Capital Adequacy Tests Research Articles

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.

Scalarized utility-based multi-asset risk measures

Financial institutions have to satisfy capital adequacy tests required, e.g., by the Basel Accords for banks or by Solvency II for insurers. If the financial situation of an institution is tight, then it can happen that no reallocation of the initial endowment would pass the capital adequacy test. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the scalarized utility-based multi-asset (SUBMA) risk measure which optimizes the hedging costs and the expected utility of an agent simultaneously subject to the capital adequacy test. We find that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Under further assumptions on the utility function we obtain existence and uniqueness results for the optimal hedging strategies. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark capital adequacy tests.

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 risky position acceptable. Recently, the alternative methodology of intrinsic risk measures was introduced in the literature. These ask for the minimal proportion of the risky 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 risky position by external capital as well as reallocating liquid 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 risk only under an additional but economically motivated assumption on the risky 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.

Risk sharing for capital requirements with multidimensional security markets

We consider the risk sharing problem for capital requirements induced by capital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, namely polyhedral constraints and distribution-based constraints. We prove existence of optimal risk allocations and equilibria within these frameworks and elaborate on their robustness.

Existence, uniqueness, and stability of optimal payoffs of eligible assets

AbstractIn a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of prespecified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to be able to do so at the lowest possible cost and to identify the corresponding portfolios, or, equivalently, their payoffs. We study the existence and uniqueness of such optimal payoffs as well as their behavior under a perturbation or an approximation of the underlying capital position. This behavior is naturally linked to the continuity properties of the set‐valued map that associates to each capital position the corresponding set of optimal eligible payoffs. Upper continuity can be ensured under fairly natural assumptions. Lower continuity is typically less easy to establish. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower continuity can often be established for eligible payoffs that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.

Surplus-Invariant, Law-Invariant, and Positively Homogeneous Acceptance Sets Must Be Induced by Value-at-Risk

The regulator is interested in proposing a capital adequacy test by specifying an acceptance set for firms' capital positions at the end of a given period. This set needs to be surplus-invariant, i.e., not to depend on the surplus of firms' shareholders, because the test means to protect firms' liability holders. We prove that any surplus-invariant, law-invariant, and positively homogeneous acceptance set must be the set of capital positions whose value-at-risk at a given level is less than zero. The result still holds if we replace positive homogeneity with numeraire-invariance, a property stipulating that whether a firm passes the test should not depend on the currency used to denominate its assets.

Surplus-Invariant, Law-Invariant, and Conic Acceptance Sets Must Be the Sets Induced by Value at Risk

The regulator is interested in proposing a capital adequacy test by specifying an acceptance set for firms’ capital positions at the end of a given period. This set needs to be surplus invariant; i.e., not to depend on the surplus of firms’ shareholders, because the test means to protect firms’ liability holders. We prove that any surplus-invariant, law-invariant, and conic acceptance set must be the set of capital positions whose value at risk at a given level is less than zero. The result still holds if we replace conicity with numéraire invariance, a property stipulating that whether a firm passes the test should not depend on the currency used to denominate its assets.

Diversification, protection of liability holders and regulatory arbitrage

Any solvency regime for financial institutions should be aligned with the fundamental objectives of regulation: protecting liability holders and securing the stability of the financial system. The first objective leads to consider surplus-invariant capital adequacy tests, i.e. tests that do not depend on the surplus of a financial institution. We provide a complete characterization of closed, convex, surplus-invariant capital adequacy tests that highlights an inherent tension between surplus-invariance and the desire to give credit for diversification. The second objective leads to requiring consistency of capital adequacy tests across jurisdictions. Of particular importance in this respect are capital adequacy tests that remain invariant under a change of numeraire. We establish an intimate link between surplus- and numeraire invariant tests.

Capital adequacy tests and limited liability of financial institutions

The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper is to investigate the class of surplus-invariant acceptance sets. We argue that surplus invariance is a reasonable requirement from a regulatory perspective, since the corresponding capital adequacy tests do not depend on the surplus of a financial institution, which benefits exclusively its shareholders, but only on the default profile, which affects its liability holders. We provide a detailed analysis of surplus-invariant acceptance sets and their associated risk measures and we discuss the link with loss-based and excess-invariant risk measures, recently studied by Cont et al. (2013) and by Staum (2013), respectively.

Value-at-Risk vs. Expected Shortfall: Beware of the Aggregation Currency!

This note highlights the impact of changing the aggregation currency when testing for capital adequacy. This is exemplified using the two most prominent capital adequacy tests used in the financial industry: Value-at-Risk and Expected Shortfall. The test consists in establishing whether the Value-at-Risk or, respectively, the Expected Shortfall of the capital position is negative or not. We show that the sign of the Expected Shortfall of a capital position -- and, hence, the corresponding capital adequacy test -- critically depends on the aggregation currency. This is in contrast to Value-at-Risk, whose sign is independent of the chosen aggregation currency. This implies that due caution is needed when using Expected Shortfall to measure capital adequacy in a multi-currency environment.

Surplus-Invariant Capital Adequacy Tests and Their Risk Measures

The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper is to investigate, in the context of bounded financial positions, the class of surplus-invariant acceptance sets. These are characterized by the fact that acceptability does not depend on the positive part, or surplus, of a capital position. We argue that surplus invariance is a reasonable requirement from a regulatory perspective, because it focuses on the interests of liability holders of a financial institution. We provide a dual characterization of surplus-invariant, convex acceptance sets, and show that the combination of surplus invariance and coherence leads to a narrow range of capital adequacy tests, essentially limited to scenario-based tests. Finally, we emphasize the advantages of dealing with surplus-invariant acceptance sets as the primary object rather than directly with risk measures, such as loss-based and excess-invariant risk measures, which have been recently studied by Cont, Deguest & He and by Staum, respectively.

Capital Adequacy Tests and Limited Liability of Fiancial Institutions

Capital Adequacy Tests and Limited Liability of Fiancial Institutions

Risk Management and Balance Sheet Volatility During Turbulent Times

Systemic risk is the protagonist of the recent financial crisis. This paper proposes a definition and a propagation mechanism for systemic risk. Risk management has a direct linkage with capital management, when addressing the question that the risk handled by a financial institution is compatible with the amount of equity available. This paper proposes a risk management of liquid market variables, which compose the assets of a bank, based on the statistical tool of PCA. The principal component analysis will define the PCR, or Principal Components of Risk. Such definition of Risk will be adopted to test if the risk represented by PCR is explanatory of the movements of equity and/or debt for the banks included in the index Itraxx financial senior: the results of these regressions will be compared with a formal Capital Adequacy test in order to assess the financial soundness of the main financial European institutions.

Regulation, Risk and Performance of U.K. Clearing Banks 1965-75

Section II identifies three distinct phases of regulation and puts forward four different hypotheses regarding the effects of these regulatory changes on bank performance. The risks associated with banking are then discussed and some conventional capital adequacy tests are performed. Section III establishes an alternative framework of analysis, based on Jensen's [I 7] formulation of the capital asset pricing model. Using this approach the validity of each hypothesis established in section II is examined. Finally, some conclusions are reached as to the efficacy of the different types of regulatory policy.