RiskMeasuresandanApplicationtotheWithdrawalsofDepositsintheBolivianFinancialSystem Gonzales-Martínez,Rolando MunichPersonalRePEcArchive

(1)

Munich Personal RePEc Archive

Risk Measures and an Application to the Withdrawals of Deposits in the Bolivian Financial System

Gonzales-Martínez, Rolando

Superintendencia de Bancos y Entidades Financieras de Bolivia

September 2008

Online at https://mpra.ub.uni-muenchen.de/14700/

MPRA Paper No. 14700, posted 17 Apr 2009 06:49 UTC

(2)

▼❡❞✐❞❛% ❞❡ ❘✐❡%❣♦ ❋✐♥❛♥❝✐❡,♦ ② ✉♥❛ ❆♣❧✐❝❛❝✐2♥ ❛ ❧❛%

❱❛,✐❛❝✐♦♥❡% ❞❡ ❉❡♣2%✐5♦% ❞❡❧ ❙✐%5❡♠❛ ❋✐♥❛♥❝✐❡,♦ ❇♦❧✐✈✐❛♥♦

❘♦❧❛♥❞♦ ●♦♥③❛❧❡) ▼❛+,-♥❡③

❘❡"✉♠❡♥

❊!"❡ ❡!"✉❞✐♦ ❞❡!❝)✐❜❡ ")❡! ♠❡❞✐❞❛! ❞❡ )✐❡!❣♦ ✜♥❛♥❝✐❡)♦ ✕❱❛❧♦) ❡♥ ❘✐❡!❣♦ ✭❱❛❘✮ ❜❛!❛❞♦ ❡♥ ❧❛

❞✐!")✐❜✉❝✐6♥ ❞❡ ●❛✉!!✱ ❱❛❘ ❜❛!❛❞♦ ❡♥ ❚❡♦):❛ ❞❡ ❱❛❧♦)❡! ❊①")❡♠♦! ② ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧ ✭❊①♣❡❝%❡❞

❙❤♦*%❢❛❧❧✮✕ ② ❡❥❡♠♣❧✐✜❝❛ !✉ ✉!♦ ❝♦♥ ✉♥❛ ❛♣❧✐❝❛❝✐6♥ ❛ ❧❛! ✈❛)✐❛❝✐♦♥❡! ❞❡ ❞❡♣6!✐"♦! ❞❡❧ !✐!"❡♠❛

✜♥❛♥❝✐❡)♦✳ ▲♦! )❡!✉❧"❛❞♦! !✉❣✐❡)❡♥ B✉❡ ❡! ✐♠♣♦)"❛♥"❡ ❝♦♥!✐❞❡)❛) ❧♦! !✉♣✉❡!"♦! ❡!"❛❞:!"✐❝♦! ❞❡ ❡!"❛!

♠❡❞✐❞❛!✱ ♣❛)❛ ♥♦ !✉❜❡!"✐♠❛) ♦ !♦❜)❡!"✐♠❛) ❧♦! ✈❡)❞❛❞❡)♦! )✐❡!❣♦! ✜♥❛♥❝✐❡)♦!✳

❈❧❛#✐❢✐❝❛❝✐'♥ ❏❊▲✿ ●✵✶✱ ●✸✷✱ ❈✻✺

-❛❧❛❜/❛# ❈❧❛✈❡✿ ❱❛❧♦* ❡♥ ❘✐❡;❣♦✱ ❘✐❡;❣♦ ❞❡ ▲✐>✉✐❞❡③✱ ❈♦**✐❞❛; ❞❡ ❉❡♣B;✐%♦;

✶✳ ■♥$%♦❞✉❝❝✐+♥

▲❛ ❝#✐%✐% ✜♥❛♥❝✐❡#❛ ✐♥)❡#♥❛❝✐♦♥❛❧ ❤❛ ♣✉❡%)♦

❡♥ ❡✈✐❞❡♥❝✐❛ ❧❛ ♥❡❝❡%✐❞❛❞ ❞❡ ❝♦♥)❛# ❝♦♥ ♠❡❞✐✲

❞❛% ❛❞❡❝✉❛❞❛% ❞❡ #✐❡%❣♦ ✜♥❛♥❝✐❡#♦ ♣❛#❛ ❝✉❛♥)✐✜❝❛#

❧❛% ♣4#❞✐❞❛% ♣♦)❡♥❝✐❛❧❡% 5✉❡ #❡%✉❧)❛♥ ❞❡ ❧❛% ❛❝✲

)✐✈✐❞❛❞❡% ✜♥❛♥❝✐❡#❛%✳ 7♦# ❡%)❡ ♠♦)✐✈♦✱ ❡% ✐♠♣♦#✲

)❛♥)❡ 5✉❡ ❧❛% ❡♥)✐❞❛❞❡% ✜♥❛♥❝✐❡#❛% ❝♦♥♦③❝❛♥ ②

❝♦♠♣#❡♥❞❛♥ ❛❞❡❝✉❛❞❛♠❡♥)❡ ❧❛% ♠❡❞✐❞❛% ❞❡ #✐❡%✲

❣♦ ❞✐%♣♦♥✐❜❧❡% ♣❛#❛ ❝✉❛♥)✐✜❝❛# ♥✉♠4#✐❝❛♠❡♥)❡ %✉%

#✐❡%❣♦% ✜♥❛♥❝✐❡#♦%✱ ❞❡ ❢♦#♠❛ 5✉❡ ❛❥✉%)❡♥ %✉% ❛❝✲

)✐✈✐❞❛❞❡% ❝♦)✐❞✐❛♥❛% ② %✉% ♣❧❛♥❡% ❞❡ ❝♦♥)✐♥❣❡♥✲

❝✐❛ ♣❛#❛ #❡%♣♦♥❞❡# ❛ ❡%)♦% #✐❡%❣♦%✳ ❙✐♥ ❡♠❜❛#❣♦✱

♠✉❝❤❛% ❡♥)✐❞❛❞❡% ✜♥❛♥❝✐❡#❛% ♥♦ )✐❡♥❡♥ ♣#❡%❡♥)❡

❧♦% %✉♣✉❡%)♦% ❡%)❛❞?%)✐❝♦% ❞❡ ❧❛% ♠❡❞✐❞❛% ❞❡ #✐❡%✲

❣♦✱ ♣♦# ❧♦ 5✉❡ ♣✉❡❞❡♥ %✉❜❡%)✐♠❛# ♦ %♦❜#❡%)✐♠❛#

❧♦% ✈❡#❞❛❞❡#♦% #✐❡%❣♦% ✜♥❛♥❝✐❡#♦%✳

❊❧ ♦❜❥❡)✐✈♦ ❞❡ ❡%)❡ ❡%)✉❞✐♦ ❡% ❞❡%❝#✐❜✐# ② ❡❥❡♠✲

♣❧✐✜❝❛# )#❡% ♠❡❞✐❞❛% ❡%)❛❞?%)✐❝❛% ❞❡ #✐❡%❣♦ ♠♦❞❡#✲

♥❛% ② ❛♠♣❧✐❛♠❡♥)❡ ✉)✐❧✐③❛❞❛%✱ ♣❛#❛ 5✉❡ ❧❛% ❡♥)✐✲

❞❛❞❡% ✜♥❛♥❝✐❡#❛% ❝♦♥♦③❝❛♥✱ ❡♥)✐❡♥❞❛♥ ② ✉)✐❧✐❝❡♥

❝♦##❡❝)❛♠❡♥)❡ ❡%)❛% ❤❡##❛♠✐❡♥)❛% ♣❛#❛ ❝✉❛♥)✐✜❝❛#

❧♦% #✐❡%❣♦% ✐♥❤❡#❡♥)❡% ❛ %✉% ❛❝)✐✈✐❞❛❞❡%✳ ❙❡ ♣❧❛♥)❡❛

5✉❡ ❧❛ ♠❡❞✐❞❛ ♠A% ✉)✐❧✐③❛❞❛ ♣❛#❛ ❝✉❛♥)✐✜❝❛# ❧♦%

#✐❡%❣♦% ✜♥❛♥❝✐❡#♦%✱ ❡❧ ❱❛❘ ❜❛%❛❞♦ ❡♥ ❧❛ ❞✐%)#✐❜✉✲

❝✐D♥ ●❛✉%%✲▲❛♣❧❛❝❡ ✭♥♦#♠❛❧✮✱ ❡% ❡♥ ❣❡♥❡#❛❧ ✐♥✲

❛♣#♦♣✐❛❞♦ ♣❛#❛ ❝✉❛♥)✐✜❝❛# ❧♦% #✐❡%❣♦% ❞❡❜✐❞♦ ❛ 5✉❡

❧❛ ✐♥❢♦#♠❛❝✐D♥ ✜♥❛♥❝✐❡#❛ ❡%❝❛%❛♠❡♥)❡ ❛♣#♦①✐♠❛ ❧❛

❞✐%)#✐❜✉❝✐D♥ ♥♦#♠❛❧✳

▲❛ %❡❝❝✐D♥ ✷ ❞❡%❝#✐❜❡ ❧❛% ♠❡❞✐❞❛% ❞❡ #✐❡%❣♦ ✜✲

♥❛♥❝✐❡#♦ ✉)✐❧✐③❛❞❛% ❡♥ ❧❛ ✐♥✈❡%)✐❣❛❝✐D♥✱ ❧❛ %❡❝❝✐D♥

✸ ❛♣❧✐❝❛ ❡%)❛% ♠❡❞✐❞❛% ❛ ❧❛ %❡#✐❡ ❞❡ )✐❡♠♣♦ ❞✐❛#✐❛

❞❡ ❧♦% #❡)✐#♦% ❞❡❧ ❞❡♣D%✐)♦% ❞❡❧ %✐%)❡♠❛ ✜♥❛♥❝✐❡#♦

❜♦❧✐✈✐❛♥♦ ❞❡%❞❡ ❡❧ ❛K♦ ✷✵✵✷ ❛ ✷✵✵✽✳ ❊%)❛ ❡% ✉♥❛

%❡#✐❡ ❞❡ )✐❡♠♣♦ ✐♥)❡#❡%❛♥)❡ ♣❛#❛ ❝❛❧❝✉❧❛# ♠❡❞✐❞❛%

❞❡ #✐❡%❣♦ ♣♦# ❧❛% ❝♦##✐❞❛% ❞❡ ❞❡♣D%✐)♦% 5✉❡ %❡ ♣#❡✲

%❡♥)❛#♦♥ ❡♥ ❡❧ %✐%)❡♠❛ ✜♥❛♥❝✐❡#♦ ❜♦❧✐✈✐❛♥♦ ❞✉#❛♥)❡

❧♦% ❛K♦% ✷✵✵✷ ② ✷✵✵✸✳ ▲❛ %❡❝❝✐D♥ ✹ ❝♦♥❝❧✉②❡✳

✷✳ ▼.❞✐❞❛0 ❞❡ ❘✐❡0❣♦

❊❧ ❱❛❧♦# ❡♥ ❘✐❡%❣♦ ✭❱❛❧✉❡✲❛&✲❘✐)❦✱ ❱❛❘✮ #❡✲

%✉♠❡ ❡♥ ✉♥ ♥O♠❡#♦ ❧❛ ♣❡♦# ♣4#❞✐❞❛ ❡♥ ✉♥ ❤♦✲

#✐③♦♥)❡ )❡♠♣♦#❛❧ ❝♦♥ ✉♥ ♥✐✈❡❧ ❞❡ ❝♦♥✜❛♥③❛ ❞❛❞♦

✭❏❤♦♥%♦♥✱ ✷✵✵✶✮✳ ❙❡ ❞❡✜♥❡ ♣♦# ❡❧ ❧?♠✐)❡ %✉♣❡#✐♦#

❞❡ ❧❛ ✐♥)❡❣#❛❧ ❞❡ #❡)♦#♥♦% ❡%♣❡#❛❞♦%✱

E(r)Z−V aR

−∞

r(s)ds=α

❉❡❜✐❞♦ ❛ 5✉❡ ✉%✉❛❧♠❡♥)❡ %❡ ❛%✉♠❡ E(r) = 0✱ ❧❛

❛♥)❡#✐♦# ❡①♣#❡%✐D♥ %❡ )#❛♥%❢♦#♠❛ ❡♥✿

−V aRZ

−∞

r(s)ds=α

❊①✐%)❡♥ ♠✉❝❤♦% ♠4)♦❞♦% ❞❡ ❝❛❧❝✉❧❛# ❡❧ ❱❛❘✳ ❊♥

❡%)❡ ❡%)✉❞✐♦ %❡ ❛♥❛❧✐③❛#❛♥✿ ✭✶✮ ❡❧ ❱❛❘ ❜❛%❛❞♦ ❡♥

❧❛ ❞✐%)#✐❜✉❝✐D♥ ●❛✉%%✲▲❛♣❧❛❝❡✱ ✭✷✮ ❡❧ ❱❛❘ ❝❛❧❝✉❧❛✲

❞♦ ♠❡❞✐❛♥)❡ ❧♦% ♠4)♦❞♦% ❞❡ ❧❛ ❚❡♦#?❛ ❞❡ ❱❛❧♦#❡%

❊①)#❡♠♦%✱ ② ✭✸✮ ❡❧ ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧✳

✶

(3)

❊❧ ❱❛❘ ❜❛$❛❞♦ ❡♥ ❧❛ ❞✐$+,✐❜✉❝✐/♥ ●❛✉$$✲

▲❛♣❧❛❝❡ ✭♥♦,♠❛❧✮ "❡ ♦❜&✐❡♥❡ ♠✉❧&✐♣❧✐❝❛♥❞♦ ❧❛

σ✲❞❡"✈✐❛❝✐1♥ ❡"&2♥❞❛3 ❞❡ ❧♦" 3❡&♦3♥♦" ♣♦3 ❧❛ α✲

♣✉♥&✉❛❝✐1♥ ❡♥ ❧❛ ❞✐"&3✐❜✉❝✐1♥ ♥♦3♠❛❧ ❡"&2♥❞❛3✱

V aRG=α·σi·√

t, ✭✶✮

❞♦♥❞❡ ❡❧ ❢❛❝&♦3 ❞❡ ❡"❝❛❧❛ ❞❡ ❧❛ ✈♦❧❛&✐❧✐❞❛❞√ t❣❡♥❡✲

3❛❧✐③❛ ❡❧ ❱❛❘ ❛ ♦&3♦"t✲❤♦3✐③♦♥&❡" &❡♠♣♦3❛❧❡"✱ ❜❛❥♦

❡❧ "✉♣✉❡"&♦ ❞❡ 3❡&♦3♥♦" ♥♦ ❝♦33❡❧❛❝✐♦♥❛❞♦"✳

❊❧ ❱❛❘ ❜❛$❛❞♦ ❡♥ ❧❛ ❚❡♦,8❛ ❞❡ ❱❛❧✲

♦,❡$ ❊①+,❡♠♦$ ❞❡♣❡♥❞❡ ❞❡ ❧❛ ❡"&✐♠❛❝✐1♥ ❞❡ ❧♦"

♣❛32♠❡&3♦"ξ②β ❞❡ ❧❛ ❢✉♥❝✐1♥ ❞❡ ❞✐"&3✐❜✉❝✐1♥ ❣❡✲

♥❡3❛❧✐③❛❞❛ ❞❡ A❛3❡&♦^✶✳ ❊"&❛ ❞✐"&3✐❜✉❝✐1♥ "❡ ❛❥✉"&❛

❝♦♥ ❧❛"nu ♦❜"❡3✈❛❝✐♦♥❡" ❡①&3❡♠❛" ♣♦3 ❡♥❝✐♠❛ ❞❡

✉♥ ✉♠❜3❛❧ u ② "❡ 3❡❡♠♣❧❛③❛ ❧♦" 3❡"✉❧&❛❞♦" ❞❡ ❧❛

❡"&✐♠❛❝✐1♥ ❡♥ ❧❛ ❡①♣3❡"✐1♥✱

V aR\E =u+βb ξb

(N nu

p ⁻ξ^b

−1 )

✭✷✮

♣❛3❛ ♦❜&❡♥❡3 ❡❧ ❱❛❘ ❜❛"❛❞♦ ❡♥ ❧❛ &❡♦3D❛ ❞❡ ✈❛❧♦3❡"

❡①&3❡♠♦" ✭V aRE✮✳ ❊❧ ✈❛❧♦3 ❞❡❧V aRE❞❡♣❡♥❞❡ ❞❡

❧❛ ❡❧❡❝❝✐1♥ ❞❡u✱ ♣❡3♦ ♥♦ ❡①✐"&❡♥ ❛❝&✉❛❧♠❡♥&❡ ♠E&♦✲

❞♦" ❡"&❛❞D"&✐❝♦" ♣❛3❛ ❝❛❧❝✉❧❛3 ❝♦♥ ♣3❡❝✐"✐1♥ ❡"&❡

u✲✉♠❜3❛❧✱ ♣♦3 ❧♦ F✉❡ ❡"&❡ ✈❛❧♦3 ❞❡❜❡ "❡3 ❡❧❡❣✐❞♦

♣♦3 ❡❧ ✐♥✈❡"&✐❣❛❞♦3 ❞❡ ❢♦3♠❛ F✉❡ "❡❛ ❧♦ "✉✜❝✐❡♥&❡✲

♠❡♥&❡ ❛❧&♦ ❝♦♠♦ ♣❛3❛ F✉❡ "❡ ❝✉♠♣❧❛ ❡❧ &❡♦3❡♠❛

A✐❝❦❛♥❞"✱ ❇❛❧❦❡♠❛ ② ❍❛❛♥ ❞❡ ✈❛❧♦3❡" ❡①&3❡♠♦"✱

♣❡3♦ ❧♦ "✉✜❝✐❡♥&❡♠❡♥&❡ ❜❛❥♦ ❝♦♠♦ ♣❛3❛ F✉❡ ❡①✐"✲

&❛♥ "✉✜❝✐❡♥&❡" ♦❜"❡3✈❛❝✐♦♥❡" ♣❛3❛ ❡"&✐♠❛3 ♣♦3 ♠2✲

①✐♠❛ ✈❡3♦"✐♠✐❧✐&✉❞ ❧♦" ♣❛32♠❡&3♦" ❞❡ ❧❛ ❞✐"&3✐❜✉✲

❝✐1♥ ❣❡♥❡3❛❧✐③❛❞❛ ❞❡ A❛3❡&♦ ✭●❡♥L❛② ❡& ❛❧✳✱ ✷✵✵✷✱

●❡♥L❛② ② ❙❡❧L✉❦✱ ✷✵✵✶✮✳ ❯♥❛ ❤❡33❛♠✐❡♥&❛ F✉❡ "❡

✉&✐❧✐③❛ ♣❛3❛ ❡❧❡❣✐3 ❡❧ ✈❛❧♦3 ❞❡❧ ✉♠❜3❛❧u❡" ❡❧ ❣32✜❝♦

❞❡ ❡①❝❡"♦" "♦❜3❡ ❡❧ ✉♠❜3❛❧ ❞❡✜♥✐❞♦ ♣♦3 ❧♦" ♣✉♥&♦"✱

(u, en(u)), xⁿ₁ < u < xⁿ_n,

❞♦♥❞❡en(u)❡" ❧❛ ❢✉♥❝✐1♥ ❞❡ ❡①❝❡"♦" ❞❡✜♥✐❞❛ ❝♦✲

♠♦✱

en(u) = Pn

i=k(xⁿ_i −u)

n−k+ 1 , k=♠✐♥{i|xⁿ_i > u},

②n−k+ 1❡" ❡❧ ♥P♠❡3♦ ❞❡ ♦❜"❡3✈❛❝✐♦♥❡" F✉❡ ❡①✲

❝❡❞❡ ❡❧ ✉♠❜3❛❧u✭●✐❧❧✐ ② ❑R❧❧❡③✐✱ ✷✵✵✻✮✳

❊❧ ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧ ✭❈❱❛❘✮✱ &❛♠❜✐E♥ ❞❡♥♦♠✐✲

♥❛❞♦ ❡♥ ❧❛ ❧✐&❡3❛&✉3❛ ❊①♣❡❝%❡❞ ❙❤♦*%❢❛❧❧✱ ♣✉❡❞❡

♦❜&❡♥❡3"❡ ❛ ♣❛3&✐3 ❞❡❧ V aRE✱ CV aR\ = V aR\E

1−ξb +βb−ξub

1−ξb ✭✸✮

❊"&❛ ♠❡❞✐❞❛ ❞❡ 3✐❡"❣♦ ❡" ✐♥&❡3❡"❛♥&❡ ♣♦3F✉❡ ❡"&✐✲

♠❛ ❡❧ ✈❛❧♦3 ♣♦&❡♥❝✐❛❧ ❞❡ ❧❛ ♣E3❞✐❞❛ F✉❡ ❡①❝❡❞❡ ❡❧

❱❛❘✱

CV aR=E(X|X > V aR).

✸✳ ❯♥❛ ❛♣❧✐❝❛❝✐)♥ ❛ ❧❛* ❱❛,✐❛✲

❝✐♦♥❡* ❞❡ ❉❡♣)✐2♦ ❞❡❧ ❙✐*✲

2❡♠❛ ❋✐♥❛♥❝✐❡,♦

▲❛" ♠❡❞✐❛" ❞❡ 3✐❡"❣♦ ❢✉❡3♦♥ ❛♣❧✐❝❛❞❛" ❛ ❧❛ ✐♥✲

❢♦3♠❛❝✐1♥ ❤✐"&13✐❝❛ ❞❡ ❧♦" 3❡&✐3♦" ❞✐❛3✐♦" ❞❡ ❞❡✲

♣1"✐&♦" ❡♥ ❡❧ "✐"&❡♠❛ ✜♥❛♥❝✐❡3♦ ❜♦❧✐✈✐❛♥♦ ❞❡"❞❡ ❡❧

❛W♦ ✷✵✵✶✳ ❊"&❛ ✐♥❢♦3♠❛❝✐1♥ ❡" ✐♥&❡3❡"❛♥&❡ ♣♦3F✉❡

❡❧ "✐"&❡♠❛ ✜♥❛♥❝✐❡3♦ ❜♦❧✐✈✐❛♥♦ ♣3❡"❡♥&1 ❡✈❡♥&♦" ❡①✲

&3❡♠♦" ❡♥ ❞♦" ❝♦33✐❞❛" ❜❛♥❝❛3✐❛"✱ ❧❛ ♣3✐♠❡3❛ ❡♥ ♦❝✲

&✉❜3❡ ❞❡ ✷✵✵✷ ② ❧❛ "❡❣✉♥❞❛ ❡♥ ❢❡❜3❡3♦ ✷✵✵✸✱ ❛♠❜❛"

3❡❧❛❝✐♦♥❛❞❛" ❝♦♥ ❧❛ ✐♥❡"&❛❜✐❧✐❞❛❞ ♣♦❧D&✐❝❛ ② "♦❝✐❛❧

❞❡ ❡"♦" ♣❡3✐♦❞♦"✳ ❆❞❡♠2"✱ ❞❡"❞❡ ❡❧ ❛W♦ ✷✵✵✺ "❡ ♥♦✲

&❛ ✉♥ ❛✉♠❡♥&♦ ❡♥ ❧❛ ✈♦❧❛&✐❧✐❞❛❞✱ 3❡❧❛❝✐♦♥❛❞❛ ❝♦♥

❧♦" 3❡&✐3♦" ② ❞❡♣1"✐&♦" ❞❡ ❧❛" ❝✉❡♥&❛" ❞❡ ❡♠♣3❡"❛"

♣❡&3♦❧❡3❛"✳

▲❛" ♠❡❞✐❞❛" V aR✱ V aRE ②CV aR ❝❛❧❝✉❧❛❞❛"

❝✉❛♥&✐✜❝❛♥ ❡❧ 3✐❡"❣♦ ❞❡ ❧✐F✉✐❞❡③ ❞❡ 3❡&✐3♦" ❞❡ ❞❡✲

♣1"✐&♦"✳ ❊"&❡ 3✐❡"❣♦ ♣✉❡❞❡ ❞❡✜♥✐3"❡ ❝♦♠♦ ❧❛ ♣3♦✲

❜❛❜✐❧✐❞❛❞ ❞❡ &❡♥❡3 ❝♦♥"❡❝✉❡♥❝✐❛" ❛❞✈❡3"❛" ♣♦3 ♥♦

❝♦♥&❛3 ❝♦♥ ❧❛ ❧✐F✉✐❞❡③ ♥❡❝❡"❛3✐❛ ♣❛3❛ 3❡"♣♦♥❞❡3 ❛

❧♦" 3❡&✐3♦" ❞❡ ❞❡♣1"✐&♦" ❞❡❧ ♣P❜❧✐❝♦✳ ▲♦" 3❡"✉❧&❛❞♦"

"❡ ♦❜"❡3✈❛♥ ❡♥ ❡❧ ❈✉❛❞3♦ ✶✳

A❛3❛ ❡❧ "✐"&❡♠❛ ✜♥❛♥❝✐❡3♦ ❜♦❧✐✈✐❛♥♦✱ ❡❧ ❱❛❧♦3

❡♥ ❘✐❡"❣♦ ♦❜&❡♥✐❞♦ ❡♥ ❜❛"❡ ❛ ❧❛ ♣✉♥&✉❛❝✐1♥ ❞❡ ❧❛

❞✐"&3✐❜✉❝✐1♥ ●❛✉""✲▲❛♣❧❛❝❡ ❡" ❞❡ ✵✳✾✼ ✪✳ ❊"&❡ ✈❛✲

❧♦3 ✐♥❞✐❝❛ F✉❡ ❧♦" 3❡&✐3♦" ❞❡ ❞❡♣1"✐&♦" "✉♣❡3❛3❛♥ ❡❧

✵✳✾✼ ✪ ❞❡❧ &♦&❛❧ ❞❡ ❞❡♣1"✐&♦" ❝♦♥ ✉♥❛ ♣3♦❜❛❜✐❧✐❞❛❞

❞❡ ✵✳✶ ✪✳ ❊"&❛ ❢♦3♠❛ ❞❡ ❝❛❧❝✉❧❛3 ❡❧ ❱❛❘ ❡" ❛♠♣❧✐❛✲

♠❡♥&❡ ✉&✐❧✐③❛❞❛ ♣♦3 "✉ "✐♠♣❧✐❝✐❞❛❞✱ "✐♥ ❡♠❜❛3❣♦ ❡"

✐♥❛❞❡❝✉❛❞❛ "✐ ❧♦" ❞❛&♦" ♥♦ ❛♣3♦①✐♠❛♥ ❧❛ ❞✐"&3✐❜✉✲

❝✐1♥ ♥♦3♠❛❧✱ ❝♦♠♦ "✉❝❡❞❡ ❞❡ ❤❡❝❤♦ ❝♦♥ ❧❛" ✈❛3✐❛✲

❝✐♦♥❡" ❞❡ ❞❡♣1"✐&♦" ❞❡❧ "✐"&❡♠❛ ✜♥❛♥❝✐❡3♦✱ ❞❛&♦"

✶❊❧ ❚❡♦%❡♠❛ (✐❝❦❛♥❞.✱ ❇❛❧❦❡♠❛ ② ❍❛❛♥ ❡.3❛❜❧❡❝❡ 5✉❡ ✉♥❛ ❢✉♥❝✐8♥ ❞❡ ❞✐.3%✐❜✉❝✐8♥ ❞❡ ❡①❝❡.♦.F_u(y)✱ ♣❛%❛ ✉♥u❧❛%❣♦✱

.❡ ❛♣%♦①✐♠❛ ❜✐❡♥ ❝♦♥✱

Fu(y)≈G_ξ,β(y), u→ ∞

❞♦♥❞❡

G_ξ,β(y) = (

1−

1 +_β^ξy−1/ξ

si ξ6= 0 1−e^−y/β si ξ= 0

♣❛%❛y∈[0,(xF−u)].✐ξ≥0②y∈[0,−^β_ξ].✐ξ <0✳Gξ,β❡. ❧❛ ❞✐.3%✐❜✉❝✐8♥ ❣❡♥❡%❛❧✐③❛❞❛ ❞❡ (❛%❡3♦✳

✷

(4)

♣❛"❛ ❧♦% &✉❡ %❡ ♣✉❡❞❡ "❡❝❤❛③❛" ❧❛ ❤✐♣./❡%✐% ♥✉❧❛ ❞❡

♥♦"♠❛❧✐❞❛❞ ❛ ♥✐✈❡❧❡% ❝♦♥✈❡♥❝✐♦♥❛❧❡% ✭❝♦♥ ✉♥ ♥✐✈❡❧

❞❡ ❝♦♥✜❛♥③❛ ❛❧❢❛ ♠❡♥♦" ❛❧ ✶ ✪✮ /❛♥/♦ ❝♦♥ ❡❧ /❡%/ ❞❡

♥♦"♠❛❧✐❞❛❞ ❜❛%❛❞♦ ❡♥ ❡❧ ❡%/❛❞:❣"❛❢♦ ❏❛"&✉❡ ❇❡"❛

❝♦♠♦ ❡♥ ❡❧ ❜❛%❛❞♦ ❡♥ ❡❧ ❡%/❛❞:❣"❛❢♦ ❑♦❧❣♦♠♦"♦✈✲

❙♠✐"♥♦✈ ✭✈A❛%❡ ❡❧ ❈✉❛❞"♦ ✷✮✳ ❊%/♦% "❡%✉❧/❛❞♦% %♦♥

♣"❡❞❡❝✐❜❧❡% %✐ %❡ ❝♦♥%✐❞❡"❛ &✉❡ ❧♦% ❞❛/♦% ♣"❡%❡♥✲

/❛♥ ❧❡♣/♦❝✉"/♦%✐% ✭❧❛ ❝✉"/♦%✐% ❡% ✐❣✉❛❧ ❛ ✻✳✾✹✼✸✱ %✉✲

♣❡"❛♥❞♦ ♣♦" ❝❛%✐ ✹ ♣✉♥/♦% ❡❧ ✈❛❧♦" /❡."✐❝♦ ❞❡ ❧❛ ❝✉"✲

/♦%✐% ❞❡ ❧❛ ❞✐%/"✐❜✉❝✐.♥ ♥♦"♠❛❧✮✳ ▲♦% ❞❛/♦% ❞❡ ❧❛%

✈❛"✐❛❝✐♦♥❡% ❞❡ ❞❡♣.%✐/♦% ♠✉❡%/"❛♥ ❞❡ ❤❡❝❤♦ ❝♦❧❛%

❧❛"❣❛% ✭✈A❛%❡ ❡❧ ❤✐%/♦❣"❛♠❛ ❛❧ ✜♥❛❧ ❞❡❧ ❞♦❝✉♠❡♥✲

/♦✮✱ ♣♦" ❧♦ &✉❡ ❡❧ ❱❛❘ ❝❛❧❝✉❧❛❞♦ ❝♦♥ ❧❛ ❞✐%/"✐❜✉✲

❝✐.♥ ●❛✉%%✲▲❛♣❧❛❝❡ %✉❜❡%/✐♠❛ ❡❧ ✈❡"❞❛❞❡"♦ "✐❡%❣♦

❞❡ "❡/✐"♦% ❞❡ ❞❡♣.%✐/♦%✳

❊❧ ❱❛❘ ❝❛❧❝✉❧❛❞♦ ❡♥ ❜❛%❡ ❛ ❧❛ ❚❡♦":❛ ❞❡ ❱❛❧♦✲

"❡% ❊①/"❡♠♦% ✭V aRE✮ ❡% ✐❣✉❛❧ ❛ ✶✳✶✽ ✪✳ ❊%/❡ ✈❛✲

❧♦" ❡% ♠❛②♦" ❛❧ ❱❛❘ ❣❛✉%%✐❛♥♦ ② ♣❛"❡❝❡ ♠❡❞✐" ♠T%

❛♣"♦♣✐❛❞❛♠❡♥/❡ ❧❛ ❞✐♥T♠✐❝❛ &✉❡ ♣"❡%❡♥/❛♥ ❧♦% "❡✲

/✐"♦% ❞❡ ❞❡♣.%✐/♦% ❞❡❧ %✐%/❡♠❛ ✜♥❛♥❝✐❡"♦✳

❊❧ ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧ ✭CV aR✮ ♣"❡%❡♥/❛ ✉♥ "❡%✉❧✲

/❛❞♦ ✐♥/❡"❡%❛♥/❡✱ %✉❣✐"✐❡♥❞♦ &✉❡✱ %✐ ❡% &✉❡ %❡ %✉✲

♣❡"❛"❛ ❡❧ ✉♠❜"❛❧ ❞❡ ♣A"❞✐❞❛% &✉❡ ♣"❡❞✐❝❡ ❡❧ ❱❛❘

❜❛%❛❞♦ ❡♥ ❧❛ ❚❡♦":❛ ❞❡ ❱❛❧♦"❡% ❊①/"❡♠♦%✱ ❧♦% "❡✲

/✐"♦% ♣♦❞":❛♥ %✐❣♥✐✜❝❛" ❤❛%/❛ ✉♥ ✶✳✹✷ ✪ ❞❡❧ %❛❧❞♦

❞❡ ❞❡♣.%✐/♦% ❡♥ ❡❧ %✐%/❡♠❛ ✜♥❛♥❝✐❡"♦✳ ❚A♥❣❛%❡ ♣"❡✲

%❡♥/❡ &✉❡ ❡❧ ♠T①✐♠♦ "❡/✐"♦ ❤✐%/."✐❝♦ ❢✉❡ ❞❡ ✶✳✾✷ ✪✳

❘❡%♣❡❝/♦ ❛ ❧♦% %✉❜%✐%/❡♠❛%✱ ❧❛ ❞❡%✈✐❛❝✐.♥ ❡%/T♥✲

❞❛" ❞❡ ✵✳✺✺ ✪ ♠✉❡%/"❛ &✉❡ ❧❛% ❡♥/✐❞❛❞❡% ❜❛♥❝❛"✐❛%

/✐❡♥❡♥ ✉♥❛ ♠❛②♦" ✈♦❧❛/✐❧✐❞❛❞ ❡♥ %✉% ❞❡♣.%✐/♦% ❡♥

❝♦♠♣❛"❛❝✐.♥ ❝♦♥ ❧♦% ❢♦♥❞♦% ✜♥❛♥❝✐❡"♦% ♣"✐✈❛❞♦%

✭❢❢♣"✱ ✵✳✸✷ ✪✮✱ ❧❛% ♠✉/✉❛❧❡% ❞❡ ❛❤♦""♦ ② ♣"A%/❛✲

♠♦ ✭♠❛♣"✱ ✵✳✸✸ ✪✮ ② ❧❛% ❝♦♦♣❡"❛/✐✈❛% ❞❡ ❛❤♦""♦ ②

❝"A❞✐/♦ ✭❝❛❝%✱ ✵✳✸✺ ✪✮✳ ❉❡❜✐❞♦ ❛ %✉ ♠❛②♦" ✈♦❧❛/✐❧✐✲

❞❛❞✱ ❡❧ ❱❛❘ ❣❛✉%%✐❛♥♦ ❡% ♠❛②♦" ♣❛"❛ ❧❛% ❡♥/✐❞❛❞❡%

❜❛♥❝❛"✐❛% &✉❡ ♣❛"❛ ❧❛% ❡♥/✐❞❛❞❡% ♥♦ ❜❛♥❝❛"✐❛%✳ ❙✐♥

❡♠❜❛"❣♦ ✉♥❛ ✈❡③ ♠T% ❡❧ ❱❛❘ ♥♦"♠❛❧ ♥♦ ❛♣"♦①✐✲

♠❛ ❜✐❡♥ ❧❛ ♥❛/✉"❛❧❡③❛ ❞❡ ❧♦% ❞❛/♦% ❞❡❜✐❞♦ ❛ &✉❡

❡♥ /♦❞♦% ❧♦% ❝❛%♦% %❡ "❡❝❤❛③❛ ❧❛ ❤✐♣./❡%✐% ♥✉❧❛

❞❡ ❣❛✉%%✐❛♥✐❞❛❞ ❝♦♥ ✉♥ ♥✐✈❡❧ ❞❡ ♣"♦❜❛❜✐❧✐❞❛❞ ❛❧❢❛

♠❡♥♦" ❛❧ ✉♥♦ ♣♦" ❝✐❡♥/♦✱ /❛♥/♦ ❝♦♥ ❡❧ ❡%/❛❞:❣"❛❢♦

❏❛"&✉❡✲❇❡"❛ ❝♦♠♦ ❝♦♥ ❡❧ ❡%/❛❞:❣"❛❢♦ ❑♦❧❣♦♠♦"♦✈✲

❙♠✐"♥♦✈✳ ▲♦% ❤✐%/♦❣"❛♠❛% ❞❡ ❧❛% ❡♥/✐❞❛❞❡% ❜❛♥✲

❝❛"✐❛% ② ♥♦ ❜❛♥❝❛"✐❛% %♦♥ ❧❡♣/♦❝X"/✐❝❛%✱ ② ❡❧ ❤✐%✲

/♦❣"❛♠❛ ❞❡ ❧❛% ♠❛♣" /✐❡♥❡ ✉♥ %❡%❣♦ ♥❡❣❛/✐✈♦✱ &✉❡

%❡ ❞❡❜❡ ❛ &✉❡ ❡①✐%/✐❡"♦♥ ♠T% "❡/✐"♦% &✉❡ ❞❡♣.%✐/♦%

❡♥ ❡%/❛% ❡♥/✐❞❛❞❡%✳ ❊♥ ❝♦♥/"❛%/❡✱ ❧❛% ❝❛❝" /✐❡♥❡♥

✉♥ %❡%❣♦ ♣♦%✐/✐✈♦ "❡❧❛❝✐♦♥❛❞♦ ❝♦♥ ❡❧ ❛✉♠❡♥/♦ ❞❡

❧♦% ❞❡♣.%✐/♦% ❡♥ ❡%/❛% ✐♥%/✐/✉❝✐♦♥❡% ✜♥❛♥❝✐❡"❛%✳

❊❧V aRE ② ❡❧CV aR%♦♥ ♠❛②♦"❡% ❛❧ ❱❛❘ ❣❛✉✲

%%✐❛♥♦ ❡♥ ❡❧ %✐%/❡♠❛ ❜❛♥❝❛"✐♦ ❡ ✐♥❞✐❝❛♥ &✉❡ ❝♦♥ ✉♥

♣"♦❜❛❜✐❧✐❞❛❞ ❞❡ ✶ ✪ ❧♦% "❡/✐"♦% ❞❡ ❞❡♣.%✐/♦% %❡"T♥

♠❛②♦"❡% ❛ ✶✳✺✽ ✪^✷✱ ② %✐ ❡% &✉❡ %✉♣❡"❛♥ ❡%/❡ ✈❛❧♦"✱

♣♦❞":❛♥ ❧❧❡❣❛" ❛ "❡♣"❡%❡♥/❛" ✉♥ ✶✳✽✽ ✪ ❞❡❧ %❛❧❞♦

❞❡ ❞❡♣.%✐/♦% ❞❡❧ %✐%/❡♠❛ ❜❛♥❝❛"✐♦✳ ❊♥ ❧❛% ♠❛♣"

❧♦% "❡/✐"♦% %❡"T♥ ♠❛②♦"❡% ❛ ✵✳✾✼ ✪ ❝♦♥ ✉♥❛ ♣"♦❜❛✲

❜✐❧✐❞❛❞ ❞❡ ✶ ✪✱ ♣❡"♦ %✐ %✉♣❡"❛♥ ❡%/❡ ✈❛❧♦"✱ ♣♦❞":❛♥

%✐❣♥✐✜❝❛" ❤❛%/❛ ✉♥ ✷✳✷✽ ✪ ❞❡❧ /♦/❛❧ ❞❡ ❞❡♣.%✐/♦% ❡♥

❡%/❛% ❡♥/✐❞❛❞❡%✳ ❊❧ ❛❧/♦ ✈❛❧♦" ❞❡❧ ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧

"❡✢❡❥❛ ❡❧ ❤❡❝❤♦ ❞❡ &✉❡ ❧❛% ♠❛♣" ❢✉❡"♦♥ ❧❛% ♠T%

❛❢❡❝/❛❞❛% ❡♥ A♣♦❝❛% ❞❡ ❝♦""✐❞❛% ❞❡ ❞❡♣.%✐/♦% ② ♣♦"

/❛♥/♦ %♦♥ %✉%❝❡♣/✐❜❧❡% ❛ ♠❛②♦"❡% ♣A"❞✐❞❛%✳

❊♥ ❡❧ ❝❛%♦ ❞❡ ❧♦% ❢❢♣" ② ❧❛% ❝❛❝% ❡❧ ❱❛❘ ❞❡

✈❛❧♦"❡% ❡①/"❡♠♦% ❡% ♠❡♥♦" ❛❧ ❱❛❘ ❣❛✉%%✐❛♥♦✱ "❡✲

%✉❧/❛❞♦ &✉❡ %✉❣❡"✐":❛ &✉❡ ❡❧ ❱❛❘ ❣❛✉%%✐❛♥♦ ❡%/T

%♦❜"❡❡%/✐♠❛♥❞♦ ❡❧ "✐❡%❣♦ ❞❡ "❡/✐"♦% ❡♥ ❡%/❛% ❡♥/✐✲

❞❛❞❡%✳ ❊%/❡ "❡%✉❧/❛❞♦ ♥♦ ❡% ✐♥❝♦♥%✐%/❡♥/❡ %✐ %❡ ❝♦♥✲

%✐❞❡"❛ &✉❡ ❧❛ ✈♦❧❛/✐❧✐❞❛❞ ✕② ♣♦" /❛♥/♦ ❡❧ "✐❡%❣♦✕ ❞❡

❧♦% ❢❢♣" %❡ ❤❛ "❡❞✉❝✐❞♦ ❞❡%❞❡ ❡❧ ❛\♦ ✷✵✵✷✳ ❘❡%♣❡❝✲

/♦ ❛ ❧❛% ❝❛❝"✱ ❡♥ ❡%/❛% ✐♥%/✐/✉❝✐♦♥❡% ❧♦% ✈❛❧♦"❡% ❡①✲

/"❡♠♦% &✉❡ ❝❛✉%❛♥ ✉♥ ✈❛❧♦" ❞❡ ❝✉"/♦%✐% ❞❡ ✶✻✾✳✾✺

%♦♥ ❛✐%❧❛❞♦% ② ♣♦" /❛♥/♦ /✐❡♥❡♥ ✉♥❛ ♠❡♥♦" ♣"♦❜❛✲

❜✐❧✐❞❛❞ ❞❡ ♦❝✉""❡♥❝✐❛✳ ◆♦/A%❡ %✐♥ ❡♠❜❛"❣♦ &✉❡ ❡❧

❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧ ❞❡ ❧♦% ❢❢♣" %✉❣✐❡"❡ &✉❡ %✐ ❧♦% "❡✲

/✐"♦% %✉♣❡"❛♥ ❡❧ ✵✳✼✷ ✪ ❧❛% ♣A"❞✐❞❛% ❞❡ ❞❡♣.%✐/♦%

♣♦❞":❛♥ ❛❧❝❛♥③❛" ✉♥ ✵✳✾✹ ✪✱ ❡♥ /❛♥/♦ &✉❡ ♣❛"❛ ❧❛%

❝❛❝% %✐ ❧♦% "❡/✐"♦% %✉♣❡"❛♥ ❡❧ ✵✳✾✼ ✪ ❧❛% ♣A"❞✐❞❛%

♣♦❞":❛♥ ❛❧❝❛♥③❛" ✉♥ ✶✳✸✵ ✪✳ ❊%/❡ "❡%✉❧/❛❞♦ ✐♥❞✐❝❛

&✉❡ ❧❛% ❝❛❝" ❡♥❢"❡♥/❛♥ ✉♥ ♠❛②♦" "✐❡%❣♦ ❡♥ A♣♦❝❛%

❞❡ ❝"✐%✐% ② ❞❡❜❡♥ ❛❥✉%/❛" %✉% ♣❧❛♥❡% ❞❡ ❧✐&✉✐❞❡③ ❞❡

❝♦♥/✐♥❣❡♥❝✐❛ ♣❛"❛ ❡♥❢"❡♥/❛" ❡%/❛% ♣A"❞✐❞❛% ♣♦/❡♥✲

❝✐❛❧❡%✳

✹✳ ❈♦♥❝❧✉(✐♦♥❡(

▲❛ ♠❡❞✐❝✐.♥ ❞❡ ❧♦% "✐❡%❣♦% ✜♥❛♥❝✐❡"♦% ❡% ✈✐/❛❧

/❛♥/♦ ♣❛"❛ &✉❡ ❧❛% ❡♥/✐❞❛❞❡% ✜♥❛♥❝✐❡"❛% ♦♣/✐♠✐❝❡♥

%✉% "❡♥❞✐♠✐❡♥/♦% ❡♥ ❜❛%❡ ❛ ❧❛ ❝✉❛♥/✐✜❝❛❝✐.♥ ❞❡ %✉%

♣A"❞✐❞❛% ♣"♦❜❛❜❧❡%✱ ❝♦♠♦ ♣❛"❛ &✉❡ ❡❧❛❜♦"❡♥ ♣❧❛♥❡%

❞❡ ❝♦♥/✐❣❡♥❝✐❛ ♣❛"❛ ❧❛% ♣A"❞✐❞❛% ❞❡ ♠❛②♦" ♠❛❣✲

♥✐/✉❞✱ &✉❡ /✐❡♥❡♥ ♠❡♥♦" ♣"♦❜❛❜✐❧✐❞❛❞ ❞❡ ♦❝✉""✐"✱

♣❡"♦ &✉❡ ❡①✐%/❡♥ %✐♥ ❡♠❜❛"❣♦ ❝♦♠♦ ✉♥❛ ♣♦%✐❜✐✲

❧✐❞❛❞ &✉❡ ♣✉❡❞❡ ❛❢❡❝/❛" %❡✈❡"❛♠❡♥/❡ ❧❛ %✐/✉❛❝✐.♥

✜♥❛♥❝✐❡"❛ ❞❡ ✉♥❛ ❡♥/✐❞❛❞ ❡ ✐♥❝❧✉%♦ ❧❛ ❝♦♥/✐♥✉✐❞❛❞

❞❡ %✉% ❛❝/✐✈✐❞❛❞❡%✳

❊%/❡ ❡%/✉❞✐♦ ❡❥❡♠♣❧✐✜❝.✱ ❡♥ ❜❛%❡ ❛ ❧❛ ✐♥❢♦"✲

♠❛❝✐.♥ ❞❡ "❡/✐"♦% ❞❡ ❞❡♣.%✐/♦% ❞❡❧ %✐%/❡♠❛ ✜✲

♥❛♥❝✐❡"♦ ❜♦❧✐✈✐❛♥♦✱ /"❡% ♠❡❞✐❞❛% ❞❡ "✐❡%❣♦ ✜✲

♥❛♥❝✐❡"♦ ❜❛%❛❞❛% ❜❛❥♦ ❡❧ ❝♦♥❝❡♣/♦ ❞❡ ❱❛❧♦" ❡♥

❘✐❡%❣♦ ✭❱❛❘✮✿ ❱❛❘ ❜❛%❛❞♦ ❡♥ ❧❛ ❞✐%"✐❜✉❝✐.♥ ❞❡

●❛✉%%✲▲❛♣❧❛❝❡✱ &✉❡ ❡% ✉♥❛ ♠❡❞✐❞❛ ♣♦♣✉❧❛" ♣❛"❛

❝✉❛♥/✐✜❝❛" ❡❧ "✐❡%❣♦ ♣♦" %✉ ❢❛❝✐❧✐❞❛❞ ❞❡ ❝T❧❝✉❧♦✱

✷❚❛♠❜✐%♥ ♣✉❡❞❡ ❞❡❝✐,-❡ .✉❡ ❧♦- ,❡1✐,♦- ❞❡ ❞❡♣2-✐1♦- ♣♦❞,3❛♥ -❡, ♠❛②♦,❡- ❛ ✶✳✺✽ ✪ ❝❛❞❛ ❝✐❡♥ ❞3❛-

✸

(5)

❚❛❜❧❛ ✶✿ ▼❡❞✐❞❛* ❞❡ ❘✐❡*❣♦ ❋✐♥❛♥❝✐❡1♦ ✭❡♥ ♣♦1❝❡♥4❛❥❡✮

❯♠❜1❛❧ nu V aRG V aRE CV aR

❙✐*4❡♠❛ ❋✐♥❛♥❝✐❡1♦ ✶✳✵✵✶✽ ✸✺ ✵✳✾✼✸✵ ✶✳✶✽✽✸ ✶✳✹✶✽✵

❙✐*4❡♠❛ ❇❛♥❝❛1✐♦ ✶✳✹✶✾✻ ✷✺ ✶✳✷✼✵✺ ✶✳✺✽✶✸ ✶✳✽✽✺✵

❋❋E* ✵✳✼✵✹✹ ✷✵ ✵✳✼✹✷✵ ✵✳✼✷✵✺ ✵✳✾✸✾✻

▼❆E* ✵✳✽✽✺✽ ✷✵ ✵✳✼✼✶✺ ✵✳✾✻✻✹ ✷✳✷✽✸✷

❈❆❈* ✵✳✺✵✼✹ ✷✺ ✵✳✽✵✽✹ ✵✳✻✵✵✹ ✶✳✸✵✸✾

❚❛❜❧❛ ✷✿ ❊*4❛❞I*4✐❝❛* ❉❡*❝1✐♣4✐✈❛* ② ❚❡*4 ❞❡ ◆♦1♠❛❧✐❞❛❞

❙❋■◆ ❇❈❖* ❋❋E* ▼❆E* ❈❆❈*

❉❡*✈✐❛❝✐P♥ ❊*4Q♥❞❛1 ✵✳✹✶✽✽ ✵✳✺✹✻✾ ✵✳✸✶✾✹ ✵✳✸✸✷✶ ✵✳✸✹✽✵

▼I♥✐♠♦ ✲✶✳✾✷✷✵ ✲✷✳✸✷✸✷ ✲✶✳✻✽✶✶ ✲✽✳✹✷✵✵ ✲✺✳✹✵✽✹

▼Q①✐♠♦ ✷✳✼✽✵✷ ✸✳✽✵✵✶ ✶✳✾✷✵✻ ✹✳✶✵✻✼ ✼✳✸✼✽✹

❈✉14♦*✐* ✻✳✾✹✼✸ ✼✳✷✶✹✹ ✼✳✼✸✶✸ ✷✺✾✳✶✶✸✻ ✶✻✾✳✾✺✶✵

❙❡*❣♦ ✲✵✳✵✵✾✵ ✵✳✵✼✼✵ ✵✳✻✼✾✾ ✲✶✵✳✻✺✸✻ ✹✳✷✹✾✾

❏❛1V✉❡ ❇❡1❛ ✶✾✾✶✳✸ ✶✸✻✽✳✽ ✶✽✻✺✳✽ ✺✵✾✶✶✵✵ ✷✶✺✹✵✵✵

✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮

❑♦❧❣♦♠♦1♦✈✲❙♠✐1♥♦✈ ✵✳✷✷✼✵ ✵✳✶✽✸✽ ✵✳✸✶✸✽ ✵✳✸✻✻✵ ✵✳✸✹✸✷

✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮ ✭✵✳✵✵✵✵✮

❱❛❘ ❜❛*❛❞♦ ❡♥ ❧❛ ❚❡♦1I❛ ❞❡ ❱❛❧♦1❡* ❊①41❡♠♦* ②

❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧✳ ▲♦* 1❡*✉❧4❛❞♦* ♠✉❡*41❛♥ V✉❡ ❡❧

❱❛❘ ❣❛✉**✐❛♥♦ ♣✉❡❞❡ *✉❜❡*4✐♠❛1 ♦ *♦❜1❡❡*4✐♠❛1

❧♦* ✈❡1❞❛❞❡1♦* 1✐❡*❣♦* ✜♥❛♥❝✐❡1♦* ❝✉❛♥❞♦ ❧❛* ✈❛1✐❛✲

❜❧❡* ❞❡ ✐♥4❡1[* ♥♦ ❛♣1♦①✐♠❛♥ ❧❛ ❞✐*41✐❜✉❝✐P♥ ♥♦1✲

♠❛❧✳ ❊*4♦ ❡* ❞❡ ❤❡❝❤♦ ✐♠♣♦14❛♥4❡ ♣♦1V✉❡ ❧♦* ❞❛4♦*

✜♥❛♥❝✐❡1♦* ♣1❡*❡♥4❛♥ ❢1❡❝✉❡♥4❡♠❡♥4❡ ❧❡♣4♦❝✉14♦*✐*

② ❝♦❧❛* ❛♥❝❤❛*✳

❯♥❛ ♠❡❞✐❞❛ ♠Q* ❛❞❡❝✉❛❞❛ ♣❛1❛ ❞❛4♦* V✉❡ ♣1❡✲

*❡♥4❡♥ ♦❜*❡1✈❛❝✐♦♥❡* ❡①41❡♠❛* ✕❝♦♠♦ ❧❛* ❞❡ ❧❛*

✈❛1✐❛❝✐♦♥❡* ❞❡ ❞❡♣P*✐4♦* V✉❡ *❡ ✉4✐❧✐③❛1♦♥ ❡♥ ❡*4❡

❡*4✉❞✐♦✕ ❡* ❡❧ ❱❛❘ ❝❛❧❝✉❧❛❞♦ ♠❡❞✐❛♥4❡ ❧❛ 4❡♦1I❛ ❞❡

✈❛❧♦1❡* ❡①41❡♠♦*✳ ❉❡❜❡ 4❡♥❡1*❡ ♣1❡*❡♥4❡ *✐♥ ❡♠✲

❜❛1❣♦ V✉❡ ❡❧ ✈❛❧♦1 ❞❡❧ ❱❛❘ ❝❛❧❝✉❧❛❞♦ ❡♥ ❜❛*❡ ❛ ❧❛

❚❡♦1I❛ ❞❡ ❱❛❧♦1❡* ❊①41❡♠♦* ❡* *❡♥*✐❜❧❡ ❛ ❧❛ ❡❧❡❝✲

❝✐P♥ ❞❡❧ ✉♠❜1❛❧✱ ② ❡❧ ❡*4❛❞♦ ❞❡❧ ❛14❡ ♥♦ ♣❡1♠✐4❡

✐❞❡♥4✐✜❝❛1 ❝♦♥ ♣1❡❝✐*✐P♥ ❡❧ ✈❛❧♦1 ❞❡ ❡*4❡ ✉♠❜1❛❧✳

❆❞❡♠Q* ❞❡ ❧♦* ❝1✐4❡1✐♦* ❝✉❛♥4✐4❛4✐✈♦*✱ ❧❛ ❡❧❡❝❝✐P♥

❞❡❧ ✉♠❜1❛❧ ♣✉❡❞❡ *❡1 ❡♥ a❧4✐♠❛ ✐♥*4❛♥❝✐❛ ✉♥❛ ❞❡✲

❝✐*✐P♥ ✜♥❛♥❝✐❡1❛✱ ②❛ V✉❡ ❞❡♣❡♥❞❡ ❞❡ ❧❛ ❛✈❡1*✐P♥

❛❧ 1✐❡*❣♦ V✉❡ ♣♦*❡❛ ❧❛ ❡♥4✐❞❛❞ ✜♥❛♥❝✐❡1❛ ♦ ❞❡ ❧❛

4♦❧❡1❛♥❝✐❛ ❛❧ 1✐❡*❣♦ V✉❡ *❡ ❤❛②❛ ❞❡✜♥✐❞♦ ❡♥ ❡❧ ❡*✲

4❛❜❧❡❝✐♠✐❡♥4♦ ❞❡❧ ❝♦♥4❡①4♦ ❞❡❧ ♣1♦❝❡*♦* ❞❡ ❣❡*4✐P♥

❞❡ 1✐❡*❣♦*✳

❊❧ ❱❛❘ ❝♦♥❞✐❝✐♦♥❛❧ ❡* ♣❛14✐❝✉❧❛1♠❡♥4❡ ✐♥4❡1❡✲

*❛♥4❡✱ ♣♦1V✉❡ ♣✉❡❞❡ *❡1 ✉4✐❧✐③❛❞♦ ♣❛1❛ ❢✉♥❞❛♠❡♥✲

4❛1 ❧♦* ♣❧❛♥❡* ❞❡ ❝♦♥4✐❣❡♥❝✐❛ ❡♥ ❝❛*♦* ❞❡ ♣[1❞✐❞❛*

*❡✈❡1❛* ♣❡1♦ ♠❡♥♦* ♣1♦❜❛❜❧❡* V✉❡ ❧❛* V✉❡ ✐♠♣❧✐✲

❝❛♥ ❧❛* ❛❝4✐✈✐❞❛❞❡* ❝♦4✐❞✐❛♥❛* ❞❡ ✉♥❛ ❡♥4✐❞❛❞ ✜✲

♥❛♥❝✐❡1❛✳

❊*4✉❞✐♦* ♣♦*4❡1✐♦1❡* ♣✉❡❞❡♥ ♦1✐❡♥4❛1*❡ ❛ ❝❛❧❝✉✲

❧❛1 ✐♥4[1✈❛❧♦* ❞❡ ❝♦♥✜❛♥③❛ ♣❛1❛ ❧♦* ✈❛❧♦1❡* ❞❡❧ ❱❛❘

❜❛*❛❞♦ ❡♥ ❧❛ 4❡♦1I❛ ❞❡ ✈❛❧♦1❡* ❡①41❡♠♦* ② ❡❧ ❱❛❘

❝♦♥❞✐❝✐♦♥❛❧✳

❘❡❢❡#❡♥❝✐❛(

●✐❧❧✐✱ ▼❛♥❢(❡❞✱ ❊✈✐- ❑/❧❧❡③✐ ✭✷✵✵✻✮✱ ❆♥

❆♣♣❧✐❝❛'✐♦♥ ♦❢ ❊①',❡♠❡ ❱❛❧✉❡ ❚❤❡♦,② ❢♦, ▼❡❛✲

6✉,✐♥❣ ❋✐♥❛♥❝✐❛❧ ❘✐6❦✱ ❈♦♠♣✉4❛4✐♦♥❛❧ ❊❝♦♥♦♠✐❝*

✷✼✭✶✮✱ ♣♣✳ ✶✲✷✸✳

●❡♥1❛②✱ ❘❛♠❛③❛♥✱ ❋❛(✉❦ ❙❡❧1✉❦✱ ❆❜✲

❞✉((❛❤♠❛♥ ❯❧✉❣?❧②❛❣❝✐ ✭✷✵✵✷✮✱ ❊❱■▼✿ ❆

❙♦❢'✇❛,❡ ?❛❝❦❛❣❡ ❢♦, ❊①',❡♠❡ ❱❛❧✉❡ ❆♥❛❧②6✐6

✐♥ ▼❆❚▲❆❇✱ ❙4✉❞✐❡* ✐♥ ◆♦♥❧✐♥❡❛1 ❉②♥❛♠✐❝* ✐♥

❊❝♦♥♦♠❡41✐❝*✱ ❱♦❧✳ ✺✱ ■**✉❡ ✸✱ ❆❧❣♦1✐4❤♠ ✶✱ ♣♣✳

✷✶✸✲✷✸✾✳

●❡♥1❛②✱ ❘❛♠❛③❛♥✱ ❋❛(✉❦ ❙❡❧1✉❦ ✭✷✵✵✶✮✱

❖✈❡,♥✐❣❤' ❇♦,,♦✇✐♥❣✱ ■♥'❡,❡6' ❘❛'❡6 ❛♥ ❊①',❡♠❡

❱❛❧✉❡ ❚❤❡♦,②✱ ❊✉1♦♣❡❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇✳

❏♦❤♥-♦♥✱ ❈❤(✐-D✐❛♥ ✭✷✵✵✶✮✱ ❱❛❧✉❡ ❛' ❘✐6❦✿

❚❡♦,E❛ ② ❆♣❧✐❝❛❝✐♦♥❡6✱ ❊*4✉❞✐♦* ❞❡ ❊❝♦♥♦♠I❛ ❱♦❧✳

✷✽✱ ◆♦✳ ✷✱ ♣♣✳ ✷✶✼✲✷✹✼✳

✹

(6)

2001 2002 2003 2004 2005 2006 2007 2008 2000

2500 3000 3500 4000 4500

Observaciones (evolucion en el tiempo)

MMUS$

2001 2002 2003 2004 2005 2006 2007 20080 200

400 600 800 1000

MMUS$

2001 2002 2003 2004 2005 2006 2007 2008 150

200 250 300 350

MUS$

2001 2002 2003 2004 2005 2006 2007 2008 300

350 400 450 500

MMUS$

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

0 50 100 150 200 250 300

VaR Gaussiano VaR

(EVT)

Variaciones de los Depósitos y Valor-en-Riesgo de Retiros de Depósitos

SISTEMA FINANCIERO BOLIVIANO

Variaciones de depósitos del sistema finaciero Histograma de las variaciones de depósitos del

sistema finaciero y medidas de riesgo VaR

Sistema Bancario Fondos Financieros Privados

Cooperativas de Ahorro y Crédito Mutuales de Ahorro y Préstamo

Evolución de Depósitos

(7)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 -3

-2 -1 0 1 2 3 4

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-6 -4 -2 0 2 4 6 8

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-10 -5 0 5

-3 -2 -1 0 1 2 3 4

0 50 100 150 200 250 300

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 50 100 150 200 250 300

-10 -5 0 5

0 200 400 600 800 1000 1200 1400

-6 -4 -2 0 2 4 6 8

0 200 400 600 800 1000 1200 1400

-4 -3 -2 -1 0 1 2 3

0 0.5 1 1.5 2 2.5 3 3.5 4

Umbral

Exceso de la Media (Mean Excess)

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

0 1 2 3 4 5 6 7 8

Umbral

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Umbral

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Umbral

RiskMeasuresandanApplicationtotheWithdrawalsofDepositsintheBolivianFinancialSystem Gonzales-Martínez,Rolando MunichPersonalRePEcArchive

Munich Personal RePEc Archive