Capital requirements with defaultable securities

Abstract

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and TailValue-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.

Risk sharing for capital requirements with multidimensional security markets

“…Monetary risk measures have been widely studied, see Föllmer & Schied [23] and the references therein. As observed in Farkas et al [18,19,20] and Munari [29], there are good reasons for revisiting the original approach to risk measures of Artzner et al [5]:…”

“…(2) Even if securitisation is constrained to buying a single asset, discounting with this asset may be impossible because it is not a numéraire; c.f. Farkas et al [19]. Also, as risk is measured after discounting, the discounting procedure is implicitly assumed not to add additional risk, which is questionable in view of risk factors such as uncertain future interest rates.…”

Intrinsic Risk Measures

“…However, stochastic interest rates can be directly addressed with risk measures of the form introduced and treated in [FKMM14a] and [FKMM14b],…”

“…More recently, this approach has been re-linked to the original idea of using eligible assets with random return rate r : Ω → R >0 by P. Artzner, F. Delbaen, and P. Koch-Medina [ADKM09] and D. Konstantinides and C. Kountzakis [KK11] among others, or has even been extended to processes by M. Fritelli and G. Scandolo [FS06]. W. Farkas, P. Koch-Medina, and C. Munari in [FKMM14a] and [FKMM14b] proposed to investigate general eligible assets r : Ω → R ≥0 and acceptance sets, revealing significant shortcomings of the simplified constant approach and pointing out the close interplay between eligible assets and acceptance sets. They work with a traded asset S = (S 0 , S T ) defined by its initial unitary price S 0 ∈ R >0 and its random payoff S T : Ω → R ≥0 and replace r in Equation (1.1) by the random return S T S 0 to arrive at an extended definition…”

Model spaces for risk measures

“…U ≥ δ for some constant δ > 0. In [11,12], such securities are called non-defaultable. We will show that if the acceptance set is "nice enough", then any finite risk measure arising from it in an a priori model-free framework like L ∞ indeed implies a probabilistic model, a so-called weak reference model; see Theorem 3.3.…”