The CAPM, the Index Model and the APT

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The CAPM, the Index Model and the APT
B, K M Chapters 9-11 Group Project III
Chapter 9 introduces the theory of the CAPM. You
are not responsible for the appendix. Instead,
read the excellent article by Andre Perold from
the Journal of Economic Perspectives (see the
syllabus). Chapter 10 provides an in-depth
treatment of topics that will be very helpful in
completing and understanding the second part of
the group project. I will only briefly introduce
the material in Chapter 11.
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The CAPM, the Index Model, and the APT

• The Markowitz portfolio selection models derives
  the efficient frontier of risky assets and
  provides a useful framework for optimally
  combining risky funds
• It does not however provide guidance with
  respect to the risk-return relationship for
  individual assets
• The CAPM (capital asset pricing model) is a
  theoretical model of equilibrium ex ante or
  expected returns on risky assets

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The CAPM, the Index Model, and the APT

• Recall the simplifying assumptions that lead to
  the basic version of the CAPM
• Investors are price takers and act as if
  security prices are unaffected by their own
  trades
• All investors have the same identical
  single-period planning horizon
• Investments are limited to publicly traded
  financial assets and to risk-free borrowing and
  lending
• Investors pay no taxes and no transaction costs
• All investors are rational mean-variance
  optimizers
• Symmetric information and identical information

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The CAPM, the Index Model, and the APT

• Below is a summary of the equilibrium that will
  prevail in this hypothetical world of securities
  and investors
• All investors will choose to hold a portfolio of
  risky assets in proportions that duplicate
  representation of the assets in the market
  portfolio (M), which includes all traded assets
• The market portfolio will be the tangency
  portfolio to the optimal capital allocation line.
  (This CAL is referred to as the capital market
  line, or CML.) All investors therefore hold M as
  their optimal risky portfolio

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The CAPM, the Index Model, and the APT
C) The risk premium on the market portfolio will
be proportional to its risk, and the degree of
risk aversion of the representative investor
Here A is the average level of risk
aversion across investors D) The risk premium on
individual assets will be proportional to the
risk premium on the market portfolio (M) and the
? coefficient of the security where
and

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All Investors Hold the Market Portfolio

• Given the assumptions above, it is not
  surprising that all investors hold the same risky
  portfolio (all investors use Markowitz analysis
  applied to the same universe of securities, for
  the same time horizon, with the same input list)
• All assets will be included in M, because there
  is a sufficiently low price at which any asset is
  a fair investment
• The contribution of the CAPM is to derive the
  fair price (in terms of the expected return) at
  which investors are willing to hold each asset in
  the optimal risky portfolio

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Expected Returns on Individual Securities

• The CAPM is built on the insight that the
  appropriate risk premium on an individual asset
  will be determined by its contribution to the
  risk of the investors overall portfolios
• Suppose we want to gauge the portfolio risk of GM
  stock. We do so by using the covariance matrix
  we developed previously

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Expected Returns on Individual Securities
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• In Class Exercise Does GMs variance or its
  covariance (with other assets in the portfolio)
  contribute more significantly to the variance of
  the market portfolio?

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Expected Returns on Individual Securities

• Note that for a portfolio with N assets, there
  are N variance terms and N2 - N covariance terms!
• Since with many stocks in the economy the
  variance is of negligible importance compared
  with the covariance terms, it is not surprising
  that we can prove the contribution of GM to the
  market portfolios variance is given as follows
  
• - GMs contribution to variance WGMCOV(rGM,rM)
• Now that we know the contribution of GM to
  market variance, what is the appropriate risk
  premium for GM?

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Expected Returns on Individual Securities

• Recall first that the market portfolio has a
  risk premium of ErM - rf , and a risk-reward
  ratio of
• We call this ratio the market price of risk.
• The intuition of the CAPM pricing relation is as
  follows If we define the marginal price of risk
  to be the change in the market price of risk if
  we go from 100 long in the market portfolio to
  100 ? long (financing the incremental
  investment by borrowing at the risk-free rate),
  then the marginal price of risk for GM must equal
  that of the market portfolio.

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Expected Returns on Individual Securities

• If it did not, then investors would either shift
  money into or out of GM until it did, because it
  would represent an unusually good or bad
  investment opportunity. 
• For example, if the marginal price of risk for
  GM exceeded that of the market, investors would
  shift money into GM until the price is bid
  sufficiently high to remove this bargain
  opportunity! 
• Consider an investor 100 invested in the M.
  Suppose she were to increase her position in M by
  a tiny fraction, ?, financed by borrowing at the
  risk-free rate.

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Expected Returns on Individual Securities

• The portfolio rate of return will be rM ?(rM -
  rf). Comparing with the original expected
  return, the incremental expected return will
  be
• Now compute the new value of portfolio variance.
• The new portfolio has a weight of (1 ?) in the
  market and -? in the risk-free asset.

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Expected Returns on Individual Securities

• Therefore the variance of the adjusted portfolio
  is
• Note however that if ? is very small, ?2 will be
  negligible compared with 2?, so we may ignore
  this term. Therefore the variance of the
  adjusted portfolio is
• and portfolio variance has increased by

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Expected Returns on Individual Securities

• Summarizing, the marginal price of risk is given
  by the ratio 
• Now suppose instead that investors invest the
  increment ? in GM stock, also financed by
  borrowing at the risk-free rate. The increase in
  mean excess return is
• The portfolio has a weight of 1.0 in the market,
  ? in GM, and -? in the risk-free asset.

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Expected Returns on Individual Securities

• The variance is
• Again, dropping the negligible term involving,
  ?2 the marginal price of risk of GM is
• In equilibrium, the marginal price of GM stock
  must equal that of the market portfolio

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Expected Returns on Individual Securities

• If the marginal price of risk of GM is greater
  than the markets, investors can increase their
  portfolio average price of risk by increasing the
  weight of GM in their portfolio.
• Until the price of GM stock rises relative to
  the market, investors will keep buying GM stock.
  This process continues until the marginal price
  of risk of GM stock equals that of the market.
• The same process would work in reverse if GMs
  marginal price of risk is less than that of the
  market.

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Expected Returns on Individual Securities

• Equating the marginal price of risk of GM to
  that of the market
• This is the CAPM pricing relation. Rearranging
  yields the more familiar
• The expected return-beta relationship is a
  reward-risk equation.
• The beta of a security is the appropriate
  measure of risk because beta is proportional to
  the risk that the security contributes to the
  optimal risky portfolio.

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Expected Returns on Individual Securities

• The expected return-beta relationship can be
  portrayed graphically as the security market line.

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Expected Returns on Individual Securities

• Note that while the CML graphs the risk premiums
  of efficient portfolios (portfolios of M and the
  risk-free asset), the SML graphs individual asset
  risk premiums as a function of asset risk.
• The SML provides a benchmark
• 1) Many firms rate the performance of portfolio
  managers according to the reward-to-variability
  ratios they maintain, and the average rates of
  return they realize relative to the SML.
• - Note that the ? of a portfolio is simply the
  weighted average of the ?s of the components.
• - A portfolio (or asset) with a higher realized
  return relative to the market than that predicted
  by its ? has a positive alpha (denoted ?)

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Expected Returns on Individual Securities
2) Many firms use the SML to obtain a benchmark
hurdle rate for capital budgeting decisions 3)
Regulatory commissions use the expected
return-beta relationship along with forecasts of
the market index return as one factor in
determining the cost of capital for regulated
firms 4) Court rulings on torts cases sometimes
use the expected return-bets relationship to
determine discount rates to evaluate claims of
lost future income
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Implementing the CAPM

• In order to implement the CAPM we need to
  estimate the risk-free return, beta, the market
  risk premium and the market portfolio
• Risk-free rate Most studies of the CAPM use
  short-term T-Bills as a proxy for the risk-free
  rate
• Beta Recall that beta is the ratio of the
  covariance of returns between a stock and the
  market, divided by the variance of the market.
  This is the slope coefficient from a regression
  of the return of a stock on the return of the
  market.
• There are many sources for such regression
  results. One widely used source is Research
  Computer Services Department of Merrill Lynch,
  Pierce, Fenner and Smith, Inc., which publishes a
  monthly Security Risk Evaluation book, commonly
  called the beta book

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Implementing the CAPM

• You can also calculate them directly using
  online services such as Bloomberg
• The raw beta is the slope coefficient from the
  regression of the return of stock i on the market
  index return. The Adjusted Beta is the raw beta
  moved 1/3 of the distance to 1. This helps
  correct for estimation error.
• Adjusted beta .66 (raw beta) .34
• R-SQR shows the square of the correlation
  between ri and rM. The R-square statistic, which
  is sometimes called the coefficient of
  determination, gives the fraction of the variance
  of return on the stock that is explained by
  movements in the return on the SP 5000 index.

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Implementing the CAPM

• Underneath the R2 is the standard deviation of
  the nonsystematic component, ?(e). This is the
  estimate of firm-specific risk.

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Implementing the CAPM

• The standard error of ? is the standard
  deviation of the possible estimation error of the
  coefficient, which is a measure of the precision
  of the estimate.
• Recall that a rule of thumb is that if an
  estimated coefficient is less than twice its
  standard error, we cannot reject that the true
  coefficient is zero (t-statistic)

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Beta Adjustments

• Small company stocks
• Since the stocks of small companies tend to
  react to the market with a lag, only part of the
  effect of market movements is captured in their
  contemporaneous covariances with the market
• This biases downward the beta estimates
  calculated from daily returns
• Therefore use at least weekly returns, and
  include the lagged index level as a right-hand
  side variable
• The beta estimate is then the sum of the
  coefficients on the contemporaneous and lagged
  market returns

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An Implication of the CAPM

• The passive strategy is efficient
• Note that the CAPM implies M is the optimal
  risky portfolio
• We already know that indexing is a low cost way
  to invest in equities. The CAPM implies it is
  also the optimal way!
• The CAPM provided the intellectual justification
  for the rise of companies like Vanguard, and the
  proliferation of index funds

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Is the CAPM Valid?

• While the predictions of the CAPM are
  qualitatively supported, empirical tests do not
  support its quantitative predictions
• Note that the CAPM is derived using expected
  returns (which are not observed)

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Violations of the CAPM

• Tests of the CAPM find evidence that is not
  supportive. Following are noteworthy
  violations
• The relation between estimated beta and average
  historical return is much weaker than the CAPM
  suggests
• The market capitalization (size) of a firm is a
  predictor of its average historical return even
  after accounting for beta.
• Stocks with low market-to-book ratios tend to
  have higher returns than stocks with high
  market-to-book ratios, again, after controlling
  for beta
• Stocks that have performed well over the past 6
  months tend to have high expected returns over
  the following six months
• Firms with high P/E ratios have lower return
• Stocks with high dividend yield have higher
  returns

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Potential Explanations for the CAPMs Shortcomings

• 1) Various proxies for the market portfolio do
  not fully capture all of the relevant risk
  factors in the economy
• - For example, human capital is excluded from the
  various proxies (SP500) for the market portfolio.
  (Large firms may be perceived to be less
  vulnerable to the economic downturns that
  diminish the value of human capital. They
  therefore may command a lower risk-premium.)
• 2) There may be behavioral biases against classes
  of stocks that have nothing to do with the
  marginal prices of risk on stocks 
• For example, portfolio managers dont lose their
  jobs for investing in GM, but they may if they
  invest in Chrysler when it is selling for
  4/share.

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Potential Explanations for the CAPMs Shortcomings

• Despite its shortcomings, the CAPM is widely
  employed, as discussed above
• Most recent research admits multiple factors as
  determinants of risk

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Details on Implementing the Index Model

• Suppose that we observe the excess return on the
  market index and a specific asset over a number
  of holding periods
• We use as an example monthly excess returns on
  the SP 500 index and GM stock for 1985
• We can summarize the results for sample period
  in a scatter diagram

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Estimating the Index Model
Security Characteristic Line (SCL) for GM
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Estimating the Index Model

• The horizontal axis measures the excess return
  on the market index, whereas the vertical axis
  measures the excess return on the asset in
  question (GM stock in our example)
• Estimating the regression equation of the
  single-index model gives us the security
  characteristic line (SCL)

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Data for Calculation of Characteristic Line for
GM Stock
Monthly Excess Excess
GM Market T-Bill GM Market Month Return Ret
urn Rate Return Return January 6.06
7.89 0.65 5.41 7.24 February -2.86
1.51 0.58 -3.44 0.93 March -8.18
0.23 0.62 -8.79 -0.38 April -7.36 -0.29 0.7
2 -8.08 -1.01 May 7.76 5.58 0.66 7.10
4.92 June 0.52 1.73 0.55 -0.03
1.18 July -1.74 -0.21 0.62 -2.36 -0.83 Augus
t -3.00 -0.36 0.55 -3.55 -0.91 September -0
.56 -3.58 0.60 -1.16 -4.18 October -0.37
4.62 0.65 -1.02 3.97 November 6.93
6.85 0.61 6.32 6.25 December 3.08
4.55 0.65 2.43 3.90   Mean 0.02
2.38 0.62 -0.60 1.75 Standard deviation
4.97 3.33 0.05 4.97 3.32   Regression
results rGM rf ? ?(rM rf)  
? ? Estimated coefficient -2.590
1.1357 Standard error of estimate (1.547) (0.309
) Variance of residuals 12.601 Standard
deviation of residuals 3.550 R2 0.575
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The Industry Version of the Index Model

• Security Risk Evaluation uses the SP 500 index
  as the proxy for the market portfolio
• It relies on the 60 most recent monthly
  observations to calculate regression parameters.
  
• Merrill Lynch and most services, including
  Value-Line, use total returns, rather than excess
  returns (deviations from T-bill rates), in the
  regressions
• Following is the information from the beta book
  showing the estimates for GM

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Market Sensitivity Statistics
June 1994 Ticker
Closing
Adjusted Symbol Security
Name Price Beta Alpha
R2 Beta   GM
General MTRS Corp
50.250 0.80 0.14 0.11
0.87 

• R-SQR, shows the square of the correlation
  between ri and rM.
• The R-square statistic, which is sometimes
  called the coefficient of determination, gives
  the fraction of the variance of the dependent
  variable (the return on the stock) that is
  explained by movements in the independent
  variable (the return on the SP 500 index)
• Recall that the part of the total variance of
  the rate of return on an asset, ?2, that is
  explained by market returns is the systematic
  variance, ?2?M2
• Hence the R-squared is systematic variance over
  total variance, which tells us what fraction of a
  firm's volatility is attributable to market
  movements

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Market Sensitivity Statistics

• The firm-specific variance, ?2(e), is the part
  of the asset variance that is unexplained by the
  market index
• The column following R-SQR reports the standard
  deviation of the nonsystematic component, ?(e),
  calling it RESID STD DEV-N. This variable is an
  estimate of firm-specific risk.
• The standard error of an estimate is the
  standard deviation of the possible estimation
  error of the coefficient, which is a measure of
  the precision of the estimate
• A rule of thumb is that if an estimated
  coefficient is less than twice its standard
  error, we cannot reject the hypothesis that the
  true coefficient is zero. (t-statistic)

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Market Sensitivity Statistics

• The next-to-last column is called Adjusted Beta.
• Merrill Lunch adjusts beta estimates in a simple
  way. They take the sample estimate of beta and
  average it with 1, using the weights of two
  thirds and one third
• Adjusted beta 2/3 sample beta 1/3 (1)
• Finally, the last column shows the number of
  observations, which is 60 months, unless the
  stock is newly listed and fewer observations are
  available

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Market Sensitivity Statistics

• GM's beta was estimated at 0.83. Note that the
  adjusted beta for GM is 0.89, taking it a third
  of the way toward 1.
• Note that GM's RESID STD DEV-N is 5.66 percent
  per month and its R-SQR is 0.25. This tells us
  that ?GM2 (e) 5.662 32.04 and R-SQR 1 -
  ?2(e)/?2
• This is GM's monthly standard deviation for the
  sample period. Therefore, the annualized
  standard deviation for that period was 6.54 (12)½
  22.64 percent

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Arbitrage Price Theory
41
Arbitrage Price Theory

• The Arbitrage Pricing Theory (APT) is a
  relatively new theory of expected asset returns
  due to Ross (1976). The APT explicitly accounts
  for multiple factors. 
• The APT requires three assumptions
• 1) Returns can be described by a factor model
• 2) There are no arbitrage opportunities
• 3) There are large numbers of securities that
  permit the formation of portfolios that diversify
  the firm-specific risk of individual stocks

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Arbitrage Price Theory

• If there are K factors, then the return
  generating process is
• ri ai ?i1F1 ?i2F2 . ?iKFK ei
• The expected returns of each security will be a
  function of its factor ?s
• The model is derived by showing that for well
  diversified portfolios, if the portfolios
  expected return (price) is not equal to the
  expected return predicted by the portfolios ?s,
  then there will be an arbitrage opportunity
• Note that fewer assumptions are necessary to
  derive the APT (than are necessary to derive the
  CAPM)

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Arbitrage Price Theory

• However
• The APT only holds exactly for well-diversified
  portfolios.
• Individual stocks with firm specific risk can
  violate the PAT pricing relation

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Arbitrage Price Theory

• In order to implement the APT we need to know
  what the factors are! Here the theory gives no
  guidance. There is some evidence that the
  following macroeconomic variables may be risk
  factors
• Changes in monthly GDP
• Changes in the default risk premium
• The slope of the yield curve
• Unexpected changes in the price level
• Changes in expected inflation
• Note that the difficulty of measuring expected
  inflation makes the estimation of 4 5 difficult

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Arbitrage Price Theory

• Does the APT help explain the anomalies we noted
  in the discussion of the CAPM?
• In fact, firm characteristics such as size and
  market-to-book directly explain historical
  returns better than a multi factor APT
• Because of the difficulty of determining what
  the factors are, the APT is likely to remain
  controversial