Risk Measures Based on Benchmark Loss Distributions

• Published date: 01 June 2020
• Author: Valeria Bignozzi,Matteo Burzoni,Cosimo Munari
• Date: 01 June 2020
• DOI: http://doi.org/10.1111/jori.12285

©2019 The Journal of Risk and Insurance (2019).

DOI: 10.1111/jori.12285

Risk Measures Based on Benchmark Loss

Distributions

Valeria Bignozzi

Matteo Burzoni

Cosimo Munari

Abstract

We introduce a class of quantile-based risk measures that generalize Value

at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both

the frequency and the severity of losses. Under VaR a single confidence

level is assigned regardless of the size of potential losses. We allow for

a range of confidence levels that depend on the loss magnitude. The key

ingredient is a benchmark loss distribution (BLD), that is, a function that

associates to each potential loss a maximal acceptable probability of oc-

currence. The corresponding risk measure, called Loss VaR (LVaR), deter-

mines the minimal capital injection that is required to align the loss dis-

tribution of a risky position to the target BLD. By design, one has full

flexibility in the choice of the BLD profile and, therefore, in the range of

relevant quantiles. Special attention is given to piecewise constant func-

tions and to tail distributions of benchmark random losses, in which case

the acceptability condition imposed by the BLD boils down to first-order

stochastic dominance. We investigate the main theoretical properties of

LVaR with a focus on their comparison with VaR and ES and discuss ap-

plications to capital adequacy, portfolio risk management, and catastrophic

risk.

Valeria Bignozzi is at the Department of Statistics and Quantitative Methods, Univer-

sity of Milano-Bicocca. Bignozzi can be contacted via e-mail: valeria.bignozzi@unimib.it.

Matteo Burzoni is at the Department of Mathematics, ETH Zurich. Burzoni can be con-

tacted via e-mail: matteo.burzoni@math.ethz.ch. Cosimo Munari is at Center for Finance and

Insurance and Swiss Finance Institute, University of Zurich. Munari can be contacted via

e-mail: cosimo.munari@bf.uzh.ch. This research started while the second and third author

were visiting the Department of Mathematics of the University of Milan and the Depart-

ment of Statistics and Quantitative Methods of the University of Milano-Bicocca supported

by the Associazione Casse Risparmio Italiane Research Prize 2017. The second author ac-

knowledges financial support from the ETH Foundation. The authors would like to thank

Pablo Koch-Medina, Andreas Tsanakas, and Ruodu Wang for useful discussions on an ear-

lier version of this article, and two anonymous referees for their insightful comments and

suggestions. The article is Swiss Finance Institute Research Paper No. 18-48 and received a

Best Paper Award at the 10th Conference in Actuarial Science & Finance on Samos held in

May–June 2018.

1

437

437

. Vol. 87, No. 2, 437–475 (2020).

2The Journal of Risk and Insurance

Introduction

Much of the debate about capital adequacy and solvency regulation over the last years

has been dominated by Value at Risk (VaR) and Expected Shortfall (ES). The discus-

sion is, nowadays, more heated than ever since VaR and ES are currently adopted

in different regulatory regimes and their relative merits and drawbacks, which have

been mainly investigated from a theoretical perspective, can be eventually evaluated

against their concrete implementation.

Value at Risk

As is well-known, VaR was introduced as part of the market risk management tool-

box developed by JP Morgan and released under the name RiskMetrics in 1994. A

personal retrospective on the birth of VaR is provided in Guldimann (2000). The new

risk measure soon became the industry standard and was eventually recognized at a

regulatory level when the Basel Committee on Banking Supervision (BCBS) adopted

it as one of the key ingredients of the second Basel Accord in 2004, usually known as

Basel 2. The centrality of VaR was confirmed by Basel 3 in 2010–2011. The transition

to a risk-sensitive solvency framework in the insurance industry, which culminated

in the enforcement of Solvency 2 within the European Union in 2009, has assigned to

VaR a central role also in the insurance regulatory world.

To recall the definition of VaR, fix a probability space (,F,P) and let Xbe a random

variable representing a financial loss at a given future point in time (positive values

of Xare understood as losses and negative values as profits). The VaR of Xat level

p∈(0, 1] is defined by

VaR p(X):=inf{m∈R;P(X≤m)≥p}.

Being equal to the (left) p-quantile of X, the quantity VaRp(X) can be naturally in-

terpreted as the best outcome of Xout of the worst 100(1 −p) percent of outcomes.

Equivalently,VaRp(X) can be interpreted as the minimal amount of capital that has to

be raised and held in cash in order to absorb future losses in at least 100ppercent of

cases. In this sense, one often says that VaRp(X) quantifies the risk of the position X

as the cost of making the risk profile of X“acceptable” in the following sense:

Xis acceptable ⇐⇒ VaR p(X)≤0⇐⇒ P(X>0) ≤1−p.

In spite of being the most widely used risk measure in practice, VaRhas been strongly

criticized in the literature for two main reasons. On the one hand, VaR depends only

on the frequency of losses and not on their severity and, hence, may fail to capture tail

risk in an appropriate way. On the other hand, VaR is neither convex nor subadditive

and thus fails to be a coherent risk measure in the sense of Artzner et al. (1999). In

particular,this implies that VaR may penalize diversification and risk decentralization

by not allowing to control the risk of an aggregate position in terms of the risk of its

individual components.

Expected Shortfall

Among the coherent risk measures that have been proposed as alternatives to VaR,

the most successful one is certainly ES as defined and first comprehensively stud-

2The Journal of Risk and Insurance

438

Risk Measures Based on Benchmark Loss Distributions 3

ied by Acerbi and Tasche (2002). After some initial skepticism, ES has eventually

encountered the favor of many practitioners and obtained the seal of approval also

by regulators. In the insurance world, ES has been adopted since 2011 as the ref-

erence risk metric of the Swiss Solvency Test, the regulatory framework for in-

surance companies in Switzerland. In the banking world, the BCBS announced a

shift from VaR to ES in the assessment of market risk to be implemented in the

forthcoming Basel 4. Recall that the ES of a position Xat level p∈(0, 1) is given

by

ESp(X):=1

1−p1

p

VaR q(X)dq.

It is well-known that, up to a correction term that accounts for a disconti-

nuity in the distribution of X, the quantity ESp(X) can be expressed as the

conditional expectation of Xbeyond the (left) p-quantile. As we can always

write

ESp(X)=inf{m∈R;ES

p(X−m)≤0},

the quantity ESp(X) can also be interpreted as the minimal amount of capital that has

to be raised and held in cash in order to ensure that, on average, Xdoes not incur

a loss in the worst 100(1 −p) percent of outcomes. As a result, ESp(X) quantifies the

risk of the position Xas the cost of making the risk profile of X“acceptable” in the

following sense (FXis the distribution function of X):

Xis acceptable ⇐⇒ ESp(X)≤0FXcontinuous

⇐⇒ E[X|X≥VaR p(X)] ≤0.

In words, a position is acceptable under ES if, on average, it does not incur a loss in

the tail beyond a certain prespecified quantile.

Tail Risk Under VaR and ES

As said above, tail risk is captured by VaR by imposing an upper bound on

the loss probability. As, by definition, VaR depends only on the frequency of

losses but not on their severity, one could in principle accumulate arbitrary

loss peaks beyond the chosen quantile without being detected by VaR. This

“blindness” of VaR to the tail of the loss distribution arguably constitutes the

most fundamental deficiency of VaR and its undesirable financial implications

have been analyzed by a vast literature, see, for example, Artzner et al. (1999),

Dan´

ıelsson et al. (2001), Acerbi and Tasche (2002), Albanese and Lawi (2004),

Galichon (2010), Jarrow (2013), Wang (2016), Embrechts, Liu, and Wang (2018),

Weber (2018).

Differently from VaR, ES depends both on the frequency and the severity of losses

and captures tail risk by controlling the average loss in the tail of the distribution

beyond a prespecified quantile threshold. As illustrated by the following quote from

the Consultative Document issued in May 2012 (see p. 20 in BCBS, 2012), tail risk

was the key concern that ultimately led the BCBS to move away from VaR and

adopt ES:

Risk Measures Based on Benchmark Loss Distributions 3

439