Short Selling Portfolio optimization
Optimization technique in Portfolio creation
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
Short Selling Portfolio optimization
Content
1 Short Selling
2 Portfolio optimization
Short Selling Portfolio optimization
Content
1 Short Selling
2 Portfolio optimization
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
The procedure by short selling:
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
The procedure by short selling:
The investor borrows a security from the owner, which he
then sells when building the portfolio
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
The procedure by short selling:
The investor borrows a security from the owner, which he
then sells when building the portfolio
The Ownership remains with the original owner, to whom the
security will be returned in the future
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
The procedure by short selling:
The investor borrows a security from the owner, which he
then sells when building the portfolio
The Ownership remains with the original owner, to whom the
security will be returned in the future
The original owner is entitled to all cash flows from the
security that occur during the time of borrowing the security
Short Selling Portfolio optimization
. . . going Short
Short selling will be applied by an investor who assumes that the
future price of the asset will decrease, i.e. Pt+k < Pt
It is assumed, there exists an opportunity for short selling and it is
not banned . . .
The procedure by short selling:
The investor borrows a security from the owner, which he
then sells when building the portfolio
The Ownership remains with the original owner, to whom the
security will be returned in the future
The original owner is entitled to all cash flows from the
security that occur during the time of borrowing the security
An illustrative example with a long and a short position . . .
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short selling also increase the set of available portfolios
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short selling also increase the set of available portfolios
Short selling is possible/suitable to use even if:
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short selling also increase the set of available portfolios
Short selling is possible/suitable to use even if:
¯ri > 0
↓
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short selling also increase the set of available portfolios
Short selling is possible/suitable to use even if:
¯ri > 0
↓
. . . , but loss of the diversification effect
Short Selling Portfolio optimization
The effect of short selling on the available set of portfolios
Short selling expands the allowable set of portfolios
Short selling also increase the set of available portfolios
Short selling is possible/suitable to use even if:
¯ri > 0
↓
. . . , but loss of the diversification effect
Graphical representation of the portfolio’s effective frontier when
applying short selling . . .
Short Selling Portfolio optimization
Content
1 Short Selling
2 Portfolio optimization
Short Selling Portfolio optimization
Portfolio optimization - calculation procedure
The task of calculation is to find the extremum of a function
Short Selling Portfolio optimization
Portfolio optimization - calculation procedure
The task of calculation is to find the extremum of a function
Two approaches applied to the efficient frontier are possible:
Short Selling Portfolio optimization
Portfolio optimization - calculation procedure
The task of calculation is to find the extremum of a function
Two approaches applied to the efficient frontier are possible:
Maximizing of E(Rp) =
n
i=1 wi ∗ ¯ri
Short Selling Portfolio optimization
Portfolio optimization - calculation procedure
The task of calculation is to find the extremum of a function
Two approaches applied to the efficient frontier are possible:
Maximizing of E(Rp) =
n
i=1 wi ∗ ¯ri
Risk minimization of the expected portfolio return:
Short Selling Portfolio optimization
Portfolio optimization - calculation procedure
The task of calculation is to find the extremum of a function
Two approaches applied to the efficient frontier are possible:
Maximizing of E(Rp) =
n
i=1 wi ∗ ¯ri
Risk minimization of the expected portfolio return:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
Short Selling Portfolio optimization
Finding a portfolio with minimal risk
Two approaches can be applied to minimize the risk of portfolio:
1 Finding the absolute/Global minimum risk
Short Selling Portfolio optimization
Finding a portfolio with minimal risk
Two approaches can be applied to minimize the risk of portfolio:
1 Finding the absolute/Global minimum risk
2 Finding the minimum risk at the desired return
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
The objective function:
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
The objective function:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
The objective function:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j → min
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
The objective function:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j → min
The constraint:
Short Selling Portfolio optimization
Minimum variance portfolio
Solving the optimization problem in order to obtain weights
The proportion of individual asset (weights) for the optimal
portfolio are the solution to the minimization problem of a
quadratic objective function with one linear constraint.
The objective function:
σ2
p =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j → min
The constraint:
n
i=1
wi = 1
Short Selling Portfolio optimization
Finding the extreme values
f (⃗w) → min
Short Selling Portfolio optimization
Finding the extreme values
f (⃗w) → min
Lagrangian function: L(⃗y) = L(⃗w,⃗λ)
Short Selling Portfolio optimization
Finding the extreme values
f (⃗w) → min
Lagrangian function: L(⃗y) = L(⃗w,⃗λ)
We will use Lagrange multipliers to find the extrema of the
function:
Short Selling Portfolio optimization
Finding the extreme values
f (⃗w) → min
Lagrangian function: L(⃗y) = L(⃗w,⃗λ)
We will use Lagrange multipliers to find the extrema of the
function:
∂L(⃗y)
∂yi
for i = 1, 2, . . . , n
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Set the first n equations to be equal 0
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Set the first n equations to be equal 0
2C ⃗W + λ1⃗e = 0
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Set the first n equations to be equal 0
2C ⃗W + λ1⃗e = 0
For the last equation transfer 1 to the right side
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Set the first n equations to be equal 0
2C ⃗W + λ1⃗e = 0
For the last equation transfer 1 to the right side
⃗wT
⃗e = 1
Short Selling Portfolio optimization
Lagrangian function - calculation procedure
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1)
Applying partial derivation with respect to individual variables:
→ n + 1 eq.
Set the first n equations to be equal 0
2C ⃗W + λ1⃗e = 0
For the last equation transfer 1 to the right side
⃗wT
⃗e = 1
Apply matrix calculation . . .
Short Selling Portfolio optimization
Minimization of the risk at desired portfolio return
The constraints:
Short Selling Portfolio optimization
Minimization of the risk at desired portfolio return
The constraints:
n
i=1
wi = 1
Short Selling Portfolio optimization
Minimization of the risk at desired portfolio return
The constraints:
n
i=1
wi = 1
E(Rp) =
n
i=1
wi ∗ ¯ri
Short Selling Portfolio optimization
Minimization of the risk at desired portfolio return
The constraints:
n
i=1
wi = 1
E(Rp) =
n
i=1
wi ∗ ¯ri
Lagrangian function:
Short Selling Portfolio optimization
Minimization of the risk at desired portfolio return
The constraints:
n
i=1
wi = 1
E(Rp) =
n
i=1
wi ∗ ¯ri
Lagrangian function:
L(⃗y) = σ2
p(⃗w) + λ1(
n
i=1
wi − 1) + λ2(
n
i=1
wi ∗ ¯ri − E(Rp))