risco de crédito 2: creditrisk+
TRANSCRIPT
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Modelos para Risco de Crédito 2:Credit Risk+
Análise de Risco (10)R.Vicente
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Resumo
Introdução Taxas de default constantes e setor único
Freqüência de DefaultsSeveridade das Perdas
Taxas de default variáveis e múltiplo setorFreqüência de DefaultsSeveridade das Perdas
Bibliografia
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Panorama Geral
FREQÜÊNCIADE PERDAS
SEVERIDADEDAS PERDAS
DISTRIBUIÇÃO DE PERDAS POR DEFAULT
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Inputs 1: Rating
Cre dit Me an S tandardRating De fault ra te De viation
A 1.50% 0.75%B 1.60% 0.80%C 3.00% 1.50%D 5.00% 2.50%E 7.50% 3.75%F 10.00% 5.00%G 15.00% 7.50%H 30.00% 15.00%
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Inputs 2: ExposiçõesCre dit
Name Exposure Rating1 358,475 H2 1,089,819 H3 1,799,710 F4 1,933,116 G5 2,317,327 G6 2,410,929 G7 2,652,184 H8 2,957,685 G9 3,137,989 D10 3,204,044 D11 4,727,724 A12 4,830,517 D13 4,912,097 D14 4,928,989 H15 5,042,312 F
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Output: Distribuição de Perdas
Cre dit Lo s s Dis tributio n
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
0 5,000,000 10,000,000 15,000,000 20,000,000 25,000,000 30,000,000
Lo s s
Mar
gina
l Pro
babi
lity
Cre ditLoss
Pe rce ntile AmountMean 11,162,85650.00 9,191,51175.00 16,114,27495.00 28,823,66997.50 33,733,87199.00 39,946,85799.50 44,482,66099.75 48,915,92299.90 54,644,673
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TAXAS DE DEFAULT CONSTANTES E SETOR
ÚNICO
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Freqüência de Defaults
Probabilidade de default de na janela jq j=Seja , assumindo que: a) defaults são eventos independentes
Introduzamos a função auxiliar0
( ) ( )N
n
n
F z p n z=
=∑
Qual é a probabilidade de n defaults em uma janela de tempo especificada ?
( )p nSuponhamos uma carteira contendo N contrapartes.
;
b) taxas não variam no tempo, teremos:
( ) ( )1 1
( ) 1 1 1N N
j j jj j
F z q q z q z= =
⎡ ⎤ ⎡ ⎤= − + = + −⎢ ⎥ ⎣ ⎦⎣ ⎦∏ ∏
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Freqüência de Defaults
( )
( )
1
1
( ) 1 1
ln ( ) ln 1 1
N
jj
N
jj
F z q z
F z q z
=
=
⎡ ⎤= + −⎣ ⎦
⎡ ⎤= + −⎣ ⎦
∏
∑
Assumindo que , utilizamos e obtemos: 0 1jq<
( )
( )
1
1
ln ( ) 1
( ) exp 1
N
jj
N
jj
F z z q
F z z q
=
=
= −
⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦
∑
∑
( )ln 1 ε ε+ ≈
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Freqüência de Defaults
Identificando o número médio de defaults :
( ) zF z e eμ μ−=
1
N
jj
qμ=
=∑
Expandindo em série de Taylor:
0 0
( ) ( )!
nn n
n n
eF z z p n zn
μμ−∞ ∞
= =
= =∑ ∑Lembrando da definição da função auxiliar.
( )!
nep nn
μμ−
=
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Severidade dos Defaults
A exposição a risco de crédito é a perda incorrida em um default.
j jL Lν=
A perda esperada é a perda agregada esperada dada por:
j j j j j jL L Lλ μ ν μ ε= = =
Probabilidade de default de acordo
com a classificação de
crédito
Para cada contraparte j.
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Severidade dos Defaults
As exposições são agrupadas em bandas:
1Lν 2Lν 3Lν 4Lν ... mLν
Cada banda com uma perda esperada associada:
1a a a a mε ν μ= ≤ ≤
O número esperado de defaults em cada banda dado por :
: j a
ja
j jν ν
εμ
ν=
= ∑
aν ∈
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Severidade dos Defaults
Introduzindo uma função geratriz para as perdas agregadas:
( )0
( ) n
n
G z p nL z∞
=
=∑Assumindo as bandas são independentes (as exposições são independentes):
m
1
( ) ( )aa
G z G z=
=∏Tratando cada banda como uma carteira :
1 1
( ) ( )!
aa
a a a a
nn n za
an n
eG z p n z z en
νμ
ν ν μ μμ−∞ ∞− +
= =
= = =∑ ∑
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Severidade dos Defaults
1 1
1 1
( ) ( )
exp
aa a
a
m mz
aa a
m m
a aa a
G z G z e
z
νμ μ
νμ μ
− +
= =
= =
= =
⎡ ⎤⎢ ⎥= − +⎢ ⎥⎣ ⎦
∏ ∏
∑ ∑
1 1
( ) ( )!
aa
a a a a
nn n za
an n
eG z p n z z en
νμ
ν ν μ μμ−∞ ∞− +
= =
= = =∑ ∑
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Severidade dos Defaults
0 0
1 ( )( ) ( ) ( )!
nl
nl z
d G zG z p lL z p nLn dz
∞
= =
= =∑
Derivadas n-ésimas da função geratriz em z=0 fornecem as probabilidades p(n)
1 10 0
1
1100
1 11
1 10 100
1 ( ) 1 exp! !
1 ( )!
11 ( )!
a
a
a
n n m m
a an na az z
n m
anazz
n k kn m
an k kk azz
d G z d zn dz n dz
d dG z zn dz dz
n d dG z zkn dz dz
ν
ν
ν
μ μ
μ
μ
= == =
−
−===
− − +−
− − += ===
⎡ ⎤⎢ ⎥= − +⎢ ⎥⎣ ⎦
=
⎛ ⎞− ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
∑ ∑
∑
∑ ∑
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Severidade dos Defaults
1
10
( ) ( 1)! (( 1)! )n k
n kz
d G z n k p n k Ldz
− −
− −=
= − − − −
1
110
( 1)!, se : 10,
a
k ma a
akaz
k a kd zccdz
ν μ νμ
+
+==
⎧ + ∃ = +⎪⎪=⎨⎪⎪⎩∑
1
0: 1
11( ) ( 1)!( 1)! (( 1) )!
a
n
aka k
np nL n k k p n k L
knν
μ−
=∃ = +
⎛ ⎞− ⎟⎜= ⎟ − − + − −⎜ ⎟⎜ ⎟⎜⎝ ⎠∑
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Severidade dos Defaults
1
0: 1
11( ) ( 1)!( 1)! (( 1) )!
a
n
aka k
np nL n k k p n k L
knν
μ−
=∃ = +
⎛ ⎞− ⎟⎜= ⎟ − − + − −⎜ ⎟⎜ ⎟⎜⎝ ⎠∑
: :
( ) (( ) ) (( ) )a a
a a aa a
a n a n
p nL p n L p n Ln nν ν
ν μ εν ν≤ ≤
= − = −∑ ∑
1
(0) expm
aa
p μ=
⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦∑
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TAXAS DE DEFAULT VARIÁVEIS E MÚLTIPLOS
SETORES
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Setores
1S 2S kS nS
~ ( , )k k kx p μ σ
( )22
k k
k k k
x
x
μ
σ μ
=
= −
Número médio de defaultspor unidade de tempo
Variância de defaults por unidade de tempo
Número de defaults por unidade de tempo = variável aleatória
SETOR = FATOR DE RISCO
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VolatilidadesVolatilidades e taxas de default dependem primordialmente da qualidade da contraparte. Volatilidades e taxas de cada setor são obtidas a partir de dados para cada rating.
Cre dit Me an S tandardRating De fault ra te De viation
A 1.50% 0.75%B 1.60% 0.80%C 3.00% 1.50%D 5.00% 2.50%E 7.50% 3.75%F 10.00% 5.00%G 15.00% 7.50%H 30.00% 15.00%
k aa
μ μ=∑aa k
k
x xμμ
=
( )22
2 22
a a a
a k aa k
k k
x
x
σ μ
μ μμ σμ μ
= −
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= − =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
ka a
k
σσ μμ
=
FATOR DE RISCO
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Volatilidades
k aa
μ μ=∑a
a kk
x xμμ
= ka a
k
σσ μμ
=
ka a k
a ak
σσ μ σμ
= =∑ ∑Cre dit Me an S tandardRating De fault ra te De viation
A 1.50% 0.75%B 1.60% 0.80%C 3.00% 1.50%D 5.00% 2.50%E 7.50% 3.75%F 10.00% 5.00%G 15.00% 7.50%H 30.00% 15.00%
Ex: SETOR = A+B+H
1,5% 1,6% 30% 33, 2%
0,75% 0,8% 15 16,55%
setor A B H
setor A B H
μ μ μ μ
σ σ σ σ
= + += + + == + += + + =
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Número de Defaults com Taxa de Defaults Estocástica
1
( ) ( )n
kk
F z F z=
=∏
( ) ( )1x zk kF z x x e −= =
SETORES
INDEPENDENTES
( ) ( ) ( )
( ) ( )
0
1
0
k k kx
x z
x
F z dx F z x x f x
dx e f x
∞
=∞
−
=
= =
=
∫
∫
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Taxa de Defaults Estocástica: Distribuição Gama
( ) ( ) ( )1
0
x zk
x
F z dx e f x∞
−
=
= ∫11( )
( )
x
f x e xαβαβ α
−−=
Γ1
0
( ) x
x
dx e xαα∞
− −
=
Γ = ∫2 2μ αβ σ αβ= =
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
70 80 90 100 110 120 130 1400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
1α β= =
51
αβ==
1001
αβ==
2 2
2k k
k kk k
μ σα βσ μ
= =PARA O SETOR k
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Taxa de Defaults Estocástica
( ) ( ) ( ) ( )1 1 1
0 0
1
110
1
0
( )
1( )
1( )
1 11 1( ) 1 1
xx z x z
kx x
xxz x
y x zx
y
y
F z dx e f x dx e e x
dx e x
dy e yz z
αβα
αβα
β
αα α
α α
α
β α
β α
β α ββ β
∞ ∞ −− − −
= =∞ − −
−⎛ ⎞⎟⎜ ⎟=− − −⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
∞− −
=
Γ
= =Γ
= =Γ
= =⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜Γ + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
∫ ∫
∫
∫
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Taxa de Defaults Estocástica
( )1
11111
k
k kk
k k
kk
kk
k
F zz
z
α
α βλα β
λλ
ββ
=+
⎛ ⎞− ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ −⎛ ⎞ ⎝ ⎠⎟⎜ ⎟+ −⎜ ⎟⎜ ⎟⎜⎝ ⎠
( )
( ) ( )
( )
2 2
0
11
11 1
2!
11
k
k
k
kk
k
k kk k k k
k n nk k
n
F zz
z z
nz
n
α
α
α
λλ
α αλ α λ λ
αλ λ
∞
=
⎛ ⎞− ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ −⎝ ⎠⎡ ⎤−⎢ ⎥= − + + +⎢ ⎥⎣ ⎦
⎛ ⎞+ − ⎟⎜= − ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑
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Frequência de Defaults
( )0
( ) nk
n
F z p n z∞
=
=∑
( )1( ) 1 kk n
k k
np n
nαα
λ λ⎛ ⎞+ − ⎟⎜= ⎟ −⎜ ⎟⎜ ⎟⎜⎝ ⎠
Distribuição Binomial Negativa
(Pascal)
0 1 2 3 4 5 6 7 8 9 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
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Bibliografia
•Crouhy M. Galai D. e Mark R., A comparative analysis of current credit risk models, Journal of Banking and Finance 24 (2000) 59-117.
•Jorion P., Value at Risk, Irwin, 1997.
• Saunders A., Credit Risk Measurement, John Wiley, 1999
•CreditRisk+,CSFB,1997 (http://www.csfb.com/creditrisk/)
Leitura ComplementarBasle Committee on Banking Supervision, Credit Risk Modelling: Current Practices and Applications, April 1999. (www.bis.org)