Other portfolio creation techniques Sharpe ratio Tangency portfolio
Tangency portfolio
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Content
1 Other portfolio creation techniques
2 Sharpe ratio
3 Tangency portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Content
1 Other portfolio creation techniques
2 Sharpe ratio
3 Tangency portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Short sell is allowed, but any rf is not available
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Short sell is allowed, but any rf is not available
Mean-variance portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Short sell is allowed, but any rf is not available
Mean-variance portfolio
Short sell is banned and there exists a rf
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Short sell is allowed, but any rf is not available
Mean-variance portfolio
Short sell is banned and there exists a rf
Cut-off portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Portfolio creation considering a risk-free asset
Four possible scenarios for creating a portfolio:
Short sell is allowed and there exists a rf
Tangency portfolio
Short sell is allowed, but any rf is not available
Mean-variance portfolio
Short sell is banned and there exists a rf
Cut-off portfolio
Short sell is banned and a rf is not available
Other portfolio creation techniques Sharpe ratio Tangency portfolio
The shape of efficient frontier by considering a rf
Risk-free asset is changing the shape of affordable and
efficient portfolios
Other portfolio creation techniques Sharpe ratio Tangency portfolio
The shape of efficient frontier by considering a rf
Risk-free asset is changing the shape of affordable and
efficient portfolios
The incorporation of rf is consistent with the preferences of a
rational investor
Other portfolio creation techniques Sharpe ratio Tangency portfolio
The shape of efficient frontier by considering a rf
Risk-free asset is changing the shape of affordable and
efficient portfolios
The incorporation of rf is consistent with the preferences of a
rational investor
Lending vs. borrowing and the efficient frontier . . .
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Content
1 Other portfolio creation techniques
2 Sharpe ratio
3 Tangency portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Sharpe ratio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Sharpe ratio
Reward to variability ratio:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Sharpe ratio
Reward to variability ratio:
SR =
Rp − rf
σp
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Sharpe ratio
Reward to variability ratio:
SR =
Rp − rf
σp
The relation between SR and efficient frontier . . .
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Content
1 Other portfolio creation techniques
2 Sharpe ratio
3 Tangency portfolio
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Tangency portfolio or max. SR portfolio
Objective function:
SR(w) =
wT µ − rf 1
wT w
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Tangency portfolio or max. SR portfolio
Objective function:
SR(w) =
wT µ − rf 1
wT w
Optimization problem:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Tangency portfolio or max. SR portfolio
Objective function:
SR(w) =
wT µ − rf 1
wT w
Optimization problem:
max SR(w) with respect to w
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
w =
−1
(µ − rf 1)
1T −1
(µ − rf 1)
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
w =
−1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio return:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
w =
−1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio return:
Rp = rf +
(µ − rf 1)T −1
(µ − rf 1)
1T −1
(µ − rf 1)
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
w =
−1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio return:
Rp = rf +
(µ − rf 1)T −1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio variance:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Calculation procedure
Structure of the portfolio:
w =
−1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio return:
Rp = rf +
(µ − rf 1)T −1
(µ − rf 1)
1T −1
(µ − rf 1)
Portfolio variance:
σ2
p =
(µ − rf 1)T −1
(µ − rf 1)
(1T −1
(µ − rf 1))2
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Finding the extremum of a function:
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Finding the extremum of a function:
f (⃗w) =
Rp − rf
σp
→ max
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Finding the extremum of a function:
f (⃗w) =
Rp − rf
σp
→ max
under weight restriction:
n
i=1
wi = 1
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Finding the extremum of a function:
f (⃗w) =
Rp − rf
σp
→ max
under weight restriction:
n
i=1
wi = 1
if we know:
Rp − rf =
n
i=1
wi (ri − rf )
Other portfolio creation techniques Sharpe ratio Tangency portfolio
More intuitive calculation approach
Finding the extremum of a function:
f (⃗w) =
Rp − rf
σp
→ max
under weight restriction:
n
i=1
wi = 1
if we know:
Rp − rf =
n
i=1
wi (ri − rf )
σp =
n
i=1
w2
i ∗ σ2
i +
n
i=1
n
j=1|i̸=j
wi ∗ wj ∗ σi,j
Other portfolio creation techniques Sharpe ratio Tangency portfolio
Getting the weights
Let´s assume:
λ =
Rp − rf
σ2
p
and further, substitute:
Zk = λ ∗ wk
then we can write:
ri −rf = Z1σi,1+Z2σi,2+Z3σi,3+· · ·+Zi σ2
i +· · ·+Zn−1σi,n−1+Znσi,n
and finally:
wk =
Zk
n
k=1 Zk