INDEX MODELS

1
INDEX MODELS

• CHAPTER 8

2
Outline of the Chapter

• Single factor model
• Common factor for the macroeconomic changes
• Single-index model
• SP 500 index is employed as the common factor
• Advantages and disadvantages of the single-index
  model
• Estimate a singel-index model
• Constructing portfolios with single-index models
• Practical Aspects
• Index vs Markowitz
• Tracking Portfolios

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A Single-Factor Security Market

• The method of finding the optimal risky portfolio
  in the last chapter (Markowitz Model) depends on
  the quality of estimates of expected security
  returns, and the covariance matrix.
• The Markowitz Model has two main difficulties
• The number of estimates for the expected returns
  and covariances matrix is very high even for the
  portfolios with small number of securities
• The errors in the estimation of correlation
  coefficients
• In order to solve the problem of estimating
  covariance and correlation uncertainty is
  decomposed into system-wide versus firm-specific
  sources.

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A Single-Factor Security Market (Continued)

• The rate of return on any security, i, can be
  divided into two as the sum of its expected
  return and the unanticipated components
  (unexpected return)
• riE(ri)ei
• E(ei)0
• s(ei) measures the uncertainty about the security
  return
• We assume that the security returns are affected
  by one or more common variables
• We assume in this chapter that security returns
  are affected from a single factor
• Single-factor security market

5
A Single-Factor Security Market (Continued)

• Assume that there is a common factor, m
• m a macroeconomic variable that affects all
  firms
• riE(ri)m ei
• Where m is the uncertainty about the economy as a
  whole and ei is the uncertainty about the
  specific firm
• m measures unanticipated macro surprises
• E(m)0 (over time surprises will average out to
  0)
• Sandard deviation sm
• has no subscript, common factor for all
  securities
• ei measures the firm-specific surprises

6
A Single-Factor Security Market (Continued)

• m and ei are uncorrelated
• since ei is firm specific, independent of shocks
  to the common factor that affect the entire
  economy
• s2i s2m s2(ei)
• The variance of ri is the total of two
  uncorrelated sources, systematic and firm
  specific
• Cov(ri ,rj)Cov(mei,mej) s2m
• m generates correlation across securities
• all securities respond to the same macroeconomic
  news
• Firm-specific surprises (ei) are uncorrelated
  across firms
• m is uncorrelated with any of the firm-specific
  surprises

7
A Single-Factor Security Market (Continued)

• The sensitivity of the securities to the
  macroeconomic shocks differ
• riE(ri)ßim ei
• Each firm assigned a sensitivity coefficient to
  macro conditions (ßi)
• We obtain single-factor model
• Systematic risk of the security i is determined
  by its beta coefficient

8
A Single-Factor Security Market (Continued)

• Total risk of the security i
• s2i ß2i s2m s2(ei)
• The systematic risk of the security i ß2i s2m
• The unsystematic risk of the security i s2(ei)
• Cov(ri ,rj)Cov(ßimei, ßjmej) ßi ßjs2m
• The covariance between any pair of securities
  also determined by their betas
• Equivalent beta securities gives equivalent
  market positions
• Show the same sensitivity to the changes in the
  macroeconomy

9
A Single-Factor Security Market (Continued)

• There is a linear relationship between security
  returns and the comon factor
• However these models do not identify the common
  factor
• We are looking for a variable that we can employ
  as a common factor in the specified models
• This variable should be observable so that the
  volatility and the sensitivity of the securities
  returns to variation of this common factor can be
  estimated

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The Single-Index Model

• The valid proxy for the common factor model is
  the rate of return on a broad index of securities
  such as the SP 500
• Single-Index Model
• Similar to single-factor model
• Uses market index as a proxy for the common
  factor
• The regression equation
• M market index (SP 500)
• RM excess return
• RMrM-rf
• sM standard deviation

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The Single-Index Model (Continued)

• Ri excess return of a security i
• Riri-rf
• We can estimate the sensitivity (beta)
  coefficient of a security on the index using a
  single-variable linear regression

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The Single-Index Model (Continued)

• The regression equation
• Ri(t)aißiRM(t)ei(t)
• ai intercept
• The security is expected excess return when the
  market excess return is 0.
• ßi the slope
• The security is sensitivity to the index
• The amount by which the security is return tends
  to increase or decrease for every 1 increase or
  decrease in the return on the index
• ei residual
• Firm-specific surprise in the security return

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The Single-Index Model (Continued)

• The Expected Return-Beta Relationship
• E(Ri)aißiE(RM)
• E(ei)0
• ßiE(RM)
• Part of the security is risk premium is due to
  the risk premium of the index (market)
• The market risk premium is multiplied by the
  relative sensitivity of the individual security
• Systematic risk premium
• ai
• Nonmarket premium
• For example a maybe larger if you think a
  security is underpriced and so offers an
  attractive expected return

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The Single-Index Model (Continued)

• Risk and Covariance in the Single-Index Model
• Ri(t)aißiRM(t)ei(t)
• E(Ri)aißiE(RM)
• Total risksystematic riskfirm-specific risk
• s2i ß2i s2M s2(ei)
• Covarianceproduct of betasmarket index risk
• Cov(ri ,rj)ßi ßjs2M
• Correlationproduct of correlations with the
  market index
• Corr(ri ,rj) ßi ßjs2M/sisj
• ßi s2Mßjs2M/sisMsjsM Corr(ri ,rM)Corr(rj
  ,rM)

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The Single-Index Model (Continued)

• Different than Markowitz model we only need to
  estimate a, ß, and s(e) for the individual
  securities and risk premium (RM) and variance of
  the market index (s2M)
• The Set of Estimates Needed for the Single-Index
  Model
• ai
• The stocks expected return if the market is
  neutral, if the markets excess return, rM-rf, is
  zero
• ßi (rM-rf)
• The component of return due to movements in the
  overall market
• ßi is the securitys responsiveness to market
  movements

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The Single-Index Model (Continued)

• ei
• The unexpected component of return due to
  unexpected events that are relevant only to this
  security (firm-specific)
• ß2i s2M
• The variance attributable to the uncertainty of
  the common macroeconomic factor
• s2(ei)
• The variance attributable to firm-specific
  uncertainty
• The advantages of the single-index model
• Decrease the number of inputs to be estimated for
  the model
• Suggests a simple way to compute covariances

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The Single-Index Model (Continued)

• Disadvantage of the single-index model
• The model divides the uncertainty into two as
  macro and micro
• Ignores the other sources of dependence in stock
  returns
• Events that affect the firms in the same industry
  without affecting the broad macroeconomy
• Ignores the correlation between residuals (ei) of
  the firms in the same industry
• Multi-index models models which includes
  additional factors to capture those extra sources
  of cross-security correlation

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The Single-Index Model (Continued)

• The Index Model and Diversification
• Sharpe
• First suggested the index model
• Offers insight into portfolio diversification

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The Single-Index Model (Continued)

• As the number of stocks included in this
  portfolio increases, the part of the portfolio
  risk attributable to nonmarket factors become
  ever smaller
• Risk attributable to nonmarket factors is
  diversified away
• Risk attributable to market factors remains
• The portfolios variance
• s2P ß2P s2M s2(eP)

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The Single-Index Model (Continued)

• ß2P s2M systematic risk component
• component that depends on marketwide movements
• depends on the sensitivity coefficients of
  individual securities
• this part of risk depends on portfolio beta
  and variance of market return and persist
  regardless of diversification
• s2(eP) nonsystematic risk component
• attributable to firm-specific component
  ei
• as number of stocks increase the
  firm-specific risk component cancells out, so
  there is smaller nonmarket risk
• diversifiable risk

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The Single-Index Model (Continued)

• As diversification increase the total variance
  of a portfolio approaches the systematic variance
• The variance decrease because of the decrease in
  firm-specific risk

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Estimating the Single-Index Model

• Example
• Estimate the equation
• Ri(t)aißiRM(t)ei(t)
• Data
• Six large US corporations and their stocks
• Hewlett-Packard, Dell (IT), Target, Wal-Mart
  (retail), British Petroleum, and Royal Dutch
  (energy)
• SP 500 portfolio
• T-bills rate
• April 2001-March 2006 (monthly data-60
  observations)

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Estimating the Single-Index Model (Continued)

• Lets look at the index model regression for
  Hewlett-Packard (HP)
• RHP(t)aHPßHPRSP500(t)eHP(t)
• Equation describes the linear dependence of HPs
  excess return on changes in the state of the
  economy (excess return of the SP500 index
  portfolio)
• The regression estimates a straight line with the
  intercept (aHP) and slope (ßHP)
• It is called the security characteristic line
  (SCL) for HP

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Estimating the Single-Index Model (Continued)

• The HP returns generally follow those of the
  index but with much larger swings
• The swings in HPs excess return suggests a
  greater-than-average sensitivity to the index
• Beta greater than 1

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Estimating the Single-Index Model (Continued)
26
Estimating the Single-Index Model (Continued)

• Regression Statistics
• Multiple R the correlation of HP with the SP
  500
• R-square the variation in the SP 500 excess
  returns explains what portion of the variation in
  the HP series
• Adjusted R-square corrects for an upward bias in
  R-square
• Standard error the standard deviation of the
  residual (s(eHP))

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Estimating the Single-Index Model (Continued)

• Analysis of Variance (ANOVA)
• Sum of squares (SS) of the regression the
  portion of the variance of the HPs return that
  is explained by the SP 500 return
• MS for the residual the variance of the
  unexplained portion of HPs return, portion of
  return that is independent of the market index
• s2(eHP)
• SS(residual)/58
• Square root is equal to standard error of the
  regression

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Estimating the Single-Index Model (Continued)

• Total SS of the regression/(n-1) the estimate of
  the variance of the dependent variable
• s2HP
• Note annualized standard deviationmonthly
  standard deviation(12)1/2
• R-square explained to total variance
• the explained (regression) SS/total SS
• Estimate of a
• The average value of HPs return net of the
  impact of market movements
• The nonmarket component of HPs return is the
  actual return minus the return attributable to
  market movements

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Estimating the Single-Index Model (Continued)

• Rfirm-specificRfsRHP-ßHPRSP500
• The intercept term (a) is insignificant
• The value of the intercept is not statistically
  different than 0
• Estimate of ß
• The beta coefficient for the HP indicates high
  market sensitivity (greater than market beta,
  which is equal to 1)
• The slope term (ß) is significant
• The slope is statistically different than 0
• The slope is statistically different than 1

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Estimating the Single-Index Model (Continued)

• Firm-specific risk
• The annual standard deviation of HPs residual is
  26.6 and the standard deviation of systematic
  risk is ßs(SP 500)27.57
• ß2HPs2SP500SS Regression/n-1
• HPs firm specific risk is as large as its
  systematic risk, a comon result for individual
  stocks

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Portfolio Construction and the Single-Index Model

• How to construct portfolios by employing
  single-index models
• The biggest advantage of the single-index model
  is the simple framework it provides for
  macroeconomic and security analysis when
  preparing the input list (needed estimates) to
  have efficient optimal portfolio
• The Markowitz model requires estimates of risk
  premiums for each security
• The estimate of expected return depends on both
  macroeconomic and individual-firm forecasts

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Portfolio Construction and the Single-Index Model
(Continued)

• Different analysts may have different forecasts
  related to macroeconomy and have different
  expected return estimates for the same securities
• The underlying assumption for market-index risk
  and return often are not explicit in the analysis
  of individual securities
• The single-index model helps to seperate sources
  of return variation (macroeconomic and
  firm-specific sources of variation) and makes it
  easier to ensure consistency across analysts

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Portfolio Construction and the Single-Index Model
(Continued)

• Steps in the single-index model
• Macroeconomic analysis is used to estimate the
  risk premium and risk of the market index
• Statistical analysis is used to estimate the beta
  coefficients of all securities and their residual
  variances
• The portfolio manager uses the estimates for the
  market-index risk premium and the beta
  coefficient of a security to establish the
  expected return of that security absent any
  contribution from security analysis
• Security-specific expected return forecasts
  (security alphas) are derived from securtiy
  valuation models

34
Portfolio Construction and the Single-Index Model
(Continued)

• Thus, the risk premium on a security (ßiE(RM)),
  risk premium derived from the securitys tendency
  to follow the market index, is not subject to
  security analysis. However any expected return
  beyond this benchmark risk premium (the security
  alpha) would be due to some nonmarket factor and
  is subject to security analysis
• Alpha is very important for the portfolio manager
• Tells whether a security is a good or bad buy
• There may be many other securities with the same
  systematic component (beta coefficient)
• The positive alpha security provides premium and
  should have a higher weight in the portfolio. On
  the other hand the negative alpha security is
  overpriced and should have less weight in the
  portfolio

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Portfolio Construction and the Single-Index Model
(Continued)

• Assume a portfolio manager plans to compile a
  portfolio from a list of n actively researched
  firms and a passive market index portfolio.
• The input list for n actively researched firms
  and a passive market index portfolio
• Risk premium on the SP 500 portfolio
• Estimate of the standard deviation of the SP 500
  portfolio
• n sets of estimates for beta coefficients, stock
  residual variances, and alpha values

36
Portfolio Construction and the Single-Index Model
(Continued)

• Ri(t)aißiRM(t)ei(t)
• E(Ri)aißiE(RM)
• Cov(ri ,rj)ßi ßjs2M
• If the SP 500 index is treated as the market
  index,
• it will have a beta of 1 (its sensitivity to
  itself)
• no firm-specific risk
• an alpha of 0 (there is no nonmarket component in
  the expected return of the market)
• covariance of any security i with the index is
  ßis2M (since ßM1)

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Portfolio Construction and the Single-Index Model
(Continued)

• Optimal risky portfolio with single-index model
• Generate (n1) expected returns by employing
  estimates of beta and alpha coefficients and risk
  premium of the index portfolio
• Construct the covariance matrix by employing the
  estimates of the beta coefficients, residual
  variances and variance of the index portfolio
• Then maximize the Sharpe Ratio for the
  constructed portfolio (find the steepest CAL) to
  get the optimum risky portfolio

38
Portfolio Construction and the Single-Index Model
(Continued)

• The optimal risky portfolio is composed of an
  active portfolio (A) and a passive portfolio (M)
• A composed of n analysed securities (formed as a
  result of active security analysis)
• M the (n1)th asset used for diversification

39
Portfolio Construction and the Single-Index Model
(Continued)
40
Portfolio Construction and the Single-Index Model
(Continued)

• The Sharpe ratio of an optimally constructed
  risky portfolio will exceed that of the index
  portfolio (the passive strategy)
• S2P S2MaA/s(eA)2
• The contribution of active portfolio (when held
  in its optimal weight, wA) to the Sharpe ratio
  of the overall risky portfolio is determined by
  the ratio of its alpha to its residual standard
  deviation
• aA/s(eA) is called information ratio
• Measures the extra return we can obtain from
  securtiy analysis (aA) compared to the
  firm-specific risk

41
Portfolio Construction and the Single-Index Model
(Continued)

• The information ratio reveals that the positive
  contribution of the security to its portfolio is
  made by its addition to the nonmarket risk
  premium (its alpha) and its negative contribution
  is made by increasing the portfolio variance by
  its firm-specific risk (residual variance)

42
Practical Aspects of Portfolio Management with
the Index Model

• Index vs Markowitz model
• The Markowitz model allows flexibility in our
  modelling of asset covariance structure but the
  problem is the errors in the covariance estimates
• The single-index model is more practical and
  helps us to seperate macro and security analysis

43
Practical Aspects of Portfolio Management with
the Index Model (Continued)

• The single-index model provides a benchmark for
  security analysis
• A portfolio manager who has no info about the
  security will assume alpha value as 0 and
  forecast a risk premium for the security (ßiRM)
• If we restate the model in terms of total returns
• E(rHP)rfßHPE(rM)-rf
• A portfolio manager who has a forecast for the
  market index, E(rM), and observes the risk-free
  T-bill rate, rf,can use the model to determine
  the benchmark expected return for any stock

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Practical Aspects of Portfolio Management with
the Index Model (Continued)

• The beta coefficient, the market risk, s2M, and
  the firm-specific risk, s2(e), can be estimated
  from historical SCLs, from regressions of
  security excess returns on market index excess
  returns

45
Practical Aspects of Portfolio Management with
the Index Model (Continued)

• Tracking Portfolios
• Example
• A portfolio manager believes that she identified
  an underpriced portfolio (positive alpha)
• RP0.041.4RSP500eP
• The alpha value of P is 4 and the beta is 1.4
• The downfall of the market will affect the value
  of the portfolio negatively if she buys this
  portfolio even if it is relatively underpriced,
  since the beta coefficient is larger than 1

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Practical Aspects of Portfolio Management with
the Index Model (Continued)

• She would like to take a position that takes
  advantage of her teams analysis but is
  independent of the performance of the overall
  market
• She constructed a tracking portfolio (T)
• A tracking portfolio for portfolio P is a
  portfolio designed to match the systematic
  components of Ps return
• The tracking portfolio must have the same beta on
  the index portfolio as P and as little
  nonsystematic risk as possible
• This procedure is also called beta capture

47
Practical Aspects of Portfolio Management with
the Index Model (Continued)

• T will have a levered position in SP 500 to have
  a beta of 1.4
• T includes position of 1.4 in the SP 500 and
  -0.4 in T-bills.
• T is constructed from the index and T-bills so
  alpha is 0
• Buy portfolio P and short sell portfolio T (you
  offset the systematic risk)
• RCRP RT (0.041.4RSP500eP)-1.4RSP500
• 0.04eP
• C is still risky due to residual risk, eP, the
  systematic risk has been eliminated
• If P is well diversified the remaining
  non-systematic risk will be small as well
• The aim is achieved. The manager can take the
  advantage of the 4 alpha without taking on
  market risk