Transcript Measuring the Beta

Measuring the Ex Ante Beta
2039
Calculating a Beta Coefficient Using Ex
Ante Returns
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Ex Ante means forecast…
You would use ex ante return data if historical rates
of return are somehow not indicative of the kinds of
returns the company will produce in the future.
A good example of this is Air Canada or American
Airlines, before and after September 11, 2001. After
the World Trade Centre terrorist attacks, a
fundamental shift in demand for air travel occurred.
The historical returns on airlines are not useful in
estimating future returns.
In this slide set
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The beta coefficient
The formula approach to beta measurement
using ex ante returns
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Ex ante returns
Finding the expected return
Determining variance and standard deviation
Finding covariance
Calculating and interpreting the beta coefficient
The Beta Coefficient
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Under the theory of the Capital Asset Pricing Model total risk is
partitioned into two parts:
–
–
Systematic risk
Unsystematic risk
Total Risk of the Investment
Systematic Risk
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Unsystematic Risk
Systematic risk is the only relevant risk to the diversified
investor
The beta coefficient measures systematic risk
The Beta Coefficient – the formula
Covariance of Returns between th e stock and the market
Variance of the Market Returns
Cov(R s R M )
Beta 
Var(R M )
Beta 
The Term – “Relevant Risk”
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What does the term “relevant risk” mean in the context of the CAPM?
– It is generally assumed that all investors are wealth maximizing
risk averse people
– It is also assumed that the markets where these people trade are
highly efficient
– In a highly efficient market, the prices of all the securities adjust
instantly to cause the expected return of the investment to equal
the required return
– When E(r) = R(r) then the market price of the stock equals its
inherent worth (intrinsic value)
– In this perfect world, the R(r) then will justly and appropriately
compensate the investor only for the risk that they perceive as
relevant…hence investors are only rewarded for systematic
risk…risk that can be diversified away IS…and prices and returns
reflect ONLY systematic risk.
The Proportion of Total Risk that is
Systematic
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Each investor varies in the percentage of total risk that is
systematic
Some stocks have virtually no systematic risk.
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Such stocks are not influenced by the health of the economy in
general…their financial results are predominantly influenced by
company-specific factors
An example is cigarette companies…people consume cigarettes
because they are addicted…so it doesn’t matter whether the
economy is healthy or not…they just continue to smoke
Some stocks have a high proportion of their total risk that is
systematic
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Returns on these stocks are strongly influenced by the health of
the economy
Durable goods manufacturers tend to have a high degree of
systematic risk
The Formula Approach to Measuring
the Beta
Cov(R s R M )
Beta 
Var(R M )
You need to calculate the covariance of the returns between the stock
and the market…as well as the variance of the market returns. To
do this you must follow these steps:
• Calculate the expected returns for the stock and the market
• Using the expected returns for each, measure the variance
and standard deviation of both return distributions
• Now calculate the covariance
• Use the results to calculate the beta
Ex ante return data (a sample)
An set of estimates of possible returns and their respective
probabilities looks as follows:
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.5
0.25
Possible
Possible
Returns on Returns on
the Stock the Market
0.28
0.17
-0.14
0.2
0.11
-0.04
The Total of the Probabilities must
equal 100%
This means that we have considered all of the possible outcomes in
this discrete probability distribution
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.50
0.25
1.00
Possible
Possible
Returns on Returns on
the Stock
the Market
0.28
0.17
-0.14
0.2
0.11
-0.04
Measuring Expected Return on the
stock From Ex Ante Return Data
The expected return is weighted average returns from
the given ex ante data
(1)
(2)
(3)
(4)
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.50
0.25
Possible
Returns on
the Stock (4) = (2)*(3)
0.28
0.17
-0.14
Expected return on the stock =
0.07
0.085
-0.035
0.12
Measuring Expected Return on the
market From Ex Ante Return Data
The expected return is weighted average returns from
the given ex ante data
(1)
(2)
(3)
(4)
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.50
0.25
Possible
Returns on
the Market (4) = (2)*(3)
0.2
0.11
-0.04
0.05
0.055
-0.01
Expected return on the market =
0.095
Measuring Variances, Standard
Deviations from Ex Ante Return Data
Using the expected return, calculate the deviations away from the mean, square
those deviations and then weight the squared deviations by the probability of
their occurrence. Add up the weighted and squared deviations from the mean
and you have found the variance!
(1)
(2)
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.50
0.25
(3)
(4)
Possible
Returns on
the Stock (4) = (2)*(3)
0.28
0.17
-0.14
Expected return on the stock =
(5)
Deviations
(7)
Squared
Deviations
Weighted
and
Squared
Deviations
0.0256
0.0064
0.0025 0.00125
0.0676
0.0169
0.12
Variance = 0.02455
Standard Deviation = 0.156684
0.07
0.085
-0.035
0.16
0.05
-0.26
(6)
Measuring Variances, Standard
Deviations from Ex Ante Return Data
Now do this for the possible returns on the market
(1)
(2)
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
0.25
0.50
0.25
(3)
(4)
Possible
Returns on
the Market (4) = (2)*(3)
0.2
0.11
-0.04
Expected return on the market
(5)
Deviations
(6)
(7)
Squared
Deviations
Weighted
and
Squared
Deviations
0.105 0.011025 0.002756
0.015 0.000225 0.000113
-0.135 0.018225 0.004556
0.095
Variance = 0.007425
Standard Deviation = 0.086168
0.05
0.055
-0.01
Covariance
The formula for the covariance between the returns on the stock and the
returns on the market is:
n
Cov( Rs RM )   Pt ( Rs  R s )( RM  R m )
t 1
Covariance is an absolute measure of the degree of ‘co-movement’ of
returns. The correlation coefficient is also a measure of the degree of
co-movement of returns…but it is a relative measure…this is why it is
on a scale from +1 to -1.
Correlation Coefficient
The formula for the correlation coefficient between the returns on the
stock and the returns on the market is:
Corr ( Rs RM ) 
Cov( Rs RM )
s M
The correlation coefficient will always have a value in the range of +1 to 1.
Measuring Covariances and Correlation
Coefficients from Ex Ante Return Data
Using the expected return (mean return) and given data measure the
deviations for both the market and the stock and multiply them
together with the probability of occurrence…then add the products up.
(1)
Possible
Future
State of the
Economy
Boom
Normal
Recession
(2)
(3)
Prob.
Possible
Returns
on the
Stock
0.25 28.0%
0.50 17.0%
0.25 -14.0%
Expected return on the stock =
(4)
(4) =
(2)*(3)
0.07
0.085
-0.035
12.0%
(5)
(6)
Possible
Returns
on the
Market (6)=(2)*(5)
20.0%
11.0%
-4.0%
0.05
0.055
-0.01
9.5%
(7)
(8)
"(9)
Deviations
from the
mean for
the stock
Deviations
from the
mean for
the market
(8)=(2)(6)(7
)
16.0%
10.5%
5.0%
1.5%
-26.0%
-13.5%
Covariance =
0.0042
0.000375
0.008775
0.01335
The Beta Measured Using
Ex Ante Return Data
Now you can plug in the covariance and the variance of the
returns on the market to find the beta of the stock:
Cov(R s R M ) .01335
Beta 

 1.8
Var(R M )
.007425
A beta that is greater than 1 means that the investment is
aggressive…its returns are more volatile than the market as a whole.
If the market returns were expected to go up by 10%, then the stock
returns are expected to rise by 18%. If the market returns are
expected to fall by 10%, then the stock returns are expected to fall by
18%.
Lets Prove the Beta of the Market is 1.0
Let us assume we are comparing the possible market
returns against itself…what will the beta be?
(1)
(2)
(3)
Possible
Possible
Future
Returns
State of the
on the
Economy
Prob. Market
Boom
Normal
Recession
0.25
0.50
0.25
20.0%
11.0%
-4.0%
Expected return on the market =
(4)
(4) =
(2)*(3)
0.05
0.055
-0.01
9.5%
(5)
(6)
Possible
Returns
on the
Market (6)=(2)*(5)
20.0%
11.0%
-4.0%
0.05
0.055
-0.01
9.5%
(6)
(7)
(8)
Deviations
from the
mean for
the stock
Deviations
from the
mean for
the market
(8)=(2)(6)(7
)
10.5%
10.5%
1.5%
1.5%
-13.5%
-13.5%
Covariance =
0.002756
0.000113
0.004556
0.007425
Since the variance of the returns on the market is = .007425 …the beta
for the market is indeed equal to 1.0 !!!
Proving the Beta of Market = 1
If you now place the covariance of the market with
itself value in the beta formula you get:
Cov(R M R M ) .007425
Beta 

 1 .0
Var(R M )
.007425
How Do We use Expected and
Required Rates of Return?
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Once you have estimated the expected and required rates of
return, you can plot them on a SML and see if the stock is
under or overpriced.
% Return
E(R) = 5.0%
R(RX) = 4.76%
SML
E(RM)= 4.2%
Risk-free Rate = 3%
BM= 1.0
BX = 1.464
Since E(r)>R(r) the stock is underpriced.
How Do We use Expected and
Required Rates of Return?
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The stock is fairly priced if the expected return = the required
return.
This is what we would expect to see ‘normally’ or most of the
time.
% Return
E(RX) = R(RX) 4.76%
SML
E(RM)= 4.2%
Risk-free Rate = 3%
BM= 1.0
BX = 1.464
Use of the Forecast Beta
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We can use the forecast beta, together with an estimate of the riskfree rate and the market premium for risk to calculate the investor’s
required return on the stock using the CAPM:
Required Return  R f  β j [E(rM )  R f ]
Conclusions
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Analysts can make estimates or forecasts for the
returns on stock and returns on the market portfolio.
Those forecasts can be analyzed to estimate the
beta coefficient for the stock.
The required return on a stock can be calculated
using the CAPM – but you will need the stock’s beta
coefficient, the expected return on the market
portfolio and the risk-free rate.