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formula sheet RISK MANAGEMENT .docx, Exams of Credit and Risk Management

Abertay UniversityCredit and Risk Management

formula sheet.docx, HELPFUL FOR STUDENTS

Typology: Exams

2019/2020

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bg1
h=ρσSS
σSF
¿contr=htotal position
asets per contract
Δ
∏
¿∂
∏
¿
∂SΔS+ ∂
∏
¿
∂tΔt+1
2∂
2
∏
¿
∂S
2
ΔS
2
+1
2∂
2
∏
¿
∂t
2
Δt
2
+∂
2
∏
¿
∂S∂tΔSΔt+. . . ¿¿¿ ¿ ¿
(delta,vega,theta
nonstoch.,gamma,gamma of vega) delta neutral:
Δ
∏
¿∂
∏
¿
∂SΔS+ ∂
∏
¿
∂σSΔσS+ ∂
∏
¿
∂tΔt+1
2∂
2
∏
¿
∂S
2
ΔS
2
+1
2∂
2
∏
¿
∂σS
2
ΔσS
2
.. . . . ¿¿¿¿¿
Δ
∏
¿ΘΔt+1
2ΓΔS
2
c=S
0
e
−qT
N(d
1
)−Ke
−rT
N(d
2
)
p=Ke
−rT
N(−d
2
)−S
0
e
−qT
N(−d
1
)
d
1
=ln(S
0
/K)+( r−q+σS
2
/2)T
σS
√
T
Normal:
A(1+R
m)
mn
vs. Cc:
Ae
Rn
;
R
c
=mln(1+R
m
m)
R
m
=m(e
R
c
/m
−1)
bond-
price:
Ce
−6 mrate∗0,5
+Ce 1
−1yrate∗1
+C+Pe
−18 mrate∗1,5
Duration of bond mod. Duration (ann.c.) key dur. Relation Convexity key total relationship
D=
∑
i=1
n
t
i
c
i
e
−yt
i
B
D∗¿D
1+y/m
ΔB=−BD Δy
C=1d
2
B
Bdy
2
=
∑
i=1
n
c
i
t
i2
e
−yt
i
B
ΔB/B = -
DΔy+1/2*C(Δy)2
Duration of the portfolio;
ΔX
i
= change in ith asset arising from the yield curve shift in
Δy
D=
∑
i=1
n
X
i
PD
i
stays the same for total relationship
−1
P
ΔP
i
Δx
i
partial duration Principle Component Analysis To calculate deltas: PCx*x-year
rate+PCy*y-year rate
σyr= σday*√252 s2=(1/(n-1)) ∑(ui-u)2
s=
√
1
n−1
∑
i
n
u
i
2
−1
n(n−1)(
∑
i=1
n
u
i
)
2
σ=s/√t st.error: σ/√2n
The Power Law: for many variables it is approx true that the value v has the property that when x is large
Pr ob(v>x)= Kx−α
u
i
=S
i
−S
i−1
S
i−1
σS
n
2
=1
m
∑
i=1
m
u
n−1
2
Forecasting Future vol. Vol. Term structure yVl=w y+ß+a =1
E[σS
n+t
2
]=V
L
+( α+β)
t
(σS
n
2
−V
L
)
V(t)=E(σS
n+t
2
)
a=ln 1
α+β
V(t)=V
L
+e
−at
[V(0)−V
L
]
σS(T)
2
=252 {V
L
+1−e
−aT
aT [V(0)−V
L
]}
Impact of Vol. Changes
When
σS(0)
by
ΔσS (0)
,
σS(T)
by
1−e−aT
aT
σS(0)
σS(T)ΔσS (0)
ρ= E
(
V
1
V
2
)
- E
(
V
1
)
E
(
V
2
)
SD(V
1
)SD( V
2
)
cov (V
1
, V
2
)=E(V
1
V
2
)−E(V
1
)E(V
2
)
cov rate = correlation x % change X x % change Y
Monitoring correlation:
cov=1
m
∑
i=1
m
x
n−i
y
n−i
var
y ,n
=1
m
∑
i=1
m
y
n−1
2
cor=cov
n
√
var
x , n
var
y ,n
Using EWMA:
cov
n
=λcov
n−1
+(1−λ)x
n−1
y
n−1
Using GARCH:
cov
n
=γ VL+αx
n−1
y
n−1
+βcov
n−1
Vasicek (one-factor Gaussian copula): We can be X per cent certain that the default rate by time T will not be worse than:
V(T , X )=N(N
−1
(Q[T])+
√
ρ N
−1
(X)
√
1−ρ
*avg exposure per loan*avg LGD=T-year VaR
Cooke ratio: total risk-weighted assets=
∑
i=1
N
w
i
L
i
+
∑
j=1
M
w
j
¿
C
j
CEA:
C
i
=max (V.,0)+ aL
Netting:
NRR=max (
∑
i=1
N
V
i
,0)
∑
i=1
N
max (V
i
,0)
Credit eq.am.=
max(
∑
i=1
N
V
i
,0)+(0.4+0 . 6∗NRR)
∑
i=1
N
a
i
L
i
market risk cap.req.=k*VaR*SRC
EWMA
σS
n
2
=λσS
n−1
2
+(1−λ)u
n−1
2
GARCH
σS
n
2
=γ VL+αu
n−1
2
+βσS
n−1
2
Autocorrelation (Lijung-Box sta.)
m
∑
k=1
K
ω
k
η
k
2
ω=m+2
m−k
Positive semidefinite
ω
Τ
Ωω≥0
Greek Call option Put option
Delta
e−qT N(d1)
e
−qT
(N(d
1
)−1)
Gamma and
gamma neutral
N'(d1)e−qT
S0σS
√
T
ω
T
=−Γ/Γ
T
N
'
(d
1
)e
−qT
S
0
σS
√
T
Theta
−S
0
N
'
(d
1
)σSe
−qT
/(2
√
T)
+qS
0
N(d
1
)e
−qT
−rKe
−rT
N(d
2
)
−S
0
N
'
(d
1
)σSe
−qT
/(2
√
T)
−qS
0
N(−d
1
)e
−qT
+rKe
−rT
N(−d
2
)
pf3

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h=ρ

σSS

σSF

¿ contr=h

total position

asets per contract

Δ∏ ¿ ∂ ∏ ∂S^ ¿ ΔS+∂ ∏ ∂^ t ¿Δt+ 12 ∂^2 ∏^ ¿

∂ S^2

ΔS^2 + 12 ∂^2 ∏^ ¿

∂t 2

Δt^2 +∂^2 ∂∏ S ∂^ ¿ t ΔSΔt +. .. ¿ ¿¿ ¿ ¿

(delta,vega,theta

nonstoch.,gamma,gamma of vega) delta neutral:

Δ (^) ∏ ¿ ∂ ∏^

∂ S ΔS^ +∂^

∏ ¿

∂ σS ΔσS^ +∂^

∏ ¿

∂t Δt+^

2 ∏ ¿

∂ S^2

ΔS^2 +

2 ∏ ¿

∂σS 2

ΔσS 2 ... .. ¿ ¿ ¿ ¿ ¿ (^) Δ (^) ∏ ¿ ΘΔt + 1 2 ΓΔ^ S 2 c=S 0 e−qT^ N (d 1 )−Ke−rT^ N ( d 2 ) p=Ke−rT^ N (−d 2 )−S 0 e−qT^ N (−d 1 ) d^1 =

ln ( S 0 / K )+( r−q +σS^2 / 2 ) T

σS √ T d 2 =d 1 −σS^ √T

Normal: A ( 1 + R m )mn

vs. Cc: Ae

Rn ; Rc =m ln ( 1 + Rm m ) (^) R m=m(e^ Rc/m − 1 ) (^) bond- price: Ce

− 6 mrate∗0,

+Ce 1

− 1 yrate∗ 1

+C+ Pe

− 18 mrate∗1,

Duration of bond mod. Duration (ann.c.) key dur. Relation Convexity key total relationship

D=

∑i= 1 n

ti ci e

− yt (^) i

B

D∗¿ D 1 + y /m ΔB=−BD Δy C= 1 d^2 B Bdy^2 = ∑i = 1 n ci t i^2 e − yt (^) i B (^) ΔB/B = - DΔy+1/2*C(Δy)^2 Duration of the portfolio; ΔX^ i^ = change in ith asset arising from the yield curve shift in Δy D=∑ i= 1

n X

i

P

Di

stays the same for total relationship − 1 P ΔPi Δxi (^) partial duration Principle Component Analysis To calculate deltas: PCxx-year rate+PCyy-year rate σyr= σday*√252 s^2 =(1/(n-1)) ∑(ui-u)^2

s=

√

n− 1

∑i n

ui^2 − 1

n(n− 1 )

(∑i= 1 n

ui )^2

σ=s/√t st.error: σ/√2n The Power Law : for many variables it is approx true that the value v has the property that when x is large Pr^ ob(^ v^ >^ x^ )=Kx −α

ui=

Si −Si − 1

Si − 1

σSn^2 = 1 m ∑ i= 1 m u^2 n− 1 Forecasting Future vol. Vol. Term structure yVl=w y+ß+a = E[ σSn^2 +^ t]=V (^) L +( α + β )t^ (σS (^) n^2 −V (^) L ) V ( t )=E (σSn^2 +^ t) a=ln^ 1 α+β V^ (t^ )=V^ L +e −at [ V ( 0 )−V (^) L ]

σS (T )^2 = 252 {V L +

1 −e

−aT

aT

[ V ( 0 )−V L ]}

Impact of Vol. Changes When σS^ (^0 )^ by ΔσS^ (^0 )^ , σS^ (^ T^ )^ by 1 −e −aT aT σS ( 0 ) σS (T ) ΔσS ( 0 )

E( V 1 V (^2) ) - E (^) ( V (^1) ) E( V (^2) )

SD(V 1 )SD( V 2 )

cov (V 1 , V 2 )=E( V 1 V 2 )−E ( V 1 ) E( V 2 ) cov rate = correlation x % change X x % change Y Monitoring correlation: cov= 1 m ∑ i = 1 m xn−i yn−i var (^) y ,n= 1 m ∑ i= 1 m yn^2 −^1 cor =

covn

√ varx , n var^ y ,n (^) Using EWMA: covn=λ covn− 1 +( 1 − λ) xn− 1 yn− 1 Using GARCH: covn=γ VL+αx (^) n− 1 yn− 1 + β covn− 1 Vasicek (one-factor Gaussian copula): We can be X per cent certain that the default rate by time T will not be worse than:

V ( T , X )=N (

N

( Q[ T ])+√ ρ N

( X )

√ 1 − ρ (^) avg exposure per loanavg LGD=T-year VaR Cooke ratio: total risk-weighted assets= ∑ i= 1 N wi Li+∑ j= 1 M w (^) j ¿ C (^) j CEA: Ci=max^ (V^.^ ,^0 )+aL^ Netting:

NRR=

max ( (^) ∑i= 1 N

V i , 0 )

∑i = 1 N

max (V i , 0 )

Credit eq.am.= max(∑ i= 1 N V (^) i , 0 )+( 0. 4 + 0. 6 ∗NRR )∑ i= 1 N ai Li market risk cap.req.=kVaRSRC

EWMA

σSn

= λσSn− 1

+( 1 − λ)un− 1

2 GARCH

σSn 2 =γ VL+ αun− 1 2

  • βσS (^) n− 1 2 Autocorrelation (Lijung-Box sta.) m (^) ∑ k = 1 K ωk ηk^2 ω= m+^2 m−k Positive semidefinite

Greek Call option Put option

Delta e−qT^ N ( d

1 )^ e

−qT

( N ( d 1 )− 1 )

Gamma and

gamma neutral

N'^ (d 1 )e−qT

S 0 σS √T

ωT =−Γ /Γ T

N'^ (d 1 )e−qT

S 0 σS √T

Theta (^) −S 0 N ' (^) (d 1 )σSe −qT (^) /( 2 √T^ ) +qS 0 N (d 1 )e−qT−rKe−rT^ N (d 2 ) −S 0 N'^ (d 1 )σSe−qT^ /( (^2) √T ) −qS 0 N (−d 1 )e−qT^ +rKe−rT^ N (−d 2 )

total cap.requirement = 0.08(credit risk RWA+market risk RWA+operational risk RWA) WCDR=V(T,X) X=99.9%  99.9% chance that loss on portf.<NEADLGDWCDR p= 0. 12 ( 1 + e

− 50 PD

) (^) Capital required: EADLGD(WCDR-PD)*MA MA=^

1 +( M − 2. 5 )∗b

1 − 1. 5 ∗b b=(^0.^11852 −^0.^05478 *ln^ (^ PD^ ))

RWA=12.5EADLGD(WCDR-PD)MA Retail exposures (No MA!): EADLGD(WCDR-PD) RWA=12.5EADLGD*(WCDR-PD) p=^0.^03 +^0.^13 e

− 35 PD

guarantees&credit derivatives: cap.req. without guarantee0.15+160PDg Ch. VaR=σSN − 1

( X ) impact of autocorrelation: σS

[ Nday + 2 ( N− 1 )ρ+ 2 ( N − 2 ) ρ

+ 2 ( N − 3 ) ρ

N− 1

]

VaR (^ X ¿ (^) )

=VaR ( X )

N

− 1

(X

¿

N

− 1

( X )

N-day VaR=1-day VaR x SR(N)

k =m n (^) n! k! (n−k )! pk^ ( 1 − p )n−k − 2 ln[( 1 − p ) n−m p m ]+ 2 ln[( 1 −m/n ) n−m (m/n) m ] (^) >3. bunching:

− 2 ln[( 1 −π )

u 00 +u 10

u 01 +u 11

]+ 2 ln [( 1 −π 01 )

u 00

π 01 u 01 ( 1 −π 11 )

u 10

u 11

] π^ =^

u 01 +u 11

u 00 +u 10 +u 10 +u 11 π^01 =^

u 01 u 00 +u 10 π 11 = u 11 u 10 +u 11 Ch. 9 Market risk VaR historical simulation approach historical simulation approach; value of market variable tomorrow =

vn

vi

vi − 1 accuracy; st.error =

f ( x ) √

q ( 1 −q)

n weight given to change between day

n-i and n-i+1 =

λi−^1 ( 1 −λ )

1 −λn incorporating volatility updating:

vn vi− 1 +( vi−vi− 1 ) σSn+ 1 /σSi vi− 1 Extreme value theory (est.tails of dist): Fu(y)=right tail of prob.dist= Probability that x lies between u and u+y given that x does exceed u

Fu ( y )=

F ( u+ y )−F( u )

1 −F (u)

generalized Pareto distribution: Gξ , β ( y )= 1 −( 1 +ξ y β )−^1 /ξ^ F( x )= 1 − nu n ( 1 +ξ x−u β )−^1 /ξ Power law: K= nu n ( ξ β )−^1 /^ ξ

Estimating the tail of the distribution: Unconditional prob that x>u+y; n=total number of obs. Nu=exceptions If estimating left tail x=-x VaR : F(VaR)=q ; q= 1 − nu n ( 1 +ξ VaR−u β

)−^1 /^ ξ^ VaR=u+^

[(

n

nu

( 1 −q ))−ξ− 1 ]

Ch. 10 Market risk VaR Model-building approach

st.dev. of X+Y= σS X + Y =√σS 2 X^ + σSY^2 + 2 ρ σS X σSY

Linear Model: ΔP=∑ i= 1 n α (^) i Δxi

Δxi = return on asset i in one day αi = amount invested in the ith investment; αi Δxi =value of invest. in asset i

in 1 day Variance of ΔP^ = σS (^) P^2 =∑ i= 1 n αi^2 σS (^) i^2 + (^2) ∑ i= 1 n ∑ j<i pij αi α (^) j σSi σS (^) j ΔP=−DP Δy (^) CF-mapping:

calculatedvol^2 = 1 stvol^2 α^2 + 2 ndvol^2 ( 1 −α )^2 + 2 ∗p∗vol 1 ∗vol 2 α ( 1 −α ) Principle Component Analysis ΔP=−x∗f^1 −^ y∗f^2 With

importance weights √std^

(^2) ∗delta (^2) +std 1 (^2) ∗delta 1 (^2) ∗N− (^1) ( 0 , 99 )=VaR

Linear model and option ΔP=δΔS^

Δx = ΔS

s > ΔP=SδΔx

ΔP=∑ i= 1 n

Si δi Δxi

Options √(vol∗delta^ )

(^2) +(vol 1 ∗delta 1 ) (^2) + 2 ∗vol∗delta∗vol 1 ∗delta 1 ∗p∗N − (^1) ( 0 , 99 )=VaR

Quadratic Model: ΔP=SδΔx+^1 /^2 S

γ ( Δx )

Mean= E(^ ΔP^ )=^0.^5 S (^2) γσS 2

sdev = E(^ ΔP

)=S

σS

+ 0. 75 S

σS

skewness: =

E( ΔP

)= 4. 5 S

γσS

+ 1. 875 S

σS

Dependent on one market variable Dependent on more than one variable

ΔP=∑

i= 1 n

Si δi Δxi+ ∑

i= 1 n

1 / 2 Si^2 γ i( Δxi )^2 ΔP=∑

i= 1 n

Si δ i Δxi+∑

i= 1 n

j= 1 n 1 / 2 Si S (^) j γ (^) ij Δxi Δx (^) j Cornish Fisher Expansion Estimate the expansion μp=E( ΔP ) σS (^) P 2 =E [( ΔP 2 )]−[ E ( ΔP )] 2 ξP=^ E [( ΔP^3 )]− 3 E [( ΔP^2 )]μP + 2 μP^3 σS (^3) P^ μP+^ ωq σS^ P ωP=zq + 1 / 6 ( zq

− 1 )ξ (^) P Ch. 14 Operational risk