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F303 Intermediate Investments

• Class 5
• Asset Pricing Models
• Capital Asset Pricing Model
• Andrey Ukhov
• Kelley School of Business
• Indiana University

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Outline of This Class

• Why we need asset pricing models
• Capital Asset Pricing Model (CAPM)
• Implications of CAPM for investors
• Empirical evidence on CAPM

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Minimum-Variance Frontier
E(R)
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Asset Pricing Models

• These are equilibrium models that describe why
  assets having different characteristics generate
  different expected returns.
• Useful to generate expected returns investors
  will want to have models that generate benchmark
  returns.
• Examples Capital Asset Pricing Model (CAPM)
  Arbitrage Pricing Theory (APT).

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Brief Overview of the CAPM

• The Capital Asset Pricing Model (CAPM) is a
  centerpiece of finance.
• This model generates an exact prediction of the
  risk-return relationship.
• Why is this important?
• CAPM serves as a benchmarkfor any asset we
  would need to have a view of the fair return
  given the assets risk.

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CAPM The Assumptions

• Remember that the CAPM is about the equilibrium
  expected returns of risky assets.
• We have a hypothetical world and ask the
  question what if?
• Assumptions
• Mean and variance are the only factors that
  matter
• All assets are divisible
• All investors plan for one holding period
• No transaction costs and no taxes
• Perfect competition
• All investors are rational mean-variance
  optimizers
• Risk-free asset available.

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Equilibrium Returns

• We observe that average returns vary across
  assets.
• Why is it that GM generates a different return
  that Lucent and Delta Airlines? Can we explain
  this?
• The CAPM argues that these variations occur
  because of the assets Betas.
• What does it mean to be in equilibrium?
• In such a state, the risk premium per unit of
  risk is the same for all assets.

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Deriving the CAPM

• If all investors use the same Markowitz analysis
  and apply it to the same universe of securities,
  using the same inputs and economic information
  for the analysis and do such a task for the same
  time period
• then they must reach the same conclusion in
  identifying the optimal risky portfolio which
  we call the Market Portfolio which is the
  tangent portfolio between the Capital Market Line
  and the Efficient Frontier.

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Mean-Standard Deviation Frontier
Capital Allocation Line (CAL)
E(R)
Market Portfolio
M
Efficient Frontier
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Security Market Line
E(R)
Security Market Line (SML)
Market Portfolio
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Beta and Expected Return
E(R)
SML
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CAPMs Major Conclusions

• The Expected Return on an asset can be expressed
  as
• giving us a linear relationship between the
  Expected Return and Beta.
• Not all risk is compensatedan assets expected
  return is ONLY related to its level of systematic
  risk, given by its Beta.

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Expected Return on StocksE(R)5 Beta(13.6 -
5)
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The CAPM Its Implications

• Investors will remove firm-specific risks by
  diversifying across different industrial sectors
• But, even the most diversified portfolio will be
  risky (Market Risk cannot be diversified away)
• Investors will be rewarded for investing in such
  a risky portfolio by earning excessive returns
  (portfolio returns less risk free rate)
• The returns from a specific investment (or asset)
  depend exclusively on the extent to which that
  investment (or asset) affects the Market Riskand
  that is captured by Beta.

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Market Portfolio

• What is the Market Portfolio?
• Summing over all the portfolios of all the
  investors will give you the aggregate risky
  portfoliowhich is equal to the entire wealth of
  the economy. This should give you the market
  portfolio referred to as M.
• What is the presence of individual stocks in M?
• The proportion of each stock in M is equivalent
  to the stocks market value divided by the entire
  market capitalization.

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Market Price of Risk

• The Market Portfolio has a risk premium
  and a variance of , giving us the
  Reward-to-Risk ratio of
• This is the market price of risk.
• It quantifies the excessive return that investors
  demand to take on the portfolio risk.
• This number will give you the risk premium that
  should be earned per unit of portfolio risk.

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Security Market Line

• The expected return-beta relationship is captured
  graphically by the Security Market Line (SML).
• Remember fairly priced assets must fall
  exactly on the SML!
• The market beta is equal to 1 hence from the SML
  (where Beta1) we can get the expected return
  from the Market Portfolio.
• The SML provides a benchmark for the evaluation
  of investment performances.

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Mispriced Securities What Should Happen?
E(R)
Security B
SML
Security A
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How Do We Get Beta?

• The main insight the correct risk premium on an
  asset is determined by its contribution to the
  risk of the Market Portfolio. This contribution
  is the assets Beta.
• Starting point Let us consider one stock, Delta.
  What is Deltas contribution to the variance of
  Market Portfolio?

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How Do We Get Beta?

• Now suppose that the investor, who is invested
  100 in the Market Portfolio, decides to increase
  his position in Delta by a very small fraction
  .
• He/She finances his/her purchase of by
  borrowing at the risk-free rate.
• What is he/she getting in terms of returns?
• There is the original position in the Market
  Portfolio, plus a negative position of size
  in the risk-free asset giving , and a long
  position of size in Delta that will return .

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How Do We Get Beta?

• What is the new portfolios excess returns?
• What is the variance of the new portfolio?
• And what is the increase in the variance?

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How Do We Get Beta?

• Hence, the marginal price of risk of Delta is
  given by
• In equilibrium, the marginal price of risk of
  Delta must be equal to that of the Market
  Portfolio. This gives us the following

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How Do We Get Beta?

• To get the fair risk premium of Delta, we have
• The ratio is the contribution of Delta
  to the variance of the Market Portfolio.
• This is Beta!
• The expected return-beta relationship of the CAPM

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Estimating Betas

• But in practice, how can we estimate betas? How
  do we use them for security analysis?
• One possible answer is a regression analysis.
• Consider a sample of returns observed for a
  period of months (or weeks, etc) for t1,2,3,T
• Let us denote the returns on
  security i, the market and the risk free asset
  respectively.

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Estimating Betas

• One standard method is to estimate Beta through
  the characteristic line as follows
• You would need to use monthly data spanning 5
  yearsgiving you a total of 60 observations.
• Then you use the excess returns of an individual
  security as the dependent variable and the excess
  return from the market as the independent
  variable as inputs in the regression model.

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Estimating Betas An Example

• Consider, as one example, the GM data in the book
  BKM (chapter 8) to estimate Beta.
• A simple regression in Excel would generate the
  following results
• Where the R-squared is 0.575.
• What does it mean to have a Beta of 1.1355? What
  is the implication of such a result to investors?

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Regression Analysis to Get Beta
Return on Individual Security
Slope Beta
February 1996
March 1996
April 1996
Return on the Market
January 1996
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Industry Asset BetasObtained from D. Mullins,
Does the CAPM Work?, Harvard Business Review,
vol. 60, pp. 105-114
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Estimating Betas Commercial Supplies

• Value Line
• employs 5 years of weekly data and
  value-weighted NYSE as the market.
• Bloomberg
• employs 5 years of monthly data and SP 500 as
  the market.
• BARRA

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Securities Alphas

• The securitys Alpha, , is the difference
  between the expected returns predicted by the
  CAPM and the actual returns. Nonzero alphas mean
  that securities do not plot on the SML.
• Example Let us say that the expected market
  return is 14, risk free rate is 6 and that a
  stock has a beta of 1.2.
• Then the SML would predict the stocks return to
  be
• 6 1.2(14 - 6) 15.6
• If during the holding period, the stock produced
  a actual return of 18, then the securitys alpha
  is 2.4

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Identifying Mispriced Assets

• One possible use of the CAPM is security
  analysis uncovering securities with nonzero
  alphas.
• If , then the assets expected
  return is too high (low) according to the CAPM
  and is under priced (overpriced).
• This is referred to as an abnormal or
  risk-adjusted return.
• The problem different than zero could be either
  produced by a mis-specified CAPM or an
  inefficient market (this is the so-called joint
  hypothesis problem)

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Example of Mispriced Assets

• Have a look at these average annualized returns
  for the last 15 years for the following three
  portfolios
• Franklin Income Fund 12.9
• Dow Jones Industrial Average 11.1
• Salomons High Grade Bond Index 9.2
• Now assume that the expected return on the Market
  Portfolio is 13 and that the risk free rate is
  7. In addition, suppose that the funds Betas
  are as follows
• Franklin Income Fund 1.0
• Dow Jones Industrial Average 0.683
• Salomons High Grade Bond Index 0.367

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Example of Mispriced Assets

• Now, let us calculate, using the information
  given before, the Expected Return using the CAPM
• Franklin Income Fund 13.071.0 x (13-7)
• Dow Jones Industrial Average 11.170.683 x
  (13-7)
• Salomons High Grade Bond Index 9.270.367 x
  (13-7)
• Interpretation For the Dow Jones and the
  Salomons Bond, the returns implies from the CAPM
  are in line with those observed in the last 15
  years. For the Franklin Fund, the market was
  expecting 13 and got less than that this means
  that there was an abnormal return. On a
  risk-adjusted basis, the fund has underperformed.

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How Good is the CAPM in Predicting Returns?

• Let us have a look at the literature on anomalies
  in the stock marketthat is patterns of returns
  that cannot be explained by the CAPM
• Then we introduce the CAPM debate, which was
  started by Roll (1977). Here, there are
  theoretical issues and empirical issues.

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Stock Anomalies Small Firm Effect

• Small market capitalization firms have produced
  higher average returns than was predicted by the
  CAPM.
• Banz (1981) and Reinganum (1981) use monthly
  data and daily data respectively and find this
  result.
• The effect is strongest for the month of
  January.
• What could explain this? Liquidity?
• Small firms are less liquid (the ability to buy
  or sell at reasonable prices and time) than large
  firms and this could be driving these higher
  returns.

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Stock Anomalies January Effect

• January has historically produced higher returns
  than other months during the year. The effect is
  particularly strong for small firms.
• Tax-loss selling could be one possible
  explanation. But the effect persists in
  international markets where capital gains tax
  does not exist.
• Window dressing by fund managers and
  institutional investors.
• There is also the Day-of-the-Week effect
  stock returns are lower over the weekend (returns
  are negative on Mondays, but they are positive
  from Wednesday to Friday).

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Average Daily Returns (1928 1982 on NYSE)
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Price/Earnings Ratio

• Evidence that securities with low Price/Earnings
  (P/E) ratio have higher average returns.
• Basu (1977) explained violations from the CAPM by
  using P/E ratiosfor a sample of NYSE securities
  there was a clear negative relationship between
  P/E ratios and the average returns in excess of
  those predicted by the CAPM.
• Following a very simple strategy of buying the
  quintile of smallest P/E securities and selling
  short the top quintile, would have produced an
  average abnormal return of 6.75 (annual, from
  1957 to 1971).

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The CAPM Debate

• Remember what we are trying to verify The CAPM
  gives us a linear relationship between an assets
  expected return and its Beta. The question Is
  this what we get in real life?
• We first review Rolls critique and then proceed
  to review empirical evidence in favor of CAPM
  (early 1970s) and against it (starting in the
  1980s and getting big in 1990s).

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Rolls Critique (1977)

• Roll states that the only acceptable test of the
  CAPM is whether the market portfolio is
  mean-variance efficient.
• But, the market portfolio is, technically
  speaking, a portfolio that includes ALL the
  assets in the economy (listed and unlisted
  stocks, listed and unlisted bonds, property,
  human capital, etc.).

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Rolls Critique (1977)

• In empirical tests, we do not use the true market
  portfolio because there is no datawe have to
  settle for a proxy, like the Dow Jones Industrial
  Index, the FTSE All Share Index, etc.
• If performance is measured relative to a proxy
  that is ex post efficient, then no security will
  produce abnormal performance
• On the other hand, if performance is measured
  relative to an ex post inefficient proxy, then
  any ranking could be possible

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Rolls Critique (1977)

• What does this mean? Is this just a quibble?
• No! Actually, it is very important
• A small change in the proxy of the market
  portfolio for example, going from SP 500 to
  the Wilshire list of 5,000 listed securities
  can alter dramatically the expected returns!
• Since no one knows the true market portfolio then
  nobody can conclude whether the CAPM holds or
  not.

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First Empirical Evidence

• Three researchers, Black, Jensen and Scholes, way
  back in 1972, wanted to test the CAPMs
  predictions.
• They divided the NYSE stocks into ten
  portfolios. The first portfolio was formed from
  securities with the lowest Betas, the second
  contained the next 10 with the next lower Betas,
  so on and so forth, up to the tenth portfolio.
• The study was carried out over a 35 year period.

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Black, Jensen and Scholes (1972)

• The evidence shows that there was an exact
  straight-line relationship between a portfolios
  Beta and the average return.
• This would be in line with the CAPMs
  predictions.
• The same was found in empirical studies by Fama
  and MacBeth (1973) and Blume and Friend (1973).

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What Happened Next?

• Toward the end of the 1970s we had some less
  favorable evidence coming out against the CAPM.
  This gave rise to the anomalies literature,
  mentioned before.
• Basu (1977) reported the Price/Earnings effect.
• Then Banz (1981) found the small size effect
  where firms with low market capitalization
  generate higher returns than predicted by the
  CAPM.

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The Death of Beta

• Fama and French (1992 and 1993) show that Beta is
  flathas no power. This has led people to declare
  that Beta is Dead!
• Fama and French show that Beta cannot explain the
  difference in returns formed on the basis of the
  Book-to-Market ratio.
• Firms with high book-to-market ratios generate
  higher returns than predicted by the CAPM.

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The Death of Beta

• In conclusion, Fama and French (1992) find that
  just two variables market equity (the firms
  size) and the ratio of the book equity (the book
  value of the equity) to market equity (equitys
  value on the market) capture much of the
  cross-section of average stock returns.
• There seems to be no role for Beta to explain
  returns!

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Something to Remember

• Let us say that you hear a fund manager saying
  "I have followed the CAPM and purchased high Beta
  securities last year but they did worse than low
  Beta securities last year! I say that, based on
  this evidence, the CAPM is dead.
• Would you agree with the fund manager?
• Is this a valid test of the CAPMs validity?

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Defending the CAPM

• Problems with data snooping and sample selection
  biases
• If Beta diedit died very recently (earlier work
  shows that the CAPM holds)
• The results from the anomalies literature could
  indicate significant deviations from the CAPM,
  but there is little theoretical motivations for
  these resultswe do not have any model for the
  behavior found in the anomalies literature.

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Key Points to Remember

• Major assumptions and conclusions from the
  CAPMremember that the model gives us a linear
  relationship between the Expected Return and
  Beta.
• Not all risk is compensatedan assets expected
  return is ONLY related to its level of systematic
  risk, given by its Beta.
• Meaning of Beta Significance of the Market
  Portfolio.

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Key Points to Remember

• The Security Market Line (SML) and how mispriced
  securities behavewhy in equilibrium assets
  should plot on the SML.
• Securitys Alpha and its meaning.
• Empirical evidence against and in favor of the
  CAPM. The main result is that some studies have
  recently found that Beta could have no
  explanatory powerbut this has been disputed by
  other studies.