Measuring the Ex Ante Beta

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Measuring the Ex Ante Beta

• 2039

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Calculating a Beta Coefficient Using Ex Ante
Returns

• 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.

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In this slide set

• The beta coefficient
• The formula approach to beta measurement using ex
  ante returns
• Ex ante returns
• Finding the expected return
• Determining variance and standard deviation
• Finding covariance
• Calculating and interpreting the beta coefficient

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The Beta Coefficient

• Under the theory of the Capital Asset Pricing
  Model total risk is partitioned into two parts
• Systematic risk
• Unsystematic risk
• Systematic risk is the only relevant risk to the
  diversified investor
• The beta coefficient measures systematic risk

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The Beta Coefficient the formula
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The Term Relevant Risk

• 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 relevanthence
  investors are only rewarded for systematic
  riskrisk that can be diversified away ISand
  prices and returns reflect ONLY systematic risk.

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The Proportion of Total Risk that is Systematic

• Each investor varies in the percentage of total
  risk that is systematic
• Some stocks have virtually no systematic risk.
• Such stocks are not influenced by the health of
  the economy in generaltheir financial results
  are predominantly influenced by company-specific
  factors
• An example is cigarette companiespeople consume
  cigarettes because they are addictedso it
  doesnt matter whether the economy is healthy or
  notthey just continue to smoke
• Some stocks have a high proportion of their total
  risk that is systematic
• 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

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The Formula Approach to Measuring the Beta

• You need to calculate the covariance of the
  returns between the stock and the marketas 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

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Ex ante return data (a sample)

• An set of estimates of possible returns and their
  respective probabilities looks as follows

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The Total of the Probabilities must equal 100

• This means that we have considered all of the
  possible outcomes in this discrete probability
  distribution

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Measuring Expected Return on the stock From Ex
Ante Return Data

• The expected return is weighted average returns
  from the given ex ante data

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Measuring Expected Return on the market From Ex
Ante Return Data

• The expected return is weighted average returns
  from the given ex ante data

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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!

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Measuring Variances, Standard Deviations from Ex
Ante Return Data

• Now do this for the possible returns on the market

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Covariance

• The formula for the covariance between the
  returns on the stock and the returns on the
  market is
• 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 returnsbut it is a relative
  measurethis is why it is on a scale from 1 to
  -1.

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Correlation Coefficient

• The formula for the correlation coefficient
  between the returns on the stock and the returns
  on the market is
• The correlation coefficient will always have a
  value in the range of 1 to -1.

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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 occurrencethen add the products
  up.

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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

A beta that is greater than 1 means that the
investment is aggressiveits 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.
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Lets Prove the Beta of the Market is 1.0

• Let us assume we are comparing the possible
  market returns against itselfwhat will the beta
  be?

Since the variance of the returns on the market
is .007425 the beta for the market is indeed
equal to 1.0 !!!
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Proving the Beta of Market 1

• If you now place the covariance of the market
  with itself value in the beta formula you get

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How Do We use Expected and Required Rates of
Return?

• 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.

Since E(r)gtR(r) the stock is underpriced.
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How Do We use Expected and Required Rates of
Return?

• 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.

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Use of the Forecast Beta

• We can use the forecast beta, together with an
  estimate of the risk-free rate and the market
  premium for risk to calculate the investors
  required return on the stock using the CAPM

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Conclusions

• 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 stocks
  beta coefficient, the expected return on the
  market portfolio and the risk-free rate.