Development of the CAPM

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Development of the CAPM

• The purpose of the CAPM is two-fold.
• First, the development of the CAPM gave us a risk
  measure, that all investors can agree on, to
  measure the riskiness of an individual asset.
• Second, the CAPM gives us a way to measure the
  required rate of return on an individual asset.

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Development of the CAPM

• Steps in the Derivation of the CAPM
• 1. Create a portfolio containing an individual
  asset I and the market portfolio (M) is created,
  where percentage invested in asset is wi and the
  percentage invested in the market portfolio is
  1 - wi.

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Development of the CAPM

• Steps in the Derivation of the CAPM
• 2. If we allow for investment in a risk-free
  asset, we see that, as we expect, we can arrive
  at the CML.
• 3. Find the slope of the CML. The twist is that
  our calculation of the slope will contain terms
  involving the individual asset I.

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Development of the CAPM

• Steps in the Derivation of the CAPM
• 4. If markets are in equilibrium, we already
  have as much of asset I in the market portfolio
  as we want. Thus, we would not want to invest
  any more in asset I. Hence, w1 0. This is the
  major insight that allowed for the development of
  the CAPM.

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Development of the CAPM

• 5. We know the slope of the CML to be .

This should be equal to the slope of the CML that
we found in step 3, that contains terms for the
individual asset.
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Development of the CAPM

• Steps in the Derivation of the CAPM
• 6. Solve this equality for the expected return
  of asset I.
• 7. This gives us the CAPM

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Development of the CAPM

• is the standardized risk measure for asset I.
  All investors agree on this risk measure.

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The CAPM and Beta
CAPM
Beta
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The CAPM and Beta

• Facts about beta
• If ? gt 1.0 , the security moves more than the
  market when the market moves
• If ? lt 1.0, the security moves less than the
  market when the market moves.
• So, if ? gt 1.0, the asset has more risk relative
  to the market portfolio and if ? lt 1.0, the
  asset is has less risk relative to the market
  portfolio.
• Since all risk is measured relative to the market
  portfolio, the beta of the market portfolio must
  be 1.0.

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The CAPM and Beta

• Estimating beta
• Estimation using formula

Gather a lengthy time-series of observations for
the market return and the individual asset
return. Use Excel to find the market variance
and the covariance of the individual asset with
the market.
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The CAPM and Beta

• Estimating beta
• 2. You can use the following regression model

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The CAPM and Beta

• Estimating beta for a portfolio of assets
• Given that beta is a linear risk measure, the
  beta of a portfolio of assets as simply the
  weighted average of all the individual betas that
  comprise the portfolio. Thus

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Beta

• Examples
• Suppose that we have three securities whose
  covariances with the market portfolio are
• ?1,M 153, ?2,M 257, ?3,M 236
• ?M 15.2
• ?1 153/(15.2)2 0.66
• ?2 257/(15.2)2 1.11
• ?3 236/(15.2)2 1.02

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Beta

• Examples
• Suppose that we have the following information
• ri,M 0.45, ?i 0.25, ?M 0.10
• What is the beta for security i?
• ?i,M 0.450.250.10 0.01125
• ?i 0.01125/(0.10)2 1.125

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Beta

• Examples
• Suppose that we estimate the following model
• If ?I 2.3 and ?I 0.05, what is the beta for
  security I?

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The CAPM and Beta

• Problems with the estimation of beta
• 1. Choice of market proxy
• 2. Time interval

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The CAPM and Beta

• Estimation of beta in practice
• -- What do major financial research companies do
  when estimating betas?
• Value line Uses 260 weekly return observations
  and uses the NYSE composite index as their market
  proxy.
• Merrill Lynch Uses 60 monthly return
  observations and uses the SP 500 index as their
  market proxy.

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The Security Market Line

• Identifying undervalued and overvalued assets
• In equilibrium, all assets and portfolios of
  assets should fall on the SML.
• Therefore, we can compare a securitys estimated
  (or expected) return with its required return
  from the SML (CAPM) to determine if the asset is
  overvalued or undervalued.

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The Security Market Line

• Identifying undervalued and overvalued assets
• If a securitys expected return is below its
  required return, based upon the SML, it is
  overvalued and if a securitys estimated return
  is above its required return, based upon the SML,
  it is undervalued.

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The Security Market Line

• Identifying undervalued and overvalued assets
• In terms of the SML, this means that securities
  that plot above the SML are undervalued and
  securities that plot below the SML are overvalued.

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The Security Market Line
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The Security Market Line

• CAPM Example

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The Security Market Line

• CAPM Example
• Rm 14, Rf 5
• Why?
• C 5 .75(14 - 5) 11.75
• D 5 2.3(14 - 5) 25.7
• E 5 1.2(14 - 5) 15.8

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The Security Market Line

• CAPM Example
• If we compare required returns to expected
  returns, investments A and E are undervalued and
  investments B, C, and D are overvalued.
• Graphically, this means investments A and E plot
  above the security market line and investments B,
  C, and D plot below the security market line.

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Example of using the SML to identify overvalued
and undervalued assets
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Testing the CAPM

• To test the CAPM we use realized (actual) returns
  and formulate the CAPM as

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Testing the CAPM

• What should be true about the CAPM?
• Beta should completely describe the risk return
  tradeoff for an individual asset. Additionally,
  the relationship between beta and returns should
  be linear.

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Testing the CAPM

• How do we test these things?
• 1. Estimate betas
• 2. Estimate the following regression model

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Testing the CAPM
What should be true about CAPM tests?
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Testing the CAPM

• What is typically found?

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Theoretical problems with the CAPM

• Applied tests of the CAPM yield mixed results.
  It is generally conceded that there must be some
  other factors than beta that explain how asset
  returns change. There is a great deal of debate,
  however, about the significance of beta itself.

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Theoretical problems with the CAPM

• Although applied tests of the CAPM still breathe
  some life into it, theoretically there are many
  problems with the CAPM.
• 1. In theory, the CAPM is untestable. If you
  have the true market portfolio, mathematically
  the CAPM must be true.

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Theoretical problems with the CAPM

• 2. The CAPM may appear to be true even if you do
  not have the true market portfolio. The CAPM
  will appear to hold for any market portfolio that
  is on the efficient frontier of assets.

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Theoretical problems with the CAPM

• 3. The only real way to test the CAPM is to find
  out if the true market portfolio is efficient.
  As we have mentioned, it is impossible to observe
  the true market portfolio.

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Theoretical problems with the CAPM

• Beta, however, is probably one of the best
  measure of risk in finance.
• Beta is intuitively pleasing.
• It encompasses the concept of diversification and
  the idea systematic risk being the important risk
  of an asset. Beta is also easy to calculate and
  it is easy to explain what it measures.

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The CAPM in practice today

• The market model

This is also known as a securitys characteristic
line. The characteristic line can also be used
to develop an expected return for an asset.
Here you need to only estimate the slope (beta)
and intercept terms.
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The CAPM in practice today

• Using the characteristic line to determine
  deviations from expectations
• While the characteristic line may not give us the
  exact returns that a security should earn, they
  at least give us an estimate.
• As long as we apply the same model for estimating
  returns for all securities, we can use the
  characteristic line to estimate expected returns
  and determine if a group of securities is
  experiencing a significant deviation from what is
  expected.

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The CAPM in practice today

• Measuring deviations from expectations
• Thus, the characteristic line gives us a way of
  gauging the markets reaction to a certain event,
  such as a merger.
• For example, suppose that we observe merger
  announcements by 100 companies at different
  points in time. We use the characteristic line
  to develop a measure of expected returns on the
  announcement day for all companies.

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The CAPM in practice today

• Measuring deviations from expectations
• By comparing the actual return to the estimated
  return from the characteristic line we can gauge
  the markets reaction to the announcement.
• This methodology is known as an "event study."
  This methodology allows us to determine the
  market reaction to a particular financial
  decision. This type of examination can tell us a
  great deal about both market perception and
  manager behavior.

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Alternatives to the CAPM

• The arbitrage pricing theory (APT)
• The APT was developed as an alternative to the
  CAPM
• The APT does not require
• Normally distributed returns
• A market portfolio that contains all risky assets

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Alternatives to the CAPM
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Alternatives to the CAPM

• Factor models
• Factor models assume that the return generating
  process on a security is sensitive to the
  movement of various factors or indices.
• A factor model attempts to capture the major
  economic forces that systematically move the
  prices of all securities.

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Alternatives to the CAPM

• Factor models
• Factor models are not equilibrium models,
  however. An equilibrium model is an exact
  pricing relationship that must be true for all
  securities, like the CAPM.
• Factor models are like applied equilibrium
  models. Factor models simply try to explain most
  of the movement in a security, however, the
  amount of movement in a particular security that
  a factor model explains will vary.

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Alternatives to the CAPM

• One factor models
• The simplest factor model contains only one
  factor. The most popular factor model used for
  asset pricing is known as the market model or the
  characteristic line for a security

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Alternatives to the CAPM

• Fama and French (1993) three factor model
• This model is roughly based on the APT. The idea
  is that there are some common factors that
  explain differences in returns. Fama and French
  just happen to pick a couple of factors that work
  pretty well.

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Alternatives to the CAPM

• Fama and French three factor model
• -- This model seems to do a fairly good job of
  describing differences in returns among
  securities and accounting for some time-series
  patterns in returns.
• -- The practical application of this model is to
  estimate the betas for the three factors and then
  use them to predict where returns should fall,
  much like the CAPM.