An axiomatic approach to default risk and model uncertainty in rating systems

Nendel, Max; S eiche , Jan Maximilian
Wo king Pape
An axioma ic app oach o de aul isk and model
unce ain y in a ing sys ems
Cen e o Ma hema ical Economics Wo king Pape s, No. 725
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Nendel, Max; S eiche , Jan Maximilian (2023) : An axioma ic app oach o de aul
isk and model unce ain y in a ing sys ems, Cen e o Ma hema ical Economics Wo king Pape s,
No. 725, Biele eld Uni e si y, Cen e o Ma hema ical Economics (IMW), Biele eld,
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725
Sep embe 2023
An axioma ic app oach o de aul isk and
model unce ain y in a ing sys ems
Max Nendel and Jan S eiche
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
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2 MAX NENDEL AND JAN STREICHER
compe i i eness, and an imp o emen in c edi decisions. Fo a gene al in oduc ion o
c edi isk modeling, we e e o Bluhm e al. [8] and Lando [30]. We also e e o Guo
e al. [26] o an axioma ic s udy o c edi a ing c i e ia.
The ocus o a a ing is on he p obabili y o de aul (PD), which e e s o he c edi -
wo hiness o he bo owe , and no o c edi -speci ic e ms, such as exposu e a de aul
(EaD) o loss gi en de aul (LGD). Ne e heless, all h ee e ms play an impo an ole
in he con ex o de aul isk, a e included in he calcula ion o expec ed losses (EL)
and isk-weigh ed asse s (RWAs), and a e he e o e pa o supe iso y equi emen s
o isk-di e en ia ed capi al backing. Since a ing sys ems a e ma hema ical s a is ical
models ha ans o m a bo owe ’s de aul - ele an cha ac e is ics in o a s a emen o
c edi wo hiness, hey a e subjec o model isks and model unce ain ies, which can
lead o majo disc epancies in c edi isk managemen i neglec ed. Addi ionally, he
Eu opean Banking Au ho i y’s (EBA) guidelines on PD and LGD es ima ion (EBA-
GL-2017-16) [20] se e o educe luc ua ions in isk pa ame e s, and ocus on modeling
echniques used in he es ima ion o isk pa ame e s. In pa icula , he PD es ima ion
in low-de aul po olios is a sub le issue, c . Plu o and Tasche [37] and Tasche [41].
In he a e ma h o he subp ime mo gage c isis, he opic o model unce ain y
o Knigh ian unce ain y has become inc easingly signi ican o inancial ins i u ions
and ound i s way in o egula o y equi emen s in a ious o ms. As a consequence,
his classical and al eady e y p ominen opic in economic heo y has ecei ed e en
mo e a en ion in he li e a u e on heo e ical economics, ma hema ical inance, and
ac ua ial sciences. Model unce ain y appea s in he economic li e a u e, o example,
in he con ex o p e e ence ela ions, c . Gilboa and Schmeidle [25] and Macche oni
e al. [33], gene al equilib ium heo y, c . Beissne and Riedel [4], insu ance p icing, c .
Cas agnoli e al. [13], Nendel e al. [35], and Wang e al. [44], as well as hedging and
no-a bi age condi ions, see, o ins ance, Boucha d and Nu z [9] and Bu zoni e al.
[12]. Howe e , o he bes o ou knowledge, a de ailed s udy o model unce ain y in
c edi isk managemen is no p esen in he li e a u e.
This pape he e o e aims o p o ide a decision- heo e ic ounda ion o he ea -
men o de aul isk and model unce ain y in a ing sys ems. While he wo k o Guo
e al. [26] ocuses on an axioma ic s udy o c edi a ing c i e ia, including, among o h-
e s, gene alized PD c i e ia, ou axioma ic app oach in oduces he no ion o a de aul
isk measu e, which aims o p o ide a mo e gene al pe spec i e on PDs, and allows o
include, o example, model unce ain y in he o m o wo s -case PDs and dis o ed
PDs, wa ning signals, and de aul isk a ising om egula o y isk measu es. In pa -
icula , we explo e he use o gene alized e sions o PDs in he con ex o c edi isk,
bu do no aim o iden i y ele an c i e ia o c edi a ings. In con as o mone a y
isk measu es, c . A zne e al. [3], F i elli and Rosazza Gianin [23], and F¨ollme and
Schied [22], and nonlinea expec a ions, c . Coque e al. [16] and Peng [36], de aul
isk measu es do no beha e linea ly along cons an s bu only ake he alues ze o (no
de aul ) o one (de aul ) o cons an unc ions.
Th oughou , we conside a se Co cus ome s, i.e., a se o bounded measu able
unc ions on a gi en measu able space (Ω,F) con aining all cons an unc ions. A de-
aul isk measu e ϱis a mono one unc ional ha assigns o each cus ome X∈Ca
de aul isk ϱ(X)∈[0,1]. He e, posi i e alues o X ep esen a nega i e o al cash low
o , loosely speaking, a de aul . In a i s s ep, we show ha e e y de aul isk measu e
can be ex ended om he se C o he space Bbo all bounded measu able unc ions,
DEFAULT RISK MEASURES 3
c . Theo em 10. The ex ension p ocedu e is cons uc i e, and shows how de aul isk
can be assigned consis en ly o new cus ome s based on a inancial ins i u ion’s s ock
o exis ing clien s.
In a second s ep, we conside ail isk measu es ela ed o de aul isk measu es. As
no ed by Liu and Wang [32], he conside a ion o ail isk, i.e., he isk beyond a gi en
h eshold, is c ucial in oday’s inancial egula ion. We also e e o Bignozzi e al. [7]
o a gene aliza ion o he alue a isk ha depends on he size o po en ial losses in
he o m o quan ile-based isk measu es, o Fadina e al. [21] o an axioma ic s udy
o quan iles, and o Bu zoni e al. [11] o a s udy o adjus ed expec ed sho all. In
Sec ion 3, we show ha each de aul isk measu e induces i s own no ion o a alue a
isk, and es ablish a ono- o-one ela ion be ween de aul isk measu es and so-called
gene alized quan ile unc ions. Fo pa icula choices o de aul isk measu es, e.g.,
dis o ed PDs and wo s -case PDs, we p o ide explici ep esen a ions o he ela ed
alue a isk. We poin ou ha ou no ion o a gene alized quan ile unc ion ollows a
di e en philosophy han he no ion o a ail isk measu e in oduced by Liu and Wang
[32].
A key p ope y o he PD is ha i is speci ied only by he s a es o he wo ld whe e a
nega i e o al cash low is ealized, independen o he amoun o capi al gi en liquidi y
o illiquidi y. In ma hema ical e ms, his means ha he PD o Xis he same as he PD
o 1{X>0}. In Sec ion 4, we cha ac e ize de aul isk measu es ha ha e his p ope y,
and connec hem o Choque capaci ies, c . Dellache ie and Meye [18]. In his con ex ,
he no ions o de aul scaling in a iance, liquidi y in a iance, and illiquidi y in a iance
play a undamen al ole. Using con inui y p ope ies o Choque in eg als, we de i e
su icien and necessa y condi ions o de aul isk measu es o admi a ep esen a ion
ia p obabili y measu es o , in o he wo ds, as wo s -case PDs.
Fo he calcula ion o RWAs, he concep o a ma gin o conse a ism (MoC) is
used, in p ac ice, o quan i y he amoun o model unce ain y. The egula o y need o
conside model unce ain y ega ding de aul isks by calcula ing a MoC ha e lec s
he expec ed ange o es ima ion e o s can be ound in A icle 179 ( ) o , PD-speci ic,
in A icle 180 (e) o he CRR [17], among o he s. In Sec ion 6, we cha ac e ize de aul
isk measu es ha a e gi en in e ms o a MoC o , equi alen ly, as dis o ed PDs.
The cha ac e iza ion gene alizes he ac ha law-in a ian capaci ies on an a omless
p obabili y space can be ep esen ed as dis o ed p obabili ies, c . Wang e al. [44],
whe e his esul is es ablished in he con ex o insu ance p emia ha a e gi en as
Choque in eg als and Ama an e and Lieb ich [1] o a de ailed s udy o dis o ion isk
measu es, i.e., law-in a ian and comono onically addi i e isk measu es.
Mo eo e , we es ablish a connec ion be ween dis o ed PDs and wo s -case PDs o ,
equi alen ly, he ma gin o conse a ism and a sui able se o p obabili y measu es,
based on he Kusuoka ep esen a ion o law-in a ian isk measu es, c . Kusuoka [29],
and he well-known F ´eche -Hoe ding bounds o join dis ibu ions, c . Bu ge and
R¨uschendo [10], which also play a undamen al ole o he collapse o he mean o
law-in a ian isk unc ionals, c . Bellini e al. [5] and Lieb ich and Muna i [31].
In Sec ion 7, we use he esul s on dis o ed PDs in a case s udy on capi al equi e-
men s as demanded by cu en egula ions. The e, we discuss he impac o model
unce ain y in a ing sys ems on inancial ins i u ions’ RWAs. Since, om a egula o y
pe spec i e, model unce ain y only has o be conside ed o unexpec ed losses and no

4 MAX NENDEL AND JAN STREICHER
o expec ed losses (EL), i onically, a high deg ee o model unce ain y can ac ually e-
duce he amoun o capi al equi emen o badly a ed cus ome s, since i ans o ms
unexpec ed losses in o expec ed ones. We e e o Example 53 o he de ails.
The es o he pape is o ganized as ollows. In Sec ion 2, we de ine he no ion
o a de aul isk measu e, illus a e he de ini ion in se e al examples, and s a e ou
ex ension esul o de aul isk measu es (Theo em 10). The link be ween de aul isk
measu es and gene alized quan ile unc ions is discussed in Sec ion 3. Sec ion 4is
de o ed o de aul isk measu es ha a e gi en only in e ms o de aul scena ios, see
Theo em 31. Sec ion 5con ains se e al esul s on obus ep esen a ions as wo s -case
PDs. In Sec ion 6, we ocus on law-in a ian isk measu es and dis o ed PDs. The e,
we connec dis o ion unc ions wi h ce ain p ope ies o se s o absolu ely con inuous
p obabili y measu es based on he alue a isk and expec ed sho all o p obabili y
densi ies. In Sec ion 7, we discuss capi al equi emen s o a ing sys ems, and illus a e
he impac o di e en de aul isk measu es on he amoun o inancial ins i u ions’ isk
weigh ed asse s. In Appendix A, we p o ide a sho p oo o a cha ac e iza ion o exac
capaci ies and dis o ed p obabili ies, c . Aouani and Cha eauneu [2] and Kadane and
Wasse mann [28]. The p oo s o Sec ion 2a e con ained in Appendix B. The p oo s
o Sec ion 3can be ound in Appendix C. The p oo s o Sec ion 4a e collec ed in
Appendix D. The p oo s o Sec ion 5a e gi en in Appendix Eand he p oo s o Sec ion
6in Appendix F.
2. De aul Risk Measu es: De ini ion and Examples
In his sec ion, we in oduce he concep o a de aul isk measu e, which is s ongly
mo i a ed by he p obabili y o de aul (PD) as a p ime example. Like mone a y isk
measu es, de aul isk measu es a e mono one unc ionals de ined on sui able se s o
measu able unc ions, c . [22]. Howe e , hey exhibi a comple ely di e en beha iou
along cons an s.
Th oughou , le (Ω,F) be a measu able space and Bb= Bb(Ω,F) deno e he space
o all bounded measu able unc ions Ω →R. We conside a se C⊂Bb, con aining
he se o all cons an unc ions. A unc ion X∈Ccan be in e p e ed as a cus ome
o a inancial ins i u ion wi h −X(ω) being he sum o all inancial lows (ea nings,
spendings, and ma u i ies combined) a he end o he espec i e obse a ion pe iod i
a scena io ω∈Ω is ealized. Thus, posi i e alues o X esemble a nega i e sum o all
inancial lows, which we will, loosely speaking, e e o as a de aul . Choosing his, in
compa ison o he li e a u e on mone a y isk measu es, in e ed sign con en ion leads
o an easie exposi ion since i a oids con usion a ising om epea ed sign changes on
se e al occasions. As in he heo y o mone a y isk measu es, we do no di e en ia e
be ween a eal cons an m∈Rand he cons an unc ion X: Ω →Rwi h X(ω) = m
o all ω∈Ω, and w i e X=m, hinking o i as cash. Fo X, Y ∈Bb, we w i e X≤Y
i X(ω)≤Y(ω) o all ω∈Ω. Mo eo e , we de ine
in X:= in
ω∈ΩX(ω) and sup X:= sup
ω∈Ω
X(ω) o all X∈Bb.
Addi ionally, o X∈Bb, we conside he s anda d decomposi ion X=X+−X−wi h
X+:= X1{X>0}and X−:= −X1{X<0}.
Fo any wo eal numbe s x, y ∈R, we use he no a ion x∨y:= max{x, y}and
x∧y:= min{x, y}. In a simila ashion, we w i e X∨Yand X∧Y o he poin wise
DEFAULT RISK MEASURES 5
maximum and minimum o X, Y ∈Bb, espec i ely. Th oughou , we use he ollowing
sligh ly modi ied no ion o a (mone a y) isk measu e, and e e o [22] o a de ailed
discussion on his opic.
De ini ion 1. We say ha a map R: Bb→Ris a (mone a y) isk measu e i
(i) R(X)≤R(Y) o all X, Y ∈Bbwi h X≤Y,
(ii) R(0) = 0 and R(X+m) = R(X) + m o all X∈Bband m∈R.
We now in oduce he cen al objec o ou s udy.
De ini ion 2. A map ϱ:C→[0,1] is called a de aul isk measu e i
(i) ϱ(X)≤ϱ(Y) o all X, Y ∈Cwi h X≤Y,
(ii) ϱ(0) = 0 and ϱ(m) = 1 o all m∈Rwi h m > 0.
Thinking o PDs, he espec i e p ope ies seem o be e y canonical. Fo ins ance,
compa ing wo cus ome s i is ob ious ha he one wi h he highe o al cash low in
all scena ios exhibi s a lowe isk o de aul (P ope y (i)). Mo eo e , o all X∈C
wi h X≤0,
0≤ϱ(X)≤ϱ(0) = 0,
i.e., i all obliga ions can be payed in any scena io he cus ome ’s de aul isk will be
ze o, and a cons an nega i e o al cash low (m > 0) leads a leas o an unlikely
epaymen , and hence o a su e de aul (P ope y (ii)). Al hough P ope y (i) in he
de ini ion o a de aul isk measu e is analogous o he mono onici y o mone a y isk
measu es, P ope y (ii) is subs an ially di e en om he s anda d cash addi i i y o
ansla ion in a iance. To ha end, conside a de aul isk measu e ϱ: Bb→[0,1] and
obse e ha
ϱ(X−sup X) = 0 o all X∈Bb.
By de ini ion, nei he con exi y no posi i e homogenei y (o deg ee 1) a e meaning ul
p ope ies o de aul isk measu es, since
ϱ(λ)=1> λ =λϱ(1) o all λ∈(0,1).
We hus obse e ha he p ope ies o mone a y isk measu es di e subs an ially om
hose o de aul isk measu es despi e he simila i y o hei e y gene al de ini ions.
Ne e heless, he e is he possibili y o cons uc de aul isk measu es om mone a y
isk measu es as he ollowing example illus a es.
Example 3 (De aul isk measu e de ined by a mone a y isk measu e).Gi en a mon-
e a y isk measu e R: Bb→Ras in De ini ion 1, we a e able o cons uc a de aul
isk measu e ia
ϱ(X) := R1{X>0} o all X∈Bb.
Fo X, Y ∈Bbwi h X≤Y, we ha e 1{X>0}≤1{Y >0}, so ha
0≤R1{X>0}≤R1{Y >0}≤1.
Mo eo e , ϱ(0) = R1∅=R(0) = 0 and ϱ(m) = R(1) = 1 o all m∈Rwi h m > 0.
We con inue wi h se e al examples o de aul isk measu es, and begin wi h he mos
p ominen one.
Example 4 (P obabili y o de aul ).We ix a e e ence p obabili y measu e Pon F,
and conside he p obabili y o de aul (PD), gi en by
PDP(X) := P(X > 0) o all X∈Bb.
6 MAX NENDEL AND JAN STREICHER
I he p obabili y measu e Pis disc e e, his de aul isk measu e could be in e p e ed
as a mapping o common a ing classes. Clea ly, PDPsa is ies all p ope ies o a de aul
isk measu e. Fo X, Y ∈Bbwi h X≤Y,{X > 0}⊂{Y > 0}, so ha
0≤P(X > 0) ≤P(Y > 0) ≤1.
Fu he mo e, PDP(0) = P(∅) = 0 and PDP(m) = P(Ω) = 1 o all m∈Rwi h m > 0.
Building on his example, we can also conside he case, whe e model unce ain y is
aken in o accoun ia a dis o ion unc ion.
Example 5 (Dis o ed PD).Again, we ix a e e ence p obabili y measu e Pon F.
Due o a lack o da a, bad da a quali y, o changing economic en i onmen s, he consid-
e a ion o unce ain ies in o m o a ma gin o conse a ism (MoC) becomes mo e and
mo e impo an o inancial ins i u ions. Since such model unce ain ies a e pa o any
model, including a ing models, i is possible ha he e e ence p obabili y measu e P
is no he ‘p ecise’ p obabili y measu e ha ep esen s he de aul isk o cos ume s
o e a one-yea ime ho izon. We he e o e conside a nondec easing dis o ion unc ion
T: [0,1] →[0,1] wi h T(0) = 0 and T(1) = 1. We de ine
ϱ(X) := TP(X > 0)=TPDP(X) o all X∈Bb.
In his case, he dis o ion unc ion Tcan be ega ded as a benchma k o model
unce ain y, and he ma gin o conse a ism is gi en by
MoC(p) := T(p)
p−1 o all p∈(0,1].
Clea ly, he wo p ope ies o a de aul isk measu e ca y o e om he classical PD, c .
Example 4, o he dis o ed PD o any p obabili y measu e Pand any nondec easing
dis o ion unc ion T: [0,1] →[0,1] wi h T(0) = 0 and T(1) = 1.
Apa om dis o ing a e e ence p obabili y measu e as in he p e ious example,
he e is also he possibili y o inco po a ing model unce ain y ia wo s -case conside -
a ions among se s o p obabili y measu es.
Example 6 (Wo s -case PD).In o de o p ope ly accoun o unce ain ies w. . .
model speci ica ions, i is o en necessa y o conside a ious models a he same ime.
This becomes pa icula ly ele an , i he models ha e di e en se s o measu e ze o,
since hen one model neglec s ce ain e en s ha occu wi h posi i e p obabili y unde
a di e en model. We he e o e conside he ollowing gene aliza ion o Example 4. Le
Pbe a nonemp y se o p obabili y measu es and
ϱ(X) := sup
Q∈P
Q(X > 0) = sup
Q∈P
PDQ(X) o all X∈Bb.
As be o e, he p ope ies (i) and (ii) ha e been shown o classical PDs in Example 4,
and emain alid when aking he sup emum o e PDs. Following [2], we will see ha ,
in many cases, dis o ed PDs allow o a ep esen a ion as wo s -case PDs and ice
e sa, see Sec ion 6.
Up o now, all examples o de aul isk measu es ha e been o he o m
ϱ(X) = ϱ1{X>0} o all X∈Bb,(1)
i.e., ϱ(X) only depends on he se whe e X∈Bbis la ge han ze o, comple ely
independen o i s alues. Thinking o PDs om a ing sys ems, his is a e y desi -
able p ope y, since i implies ha he inancial ins i u ion is only in e es ed in he
DEFAULT RISK MEASURES 7
cus ome s’ abili y o pay hei dues. In Sec ion 4, we de i e su icien and necessa y
condi ions o de aul isk measu es in o de o sa is y (1). The ollowing wo examples
show ha , howe e , no e e y de aul isk measu e needs o allow o such a ep e-
sen a ion as i is also possible o de ine de aul isk measu es ha depend on speci ic
alues o X, o ins ance, using a wa ning signal ha leads o a mo e conse a i e isk
assessmen .
Example 7 (Wa ning signal).Conside wo de aul isk measu es ρ0and ρcwi h
ρ0≤ρc, i.e., ρ0is less conse a i e hen ρc. The idea is ha ρcac s as a wa ning signal
i scena ios a e possible whe e he loss exceeds a gi en maximum le el. Fo X∈Bb
and γ > 0, we de ine
ϱγ(X) := (ϱ0(X), ϱcX−1
γ= 0,
ϱc(X), ϱcX−1
γ>0.
Thus, o γ > 0, we change om ϱ0 o he mo e conse a i e de aul isk measu e ϱc
when he po en ial loss exceeds he le el 1
γunde ϱc. We obse e ha
lim
γ→0ϱγ(X) = ϱ0(X) o all X∈Bb.
In ac , ϱγ(X)=0=ϱ0(X) o all X∈Bbwi h X≤0, and ϱc(X−sup X) = 0 o all
X∈Bbwi h sup X > 0. A conc e e choice o ϱcand ϱ0a e, o example, ϱ0= PDP
and ϱc(x) = supQ∈P PDQ(X) (wo s -case PD) o all X∈Bb, whe e Pis a nonemp y
se o p obabili y measu es con aining P. Fo example, i is concei able ha cus ome s,
o whom he possible loss exceeds he limi 1
γ, migh exhibi addi ional isk ac o s
ha inc ease hei p obabili y o de aul .
A i s glance, he ollowing example is eminiscen o Example 6, in which he wo s -
case PD was conside ed. As an addi ional c i e ion, he isk o X∈Bb, desc ibed by a
mone a y isk measu e, de e mines how many models a e conside ed in he calcula ion
o he sup emum.
Example 8 (Inc easing conse a ism).Le Pbe a se o p obabili y measu es on F,
R: Bb→Rbe a mone a y isk measu e, and α:P → [0,∞) wi h in Q∈P α(Q) = 0.
Fo X∈Bb, le ϱ(X) := 0 i R(X)≤0 and
ϱ(X) := sup PDQ(X)Q∈ P, α(Q)≤R(X)i R(X)>0,
i.e., he la ge he isk associa ed o X∈Bb, he mo e models a e aken in o accoun ,
when assessing he de aul isk. In his case, α(Q) measu es he deg ee o con idence
ha he model Q∈ P is he ’co ec ’ model, whe e α(Q) = 0 co esponds o maximal
con idence. The (mone a y) isk measu e Rcan be in e p e ed as he ou come o some
in e nal isk assessmen . Since, by assump ion, in Q∈P α(Q) = 0, i ollows ha
PDQ(X)Q∈ P, α(Q)≤R(X)=∅ o all X∈Bbwi h R(X)>0.
In pa icula , ϱ(X)∈[0,1] o all X∈Bband ϱ(m) = 1 o all m∈Rwi h m > 0. Fo
X, Y ∈Bbwi h X≤Y, i ollows ha R(X)≤R(Y) and PDQ(X)≤PDQ(Y) o all
Q∈ P, so ha ϱ(X)≤ϱ(Y).
A e some examples ha e been discussed, we now u n ou ocus on ex ensions o
a gi en de aul isk measu e ϱ:C→[0,1] om he se o exis ing cus ome s C o he
space Bbo all bounded measu able unc ions. We s a wi h ollowing de ini ion.
De ini ion 9. Le ϱ:C→[0,1] be a de aul isk measu e and F: Bb→R.
14 MAX NENDEL AND JAN STREICHER
c) In o de o explain why addi ional con inui y assump ions a e needed in P oposi ion
26 below, we ega d, o a gi en p obabili y measu e Pon F, he de aul isk measu e
ϱ: Bb→[0,1], gi en by
ϱ(X) := (1,i P({X > ε}) = 1 o some ε > 0,
P({X>0})
2,o he wise.
As such ϱis de aul scaling in a ian and does no depend on he pa , whe e X∈Bb
is less o equal o 0. Howe e , i (Ω,F,P) is a omless and X∈Bbis uni o mly
dis ibu ed on [0,1] unde P, hen
PX∈(0, ε)=ε > 0 o all ε > 0,
so ha
ϱ(X) = 1
2=1=ϱ1{X>0}.
The ollowing p oposi ion cla i ies his elemen a y example.
P oposi ion 26. Le ϱ: Bb→[0,1] be a de aul scaling in a ian de aul isk measu e.
a) Fo all X∈Bbwi h X≥0,
ϱ(X)≤ϱ1{X>0}≤in
ε>0ϱX+ε1{X>0}.(9)
b) Fo all X∈Bbwi h X≥0,
sup
ε>0
ϱ1{X>ε}≤ϱ(X)≤ϱ1{X>0}.(10)
The p e ious p oposi ion shows ha de aul scaling in a iance oge he wi h ϱ(X) =
ϱ(X+), o all X∈Bb, implies ha he de aul isk measu e can be ep esen ed ia a
capaci y excep o some addi ional con inui y p ope ies. The inequali y in pa a) is
qui e mild om a heo e ical pe spec i e, because educing a cus ome ’s cash lows by
an amoun o money ε > 0 in scena ios, whe e he cus ome is de aul ing anyways, has
no impac on he de aul scena ios. In con as o his, he con inui y assump ion
sup
ε>0
ϱ1{X>ε}=ϱ1{X>0} o all X∈Bb
is no o ee since, in ma hema ical e ms, i equi es some mild o m o con inui y
om below o he ela ed capaci y. F om an economic poin o iew, howe e , i is
qui e in ui i e since ε > 0 can be chosen a bi a ily small, e.g., in such a way ha i
alls below he smalles possible amoun o money. A heo e ical inc ease in capi al
by 10−10 eu os should no change he de aul isk o he espec i e cus ome . This
mo i a es o in oduce he ollowing de ini ion.
De ini ion 27. Le ϱ: Bb→[0,1] be a de aul isk measu e. We say ha ϱis illiquidi y
in a ian , i , o all m > 0 and X∈Bbwi h X≥0,
ϱ(X) = ϱX+m1{X>0}.
In o he wo ds, illiquidi y in a iance means ha cus ome s ha a e in de aul o ,
loosely speaking, illiquid emain illiquid i hei deb due inc eases o hei capi al
dec eases in he sense ha hei de aul isk does no change. The ollowing p oposi ion
es ablishes a i s connec ion be ween illiquidi y in a iance, de aul scaling in a iance,
and a ep esen a ion ia indica o unc ions.
P oposi ion 28. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, he ollowing
s a emen s a e equi alen .

DEFAULT RISK MEASURES 15
(i) ϱis de aul scaling in a ian and
ϱ(X) = in
ε>0ϱ(X+ε1{X>0}) o all X∈Bbwi h X≥0.
(ii) ϱis illiquidi y in a ian .
(iii) Fo all X∈Bbwi h X≥0,
ϱ(X) = ϱ1{X>0}.
In analogy o De ini ion 27, we speak o liquidi y in a iance, when he de aul isk
emains he same i capi al is added in scena ios whe e he cus ome is able o make
all paymen s due.
De ini ion 29. Le ϱ: Bb→[0,1] be a de aul isk measu e. We say ha ϱis liquidi y
in a ian , i , o all m > 0 and X∈Bbwi h X≥0,
ϱ(X) = ϱX−m1{X=0}.
Lemma 30. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, he ollowing condi-
ions a e equi alen .
(i) ϱ(X) = ϱ(X+) o all X∈Bb.
(ii) ϱ(X) = ϱX−m1{X≤0} o all m > 0and X∈Bb.
(iii) ϱis liquidi y in a ian .
In he con ex o isk unc ionals, P ope y (i) in Lemma 30 (wi h a di e en sign
con en ion) is known unde he names su plus in a iance, c . [24], loss dependence, c .
[15], and excess in a iance, c . [40]. A combina ion o P oposi ion 28 and Lemma 30
leads o he ollowing heo em, which is he main esul o his sec ion.
Theo em 31. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, he ollowing condi-
ions a e equi alen .
(i) Fo all X∈Bb,
ϱ(X) = ϱ1{X>0}.
(ii) ϱis de aul scaling in a ian wi h
ϱ(X) = ϱ(X+) = in
ε>0ϱX+ε1{X>0} o all X∈Bb.
(iii) ϱis liquidi y in a ian and illiquidi y in a ian .
To ge a link o Sec ion 5, whe e he ocus lies on he ela ed capaci ies and hei
Choque in eg als, we in oduce he no ion o submodula i y o de aul isk measu es.
In heo e ical economics, submodula i y is a classical p ope y, which is closely ela ed
o subs i u e goods, c . [42].
De ini ion 32. Le Cbe a subla ice o Bb, which con ains all cons an unc ions. A
de aul isk measu e ϱ:C→[0,1] is called submodula i
ϱ(X∧Y) + ϱ(X∨Y)≤ϱ(X) + ϱ(Y) o all X, Y ∈C.
The na u al ques ion a ises, whe he submodula i y is a sensible no ion o de aul
isk measu es. Fo classical PDs, we ha e equali y, i.e.,
PDP(X∧Y) + PDP(X∨Y) = PDP(X) + PDP(Y) o all X, Y ∈Bb.
Mo eo e , submodula i y is gi en o ce ain classes o dis o ed PDs, o ins ance, i
he dis o ion unc ion is conca e, see Sec ion 6.
16 MAX NENDEL AND JAN STREICHER
Rema k 33. Le Cbe a subla ice o Bb, which con ains all cons an unc ions, and
ϱ:C→[0,1] be a submodula de aul isk measu e. Then,
ϱ(X) = ϱ(X+) o all X∈C.
In ac , le X∈C. The submodula i y o ϱ oge he wi h 0 ∈Cimplies ha
ϱ(X+) + ϱ(−X−)≤ϱ(X).
Since −X−≤0, i ollows ha ϱ(−X−) = 0 by he de ining p ope ies o a de aul isk
measu e. The e o e,
ϱ(X)≤ϱ(X+) = ϱ(X+) + ϱ(−X−)≤ϱ(X).
As a consequence o Theo em 31 and Rema k 33, we ob ain he ollowing co olla y.
Co olla y 34. Le ϱ: Bb→[0,1] be a submodula de aul isk measu e. Then, he
ollowing s a emen s a e equi alen .
(i) ϱis illiquidi y in a ian .
(ii) Fo all X∈Bb,
ϱ(X) = ϱ1{X>0}.
5. Choque in eg als and obus ep esen a ions
In Sec ion 4, we discussed equi alen condi ions o a de aul isk measu e ϱ: Bb→
[0,1] in o de o be he o m (7). Building on his ep esen a ion, he aim o his sec ion
is o connec de aul isk measu es wi h Choque in eg als and mone a y isk measu es
and use his connec ion o a ain a ep esen a ion o ϱ ia p obabili y measu es. Recall
ha a capaci y is a map c:F → [0,1] wi h c(∅) = 0, c(Ω) = 1, and c(A)≤c(B) o all
A, B ∈ F wi h A⊂B. We s a wi h he ollowing obse a ion.
Rema k 35. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, we can de ine a
capaci y c:F → [0,1] by
c(A) := ϱ(1A) o all A∈ F.(11)
By de ini ion o a de aul isk measu e,
c(∅) = c(1∅) = ϱ(0) = 0 and c(Ω) = c(1Ω) = ϱ(1) = 1.
Mo eo e , o all A, B ∈ F wi h A⊂B,
c(A) = ϱ(1A)≤ϱ(1B) = c(B)
due o he mono onici y o ϱ. Fo X∈Bb, he Choque in eg al wi h espec o cis
de ined as
ZXdc:= Z0
−∞ c({X > s})−1ds+Z∞
0
c({X > s}) ds.
Al hough he Choque in eg al is, in gene al, no a linea unc ional, i de ines a mon-
e a y isk measu e R: Bb→R ia
R(X) := ZXdc o all X∈Bb.
By de ini ion o he Choque in eg al, he mone a y isk measu e Ris posi i ely homo-
geneous, i.e., R(λX) = λR(X) o all X∈Bband λ > 0. I ϱsa is ies (7), hen
ϱ(X) = R1{X>0} o all X∈Bb,
DEFAULT RISK MEASURES 17
which leads o he ollowing p oposi ion.
P oposi ion 36. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, he ollowing wo
s a emen s a e equi alen .
(i) Fo all X∈Bb,
ϱ(X) = ϱ1{X>0}.
(ii) The e exis s a posi i ely homogeneous mone a y isk measu e R: Bb→R, c .
De ini ion 1, wi h
ϱ(X) = R1{X>0} o all X∈Bb.
Le ϱ: Bb→[0,1] be a de aul isk measu e, which sa is ies (7). Then, ϱis submod-
ula i and only i he ela ed capaci y c:F → [0,1], gi en by (11), is 2-al e na ing,
i.e.,
c(A∪B) + c(A∩B)≤c(A) + c(B) o all A, B ∈ F.
A well-known ac is ha a capaci y is 2-al e na ing i and only i he ela ed Cho-
que in eg al de ines a cohe en isk measu e. We ecall ha a mone a y isk measu e
R: Bb→Ris cohe en i and only i he e exis s a nonemp y se Po ini ely addi i e
p obabili y measu es such ha
R(X) = sup
Q∈P
EQ(X) o all X∈Bb,
c . [22]. In his case, we ob ain he obus ep esen a ion
ϱ(X) = Z1{X>0}dc= sup
Q∈P
EQ(1{X>0}) = sup
Q∈P
Q(X > 0) o all X∈Bb.(12)
We now in es iga e addi ional con inui y p ope ies o ϱ ha gua an ee a ep e-
sen a ion ia coun ably addi i e p obabili y measu es. In he sequel, o a sequence
(Xn)n∈N⊂Bband X∈Bb, we w i e Xn↗Xo Xn↘Xas n→ ∞ i Xn≤Xn+1 o
Xn≥Xn+1 o all n∈Nand X(ω) = limn→∞ Xn(ω) o all ω∈Ω, espec i ely.
P oposi ion 37. Le ϱ: Bb→[0,1] be a de aul isk measu e wi h ϱ(X) = ϱ1{X>0}
o all X∈Bband c(A) := ϱ(1A) o all A∈ F. Then, he ollowing s a emen s a e
equi alen .
(i) ϱis con inuous om below, i.e., o e e y sequence (Xn)n∈N⊂Bbwi h Xn↗
X∈Bbas n→ ∞,
ϱ(X) = lim
n→∞ϱ(Xn).
(ii) Fo e e y sequence (An)n∈N⊂ F wi h An⊂An+1 o all n∈N
c[
n∈N
An= lim
n→∞c(An).
(iii) Fo e e y sequence (Xn)n∈N⊂Bbwi h Xn↗X∈Bbas n→ ∞,
ZXdc= lim
n→∞ZXndc.
Un o una ely, e en i ϱis submodula , con inui y om below o ϱis, in gene al,
no a su icien condi ion in o de o gua an ee a ep esen a ion ia coun ably addi i e
p obabili y measu es on Bbas he ollowing example shows.
Example 38. Le P:B → [0,1] be he Lebesgue measu e de ined on he Bo el σ-
algeb a Bo he closed in e al Ω := [0,1] and Fbe he powe se o Ω. Le Ldeno e
18 MAX NENDEL AND JAN STREICHER
he space o all bounded B-measu able unc ions Ω →Rand
R(X) := in {EP(X0)|X0∈ L, X0≥X}.
Then, by [19, P oposi ion 2.2 and Lemma 3.6], Ris a cohe en isk measu e, which is
con inuous om below and he maximal ex ension o EP o Bb. Conside he capaci y
c:F → [0,1], gi en by
c(A) := R(1A) o all A∈ F,
and ϱ(X) := c({X > 0}) o all X∈Bb. Le Q:F → [0,1] be a ini ely addi i e
p obabili y wi h
EQ(X)≤R(X) o all X∈Bb.(13)
Then, o all X∈Bbwi h X≥0,
ZXdc=Z∞
0
c({X > s}) ds≥Z∞
0
Q({X > s}) ds=EQ(X).
Mo eo e , o X∈ L wi h X≥0,
ZXdc=Z∞
0
c({X > s}) ds=Z∞
0
P({X > s}) ds=EP(X).
Hence, by he maximali y o R, i ollows ha RXdc≤R(X) o all X∈Bb. We ha e
he e o e shown ha R(X) = RXdc o all X∈Bb. In pa icula , ϱis submodula .
On he o he hand, assuming he con inuum hypo hesis, by [6, Sa z 1C], he e exis s
no a single coun ably addi i e p obabili y measu e Q:F → [0,1] wi h (13). In ac ,
wi h a simila cons uc ion, using a p obabili y measu e which is di e en om he
Lebesgue measu e, one can a oid in oking he con inuum hypo hesis, see [19, Example
3.7] o u he de ails.
I , howe e , ϱis a dis o ed PD, which is submodula , con inui y om below is a
su icien condi ion in o de o allow o a ep esen a ion in e ms o coun ably addi i e
p obabili y measu es as he ollowing ema k discusses.
Rema k 39. Le Pbe a p obabili y measu e o Fand T: [0,1] →[0,1] be a nonde-
c easing unc ion wi h T(0) = 0 and T(1) = 1. Le ϱ: Bb→[0,1] be gi en by
ϱ(X) = TPDP(X) o all X∈Bb,
and assume ha ϱis submodula , so ha c:F → [0,1], A 7→ ϱ(1A) is a 2-al e na ing
capaci y.
a) Assume ha Tis le -con inuous o , equi alen ly, lowe semicon inuous, i.e.,
T(p) = sup
q∈(0,p)
T(q) o all p∈(0,1].
Then, by P oposi ion 37, he Choque in eg al w. . . cis a cohe en and law-in a ian
isk measu e, which is con inuous om below. Hence, by [22, Theo em 4.33], he e
exis s a se o coun ably addi i e and (w. . . P) absolu ely con inuous p obabili y
measu es Pon Fwi h
ϱ(X) = sup
Q∈P
PDQ(X) o all X∈Bb.
b) Now, assume ha (Ω,F,P) is a omless. Then, he Choque in eg al w. . . cis a
cohe en and law-in a ian isk measu e on an a omless p obabili y space. By [27,
Theo em 2.1], i ollows ha he Choque in eg al w. . . cis con inuous om below.
Hence, by [22, Theo em 4.33], he e exis s a se o coun ably addi i e and (w. . . P)
DEFAULT RISK MEASURES 19
absolu ely con inuous p obabili y measu es Pon Fwi h
ϱ(X) = sup
Q∈P
PDQ(X) o all X∈Bb.
I Ω is a Polish space and Fis he Bo el σ-algeb a, we ha e he ollowing cha ac-
e iza ion o gene al submodula de aul isk measu es on he space Lbo all bounded
lowe semicon inuous unc ions Ω →R.
P oposi ion 40. Le Ωbe a Polish space and ϱ: Lb→[0,1] be a submodula de aul
isk measu e wi h
ϱ(X) = ϱ1{X>0} o all X∈Lbwi h X≥0.(14)
Then, he ollowing s a emen s a e equi alen .
(i) ϱis con inuous om below, i.e.,
ϱ(X) = lim
n→∞ϱ(Xn)
o e e y sequence (Xn)n∈N⊂Lbwi h Xn↗X∈Lbas n→ ∞.
(ii) The e exis s a nonemp y se Po p obabili y measu es on (Ω,F)such ha
ϱ(X) = sup
Q∈P
Q(X > 0) o all X∈Lb.
Rema k 41. Le ϱ: Bb→[0,1] be a de aul isk measu e. Then, con inui y om abo e
in he sense ha
ϱ(X) = lim
n→∞ϱ(Xn)
o e e y sequence (Xn)n∈N⊂Bbwi h Xn↘X∈Bbas n→ ∞, is no a sensible
p ope y. Conside ing a sequence (Xn)n∈N⊂Bbde ined by Xn=1
n, i ollows ha
ϱ(Xn) = 1, while ϱ(0) = 0. Howe e , by (12), i is enough o exp ess he Choque
in eg al ia a se o (coun ably addi i e) p obabili y measu es. We can he e o e weaken
he equi emen o con inui y om abo e and ob ain he ollowing p oposi ion.
P oposi ion 42. Le ϱ: Bb→[0,1] be a de aul isk measu e wi h ϱ(X) = ϱ1{X>0}
o all X∈Bband c(A) := ϱ(1A) o all A∈ F. Then, he ollowing s a emen s a e
equi alen .
(i) Fo e e y sequence (Xn)n∈N⊂Bbwi h Xn↘0as n→ ∞ and all ε > 0,
lim
n→∞ϱ(Xn−ε)=0.
(ii) Fo e e y sequence (An)n∈N⊂ F wi h An+1 ⊂An o all n∈Nand Tn∈NAn=∅,
lim
n→∞ϱ(1An)=0.
(iii) Fo e e y sequence (Xn)n∈N⊂Bbwi h Xn↘0as n→ ∞,
lim
n→∞ZXndc= 0.
I ϱis addi ionally submodula , ei he o hese condi ions implies ha he e exis s a
nonemp y se Po p obabili y measu es on (Ω,F)wi h
ϱ(X) = max
Q∈P Q(X > 0) o all X∈Bb.
Example 43. Le Pbe a p obabili y measu e on Fand T: [0,1] →[0,1] be nonde-
c easing wi h T(0) = 0 and T(1) = 1. Then, he de aul isk measu e ϱ: Bb→[0,1],

20 MAX NENDEL AND JAN STREICHER
gi en by
ϱ(X) := TPDP(X) o all X∈Bb,
sa is ies P ope y (ii) in P oposi ion 42 i in p>0T(p) = 0. In case he e exis s a sequence
(An)n∈N⊂ F wi h ∅ =An+1 ⊂An o all n∈Nand Tn∈NAn=∅, his is also a
necessa y condi ion o ϱ o sa is y P ope y (ii) in P oposi ion 42 using he con inui y
om abo e o P.
6. Law-in a ian de aul isk measu es and dis o ed PDs
Le Cbe a se o cus ome s C⊂Bb, which con ains he se o all cons an unc ions,
and sa is ies 1{X>0}∈C o all X∈C. In his sec ion, we ix a e e ence p obabili y
measu e Pon F, and specialize on law-in a ian de aul isk measu es ϱ:C→[0,1], i.e.,
ϱ(X) = ϱ(Y) whene e X∈Cand Y∈Cha e he same dis ibu ion unde P. In a ing
sys ems, cus ome s in he same a ing class a e conside ed o be s a is ically iden ical
in e ms o hei de aul beha iou . Hence, when choosing a de aul isk measu e in
o de o quan i y he p obabili y o de aul including unce ain y, i makes sense o
equi e law-in a iance. Since cus ome s a e usually di ided in o a ing classes and no
e e y de aul p obabili y is ealized, we also conside he case whe e (Ω,F,P) is no
a omless. We s a wi h se e al cha ac e iza ions o law-in a iance, which do no hinge
on he s anda d assump ion o an a omless p obabili y space, be o e we swi ch o an
a omless se ing in o de o de i e ine p ope ies and ep esen a ions o law-in a ian
de aul isk measu es.
In he sequel, we say ha a unc ion T: [0,1] →[0,1] is a dis o ion unc ion i
T(0) = 0 and T(1) = 1. The ollowing heo em adop s an a gumen om Wang e al.
[44, P oo o Theo em 2], and p o ides a cha ac e iza ion o dis o ed PDs.
Theo em 44. Le ϱ:C→[0,1] be a de aul isk measu e. Then, he ollowing s a e-
men s a e equi alen .
(i) ϱis law-in a ian and ϱ(X) = ϱ1{X>0} o all X∈C.
(ii) The e exis s a dis o ion unc ion T: [0,1] →[0,1] wi h
ϱ(X) = TPDP(X) o all X∈C.
A na u al ques ion ha a ises is whe he he dis o ion unc ion Tin Theo em 44 is
nondec easing o , in o he wo ds, i he de aul isk measu e ϱ:C→[0,1] is consis en
wi h P, i.e.,
ϱ(X)≤ϱ(Y) o all X, Y ∈Cwi h PDP(X)≤PDP(Y).
This p ope y is e y na u al since one would expec he de aul isk o X o be smalle
han he de aul isk o Yi he PD o Xis smalle han he PD o Y. In he si ua ion
o Theo em 44, i is, howe e , possible ha he dis o ion unc ion Tis no mono one
as he ollowing simple example shows.
Example 45. Le Ω = {0,1},Fbe he powe se , and Cconsis o all cons an s and
he wo unc ions X:= 1{0}and Y:= 1{1}. Assume ha
0< p := P(X > 0) <P(Y > 0) =: q < 1 and 0 < ϱ(Y)< ϱ(X)<1.
Then ϱis a de aul isk measu e, which sa is ies P ope y (i) in Theo em 44 bu , o
any dis o ion unc ion T: [0,1] →[0,1] wi h
ϱ(Z) = TPDP(Z) o all Z∈C,
DEFAULT RISK MEASURES 21
i ollows ha T(p) = ϱ(X)> ϱ(Y) = T(q), so ha Tis no mono one.
We now aim owa ds a cha ac e iza ion in e ms o a nondec easing dis o ion unc-
ion. The p oo o Theo em 10 indica es ha he se
P:= p∈[0,1] ∃X∈C:P(X > 0) = p.(15)
plays a undamen al ole o he mono onici y o he dis o ion unc ion.
De ini ion 46. We say Ccon ains an o de ed subse i he e exis s a amily Xpp∈P
wi h PDP(Xp) = p o all p∈Pand {Xp>0} ⊂ {Xq>0} o all p, q ∈Pwi h p≤q.
Clea ly, i (Ω,F,P) is a omless and C= Bb, hen Ccon ains an o de ed subse .
Theo em 47. Le ϱ:C→[0,1] be a de aul isk measu e, and assume ha Ccon ains
an o de ed subse . Then, he ollowing s a emen s a e equi alen .
(i) ϱis law-in a ian and ϱ(X) = ϱ1{X>0} o all X∈C.
(ii) The e exis s a nondec easing dis o ion unc ion T: [0,1] →[0,1] wi h
ϱ(X) = TPDP(X) o all X∈C.
Rema k 48. Unde he assump ions o Theo em 47, including one o he equi alences,
he de aul isk measu e ϱcan be ex ended o a law-in a ian de aul isk measu e
ϱ: Bb→[0,1] by means o he dis o ion unc ion T. The ex ension ϱis gi en by
ϱ(X) := TPDP(X) o all X∈Bb.
Ob iously, ϱ(X) = ϱ(X) o all X∈Cand ϱis a de aul isk measu e, c . Example 5.
No e ha a de aul isk measu e can, in gene al, no be ex ended using a nonmono one
dis o ion unc ion as in Example 45. This can be seen by conside ing, o example,
he se Ω = {0,1,2} oge he wi h he powe se Fand Cconsis ing o X:= 1{0},
Y:= 1{1}and all cons an unc ions. Le P(X > 0) = 0.5, P(Y > 0) = 0.3, ϱ(X)=0.5,
ϱ(Y)=0.7, and T: [0,1] →[0,1] be a dis o ion unc ion wi h
ϱ(Z) = TPDP(Z) o all Z∈C.
Then, o U:= 1{1,2}, i ollows ha P(U > 0) = 0.5 = P(X > 0). Hence,
TPDP(U)=T(0.5) = ϱ(X)=0.5<0.7 = ϱ(Y)
despi e he ac ha Y≤U.
Rema k 49. We ecall some well-known ac s abou dis o ed p obabili ies, c . [22].
Fo he eade ’s con enience, we p o ide sho p oo s o some o he s a emen s collec ed
in his ema k in he Appendix A. In he ollowing, le T: [0,1] →[0,1] be a dis o ion
unc ion.
a) A well-known ac is ha he capaci y c:F → [0,1], gi en by
c(A) := TP(A) o all A∈ F.(16)
is 2-al e na ing i Tis conca e. I (Ω,F,P) is a omless, also he con e se s a e-
men holds, c . [22, Sec ion 4.6].
b) A capaci y c:F → [0,1] is called exac i he e exis s a se o coun ably addi i e
p obabili y measu es Pon Fwi h
c(A) = sup
Q∈P
Q(A) o all A∈ F.
22 MAX NENDEL AND JAN STREICHER
I is well-known ha he dis o ed p obabili y c:F → [0,1], gi en by (16), is
an exac capaci y i , o all p, q ∈(0,1) wi h p<q,
T(p)
p≥T(q)−T(p)
q−p≥1−T(q)
1−q.(17)
I (Ω,F,P) is a omless, also he con e se holds ue, c . [2]. In his case,
T(p) = sup
Q∈P Z1
1−p
qQ(s) ds o all p∈[0,1],(18)
whe e qQdeno es he quan ile unc ion o he densi y dQ
dP, see, e.g., [22, Lemma
4.60]. In pa icula ,
MoC(p) = T(p)
p−1 = sup
Q∈P
ES1−p
PdQ
dP−1 o all p∈(0,1],
whe e, o α∈(0,1] and Q∈ P, ESα
PdQ
dP:= 1
1−αR1
αqQ(s) dsdeno es he
expec ed sho all o he densi y dQ
dPwi h con idence le el α. No e ha (17) is,
o example, sa is ied i Tis conca e, and obse e ha (17) implies ha he
unc ion Tis nondec easing and absolu ely con inuous as soon as
in
p∈(0,1] T(p)=0.(19)
Recall ha a mono one unc ion is a.e. di e en iable and ha absolu e con i-
nui y o Timplies ha
T(p) = Zp
0
T′(s) ds o all p∈[0,1],
whe e T′deno es he weak de i a i e o T. Mo eo e , (17) implies ha
MoC(q)≤MoC(p) o all p, q ∈(0,1] wi h p≤q.
In ac , assume ha (17) is sa is ied, and le p, q ∈(0,1]. I p=q, he s a emen
is i ial, and i p<q,
T(p)q
p−T(p)
q−p=T(p)
p≥T(q)−T(p)
q−p,
which yields ha T(p)q
p≥T(q).
We conclude his sec ion wi h he ollowing cha ac e iza ion o he mino an s o
dis o ed PDs, which can be ound in a simila ye di e en o m in [22, Theo em 4.79].
P oposi ion 50. Le T: [0,1] →[0,1] be a dis o ion unc ion, which sa is ies (17)
and (19), and Qbe a p obabili y measu e on F. I
Zp
0
qQ(1 −s) ds≤T(p) o all p∈[0,1],(20)
hen
PDQ(X)≤TPDP(X) o all X∈Bb.(21)
I (Ω,F,P)is a omless, also he con e se holds ue.
Rema k 51. Conside he si ua ion o P oposi ion 50. In iew o Rema k 49, a su -
icien condi ion o (20) and hus (21) o be sa is ied is ha qQ(1 −p)≤T′(p) o
a.a. p∈(0,1). Tha his is, howe e , no a necessa y condi ion, can easily be seen by
DEFAULT RISK MEASURES 23
conside ing T(p) = √p o all p∈[0,1] and Q=P. Then, qP(1−p) = 1 o all p∈(0,1)
and T′(p) = 1
2√p<1 o all p∈1
4,1. Howe e ,
T(p) = √p≥p=Zp
0
qP(1 −s) ds.
7. A Case S udy on Capi al Requi emen s o Financial Ins i u ions
In his sec ion, we link ou axioma ic s udy o de aul isk measu es o inancial
ins i u ions’ capi al equi emen s accoun ing o model unce ain y. Acco ding o he
guidelines o he Eu opean Banking Au ho i y [20], he PDs o a a ing sys em a e
calib a ed o a ‘bes es ima e’ le el wi hou using sys ema ically conse a i e inpu
alues o he calcula ion. Those PDs a e hen used in he isk-o ien ed g oup manage-
men , among o he s. In o de o es ablish he connec ion be ween model unce ain y
and capi al equi emen s, we a e guided by A icle 179 ( ) and A icle 180 (e) o he
CRR [17] ha de e mine ha an app op ia e ma gin o conse a ism, e lec ing he
expec ed ange o es ima ion e o s, mus be o med o he ‘bes es ima e’ PD o he
a ing sys em.
Usually, one di e en ia es be ween he expec ed loss (EL) and unexpec ed losses,
which a e co e ed by isk-weigh ed asse s (RWAs). Acco ding o CRR A icle 153 [17],
he dependence o RWAs on he p obabili y o de aul including model unce ain y is
desc ibed by he unc ion RWA: [0,1] →R≥0, ia RWA(0) = 0, RWA(1) = 1, and
RWA(p) := 1.06 ·12.5·EaD ·LGD NG(p)−pR(p)·G(0.999)
p1−R(p)−p!(22)
wi h
R(p) := 0.12 ·1−e−50p
1−e−50 + 0.24 ·1−1−e−50p
1−e−50  o all p∈(0,1),
whe e Nis he cumula i e dis ibu ion unc ion o he s anda d no mal dis ibu ion,
Gdeno es he in e se dis ibu ion unc ion o he s anda d no mal dis ibu ion, and R
can be seen as a co ela ion ac o .1On a p ac ical le el, a equen choice o he PD
including model unce ain y is he ‘bes -es ima e’ PD o he a ing sys em mul iplied
wi h sui able a ma gin o conse a ism (1 + MoC as a mul iplie ). In o de o ge a
be e unde s anding o he RWA o mula, we b ie ly explain he e ms EaD and LGD.
The exposu e a de aul (EaD) can be seen as he amoun o c edi s o a bo owe a
he ime o i s de aul , o ins ance 1 million. The loss gi en de aul (LGD) on he
o he hand is he heigh o he loss in ela ion o he amoun o exposu e a he ime
o de aul , i.e., i is a numbe be ween 0 and 1.
Risk-weigh ed asse s a e essen ial o inancial ins i u ions’ capi al equi emen s,
since hey mus hold a leas 8% o RWAs as equi y capi al. Hence, hey play an
impo an ole o inancial ins i u ions’ isk p o isions. In compa ison o he expec ed
loss (EL), which is calcula ed as he p oduc o ‘bes -es ima e’ PD, LGD, and EaD.
RWAs ocus on unexpec ed losses om exposu es, which show di e en cha ac e is ics
compa ed o expec ed losses.
In Figu e 1, we see he dependence o RWAs on p, whe e we assume he EaD o be
1 and he LGD o be 0.4. The s anda diza ion o se ing he EaD as 1 is also e e ed
1Fo simplici y, we assume he e ec i e ma u i y Min A icle 153 o he CRR [17] o be equal o 1.
30 MAX NENDEL AND JAN STREICHER
Since he map [0,1] →R, 7→ R1
1− qQ(s) dsis conca e o all Q∈ P, i ollows ha T
sa is ies (17) by Rema k 55.□
Rema k 57. Al e na i ely, Lemma 54 and Lemma 56 can also be p o ed wi hou
in oking he Kusuoka ep esen a ion o law-in a ian isk measu es. In his case, one
uses he ac ha , o e e y p obabili y measu e Q, which is absolu ely con inuous
w. . . P, e e y se C∈ F wi h P(C)>0, and e e y λ∈(0,1), he e exis s a se A∈ F
wi h A⊂C,P(A) = λP(C), and Q(A)≥λQ(C). In ac , le
:= in s > 0
PndQ
dP> so∩C≤λP(C).
Since (Ω,F,P) is a omless, he e exis s a se B∈ F wi h B⊂dQ
dP= and
PB∪ndQ
dP> o∩C=λP(C)>0.
Then, o A:= B∪dQ
dP> ∩C,
Q(C)≤ P(C A) + Q(A)≤P(C A)
P(A)EP1AdQ
dP+Q(A)
≤P(C)
P(A)Q(A) = Q(A)
λ.
No e ha P(C A) = (1 −λ)P(C) and Q(C A)≤(1 −λ)Q(C).
Appendix B. P oo s o Sec ion 2
P oo o Theo em 10.We i s show ha ϱF(X) = ϱ(X) o all X∈C. In o de o do
so, le X∈C. Since F(X−X) = F(0) = 0, i ollows ha ϱF(X)≤ϱ(X). On he
o he hand, ϱ(X)≤ϱ(X0) o all X0∈Cwi h F(X−X0)≤0, and we ob ain ha
ϱ(X)≤ϱF(X). Since Ccon ains all cons an s, we ha e al eady e i ied P ope y (ii)
in De ini ion 2 o ϱF. In o de o p o e P ope y (i), le X, Y ∈Bbwi h X≤Y. Since
Fis mono one,
X0∈CF(X−X0)≤0⊂X0∈CF(Y−X0)≤0,
which implies ha ϱF(X)≤ϱF(Y). □
P oo o Co olla y 11.Le X∈Bb. Since
ϱ(X)≤ϱ(X0) = ϱ(X0) o all X0∈Cwi h X≤X0,
i ollows ϱ(X)≤ϱsup(X). □
Appendix C. P oo s o Sec ion 3
P oo o P oposi ion 15.In o de o p o e pa a), le α∈(0,1). Then,
ϱ(X−m)≤ϱ(Y−m) o all m∈Rand X, Y ∈Bbwi h X≤Y.
Hence, VaRα
ϱ(X)≤VaRα
ϱ(Y). Since ϱ(0) = 0, i ollows ha VaRα
ϱ(0) = 0. Mo eo e ,
by de ini ion o VaRα
ϱ, VaRα
ϱ(X+m) = VaRα
ϱ(X) + m o all X∈Bband m∈R. We
ha e he e o e shown ha VaRα
ϱis a mone a y isk measu e.
Nex , we p o e pa b). To ha end, le X∈Bb. Fi s assume ha ϱ(X) = 1. Then,
VaRα
ϱ(X)>0 o all α∈(0,1), so ha
in α∈(0,1) VaRα
ϱ(X)≤0∪{1}=1=ϱ(X).

DEFAULT RISK MEASURES 31
Nex , assume ha ϱ(X)<1. Then, o all α∈(0,1), VaRα
ϱ(X)≤0 i and only i
ϱ(X)≤α. Hence,
in α∈(0,1) VaRα
ϱ(X)≤0∪{1}=ϱ(X).
We p oceed wi h he p oo o pa c). I VaRα
ϱis posi i ely homogeneous o all α∈
(0,1), hen ϱis scaling in a ian by pa b). On he o he hand, i ϱis scaling in a ian ,
hen, o all α∈(0,1), X∈Bb, and λ > 0,
m∈Rϱ(λX −m)≤α=λm ∈Rϱ(X−m)≤α.
Hence, o all α∈(0,1), X∈Bb, and λ > 0, i ollows ha
VaRα
ϱ(λX) = λVaRα
ϱ(X).
I emains o p o e pa d). I ϱis quasi-con ex, i ollows ha he le el se
Aα:= {X∈Bb|ϱ(X)≤α}
is con ex o all α∈(0,1). Hence, by [22, P oposi ion 4.7],
VaRα
ϱ(X) := in {m∈R|X−m∈ Aα}, o X∈Bb,
de ines a con ex isk measu e o all α∈(0,1). Nex , assume ha VaRα
ϱis con ex o
all α∈(0,1). Then, o all α∈(0,1), λ∈(0,1), and X, Y ∈Bbwi h ϱ(X)≤αand
ϱ(Y)≤α, i ollows ha
VaRα
ϱλX + (1 −λ)Y≤λVaRα
ϱ(X) + (1 −λ) VaRα
ϱ(Y)≤0,
so ha , by pa b),
ϱλX + (1 −λ)Y≤α.
This shows ha ϱis quasi-con ex. □
P oo o Theo em 19.Le X, Y ∈Bbwi h X≤Y. Since Rα(X)≤Rα(Y) o all
α∈(0,1), i ollows ha ϱ(X)≤ϱ(Y). Mo eo e , Rα(0) = 0 o all α∈(0,1), so
ha ϱ(0) = 0. On he o he hand, o all m∈Rwi h m > 0, Rα(m) = m > 0 o all
α∈(0,1). The e o e,
α∈(0,1) Rα(X)≤0∪{1}={1}
and i ollows ha ϱ(X) = 1. We ha e he e o e shown ha ϱis a de aul isk measu e.
I emains o show he equali y Rα= VaRα
ϱ o all α∈(0,1). To ha end, le
α∈(0,1) and X∈Bb. Fi s , obse e ha
ϱX−Rα(X)≤α.
Hence, VaRα
ϱ(X)≤Rα(X). Now, le β∈(α, 1) and m∈Rwi h ϱ(X−m)≤α.
Then, Rβ(X−m)≤0, which implies ha Rβ(X)≤m. Taking he sup emum o e
all β∈(α, 1) and he in imum o e all m∈Rwi h ϱ(X−m)≤α, i ollows ha
Rα(X)≤VaRα
ϱ(X). □
P oo o P oposi ion 20.The implica ion (ii) ⇒(i) ollows om P oposi ion 15 b), once
we ha e shown ha (i) implies (ii). To ha end, le X∈Bband α∈(0,1). Then, o
all m∈Rwi h
sup
P∈P
P(X > m) = ϱ(X−m)≤α,
32 MAX NENDEL AND JAN STREICHER
i ollows ha m≥VaRα
P(X) o all P∈ P. This implies ha
sup
P∈P
VaRα
P(X)≤VaRα
ϱ(X).
Now, le ε > 0 and
m:= sup
P∈P
VaRα
P(X) + ε.
Then, o all P∈ P,P(X > m)≤α. This implies ha
VaRα
ϱ(X)≤m= sup
P∈P
VaRα
P(X) + ε.
Taking he limi ε↓0, he claim ollows. □
P oo o P oposi ion 22.Again, he implica ion (ii) ⇒(i) ollows om P oposi ion 15
b), once we ha e shown ha (i) implies (ii). To ha end, le X∈Bband α∈(0,1).
Then, o all m∈Rwi h
TP(X > m)=ϱ(X−m)≤α,
i ollows ha P(X > m)≤T−1(α). Now, le m∈Rwi h P(X > m)≤T−1(α). Since
Tis lowe semicon inuous, i ollows ha
TP(X > m)≤α.
Hence,
m∈Rϱ(X−m)≤α=m∈RP(X > m)≤T−1(α).
Taking he in imum, bo h, on he le and he igh -hand side, he claim ollows. □
Appendix D. P oo s o Sec ion 4
P oo o P oposi ion 26.Le X∈Bbwi h X≥0. In Rema k 25 a), we ha e al eady
seen ha ϱ(X)≤ϱ(1{X>0}). On he o he hand, o all ε > 0,
ϱX+ε1{X>0}≥ϱε1{X>0}=ϱ1{X>0}
and, using again Rema k 25 a),
ϱ(X)≥ϱX1{X>ε}≥ϱε1{X>ε}=ϱ1{X>ε}.
The p oo is comple e. □
P oo o P oposi ion 28.We s a wi h he implica ion (i) ⇒(ii). Le m > 0 and X∈
Bbwi h X≥0. Then, ϱ(X)≤ϱX+m1{X>0}. Mo eo e , o e e y λ∈(0,1),
ϱX+m1{X>0}=ϱλX +λm1{X>0}≤ϱX+λm1{X>0}.
The e o e, by assump ion,
ϱX+m1{X>0}≤in
λ∈(0,1) ϱX+λm1{X>0}=ϱ(X).
In o de o p o e he implica ion (ii) ⇒(i), le X∈Bbwi h X≥0 and λ > 0. I X= 0
o λ= 1, i ollows ha ϱ(λX) = ϱ(X). The e o e, assume ha sup X > 0 and λ= 1.
Fi s , we conside he case, whe e λ > 1. Then, λX ≥X, so ha ϱ(λX)≥ϱ(X).
Mo eo e , λX ≤X+(λ−1) sup X1{X>0}. De ining m:= (λ−1) sup X, i ollows ha
ϱ(λX)≤ϱX+m1{X>0}=ϱ(X).
Now, le λ < 1. Then, ϱ(λX)≤ϱ(X). On he o he hand,
X≤λX + (1 −λ)(sup X)1{X>0}=λX + (1 −λ)(sup X)1{λX>0}.
DEFAULT RISK MEASURES 33
The e o e, de ining m:= (1 −λ) sup X, we ind ha
ϱ(X)≤ϱλX +m1{λX>0}=ϱ(λX).
By P oposi ion 26, (i) implies (iii) and, i ially, (iii) implies (i). □
P oo o Lemma 30.T i ially, (i) ⇒(ii) and (ii) ⇒(iii). In o de o p o e he emaining
implica ion (iii) ⇒(i), le X∈Bb. Then, X≤X+and he e o e ϱ(X)≤ϱ(X+). I
X≥0, hen X=X+and i ollows ha ϱ(X) = ϱ(X+). The e o e, assume ha
in X < 0 and de ine m:= −in X. Then,
X+−m1{X+=0}=X+−m1{X≤0}≤X.
By assump ion, we ob ain ha
ϱ(X+) = ϱX+−m1{X+=0}≤ϱ(X).
□
Appendix E. P oo s o Sec ion 5
P oo o P oposi ion 36.The implica ion (i) ⇒(ii) ollows om Rema k 35 and he
implica ion (ii) ⇒(i) is i ial. □
P oo o P oposi ion 37.Clea ly, (iii) implies (ii). The implica ion (ii) ⇒(i) ollows di-
ec ly om he ac ha , o any sequence (Xn)n∈N⊂Bbwi h Xn↗X∈Bbas
n→ ∞,{Xn>0} ⊂ {Xn+1 >0} o all n∈Nand
[
n∈N{Xn>0}={X > 0}.
I emains o p o e he implica ion (i) ⇒(iii). Le (Xn)n∈N⊂Bband X∈Bbwi h
Xn↗Xas n→ ∞. By po en ially adding ∥X1∥∞ o Xand Xn o all n∈Nand
using he ac ha he Choque in eg al is a mone a y isk measu e, we may w.l.o.g.
assume ha X1≥0. Then, using he mono one con e gence heo em,
lim
n→∞ZXndc= lim
n→∞Z∞
0
ϱ(Xn−s) ds=Z∞
0
ϱ(X−s) ds=ZXdc.
The p oo is comple e. □
P oo o P oposi ion 40.Clea ly (ii) implies (i). We p o e he non i ial implica ion
(i) ⇒(ii). Le Odeno e he se o all open subse s o Ω and c(B) := ϱ(1B) o all
B∈ O. Since ϱis submodula wi h (14), i ollows ha
ϱ(X) = c{X > 0} o all X∈Lb.
The con inui y om below o ϱimplies ha cSn∈NBn= limn→∞ c(Bn) o all se-
quences (Bn)n∈N⊂ O wi h Bn⊂Bn+1 o all n∈N. The s a emen now ollows om
[34, Co olla y 2.6]. □
P oo o P oposi ion 42.We i s p o e he implica ion (i) ⇒(iii). Le (Xn)n∈N⊂Bb
wi h Xn↘0 as n→ ∞. Using he mono one con e gence heo em,
lim
n→∞ZXndc= lim
n→∞Z∞
0
ϱ(Xn−s) ds= 0.
Clea ly, (iii) implies (ii), and i emains o p o e ha (ii) implies (i). To ha end,
obse e ha , o e e y sequence (Xn)n∈N⊂Bbwi h Xn↘0 as n→ ∞ and ε > 0,
34 MAX NENDEL AND JAN STREICHER
{Xn+1 > ε}⊂{Xn> ε} o all n∈Nand Tn∈N{Xn> ε}=∅. Hence,
lim
n→∞ϱ(Xn−ε) = lim
n→∞ϱ1{Xn>ε}= 0.
The ep esen a ion ia (coun ably addi i e) p obabili y measu es now ollows om he
s anda d heo y on cohe en isk measu es, c . [22]. □
Appendix F. P oo s o Sec ion 6
P oo o Theo em 44.Clea ly, (ii) implies (i). Fo he o he implica ion, le
P:= p∈[0,1] ∃X∈C:P(X > 0) = p
Thus, o any p∈P, he e exis s some Xp∈Cwi h P(Xp>0) = pand, by ou global
assump ion on C,Ip:= 1{Xp>0}∈C. Fo p∈[0,1], we de ine
T(p) := ϱ(Iqp) wi h qp:= sup [0, p]∩P.
As a esul T(p) = ϱ(Ip) o all p∈P. In pa icula , T(0) = 0 and T(1) = 1. Le
X∈Cand pX:= PDP(X) = P(X > 0). Then, by assump ion,
ϱ(X) = ϱ1{X>0}=ϱ(IpX) = T(pX) = TPDP(X).
□
P oo o Theo em 47.In iew o Theo em 44, we only ha e o p o e ha he dis o ion
unc ion in (ii) is nondec easing. To ha end, assume ha ϱ(X) = TPDP(X) o all
X∈Cwi h a dis o ion unc ion T: [0,1] →[0,1]. Le p, q ∈P. Since Ccon ains an
o de ed subse , he e exis X, Y ∈Cwi h PDP(X) = p, PDP(Y) = q, and {X > 0} ⊂
{Y > 0}. Due o he mono onici y o ϱ, i ollows ha
T(p) = ϱ(X) = ϱ1{X>0}≤ϱ1{Y >0}=ϱ(Y) = T(q).
The p oo is comple e. □
P oo o P oposi ion 50.Fi s assume ha Qsa is ies (20). Then,
T(p)≥Zp
0
qQ(1 −s) ds=Z1
1−p
qQ(s) ds o all p∈[0,1].
The e o e, using [22, Lemma 4.60],
TPDP(X)≥Z1
1−PDP(X)
qQ(s) ds≥PDQ(X) o all X∈Bb.
On he o he hand, i (Ω,F,P) is a omless and Qsa is ies (21), i ollows om (18)
ha
T(p)≥Z1
1−p
qQ(s) ds=Zp
0
qQ(1 −s) ds o all p∈[0,1].
□
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Cen e o Ma hema ical Economics, Biele eld Uni e si y, Ge many
Email add ess:[email p o ec ed]
Cen e o Ma hema ical Economics, Biele eld Uni e si y, Ge many and Landesbank
Baden-W¨
u embe g, S u ga , Ge many
Email add ess:[email p o ec ed]