CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Portfolio Theory
Lecture 7
Lud¥k Benada
Department of Finance - 402, benada.esf@gmail.com
April 5, 2016
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Stracture
1 CAPM - empirical testing
2 Systematic and unsystematic risk
3 Delta of security
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Testing of the model
Model was many times testing, but with ambiguous results
Sharpe, Lintner, Miller, Jensen, Fama, French
...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Testing of the model
Model was many times testing, but with ambiguous results
Sharpe, Lintner, Miller, Jensen, Fama, French
...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Testing of the model
Model was many times testing, but with ambiguous results
Sharpe, Lintner, Miller, Jensen, Fama, French
...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Background for testing
The model is based on expectations
The variables are expressed in future value
...but the calculation is on observed values
Prices of assets will be variing around equilibrium
The equilibrium return of an individual asset:
re
i = rf +(rM −rf )∗βi
...but the real return:
ri −rf = (rM −rf )∗β +εi
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Random component of the model
Properties of random error:
E(εi ) = 0;for ∀i = 1,23,...
Cov(εi ,ri ) = 0;for ∀i = 1,23,...
Cov(εi ,εj ) = 0;for ∀i = 1,23,...∧i = j
E [εi (rM − ¯rM)] = 0;for ∀i = 1,23,...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Random component of the model
Properties of random error:
E(εi ) = 0;for ∀i = 1,23,...
Cov(εi ,ri ) = 0;for ∀i = 1,23,...
Cov(εi ,εj ) = 0;for ∀i = 1,23,...∧i = j
E [εi (rM − ¯rM)] = 0;for ∀i = 1,23,...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Random component of the model
Properties of random error:
E(εi ) = 0;for ∀i = 1,23,...
Cov(εi ,ri ) = 0;for ∀i = 1,23,...
Cov(εi ,εj ) = 0;for ∀i = 1,23,...∧i = j
E [εi (rM − ¯rM)] = 0;for ∀i = 1,23,...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Random component of the model
Properties of random error:
E(εi ) = 0;for ∀i = 1,23,...
Cov(εi ,ri ) = 0;for ∀i = 1,23,...
Cov(εi ,εj ) = 0;for ∀i = 1,23,...∧i = j
E [εi (rM − ¯rM)] = 0;for ∀i = 1,23,...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Random component of the model
Properties of random error:
E(εi ) = 0;for ∀i = 1,23,...
Cov(εi ,ri ) = 0;for ∀i = 1,23,...
Cov(εi ,εj ) = 0;for ∀i = 1,23,...∧i = j
E [εi (rM − ¯rM)] = 0;for ∀i = 1,23,...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Parametr estimates
Parameters could be estimated:
The set of this points is described by empirical regresion
function:
ˆyi = f (a,b,x) = a+b ∗x
Residue...εi = yi − ˆyi
The methodology of OLS minimalize the errors
Thus the objective function:
Sr = ∑N
i=1
e2
i ⇒ min
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Solution
Partial derivatives with respect to a,b
⇒system of equiations:
na+b∑Xi = ∑Yi
a∑Xi +b∑X2
i = ∑Xi Yi
...matrix calculation...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Solution
Partial derivatives with respect to a,b
⇒system of equiations:
na+b∑Xi = ∑Yi
a∑Xi +b∑X2
i = ∑Xi Yi
...matrix calculation...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Solution
Partial derivatives with respect to a,b
⇒system of equiations:
na+b∑Xi = ∑Yi
a∑Xi +b∑X2
i = ∑Xi Yi
...matrix calculation...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Solution
Partial derivatives with respect to a,b
⇒system of equiations:
na+b∑Xi = ∑Yi
a∑Xi +b∑X2
i = ∑Xi Yi
...matrix calculation...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Solution
Partial derivatives with respect to a,b
⇒system of equiations:
na+b∑Xi = ∑Yi
a∑Xi +b∑X2
i = ∑Xi Yi
...matrix calculation...
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Variance of security
Variance of excess return:
var (ri −rf ) = var(ri )+var (rf ) = σ2
i
var [(rM −rf )+εi ] = β2
i ∗σ2
M +ε2
i
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Variance of security
Variance of excess return:
var (ri −rf ) = var(ri )+var (rf ) = σ2
i
var [(rM −rf )+εi ] = β2
i ∗σ2
M +ε2
i
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Variance of security
Variance of excess return:
var (ri −rf ) = var(ri )+var (rf ) = σ2
i
var [(rM −rf )+εi ] = β2
i ∗σ2
M +ε2
i
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Market and unique risk
ε2
i ...concerns only an individual company or industry, could be
diversied!
β2
i ∗σ2
M...undiversied part of risk, concerns all securities on
the market
Ratio of the systematic risk is given by R2
...is explaining how
ts the model
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Market and unique risk
ε2
i ...concerns only an individual company or industry, could be
diversied!
β2
i ∗σ2
M...undiversied part of risk, concerns all securities on
the market
Ratio of the systematic risk is given by R2
...is explaining how
ts the model
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Market and unique risk
ε2
i ...concerns only an individual company or industry, could be
diversied!
β2
i ∗σ2
M...undiversied part of risk, concerns all securities on
the market
Ratio of the systematic risk is given by R2
...is explaining how
ts the model
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Unequilibrium in the model
Investors are looking for investment opportunities ...securities
in unequilibrium
A security is undervalueted if the return is higher then the
equilibrium return
A security is overvalueted if the return is under expected
return (security is expansive)
The equilibrium return lies on the SML
δi = ri − ¯ri
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Unequilibrium in the model
Investors are looking for investment opportunities ...securities
in unequilibrium
A security is undervalueted if the return is higher then the
equilibrium return
A security is overvalueted if the return is under expected
return (security is expansive)
The equilibrium return lies on the SML
δi = ri − ¯ri
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Unequilibrium in the model
Investors are looking for investment opportunities ...securities
in unequilibrium
A security is undervalueted if the return is higher then the
equilibrium return
A security is overvalueted if the return is under expected
return (security is expansive)
The equilibrium return lies on the SML
δi = ri − ¯ri
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Unequilibrium in the model
Investors are looking for investment opportunities ...securities
in unequilibrium
A security is undervalueted if the return is higher then the
equilibrium return
A security is overvalueted if the return is under expected
return (security is expansive)
The equilibrium return lies on the SML
δi = ri − ¯ri
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Unequilibrium in the model
Investors are looking for investment opportunities ...securities
in unequilibrium
A security is undervalueted if the return is higher then the
equilibrium return
A security is overvalueted if the return is under expected
return (security is expansive)
The equilibrium return lies on the SML
δi = ri − ¯ri
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Investment decision
If δi > 0 ⇒the security is over the SML ...undervalueted
If δi < 0 ⇒the security is under the SML ...overvalueted
If δi = 0 ⇒the security is on the SML ...in equilibrium
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Investment decision
If δi > 0 ⇒the security is over the SML ...undervalueted
If δi < 0 ⇒the security is under the SML ...overvalueted
If δi = 0 ⇒the security is on the SML ...in equilibrium
Lud¥k Benada MPF_APOT
CAPM - empirical testing
Systematic and unsystematic risk
Delta of security
Investment decision
If δi > 0 ⇒the security is over the SML ...undervalueted
If δi < 0 ⇒the security is under the SML ...overvalueted
If δi = 0 ⇒the security is on the SML ...in equilibrium
Lud¥k Benada MPF_APOT