From Principle-Based Risk Management to Solvency ... - Scor

and that the mismatches Mi decompose into basket mismatches accordingly,
n�
Mi = M b i ,
b=1
with M b i denoting the mismatch from the basket Lb i .
Depending on the difference in speed of the reduction of uncertainty over
time for the different baskets, the relative weight of a given basket L b i within
Li will change as i =1, 2 ... increases, in the sense that its contribution to
the mismatch shortfall
SCR(i) =ESα[Mi]
will change with i.
For instance, a short-tail basket might have a high relative weight for
small i but will quickly be practically run-off, so that its weight will soon
reduce to zero. On the other hand, the basket L b i
with the longest tail will
eventually be the only remaining contributor to the mismatch, so that for
large enough i,
ESα[Mi] =ESα[M b i ].
Clearly, for any b, the basket “stand-alone” shortfall ESα[M b i ] provides an
upper bound for the contribution of the basket b to the shortfall ESα[Mi].
In case the ORP selected is the expected cash-flow-replicating-portfolio,
decompose into
given by
the mismatches Mi given by (2.22) for the portfolios Lb i
basket mismatches M b i
M b i = E[R b 0 |Fi+1] − E[R b 0 |Fi],
where Rb i denotes the discounted outstanding reserves for basket b at time i
defined in (2.10).
In order to calculate SCR(i) =ESα[Mi] from the distributions of the
basket mismatches ESα[Mi], assumptions on the dependencies of the random
variables M b i have to be made. These dependencies might depend on
i. Clearly,
ESα[Mi] ≤ �
L b i
b
ESα[M b i ].
We propose to measure the speed of reduction of uncertainty for basket
for i =1, 2 ... by the ratio
ξ (i)
b := ESα[M b i ]
ESα[M b 1 ].
(2.23)
Denote by 0