Credit rating and pricing: Poles apart

Blöchlinge , And eas
A icle
C edi a ing and p icing: Poles apa
Jou nal o Risk and Financial Managemen
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Blöchlinge , And eas (2018) : C edi a ing and p icing: Poles apa , Jou nal o
Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 11, Iss. 2, pp. 1-26,
h ps://doi.o g/10.3390/j m11020027
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Jou nal o
Risk and Financial
Managemen
A icle
C edi Ra ing and P icing: Poles Apa
And eas Blöchlinge 1,2,† ID
1Swisscan o In es by Zü che Kan onalbank, Ha ds asse 201, CH-8005 Zu ich, Swi ze land;
[email p o ec ed] o [email p o ec ed]; Tel.: +41-44-292-4580
2Uni e si y o Zu ich, Rämis asse 71, CH-8006 Zu ich, Swi ze land
† Cu en add ess: Ha ds asse 201, CH-8005 Zu ich, Swi ze land
Recei ed: 3 Ma ch 2018; Accep ed: 12 May 2018; Published: 23 May 2018
Abs ac :
Co po a e c edi a ings emo e he in o ma ion asymme y be ween lende s and bo owe s
o ind an equilib ium p ice. S uc u ed inance a ings, howe e , a e in o ma ionally insu icien
because he sys ema ic isk o equally a ed asse s can a y subs an ially. As I demons a e in a Mon e
Ca lo analysis, highly- a ed s uc u ed inance bonds can exhibi a highe non-linea sys ema ic
isks han lowly- a ed co po a e bonds. I alue c edi ins umen s unde a ou -momen CAPM,
be ween and wi hin some ma ke s he e is no one- o-one ela ion be ween expec ed loss ( a ing) and
c edi sp ead (p icing). The linea CAPM be a is insu icien , buye s and selle s need also he same
in o ma ion on non-linea isk o ha e an equilib ium.
Keywo ds:
asse backed secu i y (ABS); con ingen con e ible bond (CoCo); s anda d isk a e sion;
capi al asse p icing model (CAPM); UBS c isis
1. In oduc ion
The expec ed loss o a c edi ins umen comp ises an assessmen o de aul p obabili y as
well as loss expec a ion in he e en o a c edi de aul . The de aul isk is e lec ed in he a ing
assignmen s o he majo c edi a ing agencies such as S anda d and Poo ’s, Fi ch, and Moody’s.
Fo ins ance, i is Moody’s in en ion ha he expec ed loss a e associa ed wi h a gi en a ing
symbol and ime ho izon o be he same ac oss obliga ions o ensu e a consis ency o meaning
(see Moody’s In es o s Se ice (2009), p. 6). The same a ing assigned o bond obliga ions issued
by a non inancial co po a ion, bank, insu ance, so e eign, subso e eign bo owe , o a s uc u ed
inance obliga ion mus imply he same expec ed loss. O iginally, “Moody was in e ec add essing
he s abili y o he secu i y’s c edi sp ead,” Moody’s In es o s Se ice (2009) (p. 6). The idea ha
each a ing class ansla es in o a a ing-speci ic c edi sp ead can also be ound in mode n inance
(see, e.g., Ja ow e al. (1997), Figu es 5 and 6).
Ra ings educe he knowledge gap, o “in o ma ion asymme y,” be ween bo owe s (selle s)
and lende s (buye s) by p o iding an o dinal assessmen abou he expec ed loss. I will demons a e
ha he ad en o s uc u ed inance obliga ions and o he c edi de i a i es has comple ely b oken
down he mono one ela ion be ween expec ed loss ( a ing) and c edi sp ead (p icing). E en i he
a ing uly con eys an unbiased, powe ul es ima e o he expec ed loss o an unde lying obliga ion,
1
a ings alone a e in o ma ionally insu icien o he p icing o colla e alized deb obliga ions (CDOs)
and o he c edi ins umen s.
2
I will show ha in equilib ium an in es men - a ed s uc u ed inance
1
I will assume ha a ings ep esen powe ul, unbiased o ecas s e en hough he e is e idence ha agency a ings a e
no he mos powe ul p edic o s (see e.g., Blöchlinge and Leippold (2018)). Nowadays, new es s a is ics allow an easy
alida ion o he unbiasedness o de aul o ecas s (see Blöchlinge (2017)).
2
CDOs we e a he hea o he 2007–2008 inancial c isis. The CDO is he p o o ypical s uc u ed inance secu i y and is a
ype o s uc u ed asse -backed secu i y (ABS). A CDO is a p omise o pay in es o s in a p ede ined sequence, based on he
J. Risk Financial Manag. 2018,11, 27; doi:10.3390/j m11020027 www.mdpi.com/jou nal/j m
J. Risk Financial Manag. 2018,11, 27 2 o 26
obliga ion can ha e a signi ican ly highe c edi sp ead han a subin es men - a ed co po a e bond
due o sys ema ic isk. Tha is, a ing and p icing can be a apa since he e is no longe a one- o-one
ela ion be ween expec ed loss and sys ema ic isk o s uc u ed inance obliga ions.
3
Howe e ,
a ma ke in which po en ial buye s know he ue a ing bu canno co ec ly judge he sys ema ic isk
a ac s selle s o e ing c edi de i a i es o good a ing quali y bu o low sys ema ic isk quali y which
can in he end lead o a ma ke b eakdown. Thus, I will demons a e ha a necessa y p econdi ion o
a c edi ma ke o ha e an equilib ium a all is o ha e symme ic in o ma ion be ween issue s and
in es o s on a ing and sys ema ic isk.
The be a acco ding o he CAPM o T eyno (1962), Sha pe (1964), Lin ne (1965), and
Mossin (1966)
based on he mean- a iance c i e ion o Ma kowi z (1952) e lec s he sys ema ic isk o an
asse in he o m o a scaled co ela ion wi h he ma ke .
4
Howe e , as highligh ed by
Emb ech s e al. (1999), co ela ion is only a linea isk measu e and o limi ed sui abili y o measu ing
dependence. In a s ylized ma ke , I will demons a e ha he non-linea sys ema ic isk o c edi
de i a i es is highly subs an ial and e en mo e impo an han he linea CAPM be a. I he eby
o e ano he explana ion o he high expec ed e u n o highly- a ed deb secu i ies gi en he low
CAPM be a.5
Acco ding o he c i icism o K ugman (2009), inancial economis s a ely ask he seemingly
ob ious ques ion o whe he asse p ices make sense gi en eal-wo ld undamen als. Ins ead, hey ask
only whe he asse p ices make sense gi en o he asse p ices. The cen al insigh o asse p icing
is ha in he absence o a bi age he e exis s a isk-neu al measu e
Q
equi alen o he eal-wo ld
p obabili y
P
(see Ha ison and K eps (1979)). Hence, o ind he equilib ium alue o any de i a i e
you can assume a isk-neu al wo ld wi hou making s a emen s abou he eal wo ld. Howe e , o be
in equilib ium, ma ke pa icipan s need i s a consensus on all p icing- ele an a ibu es. The pape
o Collin-Du esne e al. (2012) is a case in poin ele an o he p icing o CDOs. The con ibu ion o
Collin-Du esne e al. (2012) “is o in es iga e he ela i e p icing ac oss he s ock op ion and CDO
ma ke s” and hey p o ide a model “ o join ly p ice long-da ed S&P 500 op ions and anche sp eads
on he i e-yea CDX index.” Summe s (1985) once cha ac e ized inancial economis s wi h a pa able
abou “ke chup economis s” who “ha e shown ha wo-qua bo les o ke chup in a iably sell
o exac ly wice as much as one-qua bo les o ke chup,” and conclude ha he ke chup ma ke
is pe ec ly e icien . Collin-Du esne e al. (2012) use he model o Du ie e al. (2000) in which
e u n dynamics unde he isk-neu al measu e a e speci ied and hey emphasize he impo ance o
“ca as ophic” isk-neu al jumps o he p icing o highly- a ed secu i ies. Collin-Du esne e al. (2012)
emain silen on whe he hese “ca as ophic” jumps a e also a eal-wo ld phenomenon o induced by
isk a e sion and he e o e “only” a phenomenon in he isk-neu al wo ld.
I de i e a ou -momen CAPM unde s anda d isk a e sion o cha ac e ize he quali y o a c edi
ins umen by ou undamen al eal-wo ld ac o s (i.e., h ee sys ema ic isk ac o s in addi ion o
he a ing ac o ) o p o ide an elemen a y ela ion be ween isk and p ice. I show ha a e sion
o a ails p ima ily a ec s he p icing o senio deb secu i ies whe eas a iance a e sion has a
p opo ionally highe impac on equi y. Unlike equi y, he sys ema ic isk o senio deb is domina ed
cash lows he CDO collec s om he unde lying pool o asse s. The CDO is “sliced” in o “ anches”. Each CDO anche
ecei es he cash low in sequence based on i s p io i y/senio i y. In a Financial Times a icle by Jones (2008), s uc u ed
inance was conside ed “ he single mos impo an in en ion in inance, i no economics, in he pas ew decades.”
3
Liquidi y and axa ion bo h play a ole in he p icing o deb secu i ies. I, like o he s udies, will abs ac om hese wo
quali y ac o s. El on e al. (2001) show ha he expec ed loss ( a ing) can explain less han a qua e o he a ia ion in he
c edi sp ead. Besides a ing, liquidi y, and axa ion, he sys ema ic isk (see Gabbi and Si oni (2005); Chen e al. (2007);
Longs a e al. (2005); Co al e al. (2009a); Blöchlinge (2011)) is highly p icing- ele an .
4Co al e al. (2009a) a gue ha “unlike ac ua ial claims, whose de aul p obabili y is un ela ed o he economic s a e (β=0),
bonds a e economic asse s and ha e posi i e CAPM be as (
β>
0)” (p. 638). The impo an implica ion o he CAPM is ha
he scaled co ela ion be ween indi idual asse e u ns and he ma ke e u n de ines he sys ema ic isk and ma e s o
p icing. The emaining isk is o en assumed o be idiosync a ic, can be di e si ied away and commands no p emium.
5This obse a ion is called he low be a anomaly and was epo ed by Jensen e al. (1972).
J. Risk Financial Manag. 2018,11, 27 3 o 26
by non-linea coku osis a he han linea co a iance isk, i.e., I explain he “ isk-neu al ca as ophic
jumps” a ec ing senio deb by ku osis a e sion. Ins ead o exp essing he CAPM adi ionally
in e ms o ela i e e u ns—which was also hea ily c i icized by K ugman (2009)—I exp ess he
equilib ium p ice in e ms o he p omised end-o -pe iod ace amoun wi h he expec ed loss a e
as a di ec inpu . The ou ac o s—expec ed loss a e, co a iance, coskewness, coku osis—can be
in e p e ed as quali y a ibu es in he spi i o Ake lo (1970). In my model, he ac o s o each asse
can simply be agg ega ed o ob ain he co esponding ac o s o he po olio.6
Finally, I will show in an empi ical applica ion ha he huge losses a he ading desks o wo
Wall S ee i ans, namely Mo gan S anley and UBS, we e basically ealized wi h a CAPM be a o
ze o bu none heless wi h conside able non-linea sys ema ic isk. These “hedged” s uc u ed inance
po olios had no co a ia ion wi h he ma ke and we e no economic asse s (
β>
0) in he sense
o Co al e al. (2009a) bu un o una ely also no ac ua ial claims whose isk could be di e si ied
away in la ge po olios. Mo gan S anley’s and UBS’s be a-neu al s uc u ed inance po olios we e
ne e heless ea ed as ac ua ial claims in he co esponding isk depa men s despi e he po olios’
inhe en non-linea sys ema ic isk. E en wo se, hey misin e p e ed he small p emium o non-linea
sys ema ic isk as CAPM alpha and he e o e le e aged hei posi ions. Thus, I p o ide empi ical
e idence ha e en “ oo big o ail” banks we e exposed o huge isks wi hou p ope assessmen abou
all quali y a ibu es by conside ing only a ing and co ela ion/be a.
My indings a e highly ele an since s uc u ed inance ma ke s such as he ma ke o asse
backed secu i ies (ABSs) imp o e he e iciency o esou ce alloca ion and help con ain sys emic isk by
eeing up he banks’ balance shee s. Gi en he absence o symme ic in o ma ion, my model o e s an
explana ion why public ABS issuances emain low in he EU as epo ed by he BOE and ECB (2014).
The low demand o ABSs is un o una e since “ABS can suppo he ansmission o accommoda i e
mone a y policy in condi ions whe e he bank lending channel may o he wise be impai ed” (p. 2).
I p oceed as ollows: Sec ion 2highligh s he di e ence be ween linea sys ema ic isk
(CAPM be a), non-linea sys ema ic isk and idiosync a ic isk. In Sec ion 3I de i e a simple
ou -momen equilib ium CAPM. Sec ion 4demons a es ha a c edi ma ke unde asymme ic
in o ma ion abou non-linea sys ema ic isk is in a disequilib ium. Sec ion 5shows in an illus a i e
ma ke ha some c edi p oduc s can ha e supe b a ing quali y bu also high sys ema ic isk exposu es
o exac ly he o he way a ound. I also discuss he in luence o coun e pa y isk on c edi de i a i es.
Sec ion 6in es iga es he empi ical cases o Mo gan S anley and UBS. Finally, Sec ion 7concludes.
2. CAPM Be a and P emium o Residual Risk
C edi po olio dis ibu ions a e ypically cha ac e ized by non-linea isks such as skewness
and hea y ails. Howe e , many pape s on c edi po olio isk such as Co al e al. (2009a,2009b);
Hame le e al. (2009); B ennan e al. (2009) la gely igno e he p icing o non-linea sys ema ic
isks. In hei CAPM-like models, based on he c edi po olio amewo k o Me on (1974),
an asse unco ela ed wi h he ma ke po olio
(β=
0
)
is also assumed o be s ochas ically
independen and he e o e an ac ua ial claim. Howe e , s ochas ic independence is oo s ong
an assump ion, he cash lows o a be a-neu al po olio may s ill con ain signi ican non-linea
sys ema ic isk which—unlike he isk o ac ua ial claims—canno be di e si ied away in a
6
Acco ding o MacKenzie (2011), he ma ke o CDOs would ha e been qui e limi ed i pa icipa ion in he ma ke equi ed
he p ope unde s anding o he inhe en (sys ema ic) isks. Ra ings “black boxed” hese complexi ies. Ra ings pe mi ed
p ices o di e en CDOs o be compa ed, bo h wi h each o he and wi h mo e amilia c edi ins umen s such as single-name
co po a e bonds, by compa ing he c edi sp ead o e ed by a gi en c edi ins umen o ha o e ed by o he s wi h he same
a ing. This p icing- a ing nexus was hus a con en ion in he sense o Young (1996) “economics o con en ion”: a way
o u ning unce ain y in o a o m o o de ha is s able enough o pe mi coo dina ion and (non- a ional, unsus ainable,
sho - e m) equilib ium. The subsequen ealiza ion o in es o s ha he e a e u he bu unknown quali y ac o s
besides he a ing ende ed coo dina ion impossible wi h no ( a ional, long- e m) equilib ium as p edic ed by he model o
Ake lo (1970).
J. Risk Financial Manag. 2018,11, 27 4 o 26
la ge po olio and mus be p iced. Fo ins ance, Co al e al. (2009a) eso o a kind o
Black and Scholes (1973) model in disc e e ime in he spi i o Rubins ein (1976), bu hey make
a c ucially di e en assump ion by changing he Gaussian assump ion ega ding he ma ke po olio:
7
P oposi ion 1
(
Black and Scholes (1973) in disc e e ime
)
.
I he ma ke po olio
M
and he payo o
asse P ollow a bi a ia e log-no mal dis ibu ion such ha
log M
log P!∼N µM
µP!, σ2
MσMσPρ
σMσPρ σ2
P!!,
whe e
ρ
is he co ela ion and i he on Neumann-Mo gens e n u ili y unc ion
u(·)
o he ep esen a i e agen
exhibi s cons an ela i e isk a e sion wi h isk a e sion coe icien
λ
hen he p ice
qP
o end-o -pe iod payo
P
is gi en by
qp=q0exp µp−λρσMσP+1
2σ2
P
. The Radon-Nikodym de i a i e
Z=M−λ/EM−λ
induces he isk-neu al measu e Q:
log M
log P!Q
∼N µQ
M
µQ
P!, σ2
MσMσPρ
σMσPρ σ2
P!!,
whe e
µQ
P=µP−λρσMσP=log qP−log q0−
0.5
σ2
P
and
µQ
M=µM−λσ2
M=log qM−log q0−
0.5
σ2
M
a e he shi ed means unde Q.
The p oo can be ound in he Appendix. By linea p ojec ion o he loga i hmic po olio cash
low
log P
on o he loga i hmic cash low o he ma ke po olio
log M
, one ob ains an addi i e,
mean-squa e e icien decomposi ion in o ma ke and esidual/idiosync a ic isk:
log P−µQ
P=βlog M−µQ
M+σPq1−ρ2ν,
log P−µQ
P
σP
=ρlog M−µQ
M
σM
+q1−ρ2ν, wi h ν∼N(0, 1), (1)
wi h
β=ρ σP/σM
deno ing he CAPM be a. The inclusion o u he ans o ma ions o
M
such
as polynomial expansions canno imp o e he goodness-o - i (see Hamil on (1994), p. 102). Due o
no mali y, he sys ema ic isk exposu e is comple ely desc ibed by
β
. Tha is, he esiduum
ν
ollows a
s anda dized no mal dis ibu ion unde he eal-wo ld measu e
P
as well as unde he isk-neu al
measu e
Q
, so he e is no p emium associa ed wi h his esidual isk. Hence, he alue o any inancial
de i a i e o P— he p ice EQ[g(P)]o any σ(P)-measu able payo g(·)—can be w i en as:
EQhEQ[g(P)|M]i=Z∞
−∞Z∞
−∞
g(ξ)
ξσPp1−ρ2h

log ξ−µQ
P−βlog m−µQ
M
σPp1−ρ2
dξ Q(m)dm,
whe e
Q(·)
is he p obabili y densi y unc ion (pd ) o he ma ke ac o
M
unde he isk-neu al
measu e
Q
,
h(·)
is he p obabili y densi y unc ion o
ν
unde
P
and
Q
. Unde he assump ion o
Rubins ein (1976), Q(·)and h(·)bo h co espond o he Gaussian pd φ(·).
7
Rubins ein (1976) de i es su icien condi ions unde which he op ion-p icing o mula o Black and Scholes (1973) in
con inuous ime applies also in disc e e ime. Rubins ein (1976) ema ks ha “since he ime in e al be ween da es can be
made a bi a ily small in disc e e ime models, hey a e in his espec o g ea e gene ali y [ han con inuous ime models].”
Fama (1970) p o ed ha e en hough a isk a e e maximized he expec ed u ili y om he s eam o consump ion o e his
li e ime, his choices in each pe iod would be indis inguishable om ha o a p ope ly speci ied isk a e se in es o wi h a
singe-pe iod ho izon.

J. Risk Financial Manag. 2018,11, 27 5 o 26
A common ailing when gi ing up he Gaussian assump ion is o p ese e—as you will
see la e — he easily ejec able assump ion ha he esidual
ν
is independen om
M
so ha
ν
is idiosync a ic, can be di e si ied away and has he same dis ibu ion unde
P
and
Q
.
Co al e al. (2009a); B ennan e al. (2009); Hame le e al. (2009) all analyze he CDO ma ke unde
assumed independence be ween
M
and
ν
. Howe e , unde non-no mali y, o hogonali y only implies
ha
E[νlog M]=E[ν]E[log M]=
0 (see Hamil on (1994), p. 74). In he ollowing, I will conside no
only co a iance bu also coskewness and coku osis isk because agen s unde s anda d isk a e sion
ca e abou all h ee co-momen s. Fo a gene al c edi po olio
P
, I will show ha
EνM26=
0 and
ha EνM36=0. I he e o e ex end he CAPM by wo u he s a is ical (co-)momen s.
3. Fou -Momen Valua ion Model
I conside a po olio choice p oblem aced by (buy-and-hold) in es o s/indi iduals in a gene al
equilib ium in ol ing a ini e numbe o agen s and a ini e numbe o asse s. My ul ima e goal in he
ollowing sec ions is no o ind “ he bes ” mul i-pe iod p icing model o i obse able ma ke p ices
o daily ma k- o-ma ke alua ion bu o show wi hin a simple, one-pe iod model ha he non-linea
sys ema ic isk o equally a ed c edi p oduc s can be as ly di e en , and i buye s and selle s canno
ag ee on sys ema ic isk hen his c edi ma ke has no equilib ium. Thus, o e i e he ma ke o
s uc u ed inance obliga ions equi es a pa simonious se o publicly a ailable sys ema ic isk igu es
(such as sys ema ic co a iance, coskewness and coku osis)— he c edi a ing alone is a guably only
su icien o he ma ke o co po a e bonds.
I de i e a ou -momen CAPM, bu no ia he s anda d way in e ms o ela i e e u ns such
as K aus and Li zenbe ge (1976)o Ha ey and Siddique (2000), ins ead he equilib ium p ice is a
unc ion o expec ed loss a e and sys ema ic isk con ibu ions pe uni no ional. In o he wo ds,
he isk me ics a e di ec ly measu ed pe uni a isk. Fo he ime being, I assume o be in a
ep esen a i e agen economy unde s anda d isk a e sion as de ined by Kimball (1993). La e , I will
ex end he model o an asymme ic ma ke . The iple
(Ω,F,P)
is he p obabili y space and I ha e a
ep esen a i e on Neumann-Mo gens e n maximize o expec ed u ili y whose u ili y unc ion
u(·)
exhibi s he ollowing p ope ies:
(a) posi i e ma ginal u ili y o weal h, i.e., non-sa ie y, o mono onici y u0>0,
(b) dec easing ma ginal u ili y o weal h, i.e., isk a e sion, o conca i y u00 <0,
(c) dec easing absolu e isk a e sion,
(d) dec easing absolu e p udence.
The qua ic u ili y unc ion is compa ible wi h non-sa ia ion, isk a e sion, dec easing absolu e
isk a e sion, dec easing absolu e p udence, wi h posi i e coe icien s o odd powe s and nega i e
coe icien s o e en powe s (see he Appendix A o a p oo ):
Lemma 1. S anda d isk a e sion implies u0>0, u00 <0, u000 >0, and u0000 <0.
The expec ed u ili y o a
F
-measu able payo
X
can be app oxima ed by a qua ic u ili y unc ion
ia ou h-o de Taylo se ies expanded a he poin
E[X]
by assuming ha he ou h momen o
X
and ou h de i a i e o u(·)indeed exis (see also Samuelson (1970)):
E[u(X)]≈u(E[X]) +1
2! u00 (E[X]) Eh(X−E[X])2i
+1
3! u000 (E[X]) Eh(X−E[X])3i+1
4! u0000 (E[X]) Eh(X−E[X])4i.
Fo a highe o de expansion, he se ies con e ges in he case o loga i hmic and powe
u ili y unc ions i
|X−E[X]|<E[X]
,
P
almos su ely (see, Ju czenko and Maille (2006), p. 81).
Howe e , e en o di e gen Taylo se ies Hlawi schka (1994) shows ha unca ed expansions
J. Risk Financial Manag. 2018,11, 27 6 o 26
p o ide “excellen app oxima ions o expec ed u ili y o he pu pose o po olio selec ion” e en
hough momen s do no ul ill he axioms o cohe en isk measu es acco ding o A zne e al. (1999).
Lemma 2.
A qua ic on Neumann-Mo gens e n u ili y
u(·)
exhibi ing s anda d isk a e sion implies
p e e ence o igh -skewness and p e e ence o pla yku ic dis ibu ions.
Simila as in B ennan (1979), I de i e he equilib ium p ices in a one-pe iod, one-good economy
wi h a capi al ma ke wi h
K+
1 asse s wi h end-o -pe iod payo s
{Y0, ..., YK}
as hough he e exis ed
only iden ical ep esen a i e agen s, i.a., all
N
buy-and-hold in es o s ha e he same p obabili y belie s
and he same u ili y unc ion. The payo Y0is assumed o be s ic ly posi i e and non- andom.
P oposi ion 2
(Symme ic equilib ium)
.
Today’s equilib ium p ice
qk
o payo
Yk
a he end o he pe iod o
any k ∈{1, ..., K}can be exp essed in e ms o isk- ee asse 0:
qk
q0
=Eu0(w)
E[u0(w)]
Yk
Y0=EZYk
Y0=EQYk
Y0, (2)
whe e he a iable
Z=u0(w)/E[u0(w)]
has mean one and is posi i e unde he assump ion o non-sa ia ion,
i.e.,
u0>
0,
Z
he e o e ul ills all equi emen s o a Radon-Nikodym de i a i e. The Radon-Nikodym de i a i e
Z induces a measu e change om he eal-wo ld measu e P o he isk-neu al measu e Q, i.e.,
Q{A}=E[1AZ], (3)
whe e 1Ais he indica o unc ion o any e en A ∈ F.8
The p oo can be ound in he Appendix. Wi hou loss o any gene ali y, I assume ha he p ice o
he isk-less asse
q0
is exp essed pe uni no ional, i.e.,
q07→ q0/Y0
, o app oxima e he equilib ium
ela ion in (3) wi h he i s ou s a is ical momen s and i s ou ma hema ical de i a i es o u(·):
P oposi ion 3
(4-momen CAPM)
.
Unde he assump ion o a ep esen a i e on Neumann-Mo gens e n
expec a ion maximize s unde s anda d isk a e sion whose u ili y unc ion
u(·)
is app oxima ed by a ou h
o de Taylo se ies a ound he mean end-o -pe iod weal h
w
, oday’s equilib ium p ice
qX
unde he p icing
measu e Qin (3)o any F-measu able inancial de i a i e payo X can be w i en as ollows:
qX
q0
=µX−λββX−λγγX−λδδX, (4)
wi h µX=E[X], and
βX=E[(X−E[X]) (M−E[M])]
Eh(M−E[M])2i
γX=
Eh(X−E[X]) (M−E[M])2i
Eh(M−E[M])3i
δX=
Eh(X−E[X]) (M−E[M])3i
Eh(M−E[M])4i,
8
The Radon-Nikodym de i a i e
Z
is also known as p icing ke nel in inance, he measu e
Q
is called he isk-neu al
measu e since i he ep esen a i e in es o we e isk-neu al, i.e.,
u0(w) = cons
, he eal-wo ld o physical measu e
P
would coincide wi h Q.
J. Risk Financial Manag. 2018,11, 27 7 o 26
whe e
M=
1
/K∑K
k=1Yk
deno es he a e aged payo o he ma ke po olio,
βX
cap u es he co a iance isk o
X
wi h
M
,
γX
he coskewness isk,
δX
he coku osis isk. Unde s anda d isk a e sion, he p emium
λγ
o
sys ema ic skewness isk is posi i e i
M
is le -skewed and nega i e i igh -skewed. The p emia o a iance and
ku osis isk, λβ,λδ, a e posi i e.
The p oo can be ound in he Appendix. No e,
µX
,
βX
,
γX
,
δX
a e compu ed unde he physical
isk measu e
P
bu do no depend on he deg ee o isk a e sion o he o m o he u ili y unc ion
u(·)
.
On he o he hand, he isk p emia
λβ
,
λγ
, and
λδ
depend on
u0
,
u00
,
u000
, and
u0000
. Ra ing agencies p o ide
an assessmen abou he expec ed loss
µX
bu a e silen on he o he physical isk me ics
βX
,
γX
,
δX
.
Two impo an ema ks a e in o de : Fi s , since he equilib ium p ice in
(4)
is no exp essed in ela i e
e u ns like o he capi al asse p icing models such as Sha pe (1964); K aus and Li zenbe ge (1976);
o Ha ey and Siddique (2000), he isk me ics mus be exp essed ela i e o he unde lying no ional
amoun o make hem compa able ac oss ins umen s. Since he bond p ice is by con en ion exp essed as
a pe cen age o nominal alue, o compa a i e s a is ics i is necessa y o exp ess
µX
,
βX
,
γX
, and
δX
o a
inancial de i a i e Xalso ela i e o i s no ional amoun , i.e., pe uni a isk.
Second, e en i he ela ion be ween p ice and isk me ics is only app oxima ely ue in p ac ice,
he eal-wo ld isk s a is ics o a
F
-measu able payo
X
— a ing
µX
, linea sys ema ic isk
βX
,
and non-linea sys ema ic isk
γX
,
δX
—wi h espec o a well-de ined ma ke po olio (e.g., a c edi
de aul swap index like CDX o iT axx) p o ide p icing- ele an in o ma ion abou he unde lying
c edi quali y. The absence o such publicly a ailable isk me ics may esul in a non- unc ioning
ma ke due o in o ma ion asymme y be ween buye and selle o c edi isks. The ma ke o
s uc u ed inance p oduc s may e en collapse.
4. C edi Ma ke s unde Asymme ic In o ma ion
I in oduce a ma ke unde asymme ic in o ma ion in he spi i o Ake lo (1970). The isk me ics
µX
,
βX
,
γX
, and
δX
in
(4)
play he e he ole o he unde lying quali y o a c edi de i a i e wi h payo
X
. I s ill assume homogenous p obabili y belie s and isk a e sions. Fo mally, I ha e he p obabili y
space
(Ω
,
F
,
P)
, he sigma ields
{B,S}
and he isk-neu al measu e
Q
induced by he ou -momen
CAPM ke nel in
(A8)
. So a , I implici ly assumed ha he buye ’s in o ma ion se
B
and he selle ’s
sigma algeb a
S
a e equal and bo h equal o he nai e ield
{∅,Ω}
. Now, he buye s ill s a s wi h
he nai e in o ma ion
{∅,Ω}
, bu a ing agencies make public he in o ma ion o calcula e he mean
µX=E[X|S]
o
X
unde he selle ’s in o ma ion
S ⊃ {∅,Ω}
so ha a po en ial buye has hen he
sigma algeb a gene a ed by µXa ailable o decision making, i.e., B=σ(µX).
De ini ion 1
(Asymme ic c edi ma ke )
.
In an asymme ic ma ke , he s a is ical momen s condi ional
on he selle ’s in o ma ion
S
and condi ional on he buye ’s in o ma ion
B
di e . In a symme ic ma ke ,
howe e , he sigma algeb as
B
and
S
esul in he same (scaled) condi ional momen s
µX
,
βX
,
γX
, and
δX
, e.g.,
βX=E[M(X−µX)|S]/V[M|S]=E[M(X−µX)|B]/V[M|B]
, almos su ely, and I ha e again he
equilib ium in (4). Unde asymme ic in o ma ion a leas one o he isk me ics is di e en unde Band S.
In simple wo ds, in an asymme ic ma ke some p icing- ele an quali y a ibu es a e known o
selle s bu unknown o buye s. Such a ma ke canno unc ion:
P oposi ion 4
(No equilib ium)
.
An asymme ic c edi ma ke in which selle s and buye s disag ee on a
leas one o he ou quali y ac o s µX,βX,γX,δXhas no equilib ium.
J. Risk Financial Manag. 2018,11, 27 8 o 26
P oo .
To show ha a ma ke wi h such asymme ic in o ma ion canno wo k p ope ly, I wo k wi h
a p oo by con adic ion, i.e., I s a wi h he assump ion ha he e is none heless an equilib ium.
9
By
pX
I deno e he compounded equilib ium p ice
qX/q0
and by
X
he isk p emium (c edi sp ead)
o he posi i e cash low
X
wi h
P{X=0}<
1. The posi i i y assump ion is wi hou loss o gene ali y
because a payo wi h nega i e ou comes can be spli in o a long and sho posi ion o wo posi i e
cash lows. Thus, he p ice o Xis gi en by:
pX=µX− Xwhe e pX=qX/q0and X= (βX,γX,δX). (5)
F om P oposi ion 3, I know ha
pX
is indeed he equilib ium p ice unde symme ic in o ma ion
and a ou -momen CAPM when
βX
,
γX
,
δX
a e known. Howe e , now I assume ha
µX
is known bu
a leas one o he ac o s
βX
,
γX
,
δX
is only known o he selle (
S
-measu able) bu no known o he
buye (no
B
-measu able). In pa icula , I assume he e is no
B
-measu able p icing unc ion
g(·)
such
ha g(µX) = pX, bu he buye knows ha
X≤µX. (6)
The uppe bound in
(6)
mus be
µX
, o he wise an a bi age oppo uni y would a ise o a posi i e
end-o -pe iod payo
X
mus ha e a posi i e p ice wi h p obabili y one. Now, condi ional on he
in o ma ion gene a ed by
1{ X≥µX−pX}
and
µX
, he buye o payo
X
knows ha he mean isk
p emium on o e a he assumed equilib ium p ice pXis gi en by:
¯
X:=EQh X1{ X≥µX−pX},µXi.
The expec a ion is aken unde he isk-neu al measu e
Q
induced by he Radon-Nikodym
de i a i e in
(A8)
o accoun o isk a e sion. The p ice bidden
p∗
X
by he buye gi en she knows
µX
and ha he a e age isk p emium on o e is ¯
Xis he e o e gi en by:
p∗
X=µX−EQh X1{ X≥¯
X},µXi. (7)
Since
¯
X=EQh X1{ X≥µX−pX},µXi≤EQh X1{ X≥¯
X},µXi⇔p∗
X≤pX.
Howe e , he bid p ice
p∗
X
is always smalle han he assumed equilib ium p ice
pX
. I only ha e
a s ic equali y i
pX=
0 o else i
X
is
σ(µX)
-measu able so ha
¯
X= X
. Howe e , a posi i e
end-o -pe iod cash low
X
wi h
P{X=0}<
1 mus ha e a s ic ly posi i e p ice o exclude a bi age
and i
X
is a unc ion o
µX
hen he ma ke is symme ic, i.e., he a ing is su icien o p icing. I ha e
a con adic ion ha
pX
is he equilib ium p ice unde asymme ic in o ma ion. Tha is, he e is no
equilib ium p ice pXand he e o e no isk-neu al isk measu e Q. No ade akes place.
No e, he a ing can be su icien o p icing in a ou -momen CAPM. Technically speaking,
in his special case, he ac o s
βX
,
γX
,
δX
in
(5)
a e
σ(µX)
-measu able and he a ing is in o ma ionally
su icien in o de o ha e an equilib ium. Su iciency may hold o he segmen o co po a e bonds,
bu in gene al he e is no a ing-p icing nexus. On he con a y, as I will show, he isk p emium can
e en be nega i e o de i a i es wi h high expec ed losses and signi ican ly posi i e o s uc u ed
inance obliga ions o high a ing quali y. Coun e pa y isk u he complica es he quali y assessmen
9
Wi hou loss o gene ali y, I assume he e ha a ings p o ide he selle ’s in o ma ion abou he expec ed loss so ha buye s
wo k unde a non-biased mean, i.e.,
µX=E[X|S]
. Such a a ing bias may ha e played a pa du ing he inancial c isis in
2007/08. I is s aigh o wa d o show ha unde biased a ings, i.e, E[X|B]6=E[X|S], he ma ke b eaks down as well.
J. Risk Financial Manag. 2018,11, 27 15 o 26
Table 4shows he epackaging o second loss CDO anches om Table 3. Each CDO squa ed
s uc u e consis s o
n∈{2, 4, 6, 8}
unde lying CDO anches
{T2`:`=1, ..., n}
on he asse side and
a senio deb anche and an equi y anche on he liabili y side. The mo e di e si ied he unde lying
asse pool, i.e., he g ea e
n
, he highe he possible le e age
N
o a gi en a ing and he sys ema ic
isk o deb (equi y) dec eases (inc eases) wi h inc easing
n
. Howe e , a CDO squa ed deb anche
compa ed o a simple CDO anche, such as he hi d loss anche in Table 3, he expec ed payo
µ
, sys ema ic a iance
β
, and skewness
γ
a e lowe , bu he sys ema ic ku osis
δ
is highe . In a
h ee-momen CAPM all CDO squa ed deb anches in Table 4mus ha e he highe p ice compa ed
o he hi d loss anche
T3
in Table 3. Howe e , in he ou -momen CAPM, he p icing ela ion
be ween simple CDO and CDO squa ed is also in luenced by
δ
which is highe o he la e . The CDO
squa ed example highligh s he ele ance o all ou quali y ac o s o p icing isky deb .10
Table 4. Fou di e en CDO squa ed s uc u es.
T anche nNo ional NPayo µ β γ δ
Fi s loss 4 0.3750 max nPn
1−N−N
1−N, 0o0.990978 3.8883 7.6913 8.7413
Senio A 1 −max n1−Pn
N, 0o0.999219 0.6104 1.9976 3.2683
Fi s loss 6 0.4167 max nPn
1−N−N
1−N, 0o0.990344 4.1447 8.1368 9.1597
Senio A 1 −max n1−Pn
N, 0o0.999284 0.5788 1.9427 3.2291
Fi s loss 8 0.4375 max nPn
1−N−N
1−N, 0o0.989991 4.2868 8.3818 9.3864
Senio A 1 −max n1−Pn
N, 0o0.999311 0.5663 1.9241 3.2216
Fi s loss 10 0.4500 max nPn
1−N−N
1−N, 0o0.989768 4.3766 8.5356 9.5273
Senio A 1 −max n1−Pn
N, 0o0.999326 0.5596 1.9146 3.2193
Unde lying CDO po olio Pn=1
n∑n
`=1T`20.994077 2.6560 5.5513 6.6843
The unde lying asse po olio wi h payo
Pn
consis s o
n
CDO mezzanine anches. Each mezzanine anche
T`j
,
`∈{1, ..., n}
,
j=
2, has an a achmen poin
aj−1=
0.06 and a de achmen poin
aj=
0.08, and he
unde lying digi al bond po olio o each mezzanine anche consis s o 100 c edi names. The payo o he
unde lying asse po olio Pnis he e o e gi en by:
Pn=1
n∑n
`=1T`j, wi h T`j=1{d`≤aj−1}+d`−aj−1
aj−aj−11{aj−1<d`≤aj},d`=1−1
100 ∑100
k=1Y10(k−1)+`,
whe e
Yk
deno es he bina y payo o digi al bond
k
, and
d`
he de aul a e in he unde lying bond po olio
o he CDO mezzanine anche
T`j
. The a iable
N
deno es he no ional amoun o senio deb ,
(
1
−N)
he no ional amoun o equi y ( i s loss anche). Due o linea i y, he isk con ibu ions
µ
,
β
,
γ
, and
δ
o
he unde lying CDO pool
Pn
a e independen om he numbe o unde lying CDOs
n
. Howe e , due o
non-linea payo s o deb and equi y, he isk con ibu ions o he CDO squa ed anches depend on
n
(and
N
).
The mo e di e si ied he unde lying CDO pool, he highe (wo se) is he quali y o senio deb (equi y).
5.5. C edi Linked No e (CLN) unde Coun e pa y Risk
The ela ion be ween a ing and p icing can be comple ely u ned upside down as I will
demons a e o c edi linked no es (CLNs). The issue o a CLN is no obliga ed o epay he no ional
amoun in ull i a speci ied e en occu s. Wi h he s uc u ing o a c edi linked no e, I can c ea e
a payo
X
wi h a low expec ed loss bu high sys ema ic isk con ibu ions
βX
,
γX
,
δX
o he o he
way a ound, i.e., a payo wi h low a ing quali y bu o high quali y wi h espec o sys ema ic isk.
Besides he isk me ics o he unde lying payo cha ac e is ic
X
, i is he c edi quali y o he issuing
coun e pa y ha ma e s o p icing.
10
O e all, I con i m he sugges ion o Co al e al. (2007) ha he appea ance o CDO squa ed can be explained by i s high
sys ema ic isk. All A- a ed CDOs squa ed in Table 4ha e simila sys ema ic isk quali ies as A- a ed CDOs in Table 2
which a e bo h much highe compa ed o an A- a ed single-name bond in Table 1.

J. Risk Financial Manag. 2018,11, 27 16 o 26
In Table 5I c ea e ou een s uc u ed p oduc s. The i s CLN eplica es he payo o an a e age
bond po olio so ha he sys ema ic isk con ibu ions
β
,
γ
,
δ
a e all equal o one and he expec ed
payo
µ
is 99%. CLN (2) and (3) eplica e digi al CDO anches wi h 100 and 1000 unde lying digi al
bonds in he asse pool. CLN (2) and (3) ha e a highe expec ed payo s wi h 99.93% and 99.94%
han CLN (1) wi h a mean payo o 99%, bu he sys ema ic skewness and he sys ema ic ku osis
a e highe . The mo e di e si ied CLN (3) has e en highe sys ema ic isk con ibu ions han he
less di e si ied CLN (2). CLN (3a), (3b), (3c), (3d) ake in o accoun he issue ’s coun e pa y isk.
In ac , CLN (3) can also be in e p e ed as an in es men in o a syn he ic digi al CDO anche wi h he
ma ke po olio as unde lying asse pool. The sys ema ic isk quali y o such an in es men is al eady
below a e age ega ding coskewness and coku osis isk
(γ
,
δ>
1
)
. Howe e , wi h he inhe en
coun e pa y isk in ol ed in a syn he ic CDO, i s quali y ac o s a e e en wo se. The “opposi e”
payo o CLN 3), i.e., he payo
1{M<0.91}Yk
is in e ec a digi al de aul swap (DDS) unde coun e pa y
isk
k
. Thus, he syn he ic CDO plus DDS issued by he same coun e pa y
k
yields he payo
Yk
,
i.e., he digi al bond o issue
k
. CLN (4), (5) a e s uc u ed p oduc s wi h no sys ema ic a iance and
no sys ema ic skewness isk, espec i ely. CLN (4), (5) demons a e he impo ance o conside ing
non-linea sys ema ic isk, he linea CAPM be a alone is insu icien . The p ice o CLN (4) en ails
compensa ion o non-linea sys ema ic isk and no CAPM alpha. Con e sely, CLN (5) is no a
nega i e CAPM alpha in es men bu o e s p o ec ion agains sys ema ic skewness.
Table 5. Ra ing and p icing o c edi linked no es unde coun e pa y isk.
CLN Payo µ β γ δ
(1) 0.25 ∑4
i=11{Y1+(i−1)250=1}Y00.990000 1.0000 1.0000 1.0000
(2) 1{1/100 ∑100
k=1Y(k−1)10+1≥0.90}Y00.999296 0.5000 1.5631 2.5387
(3) 1{M≥0.91}Y00.999419 0.5219 1.9003 3.3286
(3a) 1{M≥0.91}Y10.998430 0.6582 2.0397 3.4419
(3b) 1{M≥0.91}Y251 0.996447 0.8825 2.2434 3.6019
(3c) 1{M≥0.91}Y501 0.990491 1.4593 2.7198 3.9675
(3d) 1{M≥0.91}Y751 0.972540 2.8610 3.7416 4.7387
(4) 1{M6=0.990}Y00.960958 – 1.4613 0.6652
(5) 1 −0.16 ×1{M=0.975}−0.84 ×1{M=0.976}0.992046 −0.6652 −0.1250 –
(6) 1{M<1.000}Y00.953889 −4.3342 −0.1032 −1.0630
(6a) 1{M<1.000}Y10.952886 −4.1850 0.0857 −0.8574
(6b) 1{M<1.000}Y251 0.950886 −3.9445 0.3517 −0.5813
(6c) 1{M<1.000}Y501 0.944896 −3.3362 0.9447 −0.0073
(6d) 1{M<1.000}Y751 0.926877 −1.8730 2.1965 1.1796
Ma ke M=1/1000 ∑1000
k=1Yk0.990000 1.0000 1.0000 1.0000
CLN (1) eplica es he payo o an a e age po olio o single-name digi al bonds om Table 1, CLN (2) is
a bina y ha pays ou no hing i he de aul a e in an a e age po olio o 100 digi al bonds exceeds 10%.
CLN (3) pays ou no hing in case he ma ke loss exceeds 9% and one else. The c edi quali y dec eases
signi ican ly unde dec easing coun e pa y quali y in CLN (3a), (3b), (3c) (3d). CLN (4) pays ou no hing in
case he ma ke ’s de aul a e is exac ly 1% and one else, i shows no co a ia ion wi h he ma ke ye abo e
a e age coskewness isk
(>
1
)
. CLN (5) has no coku osis exposu e. CLN (6) pays ou one bu in he bes
s a e o he wo ld when he e a e no ma ke losses. I s de aul p obabili y is qui e high wi h 4.61%, bu i s
sys ema ic isk exposu e could no be be e . CLN (6a), (6b), (6c) and (6d) a e he same CLN as (6) bu unde
inc easing coun e pa y isk. Coun e pa y 0 is de aul p o ec ed.
CLN (6) has he lowes a ing quali y among ins umen s wi h no coun e pa y isk, i.e., i has he
highes expec ed loss. Howe e , CLN (6) only de aul s in he bes s a e o he economy when he e a e
no de aul s in he ma ke po olio, i.e., in a s a e when a ma ginal payo is leas bene icial. Apa om
a ing quali y, CLN (6) o e s he highes possible quali y and is in ha espec qui e he opposi e
o CLN (3) which has he bes a ing quali y bu unde qui e un a o able sys ema ic isk. CLN (6a),
6(b), 6(c), 6(d) ake in o accoun he coun e pa y isk o he CLN issue . Tha is, hose ou CLNs
unlike CLN (6) can also de aul in bad s a es when he CLN issue is no capable o ul ill i s obliga ion.
J. Risk Financial Manag. 2018,11, 27 17 o 26
Rema kably, all ou CLNs unde coun e pa y isk s ill ha e nega i e be as ye hei coskewness is
posi i e, again a clea demons a ion ha a ing and be a a e insu icien o p icing.
5.6. Dis ess-Con ingen Con e ible Bond (CoCo)
My model allows he analysis o many o he c edi ins umen s such as bank deposi insu ance and
loan gua an ees as an al e na i e o Me on (1977), ca as ophe bonds, o c edi de aul swaps (CDSs)
unde coun e pa y isk. Howe e , o p o ide ye a inal example, I will discuss dis ess-con ingen
con e ible bonds o simply CoCos (see Du ie (2010)).
Le
ω∈[0, 1]
deno e he dilu ion ac o ,
ω=
0 means ha he CoCo is a pu e w i e-down bond,
exis ing sha eholde s expe ience no dilu ion a all,
ω=
1 means ha exis ing sha eholde s a e wiped
ou comple ely as soon as he con e sion le el in he o m o an a achmen poin
a1
is igge ed.
Such a wa e all s uc u e can be hough o as a con olu ion o i s , second, and hi d loss anche
in a s anda d CDO s uc u e. The i s loss is bo ne by he equi y holde , he second loss anche is
bo ne by he CoCo bondholde , and claims o he hi d loss anche a e di ided be ween equi y and
CoCo bondholde , he ac ion
ω
belongs o he CoCo bondholde and
(
1
−ω)
o he equi y holde .
Consequen ly, by linea i y he p ices o equi y and CoCo bond a e he weigh ed a e age o a mo e
s anda d CDO s uc u e:
Payo equi y: a1T1+(a3−a2)(1−w)T3
a1+(a3−a2)(1−w)
Payo con ingen con e ible bond: (a2−a1)T2+(a3−a2)w T3
(a2−a1)+(a3−a2)w,
whe e
T1
,
T2
,
T3
a e he payo s o a i s , second, and hi d loss anche o a s anda d CDO s uc u e,
a1
,
a2
,
a3
a e he a achmen poin s. Due o linea i y, he sys ema ic isk con ibu ions as well as he
mean payo o equi y and CoCo can be ob ained om he co esponding CDO anches as lis ed
in Table 6. The highe he dilu ion
w
, he highe is he sys ema ic isk o equi y bu he lowe he
sys ema ic exposu e o he CoCo. In o he wo ds, he lowe he dilu ion
ω
, he less “ oxic” is equi y.
Unde
ω=
1, he a ing quali y o he CoCo is sligh ly be e han ha o a BBB- a ed co po a e bond
in Table 1, ye i s sys ema ic skewness and ku osis a e e en highe han ha o a BB- a ed co po a e
bond. Simila ly, he expec ed loss o he CoCo is clea ly lowe han ha o an a e age co po a e bond
po olio, ye i s sys ema ic isk exposu e is conside ably wo se.
Table 6. Re inancing o bank asse s wi h equi y and con ingen con e ible bond.
Comple e W i e down w=0/ CoCo Bondholde s Ha e No Claim o 3 d Loss T anche:
T anche Payo µ β γ δ
Equi y a1T1+(a3−a2)(1−w)T3
a1+(a3−a2)(1−w)0.901228 9.4124 8.6546 8.1667
Con ingen con e ible bond (a2−a1)T2+(a3−a2)w T3
(a2−a1)+(a3−a2)w0.994077 2.6560 5.5513 6.6843
Po olio ∑3
j=1
aj−aj−1
a3−a0Tj0.916703 8.2864 8.1374 7.9196
Comple e Dilu ion o Equi y w=1/Equi y Holde s Ha e no Claim o 3 d Loss T anche:
T anche Payo µ β γ δ
Equi y a1T1+(a3−a2)(1−w)T3
a1+(a3−a2)(1−w)0.836149 15.2050 13.0637 11.5560
Con ingen con e ible bond (a2−a1)T2+(a3−a2)w T3
(a2−a1)+(a3−a2)w0.997257 1.3677 3.2111 4.2832
Po olio ∑3
j=1
aj−aj−1
a3−a0Tj0.916703 8.2864 8.1374 7.9196
The equi y holde bea s he i s 6% o losses ( i s loss anche
T1
wi h a achmen
a0=
0 and de achmen
poin
a1=
0.06). The con ingen con e ible bondholde bea s he nex 2% o losses (second loss anche
T2
wi h de achmen poin
a2=
0.08), he ac ion
w
is he payo o he hi d loss anche
T3
wi h de achmen poin
a3=
0.12 ha goes o he con ingen con e ible bondholde and
(
1
−w)
o he equi y holde . Mo e senio
deb holde s a e only a ec ed i he losses exceed 12%. The unde lying asse pool consis s o 100 digi al
bonds. The o al no ional amoun o con ingen con e ible bonds is
(a2−a1) + (a3−a2)w
, he o al no ional
amoun o equi y is
a1+ (a3−a2)(
1
−w)
. As a consequence, he highe
w
, he lowe (highe ) he sys ema ic
isk exposu e
β
,
γ
,
δ
o con ingen con e ible bonds (equi y) pe uni no ional and he highe (lowe ) he
expec ed payo µ.
J. Risk Financial Manag. 2018,11, 27 18 o 26
6. Empi ical Cases
As poin ed ou by Collin-Du esne e al. (2012), “ ade s in he CDX ma ke a e ypically hough
o as being a he sophis ica ed. Thus, i would be su p ising o ind hem accep ing so much isk
wi hou ai compensa ion.” Longs a and Rajan (2008), Li and Zhao (2012) show ha CDX anches
a e consis en ly p iced unde isk-neu al models and conclude ha hese secu i ies a e “ easonably
e icien ly p iced.”
I p o ide wo empi ical coun e examples ha e en p o essional pa icipan s in he ma ke o
s uc u ed inance obliga ions, namely Mo gan S anley and UBS, we e seemingly no awa e o he
inhe en non-linea isk o appa en ly hedged CDO po olios. In pa icula , bo h banks neglec ed he
coskewness and coku osis isk o hei ading po olios which we e basically unco ela ed wi h he
ma ke po olio. Bo h deale banks hough ha hei s uc u ed inance po olios we e o highe
quali y han hey ac ually we e, so I doub whe he hey we e eally ai ly compensa ed o hei
unin en ional isk aking.
6.1. Mo gan S anley
Acco ding o Lewis (2011), he ixed-income ade Howie Huble a Mo gan S anley bough
insu ance on BBB- a ed CDOs by paying he CDS sp ead on a no ional amoun o oughly USD 2 bn.
To o se his unning cos he sold p o ec ion on AAA- a ed CDOs by ecei ing he insu ance ee on
a no ional amoun o a ound USD 18 bn.
11
In e ec , Mo gan S anley syn he ically cons uc ed a
le e aged s uc u ed inance po olio wi h a no ional
N
= USD 16 bn by in es ing
(
1
+
1
/
8
)N
= USD
18 bn in o AAA- a ed CDO anches and by sho ing 1
/
8
N
= USD 2 bn o BBB anches. As lis ed in
Table 3, such a long-sho s a egy (o second- and ou h-loss anches) is oughly be a-neu al and
he e o e i ually ee o linea sys ema ic isk bu signi ican ly exposed o non-linea sys ema ic isk.
F om a sys ema ic isk pe spec i e in a wo-momen CAPM, Huble ’s s uc u e is as isk- ee as Swiss
o US T easu y bonds. Indeed, Huble ’s posi ion “ egis e ed on Mo gan S anley’s in e nal epo as
i ually iskless,” Lewis (2011) (p. 207). Mo gan S anley managed o sell his le e age s uc u e in
July 2007 p io o expi a ion mainly o UBS wi h a loss o a ound USD 9 bn:
“The o he , bigge , buye was UBS—which ook $2 billion in Howie Huble ’s iple-A CDOs,
along wi h a couple o hund ed million dolla s’ wo h o his sho posi ion in iple-B- a ed bonds.
Tha is, in July, momen s be o e he ma ke c ashed, UBS looked a Howie Huble ’s ade and said,
"We wan some o ha , oo." [...] ade s a UBS who execu ed he ade we e mo i a ed mainly by
hei own models—which, a he momen o he ade, sugges ed hey had u ned a p o i o $30 million.”
Lewis (2011) (p. 215/216)
In he i s hal o 2007, buye s o CDOs we e g adually ealizing ha he e a e unknown quali y
ac o s in he sense o Ake lo (1970). UBS was a guably he las buye be o e he ma ke b oke down
comple ely: “In he second qua e o 2007 [...] The UBS leade ship con inued o be op imis ic and he
In es men Bank wen on pu chasing highly a ed subp ime pape while o he banks we e quickly
unloading hei posi ions” S aumann (2010).
6.2. UBS
UBS incen i ized i s business di isions wi h an UBS-speci ic economic alue added app oach
o Ospel-Bodme (2001) which is undamen ally based on a wo-momen CAPM. In he amewo k
o Ospel-Bodme (2001) he e is no men ioning o non-linea isk, he e is jus an alpha and a CAPM
11
“[T]he p emiums on he supposedly a less isky iple-A- a ed CDOs we e only one- en h o he p emiums on he iple-Bs,
and so o ake in he same amoun o he money as he was paying ou , he’d need o sell c edi de aul swaps in oughly en
imes he amoun he al eady owned” (Lewis (2011), p. 206).
J. Risk Financial Manag. 2018,11, 27 19 o 26
be a, so he compensa ion o non-linea sys ema ic isk is alsely in e p e ed as economic alue added,
economic p o i , o CAPM alpha.
As seen abo e, UBS ook o e some CDO isks om Mo gan S anley by buying a be a-neu al
long-sho CDO po olio. The con en ional hinking ha only be a measu es sys ema ic isk seems o
ha e been deeply ing ained a UBS ha ul ima ely epo ed ne losses o 18.7 bn. In i s sha eholde
epo , UBS (2008) called i s long-sho CDO po olio “Ampli ied Mo gage Po olio” Supe Senio s
(AMPS): “ hese we e Supe Senio posi ions whe e he isk o loss was ini ially hedged h ough he
pu chase o p o ec ion on a p opo ion o he nominal posi ion ( ypically be ween 2% and 4% hough
some imes mo e)” (p. 14).
12
Tha is, UBS was long senio CDO anches and hedged i s posi ion by
sho ing i s o second loss CDO anches. These long-sho s a egies a e oughly be a-neu al as can
be seen in Table 3and explain he majo i y o losses a UBS (2008): “As a he end o 2007, losses on
hese AMPS ades con ibu ed app oxima ely 63% o o al Supe Senio losses” (p. 14).
Howe e , e en wi h he bene i o hindsigh , UBS seems o ha e misunde s ood he p incipal
oo cause o hei losses. UBS (2008) s a es ha i s losses we e p ima ily o idiosync a ic and no
o sys ema ic na u e: “T ading losses: Insu icien accoun ing o he isk o di e gen mo emen s
be ween p e iously co ela ed asse classes o ins umen s (basis isk)” and “insu icien a en ion o
idiosync a ic isk ac o s (i.e., he isk o p ice change due o unique ci cums ances o a speci ic secu i y,
as opposed o he o e all ma ke )” (p. 30). The ac ha he emaining isk was sys ema ic and no
idiosync a ic should ha e been ob ious because a ull hedge by insu ance ia CDS on he exac ly
same unde lying (in UBS ocabula y a NegBasis ade), was mo e cos ly han be a-neu al hedging
(AMPS ade).
13
I he emaining isk o a be a-neu al po olio we e indeed pu ely idiosync a ic he e
would ha e been no isk p emium: “The cos o hedging h ough a NegBasis was app oxima ely 11 bp,
whe eas he cos o hedging h ough an AMPS ade was app oxima ely 5–6 bp. The easons o he
di e en ial p icing o hedging s a egies ha om a isk me ics pe spec i e we e deemed equi alen
appea s no o ha e been closely sc u inised,” UBS (2008) (p. 30).
In o he wo ds, he cos o ull p o ec ion agains sys ema ic isk was a ound 0.10%, oughly
hal o he p emium was o linea sys ema ic isk and he o he hal o non-linea sys ema ic isk.
Howe e , wi h hei be a-neu al AMPS po olio, UBS paid only a ound 0.05% o p o ec ion agains
linea sys ema ic isk and le he non-linea sys ema ic isk unhedged. Al eady an inc ease in he
ma ke p ice o ku osis isk la e esul ed in a signi ican ma k- o-ma ke loss. None heless,
UBS (2008)
conside ed i s posi ions as comple ely hedged: “Once hedged, ei he h ough NegBasis o AMPS
ades, he Supe Senio posi ions we e VaR and S ess Tes ing neu al (i.e., because hey we e ea ed
as ully hedged, he Supe Senio posi ions we e ne ed o ze o and he e o e did no u ilize VaR and
S ess limi s)” (p. 30).
14
Gi en UBS’s igno ance o non-linea sys ema ic isk, UBS was ha dly ai ly
compensa ed and i is no su p ising ha UBS was conside ed “ he bigges ool a he able” (p. 215)
acco ding o Lewis (2011).15
12
“AMPS p o ide a pla o m o hedging he c edi sp ead exposu e om UBS holdings in long syn he ic and cash asse s.
Typical ades would be ha UBS buys p o ec ion on a speci ied pe cen age o ma ke alue losses in a speci ied e e ence
pool o ABS asse s (CMBS, CDO, CLO) o o buy p o ec ion be ween wo p ede e mined le els” UBS (2008) (p. 45).
13
“A nega i e basis ade is a ansac ion in which UBS holds a highly a ed (gene ally Supe Senio AAA) s uc u al inancial
asse hedged wi h a c edi de aul swap on he exac same asse ou o ull legal ma u i y” UBS (2008) (p. 14).
14
UBS (2008) “conside ed a Supe Senio hedged wi h 2% o mo e o AMPS p o ec ion o be ully hedged. [...] [T]he long
and sho posi ions we e ne ed, and he in en o y o Supe Senio s was no shown [...]. Fo AMPS ades, he ze o VaR
assump ion subsequen ly p o ed o be inco ec as only a po ion o he exposu e was hedged [...], al hough i was belie ed
a he ime ha such p o ec ion was su icien ” (p. 30).
15
A he end o 2008 UBS’s s uc u ed inance po olio had o be sold a a huge discoun o a special pu pose ehicle called
“SNB S abFund”. The equi y ( i s loss anche) was injec ed by UBS, he deb capi al (second loss anche) was p o ided by
he Swiss Na ional Bank (SNB). In e ec , he bail-ou o UBS by SNB was achie ed by a CDO squa ed s uc u e since he
asse pool o “SNB S abFund” consis s o CDOs.
J. Risk Financial Manag. 2018,11, 27 20 o 26
7. Conclusions
I o e an explana ion in he spi i o Ake lo (1970) o he all o he s uc u ed c edi ma ke
a e he inancial ma ke c isis in 2007/08 and why we s ill see p oblems esusci a ing his ma ke
(see, e.g., Sego iano e al. (2015)). The sys ema ic isk o wo c edi ins umen s wi h he same a ing
can a y subs an ially, in pa icula he non-linea sys ema ic isk, bu he sys ema ic isk—unlike
he a ing—is no eadily a ailable o he a e age in es o . Tha is, s uc u ed c edi a ings
a e in o ma ionally insu icien o p icing e en i a ings p o ide unbiased, powe ul es ima es
on expec ed losses. Howe e , a ma ke in which po en ial buye s canno co ec ly assess all
p icing- ele an a ibu es o a p oduc can a ac selle s o e ing in e io goods, in pa icula , c edi
p oduc s o good a ing quali y bu low sys ema ic isk quali y. The p esence o ma ke pa icipan s
who a e willing o o e in e io goods ends o d i e he ma ke ou o exis ence.
I p opose o assess he quali y o a c edi ins umen wi h end-o -pe iod payo
X
by ou s a is ical
(co-)momen s: mean µX, co a iance βX, coskewness γX, coku osis δX, whe e he h ee la e me ics
a e exp essed wi h espec o a well-di e si ied po olio
M
. Cu en ly, a ing agencies o e an
assessmen only abou
µX
bu a e silen abou
βX
,
γX
,
δX
. In o he wo ds, besides he a ing in o ma ion
o compu e he expec ed payo
µX=E[X|S]
unde he issue ’s in o ma ion
S
, I sugges making also
public he sys ema ic isk, i.e., he con ibu ion o
X
o he a iance o
M
as well as he con ibu ions
o he hi d and ou h cen al momen o
M
(always condi ional on he selle ’s in o ma ion
S
).
I show ha a ma ke in which po en ial buye s a e igno an abou a leas one o he ou quali y
a ibu es has no equilib ium. Making public he selle ’s in o ma ion abou
µX
,
βX
,
γX
,
δX
, no only
µX
, esul s in a symme ic ma ke wi h an equilib ium. Such holis ic quali y assessmen s o mo e
complex c edi ins umen s can hen be benchma ked agains simple single-name bonds. As I illus a e,
an in es men - a ed s uc u ed inance obliga ion can ha e wo se sys ema ic isk quali ies han a
subin es men - a ed co po a e bond.
Wi h addi ional assump ions (i.a., he well-di e si ied po olio
M
is he ma ke po olio), I o e
a simple and s aigh o wa d ou -momen CAPM ha combines he quali y a ibu es o a c edi
ins umen
µX
,
βX
,
γX
, and
δX
in o an equilib ium p ice
g(µX
,
βX
,
γX
,
δX)
. The a iables
µX
,
βX
,
γX
,
and
δX
a e eal-wo ld isk me ics independen om any p e e ence assump ions. To a i e a he
p icing unc ion g(·), I assume ha ing a ep esen a i e agen unde s anda d isk a e sion, in e alia.
The ac ha single-name bonds, s uc u ed inance secu i ies and o he c edi p oduc s ca y
sys ema ic isk con ibu ions β,γ,δ ha can be so di e en om a p icing s andpoin cas s signi ican
doub on whe he some c edi ma ke s can eally smoo hly unc ion wi h only he in o ma ion p o ided
by a ing agencies abou he expec ed payo
µ
. The co po a e bond ma ke is possibly homogenous
enough bu o he c edi ma ke s—in pa icula CDOs—ce ainly no . I illus a e ha sys ema ic
isk canno solely be measu ed by a linea CAPM be a since an asse can be nega i ely co ela ed
wi h he ma ke bu can s ill be hea ily exposed o non-linea sys ema ic isk. I also demons a e
ha coun e pa y isk o c edi de i a i es—such as a c edi linked no e, syn he ic CDO, o de aul
swap—has a signi ican impac on he o e all p oduc quali y, in pa icula he p oduc ’s sys ema ic isk.
Finally, by conside ing wo empi ical cases, namely Mo gan S anley and UBS, I show ha
e en big deale banks we e unawa e abou inhe en non-linea sys ema ic isk o seemingly hedged
s uc u ed inance po olios. Tha is, c edi quali y was inadequa ely assessed only based on a ing
and co ela ion. The small compensa ion o non-linea sys ema ic isk was w ongly in e p e ed as
alue c ea ion o CAPM alpha and was he e o e ha dly a ai isk p emium o he huge losses ha
la e ma e ialized.
Au ho Con ibu ions:
All analyses a e done and he pape solely w i en by he au ho . The iews exp essed do
no necessa ily e lec he iews o Zü che Kan onalbank.
Con lic s o In e es : The au ho decla es no con lic o in e es .

J. Risk Financial Manag. 2018,11, 27 21 o 26
Appendix A. P oo s
P oo o P oposi ion 1.
By assump ion he ma ginal u ili y unc ion o he ep esen a i e in es o is
a powe unc ion,
u0(w)=w−λ
, o equi alen ly, he u ili y unc ion exhibi s cons an ela i e isk
a e sion, λ=−w u00(w)/u0(w).
Using he de ini ion o
Z
in
(3)
and no ing ha he dis ibu ion o
w−λ
and
M−λ
(wi h
M=c w
,
whe e
c
is a scaling ac o ) a e also log-no mal, I can ew i e he equilib ium ela ion in
(2)
as ollows:
qk
q0
=E"M−λ
EM−λYk#,
wi h a log-no mally dis ibu ed p icing ke nel Z:
Z=M−λ
EM−λ=M−λeλµM−λ2
2σ2
M. (A1)
By assump ion he po olios
M
and
P
ollow a bi a ia e log-no mal dis ibu ion wi h co ela ion
coe icien ρ:
log M
log P!∼N µM
µP!, σ2
MσMσPρ
σMσPρ σ2
P!!. (A2)
Reg essing log Mon o log PI ha e:
log M=µM+ρσM
σP
(log P−µP)+e,
whe e
log P
and
e
a e wo independen Gaussian a iables wi h he a iance o
e
gi en by
σ2
M1−ρ2
,
a p ope y esul ing om linea p ojec ion. Since
Z
in
(A1)
is a posi i e a iable wi h mean one unde
P, i has he p ope ies o a Radon-Nikodym de i a i e.
Hence, he momen gene a ing unc ion o he bi a ia e a iable in
(A2)
unde he ma ingale
measu e Qinduced by Zcan be w i en as ollows:
EQhe 1log P+ 2log Mi=Ehe 1log P+( 2−λ)log M−log E[M−λ]i
=e( 2−λ)µM−µPρσM
σP+1
2σ2
M(1−ρ2)( 2−λ)2−log E[M−λ]·Ee 1+( 2−λ)ρσM
σPlog P
=e( 2−λ)µM−µPρσM
σP+1
2σ2
M(1−ρ2)( 2
2−2 2λ+λ2)+λµM−1
2λ2σ2
M
·e 1+( 2−λ)ρσM
σPµP+1
2[ 2
1σ2
P+2 1( 2−λ)ρσMσP+( 2
2−2 2λ+λ2)ρ2σ2
M].
Le
1= 2=
0 and ake ad an age o he ac ha
EQhe0 log P+0 log Mi=
1, so ha all e ms in he
exponen nei he in ol ing 1no 2mus sum up o ze o, o ob ain:
EQhe 1log P+ 2log Mi=e 1(µP−λρσMσP)+ 2(µM−λσ2
M)+1
2 2
1σ2
P+1
2 2
2σ2
M+ 1 2ρσMσP. (A3)
Howe e ,
(A3)
is he momen -gene a ing unc ion o a bi a ia e Gaussian dis ibu ion. Hence,
unde he equi alen ma ingale measu e
Q
,
P
is log-no mally dis ibu ed wi h pa ame e s
µP−
λρσMσPand σ2
P. Al oge he , I ha e de i ed he ollowing equilib ium ela ion:
log PQ
∼NµP−λρσMσP,σ2
P, and log PQ
∼Nlog qP−log q0−1
2σ2
P,σ2
P,
whe e he second exp ession ollows om he ma ingale p ope y EQ[P/Y0]=qP/q0.
J. Risk Financial Manag. 2018,11, 27 22 o 26
P oo o Lemma 1.By dec easing absolu e isk a e sion, I ha e
−u00
u00=−u0u000 +(u00)2
(u0)2<0,
which equi es ha u000 >0 since u0>0.
Con e sely, dec easing absolu e p udence implies u0000 <0,
−u000
u00 0=−u00u0000 +(u000)2
(u00)2<0,
since u00 <0.
P oo o P oposi ion 2.
The decision p oblem o such a ep esen a i e agen can be w i en as a
maximiza ion o he expec ed u ili y:
max
c0,a0,...,aK{ (c0) + E[u(w)]}=max
c0,a0,...,aK( (c0) + E"u K
∑
k=0
akYk!#), (A4)
subjec o he cons ain :
w0=c0+
K
∑
k=0
akqk. (A5)
whe e
w
is end-o -pe iod weal h, he a iable
w=∑K
k=0akYk
is he he end o pe iod weal h,
Y0
is
a cons an o isk- ee asse , espec i ely,
(
.
)
he u ili y unc ion de ined o e ini ial consump ion
c0
, and
u(
.
)
he u ili y unc ion de ined o e end-o -pe iod weal h,
w0
is oday’s ini ial weal h o be
consumed and in es ed,
ak
is he numbe o uni s o inancial claim
k
pu chased,
qk
he p ice o claim
k>
0, and
Yk
is he isky payou a he end o he pe iod. Gi en he cons ain in
(A5)
, I can ew i e
he end-o -pe iod weal h as ollows:
w=w0−c0−∑K
k=1akqk
q0
Y0+
K
∑
k=1
akYk.
The i s o de condi ions o a maximum in (A4) a e he e o e gi en by:
0(c0) = 1
q0
Eu0(w)Y0
Eu0(w)Yk=qk
q0
Eu0(w)Y0, o k=1, ..., K, (A6)
whe e he p imes deno e di e en ia ion. The i s o de condi ions (A6) can be w i en as:
MRSk,0 =
∂
∂akE[u(w)]
∂
∂a0E[u(w)] =E[u0(w)Yk]
E[u0(w)Y0]=Eu0(w)
E[u0(w)]
Yk
Y0=qk
q0
.
In equilib ium he ma ginal a e o subs i u ion
MRSk,0
be ween asse
k
and he isk- ee asse 0
mus equal he quo ien o hei p ices. I ollows om he ma ke clea ing condi ions and he iden ical
cha ac e is ics o in es o s ha
c0=C0/N
,
w0=W0/N
,
w=W/N
, whe e
C0
,
W0
,
W
ep esen he
agg ega es o cu en consump ion, cu en weal h ( o cu en consump ion and o be in es ed o
u u e consump ion), and end-o -pe iod weal h. By ea anging he i s o de condi ions in
(A6)
,
I ob ain he desi ed equilib ium p ices.
J. Risk Financial Manag. 2018,11, 27 23 o 26
P oo o P oposi ion 3.
A heo e ical jus i ica ion o he mean- a iance-skewness-ku osis analysis is
o conside a ou h-o de polynomial u ili y speci ica ion. In his case (and by he assump ion ha
he ou h s a is ical momen exis s, i.e.,
Ew4<∞
), he expec ed u ili y o de ing can be ansla ed
exac ly in o a ou -momen o de ing. Thus, le me i s assume ha he ep esen a i e in es o ’s u ili y
u(w)is a qua ic polynomial unc ion o end-o -pe iod weal h w:
u(w) = a0+a1(w−µ) + a2(w−µ)2+a3(w−µ)3+a4(w−µ)4,
wi h
µ=E[w]
hen
u0(µ) = a1
,
u00(µ) =
2
a2
,
u000(µ) =
6
a3
,
u0000(µ) =
24
a4
and I can w i e he
ma ginal u ili y unc ion in (2) expanded a ound he expec ed end-o -pe iod weal h:
u0(w) = u0(µ)+u00 (µ) (w−µ)+1
2!u000 (µ) (w−µ)2+1
3!u0000 (µ) (w−µ)3=:P3(w),
whe e
P3(w)
deno es by de ini ion he hi d o de Taylo expansion o he ma ginal u ili y
u0(w)
a ound he mean end-o -pe iod weal h µ. The mean ma ginal u ili y is he e o e gi en by:
Eu0(w)=u0(µ)+1
2!u000 (µ)Eh(w−µ)2i+1
3!u0000 (µ)Eh(w−µ)3i=E[P3(w)]. (A7)
Since
u0(w) = P3(w)
by he assump ion o a qua ic u ili y unc ion
u(w)
, I ob ain an exac o m
o he Radon-Nikodym de i a i e Z:
Z=u0(w)
E[u0(w)]=P3(w)
E[P3(w)]=u0(µ)+u00 (µ) (w−µ)+1
2! u000 (µ) (w−µ)2+1
3! u0000 (µ) (w−µ)3
E[P3(w)]
=1
E[P3(w)]E[P3(w)]−1
2!u000 (µ)Eh(w−µ)2i−1
3!u0000 (µ)Eh(w−µ)3i
| {z }
=u0(µ)
+1
E[P3(w)]u00 (µ) (w−µ)+1
2!u000 (µ) (w−µ)2+1
3!u0000 (µ) (w−µ)3
=1−λβK
N
w−µ
Eh(w−µ)2i−λγK
N
(w−µ)2−Eh(w−µ)2i
Eh(w−µ)3i−λδK
N
(w−µ)3−Eh(w−µ)3i
Eh(w−µ)4i.
The second line ollows om
(A7)
, he las line ollows by de ini ion om he isk p emia
λβ
,
λγ
,
λδ
:
λβ:=−N
K
u00 (E[w])
E[P3(w)]Eh(w−E[w])2i,
λγ:=−N
K
1
2!
u000 (E[w])
E[P3(w)]Eh(w−E[w])3i,
λδ:=−N
K
1
3!
u0000 (E[w])
E[P3(w)]Eh(w−E[w])4i.
Since
w=
1
/N∑K
k=0Yk
is an a ine ans o ma ion o
M=
1
/K∑K
k=1Yk
and
Y0
a cons an ,
I inally ob ain:
Z=u0(w)
E[u0(w)]=1−λβM−E[M]
Eh(M−E[M])2i−λγ(M−E[M])2−Eh(M−E[M])2i
Eh(M−E[M])3i
−λδ(M−E[M])3−Eh(M−E[M])3i
Eh(M−E[M])4i. (A8)
J. Risk Financial Manag. 2018,11, 27 24 o 26
I he ep esen a i e in es o has a qua ic u ili y unc ion hen
u0(w)/E[u0(w)]=P3(w)/E[P3(w)]
is an exac equali y. Howe e , mo e gene ally, i I app oxima e he u ili y unc ion
u(w)
by a
ou h o de Taylo se ies a ound he expec ed end-o -pe iod weal h hen I can app oxima e he
Radon-Nikodym de i a i e
Z
by
P3(w)/E[P3(w)]
. To ob ain he p icing o mula
qX/q0=E[X Z]
in
(4)
wi h
Z
gi en in
(A8)
no e ha wi h
V= (M−E[M])m
,
m∈{1, 2, 3}
,
µV=E[V]
,
µX=E[X]
, I ha e
he equali y E[X(V−µV)]=E[(X−µX)V].
As seen in Lemma 1, s anda d isk a e sion implies
u0>
0,
u00 <
0,
u000 >
0, and
u0000 <
0. The
isk p emium
λγ
o skewness isk is posi i e i he hi d cen al momen o he weal h dis ibu ion is
nega i e, i.e., i skewed o he le . The p emia o a iance and ku osis isk,
λβ
and
λδ
, mus be posi i e
when u(·)exhibi s s anda d isk a e sion.16
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16 Z
is a polynomial unc ion o
w
o
M
, espec i ely, polynomial unc ions a e con inuously di e en iable, a con inuously
di e en iable unc ion on a closed in e al is Lipschi z, Lipschi z unc ions a e absolu ely con inuous.