Credit risk management for industrial corporates

1October 2016
Fabiano De Rosa
P&C and Risk Manager
Marco Berizzi
Chief Financial Officer
Credit Risk ManagementCredit Risk ManagementCredit Risk ManagementCredit Risk Management
for Industrialfor Industrialfor Industrialfor Industrial
CorporatesCorporatesCorporatesCorporates
From Nobel Prize Merton
Model and Basel
Committee Framework to
Pragmatic Approach for
Industrial Sector

ObjectiveObjectiveObjectiveObjective
• Presentation of Credit Risk ManagementPresentation of Credit Risk ManagementPresentation of Credit Risk ManagementPresentation of Credit Risk Management
Theoretical FrameworkTheoretical FrameworkTheoretical FrameworkTheoretical Framework
• Focus on Specific Aspects for IndustrialFocus on Specific Aspects for IndustrialFocus on Specific Aspects for IndustrialFocus on Specific Aspects for Industrial
CorporatesCorporatesCorporatesCorporates
• Impact Measurement of CreditImpact Measurement of CreditImpact Measurement of CreditImpact Measurement of Credit RiskRiskRiskRisk
ManagementManagementManagementManagement on Corporateon Corporateon Corporateon Corporate CustomerCustomerCustomerCustomer
Portfolio EfficiencyPortfolio EfficiencyPortfolio EfficiencyPortfolio Efficiency
2
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Marco BerizziMarco BerizziMarco BerizziMarco Berizzi
AgendaAgendaAgendaAgenda
• A Standard Credit Risk Model for a FinancialA Standard Credit Risk Model for a FinancialA Standard Credit Risk Model for a FinancialA Standard Credit Risk Model for a Financial
InstitutionInstitutionInstitutionInstitution
• A Credit Risk Management Model for an
Industrial Corporate
• Impact of Credit Risk Management Model on
Corporate Customer Portfolio Efficiency
• Bibliography
• Annex
3
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• D(Vt, t) = Vt e-δ(T-t) N(-d1) + BP(t,T) N(d2)33333333
Corporate Debt Value acknowledging CreditCorporate Debt Value acknowledging CreditCorporate Debt Value acknowledging CreditCorporate Debt Value acknowledging Credit
Risk in pioneering Merton ModelRisk in pioneering Merton ModelRisk in pioneering Merton ModelRisk in pioneering Merton Model
4
• Corporate Debt valuationCorporate Debt valuationCorporate Debt valuationCorporate Debt valuation
acknowledges in a structured
and scientific manner credit riskcredit riskcredit riskcredit risk
conceptconceptconceptconcept in pioneering NobleNobleNobleNoble
PrizePrizePrizePrize Merton modelMerton modelMerton modelMerton model
• Corporate Debt valueCorporate Debt valueCorporate Debt valueCorporate Debt value is notnotnotnot
the mere discounted ratediscounted ratediscounted ratediscounted rate of
future cash flowfuture cash flowfuture cash flowfuture cash flow but
incorporatesincorporatesincorporatesincorporates a put optionput optionput optionput option
modelling credit riskcredit riskcredit riskcredit risk arising
from firm default eventfirm default eventfirm default eventfirm default event
• In this way corporate debtcorporate debtcorporate debtcorporate debt
valuevaluevaluevalue D(D(D(D(VVVVtttt, t), t), t), t) is at any dateany dateany dateany date
evaluated as the sumsumsumsum of a zerozerozerozero
coupon bondcoupon bondcoupon bondcoupon bond P(P(P(P(t,Tt,Tt,Tt,T)))) and a shortshortshortshort
positionpositionpositionposition within a put optionput optionput optionput option
Put(Put(Put(Put(VVVVtttt , B), B), B), B) on firm assetfirm assetfirm assetfirm asset VVVVtttt with
strike pricestrike pricestrike pricestrike price being zero couponzero couponzero couponzero coupon
bond face valuebond face valuebond face valuebond face value BBBB::::
• Put(Vt,B) = e-r(T-t) [BN(-d2)-Vt e(r- δ )(T-t)N(-d1)]22222222
D(Vt, t) = P(t,T) - Put(Vt ,B)
11111111 2222222233333333
• P(t,T) = Be-r(T-t)11111111
Corporate Debt Value ComponentsCorporate Debt Value ComponentsCorporate Debt Value ComponentsCorporate Debt Value Components
Corporate Debt ValueCorporate Debt ValueCorporate Debt ValueCorporate Debt Value andandandand
Credit RiskCredit RiskCredit RiskCredit Risk
Corporate Debt ValueCorporate Debt ValueCorporate Debt ValueCorporate Debt Value andandandand
T – t = Time to expiration from current
time t
ZC
Bond
B=Face
Value
T - T =0
= Zero Coupon Bond Value
B V =Firm Asset Value
Put
Option
Value
B
0
= Put(Vt ,B) - Put Value at date t
= Put(VT ,B) - Put Value at expiration date T
B e - r(T-t)
For mathematical
04-05
For mathematical
derivation see
Annex 01-02-03-
04-05
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Corporate Debt Value Derivation using BlackCorporate Debt Value Derivation using BlackCorporate Debt Value Derivation using BlackCorporate Debt Value Derivation using Black
---- ScholesScholesScholesScholes ---- Merton FormulaMerton FormulaMerton FormulaMerton Formula
5
• Zero coupon bondZero coupon bondZero coupon bondZero coupon bond valuevaluevaluevalue P(P(P(P(t,Tt,Tt,Tt,T)))) is equal to face valueface valueface valueface value BBBB – capital redeemed at expiration T –
adjusted for discount factordiscount factordiscount factordiscount factor eeee----r(Tr(Tr(Tr(T----t)t)t)t) where rrrr is free risk interest ratefree risk interest ratefree risk interest ratefree risk interest rate
11111111
• OptionOptionOptionOption Put(Put(Put(Put(VVVVtttt,B,B,B,B)))) on firm assetfirm assetfirm assetfirm asset VVVVtttt with strike pricestrike pricestrike pricestrike price being zero coupon bond face valuezero coupon bond face valuezero coupon bond face valuezero coupon bond face value BBBB is
equal to:
22222222
Put(Vt , B) = e-r(T-t) E Q (max(B – V ;);0)
PutPutPutPut is evaluated as discounted averagediscounted averagediscounted averagediscounted average of possible pay offspay offspay offspay offs at expiration dateexpiration dateexpiration dateexpiration date TTTT given by
differencedifferencedifferencedifference between zero coupon bond face valuezero coupon bond face valuezero coupon bond face valuezero coupon bond face value BBBB and firm asset valuefirm asset valuefirm asset valuefirm asset value !". Payoff. Payoff. Payoff. Payoff is zerozerozerozero if
!" > B> B> B> B and is positivepositivepositivepositive if !" < B< B< B< B. Application of Black. Application of Black. Application of Black. Application of Black ---- ScholesScholesScholesScholes ---- Merton formulaMerton formulaMerton formulaMerton formula for optionoptionoptionoption
pricingpricingpricingpricing allows to expressexpressexpressexpress optionoptionoptionoption as it follows:
Put(Vt , B) = e-r(T-t) N(−d&) B - e−δ(T−t) Vt N(−d')
• Corporate debt valueCorporate debt valueCorporate debt valueCorporate debt value is evaluated as the sumsumsumsum of a zero coupon bondzero coupon bondzero coupon bondzero coupon bond and a short positionshort positionshort positionshort position
within a put optionput optionput optionput option as it follows:
33333333
D(Vt , t) = e−δ(T−t) Vt N(−d') + P(t,T) N(d&)
Corporate Debt Value DerivationCorporate Debt Value DerivationCorporate Debt Value DerivationCorporate Debt Value Derivation
where N(.)N(.)N(.)N(.) is a standard normal cumulative distribution functionis a standard normal cumulative distribution functionis a standard normal cumulative distribution functionis a standard normal cumulative distribution function, ) is the dividend rate, dddd1111 and
dddd2222 are as it follows:
d' = (
*+
,-
.
/ 012 /
3
4
54 ( 16)
5 16
) d& = (
*+
,-
.
/ 012 1
3
4
54 ( 16)
5 16
)
In dddd1111 and dddd2222 , 7 is volatilityis volatilityis volatilityis volatility of firm assetfirm assetfirm assetfirm asset which is modelledmodelledmodelledmodelled through following equationfollowing equationfollowing equationfollowing equation:
dVt = (r-δ)Vt dt + σVt dWt VT = Vt e(0121
3
4
54)( 16)/5(:; 1:- )
⇒
where WWWWtttt is a Brownian motionBrownian motionBrownian motionBrownian motion under risk neutral probabilityrisk neutral probabilityrisk neutral probabilityrisk neutral probability QQQQ
For mathematical
04-05
For mathematical
derivation see
Annex 01-02-03-
04-05
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Brownian Motion and Geometric BrownianBrownian Motion and Geometric BrownianBrownian Motion and Geometric BrownianBrownian Motion and Geometric Brownian
Motion DefinitionMotion DefinitionMotion DefinitionMotion Definition
6
Geometric BrownianGeometric BrownianGeometric BrownianGeometric Brownian MotionMotionMotionMotionBrownian MotionBrownian MotionBrownian MotionBrownian Motion
W6
Timet=0 Timet=0
V=
V6
• A standard Brownian motionstandard Brownian motionstandard Brownian motionstandard Brownian motion is describeddescribeddescribeddescribed
as a Wiener processWiener processWiener processWiener process WWWW which is a
continuous-time stochastic processstochastic processstochastic processstochastic process with
following characteristics:
- W0 = 0
- W6 is almost surely continuous
- has independent increments
- Wt - Ws ~ N 0, t − s with 0 ≤ s ≤ t
• A Geometric Brownian motionGeometric Brownian motionGeometric Brownian motionGeometric Brownian motion is a
continuous-time stochasticstochasticstochasticstochastic processprocessprocessprocess VVVV
with following characteristics:
- V6 satisfies a stochastic differential
equation defined as dVt=aVtdt+bVtdWt
- V6 is a log-normal variable which means
that ln V6~ N(ln V= + a −
'
&
b&
t; b t)
- E(V6 ) = V=eD6
E(V6 )= V=eD6
For mathematical
04-05
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derivation see
Annex 01-02-03-
04-05
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Credit Risk Definition within Merton ModelCredit Risk Definition within Merton ModelCredit Risk Definition within Merton ModelCredit Risk Definition within Merton Model
7
Credit Risk DefinitionCredit Risk DefinitionCredit Risk DefinitionCredit Risk Definition
• Credit riskCredit riskCredit riskCredit risk is assessedassessedassessedassessed in terms of default probabilitydefault probabilitydefault probabilitydefault probability and loss given defaultloss given defaultloss given defaultloss given default
• Default (Default (Default (Default (DDDD)))) is defined as the eventeventeventevent for which firm asset valuefirm asset valuefirm asset valuefirm asset value !" is lowerlowerlowerlower than debtdebtdebtdebt BBBB at
expiration dateexpiration dateexpiration dateexpiration date TTTT
• ApplicationApplicationApplicationApplication of Blackof Blackof Blackof Black / Scholes / Merton/ Scholes / Merton/ Scholes / Merton/ Scholes / Merton formulaformulaformulaformula allows to quantify default probability (default probability (default probability (default probability (PDPDPDPD)))) –
defined as unconditional probabilityunconditional probabilityunconditional probabilityunconditional probability - as it follows:
PD = P(D) = P (VT < B) = N(-d2 ) = N(
*+
.
,-
1 012 1
3
4
54 ( 16)
5 16
)
• Loss given defaultLoss given defaultLoss given defaultLoss given default is defined as 1111 ---- recovery raterecovery raterecovery raterecovery rate of debt valuedebt valuedebt valuedebt value in case of default eventdefault eventdefault eventdefault event
• Application of Black / Scholes / Merton formulaApplication of Black / Scholes / Merton formulaApplication of Black / Scholes / Merton formulaApplication of Black / Scholes / Merton formula allows to quantify loss given defaultloss given defaultloss given defaultloss given default as it
follows:
LGD = E Q (
G;
H
| V < B) = 1 -
'
H
Vt e(012)( 16) N 1J3
N 1J4
B = default
point
Time
Firm
Asset
Tt
Vt
Distribution of
firm asset at
expiration
date
Possible firm
asset value
path
Vte(r−δ)(T−t)
P (VT < B)
0
ln
B
V6
− r − δ −
1
2
σ&
(T − t)
σ T − t
N(-d2 )
Distribution of
N(0,1)
For mathematical
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Credit Risk Definition within KMV Model (1/2)Credit Risk Definition within KMV Model (1/2)Credit Risk Definition within KMV Model (1/2)Credit Risk Definition within KMV Model (1/2)
8
• KMV modelKMV modelKMV modelKMV model builds up an effective approacheffective approacheffective approacheffective approach aimed at assessing credit risk startingassessing credit risk startingassessing credit risk startingassessing credit risk starting from
Merton modelMerton modelMerton modelMerton model assumptions and main findings. With respect to MertonMertonMertonMerton model, KMVKMVKMVKMV model does
not staynot staynot staynot stay in a risk neutralrisk neutralrisk neutralrisk neutral environment (i.e. O is used and not rrrr), replacesreplacesreplacesreplaces normal distributionnormal distributionnormal distributionnormal distribution
probabilityprobabilityprobabilityprobability of defaultdefaultdefaultdefault with an empirical one basedempirical one basedempirical one basedempirical one based on distancedistancedistancedistance from default measuredefault measuredefault measuredefault measure and finefinefinefine
tune concepttune concepttune concepttune concept of default pointdefault pointdefault pointdefault point which no longer coincidesno longer coincidesno longer coincidesno longer coincides with debt valuedebt valuedebt valuedebt value BBBB but with:
dabs= EP
V − d∗
drel =
RS G; 1J∗
5
d∗
= SB +
'
&
LB
where UV = short term debt value= short term debt value= short term debt value= short term debt value and LBLBLBLB = long term debt value= long term debt value= long term debt value= long term debt value
• Distance from default measureDistance from default measureDistance from default measureDistance from default measure is calculated in absoluteabsoluteabsoluteabsolute terms and relativerelativerelativerelative ones as it follows:
⇒ dN =
*+
W∗
,-
1 X1
3
4
54 ( 16)
5 16
0
ln
d∗
V6
− r − δ −
1
2
σ&
(T − t)
σ T − t
Distribution of
N(0,1)
d* = default
point
Time
Firm
Asset
Tt
Vt
Distribution of
firm asset at
expiration
date
Possible firm
asset value
path
Vteμ(T−t)
dabs
dN
For mathematical
04-05
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derivation see
Annex 01-02-03-
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Credit Risk Definition within KMV Model (2/2)Credit Risk Definition within KMV Model (2/2)Credit Risk Definition within KMV Model (2/2)Credit Risk Definition within KMV Model (2/2)
9
Expected Default FrequencyExpected Default FrequencyExpected Default FrequencyExpected Default Frequency
• KMV model substitutes normal distribution functionKMV model substitutes normal distribution functionKMV model substitutes normal distribution functionKMV model substitutes normal distribution function NNNN used to calculate probabilityprobabilityprobabilityprobability of defaultdefaultdefaultdefault
with an empiricallyempiricallyempiricallyempirically determined distribution functiondistribution functiondistribution functiondistribution function called expected default frequencyexpected default frequencyexpected default frequencyexpected default frequency ---- EDFEDFEDFEDF
EDF
dN = Distance from default0
• EDFEDFEDFEDF isisisis a forwardforwardforwardforward----looking measurelooking measurelooking measurelooking measure of actual probabilityactual probabilityactual probabilityactual probability of defaultdefaultdefaultdefault and is firm specificis firm specificis firm specificis firm specific
• In the light of historical informationhistorical informationhistorical informationhistorical information on a large samplelarge samplelarge samplelarge sample of firmsfirmsfirmsfirms, EDFEDFEDFEDF estimateestimateestimateestimate is basedbasedbasedbased on the
proportionproportionproportionproportion of firmsfirmsfirmsfirms with a given default distancegiven default distancegiven default distancegiven default distance which actually defaulteddefaulteddefaulteddefaulted after one yearone yearone yearone year
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Credit Risk Definition within KMV Model for aCredit Risk Definition within KMV Model for aCredit Risk Definition within KMV Model for aCredit Risk Definition within KMV Model for a
Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio
10
• KMV modelKMV modelKMV modelKMV model with some integrations is able to support also credit risk managementcredit risk managementcredit risk managementcredit risk management in case of a
loan portfolioloan portfolioloan portfolioloan portfolio
• When dealing with a loan portfolioloan portfolioloan portfolioloan portfolio, the main aspectaspectaspectaspect to be focused on is default correlationdefault correlationdefault correlationdefault correlation
among single loanssingle loanssingle loanssingle loans able to concentrateconcentrateconcentrateconcentrate dramatically probabilityprobabilityprobabilityprobability on a few number of scenariosscenariosscenariosscenarios
• KMV modelKMV modelKMV modelKMV model for loan portfolioloan portfolioloan portfolioloan portfolio is based on Merton modelMerton modelMerton modelMerton model hypothesis and as for single loan
model does not staynot staynot staynot stay in a risk neutralrisk neutralrisk neutralrisk neutral environment (i.e. O is used and not )
• Firm asset is modeledmodeledmodeledmodeled through followingfollowingfollowingfollowing equationequationequationequation:
W^tn = ρ Yt + 1 − ρ ε6+ under historicalhistoricalhistoricalhistorical probabilityprobabilityprobabilityprobability a with n = 1,...,N
dVnt = μVnt dt + σnVntdW^nt with n = 1,...,N
• Risk sourceRisk sourceRisk sourceRisk source for each loaneach loaneach loaneach loan is given by a combinationcombinationcombinationcombination of a systematic risk factorsystematic risk factorsystematic risk factorsystematic risk factor (state of
economy) affecting all firmsaffecting all firmsaffecting all firmsaffecting all firms and an idiosyncratic firm risk factoridiosyncratic firm risk factoridiosyncratic firm risk factoridiosyncratic firm risk factor as it follows:
where bc ,,,, dce,...,,...,,...,,..., dcf are independent standard normallyindependent standard normallyindependent standard normallyindependent standard normally distributed variables and g ∈ i, e is the
correlation ratecorrelation ratecorrelation ratecorrelation rate among firm assetsfirm assetsfirm assetsfirm assets – «passing through» common element Y – controlling
moreover the proportionproportionproportionproportion between systematicsystematicsystematicsystematic and idiosyncraticidiosyncraticidiosyncraticidiosyncratic factorsfactorsfactorsfactors
• For a large homogeneous portfoliolarge homogeneous portfoliolarge homogeneous portfoliolarge homogeneous portfolio of loansloansloansloans with same probabilitysame probabilitysame probabilitysame probability of default p notdefault p notdefault p notdefault p not
dominateddominateddominateddominated by few loans much larger than the restloans much larger than the restloans much larger than the restloans much larger than the rest, portfolio default rateportfolio default rateportfolio default rateportfolio default rate j(f)
and its
approximative distributiondistributiondistributiondistribution P(P(P(P(j(f)
< x )< x )< x )< x ) are equal respectively to:
L(k)
= ∑ w(N)
n
k
+n' Dn ∈ 0,1 P(L(k)
< x ) = N(
('1o)kp3 q 1kp3 r
o
)
where wwww(N)(N)(N)(N)
nnnn are portfolio weightsportfolio weightsportfolio weightsportfolio weights and DDDDnnnn are default eventdefault eventdefault eventdefault event variables with possible value i or e
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
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Relationship between Default Correlation andRelationship between Default Correlation andRelationship between Default Correlation andRelationship between Default Correlation and
11
L = ∑ Dn
k
+n'
0 100L
100% =1
95%
5%
PerfectPerfectPerfectPerfect Correlation ScenarioCorrelation ScenarioCorrelation ScenarioCorrelation ScenarioNo Correlation ScenarioNo Correlation ScenarioNo Correlation ScenarioNo Correlation Scenario
• For a portfolioportfolioportfolioportfolio of f loans perfectlyloans perfectlyloans perfectlyloans perfectly
independentindependentindependentindependent ρ = 0 with same probabilitysame probabilitysame probabilitysame probability
of defaultdefaultdefaultdefault pppp, numbernumbernumbernumber of default Ldefault Ldefault Ldefault L is given
by:
where L is a binomial variablebinomial variablebinomial variablebinomial variable V(f, v) with
following probability mass functionprobability mass functionprobability mass functionprobability mass function:
f(k, N, p)=P(L=k)=
N
k
px
(1 − p)k1x
with k=0,1,...,N
• For a portfolioportfolioportfolioportfolio of f loans perfectlyloans perfectlyloans perfectlyloans perfectly
dependentdependentdependentdependent ρ = 1 with same probabilitysame probabilitysame probabilitysame probability of
defaultdefaultdefaultdefault pppp, numbernumbernumbernumber of default Ldefault Ldefault Ldefault L probabilityprobabilityprobabilityprobability
massmassmassmass functionfunctionfunctionfunction is given by:
f(k, N, p)=P(L=k)=z
p with k = N
1 − p with k = 0
0 100LE(L)=5
100% =1
18%
No Default Correlation distributes probability on a group of diversified events granting low /No Default Correlation distributes probability on a group of diversified events granting low /No Default Correlation distributes probability on a group of diversified events granting low /No Default Correlation distributes probability on a group of diversified events granting low /
null probability on extreme events while perfect correlation concentrates probability on onlynull probability on extreme events while perfect correlation concentrates probability on onlynull probability on extreme events while perfect correlation concentrates probability on onlynull probability on extreme events while perfect correlation concentrates probability on only
two extreme events respectively “default of all loans” event and “default of no loans” eventtwo extreme events respectively “default of all loans” event and “default of no loans” eventtwo extreme events respectively “default of all loans” event and “default of no loans” eventtwo extreme events respectively “default of all loans” event and “default of no loans” event
L Probability Mass Function
with N = 100 and p = 5%
L Probability Mass Function
with N = 100 and p = 5%
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Loss Definition for a single LoanLoss Definition for a single LoanLoss Definition for a single LoanLoss Definition for a single Loan
12
Firm Asset and Loan LossFirm Asset and Loan LossFirm Asset and Loan LossFirm Asset and Loan Loss
Expected Loss andExpected Loss andExpected Loss andExpected Loss and
Unexpected LossUnexpected LossUnexpected LossUnexpected Loss
B = default
point
Time
Firm
Asset
Tt
Vt
Probability
Density of
firm asset at
expiration
date
Possible firm
asset value
path
Vte(r−|)(T−t)
P (VT < B)
• Loan lossLoan lossLoan lossLoan loss is triggered by default eventdefault eventdefault eventdefault event for which
firm assetfirm assetfirm assetfirm asset is lowerlowerlowerlower than loan face valueloan face valueloan face valueloan face value at
expirationexpirationexpirationexpiration datedatedatedate
• LoanLoanLoanLoan losslosslossloss ((((jj)))) is given by product of default eventdefault eventdefault eventdefault event
((((e !"}V )))), loss given default rateloss given default rateloss given default rateloss given default rate ((((j~•)))) and exposureexposureexposureexposure
at defaultat defaultat defaultat default ((((€•• = V)))) as it follows:
LL = 1 G;}R‚ƒ * LGD * EAD
Expected Loss
- EL
Un-expected Loss
- UL
> Quant
0 EADLoan Loss
Loan Loss (LL)
Probability
Density
Expected
Loss Rate
- ELR
Un-
expected
Loss Rate
- ULR
> Quant
0 1e !"}€•• * LGD
Loan Loss rate
(LLR) Probability
Density
e !"}€•• * LGD
where
1 G;}R‚ƒ = ‡
1 if V < EAD
0 if V ˆ EAD
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Loss Definition for a Loan PortfolioLoss Definition for a Loan PortfolioLoss Definition for a Loan PortfolioLoss Definition for a Loan Portfolio
13
Firm Assets and Portfolio Loan LossFirm Assets and Portfolio Loan LossFirm Assets and Portfolio Loan LossFirm Assets and Portfolio Loan Loss
• Portfolio Loan lossPortfolio Loan lossPortfolio Loan lossPortfolio Loan loss is triggered by sumsumsumsum of defaultdefaultdefaultdefault
eventseventseventsevents for which each firm assetfirm assetfirm assetfirm asset is lowerlowerlowerlower than each
loan face valueloan face valueloan face valueloan face value at expirationexpirationexpirationexpiration datedatedatedate
• Portfolio LoanPortfolio LoanPortfolio LoanPortfolio Loan losslosslossloss ((((PLPLPLPL)))) is given by productproductproductproduct of
portfolioportfolioportfolioportfolio default rate (default rate (default rate (default rate (j(f)
)))), loss given default rateloss given default rateloss given default rateloss given default rate
((((LGDLGDLGDLGD)))) and exposure at defaultexposure at defaultexposure at defaultexposure at default ((((EADEADEADEADportportportport)))) as it follows:
PL = L(k)
* LGD * EADport
Expected Loss
- EL
Un-expected Loss
- UL
> Quant
0 EADportPortfolio Loss
Portfolio Loss
(PL) Probability
Density
d* = default
point
Time
Firm
Asset
Tt
Vt
Probability
Density of
firm asset at
expiration
date
Possible firm
asset value
path
Vteμ(T−t)
dabs
where
L(k)
= ∑ w(N)
n
k
+n' Dn ∈ 0,1
Expected
Loss Rate
- ELR
Un-
expected
Loss Rate
- ULR
> Quant
0 1L k * LGD
Portfolio Loss Rate
(PLR) Probability
Density
L(k) * LGD
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
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Expected Loss Definition for a single LoanExpected Loss Definition for a single LoanExpected Loss Definition for a single LoanExpected Loss Definition for a single Loan
14
Expected Loss and ExpectedExpected Loss and ExpectedExpected Loss and ExpectedExpected Loss and Expected
Loss RateLoss RateLoss RateLoss Rate DerivationDerivationDerivationDerivation
Loss Rate DefinitionLoss Rate DefinitionLoss Rate DefinitionLoss Rate Definition
ELR = E‰
(1 G;}R‚ƒ ∗ LGD)
EL = E‰
(LL) = E‰
(1 G;}R‚ƒ ∗ LGD ∗ EAD)
Expected Loss
- EL
Un-expected Loss
- UL
> Quant
0 EAD
Loan Loss (LL)
Probability
Density
Loan Loss
Expected LossExpected LossExpected LossExpected Loss
Expected Loss RateExpected Loss RateExpected Loss RateExpected Loss Rate
where
• 1 G;}R‚ƒ = ‡
1 if V < EAD
0 if V ˆ EAD
• Q is risk neutral probability
Expected
Loss Rate
- ELR
Un-
expected
Loss Rate
- ULR
> Quant
0 1e !"}€•• * LGD
Probability
Density of
e !"}€•• * LGD
• Expected Loss RateExpected Loss RateExpected Loss RateExpected Loss Rate is equal to:
ELR = LGD * E‰
(1 G;}R‚ƒ ) =
PD * LGDPD * LGDPD * LGDPD * LGD
• Given linearitylinearitylinearitylinearity of € , it gives:
• Expected LossExpected LossExpected LossExpected Loss is equal to:
EL = LGD * EAD * E‰
(1 G;}R‚ƒ ) =
PD * LGDPD * LGDPD * LGDPD * LGD **** EADEADEADEAD
where
• 1 G;}R‚ƒ = ‡
1 if V < EAD
0 if V ˆ EAD
• Q is risk neutral probability
For mathematical
04-05
For mathematical
derivation see
Annex 01-02-03-
04-05
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Expected LossExpected LossExpected LossExpected Loss DefinitionDefinitionDefinitionDefinition forforforfor aaaa Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio
15
ELR = EP
(L(k)
∗ LGD)
EL = EP
(PL) = EP
(L(k)
∗ LGD ∗ EADport)
Expected
Loss Rate
- ELR
Un-
expected
Loss Rate
- ULR
> Quant
0 1L k * LGD
Probability Density
of
L(k) * LGD
Expected Loss
- EL
Un-expected Loss
- UL
> Quant
0 EADportPortfolio Loss
Portfolio Loss
(PL) Probability
Density
Expected Loss RateExpected Loss RateExpected Loss RateExpected Loss Rate • Expected Loss RateExpected Loss RateExpected Loss RateExpected Loss Rate is equal to:
• Expected LossExpected LossExpected LossExpected Loss is equal to:
ELR = LGD * EP
(L(k)
) =
p * LGDp * LGDp * LGDp * LGD
EL = LGD * EADport * EP
(L(k)
) =
p * LGDp * LGDp * LGDp * LGD **** EADEADEADEADportportportport
where
• L(k)
= ∑ w(N)
n
k
+n' Dn ∈ 0,1
• P is historical probability
where
• L(k)
= ∑ w(N)
n
k
+n' Dn ∈ 0,1
• P is historical probability
For mathematical
09
For mathematical
derivation see
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09
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UnUnUnUn----Expected Loss Definition for a LoanExpected Loss Definition for a LoanExpected Loss Definition for a LoanExpected Loss Definition for a Loan
PortfolioPortfolioPortfolioPortfolio
16
UnUnUnUn----Expected Loss andExpected Loss andExpected Loss andExpected Loss and
UnUnUnUn----Exp. Loss Rate DefinitionExp. Loss Rate DefinitionExp. Loss Rate DefinitionExp. Loss Rate Definition
UnUnUnUn----Exp. Loss Rate DefinitionExp. Loss Rate DefinitionExp. Loss Rate DefinitionExp. Loss Rate Definition
Expected Loss
- EL
Un-expected Loss
- UL
> Quant
0 EADport
Portfolio Loss
(PL) Probability
Density
Portfolio Loss
UnUnUnUn----Expected LossExpected LossExpected LossExpected Loss
UnUnUnUn----Expected Loss RateExpected Loss RateExpected Loss RateExpected Loss Rate
UnUnUnUn----Exp. Loss RateExp. Loss RateExp. Loss RateExp. Loss Rate DerivationDerivationDerivationDerivation
UnUnUnUn----Exp. Loss RateExp. Loss RateExp. Loss RateExp. Loss Rate DerivationDerivationDerivationDerivation
• ReturnReturnReturnReturn to portfolio default rateportfolio default rateportfolio default rateportfolio default rate j(f)
and its
approx. distributiondistributiondistributiondistribution P(P(P(P(j(f)
< x )< x )< x )< x ) equal respectively
to:
L(k)
= ∑ w(N)
n
k
+n' Dn ∈ 0,1
P(L(k)
< x ) = N(
'1okp3 q 1kp3 r
o
)
• After definition of Š confidence levelconfidence levelconfidence levelconfidence level (i.e. α =
99.9%), we have:
P(L(k)
< q• )=α Ž N(
'1okp3 •• 1kp3 r
o
)
q• Ž N(
o kp3 • /kp3 r
'1o
)
• InversionInversionInversionInversion of the above formulaabove formulaabove formulaabove formula gives quantilequantilequantilequantile:
ULR = N(
o kp3 • /kp3 r
'1o
) * LGD – (p*LGD)
• SubtractingSubtractingSubtractingSubtracting ELRELRELRELR, brings to ULRULRULRULR::::
ELR + ULR = N(
o kp3 • /kp3 r
'1o
) ∗ LGD
• MultiplicationMultiplicationMultiplicationMultiplication for LGDLGDLGDLGD gives ((((ELR +ULRELR +ULRELR +ULRELR +ULR):):):):
• MultiplicationMultiplicationMultiplicationMultiplication for EAD brings to ULULULUL:
UL = N(
o kp3 • /kp3 r
'1o
) ∗ LGD – (p∗LGD) * EADport
Expected
Loss Rate
- ELR
Un-
expected
Loss Rate
- ULR
> Quant
0 1L k * LGD
Probability Density
of
L(k) * LGD
Note: where P is historical probability
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
fdrose14@gmail.com

Portfolio Default Rate according to LoanPortfolio Default Rate according to LoanPortfolio Default Rate according to LoanPortfolio Default Rate according to Loan
Correlation LevelCorrelation LevelCorrelation LevelCorrelation Level
17
Default correlation impact on creditDefault correlation impact on creditDefault correlation impact on creditDefault correlation impact on credit
riskriskriskrisk
Default correlation impact on creditDefault correlation impact on creditDefault correlation impact on creditDefault correlation impact on credit
riskriskriskrisk
• Probability Density FunctionProbability Density FunctionProbability Density FunctionProbability Density Function f(x) of Default RateDefault RateDefault RateDefault Rate
variable L(k)
---- parametrized to same
unconditional probability of default - assumes
different formsdifferent formsdifferent formsdifferent forms according to different defaultdifferent defaultdifferent defaultdifferent default
correlation valuescorrelation valuescorrelation valuescorrelation values g for a portfolioportfolioportfolioportfolio of loansloansloansloans::::
P(L(k)
< x) = N(
'1o kp3 q 1kp3 r
o
)
α Ž N(
'1okp3 •• 1kp3 r
o
) q• Ž N(
o kp3 • /kp3 r
'1o
)
• LeptokurtosisLeptokurtosisLeptokurtosisLeptokurtosis effect implies that for a givengivengivengiven
confidence levelconfidence levelconfidence levelconfidence level α relative quantilequantilequantilequantile q• increasesincreasesincreasesincreases
dramatically
= Default Rate Prob. Density Function with ρ= 10%
Probability Density Functions of
Default Rate variable L(k) with
unconditional probability of default
p = 5%
Portfolio Default Rate > Quant q•
Portfolio Default Rate DensityPortfolio Default Rate DensityPortfolio Default Rate DensityPortfolio Default Rate Density
with different correlationwith different correlationwith different correlationwith different correlation
Portfolio Default Rate DensityPortfolio Default Rate DensityPortfolio Default Rate DensityPortfolio Default Rate Density
with different correlationwith different correlationwith different correlationwith different correlation
f x =
1 − ρ
ρ
e
'
&
kp3(q)
4
1
'
&o
k(r)p31 '1o k(q)p3 4
where relative Distribution FunctionDistribution FunctionDistribution FunctionDistribution Function is
• High valueHigh valueHigh valueHigh value of defaultdefaultdefaultdefault correlationcorrelationcorrelationcorrelation ρ causes
leptokurtosisleptokurtosisleptokurtosisleptokurtosis effect that’s to say a shiftshiftshiftshift of
probability massprobability massprobability massprobability mass into the tailtailtailtail of density functiondensity functiondensity functiondensity function
• This means thatThis means thatThis means thatThis means that for a given confidence levelgiven confidence levelgiven confidence levelgiven confidence level α
with same value of LGDLGDLGDLGD and EADEADEADEAD, the sum of ELELELEL
and ULULULUL tendstendstendstends to increaseincreaseincreaseincrease strongly
For a given αFor a given α
0 1L(k)5% = 0.05
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
1111 ---- Š
……
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• Default correlationDefault correlationDefault correlationDefault correlation g ∈ 0,1 for a portfolioportfolioportfolioportfolio
of loansloansloansloans with same probabilitysame probabilitysame probabilitysame probability of default pdefault pdefault pdefault p
is the correlation ratecorrelation ratecorrelation ratecorrelation rate among respective
firm assetsfirm assetsfirm assetsfirm assets
• From an empirical point of view, a lowlowlowlow
probabilityprobabilityprobabilityprobability of default pdefault pdefault pdefault p implies a highhighhighhigh
default correlation ratedefault correlation ratedefault correlation ratedefault correlation rate g within portofolioportofolioportofolioportofolio
of loansloansloansloans while a highhighhighhigh probabilityprobabilityprobabilityprobability of defaultdefaultdefaultdefault pppp
implies a low defaultlow defaultlow defaultlow default correlationcorrelationcorrelationcorrelation g
• More specifically for probabilityprobabilityprobabilityprobability of defaultdefaultdefaultdefault
p = 0%, defaultdefaultdefaultdefault correlation ratecorrelation ratecorrelation ratecorrelation rate ρ Ž ρ’P =
24% and for probabilityprobabilityprobabilityprobability of defaultdefaultdefaultdefault p = 100%,
default correlation ratedefault correlation ratedefault correlation ratedefault correlation rate ρ Ž ρ”P = 12%
• From a mathematical point of view defaultdefaultdefaultdefault
correlation ratecorrelation ratecorrelation ratecorrelation rate ρ is a weighted averageweighted averageweighted averageweighted average of
ρ’P and ρ”P where the weightsweightsweightsweights are
exponential functionsexponential functionsexponential functionsexponential functions of pppp as shown below:
Default CorrelationDefault CorrelationDefault CorrelationDefault Correlation Estimate for aEstimate for aEstimate for aEstimate for a Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio
18
ρ =ρ”P ∗
('1 •p–—˜)
('1 •p–—)
+ ρ’P ∗
'1('1 •p–—˜)
('1 •p–—)
ρ =12% ∗
('1 ™p–—š)
('1 ™p–—)
+ 24% ∗
'1('1 ™p–—š)
('1 ™p–—)
0%
5%
10%
15%
20%
25%
30%
Loan CorrelationLoan CorrelationLoan CorrelationLoan Correlation ---- g
Default ProbabilityDefault ProbabilityDefault ProbabilityDefault Probability ---- pppp
Default Correlation EstimateDefault Correlation EstimateDefault Correlation EstimateDefault Correlation Estimate
DefaultDefaultDefaultDefault CorrelationCorrelationCorrelationCorrelation / Probability/ Probability/ Probability/ Probability
Relationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolio
DefaultDefaultDefaultDefault CorrelationCorrelationCorrelationCorrelation / Probability/ Probability/ Probability/ Probability
Relationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolio
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LGD EstimateLGD EstimateLGD EstimateLGD Estimate for afor afor afor a Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio
19
LGD Definition as an Exogenous ParameterLGD Definition as an Exogenous ParameterLGD Definition as an Exogenous ParameterLGD Definition as an Exogenous Parameter
EndogenousEndogenousEndogenousEndogenous
ApproachesApproachesApproachesApproaches
EndogenousEndogenousEndogenousEndogenous
ApproachesApproachesApproachesApproaches
• LGDLGDLGDLGD is considered as an exogenous parameterexogenous parameterexogenous parameterexogenous parameter with respect to assetassetassetasset
firm valuefirm valuefirm valuefirm value
• LGDLGDLGDLGD is estimatedestimatedestimatedestimated through econometriceconometriceconometriceconometric and statistical modelsstatistical modelsstatistical modelsstatistical models
• MajorityMajorityMajorityMajority of estimation modelsestimation modelsestimation modelsestimation models aim at finding a link betweenlink betweenlink betweenlink between LGDLGDLGDLGD and
pppp as shown in the graphicgraphicgraphicgraphic below:
• DownturnDownturnDownturnDownturn LGDLGDLGDLGD estimateestimateestimateestimate by FEDFEDFEDFED suggests to use following formulaformulaformulaformula:
Downturn LGD = 0.08 + 0.92 LGD
• There are a couplecouplecouplecouple
of interestinginterestinginterestinginteresting
attemptsattemptsattemptsattempts to define
LGDLGDLGDLGD endogenouslyendogenouslyendogenouslyendogenously
within asset firmasset firmasset firmasset firm
valuevaluevaluevalue and evolutionevolutionevolutionevolution
• First modelFirst modelFirst modelFirst model
conceivedconceivedconceivedconceived by
Schafer, Koivusalo
and Becker is able
to build upbuild upbuild upbuild up a closedclosedclosedclosed
formulaformulaformulaformula for LGDLGDLGDLGD
considering assetassetassetasset
firm portfoliofirm portfoliofirm portfoliofirm portfolio
performanceperformanceperformanceperformance
• Second modelSecond modelSecond modelSecond model
conceived by Frye
treatstreatstreatstreats LGDLGDLGDLGD
analogously to
default ratedefault ratedefault ratedefault rate using a
VasicekVasicekVasicekVasicek
distributiondistributiondistributiondistribution
2007
2006
20051987
2004
1993
1983
1997
1996
1992
1984
2003
2008
1991
1998
1999
2000
1986
1994
1995
1985
1982
1989
1988
1990
2001
2002
2009 (annualized)
80%
70%
60%
50%
40%
30%
20%
10%
10% 12% 14% 16% 18%8%6%4%2%0%
y = - 2.3137 x + 0.5029 with R2 = 0.5361
y = 30.255 x2 – 6.0594 x + 0.5671 with R2 = 0.6151
y = -0.1069 In x + 0.0297 with R2 = 0.6287
y = 0.1457 x-0.2801 with R2 = 0.6531
RecoveryRateRecoveryRateRecoveryRateRecoveryRate
Default RateDefault RateDefault RateDefault Rate
Recovery Rate / Default Rate AssociationRecovery Rate / Default Rate AssociationRecovery Rate / Default Rate AssociationRecovery Rate / Default Rate Association –––– US Corporate BondUS Corporate BondUS Corporate BondUS Corporate Bond
MarketMarketMarketMarket –––– from 1982 to 1H 2009from 1982 to 1H 2009from 1982 to 1H 2009from 1982 to 1H 2009
ConceivedConceivedConceivedConceived bybybyby
Altman, Brady,Altman, Brady,Altman, Brady,Altman, Brady,
SironiSironiSironiSironi andandandand RestiRestiRestiResti
ConceivedConceivedConceivedConceived bybybyby
Altman, Brady,Altman, Brady,Altman, Brady,Altman, Brady,
SironiSironiSironiSironi andandandand RestiRestiRestiResti
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LGD EstimateLGD EstimateLGD EstimateLGD Estimate for afor afor afor a Loan Portfolio within anLoan Portfolio within anLoan Portfolio within anLoan Portfolio within an
Endogenous ApproachEndogenous ApproachEndogenous ApproachEndogenous Approach
20
LGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous Approach
proposed by Fryeproposed by Fryeproposed by Fryeproposed by Frye
LGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous ApproachLGD Definition for a Portfolio Loan within an Endogenous Approach
proposed by Fryeproposed by Fryeproposed by Fryeproposed by Frye
• A VasicekVasicekVasicekVasicek variablevariablevariablevariable !! means that !! has a Vasicek distribution. A Vasicek variable is a
transformation of a normal variable as it follows:
P(L(k)
< q• )=α = N(
'1okp3 •• 1kp3 r
o
) with α ∈ (0,1) where q• = q• J•Ÿ 0D6•
Vasicek Variable = VV = N(
o ¡/kp3 ¢
'1o
) with Z~N(0,1)
j(f)
= N(
o kp3 • /kp3 r
'1o
) with α ∈ (0,1)
• In caseIn caseIn caseIn case β = p, VVVVVVVV is the conditional expected default rateconditional expected default rateconditional expected default rateconditional expected default rate EEEE j(f)
| b / default ratedefault ratedefault ratedefault rate ---- j(f)
variable:
• Suppose that the conditional expected loss rateconditional expected loss rateconditional expected loss rateconditional expected loss rate cELRcELRcELRcELR is a VasicekVasicekVasicekVasicek variablevariablevariablevariable with β = ELR so we
have that:
cELR = N(
o kp3 • /kp3 R’¥
'1o
) with α ∈ (0,1)
• Now consider that
• InsertInsertInsertInsert last equation into cELRcELRcELRcELR and we obtain:
cELR = N(N1'
q• J•Ÿ 0D6• −
kp3 r 1 kp3 R’¥
'1o
)
• Dividing by conditional expected default rateconditional expected default rateconditional expected default rateconditional expected default rate j(f)
, we obtain conditional expectedconditional expectedconditional expectedconditional expected loss givenloss givenloss givenloss given
default ratedefault ratedefault ratedefault rate cELGDRcELGDRcELGDRcELGDR::::
cELGDR = N(N1'
q• J•Ÿ 0D6• − k) /q• J•Ÿ 0D6• where k =
kp3 r 1 kp3 R’¥
'1o
• Banks have estimatesestimatesestimatesestimates of pppp and also of ELRELRELRELR. ELRELRELRELR should be part of the spread chargedspread chargedspread chargedspread charged on any
loan. All loansAll loansAll loansAll loans belonging to same portfolioportfolioportfolioportfolio have the same probabilitysame probabilitysame probabilitysame probability of defaultdefaultdefaultdefault pppp and the
same expected loss ratesame expected loss ratesame expected loss ratesame expected loss rate ELRELRELRELR
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
fdrose14@gmail.com

Expected and UnExpected and UnExpected and UnExpected and Un----ExpectedExpectedExpectedExpected Loss CoverageLoss CoverageLoss CoverageLoss Coverage
forforforfor a Loan Portfolioa Loan Portfolioa Loan Portfolioa Loan Portfolio
21
• Deployment of a pricing strategypricing strategypricing strategypricing strategy and
tacticstacticstacticstactics which acknowledges
expected loss making revenuesexpected loss making revenuesexpected loss making revenuesexpected loss making revenues able
to cover credit risk impactcover credit risk impactcover credit risk impactcover credit risk impact
Coverage of Expected Loss ELCoverage of Expected Loss ELCoverage of Expected Loss ELCoverage of Expected Loss EL
• QuantificationQuantificationQuantificationQuantification of a provisionprovisionprovisionprovision equal to
ELRELRELRELR for each single uniteach single uniteach single uniteach single unit of loanloanloanloan
portfolioportfolioportfolioportfolio expositionexpositionexpositionexposition EADEADEADEADportportportport
• ProvisionProvisionProvisionProvision for entireentireentireentire loan portfolioloan portfolioloan portfolioloan portfolio
expositionexpositionexpositionexposition is given by multiplicationmultiplicationmultiplicationmultiplication
of ELRELRELRELR by EADEADEADEADportportportport
• ProvisionProvisionProvisionProvision is inserted in loan portfolioloan portfolioloan portfolioloan portfolio
holder corporate P&Lholder corporate P&Lholder corporate P&Lholder corporate P&L
EL = p * LGD * EADport
• Expected Loss Rate (Expected Loss Rate (Expected Loss Rate (Expected Loss Rate (ELRELRELRELR) and) and) and) and
Expected Loss (Expected Loss (Expected Loss (Expected Loss (ELELELEL)))) of a loanloanloanloan
portfolioportfolioportfolioportfolio is respectively equal to:
ELR = p * LGD
Provision = p * LGD * EADport
Provision Rate = p * LGD
Coverage of UnCoverage of UnCoverage of UnCoverage of Un----expected Loss ULexpected Loss ULexpected Loss ULexpected Loss UL
• UnUnUnUn----expected Loss Rate (expected Loss Rate (expected Loss Rate (expected Loss Rate (ULRULRULRULR)))) and UnUnUnUn----
expected Loss (expected Loss (expected Loss (expected Loss (ULULULUL)))) of a loan portfolioloan portfolioloan portfolioloan portfolio is
respectively equal to:
UL = N(
o kp3 • /kp3 r
'1o
• Quantification of an equity capitalequity capitalequity capitalequity capital amountamountamountamount ((((KKKK))))
for each single uniteach single uniteach single uniteach single unit of loan portfolioloan portfolioloan portfolioloan portfolio
expositionexpositionexpositionexposition EADEADEADEADportportportport conceived to secure
business continuitybusiness continuitybusiness continuitybusiness continuity of loan portfolio holderloan portfolio holderloan portfolio holderloan portfolio holder
against severe impactssevere impactssevere impactssevere impacts deriving from unununun----
expected lossexpected lossexpected lossexpected loss
• Equity capitalEquity capitalEquity capitalEquity capital for entire loan portfolioentire loan portfolioentire loan portfolioentire loan portfolio
expositionexpositionexpositionexposition is given by multiplicationmultiplicationmultiplicationmultiplication of ULRULRULRULR by
EADEADEADEADportportportport
K = N(
o kp3 • /kp3 r
'1o
ULR = N(
o kp3 • /kp3 r
'1o
) * LGD – (p*LGD)
K Rate = N(
o kp3 • /kp3 r
'1o
) * LGD – (p*LGD)
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
fdrose14@gmail.com

Regulatory Capital required by BaselRegulatory Capital required by BaselRegulatory Capital required by BaselRegulatory Capital required by Basel
Committee vs Equity Capital amount K (1/2)Committee vs Equity Capital amount K (1/2)Committee vs Equity Capital amount K (1/2)Committee vs Equity Capital amount K (1/2)
22
Regulatory Capital Rate = RC Rate = (A+B)*C
Regulatory Capital by BaselRegulatory Capital by BaselRegulatory Capital by BaselRegulatory Capital by Basel
Committee on Banking SupervisionCommittee on Banking SupervisionCommittee on Banking SupervisionCommittee on Banking Supervision
Equity Capital Amount KEquity Capital Amount KEquity Capital Amount KEquity Capital Amount K
• Regulatory capital rateRegulatory capital rateRegulatory capital rateRegulatory capital rate for each single uniteach single uniteach single uniteach single unit
of loan portfolio expositionloan portfolio expositionloan portfolio expositionloan portfolio exposition EADEADEADEADportportportport required
by Basel CommitteeBasel CommitteeBasel CommitteeBasel Committee on BankingBankingBankingBanking
SupervisionSupervisionSupervisionSupervision for financial institutionsfinancial institutionsfinancial institutionsfinancial institutions is given
by:
A = [LGD*N[(1-R)^-0.5*G(PD)+(R/(1-
R))^0.5*G(0.999)]]
where
K Rate = (A+B)*C
• Equity capital rateEquity capital rateEquity capital rateEquity capital rate for each single uniteach single uniteach single uniteach single unit of
loan portfolio expositionloan portfolio expositionloan portfolio expositionloan portfolio exposition EADEADEADEADportportportport owned by
financial institutionfinancial institutionfinancial institutionfinancial institution is given by:
A = N(
o kp3 • /kp3 r
'1o
) * LGD
where
-R = ρ
-G = N1'
-0.999 = α
-PD = p
-^0.5 = …
-^-0.5 =
'
…
==
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
with with
R =0.12 ∗
('1R¬P 1-=Pƒ )
('1R¬P 1-= )
+ 0.24 ∗
'1('1R¬P 1-=Pƒ )
('1R¬P 1-= )
ρ =12% ∗
('1 •p–—˜)
('1 •p–—)
+ 24% ∗
'1('1 •p–—˜)
('1 •p–—)
==
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Regulatory Capital required by BaselRegulatory Capital required by BaselRegulatory Capital required by BaselRegulatory Capital required by Basel
Committee vs Equity Capital amount KCommittee vs Equity Capital amount KCommittee vs Equity Capital amount KCommittee vs Equity Capital amount K (2/2(2/2(2/2(2/2))))
23
C =(1-1.5*b(PD))^-1*(1+(M-2.5)*b(PD)
where
represents a maturity adjustment factor
equal to 1 in case of one year maturity M
with
where
C = 1
given that the approach is supposed to be
lean and straight forward
ŽŽ
b(PD)=(0.11852-0.05478*Ln(PD)))^2
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
Equity Capital Amount KEquity Capital Amount KEquity Capital Amount KEquity Capital Amount K
B = [-PD*LGD] B = – (p*LGD)
-PD = p
where where
==
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MarcoMarcoMarcoMarco BerizziBerizziBerizziBerizzi
Fabiano De RosaFabiano De RosaFabiano De RosaFabiano De Rosa
• A Standard Credit Risk Model for a Financial
Institution
• A Credit Risk Management Model for anA Credit Risk Management Model for anA Credit Risk Management Model for anA Credit Risk Management Model for an
Industrial CorporateIndustrial CorporateIndustrial CorporateIndustrial Corporate
• Impact of Credit Risk Management Model on
Corporate Customer Portfolio Efficiency
• Bibliography
• Annex
24
fdrose14@gmail.com

Credit RiskCredit RiskCredit RiskCredit Risk Management ModelManagement ModelManagement ModelManagement Model
25
ModelModelModelModel
Credit line plafond
and payment terms
Credit collecting
Expected loss
estimate and
coverage
Un-expected
loss estimate
and coverage
Credit risk
mitigation
Customer
rating
111111
22222222
333333
444444
55555555
6666
AAAAAAAA
AAAA
AAAA
AAAAAA
fdrose14@gmail.com

Organization and Risk GovernanceOrganization and Risk GovernanceOrganization and Risk GovernanceOrganization and Risk Governance----SupportSupportSupportSupport----
ControlControlControlControl
26
SalesSalesSalesSales
Chief ExecutiveChief ExecutiveChief ExecutiveChief ExecutiveChief ExecutiveChief ExecutiveChief ExecutiveChief Executive
OfficerOfficerOfficerOfficer
RiskRiskRiskRiskRiskRiskRiskRisk
ManagementManagementManagementManagement
Board of DirectorsBoard of DirectorsBoard of DirectorsBoard of Directors
Internal AuditInternal AuditInternal AuditInternal Audit
RiskRiskRiskRisk
CommitteeCommitteeCommitteeCommittee
RiskRiskRiskRisk
CommitteeCommitteeCommitteeCommittee
ICTICTICTICTFinanceFinanceFinanceFinance
GovernanceGovernanceGovernanceGovernance
2222 Management LayerManagement LayerManagement LayerManagement Layer
1111 ManagementManagementManagementManagement LayerLayerLayerLayer
SupportSupportSupportSupport
1111 Control LayerControl LayerControl LayerControl Layer
2222 Control LayerControl LayerControl LayerControl Layer
OrganizationOrganizationOrganizationOrganization
Risk Governance &Risk Governance &Risk Governance &Risk Governance &
Risk Governance &Risk Governance &Risk Governance &Risk Governance &
AAAAAA
fdrose14@gmail.com

Customer RatingCustomer RatingCustomer RatingCustomer Rating DefinitionDefinitionDefinitionDefinition
27
Customer Rating and Unconditional Probability of Default (PD)Customer Rating and Unconditional Probability of Default (PD)Customer Rating and Unconditional Probability of Default (PD)Customer Rating and Unconditional Probability of Default (PD)
Unconditional probabilityUnconditional probabilityUnconditional probabilityUnconditional probability of defaultdefaultdefaultdefault is articulated per rating graderating graderating graderating grade and gives the averageaverageaverageaverage
percentagepercentagepercentagepercentage of obligorsobligorsobligorsobligors that defaultdefaultdefaultdefault in this rating grade in the course of one yearone yearone yearone year
11111111
Rating GradeRating GradeRating GradeRating Grade
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
UnconditionalUnconditionalUnconditionalUnconditional Probability of DefaultProbability of DefaultProbability of DefaultProbability of Default
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
%%%%%%%%
++++
----
fdrose14@gmail.com

Customer Rating RecognitionCustomer Rating RecognitionCustomer Rating RecognitionCustomer Rating Recognition –––– MainMainMainMain
Components and ScaleComponents and ScaleComponents and ScaleComponents and Scale
28
11111111
Customer Rating Main ComponentsCustomer Rating Main ComponentsCustomer Rating Main ComponentsCustomer Rating Main Components Rating ScaleRating ScaleRating ScaleRating Scale
Sub-Rating from
Country Risk
Sub-Rating from
Payment Delay
Sub-Rating from Financial Statement
Group
Revenues Revenues
EBITDA in %
revenues
NFP /
EBITDA EBITDA / i NFP / BV Other ratios
Full Year Accounts Interim Accounts
Financial Statement
WeightedWeightedWeightedWeighted
Average ofAverage ofAverage ofAverage of
SubSubSubSub----RatingRatingRatingRating
CreditCreditCreditCredit
RatingRatingRatingRating
CreditCreditCreditCredit
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
fdrose14@gmail.com

• RequestRequestRequestRequest to customer of
financial statementfinancial statementfinancial statementfinancial statement
• Most fresh FinancialMost fresh FinancialMost fresh FinancialMost fresh Financial
StatementStatementStatementStatement to be used
• Financial StatementFinancial StatementFinancial StatementFinancial Statement
accepted not oldernot oldernot oldernot older than 2222
yearsyearsyearsyears
• SubSubSubSub----RatingRatingRatingRating attribution of
“CCC”“CCC”“CCC”“CCC” in case of no replyno replyno replyno reply
by customercustomercustomercustomer
Sub RatingSub RatingSub RatingSub Rating –––– Customer Financial StatementCustomer Financial StatementCustomer Financial StatementCustomer Financial Statement
29
11111111
Financial StatementFinancial StatementFinancial StatementFinancial Statement
Business andBusiness andBusiness andBusiness and
Financial RatiosFinancial RatiosFinancial RatiosFinancial Ratios
Business andBusiness andBusiness andBusiness and
• RevenuesRevenuesRevenuesRevenues
• EBITDA in % ofEBITDA in % ofEBITDA in % ofEBITDA in % of
revenuesrevenuesrevenuesrevenues
Ratio NatureRatio NatureRatio NatureRatio Nature
• Net Financial Position /Net Financial Position /Net Financial Position /Net Financial Position /
EBITDAEBITDAEBITDAEBITDA
• EBITDA / net financialEBITDA / net financialEBITDA / net financialEBITDA / net financial
interestsinterestsinterestsinterests
• Net Financial Position /Net Financial Position /Net Financial Position /Net Financial Position /
Book ValueBook ValueBook ValueBook Value
• Group ratiosGroup ratiosGroup ratiosGroup ratios
• Sub Group ratiosSub Group ratiosSub Group ratiosSub Group ratios
• Stand alone ratiosStand alone ratiosStand alone ratiosStand alone ratios
Business RatiosBusiness RatiosBusiness RatiosBusiness Ratios
Sub RatingSub RatingSub RatingSub Rating
RatiosRatiosRatiosRatios
SubSubSubSub----
RatingRatingRatingRatingFull Year AccountFull Year AccountFull Year AccountFull Year Account
Interim AccountInterim AccountInterim AccountInterim Account
1H Account1H Account1H Account1H Account
Quarterly AccountQuarterly AccountQuarterly AccountQuarterly Account
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
fdrose14@gmail.com

Sub RatingSub RatingSub RatingSub Rating –––– Customer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment Delay
30
11111111
Customer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment DelayCustomer Payment Delay Sub RatingSub RatingSub RatingSub Rating
RatiosRatiosRatiosRatios
SubSubSubSub----
ArithmeticArithmeticArithmeticArithmetic
Overdue DelaysOverdue DelaysOverdue DelaysOverdue Delays
Weighted AverageWeighted AverageWeighted AverageWeighted AverageWeighted AverageWeighted AverageWeighted AverageWeighted Average
of Overdueof Overdueof Overdueof Overdue
DelaysDelaysDelaysDelays
Overdue AmountsOverdue AmountsOverdue AmountsOverdue AmountsOverdue AmountsOverdue AmountsOverdue AmountsOverdue Amounts
on a daily basison a daily basison a daily basison a daily basis
Overdue VolumeOverdue VolumeOverdue VolumeOverdue Volume
FrequencyFrequencyFrequencyFrequency
Overdue ValueOverdue ValueOverdue ValueOverdue Value
FrequencyFrequencyFrequencyFrequency
• n = n.° of invoices issued within a
certain time interval
• T¯ = n.° of payment delay days of
invoice j
IndicatorIndicatorIndicatorIndicatorIndicatorIndicatorIndicatorIndicator VariablesVariablesVariablesVariablesVariablesVariablesVariablesVariables
• T¯ = n.° of payment delay days of
invoice j
• I¯= amount of invoice paid in delay
• I+ = amount of invoices issued
within a certain time interval
• m = n.° of days within a certain
time interval
• n¯ = 1 if invoice j is paid with a
delay higher than 0
• n¯ = 0 if invoice j is paid with a
delay equal or lower than 0
±
T¯
n
+
¯n'
±
T¯ ∗ I¯
I+
+
¯n'
±
T¯ ∗ I¯
m
+
¯n'
±
n¯
n
+
¯n'
±
I¯
I+
+
¯n'
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
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Sub RatingSub RatingSub RatingSub Rating –––– Customer Country RiskCustomer Country RiskCustomer Country RiskCustomer Country Risk
31
11111111
Customer Country RiskCustomer Country RiskCustomer Country RiskCustomer Country Risk Sub RatingSub RatingSub RatingSub Rating
CustomerCustomerCustomerCustomer
Country RatingCountry RatingCountry RatingCountry Rating
CustomerCustomerCustomerCustomer
Country RatingCountry RatingCountry RatingCountry Rating
PoliticalPoliticalPoliticalPolitical
RiskRiskRiskRisk
ExchangeExchangeExchangeExchange
Rate RiskRate RiskRate RiskRate Risk
EconomicEconomicEconomicEconomic
RiskRiskRiskRisk
SovereignSovereignSovereignSovereign
RiskRiskRiskRisk
TransferTransferTransferTransfer
RiskRiskRiskRisk
ComponentsComponentsComponentsComponents ComponentsComponentsComponentsComponents
ComponentsComponentsComponentsComponents
SubSubSubSub----
Country meritCountry meritCountry meritCountry merit
worthinessworthinessworthinessworthiness is
affectedaffectedaffectedaffected by politicalpoliticalpoliticalpolitical
riskriskriskrisk, economic riskeconomic riskeconomic riskeconomic risk
and sovereign risksovereign risksovereign risksovereign risk
PoliticalPoliticalPoliticalPolitical
RiskRiskRiskRisk
EconomicEconomicEconomicEconomic
RiskRiskRiskRisk
SovereignSovereignSovereignSovereign
RiskRiskRiskRisk
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
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Customer Rating AnalysisCustomer Rating AnalysisCustomer Rating AnalysisCustomer Rating Analysis –––– PD and FrequencyPD and FrequencyPD and FrequencyPD and Frequency
by Rating Gradeby Rating Gradeby Rating Gradeby Rating Grade
32
11111111
Customer RatingCustomer RatingCustomer RatingCustomer Rating
and PDand PDand PDand PD
Customer RatingCustomer RatingCustomer RatingCustomer Rating
and PDand PDand PDand PD
Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.°°°° ofofofof
CustomersCustomersCustomersCustomers
Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.Frequency by Rating Grade in terms of N.°°°° ofofofof
CustomersCustomersCustomersCustomers
ChangeChangeChangeChange inininin
commercialcommercialcommercialcommercial ////
risk policyrisk policyrisk policyrisk policy cancancancan
affectaffectaffectaffect
customercustomercustomercustomer
portfolio riskportfolio riskportfolio riskportfolio risk
levellevellevellevel
ChangeChangeChangeChange inininin
commercialcommercialcommercialcommercial ////
risk policyrisk policyrisk policyrisk policy cancancancan
affectaffectaffectaffect
portfolio riskportfolio riskportfolio riskportfolio risk
levellevellevellevel
AAA
AA
A
BBB
BB
B
CCC
CC
C
R
SD
D
PDPDPDPD
0.015%
0.043%
0.110%
0.392%
1.536%
5.762%
12.129%
20.934%
32.304%
78.500%
87.400%
100.000%
Note: time of measurement around 2015
1% 1%
20%
29%
18% 18%
8%
3%
1%
1%
0% 0%
AAA AA A BBB BB B CCC CC C R SD D
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Credit Risk MitigationCredit Risk MitigationCredit Risk MitigationCredit Risk Mitigation –––– InstrumentsInstrumentsInstrumentsInstruments
33
22222222
• Payment termsPayment termsPayment termsPayment terms definition in the form of totaltotaltotaltotal / partial prepartial prepartial prepartial pre----
paymentpaymentpaymentpayment or payment upon receiptpayment upon receiptpayment upon receiptpayment upon receipt of goodsgoodsgoodsgoods at companycompanycompanycompany
warehousewarehousewarehousewarehouse
• CompensationCompensationCompensationCompensation of creditcreditcreditcredit / debt positionsdebt positionsdebt positionsdebt positions in case customercustomercustomercustomer is
also a suppliersuppliersuppliersupplier
CRM InstrumentsCRM InstrumentsCRM InstrumentsCRM Instruments
Business instrumentsBusiness instrumentsBusiness instrumentsBusiness instruments
• Set up of an escrow accountescrow accountescrow accountescrow account by a customercustomercustomercustomer as guaranteeguaranteeguaranteeguarantee of
future paymentsfuture paymentsfuture paymentsfuture payments
• Issue of a guaranteeguaranteeguaranteeguarantee / letterletterletterletter of creditcreditcreditcredit by a financial institutionfinancial institutionfinancial institutionfinancial institution
to cover customer solvencycustomer solvencycustomer solvencycustomer solvency relative to detected transactionsdetected transactionsdetected transactionsdetected transactions
• Issue of a guaranteeguaranteeguaranteeguarantee by public trade organizationspublic trade organizationspublic trade organizationspublic trade organizations to cover
customer solvencycustomer solvencycustomer solvencycustomer solvency relative to detected transactionsdetected transactionsdetected transactionsdetected transactions
• Usage of factoringfactoringfactoringfactoring (pro-soluto) and ABSABSABSABS or ABS related
instruments
Finance instrumentsFinance instrumentsFinance instrumentsFinance instrumentsFinance instrumentsFinance instruments
• Stipulation of a policypolicypolicypolicy with an insurance institutioninsurance institutioninsurance institutioninsurance institution to cover
credit riskcredit riskcredit riskcredit risk connected with an identified poolpoolpoolpool of customercustomercustomercustomer for
all transactionsall transactionsall transactionsall transactions
Insurance instrumentsInsurance instrumentsInsurance instrumentsInsurance instruments
PragmaticPragmaticPragmaticPragmatic
ApproachApproachApproachApproach
PragmaticPragmaticPragmaticPragmatic
ApproachApproachApproachApproach
• Set up of a modelmodelmodelmodel
designed to
acknowledgeacknowledgeacknowledgeacknowledge all
types of credit riskcredit riskcredit riskcredit risk
mitigationmitigationmitigationmitigation (CRM)
instrumentsinstrumentsinstrumentsinstruments and
toolstoolstoolstools
• CRM instrumentsCRM instrumentsCRM instrumentsCRM instruments /
toolstoolstoolstools are built up to
have an impactimpactimpactimpact
directlydirectlydirectlydirectly on
exposure at defaultexposure at defaultexposure at defaultexposure at default
(EAD) of customercustomercustomercustomer
to privilegeprivilegeprivilegeprivilege a more
pragmaticpragmaticpragmaticpragmatic
approach …approach …approach …approach …
• ………… also if it would be
theoretically moremoremoremore
correctcorrectcorrectcorrect that CRMCRMCRMCRM
instrumentsinstrumentsinstrumentsinstruments trigger
customercustomercustomercustomer LGDLGDLGDLGD or
require usageusageusageusage of
guarantorguarantorguarantorguarantor LGDLGDLGDLGD
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Customer Credit Line Plafond and PaymentCustomer Credit Line Plafond and PaymentCustomer Credit Line Plafond and PaymentCustomer Credit Line Plafond and Payment
TermsTermsTermsTerms –––– Recognition CriteriaRecognition CriteriaRecognition CriteriaRecognition Criteria
34
33333333
Customer Credit Line Plafond RecognitionCustomer Credit Line Plafond RecognitionCustomer Credit Line Plafond RecognitionCustomer Credit Line Plafond Recognition
Customer PaymentCustomer PaymentCustomer PaymentCustomer Payment
Terms RecognitionTerms RecognitionTerms RecognitionTerms Recognition
Customer PaymentCustomer PaymentCustomer PaymentCustomer Payment
Terms RecognitionTerms RecognitionTerms RecognitionTerms Recognition
• It would be theoretically
more correctmore correctmore correctmore correct to
recognize paymentrecognize paymentrecognize paymentrecognize payment
termstermstermsterms for a customercustomercustomercustomer
according to its specificits specificits specificits specific
rating …rating …rating …rating …
• ………… but to be moremoremoremore
pragmaticpragmaticpragmaticpragmatic….
• …. payment termspayment termspayment termspayment terms ---- for
an existing customer ––––
are maintainedmaintainedmaintainedmaintained
constantconstantconstantconstant and changedchangedchangedchanged
only according to
managerial decisionmanagerial decisionmanagerial decisionmanagerial decision
and ...
• … positive paymentpositive paymentpositive paymentpositive payment
termstermstermsterms ---- equal to 30303030
daysdaysdaysdays ---- for a newnewnewnew
customercustomercustomercustomer are
recognizedrecognizedrecognizedrecognized only after a
trial period posttrial period posttrial period posttrial period post
acquisitionacquisitionacquisitionacquisition
(0;5] (5;25] (25;50]
(50;
100]
(100;
200]
(200;
300]
(300;
400]
(400;
500]
(500;
600] > 600
AAA 210% 164% 134% 114% 100% 91% 85% 81% 77% 77%
AA 205% 159% 129% 109% 95% 86% 80% 76% 72% 72%
A 200% 154% 124% 104% 90% 81% 75% 71% 67% 67%
BBB 195% 149% 119% 99% 85% 76% 70% 66% 62% 62%
BB 190% 144% 114% 94% 80% 71% 65% 61% 57% 57%
B 185% 139% 109% 89% 75% 66% 60% 56% 52% 52%
CCC 180% 134% 104% 84% 70% 61% 55% 51% 47% 47%
CC 175% 129% 99% 79% 65% 56% 50% 46% 42% 42%
C 170% 124% 94% 74% 60% 51% 45% 41% 37% 37%
R 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%
SD 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%
D 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%
NR 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%
• Credit line plafondCredit line plafondCredit line plafondCredit line plafond (CL) recognition for a customera customera customera customer is
calculated as follows below:
CL = C * AR
where ARARARAR is an average expositionaverage expositionaverage expositionaverage exposition along a certain historicalcertain historicalcertain historicalcertain historical
time linetime linetime linetime line towards a customercustomercustomercustomer in terms of account receivablesaccount receivablesaccount receivablesaccount receivables
- corresponding conceptually to EAD - and CCCC is a factorfactorfactorfactor which
is a functionfunctionfunctionfunction of ratingratingratingrating and ARARARAR as shown in the following table
AR in K EURAR in K EUR
RatingRating
• Credit line plafondCredit line plafondCredit line plafondCredit line plafond (CL) is maintained constantmaintained constantmaintained constantmaintained constant within 6666----
month periodmonth periodmonth periodmonth period unlessunlessunlessunless strong variationstrong variationstrong variationstrong variation of ARARARAR and ratingratingratingrating occur
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Customer CreditCustomer CreditCustomer CreditCustomer Credit Line PlafondLine PlafondLine PlafondLine Plafond ---- QuantificationQuantificationQuantificationQuantification
35
33333333
Customer Credit Line PlafondCustomer Credit Line PlafondCustomer Credit Line PlafondCustomer Credit Line Plafond (CL) by(CL) by(CL) by(CL) by Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot. CLCLCLCLCustomer Credit Line PlafondCustomer Credit Line PlafondCustomer Credit Line PlafondCustomer Credit Line Plafond (CL) by(CL) by(CL) by(CL) by Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot. CLCLCLCL
(%; 2015)
StandardStandardStandardStandardAdjustedAdjustedAdjustedAdjusted
100%
tot
100%
tot
Note: time of measurement is around end of 2015
11%
1%
11%
10%
16%
19%
6%
19%
6%
1%
0% 0%
9%
0%
8%
5%
15%
21%
11%
27%
4%
0% 0% 0%
CCCCLLLL is calculated usingis calculated usingis calculated usingis calculated using
ExposureExposureExposureExposure at defaultat defaultat defaultat default
adjusted (adjusted (adjusted (adjusted (EADEADEADEAD AdjAdjAdjAdj))))
diminished by CRMdiminished by CRMdiminished by CRMdiminished by CRM
instruments /instruments /instruments /instruments / toolstoolstoolstools
CCCCLLLL is calculated usingis calculated usingis calculated usingis calculated using
diminished by CRMdiminished by CRMdiminished by CRMdiminished by CRM
instruments /instruments /instruments /instruments / toolstoolstoolstools
CCCCLLLL is calculatedis calculatedis calculatedis calculated
using standardusing standardusing standardusing standard
ExposureExposureExposureExposure atatatat
defaultdefaultdefaultdefault ((((EADEADEADEAD))))
CCCCLLLL is calculatedis calculatedis calculatedis calculated
using standardusing standardusing standardusing standard
ExposureExposureExposureExposure atatatat
defaultdefaultdefaultdefault ((((EADEADEADEAD))))
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Credit Collecting PracticeCredit Collecting PracticeCredit Collecting PracticeCredit Collecting Practice
36
44444444
Overdue InvoiceOverdue InvoiceOverdue InvoiceOverdue Invoice
SituationSituationSituationSituation
Overdue InvoiceOverdue InvoiceOverdue InvoiceOverdue Invoice
SituationSituationSituationSituation
Credit Collecting ActionCredit Collecting ActionCredit Collecting ActionCredit Collecting Action
• Risk ManagerRisk ManagerRisk ManagerRisk Manager (RM) sendssendssendssends automatically via e-mail -
putting in cc CFO and Accounting Manager (AM) - a
reminder templatereminder templatereminder templatereminder template - generated by corporate ERP - to
respective customerrespective customerrespective customerrespective customer
• RM sendssendssendssends an e-mail containing a reminder templatereminder templatereminder templatereminder template -
generated by corporate ERP – and a requestrequestrequestrequest for
explanationexplanationexplanationexplanation to respective customerrespective customerrespective customerrespective customer putting in cc
respective Key Account Manager (KAM) / Country
Manager (CM), AM and CFO
• RM asksasksasksasks referral KAM to organizeorganizeorganizeorganize a conference callconference callconference callconference call
with respective customerrespective customerrespective customerrespective customer
• RM callscallscallscalls an internal meetinginternal meetinginternal meetinginternal meeting with referral KAMKAMKAMKAM, CFOCFOCFOCFO
and CEOCEOCEOCEO to findfindfindfind a suitable solutionsuitable solutionsuitable solutionsuitable solution
• RM callscallscallscalls promptly an internal meetinginternal meetinginternal meetinginternal meeting with referral
KAMKAMKAMKAM, CFOCFOCFOCFO and CEOCEOCEOCEO in order to take a final decisionfinal decisionfinal decisionfinal decision
and to decide submissionsubmissionsubmissionsubmission of a claimclaimclaimclaim to insuranceinsuranceinsuranceinsurance
companycompanycompanycompany
DaysDaysDaysDays of delaydelaydelaydelay relative to
overdue invoicesoverdue invoicesoverdue invoicesoverdue invoices ––––
corresponding to overduecorresponding to overduecorresponding to overduecorresponding to overdue
account receivablesaccount receivablesaccount receivablesaccount receivables (AR) ----
relative to a specifica specifica specifica specific
< 10
>= 10 and < 20
>= 20 and < 30
>= 30 and < 50
>= 50 and < 70
In days
11111111
22222222
33333333
44444444
55555555
11111111
22222222
33333333
44444444
55555555
When a customer overdue invoice amountcustomer overdue invoice amountcustomer overdue invoice amountcustomer overdue invoice amount is lowerlowerlowerlower than
5555’000 EUR’000 EUR’000 EUR’000 EUR and no other overdue invoiceno other overdue invoiceno other overdue invoiceno other overdue invoice is traced, RMRMRMRM
adopts the same above actions without involving CEOsame above actions without involving CEOsame above actions without involving CEOsame above actions without involving CEO
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7%
0%
4%
3%
10%
18%
12%
35%
6%
4%
0% 0%
0% 0%
3% 3%
7% 7%
21%
52%
0%
7%
0% 0%
Exposition at DefaultExposition at DefaultExposition at DefaultExposition at Default ---- QuantificationQuantificationQuantificationQuantification
37
55555555
Exposition at Default (Exposition at Default (Exposition at Default (Exposition at Default (EADEADEADEAD ) by) by) by) by Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot. EADEADEADEADExposition at Default (Exposition at Default (Exposition at Default (Exposition at Default (EADEADEADEADportportportport) by) by) by) by Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot.Rating Grade in % of tot. EADEADEADEAD
(%; 2015)
StandardStandardStandardStandardAdjustedAdjustedAdjustedAdjusted
100%
tot
100%
tot
diminished bydiminished bydiminished bydiminished by CRMCRMCRMCRM
instruments / toolsinstruments / toolsinstruments / toolsinstruments / tools
diminished bydiminished bydiminished bydiminished by CRMCRMCRMCRM
instruments / toolsinstruments / toolsinstruments / toolsinstruments / tools
((((EADEADEADEAD) not diminished) not diminished) not diminished) not diminished
bybybyby CRM instrumentsCRM instrumentsCRM instrumentsCRM instruments
/ tools/ tools/ tools/ tools
((((EADEADEADEAD) not diminished) not diminished) not diminished) not diminished
bybybyby CRM instrumentsCRM instrumentsCRM instrumentsCRM instruments
/ tools/ tools/ tools/ tools
Note: time of measurement is around end of 2015
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Loss Given DefaultLoss Given DefaultLoss Given DefaultLoss Given Default ---- EstimateEstimateEstimateEstimate
38
55555555
Loss given DefaultLoss given DefaultLoss given DefaultLoss given Default
• This graph shows the associationassociationassociationassociation of
weighted average default ratesweighted average default ratesweighted average default ratesweighted average default rates and
recovery ratesrecovery ratesrecovery ratesrecovery rates over the period 1982198219821982----
1H20091H20091H20091H2009 within US corporate bondUS corporate bondUS corporate bondUS corporate bond
marketmarketmarketmarket using four bibibibi----variate regressionvariate regressionvariate regressionvariate regression
specificationsspecificationsspecificationsspecifications
• These regressionsregressionsregressionsregressions include linearlinearlinearlinear
quadratic logquadratic logquadratic logquadratic log---- linearlinearlinearlinear and powerpowerpowerpower
functionfunctionfunctionfunction structuresstructuresstructuresstructures
• ProxyProxyProxyProxy is given using log functionlog functionlog functionlog function :
• LGDLGDLGDLGD is estimatedestimatedestimatedestimated using econometric relationshiprelationshiprelationshiprelationship between recovery raterecovery raterecovery raterecovery rate and default ratedefault ratedefault ratedefault rate
defined by Altman, Brady, Sironi and Resti analysis
• Econometric relationshiprelationshiprelationshiprelationship between recovery raterecovery raterecovery raterecovery rate and default ratedefault ratedefault ratedefault rate is given by relationship
between bond default ratesbond default ratesbond default ratesbond default rates and recovery ratesrecovery ratesrecovery ratesrecovery rates
y = -0.1069 In x + 0.0297
• LGDLGDLGDLGD estimateestimateestimateestimate is given by:
LGD = 1 − RR = 1 − (-0.1069 In x + 0.0297)
where y = RR and x = DR Ž p
LGD = 0.9703+0.1069 In p
2007
2006
20051987
2004
1993
1983
1997
1996
1992
1984
2003
2008
1991
1998
1999
2000
1986
1994
1995
1985
1982
1989
1988
1990
2001
2002
2009 (annualized)
80%
70%
60%
50%
40%
30%
20%
10%
10% 12% 14% 16% 18%8%6%4%2%0%
y = - 2.3137 x + 0.5029 with R2 = 0.5361
y = 30.255 x2 – 6.0594 x + 0.5671 with R2 = 0.6151
y = -0.1069 In x + 0.0297 with R2 = 0.6287
y = 0.1457 x-0.2801 with R2 = 0.6531
RecoveryRate(RR)RecoveryRate(RR)RecoveryRate(RR)RecoveryRate(RR)
Default Rate (DR)Default Rate (DR)Default Rate (DR)Default Rate (DR)
Recovery Rate / Default Rate AssociationRecovery Rate / Default Rate AssociationRecovery Rate / Default Rate AssociationRecovery Rate / Default Rate Association ––––
US Corporate Bond MarketUS Corporate Bond MarketUS Corporate Bond MarketUS Corporate Bond Market –––– from 1982 tofrom 1982 tofrom 1982 tofrom 1982 to
1H 20091H 20091H 20091H 2009
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100%
tottottottot
Expected LossExpected LossExpected LossExpected Loss –––– EstimateEstimateEstimateEstimate
39
55555555
EL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. ELEL by Rating Grade in % of tot. EL
(%; 2015)
EADport AdjEADport AdjEADport AdjEADport Adj
in % tot.in % tot.in % tot.in % tot.
0% 0% 4% 4% 7% 7% 48% 30% 0% 0% 0% 0%
pppp 0.01%0.04%0.11%0.39% 1.5% 5.8% 12% 21% 32% 78% 87% 100%
LGDLGDLGDLGD 3% 14% 24% 38% 52% 67% 74% 80% 85% 94% 96% 97%
100%100%100%100%
22%22%22%22%
38%38%38%38%
• For each rating clusterrating clusterrating clusterrating cluster
(“portfolio”) Expected LossExpected LossExpected LossExpected Loss
Rate (Rate (Rate (Rate (ELRELRELRELR) and Expected) and Expected) and Expected) and Expected
Loss (Loss (Loss (Loss (ELELELEL)))) are calculated
according to:
EL = p * LGD * EADport
ELR = p * LGD
• TotalTotalTotalTotal ELELELEL is given by the sumsumsumsum
of singlesinglesinglesingle ELELELEL:
Total EL = ∑ EL¯
'&
¯n' =
± p¯ ∗ LGD¯ ∗ EADr²06 ¯
'&
¯n'
where j = 1, ..., 12
are rating clusterrating clusterrating clusterrating cluster
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
Note: time of measurement of PD and LGD is around end of 2015 while time of measurement of EAD is end of
2015 sharp
0% 0% 0% 0% 0%
2%
12%
55%
0%
31%
0% 0%
ELELELEL is calculatedis calculatedis calculatedis calculated
using Exposureusing Exposureusing Exposureusing Exposure atatatat
defaultdefaultdefaultdefault adjustedadjustedadjustedadjusted
((((EADEADEADEAD AdjAdjAdjAdj))))
ELELELEL is calculatedis calculatedis calculatedis calculated
fdrose14@gmail.com

0% 0% 0% 0% 2%
4%
20%
68%
0%
6%
0% 0%
UnUnUnUn----ExpectedExpectedExpectedExpected LossLossLossLoss –––– EstimateEstimateEstimateEstimate
40
66666666
UnUnUnUn----ExpectedExpectedExpectedExpected LossLossLossLoss
• UnUnUnUn----expected Loss Rate (expected Loss Rate (expected Loss Rate (expected Loss Rate (ULRULRULRULR) and) and) and) and
UnUnUnUn----Expected Loss (Expected Loss (Expected Loss (Expected Loss (ULULULUL)))) are
calculated for each rating clusterrating clusterrating clusterrating cluster jjjj
(“portfolio”) with j = 1, ...,12 according
to:
• Total ULTotal ULTotal ULTotal UL is given by the sumsumsumsum of
single ULsingle ULsingle ULsingle UL:
Total UL = ∑ UL¯
'&
¯n' =
ULR = N(
o kp3 • /kp3 r
'1o
) * LGD – (p*LGD)
UL = N(
o kp3 • /kp3 r
'1o
) ∗ LGD – (p∗LGD)
g
23.9%
23.7%
23.4%
21.9%
17.6%
12.7%
12.0%
12.0%
12.0%
12.0%
12.0%
12.0%
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
Note: time of measurement of PD and LGD is around end of 2015 while time of measurement of EAD is end of
2015 sharp
UUUUL by Rating Grade in % of tot.L by Rating Grade in % of tot.L by Rating Grade in % of tot.L by Rating Grade in % of tot. UUUULLLLUUUUL by Rating Grade in % of tot.L by Rating Grade in % of tot.L by Rating Grade in % of tot.L by Rating Grade in % of tot. UUUULLLL
(%; 2015)
100.0%
tottottottot
* EADport
∑ N(
o³ kp3 • /kp3 r³
'1o³
) ∗ LGD¯ – (p¯∗LGD¯)'&
¯n'
∗ EADr²06³
LoanLoanLoanLoan
correlationcorrelationcorrelationcorrelation
insideinsideinsideinside 12121212
ratingratingratingrating
clustersclustersclustersclusters
g = (ge, … , ge´))))
LoanLoanLoanLoan
correlationcorrelationcorrelationcorrelation
insideinsideinsideinside 12121212
ratingratingratingrating
clustersclustersclustersclusters
g = (ge, … , ge´))))
UUUUL is calculatedL is calculatedL is calculatedL is calculated
UUUUL is calculatedL is calculatedL is calculatedL is calculated
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Coverage of Expected LossCoverage of Expected LossCoverage of Expected LossCoverage of Expected Loss
41
66666666
• Deployment of a pricing strategypricing strategypricing strategypricing strategy and
tacticstacticstacticstactics modelmodelmodelmodel supported by corporate ERPcorporate ERPcorporate ERPcorporate ERP
or sales applicationsales applicationsales applicationsales application which acknowledges
automatically ratingratingratingrating in the formulationformulationformulationformulation of
quotationquotationquotationquotation and proposalproposalproposalproposal to a specific
customercustomercustomercustomer and adds relative expectedexpectedexpectedexpected losslosslossloss
in cost structure listcost structure listcost structure listcost structure list in order to make
revenuesrevenuesrevenuesrevenues able to cover credit risk impactcover credit risk impactcover credit risk impactcover credit risk impact
Coverage of ELCoverage of ELCoverage of ELCoverage of EL
• Quantification of a Provision (Provision (Provision (Provision (a!€j)))) –––– equalequalequalequal to expected lossexpected lossexpected lossexpected loss - for each rating clustereach rating clustereach rating clustereach rating cluster
(“portfolio) to tackle credit riskcredit riskcredit riskcredit risk brought by occurrence of standard eventsstandard eventsstandard eventsstandard events:
PVR’=EL = p * LGD * EADport
Total PVR’ =Total EL= ∑ EL¯ = ∑ p¯ ∗ LGD¯ ∗ EADr²06 ¯
'&
¯n'
'&
¯n'
where j = 1, ..., 12 are rating clusterrating clusterrating clusterrating cluster
• Annual provision (Annual provision (Annual provision (Annual provision (a!€j)))) for standard creditstandard creditstandard creditstandard credit
riskriskriskrisk is inserted in corporate Profit & Losscorporate Profit & Losscorporate Profit & Losscorporate Profit & Loss
tabletabletabletable acknowledging a possible futurepossible futurepossible futurepossible future
burdenburdenburdenburden and allowing also to gain taxationgain taxationgain taxationgain taxation
shieldshieldshieldshield
• Total PVTotal PVTotal PVTotal PV is given by the sumsumsumsum of single PVsingle PVsingle PVsingle PV:
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
Financial CoverageFinancial CoverageFinancial CoverageFinancial CoverageFinancial CoverageFinancial Coverage Business CoverageBusiness CoverageBusiness CoverageBusiness CoverageBusiness CoverageBusiness Coverage
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Coverage of UnCoverage of UnCoverage of UnCoverage of Un----Expected LossExpected LossExpected LossExpected Loss
42
66666666
Coverage of ULCoverage of ULCoverage of ULCoverage of UL
• Set up of an equity capital bufferequity capital bufferequity capital bufferequity capital buffer ((((KKKK)))) or a Provision (Provision (Provision (Provision (a!µj)))) –––– equal to un-expected loss – for
each rating clustereach rating clustereach rating clustereach rating cluster (“portfolio”) to tackle credit riskcredit riskcredit riskcredit risk brought by occurrence of extremeextremeextremeextreme
events:events:events:events:
• Initial provisionInitial provisionInitial provisionInitial provision ((((a!µj)))) for “not standard”“not standard”“not standard”“not standard”
credit riskcredit riskcredit riskcredit risk is inserted in corporate Profit &corporate Profit &corporate Profit &corporate Profit &
Loss tableLoss tableLoss tableLoss table acknowledging a possiblepossiblepossiblepossible
future burdenfuture burdenfuture burdenfuture burden and allowing also to gaingaingaingain
taxationtaxationtaxationtaxation shieldshieldshieldshield
• Subsequent annual provisionSubsequent annual provisionSubsequent annual provisionSubsequent annual provision ∆ ((((a!µj))))
instalmentsinstalmentsinstalmentsinstalments for “not standard” credit risk“not standard” credit risk“not standard” credit risk“not standard” credit risk
permits to cover annual variationcover annual variationcover annual variationcover annual variation of unununun----
expected lossexpected lossexpected lossexpected loss valuevaluevaluevalue
Total K or Total PV·’ = ∑ UL¯
'&
¯n' = ∑ N(
o³ kp3 • /kp3 r³
'1o³
) ∗ LGD¯ – (p¯∗LGD¯) ∗ EADr²06³
'&
¯n'
K or PV·’ = N(
o kp3 • /kp3 r
'1o
• TotalTotalTotalTotal KKKK orororor a!µj is given by the sumsumsumsum of singlesinglesinglesingle KKKK orororor a!µj:
• Initial EquityInitial EquityInitial EquityInitial Equity capital buffercapital buffercapital buffercapital buffer ((((KKKK))))
establishment within corporate Balancecorporate Balancecorporate Balancecorporate Balance
SheetSheetSheetSheet table permits to strengthen meritmeritmeritmerit
worthinessworthinessworthinessworthiness and relative ratingratingratingrating facilitating
relationshiprelationshiprelationshiprelationship with stakeholdersstakeholdersstakeholdersstakeholders such as
supplierssupplierssupplierssuppliers and providersprovidersprovidersproviders of financefinancefinancefinance
• Subsequent annual / periodicalSubsequent annual / periodicalSubsequent annual / periodicalSubsequent annual / periodical ∆ capitalcapitalcapitalcapital
bufferbufferbufferbuffer ((((KKKK)))) establishments permits to covercovercovercover
annual variationannual variationannual variationannual variation of unununun----expected lossexpected lossexpected lossexpected loss value
For mathematical
09
For mathematical
derivation see
Annex 06-07-08-
09
Financial Coverage (1Financial Coverage (1Financial Coverage (1Financial Coverage (1°°°° Option)Option)Option)Option) Financial CoverageFinancial CoverageFinancial CoverageFinancial Coverage (2(2(2(2°°°° Option)Option)Option)Option)
fdrose14@gmail.com

Marco BerizziMarco BerizziMarco BerizziMarco Berizzi
Fabiano De RosaFabiano De RosaFabiano De RosaFabiano De Rosa
• A Standard Credit Risk Model for a Financial
Institution
• A Credit Risk Management Model for an
Industrial Corporate
• Impact of Credit RiskImpact of Credit RiskImpact of Credit RiskImpact of Credit Risk Management ModelManagement ModelManagement ModelManagement Model onononon
Corporate Customer Portfolio EfficiencyCorporate Customer Portfolio EfficiencyCorporate Customer Portfolio EfficiencyCorporate Customer Portfolio Efficiency
• Bibliography
• Annex
43
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Customer Overdue Portfolio DecreaseCustomer Overdue Portfolio DecreaseCustomer Overdue Portfolio DecreaseCustomer Overdue Portfolio Decrease –––– GlobalGlobalGlobalGlobal
View from Jun 2014 to Dec 2015View from Jun 2014 to Dec 2015View from Jun 2014 to Dec 2015View from Jun 2014 to Dec 2015
44
Customer OverdueCustomer OverdueCustomer OverdueCustomer Overdue PtfPtfPtfPtf. Variation. Variation. Variation. VariationCustomer OverdueCustomer OverdueCustomer OverdueCustomer Overdue PtfPtfPtfPtf. Variation. Variation. Variation. Variation
(%; Jun 2014-Dec 2015)
Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio
DefinitionDefinitionDefinitionDefinition
Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio
DefinitionDefinitionDefinitionDefinition
• Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio in a certain
time intervaltime intervaltime intervaltime interval is equalequalequalequal to:
Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio decreased of
- 99%99%99%99% between Jun 14 – Dec 2015 with strong
impact of day of delay effect
accounting for –––– 98%98%98%98%
-69% ----99%99%99%99%
-98%
67%
Day of delays
effect
Cross effect Overdue
amount
effect
Global effectGlobal effectGlobal effectGlobal effect
dAr
Ar
dIr
Ir
(
J ˜
˜
*
J¹˜
¹˜
)dTr
Tr
Ar = Tr ∗ Ir
where:
- Tr = n.° of payment delay days relative to all
customer portfolio invoices issued and paid in
delay in a certain time interval
- Ir = amount of all customer portfolio invoices
issued and paid in delay in a certain time interval
• Time interval ranges from 2014 June2014 June2014 June2014 June to
December 2015December 2015December 2015December 2015
• VariationVariationVariationVariation of Ar is equal to:
dAr = IrdTr + TrdIr + dTrdIr
• VariationVariationVariationVariation of Ar in %%%% of initial valueinitial valueinitial valueinitial value is equal
to:
J‚˜
‚˜
=
J ˜
˜
+
J¹˜
¹˜
+ (
J ˜
˜
*
J¹˜
¹˜
)
• ObjectObjectObjectObject of analysisanalysisanalysisanalysis is CCCCustomer Overdueustomer Overdueustomer Overdueustomer Overdue
PortfolioPortfolioPortfolioPortfolio of an IIIIndustrial Corporatendustrial Corporatendustrial Corporatendustrial Corporate having
used credit risk management modelcredit risk management modelcredit risk management modelcredit risk management model
described in previous chapterprevious chapterprevious chapterprevious chapter
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Customer Overdue Portfolio DecreaseCustomer Overdue Portfolio DecreaseCustomer Overdue Portfolio DecreaseCustomer Overdue Portfolio Decrease –––– GranularGranularGranularGranular
View fromView fromView fromView from JunJunJunJun----AugAugAugAug 2014 to2014 to2014 to2014 to OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015
45
Days of Delay and OverdueDays of Delay and OverdueDays of Delay and OverdueDays of Delay and Overdue
Amount for Invoices JunAmount for Invoices JunAmount for Invoices JunAmount for Invoices Jun----Aug 2014Aug 2014Aug 2014Aug 2014
Amount for Invoices JunAmount for Invoices JunAmount for Invoices JunAmount for Invoices Jun----Aug 2014Aug 2014Aug 2014Aug 2014
Amount for InvoicesAmount for InvoicesAmount for InvoicesAmount for Invoices OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015
Amount for InvoicesAmount for InvoicesAmount for InvoicesAmount for Invoices OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015
Customer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio Jun----
AugAugAugAug 2014201420142014
Customer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio Jun----
AugAugAugAug 2014201420142014
Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio OctOctOctOct----
Dec 2015Dec 2015Dec 2015Dec 2015
Customer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue PortfolioCustomer Overdue Portfolio OctOctOctOct----
Dec 2015Dec 2015Dec 2015Dec 2015
"v−DelayintermsofDelayintermsofDelayintermsofDelayintermsof
DaysDaysDaysDays
ºv ---- Overdue Amount (EUR)Overdue Amount (EUR)Overdue Amount (EUR)Overdue Amount (EUR)
•v = "v ∗ ºv
DaysDaysDaysDays
DaysDaysDaysDays
DaysDaysDaysDays
•v = "v ∗ ºv
0
50
100
150
200
250
300
0
50
100
150
200
250
300
-50
0
50
100
150
200
250
300
-50
0
50
100
150
200
250
300
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