Factor Models

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Factor Models

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• Basic Equations Used to Do Factor Computations
• Estimated Average Return for Stock i
• ri (ri1 ... ri10)/10
• Estimated Variance of Return for Stock i
• var(ri) (ri1 ri)2 ... (ri10 ri)2/9
• Estimated Average Return for Index
• f (f1 ... f10)/10
• Estimated Covariance of Stock i with Index
• cov(ri,f) (ri1 ri)(f1 f) ... (ri10
  ri)(f10 f)/9
• Estimated b Term for Stock i
• bi cov(ri,f)/var(f)
• Estimated a Term for Stock i

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The CAPM as a Factor Model
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• Characteristic Line
• This line represents a single-factor model that
  has
• rMrf as the factor for the variable rirf

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avg.

15.00

14.34

10.90

15.09

13.83

5.84

var.

90.26

107.23

162.20

68.25

72.12


stdev

9.50

10.35

12.74

8.26

8.49


corel w M

0.81

0.84

0.93

0.70

1


65.09

73.62

100.79

48.99

72.12


cov. w. M

0.90

1.02

1.40

0.68



beta

1.94

0.34

-
6.11

3.82



alpha

e
-
var.

31.52

32.07

21.36

34.98




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• To compute each ?i, divide the covariance with
  the market by the variance of the market. 
•  
• To compute each ?i, compute two terms (1) the
  difference of the return for the asset and the
  risk-free return (2) ?i times the difference of
  the return for the market and the risk-free
  return.
• Then subtract the second term from the first. 
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• To compute the variance of each error, subtract
  from the variance of the return the product of ?2
  and the variance of the market.

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Data and Statistics


Basic Question



How accurately can we estimate E
r
, Var
r
?



Common Estimation Approaches


Use historical data, e.g.,




monthly return rates for 3 years,




annual return rates for
10 years,


to compute sample average returns, sample
variances
of returns, sample correlations.

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5.01

5.88

3.21

3.81

2.98

3.24

4.66

3.55

4.12

s


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The Ta
ble 8.3 reports monthly returns simulated from
iid
normal random variables with Er 1 and
St.dev. 4.33.


Note




how much the r values vary from year to year



the overall Er estimate is 33 high



the standard deviation estimates vary less from
year
to year



the overall standard deviation estimate is not
bad.


Refer to the histogram of monthly returns, Figure
8.4. "It
is impossible to determine an accurate estimate
of the
true mean from the samples."

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Estimation of Other Parameters
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Note r is a random variable
Er Er
sample variance of r
2
2
2
2
2
s
s
(r
- r)
... (r
- r)
/(n-1)
Es

.
Þ
1
n
2
How accurate is s
? If also the r
are normally
i
distributed, then
2
4
s
var(s
) 2
/(n-1)
2
2
s
stdev(s
)
2
/
(n-1) which goes to 0 as n increases.
Ö
Ö
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5.01

5.88

3.21

3.81

2.98

3.24

4.66

3.55

4.12

s


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• Blur for Factor Models
• "The blur phenomenon applies to the parameters
  of a factor model, but mainly to the
  determination of a. In fact the presence of
  a-blur can be deduced from the mean-blur
  phenomenon, but we omit the details.
• The inherently poor accuracy of a estimates is
  reflected in the so-called Beta Book, published
  by Merrill Lynch (example in Table 8.4) .... the
  reported standard deviation for a is typically
  larger than the value of a itself. The related
  error in estimating ? is somewhat better."

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Tilting Away from Equilibrium

• Mean-variance theory suggests that the efficient
  fund of risky assets would be the market
  portfolio.
• Many investors are not satisfied with this
  conclusion and consider that a superior solution
  can be computed by solving Markowitz problem
  directly.
• Historical data may not be enough to solve the
  Markowitz problem.
• Compromise solution combine CAPM with an
  additional information

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• Equilibrium Means
• Rates of return implied by CAPM
• Erie rf ?i (ErM rf)
• ?I can be estimated from data, and ErM can be
  estimated using consensus (expert) opinions

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• Information
• CAPM rates of return may differ from true rates
• Eri Erie ei ,
• where ei has zero mean.
• Historical rates of return also differ from true
  rates
• Eri Erih ei

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• Example. Double use of data (see, Exam. 8.2)
• Average rates of return implied by CAPM and
  historical rates are not equal. Both estimates
  have errors, but they can be combined to form new
  estimates, called tilt.

Stk. 1
Stk. 2
Stk. 3
Stk. 4
Market
Riskless
avg.
15.00
14.34
10.90
15.09
13.83
5.84
var.
90.26
107.23
162.20
68.25
72.12
cov. w. M
65.09
73.62
100.79
48.99
72.12
beta
0.90
1.02
1.40
0.68
CAPM
13.05
14.00
17.01
11.27
tilt
13.82
14.14
14.17
12.57
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• Example. Double use of data (Contd)
• For example, for stock 1 rate of return implied
  by CAPM
• Er1e rf ?i (ErM rf)
• 5.84.9(13.83-5.84)13.0
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• To form a new, combined, estimates we calculate
  the variance for each estimate (errors in the
  CAPM model are ignored except error in ErM )
• sih si / v10 ,
• sie ?i sM / v10

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• Example. Double use of data (Contd)
• Tilts
• Eri Erie/(sie)2 Erih/(sih)2 /
  1/(sie)2 1/(sih)2
• Er113.82

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Multiperiod Fallacy

• Both mean-variance (Markowitz) theory and the
  CAPM are for single periods. In practice,
  however, both ... are applied to situations that
  are inherently multiperiod, such as the
  construction of portfolios of common stocks that
  can be traded at any time.
• Suppose the basic period of time is 1 month.
  Suppose we formulate the Markowitz model for this
  period, and solve it. The CAPM would imply the
  weights in W, the solution, are the same as the
  market portfolio.

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• In particular, suppose there are only 2 stocks
  in the entire market. There are 1,000 shares of
  each, and each costs 1.00 a share. We have 100
  to invest this month. The two stocks are
  uncorrelated, with the same mean return and
  variance of return. The Markowitz model will
  give W (1/2,1/2). This W corresponds to the
  market portfolio, which the total value of each
  stock is 1,000. We will buy 50 shares of each
  stock.

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Next month we can again invest. The first stock
now
sells for 2.00 a share while the second still
sells for
1.00 a share. We sell our shares our total
wealth is
now 150. The statistical properties remain
unchanged,
so the optimal solution to the Markowitz model
will
again be (1/2,1/2). This means we put 75 into
stock 1
and 75 into stock 2. This buys us 75/2 37.5
shares
of stock 1 and 75 shares of stock 2.
But because the statistical properties have
remained
unchanged, the market portfolio still has equal
shares of
each stock. Hence, we have not purchased the
market
portfolio in month 2.
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