Single index model
Cut-o ratio
Example of calculation
Portfolio Theory
Lecture 9
Lud¥k Benada
Department of Finance - 402, benada.esf@gmail.com
April 25, 2016
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Structure
1 Single index model
2 Cut-o ratio
3 Example of calculation
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Single index model
There is empirical evidence: ↑ M ⇒↑ S
Therefore the (excess) return of a security is represented in
relation to the market:
ri = ai +bi ∗rM
The return of a security consists of two parts:
Dependent on the market
Independent on the market
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
The indipendent component of the model
The model could be splited into:
Estimate
Ramdom error
ri = αi +βi ∗rM +εi
The return of the market and the error are random variable
⇒ µ,σ2
Model must garantee:
cov(εi ,rm) = 0
cov(εi ,εj ) = 0
σi,j = βi ∗βj σ2
M
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
An optimal portfolio in SIM
Suppose that the SIM is the best method of predicting a
covariace structure of returns
For creating a portfolio basket it will be useful to have a tool
to select assets
If SIM holds, then the decision making criteria:
¯ri −rf
βi
Ranking expresses favourableness of every assets included into
the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
An optimal portfolio in SIM
Suppose that the SIM is the best method of predicting a
covariace structure of returns
For creating a portfolio basket it will be useful to have a tool
to select assets
If SIM holds, then the decision making criteria:
¯ri −rf
βi
Ranking expresses favourableness of every assets included into
the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
An optimal portfolio in SIM
Suppose that the SIM is the best method of predicting a
covariace structure of returns
For creating a portfolio basket it will be useful to have a tool
to select assets
If SIM holds, then the decision making criteria:
¯ri −rf
βi
Ranking expresses favourableness of every assets included into
the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
An optimal portfolio in SIM
Suppose that the SIM is the best method of predicting a
covariace structure of returns
For creating a portfolio basket it will be useful to have a tool
to select assets
If SIM holds, then the decision making criteria:
¯ri −rf
βi
Ranking expresses favourableness of every assets included into
the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
An optimal portfolio in SIM
Suppose that the SIM is the best method of predicting a
covariace structure of returns
For creating a portfolio basket it will be useful to have a tool
to select assets
If SIM holds, then the decision making criteria:
¯ri −rf
βi
Ranking expresses favourableness of every assets included into
the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Implication of decision criteria
If a security with its ratio is in the portfolio included, then all
securities with higher ratio should be included as well
If a security with its ratio is not in the portfolio, then all
securities with lower ratio must be excluded to the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Implication of decision criteria
If a security with its ratio is in the portfolio included, then all
securities with higher ratio should be included as well
If a security with its ratio is not in the portfolio, then all
securities with lower ratio must be excluded to the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Portfolio selection with ban on short sell
It is necessary to establish the threshold C*
Subsequently selection is done:
Securities to the portfolio
Securities out of the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Portfolio selection with ban on short sell
It is necessary to establish the threshold C*
Subsequently selection is done:
Securities to the portfolio
Securities out of the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Portfolio selection with ban on short sell
It is necessary to establish the threshold C*
Subsequently selection is done:
Securities to the portfolio
Securities out of the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Portfolio selection with ban on short sell
It is necessary to establish the threshold C*
Subsequently selection is done:
Securities to the portfolio
Securities out of the portfolio
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Procedure by the selection
Ranking of every security by ¯ri −rf
βi
Include securities with:
¯ri −rf
βi
> C*
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Procedure by the selection
Ranking of every security by ¯ri −rf
βi
Include securities with:
¯ri −rf
βi
> C*
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Procedure by the selection
Ranking of every security by ¯ri −rf
βi
Include securities with:
¯ri −rf
βi
> C*
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Determining of cut-o
Ci =
σ2
M ∑N
i=1
(¯ri −rf )∗βi
σ2εi
1+σ2
M ∗∑N
i=1
β2
i
σ
ε2
i
Securities are included to the portfolio if:
¯ri −rf
βi
> Ci
C* corresponds to the last securities holding this condition
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Determining of cut-o
Ci =
σ2
M ∑N
i=1
(¯ri −rf )∗βi
σ2εi
1+σ2
M ∗∑N
i=1
β2
i
σ
ε2
i
Securities are included to the portfolio if:
¯ri −rf
βi
> Ci
C* corresponds to the last securities holding this condition
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Determining of cut-o
Ci =
σ2
M ∑N
i=1
(¯ri −rf )∗βi
σ2εi
1+σ2
M ∗∑N
i=1
β2
i
σ
ε2
i
Securities are included to the portfolio if:
¯ri −rf
βi
> Ci
C* corresponds to the last securities holding this condition
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Determining of cut-o
Ci =
σ2
M ∑N
i=1
(¯ri −rf )∗βi
σ2εi
1+σ2
M ∗∑N
i=1
β2
i
σ
ε2
i
Securities are included to the portfolio if:
¯ri −rf
βi
> Ci
C* corresponds to the last securities holding this condition
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Weights in portfolio
wi = Zi
∑N
i=1 Zi
Zi = βi
σ2
εi
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Weights in portfolio
wi = Zi
∑N
i=1 Zi
Zi = βi
σ2
εi
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Portfolio selection if SS is allowed
In this case the C* ...Cn
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Short sell is not allowed
...
Lud¥k Benada MPF_APOT
Single index model
Cut-o ratio
Example of calculation
Short sell is allowed
...
Lud¥k Benada MPF_APOT