Combining multi-asset and intrinsic risk measures

The risk of a future payoff is commonly quantified by calculating the costs of a hedging portfolio such that the resulting position is acceptable, i.e., that it passes a capital adequacy test. A multi-asset risk measure describes the minimal external capital which has to be raised into multiple eligible assets to make a future position acceptable. Recently, the alternative methodology of intrinsic risk measures was introduced in the literature. These ask for the minimal proportion of the financial position which has to be reallocated to pass the capital adequacy test, i.e., only internal capital is used. We combine these two concepts and call this new type of risk measure a multi-asset intrinsic risk measure. It allows to secure the financial position by external capital as well as reallocating parts of the portfolio as an internal rebooking. We investigate several properties to demonstrate similarities and differences to the two aforementioned classical types of risk measures. We find that diversification reduces the capital requirement only in special situations depending on the financial positions. With the help of Sion's minimax theorem we also prove a dual representation for multi-asset intrinsic risk measures. Finally, we determine capital requirements in a model motivated by the Solvency II methodology.