BKM Chapter 8 Flashcards
Drawbacks to the Markowitz Optimization Model (3)
BKM - 8
- 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
(BKM - 8)
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
(BKM - 8)
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))
(BKM - 8)
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
(BKM - 8)
Cov(r(i), r(j)) = beta(i) * beta(j) * sigma^2(M)
Difference between the single-factor model and the single-index model
(BKM - 8)
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)]
Number of estimates needed for the single-index model input list
(BKM - 8)
n estimates of alpha values
+ n estimates of beta values
+ n estimates of firm-specific variances (sigma^2(e(i)))
+ 1 estimate of market risk premium (E[R(M)])
+ 1 estimate of the variance for the common macroeconomic factor (sigma^2(M))
= 3n + 2 total estimates
Interpretation of R^2 in regression output and formula
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% of variation in excess returns that is explained by market factors
R^2 = systematic variance / total variance
Adjusted R^2
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R^2(A) = 1 - (1 - R^2) * (n - 1) / (n - k - 1)
k = # of independent variables
corrects for bias from estimated slope (beta) and intercept (alpha) parameters
Meaning of standard error for the regression and parameter estimates of regression output
(BKM - 8)
regression: high standard error means firm-specific events have a larger impact (aka residual std. dev)
parameter estimates: a measure of imprecision/estimation error
T-statistic formula and meaning of regression output
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t-stat = (estimated value - hypothesized value) / standard error
used to test null that beta = 1
reveals the # of standard errors by which the estimate exceeds 0
Interpretation of the p-value in regression output
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level of significance = probability of an estimate as large as the given coefficient if the true parameter = 0
Reasons beta tends to 1 over time (2)
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- new firms are unconventional but eventually resemble the rest of the economy
- average beta over all securities = 1
Adjusted beta
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Adjusted beta = (2/3) * estimated beta + (1/3) * 1
(kind of like a credibility weighted beta with 1/3 weight to the average beta for the market
Most predictive variables for betas (6)
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- variance of earnings
- variance of cash flow
- growth in earnings per share
- market capitalization (firm size)
- dividend yield
- debt-to-asset ratio
Components of the optimal risky portfolio for the SIM (2)
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- active portfolio (A) - made up of n analyzed securities
2. passive portfolio (M) - market-index portfolio - used to aid in diversification
Relationship between Sharpe ratio of an optimally constructed risky portfolio (S(P)) and the Sharpe ratio of the index portfolio (S(M))
(BKM - 8)
S(P)^2 = S(M)^2 + (information ratio(A))^2
Information ratio
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information ratio = alpha(i) / sigma(e(i))
the contribution of each security to the Sharpe ratio is the square of its own information ratio (e.g. information ratio(A)^2 = sum of information ratio(security)^2)
Advantages of the single-index model over the Markowitz Optimization Model (2)
(BKM - 8)
- saves time b/c fewer estimates are required
2. allows for specialization in security analysis (without calculating the covariance b/w industries)
Disadvantages of the single-index model over the Markowitz Optimization Model (2)
(BKM - 8)
- oversimplifies uncertainty by splitting it into broad categories of micro or macro risk (vs. full covariance matrix with covariances b/w all securities)
- can produce an inferior optimal portfolio if correlated stocks make up a large part of the portfolio (because of independence assumption)
Optimal weights using the Single-index model
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- calc initial position for each security in the active portfolio (assuming beta = 1) where each weight = alpha(stock) / sigma^2(e(stock))
- scale so the weights sum to 100%
- calculate the portfolio alpha, residual variance sigma^2(e(A)), and beta using the weights
- initial position in active portfolio = (portfolio risk premium / portfolio residual variance) / (market risk premium / market residual variance)
- adjust the position in the active portfolio to reflect true beta = initial position / [1 + (1 - portfolio beta) * initial position]
Initial weight for each security and initial position in the active portfolio for the single-index model
(BKM - 8)
initial weight = alpha(stock) / residual variance(stock))
*assumes beta = 1
initial position = (alpha(portfolio) / residual variance(portfolio)) / (E[R(M)] / residual variance(market))
Adjusted position for the single-index portfolio & optimal risky portfolio weights for the single-index model
(BKM - 8)
w*(A) = initial position / (1 + (1 - portfolio beta) * initial position)
invest w(A) in the risky portfolio
w(M) = 1 - w(A) in the market index
and w(A) * w(i) in each security in the risky portfolio
Goal of the single-index model (aka Treynor-Black or TB model)
(BKM - 8)
maximize the Sharpe ratio by maximizing the information ratio (max gain relative to risk)
Beta(i) in the SIM
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beta(i) = Cov(R(i),R(M)) / sigma^2(M)
i = stock i M = market
