better measure than the standard deviation of the stand-alone risk of an
individual project.
DIVERSIFICATION OF RISK
Since individuals are risk-averse, they are interested in decreasing the risk to
which they are exposed. This is accomplished through diversification. While
diversification will reduce the probability of any large losses being incurred, it will also, by
definition, reduce the probability of any large gains being realized. Nonetheless, the
risk-averse individual is willing to give up the large gains in order to avoid the large
losses. This is a consequence of utility functions. Recall from microeconomics that an
individual’s utility function is assumed to exhibit the fact that more is always preferred
over less, but at a decreasing rate (the principle of declining marginal utility).
Utility
Gain
Risk-Free
Equivalent
Loss
E(V)
Wealth
Risk-aversion is characterized by the fact that, for a gain or loss of equal size with equal
probability of occurrence, the gain in utility from winning is less than the loss of utility
from losing. If the two outcomes have equal probability of occurring, then the expected
utility from taking the risk (pink line) is less than the utility of the expected value (blue
line). The individual would be equally happy to have a risk-free (or certain) equivalent
amount that is less than the expected value and avoid the uncertainty altogether. Thus,
individuals are willing to pay money in order to avoid taking risk. Do individuals actually
do this? This is what insurance is all about.
Insurance companies use the statistical law of large numbers to reduce their
risks, whereas the individual is faced with an either/or situation – the individual either
lives or dies; the individual’s house either burns or doesn’t burn down, etc.
When it comes to investments, however, the individual can use the law of large
numbers and diversify his/her portfolio in order to reduce the overall risk of the portfolio.
The total risk of an investment (variability, measured in either absolute terms with the
standard deviation, or relative terms with the coefficient of variation) can be decomposed
into two portions:
Total Risk = Systematic Risk
(Non - diversifiable
or " Market")
+ Unsystematic Risk
(Diversifiable or
" Firm - specific )
Systematic risk is the risk related to the economic system as a whole. That is, it affects
all companies in the same manner (although not to the same extent). What’s an
example of an economic event that is bad news for all companies? A rise in interest
rates is bad for two reasons: first, any company that has any debt will have to pay more
in interest expense; secondly, an increase in interest rates leads to a decrease in the
present value of the companies’ stock prices. Variability has an upside as well, and a
fall in interest rates is good for all companies. Systematic risk is not diversifiable,
however; that is, it cannot be eliminated.
Unsystematic, or diversifiable (or firm-specific), risk is the part of the total
variability that can be eliminated through diversification. What’s an example of
unsystematic risk? The risk that a tornado destroys one of your manufacturing facilities
is an example. It has nothing to do with management’s abilities, or the industry – it just
happened. What’s an example of the upside risk of unsystematic risk? (A tornado
destroys your competition’s plant!) The diversification occurs when you own stock in
both companies. The loss sustained by having a manufacturing facility destroyed in one
company is offset (in part, at least) by the gain that occurs to the other company as a
result.
Contrary to what many think, however, you do not need to be wealthy to by welldiversified. Simply choosing your investments wisely can achieve significant
diversification with a relatively few number of stocks. Random simulations of stock
portfolios have been made to see what reduction in risk occurs through diversification.
The results are in the following table:
Number of Stocks
in Portfolio
1
2
4
8
16
32
64
128
*
*
Market
Percentage of Unsystematic
Risk Remaining in Portfolio
100%
57%
32%
16%
8%
4%
2%
1%
*
*
0%
Why is only the unsystematic portion of the risk considered? Because there is
nothing that can be done about the systematic risk.
So with only one stock in our portfolio, say General Motors, all of the
unsystematic risk remains since we have not diversified at all. By adding one more
stock, on average, we can eliminated almost one-half of the risk that is diversifiable.
Obviously, this second stock is not Ford or Daimler-Chrysler. Rather, it is a stock such
as American Airlines. Four stocks eliminates almost 70% of the diversifiable risk. So
now our portfolio contains General Motors, American Airlines, and maybe IBM and Taco
Cabana. Eight stocks will eliminate 84% of the nonsystematic risk. Note that it takes
ever increasing numbers of stocks to reduce the remaining unsystematic risk. To totally
eliminate it requires owning a little bit of everything, which is “the market”.
Diversification – The Two-Asset Portfolio
How exactly does diversification eliminate the unsystematic risk? Consider two
investment alternatives:
E ( R1 )  10%
1  2%
E ( R2 )  14%
 2  4%
If we plot these two assets in risk-return space, we get the following graph:
16%
14%
Asset 2
12%
Return
10%
Asset 1
8%
6%
4%
2%
0%
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
Risk
Notice that if investors are risk-averse, these two assets are priced appropriate since
Asset 2, which is more risky, has a higher expected return to compensate for its higher
risk.
Suppose we create a portfolio with these two assets where we put 50% of our
money in Asset 1 and the other 50% in Asset 2. What do you think the expected return
on our two-asset portfolio will be?
10% * .5 = 5%
14% * .5 = 7%
E(RP) = 12%
The two-asset portfolio has an expected return of 12%, which is just a weighted average
of the returns of each of the assets in the portfolio. The standard deviation of the
portfolio is not as simple.
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
where
x is the proportion invested in Asset 1
(1-x) is the proportion invested in Asset 2
r1,2 is the correlation between Asset 1 and Asset 2
Recall from statistics what a correlation coefficient represents. The correlation
coefficient tells you how two assets are related to one another. A correlation coefficient
ranges between two extremes: +1 (perfectly positively correlated) and –1 (perfectly
negatively correlated).
Suppose we look at two assets’ returns in different periods and plot each period
as a point. In addition let’s run a linear regression line. One possible graph and best-fit
(linear regression) line might look like this:
Asset 2
PERFECT POSITIVE CORRELATION
Asset 1
Do you recall what the square of the correlation coefficient (R2, or coefficient of
determination) interpretation is? It tells you the percentage of the variation in Asset 2
that is explained by the variation in Asset 1. In this case, every point falls on the straight
regression line. Thus, the two assets are perfectly correlated. Because the slope of the
line is positive, they are perfectly positively correlated, or the correlation coefficient is +1.
The square of of +1 is 1 or 100%. Thus, if you know what change in Asset 1 occurs, you
know exactly what the change in Asset 2 will be.
Suppose the scatter plot of asset returns and the regression line look like this:
POSITIVE CORRELATION
High
Asset 2
Estimate
Estimate
Low
Asset 1
The correlation between the two assets is still positive, but not perfectly positive. Given
the change in Asset 1 we can make a prediction about what Asset 2 will do, but we won’t
be able to estimate its exact value. As may be seen on the graph, the actual outcome of
Asset 2 may be higher or lower than our estimate.
Suppose the data points and regression line have the following appearance:
Asset 2
ZERO CORRELATION
?
Asset 1
In this case, the two assets have no (zero) correlation to one another. Given the change
in Asset 1, it is not possible to even predict the direction of the change in Asset 2, let
alone the actual value. Asset 2 is as likely to move in the opposite direction of Asset 1
as it is to move in the same direction.
What kind of correlation is this?
Asset 2
NEGATIVE CORRELATION
Asset 1
Notice that it is not perfect negative correlation because not all of the observation points
fall one the regression line.
Asset 2
PERFECT NEGATIVE CORRELATION
Asset 1
As in the case of perfect positive correlation, the square (R2) of the correlation coefficient
of –1 is still 1 or 100% explanatory power.
Returning to our two-asset portfolio, let’s calculate the standard deviation of the
portfolio when one-half of our money is invested in Asset 1 the other half is invested in
Asset 2. The one variable in the equation for the standard deviation of a two-asset
portfolio that we do not already know is the correlation coefficient. Assume that the
correlation between the two assets is perfectly positive, so that the correlation coefficient
is +1. Then the standard deviation of the portfolio is
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
 (. 5) 2 ( 2%) 2  (1.5) 2 ( 4%) 2  2 (. 5)(.5)( 1)( 2%)( 4%)
 (. 25)( 4 )  (. 25)(16)  4
 1 4  4
 9
 3%
In this case, the standard deviation is just a weighted average of the standard deviations
of the assets in the portfolio, but this is only because they are perfectly positively
correlated. Our graph now looks like
PERFECT POSITIVE CORRELATION
16%
14%
Asset 2
12%
50/50
Return
10%
Asset 1
8%
6%
4%
2%
0%
1.0%
2.0%
3.0%
Risk
4.0%
Note that the risk/return plot of our 50/50 portfolio falls on a straight line between the two
assets. In fact, we can place ourselves anywhere on the line. How?
The portfolio we have created is not an example of diversification in the financial
sense because the perfect positive correlation between the two assets results in a linear
(and proportional) reduction in return for a reduction in risk. Diversification only occurs
when two assets are less than perfectly positively correlated.
Suppose that Asset 1 and Asset 2 are positively correlated, but not perfectly
positively correlated. Assume that the correlation coefficient is +0.6 so that the standard
deviation of our 50/50 portfolio is now
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
 (. 5) 2 ( 2%) 2  (1.5) 2 ( 4%) 2  2 (. 5)(. 5)( 0. 6)( 2%)( 4%)
 (. 25)( 4 )  (. 25)(16)  2. 4
 1  4  2. 4
 7. 4
 2. 72%
The expected return on our portfolio is still 12%, but now the risk is less than the
average of the two assets. If the proportions invested in each asset are varied, we will
trace our a risk-return line that looks like the following:
PARTIALLY POSITIVE CORRELATION
16%
14%
12%
Return
10%
8%
6%
4%
2%
0%
0.0%
1.0%
2.0%
Risk
3.0%
4.0%
The effect becomes more pronounced as the correlation decreases. For zero
correlation,
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
 (. 5) 2 ( 2%) 2  (1.5) 2 ( 4%) 2  2 (. 5)(.5)( 0)( 2%)( 4%)
 (. 25)( 4 )  (. 25)(16)  0
 1 4  0
 5
 2. 24%
ZERO CORRELATION
16%
14%
12%
Return
10%
8%
6%
4%
2%
0%
0.0%
1.0%
2.0%
Risk
3.0%
4.0%
A negative correlation of –0.6 has the following standard deviation and shape:
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
 (. 5) 2 ( 2%) 2  (1.5) 2 ( 4%) 2  2 (. 5)(. 5)( 0. 6)( 2%)( 4%)
 (. 25)( 4 )  (. 25)(16)  2. 4
 1  4  2. 4
 2. 6
 1. 61%
Notice that our 50/50 portfolio now has a higher return (12%) and less risk than the least
risky of the two assets, Asset 1.
NEGATIVE CORRELATION
16%
14%
12%
Return
10%
8%
6%
4%
2%
0%
0.0%
1.0%
2.0%
3.0%
4.0%
Risk
The best diversification would occur if we could combine two assets that are perfectly
negatively correlated.
 P  x 2 12  (1  x ) 2  22  2 x (1  x ) r1,2 1 2
 (. 5) 2 ( 2%) 2  (1.5) 2 ( 4%) 2  2 (. 5)(.5)( 1)( 2%)( 4%)
 (. 25)( 4 )  (. 25)(16)  4
 1 4  4
 1
 1. 00%
If the proportions invested in Asset 1 and Asset 2 were chosen properly, all of the risk of
the portfolio could be eliminated.
CORRELATIONS
16%
14%
12%
Return
10%
8%
+1.0
+0.6
0.0
-0.6
-1.0
6%
4%
2%
0%
0.0%
1.0%
2.0%
3.0%
4.0%
Risk
Diversification can be summarized as follows: As long as two assets are less
than perfectly positively correlated, there is a chance that a loss in one asset will be
offset by a gain in the other asset. As the correlation between the two assets decreases,
the probability of a loss being offset by a gain increases. What is the probability of a loss
being offset by a gain if two assets are perfectly negatively correlated?
The Capital Asset Pricing Model
If all risky assets are plotted in terms of their risk/expected return on a graph, it
would appear as
Return
RISKY ASSETS
Risk
Since these assets are not perfectly positively correlated with one another, combining
them into portfolios yields curves like we saw before. In fact, portfolios can be combined
into larger portfolios that are not perfectly positively correlated with one another in order
Return
RISKY ASSETS
Attainable Set
Risk
to achieve even greater diversification. The universe of risky assets, and portfolios of
risky assets, that an individual can invest in is called the “Attainable Set” since investors
can put themselves anywhere within that set simply by choosing the appropriate assets
and allocating their funds in proper proportions.
From the investment opportunities available within the Attainable Set, however, a
rational investor will only consider those investments that make up the northwest edge of
the attainable set. This portion of Attainable Set is referred to as the “Efficient Frontier”.
The portfolios on the Efficient frontier “dominate” all other portfolios since they provide
the same or higher expected return and/or the same or less risk than other investment
opportunities.
RISKY ASSETS
Return
Efficient
Frontier
Attainable Set
Risk
The objective of individuals is assumed to be to maximize expected utility. This does not
necessarily mean maximize wealth, since there is value placed on non-monetary things
as well, such as leisure time, time with family, etc. In the area of finance and investing,
however, the objective of maximizing wealth is taken as a given subject to individual
tradeoffs between risk and return (i.e., individual risk-aversion). Each individual
possesses a set of parallel indifference curves, each curve corresponding to a specific
level of utility. An individual would choose their investment portfolio by finding that
portfolio that is just tangent to their highest level of utility indifference curve:
PORTFOLIO CHOICE
A
Return
B
Increasing
Utility
Attainable Set
Risk
Individual A, therefore, would choose portfolio A on the Efficient Frontier since this
maximizes their expected utility, while Individual B (who is more risk averse) would
choose portfolio B to maximize their expected level of utility.
Now suppose that individuals can also invest in a risk-free asset. Are there any
investments that are risk-free? (Treasury bills.) Then, if a portfolio on the efficient
frontier that is just tangent to a straight line drawn out of the risk-free intercept can be
identified, Investor B has an opportunity to improve their investment position.
PORTFOLIO CHOICE WITH A RISK-FREE ASSET
A
Return
M
B
Attainable Set
Rf
Risk
The combination of the risk-free asset with Portfolio M is a straight line because the
standard deviation of something that is risk-free is zero. The standard deviation of the
portfolio, then is
 P  x 2  i2  (1  x ) 2  2R  2 x (1  x ) ri , R  i  R
f
f
f
 x 2  i  (1  x ) 2 ( 0) 2  2(.5)(.5)( ri , R f )(  i )( 0)
2
 x 2  i2  0  0
 x 2  i2
 x i
which is a linear function of the proportion invested in risky asset i. Individual B could
invest in a combination of the risk-free asset and Portfolio M and have a portfolio with a
higher expected return but the same risk as Portfolio B, or the same expected return but
less risk than Portfolio B. Consequently, Individual B (and others like him/her) would sell
Portfolio B and invest part of their money in the risk-free asset and the rest in Portfolio
M. As Portfolio B is sold off, its price declines which means its expected return rises
until it sits on the straight line and there is no advantage to owning Portfolio B or a
combination of Portfolio M and the risk-free asset.
Individual A has a similar opportunity, but in this case the individual would sell off
Portfolio A and invest all of the money in Portfolio M. They would then borrow more
money at the risk-free rate and invest it in Portfolio M. This way, they could slide out the
line originating at the risk-free intercept. As did Individual B, Individual A would sell
Portfolio A, thus driving down the price of Portfolio A and driving the expected return up
(think of a bond—as the price goes down, the yield to maturity goes up), and invest the
money in Portfolio M as well as borrowing money at the risk-free rate and investing it in
Portfolio M so that the new portfolio would have the same risk as Portfolio A but a higher
expected return. In equilibrium, all portfolios below the line would experience declining
prices until the expected returns on the portfolios was equal to the returns available from
investing/borrowing at the risk-free rate and investing in Portfolio M. The resulting riskreturn tradeoff line would appear as
THE CAPITAL MARKET LINE
A
M
Return
B
Rf
Risk
If all investors have invested in Portfolio M, then this must represent all possible
investments and represents the entire marketplace. The risk-return tradeoff line that we
have just constructed is referred to as the Capital Market Line and risk has been
measured by the standard deviation, which is a measure of total risk. Since the
unsystematic portion of the total risk is diversifiable, theorists argue that only the
systematic risk needs to be considered. Moreover, the major participants in the markets
are institutions which are well-diversified (mutual funds, insurance companies) so the
only risk that matters to them is the systematic risk of an investment. (What happens to
the unsystematic risk of an individual investment when you throw it into a large
portfolio?)
The implication, then, is that we should only consider systematic risk when we
are looking at investment opportunities. By definition, all of the market’s risk is
systematic risk so the risk-free asset and Portfolio M on the Capital Market Line graph
remain the same. Portfolios A and B, however, may move to other positions if we
change the measure of risk to one that looks only at the systematic portion of the
investment’s risk.
Since the unsystematic portion of the risk can be eliminated through
diversification, what we need is a way to measure the portion that cannot be diversified
away. This systematic risk is a function of the extent to which an individual investment is
moves in the same manner as the market as a whole. A portion of the last term in the
standard deviation of a portfolio describes such a relationship if we allow one of the
assets to be the market portfolio. This term is called the covariance of the asset with the
market and is defined as
covi , M  ri , M  i  M
If asset I is the market, and since the correlation of the market with itself is +1,
the covariance of the market with itself is its own variance, which is a measure of the
total risk of the market. When the risk in the risk-return tradeoff graph is defined in terms
of systematic risk, or the covariance with the market, it is called the Securities Market
Line.
THE SECURITIES MARKET LINE
RM
Return
M
Market Risk
Premium
Rf
VARM
Covariance
We now have two points that define the straight line of the risk-return tradeoff.
The first is the Risk-Free Rate with zero covariance (no risk) while the second is the
Expected Return on the Market Portfolio with a (co)variance as a measure of its total
risk. The equation for a straight line equals the intercept plus the slope of the line times
the variable on the X-axis. In this case, we get
E ( Ri )  R f 
RM  R f
 2M
Covi , M
or
E ( Ri )  R f 
The term
Covi , M
 2M
Covi , M
 2M
( RM  R f )
measures the systematic risk of Asset I relative to the market’s risk. This term is
referred to as Beta since it is equivalent to the slope of a regression line where returns
on the market are regressed against returns on Asset i.
Return on Asset i
CALCULATION OF BETA
Slope=Beta=2
Return on Market
While covariance (it is understood in finance that covariance refers to the covariance
between a single asset and the market as a whole) is an absolute measure of systematic
risk (it is in units, typically dollars or percentage returns), Beta is a relative measure of
the systematic risk of an asset. For example, a Beta of 2 means that an asset has twice
as much risk as the market as a whole (in systematic risk terms) while a beta of one-half
means that it is only half as risky as the market. As indicated by the regression concept
of Beta, it also means that if you expect the market to go up by 5% in the coming year,
you would expect an asset with a Beta of 2 to go up 10%, or twice as much. Of course,
this also means that if the market falls 5%, the asset can be expected to decline 10%.
The risk-return tradeoff line of the Securities Market Line can now be re-written
as
E ( Ri )  R f  i ( R M  R f )
This equation is known as the Capital Asset Pricing Model and is one specific form of the
more general concept of a Risk-Adjusted Discount Rate. The concept is straight-forward
enough:
You should always get at least the risk-free rate of return plus you should get a risk
premium proportional to the amount of risk that you are taking. Thus, if you are taking
twice as much risk as the market as a whole, you should get twice the risk premium.
Similarly, if you are only taking half as much risk as the market, you should require only
half the risk premium of the market.
Why is it called the Capital Asset Pricing Model when it is determining the
required rate of return? Recall that the value of anything is the present value of the
future cash flows and the model is determining the discount rate that should be used in
calculating the present value (“price”).
In effect, Risk-Adjusted Discount Rates penalize cash flows for being more risky.
To see this, consider an expected cash flow of $100 one year from now. If your required
rate of return is 10% then the present value is $90.91
0
1
$100
.9091
$90.91
Now consider another expected cash flow of $100 one year from now, but one that is
considerably more risky where you require a 20% rate of return.
0
1
$100
.8333
$83.33
Because of the higher required rate of return, the present value is lower than for the less
risky cash flow. The risk is being adjusted for in the discounting process so that the
present values, or prices, reflect our preferences and make adjustments for varying
degrees of risk. Once we have adjusted for the risk in calculating the present values of
future cash flows, we can make direct comparisons of different alternatives with different
cash flows as well as different risks.
Numerical Calculation of Covariance and Beta
While future expectations are the relevant figures to utilize in making financial
projections, practical considerations generally rely upon past experience (objective
rather than subjective) as a basis upon which to form expectations about future
outcomes, including distributions of possible outcomes and relationships between
various economic variables. Estimation of the beta of a company, and thus its
covariance, is traditionally accomplished by utilizing historic returns of a stock versus the
historic returns of one of several market proxies. Numerous estimates of betas are
available from different sources (such as Value Line). The estimates are generally very
similar to one another, but can differ based upon which market proxy is used and
whether the returns employed are daily, weekly, monthly, etc., as well as the period of
time over which observations are collected. In general, the more observations that are
available improves the statistical validity of the results. In addition, longer periods of
time are generally preferred in order to more closely approximate the long-term trends
and avoid error that is induced by the selection of any particular beginning and ending
points.
Longer time frames have there own disadvantages. In particular, the farther one
goes back in time to begin the data analysis, the more likely that the data reflects
company-specific and economic circumstances that are no longer relevant. This is the
basis for the use of time-weighted averages rather than simple averages. A simple
average is calculated as
N
Average 
Xt
t 1
N
A time-weighted average gives proportionally more weight to more recent observations:




t 

Xt * N 
t 1 
 t 

t 1 

Average 
N
N
The actual calculation of covariance using historical data is very similar to the
calculation of an ex post variance. Letting Xt represent the return on an individual stock
during period t and Yt be the return on the market, the covariance is calculated as
N
Covariance i,M 
 (X t  X) * (Yt  Y)
t 1
N -1
and N-1 is used in the denominator rather than N in order to adjust for the degrees of
freedom. In the calculation of beta, the use of N or N-1 doesn't really matter since the
same adjustment is made in the calculation of the variance of the market:
Beta 
Covi, M
 M2
Risk Measures Summary
Standard Deviation is an absolute measure of total risk
Coefficient of Variation is a relative measure of total risk
Covariance is an absolute measure of systematic risk
Beta is a relative measure of systematic risk