Capital requirements and claims recovery: A new perspective on solvency regulation

• Published date: 01 June 2023
• Author: Cosimo Munari,Stefan Weber,Lutz Wilhelmy
• Date: 01 June 2023
• DOI: http://doi.org/10.1111/jori.12405

Received: 8 November 2021

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Revised: 23 August 2022

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Accepted: 26 August 2022

DOI: 10.1111/jori.12405

ORIGINAL ARTICLE

Capital requirements and claims recovery:

A new perspective on solvency regulation

Cosimo Munari

1

|Stefan Weber

2

|Lutz Wilhelmy

3

1

Department of Banking and Finance,

Center for Finance and Insurance and

Swiss Finance Institute, University of

Zurich, Zurich, Switzerland

2

House of Insurance & Institute of

Actuarial and Financial Mathematics,

Leibniz Universität Hannover,

Hannover, Germany

3

Group Risk Management, Swiss Re

Management Ltd, Zurich, Switzerland

Correspondence

Cosimo Munari, Department of Banking

and Finance, Center for Finance and

Insurance and Swiss Finance Institute,

University of Zurich, Plattenstrasse 14,

8032 Zurich, Switzerland.

Email: cosimo.munari@bf.uzh.ch

Abstract

Protection of creditors is a key objective of financial

regulation. Where the protection needs are high, that

is, in banking and insurance, regulatory solvency

requirements are an instrument to prevent that

creditors incur losses on their claims. The current

regulatory requirements based on value at risk (V@R)

and average value at risk (AV@R) limit the probability

of default of financial institutions, but they fail to

control the size of recovery on creditors' claims in the

case of default. We resolve this failure by developing a

novel risk measure, recovery V@R. Our conceptual

approach is flexible and allows the construction of

general recovery risk measures for various risk

management purposes. We provide detailed case

studies and applications. We show that recovery risk

measures can be used for performance‐based manage-

ment of business divisions of firms and discuss how to

calibrate recovery risk measures to historical regulatory

standards. Finally, we analyze how recovery risk

measures react to the joint distributions of assets and

liabilities on firms' balance sheets and compare the

corresponding capital requirements with the current

regulatory benchmarks based on V@R and AV@R.

J Risk Insur. 2023;90:329–380. wileyonlinelibrary.com/journal/JORI

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This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

© 2022 The Authors. Journal of Risk and Insurance published by Wiley Periodicals LLC on behalf of American Risk and Insurance

Association.

KEYWORDS

capital requirements, recovery on liabilities, risk measures,

solvency regulation

1|INTRODUCTION

Banks and insurance companies are subject to a variety of regulatory constraints. A key

objective of financial regulation is the appropriate protection of creditors, for example,

depositors, policyholders, and other counterparties. Corporate governance, reporting, and

transparency are cornerstones of regulatory schemes, but equally important is capital

regulation. Financial companies are required to respect solvency capital requirements that

define a minimum for their current net asset value. Firms that fail to meet these requirements

are subject to supervisory interventions.

The computation of solvency capital requirements is often based on some prespecified

notion of acceptable default risk. Banks and insurance companies must hold enough capital to

meet their obligations in a sufficient number of future economic scenarios. Regulators typically

focus on quantities such as the change of net asset value over a specific time horizon—for

example, 1 year—and require that a suitable risk measure applied to such quantities is below

the current level of available capital. The risk measure implicitly defines a notion of acceptable

default risk. Different risk measures are applied in practice.

The standard example are solvency capital requirements defined in terms of value at risk

(V@R). In this case, a company is adequately capitalized if its default probability is lower than a

given threshold. The upcoming regulatory framework for the internationally active insurance

groups uses a V@R at the level

0

.5

%

. In Europe, insurance companies and groups are subject to

the same requirement under the Solvency II regime. V@R has been strongly criticized due to its

tail blindness and its lack of convexity—not encouraging diversification.

An alternative to V@R is the coherent risk measure average V@R, also called conditional or

tail V@R or expected shortfall. The market risk standards in Basel III, the international

regulatory framework for banks, and the Swiss Solvency Test, the Swiss regulatory framework

for insurance companies, are both based on average V@R with levels

2

.5

%

and

1

%

, respectively.

In this case, a company or portfolio is deemed adequately capitalized, if it generates profits on

average conditional on its tail distribution below the chosen level. Average value at risk

(AV@R) is sensitive to the tail, and, being convex, it does not penalize diversification. It is also

a tractable ingredient to optimization problems in the context of asset liability management and

provides an instrument for decentralized risk management, for example, limit systems within

firms.

Despite all its merits, AV@R fails—just as V@R—at one central task: It cannot control

recovery in the case of default, that is, the probabilities that creditors recover prespecified

fractions of their claims. This goal is, of course, important from a regulatory point of view.

Recovering, say, 80% instead of 0% in the case of default makes a big difference to creditors

such as depositors or policyholders. This failure is apparent when we consider V@R. By design,

the corresponding solvency tests only limit the probability of insolvency and are incapable of

imposing any stricter bound on the loss given default.

But the same failure is shared by AV@R. In spite of being sensitive to tail losses, AV@R still

leaves too many degrees of freedom to control recovery. This is because the loss given default is

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captured by way of an average loss, which is too gross to exert a fine control on the recovery

probability. An additional key deficiency is that all monetary risk measures in current solvency

regulation focus on a residual quantity, that is, the difference between assets and liabilities, that

is owned by shareholders. This quantity is insufficient to adequately capture what will happen

in the case of default.

The goal of this paper is to address the question:

How should regulators design solvency tests in order to control the recovery on

creditors' claims in the case of default?

Our contributions are the following:

I. We demonstrate that classical monetary risk measures such as V@R and AV@R are

unable to control recovery on creditors' claims in the case of default. In fact, we argue that,

to capture this important aspect of tail risk, one has to abandon solvency tests based on the

net asset value only and consider more articulated solvency tests based on both the net

asset value and the firm's liabilities. This is discussed in Section 2.

II.We develop a novel risk measure, recovery value at risk (RecV@R), to successfully address

the goal of controlling recovery risk. We demonstrate that RecV@R can serve as the basis

of solvency tests and discuss its operational interpretation as a capital requirement rule.

This new risk measure can be applied to both external and internal risk management and

helps to quantify how far standard regulatory risk measures are from controlling recovery

risk, thereby improving our understanding of (the limitations of) these standard risk

measures. This is discussed in Section 3.

III.Our conceptual approach is flexible and leads to the construction of general recovery risk

measures that include recovery AV@R. This allows to integrate the ability to control the

recovery on creditors' claims with other desirable properties such as convexity or

subadditivity. Convexity facilitates applications to optimization problems such as portfolio

choice under risk constraints. Subadditivity provides incentives for the diversification of

positions and enables limit systems within firms for decentralized risk management. This

is discussed in Section 4.1 and in Section 4.2.

IV. We demonstrate how recovery risk measures can be applied to performance‐based

management of business divisions of firms. We define and investigate the appropriate

notion of RoRaC‐compatibility. This is discussed in Section 5.2.

V. We discuss a possible strategy to calibrate recovery risk measures consistently with

existing regulatory standards, following a common methodology chosen by regulators in

the context of classical risk measures. We refer to Section 5.3.

VI.To better understand the behavior of recovery risk measures we illustrate how they react

to changes of the joint distribution of assets and liabilities on the firm's balance sheet. We

focus on two characteristics—marginal distributions and stochastic dependence—and

compare risk measurements to the classical solvency benchmarks, that is, V@R and

AV@R. This is discussed in Section 5.4.

The paper is structured as follows. Section 2reviews solvency regulation based on V@R and

AV@R with a focus on recovery risk. In Section 3we introduce the new risk measure recovery

V@R and discuss its main properties. In the parallel Section 4.1 we introduce the convex risk

measure recovery AV@R. Section 5focuses on a number of related applications including risk

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