Single index model - SIM Cut-off ratio Example of application
Cut-off portfolio
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
Single index model - SIM Cut-off ratio Example of application
Content
1 Single index model - SIM
2 Cut-off ratio
3 Example of application
Single index model - SIM Cut-off ratio Example of application
Content
1 Single index model - SIM
2 Cut-off ratio
3 Example of application
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Was developed as easier procedure to Markowitz
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Was developed as easier procedure to Markowitz
i.e. cov. structure N2−N
2 vs. N
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Was developed as easier procedure to Markowitz
i.e. cov. structure N2−N
2 vs. N
The main argument of the model is based on the common
interlinking of assets through a reference index
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Was developed as easier procedure to Markowitz
i.e. cov. structure N2−N
2 vs. N
The main argument of the model is based on the common
interlinking of assets through a reference index
The process of determining the portfolio weights is relatively
easy
Single index model - SIM Cut-off ratio Example of application
The single index model
Basically it is simple asset pricing model
Was developed as easier procedure to Markowitz
i.e. cov. structure N2−N
2 vs. N
The main argument of the model is based on the common
interlinking of assets through a reference index
The process of determining the portfolio weights is relatively
easy
It can very easy solve the problem with short sell ban
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development (βi )
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development (βi )
Expected specific company value
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development (βi )
Expected specific company value (αi )
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development (βi )
Expected specific company value (αi )
Unexpected component of the company
Single index model - SIM Cut-off ratio Example of application
The essence of the SIM
Introduced by W. Sharpe in 1963
Mathematical expression of the SIM:
ri,t − rf = αi + βi,t(Rm,t − rf ) + ϵi,t
where:
αi . . . Abnormal return
ϵi,t . . . Residual returns with N ∼ (0, σ2
i )
Thus, the model includes following information about the company
(stock) return:
It is determined by the market development (βi )
Expected specific company value (αi )
Unexpected component of the company (ϵi,t)
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Thus:
By most stocks could be observed a positive covariance
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Thus:
By most stocks could be observed a positive covariance
→ Similar response to macroeconomic factors
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Thus:
By most stocks could be observed a positive covariance
→ Similar response to macroeconomic factors
However, some companies are more sensitive to changes in
macroeconomic fundamentals
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Thus:
By most stocks could be observed a positive covariance
→ Similar response to macroeconomic factors
However, some companies are more sensitive to changes in
macroeconomic fundamentals
→ Subsequently, their firm-specific variance is
described primarily by their beta
Single index model - SIM Cut-off ratio Example of application
Basic assumption of the SIM
Only one factor causes the systematic risk affecting all stock
returns
→ Market index
Thus:
By most stocks could be observed a positive covariance
→ Similar response to macroeconomic factors
However, some companies are more sensitive to changes in
macroeconomic fundamentals
→ Subsequently, their firm-specific variance is
described primarily by their beta
Ultimately, it can be concluded that mutual covariances
of assets are determined by their sensitivity to
macroeconomic factors
Single index model - SIM Cut-off ratio Example of application
Simplifying the computational task
The premise of the model is based on the following:
E((ri − βi Rm)(rj − βj Rm)) = 0
Single index model - SIM Cut-off ratio Example of application
Simplifying the computational task
The premise of the model is based on the following:
E((ri − βi Rm)(rj − βj Rm)) = 0
Subsequently, it leads to:
Single index model - SIM Cut-off ratio Example of application
Simplifying the computational task
The premise of the model is based on the following:
E((ri − βi Rm)(rj − βj Rm)) = 0
Subsequently, it leads to:
σi,j = βi βj σ2
m
...
Single index model - SIM Cut-off ratio Example of application
Content
1 Single index model - SIM
2 Cut-off ratio
3 Example of application
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Assume that the SIM is the best method of predicting the
covariance structure of assets
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Assume that the SIM is the best method of predicting the
covariance structure of assets
To construct a portfolio, it would be beneficial to have some
decision-making tool for asset selection
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Assume that the SIM is the best method of predicting the
covariance structure of assets
To construct a portfolio, it would be beneficial to have some
decision-making tool for asset selection
Assuming the validity of the SIM, the following ratio can be
considered as a decisive criterion:
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Assume that the SIM is the best method of predicting the
covariance structure of assets
To construct a portfolio, it would be beneficial to have some
decision-making tool for asset selection
Assuming the validity of the SIM, the following ratio can be
considered as a decisive criterion:
ri − rf
βi
Single index model - SIM Cut-off ratio Example of application
The optimal portfolio in the SIM
Assume that the SIM is the best method of predicting the
covariance structure of assets
To construct a portfolio, it would be beneficial to have some
decision-making tool for asset selection
Assuming the validity of the SIM, the following ratio can be
considered as a decisive criterion:
ri − rf
βi
The subsequent ranking of the ratio results determines the
asset’s suitability for portfolio creation
Single index model - SIM Cut-off ratio Example of application
Application of the decision criterion
If a certain asset with ri −rf
βi
is included in the portfolio, all
available assets with higher ratio values will be part of the
portfolio
Single index model - SIM Cut-off ratio Example of application
Application of the decision criterion
If a certain asset with ri −rf
βi
is included in the portfolio, all
available assets with higher ratio values will be part of the
portfolio
If a certain asset with ri −rf
βi
is not included in the portfolio, all
other assets with lower ratio values will not be part of the
portfolio
Single index model - SIM Cut-off ratio Example of application
Portfolio creation - Short selling ban
It is necessary to determine the border asset that will be the
last to be included in the portfolio
Single index model - SIM Cut-off ratio Example of application
Portfolio creation - Short selling ban
It is necessary to determine the border asset that will be the
last to be included in the portfolio C∗
Single index model - SIM Cut-off ratio Example of application
Portfolio creation - Short selling ban
It is necessary to determine the border asset that will be the
last to be included in the portfolio C∗
Making a selection of available assets:
Single index model - SIM Cut-off ratio Example of application
Portfolio creation - Short selling ban
It is necessary to determine the border asset that will be the
last to be included in the portfolio C∗
Making a selection of available assets:
Included in the portfolio
Single index model - SIM Cut-off ratio Example of application
Portfolio creation - Short selling ban
It is necessary to determine the border asset that will be the
last to be included in the portfolio C∗
Making a selection of available assets:
Included in the portfolio
Not included in the portfolio
Single index model - SIM Cut-off ratio Example of application
Procedure for choosing the optimal portfolio
1 Calculate the ratios . . .
Single index model - SIM Cut-off ratio Example of application
Procedure for choosing the optimal portfolio
1 Calculate the ratios . . . ri −rf
βi
Single index model - SIM Cut-off ratio Example of application
Procedure for choosing the optimal portfolio
1 Calculate the ratios . . . ri −rf
βi
2 Make raking of the ratios
Single index model - SIM Cut-off ratio Example of application
Procedure for choosing the optimal portfolio
1 Calculate the ratios . . . ri −rf
βi
2 Make raking of the ratios
3 All assets that satisfy the following condition will be included
to the portfolio:
ri − rf
βi
> C∗
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
At the beginning of building the portfolio, the number of
assets that will be part of it is not known
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
At the beginning of building the portfolio, the number of
assets that will be part of it is not known
For each asset, it is necessary to calculate the parameter Ci
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
At the beginning of building the portfolio, the number of
assets that will be part of it is not known
For each asset, it is necessary to calculate the parameter Ci
Ci =
σ2
M
n
j=1
(¯rj −rf )βj
σ2
ϵj
1 + σ2
M
n
j=1(
β2
j
σ2
ϵj
)
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
At the beginning of building the portfolio, the number of
assets that will be part of it is not known
For each asset, it is necessary to calculate the parameter Ci
Ci =
σ2
M
n
j=1
(¯rj −rf )βj
σ2
ϵj
1 + σ2
M
n
j=1(
β2
j
σ2
ϵj
)
Assets are included in the portfolio if they meet the condition
ri − rf
βi
> Ci
Single index model - SIM Cut-off ratio Example of application
Determining the Cut-off
C∗ . . . is calculated from the characteristics of the asset
included in the portfolio
At the beginning of building the portfolio, the number of
assets that will be part of it is not known
For each asset, it is necessary to calculate the parameter Ci
Ci =
σ2
M
n
j=1
(¯rj −rf )βj
σ2
ϵj
1 + σ2
M
n
j=1(
β2
j
σ2
ϵj
)
Assets are included in the portfolio if they meet the condition
ri − rf
βi
> Ci
Single index model - SIM Cut-off ratio Example of application
Determining C∗
The cut-off, or the last included asset according to the previous
condition will with its Ci serve as the C∗
Single index model - SIM Cut-off ratio Example of application
Determination of weights
1 Calculate parameters Zi
Single index model - SIM Cut-off ratio Example of application
Determination of weights
1 Calculate parameters Zi
Zi =
βi
σ2
ϵi
(
¯ri − rf
βi
− C∗
)
Single index model - SIM Cut-off ratio Example of application
Determination of weights
1 Calculate parameters Zi
Zi =
βi
σ2
ϵi
(
¯ri − rf
βi
− C∗
)
2 Calculate weights
Single index model - SIM Cut-off ratio Example of application
Determination of weights
1 Calculate parameters Zi
Zi =
βi
σ2
ϵi
(
¯ri − rf
βi
− C∗
)
2 Calculate weights
wi =
Zi
n
i=1 Zi
Single index model - SIM Cut-off ratio Example of application
Cut-off also for portfolio with allowed short selling
The whole procedure could be applied also for shorted positions
Single index model - SIM Cut-off ratio Example of application
Cut-off also for portfolio with allowed short selling
The whole procedure could be applied also for shorted positions
Thus:
Single index model - SIM Cut-off ratio Example of application
Cut-off also for portfolio with allowed short selling
The whole procedure could be applied also for shorted positions
Thus:
C∗
. . . Cn
Single index model - SIM Cut-off ratio Example of application
Content
1 Single index model - SIM
2 Cut-off ratio
3 Example of application