BKM Chapter 8 Flashcards
Drawbacks to the Markowitz Optimization Model (3)
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- model requires a large number of estimates for the input list
- requires accurate correlations for covariance calculations - poor estimates could lead to nonsensical results
- Does not provide guidance for forecasting security risk premiums to construct the efficient frontier of risky assets
Total number of estimates needed for the Markowitz Optimization Model input list
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n estimates of expected return
+ n estimates of variances
+ (n^2 - n) / 2 estimates of covariances
n = # of securities in the portfolio
Single-factor model return (r(i))
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r(i) = E[r(i)] + beta(i) * M + e(i)
beta(i) = firm-sensitivity to market index
M = uncertainty about the economy (systematic uncertainty)
e(i) = firm-specific (non-systematic) uncertainty
Relationship between market uncertainty (M) and firm-specific uncertainty (e(i)) in a single-factor model
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M and e(i) are assumed to be uncorrelated and E[M] = E[e(i)] = 0
Variance of the single-factor model and single-index model (sigma^2(i))
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sigma^2(i) = beta(i)^2 * sigma^2(M) + sigma^2(e(i))
where beta(i)^2 * sigma^2(M) = systematic risk and sigma^2(e(i)) = firm-specific risk
firm-specific risk goes to 0 with diversification
Cov(r(i), r(j)) in a single-factor model and single index model
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Cov(r(i), r(j)) = beta(i) * beta(j) * sigma^2(M)
Difference between the single-factor model and the single-index model
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the single-index model uses the rate of return on a broad market index as a proxy for the systematic factor (M)
Single-index model (R-i(t))
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R-i(t) = alpha-i + beta-i * R-M(t) + e-i(t)
R-i(t) = firm's excess returns t = month of return
Independence assumption of the single-index model
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assumes securities are independent
Security characteristic line (SCL)
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regression line that plots excess monthly returns for the security on the y-axis and excess monthly returns for the market index on the x-axis
the line has slope = beta-i and intercept = alpha-i
Interpretation of alpha
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security’s expected excess return when the market return = 0 (e.g. non-market risk premium)
alpha > 0 means the security is underpriced
positive alpha values are more attractive
negative alpha values should be shorted (if allowed)
Interpretation of beta
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amount security return changes for every 1% increase in return on the index (e.g. sensitivity to the market index)
Interpretation of the firm-specific surprise (e(i))
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firm-specific unexpected variation in security return (aka residual)
Beta values
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beta > 1 = cyclical/aggressive stocks (high sensitivity to macroeconomy)
beta < 1 = defensive stocks (low sensitivity to macroeconomy)
avg beta of all stocks = beta of market index = 1
Expected excess return for the single-index model (E[R(i)])
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E[R(i)] = alpha(i) + beta(i) * E[R(M)]