Flaig, Sol eig; Junike, Ge o
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P o i and loss a ibu ion: an empi ical s udy
Eu opean Ac ua ial Jou nal
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LETTER
P o i and loss a ibu ion: an empi ical s udy
Sol eig Flaig1,2 ·Ge o Junike2
Recei ed: 18 Decembe 2023 / Re ised: 4 Ma ch 2024 / Accep ed: 8 Ma ch 2024 /
Published online: 24 May 2024
© The Au ho (s) 2024
Abs ac
The p o i and loss (P&L) a ibu ion o each business yea in o di e en isk ac o s
(e.g., in e es a es, c edi sp eads, o eign exchange a e e c.) is a egula o y equi e-
men , e.g., unde Sol ency 2. Th ee di e en decomposi ion p inciples a e p e alen :
one-a -a- ime (OAT), sequen ial upda ing (SU) and a e age sequen ial upda ing (ASU)
decomposi ions. In his esea ch, using inancial ma ke da a om 2003 o 2022, we
demons a e ha he OAT decomposi ion can gene a e signi ican unexplained P&L
and ha he SU decomposi ions depends signi ican ly on he o de o labeling o he
isk ac o s. On he basis o an in es men in a o eign s ock, we u he explain ha
he SU decomposi ion is no able o iden i y all ele an isk ac o s. This po en ially
e ec s he hedging s a egy o he po olio manage . In conclusion, we sugges o use
he ASU decomposi ion in p ac ice.
Keywo ds P o i and loss a ibu ion ·Change analysis ·Sequen ial decomposi ions ·
Shapley alue ·Sol ency 2
JEL Classi ica ion D53 ·C58 ·G22
1 In oduc ion
The analysis o he p o i s and losses (P&L) be ween wo epo ing da es is a common
ask in isk managemen . The isk ac o s, mus be iden i ied and hei con ibu ion
o he P&L has o be quan i ied. The cu en egula ion o insu ance companies in
Eu ope, Sol ency 2, equi es ha when an in e nal model is used, a p o i and loss a i-
BGe o Junike
[email p o ec ed]
Sol eig Flaig
[email p o ec ed]
1Deu sche Rück e siche ung AG, Kapi alanlage / Ma ke Risk Managemen , Hansaallee 177,
40549 Düsseldo , Ge many
2Ca l on Ossie zky Uni e si ä , Ins i u ü Ma hema ik, 26111 Oldenbu g, Ge many
123
1014 S. Flaig, G. Junike
bu ion o change analysis mus be pe o med in su icien de ail acco ding o he isk
ca ego iza ion selec ed in he in e nal model, see A icle 240 in Sol ency 2. Typically,
o he ma ke isk module, he in es men P&L is alloca ed o he Sol ency Capi al
Requi emen (SCR) sub-modules: in e es a e, c edi sp ead, equi ies, eal es a e and
o eign exchange. The e a e a leas wo basic equi emen s: a P&L a ibu ion should
be able o ully explain he P&L, i.e., he e should no be any unexplained P&L and a
change analysis should iden i y all ele an isk ac o s.
The P&L a ibu ion can also ha e an economic impac when ading decisions,
e.g., he amoun o he o eign exchange hedge, a e based on his analysis.
In p ac ice, he e a e many well-known decomposi ions ha can be used o iden-
i y he con ibu ion o di e en isk ac o s: The e is he sequen ial upda ing (SU)
decomposi ion, which is also known as wa e all, see Cadoni [1]. The SU decompo-
si ion depends on he o de o labeling o he isk ac o s: i he e a e d isk ac o s,
he e a e d!SU decomposi ions.
The a e age o he d!possible SU decomposi ions is called he a e age sequen-
ial upda ing (ASU) decomposi ion. The ASU decomposi ion is also known as he
Shapley alue o Shapley-Shubik decomposi ion and has many desi able p ope ies,
see F iedman and Moulin [4] and e e ences he ein.
The one-a -a- ime (OAT) decomposi ion is also known as bump and ese ,see
Cadoni [1]. In gene al, he OAT decomposi ion is no able o ully explain he P&L.
All decomposi ions a e linea ; he e o e, o decompose a po olio i is possi-
ble o decompose he ins umen s o he po olio indi idually. F ei [3], Je ses and
Ch is iansen [5] and Ch is iansen [2] applied he OAT, SU and ASU decomposi ions
ecu si ely o mul iple sub-in e als. In p ac ice, one would apply a decomposi ion on
annual, qua e ly, mon hly o mo e g anula sub-in e als.
Fo example, a isk manage wo king in an insu ance company has o make many
decisions o implemen a change analysis: which o he d!+2 decomposi ions (OAT,
d!SU o ASU) is he mos app op ia e? How o choose he size o he sub-in e als?
In Sec .2, we apply hese decomposi ions o a ypical in es men o an insu ance
company; a co po a e bond. We show empi ically ha he unexplained P&L o he OAT
decomposi ion is signi ican and ha he P&L a ibu ion di e s signi ican ly o he
di e en SU decomposi ions. A p io i, i is unclea , which o he d!SU decomposi ion
wo ks bes .
Fu he , in Sec . 3, we p o ide an example o a pa ially hedged po olio wi h
signi ican o eign exchange exposu e, bu he OAT and some SU decomposi ions
assign ze o P&L o he isk ac o desc ibing he o eign exchange a e. This po en ially
e ec s he hedging s a egy o he po olio manage : a nai e SU o OAT decomposi ion
may lead o w ong ading and hedging decisions. The ASU decomposi ion does no
ha e his d awback. The e o e, bo h he OAT and he SU decomposi ions a e unsui able
o pe o ming a change analysis.
F om a heo e ical poin o iew, he ASU decomposi ion is p e e ed because i is
able o ully explain he P&L and does no depend on he o de o labeling o he isk
ac o s. Mo eo e , i seems easonable o keep he sub-in e als as small as possible
in o de o ake in o accoun he whole pa hs o he isk ac o s, see Mai [6], and o
p e en inconsis encies when using con lic ing sub-in e als o di e en pu poses.
Howe e , he a ailable compu a ional powe limi s he numbe o he sub-in e als.
123
P o i and loss a ibu ion... 1015
Ou empi ical expe imen s based on a co po a e bond indica e ha he ASU decom-
posi ion changes signi ican ly using annual e sus daily da a. Howe e , he di e ence
be ween he ASU decomposi ion based on mon hly and daily da a is less han 0.2%
ac oss wen y business yea s and all isk ac o s.
In conclusion, we sugges o use he ASU decomposi ion and mon hly o mo e
g anula sub-in e als o ob ain a change analysis o a (co po a e) bond po olio.
2 Change analysis o a bond
Le us look a he e olu ion o he isk ac o s ele an o p icing a USD A- a ed
co po a e bond wi h a ma u i y o en yea s. We conside a Eu opean in es o , so
he P&L has o be con e ed in o EUR and is exp essed in pe cen age poin s o he
nominal in EUR. We ake in o accoun he ime pe iod 2002/12/31 −2022/12/31,
i.e., wen y business yea s. The mos g anula a ailable da a is daily da a.
The bond is modeled as a cons an ma u i y bond, i.e., a e e y day he ma u i y o
he bond is assumed o be T=10 yea s. This is he usual assump ion i he insu ance
po olio is g ouped in o benchma k po olios ha a e ep esen ed by a bond wi h an
a e age a ing and du a ion. The isk ac o s a e: in e es a es (IR), c edi sp ead (CS)
and o eign exchange a e (FX).
The ollowing Bloombe g icke s a e used: USSW10 Index, C40410Y Index and
USDEUR Cu ncy, which desc ibe he US 10-yea in e es a es, he US co po a e
c edi sp eads o an A- a ed class and a ma u i y o en yea s and he USDEUR
exchange a e, espec i ely. The p ice o he bond a ime ≥0 wi h ma u i y Ta
ime is gi en by
P( ( ), s( ), x( )) =x( )
(1+ ( )+s( ))T,(1)
whe e ( )and s( )deno e a ealiza ion o IR and CS a ime o ime ho izon T.
The e m x( )deno es a ealiza ion o he o eign exchange a e be ween he o eign
cu ency and he domes ic cu ency a ime . In his sec ion, we conside only a bond.
O he ins umen s could be ea ed simila ly by de ining Papp op ia ely.
Unde Sol ency 2, an insu ance company has o explain he P&L
P=P( (1), s(1), x(1)) −P( (0), s(0), x(0))
o e e y business yea , i.e., be ween wo epo ing da es 0 and 1.
The OAT decomposi ion decomposes he P&L Pby
P={P( (0), s(1), x(0)) −P( (0), s(0), x(0))}
+{P( (1), s(0), x(0)) −P( (0), s(0), x(0))}
+{P( (0), s(0), x(1)) −P( (0), s(0), x(0))}+R,(2)
whe e R∈Ris he unexplained P&L.The i s , second and hi d e m in (2) co -
esponds o he con ibu ion o CS, IR and FX o he P&L, espec i ely. Tha is, o
123
1016 S. Flaig, G. Junike
Table 1 De ini ion o he six possible upda e o de s o he isk ac o s IR, CS and FX o ob ain six SU
decomposi ions
Upda eo de 123456
CS IR FX IR CS FX IR FX CS CS FX IR FX IR CS FX CS IR
ob ain he con ibu ion o a isk ac o , ix all o he isk ac o s a he o igin, i.e., a
ime 0 and allow only he isk ac o o in e es o mo e om ime 0 o ime 1.
The SU decomposi ion is simila ly de ined bu upda es he isk ac o s sequen ially:
a e upda ing one isk ac o , i is no ese o ime 0 bu kep a ime 1, see Cadoni [1].
I he e a e h ee isk ac o s, he e a e 3!=6 SU decomposi ions. Fo example, he
h ee isk ac o s IR, CS and FX can be o de ed acco ding o Table 1. Fo example,
he upda e o de (CS, IR, FX) esul s in o he ollowing SU decomposi ion:
P={P( (0), s(1), x(0)) −P( (0), s(0), x(0))}
+{P( (1), s(1), x(0)) −P( (0), s(1), x(0))}
+{P( (1), s(1), x(1)) −P( (1), s(1), x(0))}.(3)
The i s , second and hi d e m in Eq. (3) is in e p e ed as he con ibu ions o he isk
ac o s CS, IR and FX, espec i ely. The SU decomposi ion ully explains he P&L,
i.e., he e is no unexplained P&L, because he igh -hand side o Eq. (3) is ac ually a
elescoping se ies. The ASU decomposi ion is de ined as he a e age o all possible
SU decomposi ions.
One may also di ide he business yea in o msub-in e als and apply a s a ic
decomposi ion ecu si ely along he sub-in e als o de ine a decomposi ion in a mul i-
pe iod se ing, see F ei [3], Je ses and Ch is iansen [5] and Ch is iansen [2]. Fo
example, hese sub-in e als can be chosen annual, qua e ly, mon hly, weekly o
daily.
In he business yea 2020 (2003), he OAT decomposi ion has an unexplained
P&L o −2.4% (−0.8%)using qua e ly (daily) sub-in e als. The absolu e a e age
unexplained P&L o e all wen y business yea s a e 0.4% o annual and qua e ly sub-
in e als and 0.3% o mon hly and daily sub-in e als. No ma e which sub-in e als
we choose, he unexplained P&L can be signi ican .
Figu e 1shows he six SU decomposi ions o he upda e o de s de ined in Table 1
and he business yea 2022. The OAT decomposi ion is shown on he le . The ASU
decomposi ion is shown on he igh . I can be seen ha he SU decomposi ion clea ly
depends on he upda ing o de o he isk ac o s, a pa e n ha can also be obse ed
in o he yea s. The ange (i.e., he di e ence be ween he la ges and smalles alue)
o he isk ac o FX (IR) in 2022 (2003) ac oss he six SU decomposi ions using
qua e ly (daily) sub-in e als is 1.9% (0.9%).
On a e age o e all wen y business yea s, he ange ac oss he six possible pe mu-
a ions o he FX, IR, CS isk ac o s o mon hly sub-in e als a e 0.4%, 0.3% and
0.2%, espec i ely. No ma e which sub-in e als we choose, he SU decomposi ion
may depend signi ican ly on he o de o labeling o he isk ac o s.
123
P o i and loss a ibu ion... 1017
Fig. 1 OAT, SU and ASU decomposi ions o business yea 2022 o a US co po a e bond
The ASU decomposi ion is de ined by he a e age o he six SU decomposi ions.
The ASU decomposi ion is independen o he o de o he isk ac o s and is able o
ully explain he P&L. Like he SU o he OAT decomposi ion, he ASU decomposi ion
depends on he size o he sub-in e als. In 2022, he ange ac oss annual, qua e ly,
mon hly, weekly and daily sub-in e als o he ASU decomposi ion o he con ibu ion
o he isk ac o FX is 1.0%.
While i seems economically easonable o use daily sub-in e als o ob ain he ASU
decomposi ion in o de o ake in o accoun he whole pa hs o he isk ac o s, o
he co po a e bond we conside ed in his sec ion, i may be su icien o use mon hly
sub-in e als since he di e ence o he ASU decomposi ion based on daily e sus
mon hly sub-in e als is less han 0.2% ac oss all wen y business yea s and he h ee
isk ac o s FX, IR and CS.
123
1018 S. Flaig, G. Junike
3 SU decomposi ion o a US s ock
In his sec ion, we p o ide an example o illus a e ha a SU decomposi ion may
assign ze o P&L o he FX isk ac o e en hough he po olio has signi ican FX
exposu e. In pa icula , we conside an in es men by a Eu opean in es o in he US
s ock ma ke such ha he FX isk is hedged by an FX- u u e. The FX-hedge is no
pe ec ; some FX- isk emains. Howe e , he e is a SU decomposi ion which indica es
ha he P&L is only d i en by he mo emen s o he s ock ma ke . We will also wo k
wi h eal ma ke da a.
Conside a s a ic se ing and a pa ially hedged po olio: Suppose he e a e wo
isk ac o s, say a o eign exchange a e Xand o eign s ock Y, and some po olio P
such ha X( ), Y( )∈R o ∈{0,1}and
P(X(1), Y(1)) = P(X(0), Y(1)), (4)
P(X(1), Y(0)) =P(X(0), Y(0)). (5)
Equa ion (4) says ha he P&L o Pdepends on X(1). Equa ion (5) says ha he
po olio is pa ially hedged wi h espec o X: i only Xchanges and Y emains
cons an , he alue o he po olio does no change.
We apply he decomposi ions OAT, SU and ASU o ob ain he con ibu ion o he
isk ac o X o a po olio ha sa is y Equa ions (4,5): Equa ion (6) is he con ibu ion
o X o he P&L acco ding o he OAT and he SU decomposi ion upda ing he isk
ac o X i s . Equa ion (7) is he con ibu ion o Xacco ding o he SU decomposi ion
upda ing he isk ac o Y i s . Equa ion (8) is he con ibu ion o Xacco ding o he
ASU decomposi ion:
P(X(1), Y(0)) −P(X(0), Y(0)) =0,(6)
P(X(1), Y(1)) −P(X(0), Y(1)) = 0,(7)
1
2(P(X(1), Y(0)) −P(X(0), Y(0)) +P(X(1), Y(1)) −P(X(0), Y(1)))= 0.(8)
Example 1 We conside a po olio Pin EUR consis ing in a long posi ion in he S&P
500, Y o sho , and a sho posi ion in Y(0)FX- o wa ds wi h s ike X(0), whe e X
models he USDEUR exchange a e, i.e.,
P(X( ), Y( )) =X( )Y( )+Y(0)(X(0)−X( )), ∈{0,1}.(9)
Psa is ies Eqs. (4,5). The S&P 500 changed in he business yea 2003 om Y(0)=
880 o Y(1)=1110 poin s. The USDEUR exchange a e changed om X(0)=0.95
o X(1)=0.79. The po olio in Eq. (9) changed om P(0)=836.0EUR o
P(1)=1017.7 EUR, i.e., by 22%, while he S&P 500 inc eased by 26%. The e o e,
he mo emen s o Xmus also ha e a signi ican impac on he P&L o P. Howe e , i
we apply he SU decomposi ion, which upda es X i s , o he OAT decomposi ion, he
con ibu ion o Xis de ined as ze o, see Eq. (6). Acco ding o he ASU decomposi ion,
123
P o i and loss a ibu ion... 1019
he con ibu ion o Xis gi en by 20.45 EUR and he con ibu ion o Yis gi en by
161.25 EUR, see Eq. (8).
In summa y, he OAT decomposi ion and some SU decomposi ions may assign a
ze o con ibu ion o he isk ac o X, e en hough Xmay ha e signi ican impac
on he P&L. The ASU decomposi ion assigns a non-ze o con ibu ion o X.Thisis
po en ially ele an o an insu ance company because he decomposi ion o he P&L
usually has some impac on he po olio manage ’s ading and hedging s a egy.
Acknowledgemen s We hank Ma cus Ch is iansen o e y ui ul discussions ha helped o imp o e
his le e . S. Flaig would like o hank Deu sche Rück e siche ung AG o he unding o his esea ch.
Opinions, e o s and omissions a e solely hose o he au ho s and do e lec on Deu sche Rück e siche ung
AG o i s a ilia es.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Da a a ailabili y All inancial da a used in his esea ch a e a ailable ia Bloombe g.
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Re e ences
1. Cadoni P (2014) In e nal models and Sol ency II. Risk Books, London
2. Ch is iansen MC (2022) On he decomposi ion o an insu e ’s p o i s and losses. Scand Ac J pp 1–20
3. F ei C (2020) A new app oach o isk a ibu ion and i s applica ion in c edi isk analysis. Risks 8(2):65
4. F iedman E, Moulin H (1999) Th ee me hods o sha e join cos s o su plus. J Econ Theo y 87(2):275–
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5. Je ses J, Ch is iansen MC (2022) A gene al su plus decomposi ion p inciple in li e insu ance. Scand
Ac ua J 2022(10):901–925
6. Mai J-F (2023) Pe o mance a ibu ion wi h espec o in e es a es, FX, ca y, and esidual ma ke
isks. a Xi :2302.01010
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
123
