értekezés - Budapesti Corvinus Egyetem

Symbols and associations:
N - the number of iterations (i = 1 to N)
M - the number of building blocks (j = 1 to M)
k
x
i
- the simulated value of the k th building block in iteration i
x
k
=
1
N
N
∑
i=
1
x
k
i
- the estimated expected value of the k th building block
M
∑
p i
= x i
j=
1
j
- the simulated value of portfolio p in iteration i
p =
1
N
1
N
N M
l
∑pi = ∑∑x i
i= 1 N i= 1 l=
1
- the estimated expected value of portfolio p
i ∈ S(
l)
if l ≤ p < l
down
i
up
The relative size of the k th building block’s contribution to the portfolio risk in partition
S(l) is then (derived from the output data table of the N iterations):
Contr(
k
k k
( x − x )( p − p)
i=
1
( )
) =
for i ∈ S(
l)
S l
M
N
∑
N
∑∑
j= 1 i=
1
i
j j
( x − x )( p − p)
i
i
i
This solution can be corresponded to the Covariance-method applied for normally
distributed variables if the risk contribution of the k th building block is calculated for the
entire range of the portfolio’s distribution (no partition applied, showing the building
block’s average contribution to risk):
p i
− p =
=
M
∑
l=
1
x
l
i
1
−
N
N
M
∑∑
i= 1 l=
1
x
l
i
=
M
∑
l=
1
x
l
i
−
M
N
∑∑
l= 1 i=
1
1
N
x
l
i
=
=
M
∑
l=
1
M
∑
l
x i
− x
l=
1
M
l l
∑( x i
− x )
l=
1
l
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