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1

Drawbacks to the Markowitz Optimization Model (3)

(BKM - 8)

1. model requires a large number of estimates for the input list

2. requires accurate correlations for covariance calculations - poor estimates could lead to nonsensical results

3. Does not provide guidance for forecasting security risk premiums to construct the efficient frontier of risky assets

2

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

3

Single-factor model return (r(i))

(BKM - 8)

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

4

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

5

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

6

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)

7

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)

8

Single-index model (R-i(t))

(BKM - 8)

R-i(t) = alpha-i + beta-i * R-M(t) + e-i(t)

R-i(t) = firm's excess returns
t = month of return

9

Independence assumption of the single-index model

(BKM - 8)

assumes securities are independent

10

Security characteristic line (SCL)

(BKM - 8)

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

11

Interpretation of alpha

(BKM - 8)

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)

12

Interpretation of beta

(BKM - 8)

amount security return changes for every 1% increase in return on the index (e.g. sensitivity to the market index)

13

Interpretation of the firm-specific surprise (e(i))

(BKM - 8)

firm-specific unexpected variation in security return (aka residual)

14

Beta values

(BKM - 8)

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

15

Expected excess return for the single-index model (E[R(i)])

(BKM - 8)

E[R(i)] = alpha(i) + beta(i) * E[R(M)]

16

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

17

Interpretation of R^2 in regression output and formula

(BKM - 8)

% of variation in excess returns that is explained by market factors

R^2 = systematic variance / total variance

18

Adjusted R^2

(BKM - 8)

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

19

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

20

T-statistic formula and meaning of regression output

(BKM - 8)

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

21

Interpretation of the p-value in regression output

(BKM - 8)

level of significance = probability of an estimate as large as the given coefficient if the true parameter = 0

22

Reasons beta tends to 1 over time (2)

(BKM - 8)

1. new firms are unconventional but eventually resemble the rest of the economy

2. average beta over all securities = 1

23

Adjusted beta

(BKM - 8)

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

24

Most predictive variables for betas (6)

(BKM - 8)

1. variance of earnings
2. variance of cash flow
3. growth in earnings per share
4. market capitalization (firm size)
5. dividend yield
6. debt-to-asset ratio

25

Components of the optimal risky portfolio (2)

(BKM - 8)

1. active portfolio (A) - made up of n analyzed securities

2. passive portfolio (M) - market-index portfolio - used to aid in diversification

26

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

27

Information ratio

(BKM - 8)

information ratio = alpha(i) / sigma(e(i))

the contribution of each security is the square of its own information ratio (e.g. information ratio(A)^2 = sum of information ratio(security)^2)

28

Advantages of the single-index model over the Markowitz Optimization Model (2)

(BKM - 8)

1. saves time b/c fewer estimates are required

2. allows for specialization in security analysis (without calculating the covariance b/w industries)

29

Disadvantages of the single-index model over the Markowitz Optimization Model (2)

(BKM - 8)

1. oversimplifies uncertainty by splitting it into broad categories of micro or macro risk (vs. full covariance matrix with covariances b/w all securities)

2. can produce an inferior optimal portfolio if correlated stocks make up a large part of the portfolio (because of independence assumption)

30

Optimal weights using the Single-index model

(BKM - 8)

1. calc initial position for each security in the active portfolio assuming beta = 1 - each weight = alpha(stock) / sigma^2(e(stock))

2. scale so the weights sum to 100%

3. calculate the portfolio alpha, residual variance sigma^2(e(A)), and beta using the weights

4. initial position in active portfolio = (portfolio risk premium / portfolio residual variance) / (maarket risk premium / market residual variance)

5. adjust the position in the active portfolio to reflect true beta = initial position / [1 + (1 - portfolio beta) * initial position]