Credit risk management for industrial corporates

1October 2016

Fabiano De RosaP&C and Risk Manager

Marco BerizziChief Financial Officer

Credit Risk Management Credit Risk Management Credit Risk Management Credit Risk Management for Industrial for Industrial for Industrial for Industrial CorporatesCorporatesCorporatesCorporates

From Nobel Prize Merton Model and Basel Committee Framework to Pragmatic Approach for Industrial Sector

ObjectiveObjectiveObjectiveObjective

• Presentation of Credit Risk Management Presentation of Credit Risk Management Presentation of Credit Risk Management Presentation of Credit Risk Management

Theoretical FrameworkTheoretical FrameworkTheoretical FrameworkTheoretical Framework

• Focus on Specific Aspects for Industrial Focus on Specific Aspects for Industrial Focus on Specific Aspects for Industrial Focus on Specific Aspects for Industrial

CorporatesCorporatesCorporatesCorporates

• Impact Measurement of Credit Impact Measurement of Credit Impact Measurement of Credit Impact Measurement of Credit Risk Risk Risk Risk

Management Management Management Management on Corporate on Corporate on Corporate on Corporate Customer Customer Customer Customer

Portfolio EfficiencyPortfolio EfficiencyPortfolio EfficiencyPortfolio Efficiency

2

[email protected]@gmail.com

Marco BerizziMarco BerizziMarco BerizziMarco Berizzi

AgendaAgendaAgendaAgenda

• A Standard Credit Risk Model for a Financial A Standard Credit Risk Model for a Financial A Standard Credit Risk Model for a Financial A 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


• D(Vt, t) = Vt e-δ(T-t) N(-d1) + BP(t,T) N(d2)33333333

Corporate Debt Value acknowledging Credit Corporate Debt Value acknowledging Credit Corporate Debt Value acknowledging Credit Corporate Debt Value acknowledging Credit Risk in pioneering Merton ModelRisk in pioneering Merton ModelRisk in pioneering Merton ModelRisk in pioneering Merton Model

4

• Corporate Debt valuation Corporate Debt valuation Corporate Debt valuation Corporate Debt valuation acknowledges in a structured and scientific manner credit risk credit risk credit risk credit risk concept concept concept concept in pioneering Noble Noble Noble Noble PrizePrizePrizePrize Merton modelMerton modelMerton modelMerton model

• Corporate Debt value Corporate Debt value Corporate Debt value Corporate Debt value is not not not not the mere discounted rate discounted rate discounted rate discounted rate of future cash flow future cash flow future cash flow future cash flow but incorporates incorporates incorporates incorporates a put option put option put option put option modelling credit risk credit risk credit risk credit risk arising from firm default eventfirm default eventfirm default eventfirm default event

• In this way corporate debt corporate debt corporate debt corporate debt

value value value value D(D(D(D(VVVVtttt, t), t), t), t) is at any dateany dateany dateany date

evaluated as the sumsumsumsum of a zero zero zero zero

coupon bondcoupon bondcoupon bondcoupon bond P(P(P(P(t,Tt,Tt,Tt,T)))) and a short short short short

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 coupon zero coupon zero coupon zero coupon

bond face value bond face value bond face value bond 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 ComponentsCorporate Debt Value Corporate Debt Value Corporate Debt Value Corporate Debt Value and and and and

Credit RiskCredit RiskCredit RiskCredit RiskCorporate Debt Value Corporate Debt Value Corporate Debt Value Corporate Debt Value and and and and

Credit RiskCredit RiskCredit RiskCredit Risk

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-

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[email protected]@[email protected]

Corporate Debt Value Derivation using Black Corporate Debt Value Derivation using Black Corporate Debt Value Derivation using Black Corporate Debt Value Derivation using Black ---- Scholes Scholes Scholes Scholes ---- Merton Formula Merton Formula Merton Formula Merton 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 factor discount factor discount factor discount 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 value zero coupon bond face value zero coupon bond face value zero coupon bond face value BBBB is equal to:

22222222

Put(Vt , B) = e-r(T-t) E Q (max(B – V�;);0)Put Put Put Put is evaluated as discounted average discounted average discounted average discounted average of possible pay offs pay offs pay offs pay offs at expiration dateexpiration dateexpiration dateexpiration date T T T T given by difference difference difference difference between zero coupon bond face value zero coupon bond face value zero coupon bond face value zero coupon bond face value BBBB and firm asset value firm asset value firm asset value firm asset value !" . Payoff . Payoff . Payoff . Payoff is zero zero zero zero if

!" > B > B > B > B and is positive positive positive positive if !" < B< B< B< B. Application of Black . Application of Black . Application of Black . Application of Black ---- Scholes Scholes Scholes Scholes ---- Merton formula Merton formula Merton formula Merton formula for option option option option

pricing pricing pricing pricing allows to express express express express option option option option 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 sum sum sum sum of a zero coupon bondzero coupon bondzero coupon bondzero coupon bond and a short positionshort positionshort positionshort positionwithin a put option put option put option put 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 / 34 54 (�16)

5 �16 ) d& = ( *+ ,-. / 012 1 3

4 54 (�16)5 �16 )

In dddd1111 and dddd2 2 2 2 , 7 is volatility is volatility is volatility is volatility of firm asset firm asset firm asset firm asset which is modelled modelled modelled modelled through following equationfollowing equationfollowing equationfollowing equation:

dVt = (r-δ)Vt dt + σVt dWt VT = Vt e(01213454)(�16)/5(:; 1:- )⇒

where WWWWtttt is a Brownian motionBrownian motionBrownian motionBrownian motion under risk neutral probability risk neutral probability risk neutral probability risk neutral probability QQQQ

For mathematical

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Brownian Motion and Geometric Brownian Brownian Motion and Geometric Brownian Brownian Motion and Geometric Brownian Brownian Motion and Geometric Brownian Motion DefinitionMotion DefinitionMotion DefinitionMotion Definition

6

Geometric Brownian Geometric Brownian Geometric Brownian Geometric Brownian MotionMotionMotionMotionBrownian MotionBrownian MotionBrownian MotionBrownian Motion

W6

Timet=0 Timet=0

V=

V6

• A standard Brownian motion standard Brownian motion standard Brownian motion standard Brownian motion is described described described described

as a Wiener process Wiener process Wiener process Wiener process WWWW which is a

continuous-time stochastic process stochastic process stochastic process stochastic 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 motion Geometric Brownian motion Geometric Brownian motion Geometric Brownian motion is a

continuous-time stochastic stochastic stochastic stochastic process process process process VVVVwith 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

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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 risk Credit risk Credit risk Credit risk is assessed assessed assessed assessed in terms of default probability default probability default probability default probability and loss given defaultloss given defaultloss given defaultloss given default

• Default (Default (Default (Default (DDDD) ) ) ) is defined as the event event event event for which firm asset value firm asset value firm asset value firm asset value !" is lower lower lower lower than debt debt debt debt BBBB at expiration date expiration date expiration date expiration date TTTT

• Application Application Application Application of Black of Black of Black of Black / Scholes / Merton / Scholes / Merton / Scholes / Merton / Scholes / Merton formula formula formula formula 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 default Loss given default Loss given default Loss given default is defined as 1 1 1 1 ---- recovery rate recovery rate recovery rate recovery rate of debt value debt value debt value debt value in case of default eventdefault eventdefault eventdefault event

• Application of Black / Scholes / Merton formula Application of Black / Scholes / Merton formula Application of Black / Scholes / Merton formula Application of Black / Scholes / Merton formula allows to quantify loss given default loss given default loss given default loss given default as it follows:

LGD = E Q ( G;H | V� < B) = 1 - '

H Vt e(012)(�16) N 1J3N 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 BV6 − 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)

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• KMV model KMV model KMV model KMV model builds up an effective approach effective approach effective approach effective approach aimed at assessing credit risk starting assessing credit risk starting assessing credit risk starting assessing credit risk starting from Merton model Merton model Merton model Merton model assumptions and main findings. With respect to Merton Merton Merton Merton model, KMV KMV KMV KMV model does not stay not stay not stay not stay in a risk neutral risk neutral risk neutral risk neutral environment (i.e. O is used and not rrrr), replacesreplacesreplacesreplaces normal distribution normal distribution normal distribution normal distribution probabilityprobabilityprobabilityprobability of default default default default with an empirical one based empirical one based empirical one based empirical one based on distance distance distance distance from default measure default measure default measure default measure and fine fine fine fine

tune concept tune concept tune concept tune concept of default point default point default point default point which no longer coincides no longer coincides no longer coincides no longer coincides with debt value debt value debt value debt value BBBB but with:

dabs= EP V� − d∗ drel = RS G; 1J∗5

d∗= SB + '& LBwhere 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 measure Distance from default measure Distance from default measure Distance from default measure is calculated in absolute absolute absolute absolute terms and relative relative relative relative 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


Vteμ(T−t)

dabsdN

For mathematical

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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)

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Expected Default FrequencyExpected Default FrequencyExpected Default FrequencyExpected Default Frequency

• KMV model substitutes normal distribution function KMV model substitutes normal distribution function KMV model substitutes normal distribution function KMV model substitutes normal distribution function NNNN used to calculate probability probability probability probability of default default default default

with an empirically empirically empirically empirically determined distribution function distribution function distribution function distribution function called expected default frequency expected default frequency expected default frequency expected default frequency ---- EDFEDFEDFEDFEDF

dN = Distance from default0

• EDFEDFEDFEDF is is is is a forwardforwardforwardforward----looking measure looking measure looking measure looking measure of actual probability actual probability actual probability actual probability of default default default default and is firm specificis firm specificis firm specificis firm specific

• In the light of historical information historical information historical information historical information on a large sample large sample large sample large sample of firmsfirmsfirmsfirms, EDFEDFEDFEDF estimate estimate estimate estimate is based based based based on the

proportion proportion proportion proportion of firms firms firms firms with a given default distance given default distance given default distance given default distance which actually defaulted defaulted defaulted defaulted after one yearone yearone yearone year


Credit Risk Definition within KMV Model for a Credit Risk Definition within KMV Model for a Credit Risk Definition within KMV Model for a Credit Risk Definition within KMV Model for a Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio

10


• KMV model KMV model KMV model KMV model with some integrations is able to support also credit risk management credit risk management credit risk management credit risk management in case of a loan portfolio loan portfolio loan portfolio loan portfolio

• When dealing with a loan portfolioloan portfolioloan portfolioloan portfolio, the main aspect aspect aspect aspect to be focused on is default correlation default correlation default correlation default correlation among single loans single loans single loans single loans able to concentrate concentrate concentrate concentrate dramatically probability probability probability probability on a few number of scenariosscenariosscenariosscenarios

• KMV model KMV model KMV model KMV model for loan portfolio loan portfolio loan portfolio loan portfolio is based on Merton model Merton model Merton model Merton model hypothesis and as for single loan

model does not stay not stay not stay not stay in a risk neutral risk neutral risk neutral risk neutral environment (i.e. O is used and not \)• Firm asset is modeled modeled modeled modeled through following following following following equationequationequationequation:

W^tn = ρ Yt + 1 − ρ ε6+ under historical historical historical historical probabilityprobabilityprobabilityprobability a with n = 1,...,N

dVnt = μVnt dt + σnVntdW^nt with n = 1,...,N• Risk source Risk source Risk source Risk source for each loan each loan each loan each loan is given by a combination combination combination combination of a systematic risk factor systematic risk factor systematic risk factor systematic risk factor (state of economy) affecting all firms affecting all firms affecting all firms affecting all firms and an idiosyncratic firm risk factor idiosyncratic firm risk factor idiosyncratic firm risk factor idiosyncratic firm risk factor as it follows:

where bc ,,,, dce ,...,,...,,...,,..., dcf are independent standard normally independent standard normally independent standard normally independent standard normally distributed variables and g ∈ i, e is the

correlation rate correlation rate correlation rate correlation rate among firm assetsfirm assetsfirm assetsfirm assets – «passing through» common element Y – controlling moreover the proportionproportionproportionproportion between systematic systematic systematic systematic and idiosyncratic idiosyncratic idiosyncratic idiosyncratic factorsfactorsfactorsfactors

• For a large homogeneous portfolio large homogeneous portfolio large homogeneous portfolio large homogeneous portfolio of loans loans loans loans with same probability same probability same probability same probability of default p not default p not default p not default p not

dominated dominated dominated dominated 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)nk+n' Dn ∈ 0,1 P(L(k) < x ) = N( ('1o)kp3 q 1kp3 ro )

where wwww(N)(N)(N)(N)nnnn are portfolio weights portfolio weights portfolio weights portfolio weights and DDDDnnnn are default eventdefault eventdefault eventdefault event variables with possible value i or e

For mathematical

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Relationship between Default Correlation and Relationship between Default Correlation and Relationship between Default Correlation and Relationship between Default Correlation and Credit RiskCredit RiskCredit RiskCredit Risk

11

L = ∑ Dnk+n'

0 100L

100% =195%

5%

Perfect Perfect Perfect Perfect Correlation ScenarioCorrelation ScenarioCorrelation ScenarioCorrelation ScenarioNo Correlation ScenarioNo Correlation ScenarioNo Correlation ScenarioNo Correlation Scenario

• For a portfolio portfolio portfolio portfolio of f loans perfectly loans perfectly loans perfectly loans perfectly

independent independent independent independent ρ = 0 with same probability same probability same probability same probability

of default default default default pppp, number number number number of default L default L default L default L is given by:

where L is a binomial variable binomial variable binomial variable binomial variable V(f, v) with following probability mass functionprobability mass functionprobability mass functionprobability mass function:

f(k, N, p)=P(L=k)= Nk px(1 − p)k1x with k=0,1,...,N

• For a portfolio portfolio portfolio portfolio of f loans perfectly loans perfectly loans perfectly loans perfectly

dependent dependent dependent dependent ρ = 1 with same probability same probability same probability same probability of

default default default default pppp, number number number number of default L default L default L default L probability probability probability probability mass mass mass mass functionfunctionfunctionfunction is given by:

f(k, N, p)=P(L=k)=z p with k = N1 − 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 only null probability on extreme events while perfect correlation concentrates probability on only null probability on extreme events while perfect correlation concentrates probability on only null 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” event two extreme events respectively “default of all loans” event and “default of no loans” event two extreme events respectively “default of all loans” event and “default of no loans” event two 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%


Loss Definition for a single Loan Loss Definition for a single Loan Loss Definition for a single Loan Loss Definition for a single Loan

12

Firm Asset and Loan Loss Firm Asset and Loan Loss Firm Asset and Loan Loss Firm Asset and Loan Loss Expected Loss and Expected Loss and Expected Loss and Expected Loss and Unexpected LossUnexpected LossUnexpected LossUnexpected LossExpected Loss and Expected Loss and Expected Loss and Expected Loss and Unexpected LossUnexpected LossUnexpected LossUnexpected Loss

B = default point

Time

Firm Asset

Tt

Vt

Probability Density of firm asset at expiration date


Vte(r−|)(T−t)

P (VT < B)

• Loan loss Loan loss Loan loss Loan loss is triggered by default event default event default event default event for which firm asset firm asset firm asset firm asset is lower lower lower lower than loan face value loan face value loan face value loan face value at expiration expiration expiration expiration datedatedatedate

• Loan Loan Loan Loan loss loss loss loss ((((jj) ) ) ) is given by product of default event default event default event default event

((((e !"}V )))), loss given default rateloss given default rateloss given default rateloss given default rate ((((j~�)))) and exposure exposure exposure exposure

at defaultat defaultat defaultat default ((((�� = V)))) as it follows:LL = 1 G;}R�� * LGD * EAD

Expected Loss

- ELUn-expected Loss

- UL> Quant

0 EADLoan Loss

Loan Loss (LL)Probability Density

Expected Loss Rate

- ELRUn-

expected Loss Rate

- ULR

> Quant

0 1e !"}�� * LGD

Loan Loss rate

(LLR) Probability Density

e !"}�� * LGD

where

1 G;}R�� = � 1 if V� < EAD0 if V� � EAD

For mathematical

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Loss Definition for a Loan Portfolio Loss Definition for a Loan Portfolio Loss Definition for a Loan Portfolio Loss Definition for a Loan Portfolio

13

Firm Assets and Portfolio Loan Loss Firm Assets and Portfolio Loan Loss Firm Assets and Portfolio Loan Loss Firm Assets and Portfolio Loan Loss Expected Loss and Expected Loss and Expected Loss and Expected Loss and Unexpected LossUnexpected LossUnexpected LossUnexpected LossExpected Loss and Expected Loss and Expected Loss and Expected Loss and Unexpected LossUnexpected LossUnexpected LossUnexpected Loss

• Portfolio Loan loss Portfolio Loan loss Portfolio Loan loss Portfolio Loan loss is triggered by sum sum sum sum of default default default default events events events events for which each firm asset firm asset firm asset firm asset is lower lower lower lower than each loan face value loan face value loan face value loan face value at expiration expiration expiration expiration datedatedatedate

• Portfolio Loan Portfolio Loan Portfolio Loan Portfolio Loan loss loss loss loss ((((PLPLPLPL) ) ) ) is given by product product product product 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 * EADportExpected 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


Vteμ(T−t)

dabs

where

L(k) = ∑ w(N)nk+n' Dn ∈ 0,1

Expected Loss Rate

- ELRUn-

expected Loss Rate

- ULR

> Quant

0 1L k * LGD

Portfolio Loss Rate

(PLR) Probability Density

L(k) * LGD

For mathematical

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Expected Loss Definition for a single Loan Expected Loss Definition for a single Loan Expected Loss Definition for a single Loan Expected Loss Definition for a single Loan

14

Expected Loss and Expected Expected Loss and Expected Expected Loss and Expected Expected Loss and Expected Loss Rate Loss Rate Loss Rate Loss Rate DerivationDerivationDerivationDerivation


Expected Loss and Expected Expected Loss and Expected Expected Loss and Expected Expected Loss and Expected Loss Rate DefinitionLoss Rate DefinitionLoss Rate DefinitionLoss Rate Definition


ELR = E�(1 G;}R�� ∗ LGD)

EL = E�(LL) = E�(1 G;}R�� ∗ LGD ∗ EAD)

Expected Loss


- UL> Quant

0 EAD

Loan Loss (LL)Probability Density

Loan Loss

Expected LossExpected LossExpected LossExpected Loss

Expected Loss Rate Expected Loss Rate Expected Loss Rate Expected Loss Rate

where

• 1 G;}R�� = � 1 if V� < EAD0 if V� � EAD

• Q is risk neutral probabilityExpected Loss Rate

- ELRUn-

expected Loss Rate

- ULR

> Quant

0 1e !"}�� * LGD

Probability Density of

e !"}�� * LGD

• Expected Loss Rate Expected Loss Rate Expected Loss Rate Expected Loss Rate is equal to:

ELR = LGD * E�(1 G;}R�� ) = PD * LGDPD * LGDPD * LGDPD * LGD

• Given linearity linearity linearity linearity of � , it gives:

• Expected Loss Expected Loss Expected Loss Expected Loss is equal to:

EL = LGD * EAD * E�(1 G;}R�� ) =PD * LGD PD * LGD PD * LGD PD * LGD * * * * EADEADEADEAD


where

• 1 G;}R�� = � 1 if V� < EAD0 if V� � EAD

• Q is risk neutral probability

For mathematical

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Expected Loss Expected Loss Expected Loss Expected Loss Definition Definition Definition Definition for for for for a a a a Loan PortfolioLoan PortfolioLoan PortfolioLoan Portfolio

15

ELR = EP(L(k) ∗ LGD)

EL = EP(PL) = EP(L(k) ∗ LGD ∗ EADport)

Expected Loss Rate

- ELRUn-

expected Loss Rate

- ULR

> Quant

0 1L k * LGD

Probability Densityof

L(k) * LGD

Expected Loss


- UL> Quant

0 EADportPortfolio Loss

Portfolio Loss







Expected Loss Rate Expected Loss Rate Expected Loss Rate Expected Loss Rate • Expected Loss Rate Expected Loss Rate Expected Loss Rate Expected Loss Rate is equal to:


• Expected Loss Expected Loss Expected Loss Expected Loss is equal to:


ELR = LGD * EP(L(k)) = p * LGDp * LGDp * LGDp * LGD

EL = LGD * EADport * EP(L(k)) =p * LGD p * LGD p * LGD p * LGD * * * * EADEADEADEADportportportport

where

• L(k) = ∑ w(N)nk+n' Dn ∈ 0,1• P is historical probability

where

• L(k) = ∑ w(N)nk+n' Dn ∈ 0,1• P is historical probability

For mathematical

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UnUnUnUn----Expected Loss Definition for a Loan Expected Loss Definition for a Loan Expected Loss Definition for a Loan Expected Loss Definition for a Loan PortfolioPortfolioPortfolioPortfolio

16

UnUnUnUn----Expected Loss and Expected Loss and Expected Loss and Expected Loss and UnUnUnUn----Exp. Loss Rate Definition Exp. Loss Rate Definition Exp. Loss Rate Definition Exp. Loss Rate Definition UnUnUnUn----Expected Loss and Expected Loss and Expected Loss and Expected Loss and

UnUnUnUn----Exp. Loss Rate Definition Exp. Loss Rate Definition Exp. Loss Rate Definition Exp. Loss Rate Definition

Expected Loss


- UL> Quant

0 EADport

Portfolio Loss


Portfolio Loss

UnUnUnUn----Expected LossExpected LossExpected LossExpected Loss

UnUnUnUn----Expected Loss Rate Expected Loss Rate Expected Loss Rate Expected Loss Rate

UnUnUnUn----Expected Loss and Expected Loss and Expected Loss and Expected Loss and UnUnUnUn----Exp. Loss Rate Exp. Loss Rate Exp. Loss Rate Exp. Loss Rate Derivation Derivation Derivation Derivation UnUnUnUn----Expected Loss and Expected Loss and Expected Loss and Expected Loss and

UnUnUnUn----Exp. Loss Rate Exp. Loss Rate Exp. Loss Rate Exp. Loss Rate Derivation Derivation Derivation Derivation

• 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)nk+n' Dn ∈ 0,1

P(L(k) < x ) = N( '1okp3 q 1kp3 ro )

• After definition of � confidence level confidence level confidence level confidence level (i.e. α = 99.9%), we have:

P(L(k)< q� )=α � N( '1okp3 �� 1kp3 ro )

q� � N( o kp3 � /kp3 r'1o )

• Inversion Inversion Inversion Inversion of the above formula above formula above formula above formula gives quantilequantilequantilequantile:

ULR = N( o kp3 � /kp3 r'1o ) * LGD – (p*LGD)

• Subtracting Subtracting Subtracting Subtracting ELRELRELRELR, brings to ULRULRULRULR::::ELR + ULR = N( o kp3 � /kp3 r

'1o ) ∗ LGD• Multiplication Multiplication Multiplication Multiplication for LGD LGD LGD LGD gives ((((ELR +ULRELR +ULRELR +ULRELR +ULR):):):):

• Multiplication Multiplication Multiplication Multiplication for EAD brings to ULULULUL: UL = N( o kp3 � /kp3 r

'1o ) ∗ LGD – (p∗LGD) * EADport

Expected Loss Rate

- ELRUn-

expected Loss Rate

- ULR

> Quant

0 1L k * LGD

Probability Densityof

L(k) * LGD

Note: where P is historical probability

For mathematical

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Portfolio Default Rate according to Loan Portfolio Default Rate according to Loan Portfolio Default Rate according to Loan Portfolio Default Rate according to Loan Correlation LevelCorrelation LevelCorrelation LevelCorrelation Level

17

Default correlation impact on credit Default correlation impact on credit Default correlation impact on credit Default correlation impact on credit riskriskriskrisk

Default correlation impact on credit Default correlation impact on credit Default correlation impact on credit Default correlation impact on credit riskriskriskrisk

• Probability Density Function Probability Density Function Probability Density Function Probability Density Function f(x) of Default Rate Default Rate Default Rate Default Rate

variable L(k) ---- parametrized to same unconditional probability of default - assumes different forms different forms different forms different forms according to different default different default different default different default

correlation values correlation values correlation values correlation values g for a portfolioportfolioportfolioportfolio of loansloansloansloans::::

P(L(k) < x) = N( '1o kp3 q 1kp3 ro )

α � N( '1okp3 �� 1kp3 ro ) q� � N( o kp3 � /kp3 r

'1o )

• Leptokurtosis Leptokurtosis Leptokurtosis Leptokurtosis effect implies that for a given given given given

confidence level confidence level confidence level confidence level α relative quantile quantile quantile quantile q� increases increases increases increases dramatically

= Default Rate Prob. Density Function with ρ= 10%= Default Rate Prob. Density Function with ρ= 20%= Default Rate Prob. Density Function with ρ= 30%

Probability Density Functions of

Default Rate variable L(k) with unconditional probability of default

p = 5%

Portfolio Default Rate > Quant q�



Portfolio Default Rate Density Portfolio Default Rate Density Portfolio Default Rate Density Portfolio Default Rate Density with different correlationwith different correlationwith different correlationwith different correlation

Portfolio Default Rate Density Portfolio Default Rate Density Portfolio Default Rate Density Portfolio Default Rate Density with different correlationwith different correlationwith different correlationwith different correlation

f x = 1 − ρρ e

'& kp3(q) 41 '

&o k(r)p31 '1o k(q)p3 4

where relative Distribution FunctionDistribution FunctionDistribution FunctionDistribution Function is

• High value High value High value High value of default default default default correlation correlation correlation correlation ρ causes leptokurtosis leptokurtosis leptokurtosis leptokurtosis effect that’s to say a shift shift shift shift of probability mass probability mass probability mass probability mass into the tail tail tail tail of density functiondensity functiondensity functiondensity function

• This means that This means that This means that This means that for a given confidence level given confidence level given confidence level given confidence level αwith same value of LGD LGD LGD LGD and EADEADEADEAD, the sum of EL EL EL EL and UL UL UL UL tends tends tends tends to increase increase increase increase strongly

For a given αFor a given α

0 1L(k)5% = 0.05

For mathematical

09


09

1 1 1 1 ---- �……


• Default correlation Default correlation Default correlation Default correlation g ∈ 0,1 for a portfolio portfolio portfolio portfolio

of loans loans loans loans with same probability same probability same probability same probability of default p default p default p default p is the correlation rate correlation rate correlation rate correlation rate among respective firm assetsfirm assetsfirm assetsfirm assets

• From an empirical point of view, a low low low low probability probability probability probability of default p default p default p default p implies a high high high high

default correlation rate default correlation rate default correlation rate default correlation rate g within portofolio portofolio portofolio portofolio

of loans loans loans loans while a high high high high probability probability probability probability of default default default default ppppimplies a low default low default low default low default correlation correlation correlation correlation g

• More specifically for probability probability probability probability of default default default default

p = 0%, default default default default correlation ratecorrelation ratecorrelation ratecorrelation rate ρ � ρ�P =24% and for probability probability probability probability of default default default default p = 100%, default correlation ratedefault correlation ratedefault correlation ratedefault correlation rate ρ � ρ�P = 12%

• From a mathematical point of view default default default default

correlation ratecorrelation ratecorrelation ratecorrelation rate ρ is a weighted averageweighted averageweighted averageweighted average of

ρ�P and ρ�P where the weights weights weights weights are

exponential functions exponential functions exponential functions exponential functions of pppp as shown below:

Default Correlation Default Correlation Default Correlation Default Correlation Estimate for a Estimate for a Estimate for a Estimate 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��)('1 �p��) + 24% ∗ '1('1 �p��)

('1 �p��)

0%

5%

10%

15%

20%

25%

30%Loan Correlation Loan Correlation Loan Correlation Loan Correlation ---- g

Default Probability Default Probability Default Probability Default Probability ---- pppp

Default Correlation EstimateDefault Correlation EstimateDefault Correlation EstimateDefault Correlation EstimateDefault Default Default Default Correlation Correlation Correlation Correlation / Probability / Probability / Probability / Probability Relationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolioDefault Default Default Default Correlation Correlation Correlation Correlation / Probability / Probability / Probability / Probability Relationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolioRelationship for a Loan portfolio


LGD Estimate LGD Estimate LGD Estimate LGD Estimate for a for a for a for 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 ParameterEndogenous Endogenous Endogenous Endogenous ApproachesApproachesApproachesApproachesEndogenous Endogenous Endogenous Endogenous ApproachesApproachesApproachesApproaches

• LGD LGD LGD LGD is considered as an exogenous parameter exogenous parameter exogenous parameter exogenous parameter with respect to asset asset asset asset

firm value firm value firm value firm value

• LGD LGD LGD LGD is estimated estimated estimated estimated through econometric econometric econometric econometric and statistical modelsstatistical modelsstatistical modelsstatistical models

• Majority Majority Majority Majority of estimation models estimation models estimation models estimation models aim at finding a link between link between link between link between LGD LGD LGD LGD and pppp as shown in the graphic graphic graphic graphic below:

• Downturn Downturn Downturn Downturn LGD LGD LGD LGD estimate estimate estimate estimate by FED FED FED FED suggests to use following formulaformulaformulaformula:

Downturn LGD = 0.08 + 0.92 LGD

• There are a couple couple couple couple of interesting interesting interesting interesting attempts attempts attempts attempts to define

LGD LGD LGD LGD endogenously endogenously endogenously endogenously

within asset firm asset firm asset firm asset firm value value value value and evolutionevolutionevolutionevolution

• First model First model First model First model conceived conceived conceived conceived by Schafer, Koivusaloand Becker is able to build up build up build up build up a closed closed closed closed

formula formula formula formula for LGD LGD LGD LGD considering asset asset asset asset firm portfolio firm portfolio firm portfolio firm portfolio performanceperformanceperformanceperformance

• Second model Second model Second model Second model conceived by Frye

treats treats treats treats LGD LGD LGD LGD analogously to default rate default rate default rate default rate using a VasicekVasicekVasicekVasicekdistribution distribution distribution distribution

2007

2006

20051987

200419931983

19971996

19921984

20032008

19911998

19992000

1986

19941995

1985

19821989

1988

19902001

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 = 0.6151y = -0.1069 In x + 0.0297 with R2 = 0.6287y = 0.1457 x-0.2801 with R2 = 0.6531

Recovery Rate

Recovery Rate

Recovery Rate

Recovery Rate

Default RateDefault RateDefault RateDefault Rate

Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association –––– US Corporate Bond US Corporate Bond US Corporate Bond US Corporate Bond Market Market Market Market –––– from 1982 to 1H 2009from 1982 to 1H 2009from 1982 to 1H 2009from 1982 to 1H 2009

[email protected]

Conceived Conceived Conceived Conceived by by by by Altman, Brady, Altman, Brady, Altman, Brady, Altman, Brady, SironiSironiSironiSironi and and and and RestiRestiRestiResti

Conceived Conceived Conceived Conceived by by by by Altman, Brady, Altman, Brady, Altman, Brady, Altman, Brady, SironiSironiSironiSironi and and and and RestiRestiRestiResti


LGD Estimate LGD Estimate LGD Estimate LGD Estimate for a for a for a for a Loan Portfolio within an Loan Portfolio within an Loan Portfolio within an Loan Portfolio within an Endogenous ApproachEndogenous ApproachEndogenous ApproachEndogenous Approach

20

LGD Definition for a Portfolio Loan within an Endogenous Approach LGD Definition for a Portfolio Loan within an Endogenous Approach LGD Definition for a Portfolio Loan within an Endogenous Approach LGD 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 Approach LGD Definition for a Portfolio Loan within an Endogenous Approach LGD Definition for a Portfolio Loan within an Endogenous Approach LGD Definition for a Portfolio Loan within an Endogenous Approach proposed by Fryeproposed by Fryeproposed by Fryeproposed by Frye

• A VasicekVasicekVasicekVasicek variable variable variable variable !! 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 ro ) with α ∈ (0,1) where q� = q� J�� 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 case In case In case In case β = p, VVVVVVVV is the conditional expected default rateconditional expected default rateconditional expected default rateconditional expected default rate EEEE j(f)| b / default rate default rate default rate default rate ---- j(f) variable:

• Suppose that the conditional expected loss rate conditional expected loss rate conditional expected loss rate conditional expected loss rate cELRcELRcELRcELR is a VasicekVasicekVasicekVasicek variable variable variable variable with β = ELR so we have that: cELR = N( o kp3 � /kp3 R�¥

'1o ) with α ∈ (0,1)• Now consider that

• Insert Insert Insert Insert last equation into cELRcELRcELRcELRand we obtain: cELR = N(N1' q� J�� 0D6� − kp3 r 1 kp3 R�¥

'1o )• Dividing by conditional expected default rate conditional expected default rate conditional expected default rate conditional expected default rate j(f) , we obtain conditional expected conditional expected conditional expected conditional expected loss given loss given loss given loss given

default rate default rate default rate default rate cELGDRcELGDRcELGDRcELGDR: : : : cELGDR = N(N1' q� J�� 0D6� − k) /q� J�� 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 charged spread charged spread charged spread charged on any

loan. All loans All loans All loans All loans belonging to same portfolio portfolio portfolio portfolio have the same probability same probability same probability same probability of default default default default p p p p and the same expected loss rate same expected loss rate same expected loss rate same expected loss rate ELRELRELRELR

For mathematical

09


09


Expected and UnExpected and UnExpected and UnExpected and Un----Expected Expected Expected Expected Loss Coverage Loss Coverage Loss Coverage Loss Coverage for for for for a Loan Portfolioa Loan Portfolioa Loan Portfolioa Loan Portfolio

21

• Deployment of a pricing strategy pricing strategy pricing strategy pricing strategy and tacticstacticstacticstactics which acknowledges expected loss making revenues expected loss making revenues expected loss making revenues expected 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 provision provision provision provision equal to

ELR ELR ELR ELR for each single uniteach single uniteach single uniteach single unit of loan loan loan loan

portfolio portfolio portfolio portfolio exposition exposition exposition exposition EADEADEADEADportportportport• Provision Provision Provision Provision for entire entire entire entire loan portfolio loan portfolio loan portfolio loan portfolio exposition exposition exposition exposition is given by multiplication multiplication multiplication multiplication

of ELR ELR ELR ELR by EADEADEADEADportportportport• Provision Provision Provision Provision is inserted in loan portfolio loan portfolio loan portfolio loan portfolio holder corporate P&L holder corporate P&L holder corporate P&L holder 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 loan loan loan loan

portfolio portfolio portfolio portfolio 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 portfolio loan portfolio loan portfolio loan portfolio is

respectively equal to:

UL = N( o kp3 � /kp3 r'1o ) ∗ LGD – (p∗LGD) * EADport

• Quantification of an equity capitalequity capitalequity capitalequity capital amountamountamountamount ((((KKKK))))for each single uniteach single uniteach single uniteach single unit of loan portfolio loan portfolio loan portfolio loan portfolio

exposition exposition exposition exposition EADEADEADEADportportportport conceived to secure business continuity business continuity business continuity business continuity of loan portfolio holder loan portfolio holder loan portfolio holder loan portfolio holder against severe impacts severe impacts severe impacts severe impacts deriving from unununun----expected lossexpected lossexpected lossexpected loss

• Equity capital Equity capital Equity capital Equity capital for entire loan portfolio entire loan portfolio entire loan portfolio entire loan portfolio

exposition exposition exposition exposition is given by multiplication multiplication multiplication multiplication of ULR ULR ULR ULR by EADEADEADEADportportportport

K = N( o kp3 � /kp3 r'1o ) ∗ LGD – (p∗LGD) * EADport


K Rate = N( o kp3 � /kp3 r'1o ) * LGD – (p*LGD)

For mathematical

09


09


Regulatory Capital required by Basel Regulatory Capital required by Basel Regulatory Capital required by Basel Regulatory 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 Basel Regulatory Capital by Basel Regulatory Capital by Basel Regulatory Capital by Basel Committee on Banking Supervision Committee on Banking Supervision Committee on Banking Supervision Committee on Banking Supervision


Equity Capital Amount KEquity Capital Amount KEquity Capital Amount KEquity Capital Amount K

• Regulatory capital rate Regulatory capital rate Regulatory capital rate Regulatory capital rate for each single uniteach single uniteach single uniteach single unit

of loan portfolio exposition loan portfolio exposition loan portfolio exposition loan portfolio exposition EADEADEADEADportportportport required by Basel Committee Basel Committee Basel Committee Basel Committee on Banking Banking Banking Banking Supervision Supervision Supervision Supervision for financial institutions financial institutions financial institutions financial 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 rate Equity capital rate Equity capital rate Equity capital rate for each single uniteach single uniteach single uniteach single unit of

loan portfolio exposition loan portfolio exposition loan portfolio exposition loan portfolio exposition EADEADEADEADportportportport owned by financial institution financial institution financial institution financial 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


09

with with

R =0.12 ∗ ('1R¬P 1=P� )('1R¬P 1= ) + 0.24 ∗ '1('1R¬P 1=P� )

('1R¬P 1= ) ρ =12% ∗ ('1 �p��)('1 �p��) + 24% ∗ '1('1 �p��)

('1 �p��)==


Regulatory Capital required by Basel Regulatory Capital required by Basel Regulatory Capital required by Basel Regulatory Capital required by Basel Committee vs Equity Capital amount K Committee vs Equity Capital amount K Committee vs Equity Capital amount K Committee 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 = 1given that the approach is supposed to be lean and straight forward

��

b(PD)=(0.11852-0.05478*Ln(PD)))^2

For mathematical

09


09



Equity Capital Amount KEquity Capital Amount KEquity Capital Amount KEquity Capital Amount K

B = [-PD*LGD] B = – (p*LGD) - PD = p

where where

==


Marco Marco Marco Marco BerizziBerizziBerizziBerizziFabiano De RosaFabiano De RosaFabiano De RosaFabiano De Rosa


• A Standard Credit Risk Model for a Financial Institution

• A Credit Risk Management Model for an A Credit Risk Management Model for an A Credit Risk Management Model for an A Credit Risk Management Model for an

Industrial CorporateIndustrial CorporateIndustrial CorporateIndustrial Corporate


• Bibliography

• Annex

24


Credit Risk Credit Risk Credit Risk Credit 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

AAAAAAAAAA


Organization and Risk GovernanceOrganization and Risk GovernanceOrganization and Risk GovernanceOrganization and Risk Governance----SupportSupportSupportSupport----ControlControlControlControl

26

SalesSalesSalesSales

Chief Executive Chief Executive Chief Executive Chief Executive Chief Executive Chief Executive Chief Executive Chief Executive

OfficerOfficerOfficerOfficer

Risk Risk Risk Risk Risk Risk Risk Risk ManagementManagementManagementManagement

Board of DirectorsBoard of DirectorsBoard of DirectorsBoard of Directors

Internal AuditInternal AuditInternal AuditInternal Audit

Risk Risk Risk Risk CommitteeCommitteeCommitteeCommitteeRisk Risk Risk Risk

CommitteeCommitteeCommitteeCommittee

ICTICTICTICTFinanceFinanceFinanceFinance

GovernanceGovernanceGovernanceGovernance

2 2 2 2 Management LayerManagement LayerManagement LayerManagement Layer

1 1 1 1 Management Management Management Management LayerLayerLayerLayer

SupportSupportSupportSupport

1111 Control LayerControl LayerControl LayerControl Layer

2222 Control LayerControl LayerControl LayerControl Layer

OrganizationOrganizationOrganizationOrganizationRisk Governance & Risk Governance & Risk Governance & Risk Governance &

ControlControlControlControlRisk Governance & Risk Governance & Risk Governance & Risk Governance &

ControlControlControlControl

AAAAAA


Customer Rating Customer Rating Customer Rating Customer 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 probability Unconditional probability Unconditional probability Unconditional probability of default default default default is articulated per rating grade rating grade rating grade rating grade and gives the average average average average percentage percentage percentage percentage of obligors obligors obligors obligors 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

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

%%%%%%%%

++++

----


Customer Rating Recognition Customer Rating Recognition Customer Rating Recognition Customer Rating Recognition –––– Main Main Main Main 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

GroupRevenues Revenues

EBITDA in % revenues

NFP / EBITDA EBITDA / i NFP / BV Other ratios

Full Year Accounts Interim Accounts

Financial Statement

Weighted Weighted Weighted Weighted Average of Average of Average of Average of SubSubSubSub----RatingRatingRatingRating

CreditCreditCreditCreditRatingRatingRatingRating

Weighted Weighted Weighted Weighted Average of Average of Average of Average of SubSubSubSub----RatingRatingRatingRating

CreditCreditCreditCreditRatingRatingRatingRating


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

RatingRatingRatingRating

AAA

AA

A

BBB

BB

B

CCC

CC

C

R

SD

D


• Request Request Request Request to customer of financial statementfinancial statementfinancial statementfinancial statement

• Most fresh Financial Most fresh Financial Most fresh Financial Most fresh Financial Statement Statement Statement Statement to be used

• Financial Statement Financial Statement Financial Statement Financial Statement accepted not older not older not older not older than 2 2 2 2 yearsyearsyearsyears

• SubSubSubSub----Rating Rating Rating Rating attribution of “CCC” “CCC” “CCC” “CCC” in case of no reply no reply no reply no reply by customer customer customer customer

Sub Rating Sub Rating Sub Rating Sub Rating –––– Customer Financial StatementCustomer Financial StatementCustomer Financial StatementCustomer Financial Statement

29

11111111

Financial StatementFinancial StatementFinancial StatementFinancial StatementBusiness and Business and Business and Business and Financial RatiosFinancial RatiosFinancial RatiosFinancial RatiosBusiness and Business and Business and Business and Financial RatiosFinancial RatiosFinancial RatiosFinancial Ratios

• RevenuesRevenuesRevenuesRevenues• EBITDA in % of EBITDA in % of EBITDA in % of EBITDA in % of revenuesrevenuesrevenuesrevenues

Ratio NatureRatio NatureRatio NatureRatio Nature

• Net Financial Position / Net Financial Position / Net Financial Position / Net Financial Position / EBITDAEBITDAEBITDAEBITDA

• EBITDA / net financial EBITDA / net financial EBITDA / net financial EBITDA / 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

Financial RatiosFinancial RatiosFinancial RatiosFinancial Ratios

Sub RatingSub RatingSub RatingSub Rating

Weighted Weighted Weighted Weighted Average of Average of Average of Average of 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

SubSubSubSub----RatingRatingRatingRating

AAA

AA

A

BBB

BB

B

CCC

CC

C

R

SD

D


Sub Rating Sub Rating Sub Rating Sub 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 DelaySub RatingSub RatingSub RatingSub Rating

Weighted Weighted Weighted Weighted Average of Average of Average of Average of RatiosRatiosRatiosRatios


Arithmetic Arithmetic Arithmetic Arithmetic Average of Average of Average of Average of

Overdue DelaysOverdue DelaysOverdue DelaysOverdue Delays

Weighted Average Weighted Average Weighted Average Weighted Average Weighted Average Weighted Average Weighted Average Weighted Average of Overdue of Overdue of Overdue of Overdue DelaysDelaysDelaysDelays

Overdue Amounts Overdue Amounts Overdue Amounts Overdue Amounts Overdue Amounts Overdue Amounts Overdue Amounts Overdue Amounts on a daily basison a daily basison a daily basison a daily basis

Overdue Volume Overdue Volume Overdue Volume Overdue Volume FrequencyFrequencyFrequencyFrequency

Overdue Value Overdue Value Overdue Value Overdue Value FrequencyFrequencyFrequencyFrequency

• n = n.° of invoices issued within a certain time interval

• T̄ = n.° of payment delay days of invoice j

IndicatorIndicatorIndicatorIndicatorIndicatorIndicatorIndicatorIndicator VariablesVariablesVariablesVariablesVariablesVariablesVariablesVariables

• T̄ = 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

± T̄n

+

¯n'

± T̄ ∗ I¯I+

+

¯n'

± T̄ ∗ I¯m

+

¯n'

± n¯n

+

¯n'

± I¯I+

+

¯n'


AAA

AA

A

BBB

BB

B

CCC

CC

C

R

SD

D


Sub Rating Sub Rating Sub Rating Sub 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 RatingCustomer Customer Customer Customer

Country RatingCountry RatingCountry RatingCountry RatingCustomer Customer Customer Customer

Country RatingCountry RatingCountry RatingCountry Rating

Political Political Political Political RiskRiskRiskRisk

Exchange Exchange Exchange Exchange Rate RiskRate RiskRate RiskRate Risk

Economic Economic Economic Economic RiskRiskRiskRisk

Sovereign Sovereign Sovereign Sovereign RiskRiskRiskRisk

Transfer Transfer Transfer Transfer RiskRiskRiskRisk

ComponentsComponentsComponentsComponents ComponentsComponentsComponentsComponents

Weighted Weighted Weighted Weighted Average of Average of Average of Average of ComponentsComponentsComponentsComponents


Country merit Country merit Country merit Country merit worthiness worthiness worthiness worthiness is

affected affected affected affected by political political political political riskriskriskrisk, economic risk economic risk economic risk economic risk and sovereign risksovereign risksovereign risksovereign risk

Political Political Political Political RiskRiskRiskRisk

Economic Economic Economic Economic RiskRiskRiskRisk

Sovereign Sovereign Sovereign Sovereign RiskRiskRiskRisk


AAA

AA

A

BBB

BB

B

CCC

CC

C

R

SD

D


Customer Rating Analysis Customer Rating Analysis Customer Rating Analysis Customer Rating Analysis –––– PD and Frequency PD and Frequency PD and Frequency PD and Frequency by Rating Gradeby Rating Gradeby Rating Gradeby Rating Grade

32

11111111

Customer Rating Customer Rating Customer Rating Customer Rating and PDand PDand PDand PD

Customer Rating Customer Rating Customer Rating Customer 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.°°°° of of of of 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.°°°° of of of of CustomersCustomersCustomersCustomers

Change Change Change Change in in in in commercial commercial commercial commercial / / / / risk policy risk policy risk policy risk policy can can can can

affect affect affect affect customer customer customer customer portfolio risk portfolio risk portfolio risk portfolio risk

levellevellevellevel

Change Change Change Change in in in in commercial commercial commercial commercial / / / / risk policy risk policy risk policy risk policy can can can can

affect affect affect affect customer customer customer customer portfolio risk portfolio risk portfolio risk portfolio risk

levellevellevellevel

RatingRatingRatingRating

AAA

AA

A

BBB

BB

B

CCC

CC

C

R

SD

D

PDPDPDPD0.015%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


Credit Risk Mitigation Credit Risk Mitigation Credit Risk Mitigation Credit Risk Mitigation –––– Instruments Instruments Instruments Instruments

33

22222222

• Payment terms Payment terms Payment terms Payment terms definition in the form of total total total total / partial prepartial prepartial prepartial pre----payment payment payment payment or payment upon receipt payment upon receipt payment upon receipt payment upon receipt of goods goods goods goods at company company company company warehousewarehousewarehousewarehouse

• Compensation Compensation Compensation Compensation of credit credit credit credit / debt positions debt positions debt positions debt positions in case customer customer customer customer is also a suppliersuppliersuppliersupplier

CRM InstrumentsCRM InstrumentsCRM InstrumentsCRM Instruments

Business instrumentsBusiness instrumentsBusiness instrumentsBusiness instruments

• Set up of an escrow account escrow account escrow account escrow account by a customer customer customer customer as guarantee guarantee guarantee guarantee of future paymentsfuture paymentsfuture paymentsfuture payments

• Issue of a guarantee guarantee guarantee guarantee / letter letter letter letter of credit credit credit credit by a financial institution financial institution financial institution financial institution to cover customer solvency customer solvency customer solvency customer solvency relative to detected transactionsdetected transactionsdetected transactionsdetected transactions

• Issue of a guarantee guarantee guarantee guarantee by public trade organizations public trade organizations public trade organizations public trade organizations to cover customer solvency customer solvency customer solvency customer solvency relative to detected transactionsdetected transactionsdetected transactionsdetected transactions

• Usage of factoring factoring factoring factoring (pro-soluto) and ABS ABS ABS ABS 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 risk credit risk credit risk credit risk connected with an identified poolpoolpoolpool of customer customer customer customer for all transactionsall transactionsall transactionsall transactions

Insurance instrumentsInsurance instrumentsInsurance instrumentsInsurance instruments

Pragmatic Pragmatic Pragmatic Pragmatic ApproachApproachApproachApproachPragmatic Pragmatic Pragmatic Pragmatic ApproachApproachApproachApproach

• Set up of a model model model model designed to acknowledge acknowledge acknowledge acknowledge all types of credit risk credit risk credit risk credit risk mitigation mitigation mitigation mitigation (CRM) instruments instruments instruments instruments and toolstoolstoolstools

• CRM instruments CRM instruments CRM instruments CRM instruments / tools tools tools tools are built up to have an impact impact impact impact directly directly directly directly on exposure at default exposure at default exposure at default exposure at default

(EAD) of customer customer customer customer

to privilege privilege privilege privilege a more pragmatic pragmatic pragmatic pragmatic approach …approach …approach …approach …

• … … … … also if it would be theoretically more more more more correct correct correct correct that CRM CRM CRM CRM instruments instruments instruments instruments trigger

customer customer customer customer LGD LGD LGD LGD or require usage usage usage usage of

guarantor guarantor guarantor guarantor [email protected]@gmail.com

Customer Credit Line Plafond and Payment Customer Credit Line Plafond and Payment Customer Credit Line Plafond and Payment Customer Credit Line Plafond and Payment Terms Terms Terms Terms –––– Recognition CriteriaRecognition CriteriaRecognition CriteriaRecognition Criteria

34

33333333

Customer Credit Line Plafond RecognitionCustomer Credit Line Plafond RecognitionCustomer Credit Line Plafond RecognitionCustomer Credit Line Plafond RecognitionCustomer Payment Customer Payment Customer Payment Customer Payment Terms RecognitionTerms RecognitionTerms RecognitionTerms RecognitionCustomer Payment Customer Payment Customer Payment Customer Payment Terms RecognitionTerms RecognitionTerms RecognitionTerms Recognition

• It would be theoretically more correct more correct more correct more correct to recognize payment recognize payment recognize payment recognize payment terms terms terms terms for a customer customer customer customer according to its specific its specific its specific its specific rating …rating …rating …rating …

• … … … … but to be more more more more pragmaticpragmaticpragmaticpragmatic….

• …. payment terms payment terms payment terms payment terms ---- for an existing customer ––––are maintained maintained maintained maintained constant constant constant constant and changed changed changed changed only according to managerial decision managerial decision managerial decision managerial decision and ...

• … positive payment positive payment positive payment positive payment

terms terms terms terms ---- equal to 30303030days days days days ---- for a new new new new customer customer customer customer are recognized recognized recognized recognized only after a trial period post trial period post trial period post trial 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 * ARwhere ARARARAR is an average expositionaverage expositionaverage expositionaverage exposition along a certain historical certain historical certain historical certain historical

time linetime linetime linetime line towards a customer customer customer customer in terms of account receivablesaccount receivablesaccount receivablesaccount receivables

- corresponding conceptually to EAD - and CCCC is a factorfactorfactorfactor which

is a functionfunctionfunctionfunction of ratingratingratingrating and ARARARARas shown in the following tableAR in K EURAR in K EUR

Rating

Rating

• 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 variation strong variation strong variation strong variation of ARARARAR and ratingratingratingrating occur


Customer Credit Customer Credit Customer Credit Customer Credit Line Plafond Line Plafond Line Plafond Line Plafond ---- QuantificationQuantificationQuantificationQuantification

35

33333333

Customer Credit Line Plafond Customer Credit Line Plafond Customer Credit Line Plafond Customer 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 Plafond Customer Credit Line Plafond Customer Credit Line Plafond Customer 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)

Standard

Standard

Standard

Standard

Adjusted

Adjusted

Adjusted

Adjusted

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%AAA AA A BBB BB B CCC CC C R SD D

9%

0%

8%5%

15%

21%

11%

27%

4%0% 0% 0%

AAA AA A BBB BB B CCC CC C R SD DCCCCLLLL is calculated using is calculated using is calculated using is calculated using

Exposure Exposure Exposure Exposure at default at default at default at default

adjusted (adjusted (adjusted (adjusted (EAD EAD EAD EAD AdjAdjAdjAdj) ) ) ) diminished by CRM diminished by CRM diminished by CRM diminished by CRM instruments / instruments / instruments / instruments / toolstoolstoolstools

CCCCLLLL is calculated using is calculated using is calculated using is calculated using


adjusted (adjusted (adjusted (adjusted (EAD EAD EAD EAD AdjAdjAdjAdj) ) ) ) diminished by CRM diminished by CRM diminished by CRM diminished by CRM instruments / instruments / instruments / instruments / toolstoolstoolstools

CCCCLLLL is calculated is calculated is calculated is calculated

using standard using standard using standard using standard Exposure Exposure Exposure Exposure at at at at

default default default default ((((EADEADEADEAD))))

CCCCLLLL is calculated is calculated is calculated is calculated

using standard using standard using standard using standard Exposure Exposure Exposure Exposure at at at at

default default default default ((((EADEADEADEAD))))


Credit Collecting PracticeCredit Collecting PracticeCredit Collecting PracticeCredit Collecting Practice

36

44444444

Overdue Invoice Overdue Invoice Overdue Invoice Overdue Invoice SituationSituationSituationSituation

Overdue Invoice Overdue Invoice Overdue Invoice Overdue Invoice SituationSituationSituationSituation

Credit Collecting ActionCredit Collecting ActionCredit Collecting ActionCredit Collecting Action

• Risk Manager Risk Manager Risk Manager Risk Manager (RM) sends sends sends sends 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 template reminder template reminder template reminder template -generated by corporate ERP – and a request request request request for explanation explanation explanation explanation to respective customer respective customer respective customer respective customer putting in cc respective Key Account Manager (KAM) / Country Manager (CM), AM and CFO

• RM asks asks asks asks referral KAM to organize organize organize organize a conference call conference call conference call conference call with respective customerrespective customerrespective customerrespective customer

• RM calls calls calls calls an internal meeting internal meeting internal meeting internal meeting with referral KAMKAMKAMKAM, CFO CFO CFO CFO and CEO CEO CEO CEO to find find find find a suitable solutionsuitable solutionsuitable solutionsuitable solution

• RM callscallscallscalls promptly an internal meeting internal meeting internal meeting internal meeting with referral KAMKAMKAMKAM, CFO CFO CFO CFO and CEO CEO CEO CEO in order to take a final decisionfinal decisionfinal decisionfinal decisionand to decide submission submission submission submission of a claim claim claim claim to insurance insurance insurance insurance companycompanycompanycompany

Days Days Days Days of delay delay delay delay relative to overdue invoices overdue invoices overdue invoices overdue invoices ––––

corresponding to overdue corresponding to overdue corresponding to overdue corresponding to overdue account receivables account receivables account receivables account receivables (AR) ----relative to a specific a specific a specific a specific

customercustomercustomercustomer

< 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 lower lower lower lower 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, RM RM RM RM adopts the same above actions without involving CEOsame above actions without involving CEOsame above actions without involving CEOsame above actions without involving CEO


7%

0%4% 3%

10%

18%12%

35%

6% 4%0% 0%

AAA AA A BBB BB B CCC CC C R SD D

0% 0% 3% 3% 7% 7%

21%

52%

0%7%


Exposition at Default Exposition at Default Exposition at Default Exposition 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)

Standard

Standard

Standard

Standard

Adjusted

Adjusted

Adjusted

Adjusted

100%

tot

100%

tot


adjusted (adjusted (adjusted (adjusted (EAD EAD EAD EAD AdjAdjAdjAdj) ) ) ) diminished by diminished by diminished by diminished by CRM CRM CRM CRM instruments / toolsinstruments / toolsinstruments / toolsinstruments / tools


adjusted (adjusted (adjusted (adjusted (EAD EAD EAD EAD AdjAdjAdjAdj) ) ) ) diminished by diminished by diminished by diminished by CRM CRM CRM CRM instruments / toolsinstruments / toolsinstruments / toolsinstruments / tools


((((EADEADEADEAD) not diminished ) not diminished ) not diminished ) not diminished

by by by by CRM instruments CRM instruments CRM instruments CRM instruments / tools/ tools/ tools/ tools


((((EADEADEADEAD) not diminished ) not diminished ) not diminished ) not diminished

by by by by CRM instruments CRM instruments CRM instruments CRM instruments / tools/ tools/ tools/ tools

Note: time of measurement is around end of [email protected]@[email protected]

Loss Given Default Loss Given Default Loss Given Default Loss Given Default ---- EstimateEstimateEstimateEstimate

38

55555555

Loss given DefaultLoss given DefaultLoss given DefaultLoss given Default

• This graph shows the association association association association of weighted average default rates weighted average default rates weighted average default rates weighted average default rates and recovery rates recovery rates recovery rates recovery rates over the period 1982198219821982----1H2009 1H2009 1H2009 1H2009 within US corporate bond US corporate bond US corporate bond US corporate bond market market market market using four bibibibi----variate regression variate regression variate regression variate regression specificationsspecificationsspecificationsspecifications

• These regressions regressions regressions regressions include linear linear linear linear quadratic logquadratic logquadratic logquadratic log---- linear linear linear linear and power power power power function function function function structuresstructuresstructuresstructures

• Proxy Proxy Proxy Proxy is given using log function log function log function log function :

• LGD LGD LGD LGD is estimated estimated estimated estimated using econometric relationship relationship relationship relationship between recovery rate recovery rate recovery rate recovery rate and default rate default rate default rate default rate

defined by Altman, Brady, Sironi and Resti analysis• Econometric relationship relationship relationship relationship between recovery rate recovery rate recovery rate recovery rate and default rate default rate default rate default 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

• LGD LGD LGD LGD estimate estimate estimate estimate is given by:

LGD = 1 − RR = 1 − (-0.1069 In x + 0.0297)

where y = RR and x = DR � p

LGD = 0.9703+0.1069 In p

2007

2006

20051987

200419931983

19971996

19921984

20032008

19911998

19992000

1986

19941995

1985

19821989

1988

19902001

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 = 0.6151y = -0.1069 In x + 0.0297 with R2 = 0.6287y = 0.1457 x-0.2801 with R2 = 0.6531

Recovery Rate (RR)

Recovery Rate (RR)

Recovery Rate (RR)

Recovery Rate (RR)

Default Rate (DR)Default Rate (DR)Default Rate (DR)Default Rate (DR)

Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association Recovery Rate / Default Rate Association ––––US Corporate Bond Market US Corporate Bond Market US Corporate Bond Market US Corporate Bond Market –––– from 1982 to from 1982 to from 1982 to from 1982 to

1H 20091H 20091H 20091H 2009


100%

tottottottot

Expected Loss Expected Loss Expected Loss Expected 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 Adj EADport Adj EADport Adj EADport 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 Loss Expected Loss Expected Loss Expected 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

• Total Total Total Total ELELELEL is given by the sum sum sum sum

of single single single single ELELELEL:Total EL = ∑ EL¯'&¯n' =

± p¯ ∗ LGD¯ ∗ EADr²06 ¯'&

¯n'

where j = 1, ..., 12 are rating clusterrating clusterrating clusterrating cluster

For mathematical

09


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%


EL EL EL EL is calculated is calculated is calculated is calculated

using Exposure using Exposure using Exposure using Exposure at at at at default default default default adjusted adjusted adjusted adjusted

((((EAD EAD EAD EAD AdjAdjAdjAdj))))

EL EL EL EL is calculated is calculated is calculated is calculated

using Exposure using Exposure using Exposure using Exposure at at at at default default default default adjusted adjusted adjusted adjusted



0% 0% 0% 0% 2% 4%

20%

68%

0%6%


UnUnUnUn----Expected Expected Expected Expected Loss Loss Loss Loss –––– EstimateEstimateEstimateEstimate

40

66666666

UnUnUnUn----Expected Expected Expected Expected 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 j j j j

(“portfolio”) with j = 1, ...,12 according to:

• Total ULTotal ULTotal ULTotal UL is given by the sum sum sum sum of single ULsingle ULsingle ULsingle UL:

Total UL = ∑ UL¯'&¯n' =


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


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

Loan Loan Loan Loan correlation correlation correlation correlation inside inside inside inside 12121212rating rating rating rating clusters clusters clusters clusters

g = (ge, … , ge´))))

Loan Loan Loan Loan correlation correlation correlation correlation inside inside inside inside 12121212rating rating rating rating clusters clusters clusters clusters

g = (ge, … , ge´))))

UUUUL is calculated L is calculated L is calculated L is calculated using Exposure using Exposure using Exposure using Exposure at at at at default default default default adjusted adjusted adjusted adjusted


UUUUL is calculated L is calculated L is calculated L is calculated using Exposure using Exposure using Exposure using Exposure at at at at default default default default adjusted adjusted adjusted adjusted



Coverage of Expected LossCoverage of Expected LossCoverage of Expected LossCoverage of Expected Loss

41

66666666

• Deployment of a pricing strategy pricing strategy pricing strategy pricing strategy and tacticstacticstacticstactics modelmodelmodelmodel supported by corporate ERPcorporate ERPcorporate ERPcorporate ERPor sales applicationsales applicationsales applicationsales application which acknowledges automatically ratingratingratingrating in the formulationformulationformulationformulation of quotationquotationquotationquotation and proposalproposalproposalproposal to a specific customercustomercustomercustomer and adds relative expected expected expected expected loss loss loss loss in cost structure listcost structure listcost structure listcost structure list in order to make revenues revenues revenues revenues 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 risk credit risk credit risk credit 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 credit standard credit standard credit standard credit

risk risk risk risk is inserted in corporate Profit & Loss corporate Profit & Loss corporate Profit & Loss corporate Profit & Loss table table table table acknowledging a possible future possible future possible future possible future burden burden burden burden and allowing also to gain taxation gain taxation gain taxation gain taxation shield shield shield shield

• Total PVTotal PVTotal PVTotal PV is given by the sum sum sum sum of single PVsingle PVsingle PVsingle PV:

For mathematical

09


09

Financial CoverageFinancial CoverageFinancial CoverageFinancial CoverageFinancial CoverageFinancial Coverage Business CoverageBusiness CoverageBusiness CoverageBusiness CoverageBusiness CoverageBusiness Coverage


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 risk credit risk credit risk credit risk brought by occurrence of extreme extreme extreme extreme events: events: events: events:

• Initial provision Initial provision Initial provision Initial provision ((((a!µj) ) ) ) for “not standard” “not standard” “not standard” “not standard”

credit risk credit risk credit risk credit risk is inserted in corporate Profit & corporate Profit & corporate Profit & corporate Profit & Loss tableLoss tableLoss tableLoss table acknowledging a possible possible possible possible future burden future burden future burden future burden and allowing also to gain gain gain gain taxation taxation taxation taxation shieldshieldshieldshield

• Subsequent annual provision Subsequent annual provision Subsequent annual provision Subsequent 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 variation cover annual variation cover annual variation cover annual variation of unununun----expected loss expected loss expected loss expected loss value value value value

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 ) ∗ LGD – (p∗LGD) * EADport

• Total Total Total Total KKKK or or or or a!µj is given by the sum sum sum sum of single single single single KKKK or or or or a!µj:

• Initial Equity Initial Equity Initial Equity Initial Equity capital buffercapital buffercapital buffercapital buffer ((((KKKK))))establishment within corporate Balance corporate Balance corporate Balance corporate Balance Sheet Sheet Sheet Sheet table permits to strengthen merit merit merit merit worthinessworthinessworthinessworthiness and relative rating rating rating rating facilitating relationship relationship relationship relationship with stakeholders stakeholders stakeholders stakeholders such as suppliers suppliers suppliers suppliers and providers providers providers providers of financefinancefinancefinance

• Subsequent annual / periodical Subsequent annual / periodical Subsequent annual / periodical Subsequent annual / periodical ∆ capital capital capital capital

bufferbufferbufferbuffer ((((KKKK) ) ) ) establishments permits to cover cover cover cover

annual variation annual variation annual variation annual variation of unununun----expected loss expected loss expected loss expected loss value

For mathematical

09


09

Financial Coverage (1Financial Coverage (1Financial Coverage (1Financial Coverage (1°°°° Option)Option)Option)Option) Financial Coverage Financial Coverage Financial Coverage Financial Coverage (2(2(2(2°°°° Option)Option)Option)Option)


Marco BerizziMarco BerizziMarco BerizziMarco BerizziFabiano De RosaFabiano De RosaFabiano De RosaFabiano De Rosa




• Impact of Credit Risk Impact of Credit Risk Impact of Credit Risk Impact of Credit Risk Management Model Management Model Management Model Management Model on on on on

Corporate Customer Portfolio EfficiencyCorporate Customer Portfolio EfficiencyCorporate Customer Portfolio EfficiencyCorporate Customer Portfolio Efficiency

• Bibliography

• Annex

43


Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease –––– Global Global Global Global 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 Overdue Customer Overdue Customer Overdue Customer Overdue PtfPtfPtfPtf. Variation. Variation. Variation. VariationCustomer Overdue Customer Overdue Customer Overdue Customer Overdue PtfPtfPtfPtf. Variation. Variation. Variation. Variation(%; Jun 2014-Dec 2015)

Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio DefinitionDefinitionDefinitionDefinition

Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio DefinitionDefinitionDefinitionDefinition

• Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio in a certaintime interval time interval time interval time interval is equal equal equal equal to:

Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer 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

effectGlobal effectGlobal effectGlobal effectGlobal effect

dArAr

dIrIr

(J��

* J¹�¹�

)dTrTr

Ar = Tr ∗ Irwhere:- 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 June 2014 June 2014 June 2014 June to December 2015December 2015December 2015December 2015

• Variation Variation Variation Variation of Ar is equal to:dAr = IrdTr + TrdIr + dTrdIr

• Variation Variation Variation Variation of Ar in % % % % of initial value initial value initial value initial value is equal

to:J��

= J��

+ J¹�¹�

+ (J��

* J¹�¹�

)

• Object Object Object Object of analysis analysis analysis analysis is CCCCustomer Overdue ustomer Overdue ustomer Overdue ustomer Overdue Portfolio Portfolio Portfolio Portfolio of an IIIIndustrial Corporate ndustrial Corporate ndustrial Corporate ndustrial Corporate having used credit risk management model credit risk management model credit risk management model credit risk management model described in previous chapterprevious chapterprevious chapterprevious chapter


Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease Customer Overdue Portfolio Decrease –––– Granular Granular Granular Granular View from View from View from View from JunJunJunJun----Aug Aug Aug Aug 2014 to 2014 to 2014 to 2014 to OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015

45

Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue Amount for Invoices JunAmount for Invoices JunAmount for Invoices JunAmount for Invoices Jun----Aug 2014Aug 2014Aug 2014Aug 2014Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue

Amount for Invoices JunAmount for Invoices JunAmount for Invoices JunAmount for Invoices Jun----Aug 2014Aug 2014Aug 2014Aug 2014Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue

Amount for Invoices Amount for Invoices Amount for Invoices Amount for Invoices OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue Days of Delay and Overdue

Amount for Invoices Amount for Invoices Amount for Invoices Amount for Invoices OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015

Customer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio Jun----Aug Aug Aug Aug 2014201420142014

Customer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio JunCustomer Overdue Portfolio Jun----Aug Aug Aug Aug 2014201420142014

Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015

Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio Customer Overdue Portfolio OctOctOctOct----Dec 2015Dec 2015Dec 2015Dec 2015

[email protected]

" v −

Delay in terms of

Delay in terms of

Delay in terms of

Delay in terms of

Days

Days

Days

Days

ºv ---- Overdue Amount (EUR)Overdue Amount (EUR)Overdue Amount (EUR)Overdue Amount (EUR)

�v = "v ∗ ºv

" v −

Delay in terms of

Delay in terms of

Delay in terms of

Delay in terms of

Days

Days

Days

Days


" v −

Delay in terms of

Delay in terms of

Delay in terms of

Delay in terms of

Days

Days

Days

Days


" v −

Delay in terms of

Delay in terms of

Delay in terms of

Delay in terms of

Days

Days

Days

Days


�v = "v ∗ ºv

050

100150200250300

050

100150200250300

-500

50100150200250300

-500

50100150200250300


0%20%40%60%80%

100%

0%20%40%60%80%

100%

Payment Delay and Overdue Amount Decrease Payment Delay and Overdue Amount Decrease Payment Delay and Overdue Amount Decrease Payment Delay and Overdue Amount Decrease within Customer Portfoliowithin Customer Portfoliowithin Customer Portfoliowithin Customer Portfolio

46

Arithmetic Average of Delays in Arithmetic Average of Delays in Arithmetic Average of Delays in Arithmetic Average of Delays in terms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisArithmetic Average of Delays in Arithmetic Average of Delays in Arithmetic Average of Delays in Arithmetic Average of Delays in terms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basis

Weighted Average of Delays in Weighted Average of Delays in Weighted Average of Delays in Weighted Average of Delays in terms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisWeighted Average of Delays in Weighted Average of Delays in Weighted Average of Delays in Weighted Average of Delays in terms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basisterms of Days on a monthly basis

- 5

10 15 20 25 30 35 40

- 5

10 15 20 25 30 35 40

Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis Overdue Amount on a daily basis (in EUR; 2014-2015; monthly basis)

Overdue Volume / Value Overdue Volume / Value Overdue Volume / Value Overdue Volume / Value Frequency on a monthly basisFrequency on a monthly basisFrequency on a monthly basisFrequency on a monthly basisOverdue Volume / Value Overdue Volume / Value Overdue Volume / Value Overdue Volume / Value

Frequency on a monthly basisFrequency on a monthly basisFrequency on a monthly basisFrequency on a monthly basis

020'00040'00060'00080'000

100'000120'000140'000160'000

= Overdue volume frequency

= Overdue value frequency

% growth = ----98%98%98%98% % growth = ----97%97%97%97%

% growth = ----98%98%98%98% % growth = ----59%59%59%59%

% growth = ----75%75%75%75%

± T̄n

+

¯n'± T̄ ∗ I¯

I+

+

¯n'

± T̄ ∗ I¯m

+

¯n'

± n¯n

+

¯n'

± I¯I+

+

¯n'

Jun 14

Aug 14

Oct 14

Dec 14

Feb 15

Apr 15

Jun 15

Aug15

Oct15

Dec 15

Jun 14

Aug 14

Oct 14

Dec 14

Feb 15

Apr 15

Jun 15

Aug15

Oct15

Dec 15

Jun 14

Aug 14

Oct 14

Dec 14

Feb 15

Apr 15

Jun 15

Aug15

Oct15

Dec 15

Jun 14

Aug 14

Oct 14

Dec 14

Feb 15

Apr 15

Jun 15

Aug15

Oct15

Dec 15







• BibliographyBibliographyBibliographyBibliography

• Annex

47


Bibliography references (1/2)Bibliography references (1/2)Bibliography references (1/2)Bibliography references (1/2)

48

• Basel Committee on Banking Supervision - “An Explanatory Note on the Basel II IRB Risk Weight Functions” – July 2005 - Bank for International Settlements Press & Communications

• Robert C. Merton - “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates” –December 1973 - The Journal of Finance

• Suresh Sundaresan - “The Annual Review of Financial Economics” – 2013 - Journal of Credit Risk

• Alexandra Kochend orfer - “Vasicek Single Factor Model” – February 2011

• Kjersti Aas - “The Basel II IRB approach for credit portfolios: A survey” – October 2005 – NorskRegnesentral Note

• Edward I. Altman - “Default Recovery Rates and LGD in Credit Risk Modelling and Practice”

• Jon Frye - “Loss given default as a function of the default rate” - 10 September 2013 - Federal Reserve Bank of Chicago

• Edward I. Altman, Brooks Brady, Andrea Resti, Andrea Sironi - “The Link between Default and Recovery Rates: Theory, Empirical Evidence, and Implications” – 2005 - Journal of Business

• Oldrich Alfons Vasicek - “Limiting Loan Loss Probability Distribution” – 09 August 1991 – KMV Corporation

• Abel Elizalde, Rafael Repullo - “Economic and Regulatory Capital What is the Difference?” – June 2005 – Paper – CEMFI, UPNA and CEPR


Bibliography references (2/2)Bibliography references (2/2)Bibliography references (2/2)Bibliography references (2/2)

49

• John C. Hull, Izzy Nelken, Alan D. White - “Merton’s model, credit risk and volatility skews” –Winter 2004/2005 - Journal of Credit Risk

• Fabrice Douglas Rouah - “Four Derivations of the Black-Scholes Formula” – Note

• John Norstad - “Black-Scholes the Easy Way” – 3 November 2011 - Note

• Peter Crosbie, Ahmet Kocagil - “Modeling default risk” – 18 December 2003 - Moody’s KMV Company

• Rudi Schafer, Alexander F. R. Koivusalo - “Dependence of defaults and recoveries in structural credit risk models” – 30 March 2011

• Alexander Becker, Rudi Schafer, Alexander F. R. Koivusalo - “Empirical Evidence for the Structural Recovery Model” – 14 March 2012

• Youbaraj Paudel - “Minimum Capital Requirement Basel II Credit Default Model & its Application” – 21 June 2007 – Vrije Universiteit

• Jakub Seidler - “Implied Market Loss Given Default: structural-model approach ” – 2008 – IES Working Paper

• Philipp J. Schonbucher - “Taken to the Limit: Simple and Not-so-simple Loan Loss Distributions” –Wilmott Magazine







• Bibliography

• AnnexAnnexAnnexAnnex

50


Credit Risk Credit Risk Credit Risk Credit Risk –––– Annex (1/9)Annex (1/9)Annex (1/9)Annex (1/9)

51

(1) (1) (1) (1) dVdVdVdVtttt = = = = μμμμVVVVtttt dtdtdtdt + + + + σVσVσVσVtttt dW^dW^dW^dW^tttt

Mathematical DerivationMathematical DerivationMathematical DerivationMathematical Derivation

(2) W^t is a Brownian motion under Historical probability P

Firm Asset

Firm Asset

Firm Asset

Firm Asset

Value, Dynamic Equation, Probability Distribution, RR

/ LGD and Expected Value

(6) dlnVt = (μ − '& σ&)dt + σdW^

(3) Given that Ito’s lemma affirms that:dG = ( ÀÁ

ÀG μV + ÀÁÀ6 +'

&À4ÁÀG4 σ&V&)dt + ÀÁ

ÀG σVdW^(4) Define G = lnVt(5) dlnVt = (1 / Vt) μVdt − '

&'

G4 σ&V&dt + (1/ Vt)σVdW^

(8) lnVT - lnVt = (μ − '& σ&)(T − t) + σ(W^� − W^6 )

(7) Â dlnVs�

6 = Â (μ − '& σ&)ds�

6 + Â σdW^�6

(9)(9)(9)(9) VVVVTTTT = = = = VVVVtttt Ã(O1e´7´)("1c)/7(Ä^" 1Ä^c ) is a log normal is a log normal is a log normal is a log normal

distributed random variable with:distributed random variable with:distributed random variable with:distributed random variable with:• ln ln ln ln VVVVTTTT ~ f(ÅÆVVVVtttt + (O − e

´ 7´)(" − c); ; ; ; 7 " − c))

• Ç!" = = = = e!" 7 ´È("1c) Ã1e

´ÅÆ!"p(ÅÆ!cÉ( Ope

´7´ "pc )7 "pc ´

• �a((((!" ) = ) = ) = ) = VVVVtttt ÃO("1c)

• ÊÃËÌÍÃ\Î ÊÏcÃ = �a(((( VVVVTTTTV |VVVVTTTT <B)=<B)=<B)=<B)=

�a(((( VVVVTTTTV |VVVVTTTT <B)<B)<B)<B) ==== e

V VVVVtttt ÃO("1c) NNNN 1ÐeNNNN 1Ð´• Loss GivenLoss GivenLoss GivenLoss Given �ÃÇÏÑÅc = 1 = 1 = 1 = 1 ---- Recovery Rate =Recovery Rate =Recovery Rate =Recovery Rate =

1 1 1 1 ---- eV VVVVtttt ÃO("1c) NNNN 1ÐeNNNN 1Ð´

Ito LemmaIto

Lemma

(11) Wt = λt+W^t is a Brownian motion under risk neutral probability Q

(10) μ=(r-δ)+σλ is market premium

(12) dWt = λdt+dW^t(13) dVt = (r-δ)Vtdt+σλVtdt+σVdW^t(14) dVt = (r-δ)Vt dt+σVt [λdt+tdW^t](15) (15) (15) (15) dVdVdVdVtttt = (r= (r= (r= (r----δ)δ)δ)δ)VVVVtttt dtdtdtdt + + + + σVσVσVσVtttt dWdWdWdWtttt(16) Apply Ito lemma shown in (3) and following steps(17)(17)(17)(17) VVVVTTTT = = = = VVVVtttt Ã(\1)1e

´7´)("1c)/7(Ä" 1Äc ) is a log normal is a log normal is a log normal is a log normal distributed random variable distributed random variable distributed random variable distributed random variable withwithwithwith• ln ln ln ln VVVVTTTT ~ f(ÅÆVVVVtttt + (\ − ) − e

´ 7´)(" − c); ; ; ; 7 " − c)

• Ç!" = = = = e!" 7 ´È("1c) Ã1e

´ÅÆ!"p(ÅÆ!cÉ( \p)pe

´7´ "pc )7 "pc ´

• �Ó((((!") = ) = ) = ) = VVVVtttt Ã(\1))("1c)

• ÊÃËÌÍÃ\Î ÊÏcÃ = �Ó(((( VVVVTTTTV |VVVVTTTT <B<B<B<B) =) =) =) =

�Ó(((( VVVVTTTTV |VVVVTTTT <B)<B)<B)<B) ==== e

V VVVVtttt Ã(\1))("1c) NNNN 1ÐeNNNN 1Ð´• Loss GivenLoss GivenLoss GivenLoss Given �ÃÇÏÑÅc = = = = 1 1 1 1 ---- Recovery Recovery Recovery Recovery Rate Rate Rate Rate ====

1111 ---- e V VVVVtttt Ã(\1))("1c) NNNN 1ÐeNNNN 1Ð´

GirsanovTheoremGirsanovTheorem


Credit Risk Credit Risk Credit Risk Credit Risk –––– Annex Annex Annex Annex (2/9)(2/9)(2/9)(2/9)

52

Mathematical Mathematical Mathematical Mathematical DerivationDerivationDerivationDerivation

(9e) Let W^� − W^6 = T − t )Y and d(W^� −W6̂ ) = T − t dY where W^� −W^6 ~N (0,T−t) and Y~N (0,1)(9d) Then EP(VT)=Vt eX(�16) Â '

&Ô(�16) e134

Õ; pÕ- ;p- 4e1 3454(�16)/5(:^; 1:^- )Ö

1Ö d W^� − W^6

(9f) Then EP(VT) = Vt eX(�16) Â '&Ô(�16) e1 34 × 4e1 3454 �16 /5 �16× T − tÖ

1Ö dY(9g) Then EP(VT) = Vt eX(�16) Â '

&Ô e1 34 × 4e1 3454 �16 /5 �16×Ö1Ö dY

(9h) Then EP(VT) = Vt eX(�16) Â '&Ô e1 34 ×15 �16 4Ö

1Ö dY (9i) Let X = Y-σ T − t, then (9l)(9l)(9l)(9l) "hen hen hen hen �a(V(V(V(VTTTT) = ) = ) = ) = VVVVtttt ÃO("1c) Â e

&Ô Ã1 e´ Ù ´Ö1Ö ddddÙ = VVVVtttt ÃO("1c) *1 = *1 = *1 = *1 = VVVVtttt ÃO("1c)

(9b)(9b)(9b)(9b) ~ÚÍÃÆ ÅÌÛ ÆÌ\ÜÏÅÚcÎ, �a((((VVVVTTTT ) =) =) =) = Â VVVVTTTT e

!" 7 ´È("1c) Ã1e´

ÅÆ!"p(ÅÆ!cÉ( Ope´7´ "pc )

7 ;p- ´ Ö1Ö dVdVdVdVTTTT

(9c) In the light of random variable transformation for which density function is f(y)× = f¬ g1' y JÝp3(Þ)JÞ , let pose that

W^� − W^6 = *+G;1(*+G-/( X13454 �16 )

5 , then EP(VT)=Â VT

'G; 5 &Ô(�16) e13

4Õ^; pÕ^-

;p- 4Vt e(X1 3454)(�16)/5(:^; 1:^- )Ö1Ö σ d W^� − W^6

(9m) (9m) (9m) (9m) �a(((( VVVVTTTTV |VVVVTTTT < B) =< B) =< B) =< B) = eV

�a e !"ßV !"a !"}V where where where where �a e !"}V !" = = = = Â VVVVTTTT

e!" 7 ´È("1c) Ã1e

´ÅÆ!" p (ÅÆ!cÉ( Ope

´7´ "pc )7 ;p- ´ V

i dVdVdVdVTTTT

(9n) Following steps between 9b and 9i , we obtain EP 1 G;}H V� = Vt eX(�16) Â '&Ô e13

4 ¬4àá .,- p ( âp3

4ã4 ;p- )ã ;p- 15 �16

1Ö dX

=Vt eX(�16) N(- *+,-. / X1 3454 �16 /54(�16)

5 �16 ) = Vt eX(�16)N(- *+,-. / X/3

454 �165 �16 )

(9o) Given (29), EP 1 G;}H V� = Vt eX(�16) N(- d') (9p) Given (41), P V� < B = N(- d&)(9q) So(9q) So(9q) So(9q) So �a(((( VVVVTTTT V |VVVVTTTT < B< B< B< B)))) ==== e

V VVVVtttt ÃO("1c) NNNN 1ÐeNNNN 1Ð´

Firm Asset

Firm Asset

Firm Asset

Firm Asset

Recovery Rate and Expected Value under Historical

Probability



53


(17b)(17b)(17b)(17b) ~ÚÍÃÆ ÅÌÛ ÆÌ\ÜÏÅÚcÎ, �Ó((((VVVVTTTT )=)=)=)=Â VVVVTTTT e

!" 7 ´È("1c) Ã1e´

ÅÆ!"p(ÅÆ!cÉ( \p)pe´7´ "pc )

7 ;p- ´ Ö1Ö dVdVdVdVTTTT

(17c) In the light of random variable transformation for which density function is f(y)× = f¬ g1' y JÝp3(Þ)JÞ , let pose that

W� − W6 = *+G; 1(*+G-/( 01213454 �16 )

5 , then E�(VT)=Â VT

'!" 7 ´È("1c) e13

4Õ; pÕ-

;p- 4Vt e(0121 3454)(�16)/5(:; 1:- )Ö1Ö σ d W� − W6

(17e) Let W� − W6 = T − t Y and d(W� −W6 ) = T − t dY where W� −W6 ~N (0,T−t) and Y~N (0,1)(17d) Then E�(VT) = Vt e(012)(�16) Â '

&Ô(�16) e134

Õ; pÕ- ;p- 4e1 3454(�16)/5(:; 1:- )Ö

1Ö d W� − W6

(17f) Then E�(VT) = Vt e(012)(�16) Â '&Ô(�16) e1 34 × 4e1 3454 �16 /5 �16 × T − tÖ

1Ö dY(17g) Then E�(VT) = Vt e(012)(�16) Â '

&Ô e1 34 × 4e1 3454 �16 /5 �16×Ö1Ö dY

(17h) Then E�(VT) = Vt e(012)(�16) Â '&Ô e1 34 ×15 �16 4Ö

1Ö dY(17l)(17l)(17l)(17l) "hen hen hen hen �Ó(V(V(V(VTTTT ) =) =) =) = VVVVtttt Ã(012)("1c) Â e

´È Ã1 e´ Ù ´Ö1Ö ddddÙ = VVVVtttt Ã(012)("1c) *1 = *1 = *1 = *1 = VVVVtttt Ã(012)("1c)

(17i) Let X =Y-σ T − t, then

(17m) (17m) (17m) (17m) �Ó(((( VVVVTTTTV |VVVVTTTT < B) = < B) = < B) = < B) = eV

�Ó e !"ßV !"a !"}V where where where where �a e !"}V !" = = = = Â VVVVTTTT

e!" 7 ´È("1c) Ã1e

´ÅÆ!" p(ÅÆ!cÉ( äpåp e ´7´ "pc )

7 ;p- ´ Vi dVdVdVdVTTTT

(17n) Following steps between 17b and 17i , we obtain E� 1 G;}H V� = Vt e(012)(�16) Â '√&Ô e13

4 ¬4àá .,- p ( äpåp 34ã4 ;p- )

ã ;p- 15 �161Ö dX

= Vt e(012)(�16) N(- *+,-. / 01213

454 �16 /54(�16)5 �16 ) = Vt e(012)(�16)N(- *+,-

. / 012/3454 �16

5 �16 )(17o) Given (29), E� 1 G;}H V� = Vt e(012)(�16) N(- d') (17p) Given (41), P V� < B = N(- d&)(17q) So, (17q) So, (17q) So, (17q) So, �Ó(((( VVVVTTTT

V | VVVVTTTT <<<<BBBB) = ) = ) = ) = eV VVVVtttt Ã(012)("1c) NNNN 1ÐeNNNN 1Ð´

Firm Asset

Firm Asset

Firm Asset

Firm Asset

Recovery Rate and Expected Value under Risk

Neutral Probability



54


(25) Put(Vt , B) = e-r(T-t)N(y)B-e-δ(T-t)Vt Â '&Ô e1ç4

4 dxè15 �161Ö

(26) Put(Vt , B) = e-r(T-t)N(y)B - e−δ(T−t) Vt N(z − σ T − t)

(27) Put(Vt , B) = e-r(T-t)N(*+ .,- 1 (012)1 3

4 54 (�16)5 �16 )B - e−δ(T−t) Vt N(*+ .

,- 1( 012 / 34 54)(�16)

5 �16 )(28) Put((28) Put((28) Put((28) Put(VVVVtttt , B, B, B, B) = ) = ) = ) = e e e e ----r (T r (T r (T r (T ---- t) t) t) t) N(N(N(N(−Ð´) B ) B ) B ) B ---- eeee −−−−)(((( T−t)T−t)T−t)T−t) VVVVtttt N(N(N(N(−Ðe)

(29) Where d' = ( *+ ,-. / 012 / 3

4 54 (�16)5 �16 ) and d& = ( *+ ,-

. / 012 1 34 54 (�16)

5 �16 ) or d& = d' - σ T − t

(23) Put(Vt , B) = 0+e-r(T-t)N(y)B-Vte-r(T-t) e(012)(�16) Â '&Ô e1 êpã (;p- 4

4 dyè1Ö

(24) Substitute x = y − σ T − t where dy = dx

(22) Let z = point where put value is zero = *+.,- 1 0121(' &)54⁄ (�16)

5 �16

(21) Put(Vt , B) = e-r(T-t) Â max (B−Vt e 01213454 �16 /5(:; 1:- )) '

´È("1c)Ö

1Ö e134

Õ; pÕ- ;p-

& d(W� −W6 )Put(Vt , B) = e-r(T-t) Â max (B−Vt e 01213

454 �16 /5 �16×) '´È("1c) e1ê4

4Ö1Ö T − t dy

Put(Vt , B) = e-r(T-t) Â max (B−Vt e 01213454 �16 /5 �16×) '

√&Ô e1ê4 4Ö

1Ö dy

(19) Let W� − W6 = T − t Y and d(W� −W6 ) = T − t dY(18) Put ((18) Put ((18) Put ((18) Put (VVVVtttt , B, B, B, B) = ) = ) = ) = e e e e ––––rrrr (T(T(T(T---- t) t) t) t) E E E E Q Q Q Q (max (B (max (B (max (B (max (B –––– !");0));0));0));0)

(20) where W� − W6 ~N (0,T−t) and Y~N (0,1)

Put Option Value

Put Option Value

Put Option Value

Put Option Value



55


(29 b) D ((29 b) D ((29 b) D ((29 b) D (VVVVtttt , t) = P( t , T ) , t) = P( t , T ) , t) = P( t , T ) , t) = P( t , T ) –––– Put (Put (Put (Put (VVVVtttt , B) is corporate debt value, B) is corporate debt value, B) is corporate debt value, B) is corporate debt value(30) Given that P (t , T) = B e(30) Given that P (t , T) = B e(30) Given that P (t , T) = B e(30) Given that P (t , T) = B e----r (Tr (Tr (Tr (T----t)t)t)t)

(32) Then D(Vt , t) = Be-r(T-t) - e−r(T−t) (1−N(d&))B−e−δ(T−t)VtN(−d') (31) It results that D(Vt , t) = B e-r(T-t) – e−r(T−t)N( −d& )B − e−δ(T−t)Vt N(−d')

(33) Then D(Vt , t) = e−δ(T−t) Vt N(−d') + e−r(T−t) N(d&) B(34) D ((34) D ((34) D ((34) D (VVVVtttt , , , , t) t) t) t) = = = = eeee−−−−)(T−t)(T−t)(T−t)(T−t) VVVVtttt NNNN (((( −Ðe ))))+ + + + P(P(P(P(t,Tt,Tt,Tt,T)))) N(N(N(N(Ð´))))

(36) According to (17), P(VT < B) = P Vt e(01213454)(�16)/5(:; 1:- )< B

(35) Let P (V(35) Let P (V(35) Let P (V(35) Let P (VTTTT < B) probability of default< B) probability of default< B) probability of default< B) probability of default

(37) P (VT < B) = P (In Vt + (r − δ − '& σ&)(T − t) + σ(W� − W6 ) < In B)

(38) P (VT < B) = P ( (W� −W6 ) < (In (B/ Vt) - (r − δ − '& σ&)(T − t))/σ)

(39) Let (W� −W6 ) = T − t K�(40) According to (39), P (VT < B) = P (K� < (In (B/ Vt) - (r − δ − '

& σ&)(T − t))/σ T − t )(41) According to (29), P (V(41) According to (29), P (V(41) According to (29), P (V(41) According to (29), P (VTTTT < B) = N (< B) = N (< B) = N (< B) = N (----dddd2222 ))))

Corporate

Corporate

Corporate

Corporate Debt Value and

Debt Value and

Debt Value and

Debt Value and Probability

Probability

Probability

Probability of of of of

Default

Default

Default

Default



56


(42) Consider (42) Consider (42) Consider (42) Consider a a a a portfolio consisting portfolio consisting portfolio consisting portfolio consisting of of of of N loans. N loans. N loans. N loans. Let Let Let Let the probability the probability the probability the probability of default on any one loan be of default on any one loan be of default on any one loan be of default on any one loan be p p p p and and and and assume assume assume assume that the asset that the asset that the asset that the asset values of the borrowing companies values of the borrowing companies values of the borrowing companies values of the borrowing companies are correlated are correlated are correlated are correlated with with with with a a a a coefficient ρ coefficient ρ coefficient ρ coefficient ρ for for for for any two companiesany two companiesany two companiesany two companies. We . We . We . We will will will will further assume that all further assume that all further assume that all further assume that all loans have the same term loans have the same term loans have the same term loans have the same term TTTT. Let . Let . Let . Let VVVVnnnn be be be be the value of the the value of the the value of the the value of the nnnn----thththth borrower’s assets, borrower’s assets, borrower’s assets, borrower’s assets, described described described described by the by the by the by the process process process process dVdVdVdVntntntnt ==== μμμμÆVVVVn tn tn tn t dtdtdtdt + + + + σσσσnnnn VVVVnnnn t t t t dWdWdWdW^̂̂̂nnnn t t t t with n = 1,...,Nwith n = 1,...,Nwith n = 1,...,Nwith n = 1,...,N(43) or (43) or (43) or (43) or dVdVdVdVtttt ==== OVVVVtttt dtdtdtdt ++++ 7VVVVt t t t dWdWdWdW^̂̂̂tttt with with with with VVVVtttt = (V= (V= (V= (V1 t 1 t 1 t 1 t , ...,V, ...,V, ...,V, ...,VN t N t N t N t ))))(44) Where W^t is a multi-dimensional Brownian motion W^t = (W^1t , ...,W^Nt ) under Historical probability P with following properties: • E(W^T –W^t) = 0 where E(W^nT –W^nt) = 0 for n=1,...,N• V(W^T –W^t) = E((W^T –W^t)(W^T –W^t )T ) where Var = E((W^nT –W^nt)2 )=(T-t) and Var= E((W^nT –W^nt) (W^jT –W^jt))

=ρ(T-t) for n≠ j(45) Given that ((45) Given that ((45) Given that ((45) Given that (WWWW^̂̂̂nnnn t t t t ; n = 1,...,N) are jointly distributed with equal correlation, the above multi; n = 1,...,N) are jointly distributed with equal correlation, the above multi; n = 1,...,N) are jointly distributed with equal correlation, the above multi; n = 1,...,N) are jointly distributed with equal correlation, the above multi----dimensional Brownian motion dimensional Brownian motion dimensional Brownian motion dimensional Brownian motion WWWW^̂̂̂tttt = (= (= (= (WWWW^̂̂̂tttt 1 1 1 1 ,..., ,..., ,..., ,..., WWWW^̂̂̂tttt NNNN) can be represented as ) can be represented as ) can be represented as ) can be represented as WWWW^̂̂̂tttt = = = = bxbxbxbxtttt + a+ a+ a+ adc with with with with ((((WWWW^̂̂̂tttt n n n n = = = = bYbYbYbYtttt + a+ a+ a+ adc Æ ; n = 1 ,..., N); n = 1 ,..., N); n = 1 ,..., N); n = 1 ,..., N)

(47) Infact we can demonstrate that:• E(W^T –W^t) = E((bYT+aε�) − (bYt+aε6)) = bE(Y� − Y6) + aE(ε� − ε6) which is equal to 0 in the light of (46)• V(W^T -W^t) = E(((bYT+aε�) − (bYt+aε6))((bYT+aε�) − (bYt+aε6))T)

- E(((bYT+aε�+) − (bYt+aε6+))&) = b2E((Y� − Y6)&) + a2 E((ε�+ −ε6+)2)+2ab E((ε�+ −ε6+) (Y� −Y6)) which transforms to b2E((Y� − Y6)&) + a2 E((ε�+ −ε6+)2)= b2(T-t)+ a2(T-t) in the light of (46) and finally to (T-t) putting b= ρ and a= 1 − ρ in (45)

- for n≠ j, E(((bYT+aε�+) − (bYt+aε6+))((bYT+aε�¯) − (bYt+aε6¯))) = b2E((Y� − Y6)&) + a2 E((ε�+ −ε6+) (ε�¯ −ε6¯))+ab E((ε�+ −ε6+) (Y� −Y6))+ab E((ε�¯ −ε6¯) (Y� −Y6) ) which transforms to b2E((Y� − Y6)&) = b2(T-t) in the light of (46) and finally to ρ(T-t) putting b = ρ in (45)

(46) Where (YT – Yt ), (ε�' −ε6'), … , (ε�k −ε6k) are independent, normally distributed with E(.)=0, V(.)=(T-t) and V(.)=0 for n≠ j

(48) In the light of (45), it means that all assets are exposed to a common risk factor Y (such as the state of the economy)(48) In the light of (45), it means that all assets are exposed to a common risk factor Y (such as the state of the economy)(48) In the light of (45), it means that all assets are exposed to a common risk factor Y (such as the state of the economy)(48) In the light of (45), it means that all assets are exposed to a common risk factor Y (such as the state of the economy) anananand that d that d that d that term term term term εεεεnnnn represents the company’s specific risks of each assetrepresents the company’s specific risks of each assetrepresents the company’s specific risks of each assetrepresents the company’s specific risks of each asset

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Risk Factor and Specific Risk Factors

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Values, Dynamic Equations, Loan Unconditional

Probability of Default, Asset correlation, Common

Risk Factor and Specific Risk Factors



57


(49) Single loan default occurs when single asset value falls at expiration date T below (49) Single loan default occurs when single asset value falls at expiration date T below (49) Single loan default occurs when single asset value falls at expiration date T below (49) Single loan default occurs when single asset value falls at expiration date T below a certain barrier a certain barrier a certain barrier a certain barrier CCCCn n n n which is debt value which is debt value which is debt value which is debt value where default event is modelled as where default event is modelled as where default event is modelled as where default event is modelled as DDDDnnnn = = = = e !Æ íîÆ

(52) In the light of the fact that probability of default of any one loan is p as mentioned in (42), it follows that P L = k = N

k P V�' < C', … , V�x < Cx, V�x/' > Cx/', … , V�k > Ck and given definition of conditional probability, it follows that P L = k = N

k Â P V�' < C', … , V�x < Cx, V�x/' > Cx/', … , V�+ > C+ |Y� = y dP Y� < yÖ1Ö .

This means that we will evaluate the probability of k defaults within portfolio L as the expectation over the common factor Y of the conditional probability given y. This can be interpreted as assuming various scenarios for the economy, determining the probability of number of losses within portfolio under each scenario and then weighting each scenario by its likelihood(53) According to (45), (47) and (50), (53) According to (45), (47) and (50), (53) According to (45), (47) and (50), (53) According to (45), (47) and (50), PPPP !"Æ < îÆ | b" = Î = PPPP Ä^"Æ < ËÆ | b" = Î = P= P= P= P g b"Æ++++ e − gd"Æ < ËÆ | b"Æ = Î = = = = PPPP d"Æ < ËÆ 1 g b"Æ

e1g | b"Æ = Î = = = = NNNN fpe v 1 g Î

e1g = p(y) = p(y) = p(y) = p(y)

(55) So (55) So (55) So (55) So ïÃ ËÏÆ ðÏÎ cñÏc, PPPP j < Ü ==== ∑ fò Â NNNN fpe v 1 g Î

e1gò

1−N1−N1−N1−N fpe v 1 g Îe1g

f1òdP(y)dP(y)dP(y)dP(y)Ö

1ÖÜòni

(50) (50) (50) (50) Now for simplicity we assume on beyond for Now for simplicity we assume on beyond for Now for simplicity we assume on beyond for Now for simplicity we assume on beyond for ---- notation simplicity notation simplicity notation simplicity notation simplicity –––– that t = 0that t = 0that t = 0that t = 0, so default probability of each n, so default probability of each n, so default probability of each n, so default probability of each n----thththth loan is loan is loan is loan is v�Æ = a �"Æ = e = a !"Æ < î"Æ = a Ä^"Æ < ËÆ = NNNN ËÆ ïñÃ\Ã îÆ ÏÆÐ ËÆ coincide respectively with B and coincide respectively with B and coincide respectively with B and coincide respectively with B and −Ð´ ÚÆ (41) (41) (41) (41) and and and and v�Æ= p= p= p= p(51) Given L = (51) Given L = (51) Given L = (51) Given L = ∑ �ÆòÆne equal to number of defaultsequal to number of defaultsequal to number of defaultsequal to number of defaults

(54) Now given that (54) Now given that (54) Now given that (54) Now given that dcÆ are independent, are independent, are independent, are independent, PPPP j = ò = = = = fò NNNN fpe v 1 g Î

e1gò

1−N1−N1−N1−N fpe v 1 g Îe1g

f1ò

(56) Finally (56) Finally (56) Finally (56) Finally ïÃ ËÏÆ ðÏÎ cñÏc, PPPP j < Ü ==== ∑ fò Â (v(Î)kkkk (e − v(Î))N−kN−kN−kN−k)))) dP(y)dP(y)dP(y)dP(y)Ö

1ÖÜòni

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Factor Scenario Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio Default Rate, Portfolio Conditional

Probability of Defaultgiven certain Common Risk

Factor Scenario



58


(57) Consider a loan portfolio which is a «Large Homogeneous Portfolio» (LHP) having following properties:(57) Consider a loan portfolio which is a «Large Homogeneous Portfolio» (LHP) having following properties:(57) Consider a loan portfolio which is a «Large Homogeneous Portfolio» (LHP) having following properties:(57) Consider a loan portfolio which is a «Large Homogeneous Portfolio» (LHP) having following properties:- v�en v�f n v as already said before all loans have same probability of defaultas already said before all loans have same probability of defaultas already said before all loans have same probability of defaultas already said before all loans have same probability of default- portfolio weight s wportfolio weight s wportfolio weight s wportfolio weight s w(N)(N)(N)(N)1111 ,..., w,..., w,..., w,..., w(N)(N)(N)(N)N N N N where each wwhere each wwhere each wwhere each w(N)(N)(N)(N)n n n n is single loan value out of total loan portfolio are modelled such that is single loan value out of total loan portfolio are modelled such that is single loan value out of total loan portfolio are modelled such that is single loan value out of total loan portfolio are modelled such that

∑ wwww(N)(N)(N)(N)nnnnfÆne = 1 and = 1 and = 1 and = 1 and ÅÚÜf→Ö wwww(N)(N)(N)(N)nnnn ´ = i which means that portfolio is not dominated by few loans much larger than the otherswhich means that portfolio is not dominated by few loans much larger than the otherswhich means that portfolio is not dominated by few loans much larger than the otherswhich means that portfolio is not dominated by few loans much larger than the otherswith the following with the following with the following with the following portfolio default rate portfolio default rate portfolio default rate portfolio default rate j(f) = = = = ∑ wwww(N)(N)(N)(N)nnnnfÆne DDDDnnnn

(59) Portfolio default rate (59) Portfolio default rate (59) Portfolio default rate (59) Portfolio default rate j(f) in LHP converges in probability for N in LHP converges in probability for N in LHP converges in probability for N in LHP converges in probability for N → ∞ as follows as follows as follows as follows j(f) a→ p(Y) = p(Y) = p(Y) = p(Y) = f( fpe v 1 g be1g ) that’s to say ) that’s to say ) that’s to say ) that’s to say

that P (that P (that P (that P (j(f) < x ) < x ) < x ) < x ) NNNN →Ö P (P (P (P (v(b) < x < x < x < x ))))(60) It follows the proof of result contained in (59) which states that V L(k)| Y k→Ö 0. Infact given (58), V L(k)| Y =∑ w(N)nk+n' 2 p Y (1– p Y ) . Then for normal distribution property V L(k)| Y = ∑ w(N)nk+n' 2 p Y (1– p Y )≤ '

÷ ∑ w(N)nk+n' 2. Subsequently given (57), '÷ ∑ w(N)nk+n' 2 k→Ö 0. Then this provides convergence in L2 , E L(k) − p( Y) 2 k→Ö 0. In fact given (58), E L(k) − p(Y) 2 = E L(k) − E L(k)| Y 2 and given identity property of conditional expectation,E L(k) − p(Y) 2 = E L(k) − E L(k)| Y 2 = E(E (L(k)−E L(k)| Y 2 |Y)), then for independent properties of DnE L(k) − p(Y) 2 = E L(k) − E L(k)| Y 2 = E(E (L(k)−E L(k)| Y 2 |Y)) = E(Var L(k)| Y ) and given variance convergence to 0 as stated at the beginning, we have thatE L(k) − p(Y) 2 = E L(k) − E L(k)| Y 2 =E(E (L(k)−E L(k)| Y 2 |Y)) = E(Var L(k)| Y ) k→Ö 0. Convergence in L2 implies convergence in probability i.e. for all ϵ > 0 it is true that limk→ÖP( L k −p(Y) > ϵ) = 0

(58) Then for LHP, it holds that E L(k)| Y = p Y = N( kp3 r 1 g Ye1g ) and V L(k)| Y = ∑ w(N)nk+n' 2 p Y (1-p Y ) in fact for

linearity property, for independent property of Dn conditioned on y and for (53), E L(k)| Y = ∑ w(N)nk+n' E Dn| Y = ∑ w(N)nk+n' E 1 Gá íùá | Y = ∑ w(N)nk+n' P Dn| Y = ∑ w(N)nk+n' p Y = p Y ∑ w(N)nk+n' = p(Y) and V L(k)| Y =∑ w(N)nk+n' 2 V Dn| Y = ∑ w(N)nk+n' 2 (E( Dn 2|Y) – E2 Dn| Y )= ∑ w(N)nk+n' 2 (E ((1 Gá íùá )2 | Y)– p2 Y ) =∑ w(N)nk+n' 2 (P Dn| Y – p2 Y ) =∑ w(N)nk+n' 2 (p Y – p2 Y ) = ∑ w(N)nk+n' 2 p Y (1– p Y )

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Expected Value of Portfolio Default Rate

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Loan Portfolio Granularity Hypothesis, Convergence

in Probability of Portfolio Default Rate to Conditional

Expected Value of Portfolio Default Rate



59


(61) Starting from (59), P(p(Y) < x ) = P(N( kp3 r 1 g×e1g ) < x) = P(Y≤ e1g kp3 q 1kp3 r

o )direction of <> sign due to N . property and subsequently, in the light of 46 , P(p(Y) < x ) = N( e1g kp3 q 1kp3 r

o )(62) Then given (59), (62) Then given (59), (62) Then given (59), (62) Then given (59), P(P(P(P(j(f) < x ) < x ) < x ) < x ) NNNN →Ö N (N (N (N ( e1g fpe ú 1fpe v

g )(63) In case of quantiles calculation, we have for large N that P (63) In case of quantiles calculation, we have for large N that P (63) In case of quantiles calculation, we have for large N that P (63) In case of quantiles calculation, we have for large N that P ((((j(f) < < < < û� ) =) =) =) = � � N (N (N (N ( e1g fpe û� 1fpe v

g ) (with for example (with for example (with for example (with for example � =99.9%) =99.9%) =99.9%) =99.9%)

(65) This property is true: (65) This property is true: (65) This property is true: (65) This property is true: E(E(E(E(v b )))) = ÅÚÜf→Ö�(j(f)) = pppp(66) The proof of (65) is contained belowConvergence in L2 implies convergence in terms of expectations, so E(p Y ) = limk→ÖE(L(k)) is true then limk→ÖE(L(k)) = limk→ÖE ∑ w(N)nk+n' Dn = limk→Ö ∑ w(N)nk+n' E Dn = limk→Ö ∑ w(N)nk+n' E 1 Gá íùá = limk→Ö ∑ w(N)nk+n' P D+ = 1 = limk→Ö ∑ w(N)nk+n' p =

p limk→Ö ∑ w(N)nk+n' = p due to (50) and (57)

(64) If we invert (63) we obtain (64) If we invert (63) we obtain (64) If we invert (63) we obtain (64) If we invert (63) we obtain û� � NNNN (((( g fpe � /fpe ve1g )

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Portfolio of Loans

Convergence in Probability of Portfolio Default Rate

and Portfolio Default Rate Quantiles


DisclaimerDisclaimerDisclaimerDisclaimer

60

• This document (“Document”) has been prepared by the authors (“Authors”) solely forinformational purposes

• The Authors shall not have liability for the contents of, for any representation (expressed orimplied) contained in, or for omissions from, this Document or any other written or oralcommunication

• The information contained herein is being made as of the date of this Document, unlessanother time is specified, and the Authors have no obligation to update such information

• Reference to other authors (“Other Authors”) publications, analysis, economic model orformula is made solely for communication or informational purposes and it does not implyany linkage between Other Authors and content presented within this Document in termsof accuracy and completeness of information conveyed

• This Document does not contain any data or confidential information concerning thebusiness, contractual arrangements, deals, transactions or particular affairs of theCompany or his affiliates for which the Authors had or currently have an employmentagreement


AuthorsAuthorsAuthorsAuthors

61

Marco BerizziChief Financial Officer

e-mail [email protected] mobile +39 342 9204033

MSc in Probability and Application UPMC - Pierre et Marie Curie University (Paris VI Un.), Paris

MSc and BSc in Economics L. Bocconi University, Milan

Fabiano De RosaP&C and Risk Manager

e-mail [email protected] mobile +39 331 4015461

MSc in Engineering and Management Politecnico, Turin

BSc in Engineering and Management of Logistics and ProductionFederico II University, Naples


Date post:	18-Jan-2017
Category:	Economy & Finance
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Credit risk management for industrial corporates

Economy & Finance