A single-factor security market

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Topic 3 Index Models

• A single-factor security market
• The single-index model
• Estimating the single-index model

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A Single-Factor Security Market

• The success of a portfolio selection rule depends
  on the quality of the input list (i.e. the
  estimates of expected security returns and the
  covariance matrix).
• e.g. To analyze 50 stocks, the input list
  includes
• n 50 estimates of
  expected returns
• n 50 estimates of
  variances
• (n2 - n)/2 1,225 estimates of
  covariances
• 1,325 estimates

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• If n 3,000 (roughly the number of NYSE
  stocks), we need more than 4.5 million estimates.
• Errors in the assessment or estimation of
  correlation coefficients can lead to nonsensical
  results. This can happen because some sets of
  correlation coefficients are mutually
  inconsistent.
• e.g.

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• Construct a portfolio with weights -1.00
  1.00 1.00, for assets A B C, respectively, and
  calculate the portfolio variance.
• ? Portfolio variance -240!
• Covariances between security returns tend to be
  positive because the same economic forces affect
  the fortunes of many firms (e.g. business cycles,
  interest rates, technological changes, etc.).
• All these (interrelated) factors affect
  almost all firms. Thus, unexpected changes in
  these variables cause, simultaneously, unexpected
  changes in the rates of return on the entire
  stock market.

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• Suppose that we summarize all relevant economic
  factors by one macroeconomic indicator and assume
  that it moves the security market as a whole.
• We further assume that, beyond this common
  effect, all remaining uncertainty in stock
  returns is firm specific (i.e. there is no other
  source of correlation between securities).
• Firm-specific events would include new
  inventions, deaths of key employees, and other
  factors that affect the fortune of the individual
  firm without affecting the broad economy in a
  measurable way.

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• We can summarize the distinction between
  macroeconomic and firm-specific factors by
  writing the holding-period return on security i
  as
• where E(ri) expected return on the security
  i as
• of the beginning of the
  holding period
• m impact of unanticipated macro
  events
• on all securities return
  during the period
• ei impact of unanticipated
  firm-specific events.
• Note Both m and ei have 0 expected values
  because each represents the impact of
  unanticipated events, which by definition must
  average out to 0.

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• Since m and ei are uncorrelated, the variance of
  ri arises from two uncorrelated sources,
  systematic and firm specific.
• Since m is also uncorrelated with any of the
  firm-specific surprises, the covariance between
  any two securities i and j is

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• Some securities will be more sensitive than
  others to macroeconomic shocks.
• We can capture this refinement by assigning
  each firm a sensitivity coefficient to macro
  conditions.
• Thus, if we denote the sensitivity
  coefficient for firm i by ?i, we have the
  following single-factor model
• The systematic risk of security i is
  determined by its beta coefficient (?i).

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• The variance of the rate of return on each
  security includes 2 components
• variance attributable to the
  uncertainty of the common macroeconomic factor
  (i.e. systematic risk)
• variance attributable to
  firm-specific uncertainty.
• ?

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• The covariance between any pair of securities is
  determined by their betas

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The Single-Index Model

• To make the single-factor model operational, we
  use the rate of return on a broad index of
  securities (such as SP 500) as a proxy for the
  common macroeconomic factor.
• This approach leads to an equation similar to
  the single-factor model, which is called the
  single-index model, because it uses the market
  index to proxy for the common factor.

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The regression equation of the single-index model

• Denote the market index by M, with excess return
  of RM rM - rf and standard deviation of sM.
• Excess return of a securityRi ri rf

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• Collect a historical sample of paired
  observations and regress Ri(t) on RM(t), where t
  denotes the date of each pair of observations.
• The regression equation is
• Intercept
• ai the security is expected excess return
  when the
• market excess return is zero.

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• Slope coefficient
• ßi the security is sensitivity to the
  market index.
• For every (or -) 1 change in the market
  excess return, the excess return on the security
  will change by (or -)ßi.
• Residual
• ei is the zero-mean, firm specific surprise in
  the security return in time t.

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The expected return-beta relationship
nonmarket premium
part of a securitys risk premium is due to the
risk premium of the market index ? systematic
risk premium
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Risk and covariance in the single-index model

• Recall that we have the following equation
• The variance of the rate of return on each
  security includes 2 components
• variance attributable to the
  uncertainty of the
• market index

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• variance attributable to firm-specific
  uncertainty.
• ?
• (total risk systematic risk
  firm-specific risk)
• Note
• The covariance between RM and ei is zero
  because ei is defined as firm specific (i.e.
  independent of movements in the market).

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• The covariance between the rates of return on 2
  securities
• Note
• Since ?i and ?j are constants, their
  covariance with any variable is zero.
• Further, the firm-specific terms (ei, ej)
  are assumed uncorrelated with the market and with
  each other.
• ?
• Covariance Product of betas Market index
  risk

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• The covariance between the return on stock i and
  the market index
• Notes
• We can drop ?i from the covariance terms because
  ?i is a constant and thus has zero covariance
  with all variables.
• The firm-specific or nonsystematic component is
  independent of the marketwide or systematic
  component (i.e. Cov(ei, RM) 0).

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• The correlation coefficient between the rates of
  return on 2 securities

(product of correlations with the market index)
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The set of estimates needed for the single-index
model

• If we have
• n estimates of the extra-market
  expected excess returns, ai
• n estimates of the sensitivity
  coefficients, ßi
• n estimates of the firm-specific variances,
  s2(ei)
• 1 estimate for the market risk premium,
• 1 estimate for the variance of the (common)
  
• macroeconomic factor, sM2
• then these (3n 2) estimates will enable us
  to prepare the input list for this single-index
  security universe.

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For n 50 need 152 estimates (not 1,325
estimates). n 3,000 need 9,002
estimates (not 4.5 million).
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The index model and diversification

• Suppose that we choose an equally weighted
  portfolio of n securities (I.e. wi 1/n).
• The excess rate of return on each security
  is
• ? The excess return on the portfolio of
  securities
• Note

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? The portfolio has a sensitivity to the market
given by
(the average of the individual ?is) ?
It has a nonmarket return component of a constant
(intercept)
(the average of the individual alphas)
? It has a zero mean variable (the
average of the firm-specific components)
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? The portfolios variance is The
systematic risk component of the portfolio
variance (the component that depends on
marketwide movements) is and depends
on the sensitivity coefficients of the individual
securities. This part of the risk depends
on portfolio beta and , and will
persist regardless of the extent of portfolio
diversification. No matter how many
stocks are held, their common exposure to the
market will be reflected in portfolio systematic
risk.
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In contrast, the nonsystematic component of
the portfolio variance is ?2(eP) and is
attributable to firm-specific components ei.
Because the eis are uncorrelated, we have
where the average of the
firm-specific variances. Because this
average is independent of n, when n gets
large, ?2(eP) becomes negligible.
Thus, as more and more securities are added to
the portfolio, the firm-specific components tend
to cancel out, resulting in ever-smaller
nonmarket risk.
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Summary As more and more securities
are combined into a portfolio, the portfolio
variance decreases because of the diversification
of firm-specific risk. However, the power
of diversification is limited. Even for
very large n, part of the risk remains because of
the exposure of virtually all assets to the
common, or market, factor. Therefore, this
systematic risk is said to be nondiversifiable.
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Estimating the Single-Index Model

• The single-index model
• suggests how we might go about actually
  measuring market and firm-specific risk.
• 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 a one-year
  period.

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We can summarize the results for a sample
period in a scatter diagram
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• The single-index model states that the
  relationship between the excess returns on GM and
  the SP 500 is given by the following regression
  equation
• In this single-variable regression equation,
  the dependent variable plots around a straight
  line with an intercept ? and a slope ?.
• The deviations from the line (e) are assumed
  to be mutually uncorrelated and uncorrelated with
  the independent variable.

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The sensitivity of GM to the market,
measured by ?GM, is the slope of the regression
line. The intercept of the regression line
is ?GM, representing the average firm-specific
return when the markets excess return is zero.
Deviations of particular observations from
the regression line in any period are denoted
eGM, and called residuals (i.e. each of these
residuals is the difference between the actual
security return and the return that would be
predicted from the regression equation describing
the usual relationship between the security and
the market). Thus, residuals measure the
impact of firm-specific events.
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• Estimating the regression equation of the
  single-index model gives us the security
  characteristic line (SCL).
• The SCL is a plot of the typical excess
  return on a security as a function of the excess
  return on the market.
• Compute ?GM and ?GM
• Let yt excess return on GM in month t
• xt excess return on the market (SP
  500) in month t
• n the total number of months.

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? The estimate of beta coefficient (i.e. the
slope of the regression line SCL) ? The
intercept of the regression line
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• Compute residuals
• For each month t, our estimate of the
  residual is the deviation of GMs excess return
  from the prediction of the SCL
• Deviation Actual Predicted
  Return
• These residuals are estimates of the monthly
  unexpected firm-specific component of the rate of
  return on GM stock.

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• Hence, we can estimate the firm-specific
  variance
• The standard deviation of the firm-specific
  component of GMs return
• which is equal to the standard deviation of
  the regression residual.

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The Industry Version of the Index Model

• Practitioners often use a modified index model
  that uses total rather than excess returns
  (deviations from T-bill rates) in the
  regressions
• instead of

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• To see the impact of this departure
• If rf is constant over the sample period,
  both equations have the same independent variable
  rM and residual e.
• Thus, the slope coefficient will be the same
  in the two equations.

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• However, the intercept is really an estimate
  of
• The apparent justification for this procedure
  is that, on a monthly basis, rf(1 - ?) is small.
  
• But, note that for ß?1, the regression
  intercept will not equal the index model alpha.

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Predicting Betas

• Betas estimated form past data may not be the
  best estimates of future betas.
• This suggests that we might want a forecasting
  model for beta.
• One simple approach would be to collect data on
  beta in different periods and then estimate a
  regression equation
• Current beta a b (Past beta)
• Given estimates of a and b, we would then
  forecast future betas using the rule
• Forecast beta a b (Current beta)

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• However, there is no reason to limit ourselves to
  such simple forecasting rules.
• Why not also investigate the predictive power
  of other financial variables in forecasting beta?
• Rosenberg and Guy find the following variables
  help predict betas
• Variance of earnings.
• Variance of cash flow.
• Growth in earnings per share.
• Market capitalization (firm size).
• Dividend yield.
• Debt-to-asset ratio.

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• Rosenberg and Guy also find that even after
  controlling for a firms financial
  characteristics, industry group helps to predict
  beta.