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Munich Personal RePEc Archive
Factor models and the credit risk of a
loan portfolio
Palombini, Edgardo
October 2009
Online at https://mpra.ub.uni-muenchen.de/20107/
MPRA Paper No. 20107, posted 18 Jan 2010 16:49 UTC
t♦r ♠♦s ♥ t rt rs ♦ ♦♥ ♣♦rt♦♦
r♦ P♦♠♥
t♦r
strt
t♦r ♠♦s ♦r ♣♦rt♦♦ rt rs ss♠ tt ts r ♥♣♥♥t ♦♥
t♦♥ ♦♥ s♠ ♥♠r ♦ s②st♠t t♦rs s ♣♣r s♦s tt t ♦♥t♦♥
♥♣♥♥ ss♠♣t♦♥ ♠② ♦t ♥ ♦♥t♦r ♠♦s t ♦♥st♥t t
trs♦s s ♦♥t♦♥ ts ♦♠ ♥♣♥♥t ♦♥② ♥♥ st ♦ ♦sr
t♠ rs t♦rs s rst s ♦♥r♠ ♦t ♥ ♦♥sr s♠
♥♥ t rts ♥ ♦s ♦♥ s♠ r♠s ①♠♠ ♦♦ st♠ts ♦r
t s♥stt② ♦ t rts t♦ s②st♠t rs t♦rs r ♦t♥ s♦♥ ♦ t②
♠② sst♥t② r② r♦ss ♥str② st♦rs ♥② ♥ rs ♦♥trt♦♥s
r r tr♦ ♦♥t r♦ s♠t♦♥
②♦rs sst ♦rrt♦♥ t♦r ♠♦s ♦ss strt♦♥ ♣♦rt♦♦ rt rs rs
♦♥trt♦♥s
sst♦♥
rss ♦♥♦ ♥tr♥r♦ t ♣♦st ❱ Pst♦ ♦♠
t② ♠♣♦♠♥tt
t♦r ♦ t♦ t♥ ♥ ös ♥ t♦ ♥♦♥②♠♦s rrs ♦r tr ♦♠♠♥ts t♦ trst rs♦♥ ♦ ts ♣♣r s ①♣rss r♥ r t♦s ♦ t t♦r ♥ ♦ ♥♦t ♥ssr② rtt♦s ♦ t ♦♥♦ ♥tr♥r♦ t ♣♦st
♥tr♦t♦♥
st♠t♥ t rt rs ♦ ♣♦rt♦♦ ♦ ♥♥ ♥str♠♥ts rqrs t♦ t②♣s ♦ ♥♣ts
①♣♦srs♣ rs ♣r♦t② ♦ t ♦ss ♥ t ♥ ①♣♦sr t
t t ♠trt strt♦♥ ♦ tr rt ♥ts ♦r ♥ ♦tr ♦rs s♦♠
ss♠♣t♦♥s ♦♥ ♦t t ♠r♥ strt♦♥ ♥ t ♣♥♥ strtr ♦ rs
tt ♠② trr ts s♦♥ t②♣ ♦ ♥♣ts ♥s t st♠t♦♥ ♦ ♦rrt♦♥s
t♥ ts rs r ♦t♥ ♥tr♣rt s t ♦♦rs sst s s ♥ t
rt rs ♠♦ ♥tr♦ ② rt♦♥
♦rrt♦♥s r ♣rtr② ♠♣♦rt♥t s t② ♠t t ♣♦sst② t♦ ② rs②
rt rs t ♣♦rt♦♦ ♦r r ♥ rs ♣♦rt♦♦s t② r♣rs♥t t ♠♥
s♦r ♦ ♥rt♥t② ♦t tr ♦sss t tr st♠t♦♥ s qt ♥♥ t♦
t t tt sst rtr♥s r ♥♦t ♦sr ①st♥ trtr ♦rs tr tr♥ts
t♦ s♦ ts ♣r♦♠ rst t♦r strtr ♥ ss♠ ♦r sst rtr♥s ♥ sst
♦rrt♦♥s r r r♦♠ t ♦r♥ ♠tr① ♦ t ♦s♥ t♦rs ♦r ♥
♦♥ ♦♥ t r♣ ♦♣♠♥t ♦ t rt rts ♠rt ♦s t♦ r
t ♦rrt♦♥s ♠♣t ♥ t ♣rs ♦ s ♥str♠♥ts rs ♥ ❩ r
sst ♦rrt♦♥s ♥ ♦t♥ r♦♠ st♦r t ♦♥ ts ♥ rt tr♥st♦♥s
❲♥ t t r s sst ♦rrt♦♥s ♥ r r♦♠ t ♦rrt♦♥s
s♥ ♥♦♥ ♣r♠tr ♣♣r♦s s ♥ s ♦r ♥ ♦r② tr♥t② sst
♦rrt♦♥s r ♦t♥ rt② r♦♠ t t tr♦ ♠①♠♠ ♦♦ st♠t♦♥
ts ♣♣r♦ s ♥tr♦ ② ♦r② ♥ t ♥ t♥ s ♥ r♦s ♣♣rs
♠♦♥ ü♠♥♥ ♥ ♥ ös ♥ ♥ ♠♣♦rt♥t
♥t ♦ ♣r♠tr ♠t♦♦♦s s tt strtr rstrt♦♥s rs♥ r♦♠ t♦rt
ss♠♣t♦♥s ♥ s② ♥♦r♣♦rt ♥ t ♠♦ rtr♠♦r tr♠♥st ①♣♥t♦r②
rs ♠r♦♦♥♦♠ t♦rs ♥ ♦♥sr t♦tr t ♥♦sr r♥♦♠
t♦rs
♥② ♠♣r ♦rs ♥ ♣♣t♦♥s ♥♥ t s rt♦r② r♠♦r r
s ♦♥ t ♠♦ ♣r♦♣♦s ② ❱s sst rtr♥s r ss♠ t♦ ♦♥t②
ss♥ strt ♥ ts ♦r ♥ t② ♦ ♦♥st♥t trs♦ r
ss♠♣t♦♥ ♦ ♦♥t♦r ♠♦s s tt ts r ♥♣♥♥t ♦♥t♦♥ ♦♥ t s♥
♦♠♠♦♥ rs t♦r s sst ② ö♥r ts ss♠♣t♦♥ s♦ r②
s ♦t♦♥ ♦ t♦ ♥ ♥rst♠t♦♥ ♦ t ♣r♦ts
r ♠♣r rsts s♦ tt ♥ t ♦♥t♦r r♠♦r t ②♣♦tss ♦ ♦♥t♦♥
♥♣♥♥ t♥ ts s ♦t ♥ tt t ♥tr♦t♦♥ ♦ st ♦ tr♠♥st
①♣♥t♦r② rs s ♥ssr② s ♥ s ♦♥r♠ s♦ ♥ s s♠♥♥
t ♦♥ t rts ts ♦t♥♥ st♠ts s ♦♥ ♦♥sr② ♦♥r t♠ srs
♦ ts ♠r rsts r s♦ ♦♥ st♠t♥ s♣ ♠♦ ♦r s♠ ♥tr♣rss
♥ s♥stts t♦ ♦♥♦♠ ♦♥t♦♥s t♥ t♦ rs
s ♣♣r s ♦r♥③ s ♦♦s t♦♥ srs t t♦stt t♦r ♠♦ s
♥ t ♦♦♥ ♥②ss t♦♥ ♣rs♥ts ♦r ♠♣r rsts ♦ t② r② ♦r♥
t♦ t ♥str② ♥ t s③ ♦ t ♦♦r ♥ s♦ ♦r♥ t♦ t ♥t ♦ t ♦r③♦♥
♦♥sr ♥ t♦♥ ♥ ♦♥trt♦♥s t♦ ♣♦rt♦♦ rt rs r ♦t♥ tr♦
♦♥t r♦ s♠t♦♥
rt rs ♠♦ t t♦r strtr
♦♥sr ♣♦rt♦♦ ♦ ①♣♦srs r t ♥♦r♠③ rtr♥ ♦♥ t ssts ♦ ♦♦r i t
t♠ t t i ∈ , ..., nt ♥ t ∈ , ..., T s r♥ ② ♦t ♦sr t♠ ♥
♥♦sr rs t♦rs
Ri,t = α′Zt−1 + Ui,t
r t t♦r Zt−1 r♣rs♥ts st ♦ ♥♦r♠t♦♥ t t♠ t−1 t♦ st♠t t
t ♣r♦t② ♦ ♦♦r i t ♠② ♥ ♦♦rs♣ rs s s t♦rs tt
t t rt qt② ♦ t ♦ ♣♦rt♦♦ ♥♦sr r Ui,t s ♥♥
② ♦♥♠♥s♦♥ s②st♠t t♦r Yt ♥ ② r♠s♣ ♦♠♣♦♥♥t Ei,t
Ui,t =√δYt +
√1 − δEi,t
r Yt ♥ Ei,t ♦♦ st♥r ♥♦r♠ strt♦♥s rs ♥ qt♦♥s ♥
r ss♠ t♦ ♥♣♥♥t ♥ Ei,t s s♦ ♥♣♥♥t r♦ss ♦♦rs r
♦r ♦♥t♦♥ ♦♥ t r③t♦♥s ♦ ♦♠♠♦♥ rs t♦rs ts r ♥♣♥♥t r♦♠
qt♦♥ t ♦♦s tt t s♥stt② ♦ sst rtr♥s t♦ t s②st♠t ♥♦sr
rs t♦r ♣♥s ♦♥ t t♠♥r♥t ♣r♠tr δ r♣rs♥ts t s♦ sst
♦rrt♦♥
t ♦ r♠ ♦rs ♥ t rtr♥ ♦♥ ts ssts s ♦ trs♦ k ♥
t t r♥♦♠ r Di,t t ♥t♦r r tt s q t♦ ♦♥ r♠
i s ♥ t t t♠ t ♥ ③r♦ ♦trs
Di,t = 1 ⇐⇒ Ri,t < k
♦♥t♦♥ ♦♥ t r③t♦♥ ♦ ♦♠♠♦♥ rs t♦rs t t ♣r♦t② ♥
①♣rss s
pi,t (zt−1, yt) = P (Ri,t < k | zt−1, yt)
= P (α′zt−1 + Ui,t < k)
= P(
α′zt−1 +
√δyt +
√1 − δEi,t < k
)
= Φ
(
k − α′zt−1 −
√δyt√
1 − δ
)
r Φ ♥♦ts t st♥r ♥♦r♠ ♠t ♥st② ♥t♦♥
s t ♥♠r ♦ ♦rr♦rs r♦s t sr ♦ t rst ①♣♦sr ♥ss t♦ ③r♦
t t rt DRt ♦♥rs t♦ t ♦♥t♦♥ t ♣r♦t② ② t str♦♥ ♦
r ♥♠rs
DRt → [DRt | zt−1, yt] = pi,t (zt−1, yt)
Φ−1 (DRt) →k − α
′zt−1√
1 − δ−
√δ√
1 − δyt
♦r ♥ q ∈ (0, 1) t ψq(Y ) ♥♦t t qth ♣r♥t ♦ t strt♦♥ ♦ t r♥♦♠
r Y t ①♣t t rt s ♠♦♥♦t♦♥② rs♥ ♥ t s②st♠t r♥♦♠
t♦r t qth ♣r♥t ♦ ts strt♦♥ ♥ ①♣rss s ♦r②
ψq( [DRt | zt−1, Yt]) = [DRt | zt−1, ψq(Yt)]
♥ ts ♦ t t ss♠♥ tt t r♥♦♠ t♦r s ♥♣♥♥t② ♥
♥t② strt ♦r t♠ t ♣r♠trs ♥ qt♦♥ ♥ st♠t ♠①♠③♥
t ♦♦♦ ♥t♦♥
lnL =T∑
t=1
ln
+∞∫
−∞
∏
i∈Nt
pi,t (zt−1, yt)di,t (1 − pi,t (zt−1, yt))
1−di,t
dΦ (yt)
♥trs ♥ qt♦♥ ♥ s♦ ♣♣r♦①♠t② s♥ t ♣t ss♥
qrtr sr ♥ P♥r♦ ♥ ts t ♦♦s r♦♠ t ♥r t♦r② ♦
♠①♠♠ ♦♦ st♠t♦♥ tt t st♠ts ①st s②♠♣t♦t② ♥ r ♦♥sst♥t
s s s②♠♣t♦t② ♥♦r♠② strt s♦♥ ♥ ♥♥♦♥
♠♣r ♥②ss
t
s ♣♣r ss t♠ srs ♦ t t ♦♠♥ r♦♠ t s ♥♦r♠t P
P ts ♣r♦ ② t ♥ ♦ t② ♦♥t♥♥ qrtr② t♠ srs ♦r t
♥♠r ♦ ts ♥ t ♥♠r ♦ ♦♥s tt r ♥♦t ♥ t t t ♥♥♥ ♦ t
♣r♦ strt♥ ♥ ♥♠r ♦ ts ♥ ♥ qrtr s ♥ ② t ♥♠r
♦ ♦rr♦rs tt ♦♠ st t♦rs r♥ tt qrtr
♥ t P ts ♦rr♦rs ♥ st♥s ♦r♥ t♦ sr rs
♥str② st♦r ♥ ♦r♣ ♦t♦♥ ♥ ts ♣♣r ♦s ♦♥ ♥♦♥♥♥ ♦♠♣♥s
s t♥ t t♥ ♥strs st ♥ s♦ s♦s t r
♥♠r ♦ r♠s ♥ t r t rt
Pr♠tr st♠ts
rt rs ♠♦ sr ♥ t♦♥ rs ♦♥ t r ss♠♣t♦♥ tt ♦♥t♦♥
♦♥ ♦♠♠♦♥ rs t♦rs ts r ♥♣♥♥t ts r ♥♦t t s s②st♠t
t♦rs ♥♦t ♣tr t ♦rrt♦♥ t♥ ts ♠①♠③♥ t ♦♦♦
♥t♦♥ ♥ qt♦♥ ♦ t♦ ♥ ♥rst♠t♦♥ ♦ t ♣r♦ts
s tst ♦ ♦r ♠♦ ♥ s t t tt ♥r t ♦♥t♦♥ ♥♣♥♥
ss♠♣t♦♥ s t ♥♠r ♦ ♦rr♦rs r♦s t t rt DRt ♦♥rs t♦ t ♦♥
t♦♥ t ♣r♦t② ♥ t ♥♠r ♦ ♦rr♦rs ♥ ♦r tst s r ♥♦
t♦ r♣rs♥t rs♦♥ ♣♣r♦①♠t♦♥ ♦ ♥ ♥♥t② r♥r ♣♦rt♦♦ ♥ s
♦r♥ t♦ t ♥t♦♥ s ② t ♥ ♦ t② ♦rr♦r ♦♠s ♥ st t♦r r♣♦rt t♦ t t♥ ♥tr rt str s t ② t ♦♥② ♥ tt srs rt s t ② ♦♥ ♥ ♥ s ♥ ♥ ♦rs♦♦t ② t ♦♥② ♦tr ♥ ①♣♦s s t ②♦♥ ♥ ♥ t ♠♦♥t ♦ t t s t st ♦ ts ①♣♦sr t♦rs t ♥♥ s②st♠ ♦r s♥ ♦rs♦♦ts q t♦ ♦r ♠♦r t♥ ♦ ts t♦t ♦♥s ♦tst♥♥ s t ② t st t♦♥s ♦r ♠♦♥ts q t♦ ♦r ♠♦r t♥ ♦ ts t♦t ♦♥s ♦tst♥♥
rtr ♥♦r♠t♦♥ ♦t t tst ♥ t t♥ ♥tr rt str s t t ♥ ♦t②s st
♥str② st♦rsr ♥♠r ♦ ♦rr♦rs ♥r t rt
N %DR rtr ♦rstr② ♥ sr② ♣r♦ts t ♣r♦ts ①♣t tr♥s♣♦rt q♣ ♦♦ ♥ t♦♦ ♣r♦ts ①ts ♦t♥ ♥ ♦♦tr P♣r ♥ ♣♣r ♣r♦ts ♥ ♥ ♦♥strt♦♥ r srs r♦r② ♥ r♣r srs ♦♥ ♥ tr♥ srs tr ♠rt srs ♣♦r ♣r♦ts ♥ ♠ ♣r♦ts rs ♥ ♠ts ♥ ♥♦♥♠t ♠♥rs ♥ ♣r♦ts rtr ♥ ♥str ♠♥r② ♥ tr♥s♣♦rt q♣ tr ♦♦s ♦ ♥ t ♣r♦ss♥ ♠♥s tr ♠♥tr ♣r♦ts r♥s♣♦rt ♥ ♦♠♠♥t♦♥ srs
♥ r♥ rrss♦♥ ♦ Φ−1 (DRt) ♦♥ ♦♥st♥t ♥ st ♦ ♣♦t♥t ①♣♥t♦r② r
s ♥ ♥ ♥stt t ♣r♦♣rts ♦ t rss ♥ ♣rtr ♥ t
ss♠♣t♦♥s ♦ ♥♦r♠t② ♥ sr ♥♣♥♥ r ♦t
❲ strt ♦♥sr♥ t ♦♥t♦r rs♦♥ ♦ t ❱s ♠♦ ts ♠♣♦s♥ α′ =
0 s ts ♠♦ s s ♥ ♠♥② ♠♣r ♦rs ♥ ♣♣t♦♥s ♥♥ t s
rt♦r② r♠♦r ♥ t ♥ ♦rrt♦♥ st♠ts r ♦t♥ tr♦ ♥♦♥
♣r♠tr ♣♣r♦s s ♥ ♦r② ♦r r♥② ♥ ♥t sst rtr♥s
r ss♠ t♦ ♦♥t② ss♥ strt ♥ ts ♦r ♥ t② ♦
♦♥st♥t trs♦
r♥ rrss♦♥ ♦ Φ−1 (DRt) ♦♥ ♦♥st♥t t ♦rr♦r♠ ♦ rss ♣r♦s
str♦♥ ♥ ♦ sr ♦rrt♦♥ s♦s t♦♦rrt♦♥s ♥ ♣rt t♦♦rr
t♦♥s ♣ t♦ s t♦tr t t ♥♦① sttsts ♦r t ♥ ②♣♦tss ♦ ♥♦
t♦♦rrt♦♥ s str♦♥② rt s ♥ ①♣♥ ② t t tt t
rts r ② t♦ ♣♥ ♦♥ tr ♣st s t♦ t ♣rsst♥ ♦ ♦♥♦♠ s♦s ♥
s ♦ ♣♦ss ♦♥t♦♥ ts
♥ ♦rr t♦ ♠♥t ♥② sr ♦rrt♦♥ ♥ t ♣r♥t t rt
♥ t ♠♦ ♦♥t ♦ t ♥ r s ② s♥♥t t ♠♦s t ♠♣r♦s
♥r rrss♦♥ ♦r t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
♥ t ♦rr♦r♠ s♦s tt ♠♦st ♠♥t ♦rrt♦♥ t♥ rss
♦ r② rss r ♥♦r♠② strt ♣r♦r♠ t♦ sttst tsts
♣r♦❲ ♥ ♦st♥♦Prs♦♥ s ♦st♥♦ ♥ t♣♥s ♦r sss♦♥
♥ ♦t ss t ♥♦r♠t② ss♠♣t♦♥ s rt t r② ♦♥♥ s sst♥
tt s♦ ①t♥ ♦r st ♦ ①♣♥t♦r② rs
♥r rrss♦♥ ♦r t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
♥ ♦rr t♦ ♦♦s ♥ t♦♥ ♥♣♥♥t r ♦♥sr t ♦♥s ♦
r♦ st ♦ ② ♦♥♦♠ ♥t♦rs ♦r t② ♥♥ P ♥str ♣r♦t♦♥ ♥
♦rrs t♦ ♠♥tr♥ r♦ss ① ♣t ♦r♠t♦♥ ♦s♦ ♦♥♥ ♦♠♣♦st
♥ ♥t♦r ♥ r ♥trst rts s ♠sr t r♦s qrtr② s ♥ ♦r
r♥t t♠ ♦r③♦♥s r♥♥ r♦♠ t♦ t♦ ♦r qrtrs t tr♥s ♦t tt t s♠♥♥
♦♥ ♦ ♦s♦ ♦♥♥ t t t ♥ ♦ t ♣r♦s ②r s t r
tt t♦tr t ♦♥st♥t ♥ t t rt ♠①♠③s ♦r ♠♦s t ♥
♥②ss ♦ t ♣♦ss rs♦♥s ② ♦s♦ ♦♥♥ s♦s t st ♣rt ♣♦r
♠♦♥ t st ♠r♦♦♥♦♠ rs ♦s ②♦♥ t s♦♣ ♦ ts ♣♣r t ts
rst s ♣r♦② t♦ ts ♥t♣t♦r② ♥tr s ♦♥r♠ ② t t tt t ♥
♦♥♦♠ ♥t♦r s s♠r ♣r♦r♠♥
s♦s tt ♦♥ts r s♥♥t t t ①♣t s♥ t rts
♥rs ♥ ♦♥♦♠ ♦♥t♦♥s tr♦rt ♥ ♥② tt t ♥♦r♠t② ②♣♦tss
♦r t rss ♥♥♦t rt ♥ t qt ♦ ♦♥♥ s ❲ s♦ t
③r♦♠♥ ss♠♣t♦♥ ♦r ♦r rss ♣r♦r♠♥ t s♥ tst ♥ t ❲♦①♦♥ s♥
r♥ tst ♥ ♦t ss t ♥ ②♣♦tss ♥♥♦t rt t r② ♦ ♦♥♥ s
♥r rrss♦♥ ♦r t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
Houseconf t−1 sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
❲ ♥② ♠♦ tt ♠ts t ss♠♣t♦♥s sss ♥ st♦♥ ts ♥
r rs st♠ts s q♥t ♠srs ♦ ♦r ♦ss strt♦♥ s♥ qt♦♥ ♥ ♦rr
t♦ ♦ s♦ ♥ t♦ st♠t t ♦rrt♦♥ ♣r♠tr tt ♠srs t s♥stt② ♦ sst
rtr♥s t♦ t r♥♦♠ rs t♦r t s♦ sst ♦rrt♦♥ s ♥ ♦♥ tr
♦♥sr♥ t r♥ ♦ t rss ♦ ♦r rrss♦♥ ♦r ♠①♠③♥ t ♦♦♦
♥t♦♥ ♥ qt♦♥
s♦s ♦r ♠①♠♠ ♦♦ st♠ts ♦r t ♥tr ♣♦rt♦♦ ♥ ♦r
♥str② st♦r s♣rt② st ♥♦♥t♦♥ t ♣r♦ts ♥t ②
t rst t trs♦s r ♦♥ ♦r st♦rs ♥ ♦♠♣rt② ♦ t
♣r♦ts r ♦♥ ♦r st♦rs ♥ sst ♦rrt♦♥ s ①♣rss ② t s♥stt②
♦ sst rtr♥s t♦ t ♥♦sr ♦♠♠♦♥ rs t♦r s ♥r s♥♥t t t
①♣t ♦r t ♦ ♣♦rt♦♦ ♥ t s ♥♦t s♥♥t t t ♥ tr ss ts
st s r ♦r st♦r ♥ ♦r st♦r r ♣♥♥ t♥
t rts tr♥s ♦t t♦ qt r♥t ♥ ♥strs ♥ st t s ♥♦t s♥♥t
t t ♦r st♦r ♦♥ ♥ ♦s♦ ♦♥♥ ♣r♦r♠s r② s
t s s♥♥t t t ♥ st♦rs ♥ t s ♦ t ♦♥② ♦r ♥
♥ ♦♥strt♦♥ ♥ ♠r♥② ♦r ♣♦r ♣r♦ts ♥ ♠ ♣r♦ts
①♠♠ ♦♦ st♠ts♥str② ♦♥st♥t sst ♦rrt♦♥ %DRt−1 Houseconft−1
st Pr > |t| st Pr > |t| st Pr > |t| st Pr > |t|
♥ ♥r t rsts ♥ sst tt s♥stts t♦ rs t♦rs r② r♦ss
♦rr♦rs ♦r♥ t♦ t ♥str② ♥ t② ♦♣rt ts r t s t ♣♦rt♦♦
♦♠♣♦st♦♥ ♦ s♥♥t② t ♦r rs st♠ts ♥ rs ♦♥trt♦♥s ♦ r
r♦ss ♥strs s ss ♥ ♦r♠② ♥stt tr♦ sttst tsts s t
rt♦ ♦ ♥② st♠t ♥ t ts st♥r rr♦r ♣r♦s t t ♣♣r♦①♠t
r ♥♣♥♥ ♥ ♥♦r♠t② ♦ rss r s♦ t ♥str②
ttst tsts ♦r st♠tsH0 :♥str② st♠ts r q t♦ ♣♦rt♦♦ st♠ts
♥str② ♥tr♣t sst ♦rr %DRt−1 Houseconft−1
♥ts tt t ♥ ②♣♦tss ♥♥♦t rt t t
rs ♦ r♦♠ ♦♠♣t s t ♥♠r ♦ ♦srt♦♥s ♠♥s ♦♥ s♦s t
s ♦ t r♥t tsttsts ♥r t ♥ ②♣♦tss tt s♥stts st♠t t
♥str② r q t♦ t♦s st♠t ♦r t ♥tr ♣♦rt♦♦ s rt t t
♦♥♥ ♥ ♠♦st ss
♦♥sr♥ t ♠♣♦rt♥t r♦ ♣② ② ♦♠♠♦♥ rs t♦rs t ♠② s t♦ ♣r
♦r♠ s♦♠ s♥stt② ♥②ss ♦ ♣♦rt♦♦ rt rs t rs♣t t♦ tr ♦♥ts ♥
♣rtr ♦♥sr t s ♦ s ♦♥ts t t t t ♥ t ♣r♥ts
♦ tr strt♦♥ ♥ t ❱ s♥ qt♦♥ st♠t s♥stts
t♦ s②st♠t rs t♦rs ♠② s♥♥t ♠♣t ♦♥ ♣♦rt♦♦ rt rs ♥
♦♥sr s ♦r rs ♠sr ♠② ♥rs ② ♠♦r t♥ t ♦♥sr
s ts ♥rss r♥ t♥ ♥
P♦rt♦♦ rt rs ♥ t♠ ♦r③♦♥
♥ ts st♦♥ r♣t t ♠♣r ♥②ss ♦ st♦♥ s♥ s♠♥♥ t t
st♠t♥ ♣♦rt♦♦ rt rs ♦r s♦rt ♦r③♦♥s ss t♥ ♦♥ ②r ♠② s ♦r
♦r ♦sr rs t♦rs t rsts ♦ s♥stt② ♥②ss r t♠r②♥ s t② ♣♥ ♦♥ t ♦ t ♦rrs♣♦♥♥ r
♥stt② ♦ V aR99.9% t♦ st♠ts t rts ♦r
Pr♥tt t t t
♥♦♠ t♦r
%DRt−1 Houseconft−1
t♦ rs♦♥s rst t ♦s t♦ ♥rt s♦rttr♠ ♣♦rt♦♦ ♦ss strt♦♥s ♥
s t♦ ♣r ♥♥ ♥str♠♥ts t qrtr② ♦r s♠♥♥ s ♦s ♣♦rt♦♦
rt rts s♦♥ t s t ♦♣♣♦rt♥t② t♦ ♦t♥ ♦♥②r ♦ss strt♦♥s
s♥ ♣r♠trs st♠t r♦♠ ♠ rr ♥♠r ♦ ♦srt♦♥s
P♦rt♦♦ rt rs ♥rss t ♦r③♦♥ s ♦ ♥ ♥rs ♥ t ♣r♦
ts t ♦♥st♥t sst ♦rrt♦♥s ♦r ♥ ♥rs ♥ ♦t s ♥ ♦tr ♦rs
ts t♥ t♦ str ♦r ♦♥r ♦r③♦♥s s♦② t♦ ♥ ♥rs ♥ t ♠r♥ ♣r♦
ts ♦ t ♥ts ♦r s♦ s t② r ♠♦r r♠s♣ ♦r s♦rt ♦r③♦♥s t
♠♦r trr ② s②st♠t t♦rs ♦r ♦♥r t♠ ♣r♦s ♦r ♥st♥ ❩♦
s♦s tt s t ♦r③♦♥ ♥t♥s ♥ ♥rs ♥ t ♦rrt♦♥s ♠② s♦②
t♦ ♥ ♥rs ♥ t ♣r♦ts t ♦♥st♥t sst ♦rrt♦♥s
r rsts ♦r s♠♥♥ t rts ♦♥r♠ t ♠♣r ♥♥s ♦ st♦♥
t ①t sr ♦rrt♦♥ tt ♥ ♠♥t ♥♥ t rts ♥ t
♠♦ t rss r♠♥ ② ♥♦♥♥♦r♠ s ♥
❲ r♣t t st♦♥ ♣r♦ss ♠ t ♥♥ ♥ t♦♥ ①♣♥t♦r② r ♥
s ♥ st♦♥ t s♠♥♥ ♦♥ ♦ ♦s♦ ♦♥♥ tr♥s ♦t t♦
t ♦♥ tt ♠①♠③s ♦r rrss♦♥s t ♦♥t ♦ t ♥ r s ②
s♥♥t t t ①♣t s♥ ♥ ♥ ♥♦ ♦♥r rt t ②♣♦tss tt rss
r ♥♦r♠② strt
s♦s ♦r ♠①♠♠ ♦♦ st♠ts ♦r t ♥tr ♣♦rt♦♦ ♥ ♦r
♥str② st♦r s♣rt② sst ♦rrt♦♥ s ①♣rss ② t s♥stt② ♦ sst rtr♥s
t♦ t ♥♦sr ♦♠♠♦♥ rs t♦r s ♥♦ s♥♥t t t ♦r t ♥strs
♥ ♦r t ♦ ♣♦rt♦♦ ts st s ♦r st♦rs ♥ r ♣♥♥
t♥ t rts s ♥ ♠♦r r♥t s t s ②s s♥♥t t t ♥
t ♦♥trr② t ♦♥ ♥ ♦s♦ ♦♥♥ s♦s ♦r ♣rt ♣♦r s t s
r ♥♣♥♥ ♥ ♥♦r♠t② ♦ rss r s♦ t ♥str②
♥r rrss♦♥ ♦r s♠♥♥ t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
♥r rrss♦♥ ♦r s♠♥♥ t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
s♥♥t t t ♥ ♦♥② s① st♦rs ♥ t s ♦ t ♥ ♦r ss ♥
♥r ♦r t rsts ♥ ♦♥r♠ tt s♥stts t♦ rs t♦rs sst♥t②
r② r♦ss ♥strs
♥r rrss♦♥ ♦r s♠♥♥ t rt Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
Houseconft−1
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
①♠♠ ♦♦ st♠ts s♠♥♥ t♥str② ♦♥st♥t sst ♦rrt♦♥ %DRt−1 Houseconft−1
st Pr > |t| st Pr > |t| st Pr > |t| st Pr > |t|
P♦rt♦♦ rt rs ♥ r♠ s③
rsts ♦ t♦♥ r ♥♦t ♥ssr② ♦r s♠ ♦♠♣♥s s t② ♦♥
r ss rs ♣♦rt♦♦ ♦ ssts t♥ r ♦♠♣♥s ts t② ♠② s♦
♦r s♥stt② t♦ s②st♠t rs t♦rs ♥ t s ♣♣r♦ ♦r ♥st♥
sst ♦rrt♦♥ s ♥ ♥rs♥ ♥t♦♥ ♦ t s③ ♦ t ♦♦r s ♠sr ② ts
②r② tr♥♦r rsts ♥ ü♠♥♥ ♥ ♥ ♦♣③ s♦ tt
sst ♦rrt♦♥s ♥rs t r♠ s③ st ts ♥ Pt② ♥ ❯s♣
rt♦♥s♣ t♥ t t♦ rs s ♦♥s♦♥s ♦r ♥ ♥♥ ②
t t tt s♠ ♥tr♣rss ♦♣rt ♠♥② ♥ ♥strs tt r ss ② rs
r r♠s ♣r ♥ st♦rs ♠♦r ♣♥♥t ♦♥ ♠r♦♦♥♦♠ ♦♥t♦♥s ♥ s♠
♦♠♣♥s t♦ s♦ ♦r s②st♠t rs
ts s ♥ ts ♣♣r ♦s t♦ st♥s ♦♥② ♦♥s t♦ s♦ ♣r♦♣rt♦rs♣s
t ♣ t♦ ♠♣♦②s s ♣r♦①② ♦r t s♠ s♥ss st♦r ♥ s t♦r②
♦rrs♣♦♥s ♠♦r t♦ ♥t♦♥ ♦ ♠r♦♥tr♣rs t ♥ s t♦ r♣t t ♥②ss ♦
st♦♥ ♥ t rsts r② t r♠ s③
s ♦r rr r♠s r♥ rrss♦♥ ♦ Φ−1 (DRt) ♦♥ ♦♥st♥t t ♦rr♦r♠
♦ rss ♣r♦s str♦♥ ♥ ♦ sr ♦rrt♦♥ ♥ ♠♥t ♥♥
t ♣r♥t t rt ♥ t ♠♦ s ♥
♥r rrss♦♥ ♦r t rt s♠ r♠s Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
t s ♥trst♥ t♦ ♥♦t tt t rs♣t t♦ rr r♠s ♣rtrs r♦♠ ♥♦r♠t②
r ss ♥t sst♥ tt ♦r s♦♠ s♣♦rt♦♦s t ♠② ♥♦t ♥ssr② t♦ ♥
t♦♥ rs t♦ ♠t t ♦♥t♦♥ ♥♣♥♥ ss♠♣t♦♥ ♦♥t ♦ t
sst ♦rrt♦♥ s strt② ♥rs♥ ♥t♦♥ ♦ t ♦♦rs tr♥♦r ♦♥② ts s ♥♦t ♦ ♠♦♥ r♦ ♥ ♥♦t ♦ ♠♦♥ r♦
♥r rrss♦♥ ♦r t rt s♠ r♠s Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
s♠♥♥ ♦♥ ♥ ♦s♦ ♦♥♥ s ♥♦t s♥♥t t t ♦♥r♠♥
tt s♠ r♠s r ss ♥♥ ② s②st♠t rs t♦rs t t ♥♦r♠t② ②♣♦tss
♦r t rss ♥♥♦t rt ♥ t qt ♦ ♦♥♥ s
♥r rrss♦♥ ♦r t rt s♠ r♠s Pr♠tr st♠ts ♦rr♦r♠ ♦ rss
❱r st♠t Pr > |t| P tt ♣♥tr♣t %DRt−1
Houseconft−1
sts ♦r ③r♦♠♥ rss st ttst ♣ ♥
❲♦①♦♥
sts ♦r ♥♦r♠t② ♦ rss st ttst ♣
♣r♦❲ ♦st♥♦t♣♥s
s ♦♥trt♦♥s
s ♠♥rs r t②♣② ♥trst ♥ st♠t♥ ♥♦t ♦♥② t ♦r rs ♦ ♣♦rt♦♦
t s♦ t ♦♥trt♦♥ ♦ ♥ ①♣♦srs t♦ s rs ♥ t ♥ rs ♦♥
trt♦♥s r ② ♥♣t ♦r stt♥ ♣r♥ ♠ts t♥ rsst ♣r♦r♠♥s
♥ ♦r ♣t ♦t♦♥ s♦♥s
r♥ rs ♦♥trt♦♥s ♥ r♣rs♥t s ♦♥t♦♥ ①♣t ♦sss s
t Li t ♦ss r♦♠ ♦♦r i ♥ L t t♦t ♦ss ♥ t ♣♦rt♦♦ ss♠♥
P (L = V aRα) > 0 t ♠r♥ V aRα ♦♥trt♦♥ r♦♠ ♦♦r i ♥ ①♣rss s
E (Li | L = V aRα)
♥ t ♠♦ sr ♥ st♦♥ ♥ ①♣t ♦sss r q ♦r ①♣♦srs
s ♦t t trs♦s ♥ s♥stts t♦ ♦♠♠♦♥ rs t♦rs r ss♠ t♦ ♦♥
st♥t r♦ss ♦rr♦rs ♠♣r rsts ♥ st♦♥ s♦ tt ts ss♠♣t♦♥s r
♣t ♥ t♥ ♥ ♦r ♣♦rt♦♦ rs ♠sr ♥ t ♦♥trr② ♥ s
t♠t ♥ rs ♦♥trt♦♥s t s♠ ss♠♣t♦♥s s♦ r① s ♥r♠♥t
rs ♦r ♦rr♦r t r P ♦r sst ♦rrt♦♥ s trs ♣rs rtr t♥
♥r♠♥t rs ♦r ♦rr♦r t ♦r P ♦r sst ♦rrt♦♥
♥ ts st♦♥ ♦♥sr tt♦s ♣♦rt♦♦ ♦ ♦♥s r ♥strs r
q② r♣rs♥t ♥ st♠t rs ♦♥trt♦♥s ♦r st♦r ♥ ♦rr t♦ ♦ tt
s ♠①♠♠ ♦♦ st♠ts ♦ ♦rrt♦♥s ♥ t ♣r♦ts t♦ ♣♣r♦①♠t
t ♦ss strt♦♥ ② ♦♥t r♦ s♠t♦♥
❲ r ♥ ♥♣♥♥t st♥r ♥♦r♠ r ♦r ①♣♦sr ♥ t♦r r♦♠
t ♠trt ♥♦r♠ strt♦♥ ♦ ♦♠♠♦♥ rs t♦rs Y ∼ N(0,Ω) r Ω s t
♦r♥ ♠tr① ♦ t ♠♣r st♠ts ♦ s②st♠t t♦rs ♥ r♥t ♥strs ❲
t t sst rtr♥ ♦r ①♣♦sr ♦♠♣r t t♦ t t trs♦ ♥
tr♠♥ t t ♥t♦r ♦♥ ♦r t ③r♦ ♦trs ♣♦rt♦♦ ♦ss ♦r ts
r s ♥ ② t s♠ ♦ t ♥t♦rs ♦ ①♣♦srs ts ss♠♥ tt ♦ss
♥ t ♥ ①♣♦sr t t r ②s q t♦ ♦♥ ♦ st♠t t ♣♦rt♦♦ ♦ss
strt♦♥ t♦ ♥tr♣rt s t strt♦♥ ♦ t ♥♠r ♦ ts ♥ t ♣♦rt♦♦
r♣t ts ♣r♦ss ♠♥② t♠s
t♥r ♦♥t r♦ ♠t♦s r t ② sr♦s ♣r♦♠s ♥ st♠t♥ ♠r♥
rs ♦♥trt♦♥s s ts ♥ r♣rs♥t s ♦♥t♦♥ ①♣t ♦sss ♦♥ s♣♦rt♦♦s
♦♥t♦♥ ♦♥ ♦♣r♦t② ♥ts ♦ rss t ♣r♦♠s r♥ r♦♠ t rrt②
♦ r ♦sss s ♥ ♣tt♦♥ ♦ t ♠♣♦rt♥ s♠♣♥ ♣r♦r ♦♣ ②
ssr♠♥ ♥ t♦ st♠t t t ♦ ♦r ♦ss strt♦♥ s s
♠♦♥ r♦♠ t r ♣r♦t② ♠sr P t♦ ♥ ♣r♦t② ♠sr P ♥r
r ♦sss r ♠♦r rq♥t ♥ ♣rtr t ♣r♦t② ♦r t t♦t ♦ss t♦ ♦
rt♥ l ♥ ①♣rss s
P (L > l) = E(
1L>l
)
= E
(
1L>ldP
dP
)
r ♥♦ts t ①♣tt♦♥ ♥r P ♥ dP
dPs t ♦♦ rt♦ t♥ t t♦
♣r♦t② ♠srs
♣r♦r sst ② ssr♠♥ ♥ ♦r rt rs ♠♦s t t♦r
strtr ♦♥ssts ♦ t♦ st♣s st♥ t strt♦♥ ♦ s②st♠t t♦rs ♥rs♥
♦♥t♦♥ t ♣r♦ts
♥ ♦r ♠♦ t st♠t ♦♥ts ♦ t ♥♦sr ♦♠♠♦♥ rs t♦rs r r②
♦ ts ♦s ♦♥② ♦♥ t s♦♥ st♣ ♦ t ♠♣♦rt♥ s♠♣♥ ♣r♦r ♥ ♣rt
r ♦♥sr ♥ ♥rs t ♣r♦t② ♦r ♥str② j t j ∈ , ...,
p (θ, Yj) =p (Yj) e
θ
1 + p (Yj) (eθ − 1)
r θ s t s♦t♦♥ t♦ t qt♦♥
∂
∂θψ (θ,Y) = V aRα
t
ψ (θ,Y) =15∑
j=1
log(
1 + p (Yj)(
eθ − 1))
♦♦ rt♦ tt ♦s t♦ ♠♦ r♦♠ t ♣r♦t② ♠sr P t♦ t
♦r♥ ♠sr ♥ ①♣rss s
λ = exp (−θ (Y)L+ ψ (θ (Y) ,Y))
V aRα ♦♥trt♦♥ ♦r ♦♦rs ♥ st♦r j ♦♠s
V aRα,j =E(
Ljλ1L=V aRα
)
E(
λ1L=V aRα
)
r♣♦rts t V aR99.9% ♦♥trt♦♥s ♦r ♥str② ♥ ♦r ♣♦rt♦♦ rsts
♦♥r♠ t ② r♦ ♣② ② s♥stts t♦ ♦♠♠♦♥ rs t♦rs ♦t ♦sr ♥
r♥♦♠ ♥ tr♠♥♥ ♥ rs ♦♥trt♦♥s t♦ ♣♦rt♦♦ rt rs ♦r ♥st♥
st♦r s♦s t tr ♦st ♥♦♥t♦♥ P s ♥t ② t ♦♥st♥t ♥
t s♦ qt t♦rrss ♦♠♣♦♥♥t rts ♥ ts rs ♦♥trt♦♥ ♥
t ♦♥trr② st♦r s t st ♥♦♥t♦♥ P t ts rs ♦♥trt♦♥ s r
② ♦ sr ♣♥♥ s ♦r sst ♦rrt♦♥ ♥ ♥♦t tt st♦r s♦s
t ♣r♦t② t ♠ ♦r rs ♦♥trt♦♥ ♥ rt tr♠s t♦ ts ♦
s♥stt② t♦ t r♥♦♠ rs t♦r s ♥
V aR99.9% ♦♥trt♦♥s♠♣♦rt♥ s♠♣♥ st♠ts ♦r
s t rt ♦♥♥ ♥trs r ♥ ♣r♥tss
♥str② ♥♥ ♠♥♥
❲♥ ♦♥sr s♠♥♥ t ♦sr rs t♦rs r ss ♠♣♦rt♥t ♥ tr
♠♥♥ rs ♦♥trt♦♥s s t rts r ♦r t rs♣t t♦ ②r② t t
s♦ s ♥s ♥ ♠r♦♦♥♦♠ ♦♥t♦♥s ♦ ♥♦t ② s♣② tr ts ♦r s♦rt
♦r③♦♥s t t s♠ t♠ s♥stts t♦ r♥♦♠ t♦rs rt② rtr ♠♣t
♦r ♥st♥ t rs ♦♥trt♦♥ ♦ st♦r s r② ♦♠♣r t♦ ts P s ♦
s♥♥t sst ♦rrt♦♥ rs ♥strs ♥ s♣② ♦ rs ♦♥trt♦♥s ♦r
t ♦♣♣♦st rs♦♥ s ♥
♦♥s♦♥s
s ♣♣r s♦s tt ♠sr♥ ♣♦rt♦♦ rt rs s♥ t ♦♥t♦r rs♦♥ ♦ t
❱s ♠♦ t ♦♥st♥t t trs♦s ♠② t♦ ♥ ♥rst♠t♦♥ ♦ t ♣r♦
ts s s♥ s②st♠t rs t♦r ♥♥♦t ♣tr t ♦rrt♦♥ t♥ sst
rtr♥s ♥ t♦ ♦t♦♥ ♦ t ♦♥t♦♥ ♥♣♥♥ ss♠♣t♦♥ ♥st
♣r♦ tr♦ sttst tsts tt t♦r ♠♦s rqr t ♥s♦♥ ♦ tr♠♥st ①
♣♥t♦r② rs ♠r♦♦♥♦♠ t♦rs ♥ t♦rrss ♦♠♣♦♥♥ts t♦tr t
♥♦sr r♥♦♠ t♦rs ♥ t t rts r ♥♥ ② ♠r♦♦♥♦♠ ♦♥
t♦♥s t rt♥ t♠ t t② s♦ ♣♥ ♦♥ tr ♣st s t♦ t ♣rsst♥
♦ ♦♥♦♠ s♦s ♥ s ♦ ♣♦ss ♦♥t♦♥ ts
♥stts t♦ rs t♦rs r② r♦ss ♦rr♦rs ♦r♥ t♦ t ♥str② ♥ t②
♦♣rt tr♦r ♥ ♥ t ♦♠♣♦st♦♥ ♦ ♣♦rt♦♦ ♠② s♥♥t② t ♦r rs
st♠ts r ♣♥♥ t♥ t rts s qt r♥t ♥ ♥strs ♥ t
s ♥♦t s♥♥t t t ♥ ♦♥② t♦ ss r♦♦♥♦♠ ♦♥t♦♥s s ♠sr
② t ♦♥ ♥ ♦s♦ ♦♥♥ s♦ ♣② ♠♦r r♦ ♥ ♠♦st ss t t② r
♥♦t s♥♥t t t ♦r t♦ st♦rs ♥② sst ♦rrt♦♥ s ①♣rss ②
t s♥stt② ♦ sst rtr♥s t♦ t ♥♦sr ♦♠♠♦♥ rs t♦r s ♥r s♥♥t
t t ①♣t ♦r t ♦ ♣♦rt♦♦ ♥ t s ♥♦t s♥♥t t t ♥
tr ss
♠♥ rsts ♦t♥ t ♥♥ t rts r ♦♥r♠ ② s♠♥♥ t
ts s♥ ♦♥sr② ♦♥r srs ♦ ♦srt♦♥s ❲♥ ♥rs t rq♥② ♦
♦r t t t♦rrss ♦♠♣♦♥♥t ♦♠s ♠♦r ♠♣♦rt♥t t t s st ♥♦t s♥t
t♦ ①♣♥ t ♦rrt♦♥ t♥ ts
P♦rt♦♦ rt rs ♥ s♥stts t♦ s②st♠t rs t♦rs r② ♦r♥ t♦ t s③
♦ t ♦rr♦r ♦♠♣r t rsts ♦t♥ ♦r s♠ ♦♠♣♥s t t ♦♥ts
r ♦r rr r♠s ♦sr tt sr ♣♥♥ t♥ t rts r♠♥s
s♥♥t t ♠r♦♦♥♦♠ ♦♥t♦♥s ♣② ss ♠♣♦rt♥t r♦
t trs♦s ♥ s♥stts t♦ ♦♠♠♦♥ rs t♦rs ♥ ss♠ t♦ ♦♥
st♥t r♦ss ♦rr♦rs ♥ t♥ s♥ ♠sr ♦r ♣♦rt♦♦ rt rs s
ss♠♣t♦♥s ♠st r① ♥ st♠t ♥ rs ♦♥trt♦♥s r♣r
s♥t ♥♠♥t ♥♣t ♦r ♣r♥ t♥ rsst ♣r♦r♠♥s ♥ ♦r ♣t
♦t♦♥ s♦♥s ①♠♠ ♦♦ st♠ts r s t♦ r t rs ♦♥trt♦♥
♦r ♥str② st♦r tr♦ ♦♥t r♦ s♠t♦♥ ♥ ♦r rsts ♦♥r♠ t ② r♦
♣② ② s♥stts t♦ ♦♠♠♦♥ rs t♦rs ♦t ♦sr ♥ r♥♦♠
r♥s
❬❪ ♦st♥♦ t♣♥s ♦♦♥ss ♦ t t♥qs r r
❨♦r
❬❪ s♦♥ ♥♥♦♥ st♠t♦♥ ♥ ♥r♥ ♥ ♦♥♦♠trs ①
♦r ❯♥rst② Prss ❨♦r
❬❪ r♥② ♥t t ♦rrt♦♥ ♠♣r ♥ ❲♦r♥
P♣r t♥r P♦♦rs
❬❪ ts Pt② ♦ ①♣♦srs trt s rt ♦r ♦r♣♦rt
①♣♦srs ♦♠♣rt ♥②ss ♦ ♣r♦ts ♦ t ♥ sst ♦rrt♦♥s ♥
r♥ ♥ r♠♥ s ♦r♥ ♦ ♥♥ ♥ ♥♥
❬❪ ü♠♥♥ sst ♦rrt♦♥ ♦ r♠♥ ♦r♣♦rt ♦♦rs ts
st♠t♦♥ ts rrs ♥ ♠♣t♦♥s ♦r rt♦r② ♣t ❲♦r♥ P♣r
❬❪ ssr♠♥ P ♠♣♦rt♥ s♠♣♥ ♦r ♣♦rt♦♦ rt rs♥♠♥t
♥
❬❪ ♦r② ♦♠♣rt ♥t♦♠② ♦ rt rs ♠♦s ♦r♥ ♦ ♥♥ ♥
♥♥
❬❪ ♦r② st♦r ♠♦ ♦♥t♦♥ ♦r rt♥ss ♥ ♣t rs
♦r♥ ♦ ♥♥ ♥tr♠t♦♥
❬❪ ♦r② t st♠t♥ t ♦rrt♦♥s r♦♠ s♦rt ♣♥s ♦
rt rt♥ ♣r♦r♠♥ t ❲♦r♥ P♣r r sr ♦r
❬❪ ♦r ♦♥ P♦rt♦♦ ♥♠♥t ♦ t s sr P♣r
♦♦②s ❱
❬❪ ♦♣③ ♠♣r rt♦♥s♣ t♥ r sst ♦rrt♦♥ r♠
♣r♦t② ♦ t ♥ sst s③ ♦r♥ ♦ ♥♥ ♥tr♠t♦♥
❬❪ s t ♦rrt♦♥ ♥ rt ♥②ss ♦r♥ ♦ ① ♥♦♠ r
❬❪ rt♦♥ ♥ t ♣r♥ ♦ ♦r♣♦rt t t rs strtr ♦ ♥trst rts
♦r♥ ♦ ♥♥
❬❪ P♥r♦ ts ♣♣r♦①♠t♦♥s t♦ t ♦♦♦ ♥t♦♥ ♥ t
♥♦♥♥r ♠① ts ♠♦ ♦r♥ ♦ ♦♠♣tt♦♥ ♥ r♣ ttsts
❬❪ ös ♦rst♥ rt ♣♦rt♦♦ rt rs ♦r♥ ♦ s
♥♥ ❲♥tr♣r♥
❬❪ ö♥r t♦r ♠♦s ♦r ♣♦rt♦♦ rt rs ♠♠♦
❬❪ rs ❩ ♣r♥ ♦ ♣♦rt♦♦ rt rs ❲♦r♥ P♣r
♥ ♦r ♥tr♥t♦♥ tt♠♥ts
❬❪ s s ♦♥trt♦♥s ♥ ♣r♦r♠♥ ♠sr♠♥t ❲♦r♥ P♣r
♥s ❯♥rstät ü♥♥
❬❪ ❱s ♦♥ ♦ss strt♦♥ ❲♦r♥ P♣r ❱ ♦r♣♦rt♦♥
❬❪ ❩♦ t ♦rrt♦♥ ♥ ♥②t rst r sr ②st♠
♥♥ ♥ ♦♥♦♠s sss♦♥ rs