Calculations by available probabilities Investment portfolio
Probabilistic Scenario & Basic of Investment
Portfolio
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
Calculations by available probabilities Investment portfolio
Content
1 Calculations by available probabilities
2 Investment portfolio
Calculations by available probabilities Investment portfolio
Content
1 Calculations by available probabilities
2 Investment portfolio
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
The dividend payments are not considered
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
The dividend payments are not considered
The potential market prices in the future are estimated
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
The dividend payments are not considered
The potential market prices in the future are estimated
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
The dividend payments are not considered
The potential market prices in the future are estimated
Price scenarios (Pt,i)
Calculations by available probabilities Investment portfolio
Estimation of the probability
The current price of the asset under consideration is known
The dividend payments are not considered
The potential market prices in the future are estimated
Price scenarios (Pt,i)
where,
ri = ln(
Pt,i
Pt−k
)
Calculations by available probabilities Investment portfolio
Return and risk of an asset
If the future price of a considered asset is known, then could be
determined:
Expected return:
Calculations by available probabilities Investment portfolio
Return and risk of an asset
If the future price of a considered asset is known, then could be
determined:
Expected return:
E(ri ) =
n
i=1
ri ∗ pi
Calculations by available probabilities Investment portfolio
Return and risk of an asset
If the future price of a considered asset is known, then could be
determined:
Expected return:
E(ri ) =
n
i=1
ri ∗ pi
Risk of the asset:
Calculations by available probabilities Investment portfolio
Return and risk of an asset
If the future price of a considered asset is known, then could be
determined:
Expected return:
E(ri ) =
n
i=1
ri ∗ pi
Risk of the asset:
σi = (ri − E(r))2 ∗ pi
Calculations by available probabilities Investment portfolio
Calculation procedure
A larger and more plausible probability distribution for the
future prices can lead to better estimation results
Calculations by available probabilities Investment portfolio
Calculation procedure
A larger and more plausible probability distribution for the
future prices can lead to better estimation results
More probabilistic scenarios need to be condensed, so the
probability range condition would be ensured:
Calculations by available probabilities Investment portfolio
Calculation procedure
A larger and more plausible probability distribution for the
future prices can lead to better estimation results
More probabilistic scenarios need to be condensed, so the
probability range condition would be ensured:
Only one price scenario
n
i=1
pi = 100%
Calculations by available probabilities Investment portfolio
Calculation procedure
A larger and more plausible probability distribution for the
future prices can lead to better estimation results
More probabilistic scenarios need to be condensed, so the
probability range condition would be ensured:
Only one price scenario
n
i=1
pi = 100%
N prices scenarios
1
N
n
i=1
pi = 100%
Calculations by available probabilities Investment portfolio
The sequence of the calculation
Unification of prices
Calculations by available probabilities Investment portfolio
The sequence of the calculation
Unification of prices
Linking with their probabilities
Calculations by available probabilities Investment portfolio
The sequence of the calculation
Unification of prices
Linking with their probabilities
Sum and standardization of probabilities
Calculations by available probabilities Investment portfolio
The sequence of the calculation
Unification of prices
Linking with their probabilities
Sum and standardization of probabilities
Calculation of returns and their linking with probabilities
Calculations by available probabilities Investment portfolio
The sequence of the calculation
Unification of prices
Linking with their probabilities
Sum and standardization of probabilities
Calculation of returns and their linking with probabilities
Calculation of E(ri ) & σi
Calculations by available probabilities Investment portfolio
The sequence of the calculation
C o n c r e t e e x a m p l e
Calculations by available probabilities Investment portfolio
Content
1 Calculations by available probabilities
2 Investment portfolio
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Basic assumption:
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Basic assumption:
n
i=1
wi = 1
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Basic assumption:
n
i=1
wi = 1, also wi < 0
thus,
Rp =
n
i=1
wi ∗ ri
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Basic assumption:
n
i=1
wi = 1, also wi < 0
thus,
Rp =
n
i=1
wi ∗ ri
or
Calculations by available probabilities Investment portfolio
Portfolio return
Portfolio consists of n-assets
The proportion of i-th assets in the portfolio is wi
Basic assumption:
n
i=1
wi = 1, also wi < 0
thus,
Rp =
n
i=1
wi ∗ ri
or
E(Rp) =
n
i=1
wi ∗ ¯ri
Calculations by available probabilities Investment portfolio
Risk of portfolio
σp =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
Calculations by available probabilities Investment portfolio
Risk of portfolio
σp =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
thus,
σp = Varp + Covarp
Calculations by available probabilities Investment portfolio
Risk of portfolio
σp =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
thus,
σp = Varp + Covarp
σ2
p =
n
i=1
w2
i ∗ σ2
i +
n
i=1
n
j=1|i̸=j
wi ∗ wj ∗ σi,j
Calculations by available probabilities Investment portfolio
Risk of portfolio
σp =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
thus,
σp = Varp + Covarp
σ2
p =
n
i=1
w2
i ∗ σ2
i +
n
i=1
n
j=1|i̸=j
wi ∗ wj ∗ σi,j
or simply;
Calculations by available probabilities Investment portfolio
Risk of portfolio
σp =
n
i=1
n
j=1
wi ∗ wj ∗ σi,j
thus,
σp = Varp + Covarp
σ2
p =
n
i=1
w2
i ∗ σ2
i +
n
i=1
n
j=1|i̸=j
wi ∗ wj ∗ σi,j
or simply;
σp =
√
wT ∗ C ∗ w
Calculations by available probabilities Investment portfolio
Special case - Naive Portfolio
wi =
1
n
Calculations by available probabilities Investment portfolio
Special case - Naive Portfolio
wi =
1
n
σ2
p =
1
n2
n
i=1
σ2
i +
1
n2
n
i=1
n
j=1|i̸=j
σi,j
Calculations by available probabilities Investment portfolio
Special case - Naive Portfolio
wi =
1
n
σ2
p =
1
n2
n
i=1
σ2
i +
1
n2
n
i=1
n
j=1|i̸=j
σi,j
σ2
p =
1
n
¯σi
2
+ (1 −
1
n
) ¯σi,j