One of the basic ideas in finance is the trade-off between risk and return. In general, the higher the risk involved in an asset the higher the rate of return that the asset must offer to convince an investor to invest in that asset.
To find the rate of return on an asset we use the formula FV = PV x (1+r) t and solve for r. The PV is the price that we paid for that asset and the
FV is the amount that the investor has at the end of the period. That amount includes the final value of the asset plus the future value of any intermediate cash flows such as dividends or coupon payments. The dollar return is simply
$R = FV - PV.
K. D. Brewer 2008

In Table 12.1 the text tracks the performance of several classes of assets over 58 years starting in 1948 (1970 for small stocks). The table also reports the Canadian rate of inflation over that same period. The results for 1957-2005 are
(small stocks 1970-2005):
Average Geo. Avg.
Inflation 4.21%
Canadian Stocks 10.97%
T-Bills
Long Bonds
6.62%
8.88%
4.16%
9.78%
6.56%
8.42%
US Stocks ($C) 12.31% 11.00%
Small cap. 14.16% 11.52%
The average column is the arithmetic average of the returns seen over 58 years. The geometric average can be used to find the future value, as of 2005, of an investment made in 1948.
 FV = PV (1 + r) 58
K. D. Brewer 2008

Since we have the rate of inflation and the nominal returns, we can back out the effect of inflation on these investments using the Fisher
Effect. If we do we get:
Inflation
Canadian Stocks
T-Bills
Long Bonds
US Stocks ($C)
Small cap.
Real AA Real GA
0.00% 0.00%
6.76%
2.41%
4.67%
8.10%
9.95%
5.40%
2.30%
4.09%
6.57%
7.07%
The real geometric average is the future value of the investment divided by the future value of $1 due to inflation treated as a future value factor.
The real arithmetic average is simply the arithmetic average for a security minus the arithmetic average of the inflation rate.
K. D. Brewer 2008

For Canadian securities, the T-Bills had the lowest rate of return followed by the long bond index, the TSE and the small stock index.
Why do these returns follow this pattern?
From the title you can guess that the driving force for this pattern is risk. The T-Bills over that period had no negative returns, but no really high returns either. The small stock index had both the biggest gain and the biggest loss in spite of the fact that the index is tracked over the fewest years.
The less certain an investor is of receiving the average return the higher that average return must be to convince people to invest.
K. D. Brewer 2008

If we rank the various asset classes by standard deviation, we can see that, the higher the standard deviation, the higher the average return over the past 58 years. This is consistent with the theory, but can breakdown within smaller sub-samples.
Inflation
T-Bills
Long Bonds
Average
4.21%
STD Risk P
3.22% -2.41%
6.62% 3.66% 0.00%
8.88% 10.15% 2.26%
Canadian Stocks 10.97% 16.17% 4.35%
US Stocks ($C) 12.31% 16.99% 5.69%
Small cap. 14.16% 22.58% 7.54%
T-Bills are considered a risk-free investment.
Therefore, any return higher than the return on
T-Bills is considered to be a risk premium. That risk premium is the extra return offered to compensate investors for bearing the additional risk of the asset class.
K. D. Brewer 2008

The fundamental idea of market efficiency is that the market price of the asset is equal to the value of that asset. If this is the case, the NPV of purchasing any asset is zero.
Why would we expect this to hold? If the price of the asset is less than the value of the asset, many people are going to want to buy the asset and few people are going to want to sell the asset. This excess demand will drive up the price of the asset until the price equals the value.
As time goes by and new information becomes available, sales are higher than expected, a new product is launched, the company is sued for negligence, etc. then the new information changes the value of the firm and the price in the market should adjust almost immediately.
K. D. Brewer 2008

With strong form market efficiency, the price of a security is expected to equal the value of the security if all relevant information is used in this valuation, even if not publicly available. If strong form market efficiency holds, gaining insider information such as potential take-over bids will have no value to an investor because that is relevant information and will have already been incorporated into the price of the security.
In reality gaining inside information can allow an investor to make excessive gains. Trading on that inside information can also land the trader in jail. Many countries have made it a criminal offence to trade based on inside information and corporate executives have to be careful about what trades they make, and all of those trades are also reported publicly. The movie " Wall
Street " gives a dramatic representation of insider trading.
K. D. Brewer 2008

With semi-strong form efficiency the ability to make excess returns on insider information. In this form of market efficiency, any information that is available to the public will be factored into the value and therefore the price of the security.
If semi-strong form efficiency holds, the excess returns that an investor can realize should not exceed the cost of the analysis. If analysis yields higher returns, more people will engage in analysis and the competitive nature of the market will drive the excess returns to zero.
A fundamental analyst makes his living by examining the publicly available information and trying to be the first to realize what it actually means in terms of security valuation.
K. D. Brewer 2008

Considering the volume of publicly available information, weak form efficiency accepts the premise that evaluating that information can provide excess returns, but states that the current price of a security reflects its own past price.
A technical analyst tries to generate excess returns by plotting the price movements of a security and making trades on the basis of price trends. Therefore since the technical analyst believes that even the past price data of a security is not truly reflected in the current price.
A reason to doubt weak form efficiency is if you believe that the market is dominated by traders that are governed more by emotion than by any rational evaluation of value.
K. D. Brewer 2008

Several different methodologies are used to study market efficiency. Event studies look at how prices react to announcements. If the market is efficient, the price of the security should change rapidly when the announcement is made. If the prices take a long time to adjust or over adjust and then correct, the market is not efficient. Another way of studying market efficiency is to track the performance of professionally managed funds and see if they can generate a return in excess of that expected in relation to their level of risk.
The bulk of the evidence seems to be consistent with semi-strong form efficiency. There have been noticeable cases of "bubbles" or what Alan
Greenspan referred to as "irrational exuberance" where prices seemed to increase faster than value for a time before a correction.
K. D. Brewer 2008

Accounting policy choices and practices can hide the value of a company. With proper disclosure investors should be able to find the real value of the company in spite of the efforts of management. In a number of cases, although the firm has reported good results, investors have been able to see what is really happening and price the security accordingly. A couple of examples are Northland Bank in 1985 and W. T.
Grant in the early 1970s.
Accounting policy can also hide some of the firm's value due to using historical cost instead of market value. This can be a bit more difficult to find. The movies " Pretty Woman " and " Other
People's Money " give fictional accounts of what can happen then.
One company, trading at $34 per share was able to finance a $40 per share special dividend to fight off a takeover bid.
K. D. Brewer 2008

The main measure of total risk used in finance is the standard deviation of returns. This can be a forward-looking estimation of the probability distribution of returns (Ex-Ante) or it can be based on historical returns (Ex-Post). The calculation of standard deviation is slightly different in the two cases.
With historical data, the first step is to generate a list of the returns on the security over a time period. For example we have the 52-year time series in table 12.1. From there we calculate the arithmetic average return. Next we calculate the squared deviation from the mean for each of our observations. This is (the actual return minus the average return) squared. When we have this for all the years, we add them up and divide by the number of terms minus 1 in this case 51.
This is the variance. The square root of the variance is the standard deviation.
K. D. Brewer 2008

A minor note on the symbols used to denote the various statistical terms.

Variance of security a =
 a
2

Standard deviation of security a =
 a

Covariance between a and b =
 a, b

Correlation between a and b =
 a, b
K. D. Brewer 2008

If you are trying to measure the risk using ex ante estimates of the probability distribution for a security, you need to find each possible return and the probability of that return. From there you find the expected return which is simply the sum of the probability of a return times that return. The next step is to find the squared deviation for each possible return, multiply that by the probability of that return and add these numbers together. That sum is the variance of the returns. Taking the square root of the variance gives the standard distribution.
Theoretically we should only use the Ex Ante method to analyze securities, but in practice the historical data is often used. Although past performance is no guarantee of future performance, it is a good starting point.
K. D. Brewer 2008

Consider a company with possible returns of
10% - 30% with the following probabilities.
Probability
10%
20%
45%
15%
10%
Return
10%
12%
15%
20%
30%
P x r
1.00%
2.40%
6.75%
3.00%
3.00%
Sq. Dev. P x SqD
0.003782 0.0003782
0.001722 0.0003445
0.000132 0.0000595
0.001482 0.0002223
0.019182 0.0019182
16.15% Variance 0.0029228
Std Dev 5.41%
The sum of the probability times the actual return works out to be 16.15%. Using this we can find the squared deviation for each possible outcome. We then multiply these squared deviations and add them up. The sum is the variance of the security’s return. The square root of the variance is the standard deviation.
In this case we have a security with an expected return of 16.15% and a standard deviation of
5.41%.
K. D. Brewer 2008

0.25
0.2
0.15
0.1
0.05
-3.5
-2.5
0
-1.5
-0.5
0.5
1.5
Standard deviations from the mean
2.5
3.5
What does the standard deviation tell us about the risk level of the security?
The standard distribution assumes that the returns are normally distributed. If that is the case, approximately 68% of the time the returns will be within 1 standard deviation of the mean.
95% will be within 2 SD and over 99% will be within 3 SD of the mean.
K. D. Brewer 2008

T-Bills
Long Bonds
Canadian Stocks
US Stocks ($C)
Small cap.
Average STD
6.62% 3.66%
8.88% 10.15%
10.97% 16.17%
12.31% 16.99%
14.16% 22.58%
Looking at the historical record chart that we had earlier, we can see that all of the risky security classes have standard deviations that are greater than their average return. Only T-Bills have no serious chance of negative returns.
You might notice that the normal distribution would predict negative returns for T-Bills at about a 5% chance. Although the normal distribution is a reasonable assumption, it is not a perfect fit.
K. D. Brewer 2008

In many cases we will want to consider what happens when we invest in more than one security. Covariance is a measure of how two securities returns move in relation to each other.
Suppose that the returns of two securities are determined based on the overall economic conditions in the following way
GDP
+1%
+2%
+3%
Prob.
25%
40%
35%
A P x r
10% 0.025
15%
20%
0.06
0.07
15.5%
B P x r
15% 0.0375
15% 0.06
25% 0.0875
18.5%
If we want to measure how closely the returns of the two securities mimic each other, we can do something similar to what we did with variance.
Instead of calculating the squared deviation from the expected return, we find the deviation of A and B and multiply them by each other.
Multiplying by the probability and summing these numbers gives us the covariance of the two securities.
K. D. Brewer 2008

Continuing the example we can find that the covariance between the two securities is 15.8 basis points.
GDP
+1%
+2%
+3%
Dev A Dev B Dev AxB xProb
-5.500% -3.500% 0.00193 0.00048
-0.500% -3.500% 0.00017 0.00007
4.500% 6.500% 0.00293 0.00102
Covariance 0.00158
What does this mean?
The covariance between the two securities is a measure that combines how closely the two securities move together with the amount of variance involved in the returns of the two securities.
It would be nice to be able to remove the part of the covariance that reflected the variance of the two individual securities. This would leave us with a measure of how closely the security returns mirror each other.
K. D. Brewer 2008

Unlike variance, the square root of covariance is not particularly useful. However if we divide the covariance between the two securities by the product of the standard deviation of the two securities we get a statistic called correlation.
This statistic is a measure of how the securities mirror each other’s returns.

A , B



A
A

, B
B
Due to the method of calculating the correlation coefficient, it has a maximum possible value of
+1 and a minimum of –1. +1 correlation is called perfect positive correlation while –1 is called perfect negative correlation.
The securities in our example have standard deviations of 3.84% and 4.77% giving us a correlation coefficient of 0.86, which is quite high.
K. D. Brewer 2008

When an investor holds multiple assets, we refer to the bundle of assets held as a portfolio. This can include shares, bonds, real estate, etc.
Portfolio weight: The easiest way to describe the allocation of funds that the investor has in each of the assets in the portfolio is to express each of those holdings as a fraction of the total value of the portfolio. So, if you have a portfolio worth
$10,000 and you have $300 invested in Royal
Bank preferred shares, those shares have a portfolio weight ( X i
) of 0.03 or they amount to
3% of your portfolio.
Expected return: the expected return of the portfolio is simply the sum of the expected return on each asset multiplied by the portfolio weight for each of those assets.
K. D. Brewer 2008

Your portfolio consists of 500 shares of ABC Inc. that sell for $25 per share, 200 shares of DEF
Inc. worth $40 each and 100 shares of GHI Inc. that sell for $35 each, what is the weight of each of those securities in your portfolio?
Company Shares
ABC 500
DEF
GHI
Portfolio
200
100
Price Value Weight
25 12,500 52.1%
40 8,000 33.3%
35 3,500 14.6%
24,000 100.0%
If the expected return on the securities are 12% for ABC, 15% for DEF and 20% for GHI, what is the expected return on this portfolio?
Company
ABC
DEF
GHI
Portfolio
Weight Return
52.1% 12
33.3%
14.6%
100.0%
15
20
X i r
6.250%
5.000%
2.917%
14.167%
K. D. Brewer 2008

One of the major reasons to hold more than on asset in a portfolio is to protect you from some of the risk associated with an asset. The cliché of
"don't keep all of your eggs in one basket" is an example of the idea of diversification reducing risk. In finance we want to measure the amount of risk reduction that this diversification creates.
Returning to an earlier example.
GDP
+1%
+2%
+3%
 A  x r B  x r
25% 10% 0.025 15% 0.0375
40% 15% 0.06 15% 0.06
35% 20% 0.07 25% 0.0875
15.5% 18.5%
We found that

A
= 3.84%,

B
= 4.77%, and

A,B
=15.8 basis points (% 2 or 0.0001).
What is the expected return and standard deviation of a portfolio with 50% in each?
K. D. Brewer 2008

The return for each level of GDP growth, the return on the portfolio is simply the weighted average of the returns on A & B.
GDP
+1%
+2%
+3%
 P P x  Dev 2 Dev 2 x 
25% 12.5% 3.1% 0.00203 0.00051
40% 15.0% 6.0% 0.0004 0.00016
35% 22.5% 7.9% 0.00303 0.00106
E(r) 17.0% 0.00173
Standard Deviation = 4.15%
As you might expect the expected return on the portfolio is the weighted average of the expected return of the two securities.
However the weighted average of the two standard deviations is
0.5 x 3.84% + 0.5 x 4.77% = 4.31%
The portfolio standard deviation is lower. Why?
K. D. Brewer 2008

If you work through the math, you can find that the formula for the standard deviation of a portfolio of two securities is

2
P

2
P

X 2
A

2
A

X
2
A

2
A

X 2
B

2
B

X
2
B

2
B

2 X
A
X
B

A , B

2 X
A
X
B

A , B

A

B
As you might notice, the higher the correlation coefficient, the higher the variance of the portfolio.
If the correlation reaches +1 or perfect positive correlation then the right hand side is a perfect square and the standard deviation of the portfolio is simply the weighted average of the standard deviation of the two securities. Since the standard deviation has not been reduced from the average, we say that perfect positive correlation offers no benefits of diversification .
K. D. Brewer 2008

How much can risk be reduced with a portfolio of
2 securities? The amount of reduction depends on the correlation coefficient of the 2 securities.
If the correlation is perfectly positive, there is no risk reduction.
If the securities are perfectly negatively correlated, it is possible to set up a risk free portfolio.
With a correlation somewhere between those extremes, you can often set up a portfolio that has less risk than a portfolio that has a lower level of return.
K. D. Brewer 2008

Risk vs. Return
B
 = -1
-1 <  < +1
 = +1
Risk
The above graph shows the effect of the level of correlation between 2 securities on the amount by which risk can be reduced.
K. D. Brewer 2008

If we rewrite the formula for the variance of the portfolio with X
B
being replaced by 1-X
A
we can take the derivative of the expression and by setting that to zero, we can find the weight that minimizes the risk of our 2 asset portfolio.
X
*
A


2
A


2
B


2
B


A , B
2

A , B
X
B
* 
1

X
*
A
With perfect negative correlation the equation becomes
X *
A






B

2
A

2

 
B

A
A


A


B

B

2
B



B



B
2
A

2

B
A


B
K. D. Brewer 2008

What happens to the level of risk as more assets are added to the portfolio? With 3 assets we get

2
P


X 2
A

2 X
2
A

X 2
B

A
X
C

A , C
2
B

X 2
C


2 X
B
X
2
C

C

2 X
B , C
A
X
B

A , B
Each time we add a new asset we get more terms. The number of terms is equal to n 2 but the covariance terms can be halved since

A,B
is the same as

B,A
.
Using summation notation we can simplify this to

P
2  n n  i

1 j

1
X i
X j
 i , j
 
X i
2
 i
2   
 i j , j i
X i
X j
 i , j
The second part breaks the risk down into the variance of the asset and the covariance with other assets in the portfolio.
K. D. Brewer 2008

Risk
If you keep adding securities and finding the minimum level of risk for a given level of return, you will generate an efficient frontier. A portfolio can be constructed for any point under or on the curve. For any portfolio under the curve, the portfolio that lies directly above it on the curve, has a higher expected return for the same level of risk.
K. D. Brewer 2008

R f
P * the optimal portfolio
Risk
With the efficient frontier, which portfolio an individual investor would chose depends on that investor's risk preference. If we add a risk free asset to the model and allow borrowing and lending at that rate, the optimal portfolio becomes independent of the investor's risk preference. This portfolio, P * , can be combined with the risk free asset to make a portfolio that has a higher return for any given level of risk.
K. D. Brewer 2008

*
f
Any investor can create an efficient portfolio by investing in the optimal portfolio of risky assets and either borrowing or lending at the risk free rate. The risk and return trade off in this case is linear since

P
2

P
2
 p

X r
2
 r
2 
X f
2

2 f

2 X r
X f
 r , f


X
X r
2
 r
 r r
2
So the risk level of any efficient portfolio is simply the risk of the optimal risky portfolio times the weight in the risky portfolio. If the investor is risk tolerant the weight in the risk free asset can be negative, this means that the investor is borrowing in order to invest in the risky assets.
K. D. Brewer 2008

*
In a world with only 3 risky asset and a risk free asset, Henry has an efficient portfolio. He owns
$500 worth of asset A, $400 of asset B, $600 of asset C and $500 of the risk free asset. What is the composition of P * ?
P * is the optimal portfolio of risky assets so we take Henry's holdings of each of the risky assets divided by his investment in risky assets to get the appropriate weights.
X
A
= 500/1500 = 33%
X
B
= 400/1500 = 27%
X
C
= 600/1500 = 40%
If Sue has $2,000 and plans to borrow an additional $500 to invest in P * , how much of the
3 assets should she buy?
X
A
= 33% A = 2500 x .33 = $825
X
B
= 27% B = 2500 x .27 = $675
X
C
= 40% C = 2500 x .40 = $1,000
K. D. Brewer 2008

*
P * has a return of 15% and a standard deviation of 16%. The risk free rate is 6%. Find the return and risk of Henry and Sue's portfolio.
Weight in P *
Henry = 1,500/2,000 = 0.75
Sue = 2,500/2,000 = 1.25
Expected return
Henry = 0.75 x 15% + 0.25 x 6% = 12.75%
Sue = 1.25 x 15% - .25 x 6% = 17.25%
Standard Deviation
Henry = 0.75 x 16% = 12%
Sue = 1.25 x 16% = 20%
K. D. Brewer 2008

If we add an assumption that investors have homogeneous expectations to the previous model we get CAPM. This means that investors have the same beliefs about the expected return of a given asset. If this is the case, then any asset that is not included in the optimal portfolio will have no demand. Therefore the price of that asset will fall, causing the rate of return to increase until the asset is included in the optimal portfolio. In CAPM, the optimal portfolio is called the market portfolio.
The capital market line
R f
K. D. Brewer 2008
M, the market portfolio
Risk, 

The security market line
R m
M, the market portfolio
R f
Risk, 
If all assets are included in the market portfolio we can ignore the rest of the efficient frontier.
By using calculus to find the slope of the security market line. If we standardize the risk of a security by dividing by the risk of the market portfolio, we get a new measure of a security's risk called beta (

).
 i


 i , m
2 m
K. D. Brewer 2008

By examining the formula for

two special cases can be found.

For the risk free asset,

= 0

For the market portfolio,

=1
Using the SML (security market line) we can find that the return on a given asset should be equal to the rate of return on the risk free asset plus the market risk premium (R m
- R f
) times the

of the security.
E(r i
) = R f
+ (R m
- R f
)
 i
If the market portfolio has a return of 15% and the risk free rate is 6%, what is the expected return on an asset that has a beta of 1.3?
E(r) = 6 + (15 - 6) x 1.3 = 17.7%
So CAPM predicts an expected return on that asset of 17.7%
K. D. Brewer 2008

Why is the return on an asset not often equal to the expected return on that asset?
The return on a security comes in two parts, the expected return and an unexpected return.
Total Return = E(r) + Unexpected Return
When unexpected information is received during a time period, that information can have an effect on the price of an asset and therefore the return on that asset during the time period.
Are all announcements and news items things that generate unexpected returns?
No. An announcement that the earlier prediction has happened has no new information.
Announcement = Expected part + surprise
If there was no surprise there will be no impact.
K. D. Brewer 2008

Surprise announcements or news items can be relevant to either a large number of firms or a small number of firms.
Employment statistics, interest rates, or inflation numbers, etc., are relevant to the expected return on many different assets. The effect of this type of surprise is called systematic risk . This type of risk is also called market risk since it can affect every company in the market.
A lawsuit against a company, the resignation of the CEO, a successful new product launch etc., are not likely to have an impact on the general market. Since the impact of these surprises in limited in nature the are termed asset specific , unique , or unsystematic risk .
K. D. Brewer 2008

If there were no such thing as systematic risk, the ability for diversification to reduce the level of risk would be great. Consider a gamble that pays $2,000 if a coin comes up heads, $0 if it comes up tails. If that could be split into 10 bets of $200 or $0, or 100 bets of $20 or $0, or even
1,000 bets of $2 or $0, how much would this reduce the level of uncertainty?
# of flips
1
10
100
1,000 pay
2,000
200
20
2

-2

1,000 -1,000
316
100
32
368
800
937
+2

3,000
1,632
1,200
1,063
The above chart demonstrates the amount that the risk would be reduced. Increasing the number of tosses decreased the level of risk by a factor of 32. With 1,000 tosses, there is a 95% chance that you will be within $63 of the expected return.
K. D. Brewer 2008

35%
30%
25%
20%
Unsystematic risk
15%
10%
5%
0%
0
Systematic Risk
10 20
Number of Assets
30 40
In reality, the existence of systematic risk means that holding more securities can't fully protect the investor from all risk.
K. D. Brewer 2008

The basic principal of diversification is that if you split your investment amongst a number of assets you will reduce the level of risk to which you are exposed. In the previous diagram, the area between the upper curve and the lower curve is the level of risk that can be eliminated by diversification. This is the diversifiable risk , unique risk or unsystematic risk.
Since the systematic risk affects all assets in the market, that risk cannot be eliminated through diversification. Therefore systematic risk is also referred to as nondiversifiable risk .
The total risk involved in any asset is the sum of the systematic and unsystematic risk of that asset.
Total risk = systematic risk + unsystematic risk
K. D. Brewer 2008

If the unsystematic risk can be eliminated by diversification, should it increase the return on an asset?
The answer is no. If the asset was priced to reward unsystematic risk sensible investors would notice that they could get a return on risk that they could eliminate. This would be seen as money for nothing. These investors would seek to buy that asset. This increased demand would drive up the price of the asset, and reduce the return until the unsystematic risk had been priced out of the asset.
This gives us the systematic risk principle , an investor will only be rewarded for bearing the systematic risk that cannot be eliminated by diversification.
K. D. Brewer 2008

The amount of systematic risk that is present in a security is measured by beta. This is the same as the beta found under CAPM.

2
P
 
X i
2
 i
2    i j , j
 i
X i
X j
 i , j
The earlier formula for portfolio risk can be used here to show why. As the number of securities increases, the company specific portion of the portfolio risk
 2 becomes insignificant and we are left with the second term which is dependant on the covariance of the security with the other assets in the market portfolio. If we divide all of these by the variance of the market portfolio, we have a measure of the systematic risk in each asset.
 i

 i , m
 m
2
K. D. Brewer 2008

Unlike the total risk calculation, the portfolio beta can be calculated as a simple weighted average of the assets that make up the portfolio.
Joe has a portfolio that includes $500 in security
A (beta = 0.5), $1,000 in security B (beta = 1.1), and $800 in security C (beta = 0.9). What is the beta of Joe's portfolio?

P

500
2 , 300

0 .
5

1 , 000
2 , 300

1 .
1

800
2 , 300

0 .
9

0 .
9
If the risk free rate of return is 5% and the market risk premium is 9%, what is the expected return on the portfolio?
E(r
P
) = 5% + 9% x

P
= 13.1%
K. D. Brewer 2008

As with the ex-ante return and risk calculations, finding the beta of a security by estimating the probability distribution is difficult.
One method that is often used is to calculate the beta of a security from past return data. This method uses a time series of returns on a market index and the asset under consideration.
A regression analysis is then performed to find the slope of the characteristic line, which is used as an estimate of beta. The r 2 of the regression is the proportion of total risk that is market risk.
Beta can be calculated this way using standard computer packages or even some hand-held calculators. Many services also provide their estimates of various companies' beta, either for free or for a price. Some of these betas are found from historical analysis, some are estimated in other ways.
K. D. Brewer 2008

One use of

is to find the required rate of return on an asset based on the level of systematic risk involved in that asset. For a company the rate of return that investors require before they will invest in that company is the company's cost of capital. This cost of capital is the appropriate discount rate for capital budgeting for the firm.
MFM is considering a conventional investment project that has an IRR of 14%. If the risk free rate of return is 5%, the market risk premium is
7%, and MFM has a

of 1.2, should they accept this project?
E(R
MFM
) = 5 + 7 x 1.2 = 13.4%
Therefore the cost of capital is less than the IRR and MFM should accept the project.
K. D. Brewer 2008

An alternate theory to CAPM is the arbitrage pricing theory . With APT instead of comparing the asset's return to the overall return on the market, the return is measured in regards to its sensitivity to several factors such as inflation,
GDP, the price of oil, etc.
R

E
 
1
F
1
 
2
F
2
 
3
F
3
 
4
F
4
 
In this equation the various betas are the sensitivity of the asset's return to unexpected changes in the factor represented by the Fs.
The final epsilon is the unsystematic risk of the security.
In general the APT can better predict the return on an asset better than CAPM, but CAPM is more often used. One reason is that there is no real consensus on which factors to consider for use in APT. A one factor APT using the return on the market portfolio = CAPM.
K. D. Brewer 2008