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