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

is not exercised, and that fi(ORPi) denotes the pay out from the ORP in
year i.
For balance sheets which are not economic in the sense above, it might
not be possible to express the additional restriction in terms of values and
ORPs. At any rate, the restriction introduces additional frictional costs.
There is an obvious instrument to provide the one-year max covers: the
shareholders equity. The equity corresponds to capital put up by outside
investors, which in an economic balance sheet corresponds to the difference
between the market-consistent value of the assets less the market-consistent
value of the liabilities.
In this situation, the composition of the asset side is per se arbitrary,
and so the assets do not necessarily replicate the liabilities in the sense of
the ORP. This results in an additional cash flow mismatch between assets
and liabilities. However, for simplicity we assume in the following that the
assets correspond to the required super-replicating portfolio.
Valuation of the basis risk, which we reformulated as valuation of the
required coupon for the one-year max covers, is then further translated into
the cost of capital margin CoCM approach. In this approach, one needs to
determine
� The required shareholder capital ˜ Ki for all years i
� The cost of capital, i.e. the required return on the capital ηi
The required return on capital ηi is a proportionality factor applied to the
required capital ˜ Ki. Per se, the returns can be different for different years.
However, in practice, one usually assumes that the required return is prescribed
externally and is constant over the years equal to some η. The
required return on capital has to be understood as an expected value: The
expected value of the actual return has to be equal to the required return.
The required capital corresponds to the amounts Ki for the one-year max
covers. Usually, to calculate the amounts Ki, the supremum over a set Ω ⋆ t
of states of the world is expressed by choosing an appropriate, usually lawinvariant
risk measure. For instance, selecting the same law-invariant risk
measure ρ for any year i, the formula (2.7) for the amounts Ki determined
at time t becomes
Ki = ρ(Mi) (2.9)
where the distribution of the mismatch Mi given by (2.8) is conditional on
the information available at time t ≤ i.
Since the set Ω ⋆ t reflects the states of the world where the default option
is not exercised, the selection of the risk measure is directly connected to
the (value of the) default option. For instance, selecting as risk measure the
expected shortfall at a less conservative level α = 10% instead of at α =1%
21