Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multiple index models
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Content
1 Multi-factor models
2 Arbitrage Pricing Theory - APT
3 Estimation of parameters
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Content
1 Multi-factor models
2 Arbitrage Pricing Theory - APT
3 Estimation of parameters
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Captures the correlation structure of non-market influences
(Economic factors, structural groups, etc.)
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Captures the correlation structure of non-market influences
(Economic factors, structural groups, etc.)
Benjamin King demonstrated the influence of industry on
stock prices (1966)
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Captures the correlation structure of non-market influences
(Economic factors, structural groups, etc.)
Benjamin King demonstrated the influence of industry on
stock prices (1966)
→ Two models have been proposed:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Captures the correlation structure of non-market influences
(Economic factors, structural groups, etc.)
Benjamin King demonstrated the influence of industry on
stock prices (1966)
→ Two models have been proposed:
Multi index model
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Multi-factor models
Extend the single model
Captures the correlation structure of non-market influences
(Economic factors, structural groups, etc.)
Benjamin King demonstrated the influence of industry on
stock prices (1966)
→ Two models have been proposed:
Multi index model
Industrial index model
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Where,
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Where,
bi . . . sensitivity of asset to the factor F
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Where,
bi . . . sensitivity of asset to the factor F
and
bi =
σF,ri
σ2
F
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Where,
bi . . . sensitivity of asset to the factor F
and
bi =
σF,ri
σ2
F
Also we can assume:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
ri = ai + bi ∗ F + ei
Where,
bi . . . sensitivity of asset to the factor F
and
bi =
σF,ri
σ2
F
Also we can assume:
¯ri = ai + bi ∗ ¯F
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Where,
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Where,
bi1 . . . sensitivity of asset to the factor F1
bi2 . . . sensitivity of asset to the factor F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Where,
bi1 . . . sensitivity of asset to the factor F1
bi2 . . . sensitivity of asset to the factor F2
and
bi1 =
σF1,ri
σ2
F1
, bi2 =
σF2,ri
σ2
F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Where,
bi1 . . . sensitivity of asset to the factor F1
bi2 . . . sensitivity of asset to the factor F2
and
bi1 =
σF1,ri
σ2
F1
, bi2 =
σF2,ri
σ2
F2
Also we can assume:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Two-factors model:
ri = ai + bi1 ∗ F1 + bi2 ∗ F2 + ei
Where,
bi1 . . . sensitivity of asset to the factor F1
bi2 . . . sensitivity of asset to the factor F2
and
bi1 =
σF1,ri
σ2
F1
, bi2 =
σF2,ri
σ2
F2
Also we can assume:
¯ri = ai + bi1 ∗ ¯F1 + bi2 ∗ ¯F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Two-factors model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Two-factors model:
σ2
i = b2
i1 ∗ σF
2
1 + b2
i2 ∗ σF
2
2 + σ2
ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Two-factors model:
σ2
i = b2
i1 ∗ σF
2
1 + b2
i2 ∗ σF
2
2 + σ2
ei
/ + 2 ∗ bi1 ∗ bi2 ∗ σF1,F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Two-factors model:
σ2
i = b2
i1 ∗ σF
2
1 + b2
i2 ∗ σF
2
2 + σ2
ei
/ + 2 ∗ bi1 ∗ bi2 ∗ σF1,F2
and,
σi,j = bi1∗bj1∗σ2
F1
+bi2∗bj2∗σ2
F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
One-factor model:
σ2
i = b2
i ∗ σ2
F + σ2
ei
and,
σi,j = bi ∗ bj ∗ σ2
F
Two-factors model:
σ2
i = b2
i1 ∗ σF
2
1 + b2
i2 ∗ σF
2
2 + σ2
ei
/ + 2 ∗ bi1 ∗ bi2 ∗ σF1,F2
and,
σi,j = bi1∗bj1∗σ2
F1
+bi2∗bj2∗σ2
F2
/ + (bi1 ∗ bj2 + bi2 ∗ bj1) ∗ σF1,F2
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
ri = ai +
n
k=1
bik ∗ Fk + ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
ri = ai +
n
k=1
bik ∗ Fk + ei
Thus, for portfolio,
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
ri = ai +
n
k=1
bik ∗ Fk + ei
Thus, for portfolio,
Rp =
n
i=1
wi ∗ (ai + bi1 ∗ F1 + bi2 ∗ F2 + · · · + bik ∗ Fk + ei )
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
ri = ai +
n
k=1
bik ∗ Fk + ei
Thus, for portfolio,
Rp =
n
i=1
wi ∗ (ai + bi1 ∗ F1 + bi2 ∗ F2 + · · · + bik ∗ Fk + ei )
and:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Equation of models
Multi-factors model:
ri = ai +
n
k=1
bik ∗ Fk + ei
Thus, for portfolio,
Rp =
n
i=1
wi ∗ (ai + bi1 ∗ F1 + bi2 ∗ F2 + · · · + bik ∗ Fk + ei )
and:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
σ2
p =
n
i=1
w2
i (b2
i ∗ σ2
F + σ2
ei
)
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
σ2
p =
n
i=1
w2
i (b2
i ∗ σ2
F + σ2
ei
)
Factor (systematic) risk
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
σ2
p =
n
i=1
w2
i (b2
i ∗ σ2
F + σ2
ei
)
Factor (systematic) risk
b2
p ∗ σ2
F = (
n
i=1
wi ∗ bi )2
∗ σ2
F
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
σ2
p =
n
i=1
w2
i (b2
i ∗ σ2
F + σ2
ei
)
Factor (systematic) risk
b2
p ∗ σ2
F = (
n
i=1
wi ∗ bi )2
∗ σ2
F
Non-factor (unsystematic) risk
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Factor vs. Non-factor risk
One-factor model:
σ2
p =
n
i=1
w2
i (b2
i ∗ σ2
F + σ2
ei
)
Factor (systematic) risk
b2
p ∗ σ2
F = (
n
i=1
wi ∗ bi )2
∗ σ2
F
Non-factor (unsystematic) risk
σ2
ep
=
n
i=1
w2
i ∗ σ2
ei
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Content
1 Multi-factor models
2 Arbitrage Pricing Theory - APT
3 Estimation of parameters
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
σri ,rj = cov ai +
n
k=1
bik ∗ Fk + ei ; aj +
n
m=1
bjm ∗ Fm + ej
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
σri ,rj = cov ai +
n
k=1
bik ∗ Fk + ei ; aj +
n
m=1
bjm ∗ Fm + ej
Assumptions for ATP:
E(ei ) = 0, i = 1, 2, 3, . . . , n
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
σri ,rj = cov ai +
n
k=1
bik ∗ Fk + ei ; aj +
n
m=1
bjm ∗ Fm + ej
Assumptions for ATP:
E(ei ) = 0, i = 1, 2, 3, . . . , n
σFk ,Fm = 0, m̸= k, m,k=1, 2, 3, . . . , K
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
σri ,rj = cov ai +
n
k=1
bik ∗ Fk + ei ; aj +
n
m=1
bjm ∗ Fm + ej
Assumptions for ATP:
E(ei ) = 0, i = 1, 2, 3, . . . , n
σFk ,Fm = 0, m̸= k, m,k=1, 2, 3, . . . , K
σei ,ej = 0, i̸= j, i,j=1, 2, 3, . . . , n
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Model APT
This is a special case of a multi-factors model
In general:
σri ,rj = cov ai +
n
k=1
bik ∗ Fk + ei ; aj +
n
m=1
bjm ∗ Fm + ej
Assumptions for ATP:
E(ei ) = 0, i = 1, 2, 3, . . . , n
σFk ,Fm = 0, m̸= k, m,k=1, 2, 3, . . . , K
σei ,ej = 0, i̸= j, i,j=1, 2, 3, . . . , n
σei ,Fk
= 0, i = 1, 2, 3, . . . , n k = 1, 2, 3, . . . , K
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
...
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
...
Beta of factors:
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
...
Beta of factors:
βF1 =
σF1,rm
σ2
m
, . . .
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
...
Beta of factors:
βF1 =
σF1,rm
σ2
m
, . . .
Thus,
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Merging CAPM & APT
Beta coefficients and factor weights (sensitivity)
σri ,rm = σF1,rm ∗ bi1 + σF2,rm ∗ bi2 + σei ,rm
...
Beta of factors:
βF1 =
σF1,rm
σ2
m
, . . .
Thus,
βi = βF1 ∗ bi1 + βF2 ∗ bi2 + . . .
Multi-factor models Arbitrage Pricing Theory - APT Estimation of parameters
Content
1 Multi-factor models
2 Arbitrage Pricing Theory - APT
3 Estimation of parameters