03.2 homogeneous debt portfolios

Copyright © 2018 CapitaLogic Limited
This presentation file is prepared in accordance with
Chapter 3 of the text book
“Managing Credit Risk Under The Basel III Framework, 3rd ed”
Website : https://sites.google.com/site/crmbasel
E-mail : crmbasel@gmail.com
Chapter 3
Homogeneous
Debt Portfolios

Copyright © 2018 CapitaLogic Limited 2
Declaration
 Copyright © 2018 CapitaLogic Limited.
 All rights reserved. No part of this presentation file may be
reproduced, in any form or by any means, without written
permission from CapitaLogic Limited.
 Authored by Dr. LAM Yat-fai (林日辉),
Director, CapitaLogic Limited,
Adjunct Professor of Finance, City University of Hong Kong,
Doctor of Business Administration,
CFA, CAIA, CAMS FRM, PRM.

Outline
 Credit risk identification
 Independent homogeneous portfolio
 Gaussian copula
 Finite homogeneous portfolio
 Infinite homogeneous portfolio
 Appendices

Portfolio 1-year EL
 A debt portfolio comprising NOB different debts
  
  
k
NOB
k
k=1
NOB
RM
k k k k
k=1
NOB
k k k k
k=1
Portfolio 1-year EL
= 1-year EL
= EAD × LGD × Min PD , 1 - 1 - PD
EAD × LGD × PD × Min 1, RM
 
 





Portfolio 1-year EL
 Failed to incorporate diversification effect
 More borrowers with smaller EADs => lower risk
 Lower default dependency => lower risk
 Not an effective credit risk measure for a debt
portfolio
Example 3.1
Example 3.2

Credit risk identification
Credit risk
Default loss
Exposure
at default
Default chance
Loss given
default
Probability
of default
Diversification
effect
Concentration
of debts
Default
dependency
Residual
maturity

Diversification effect
 For a fixed portfolio EAD
 Concentration of borrowers
 Higher concentration among very few borrowers =>
higher credit risk
 Lower concentration among many borrowers =>
lower credit risk
 Default dependency
 Higher default dependency => higher credit risk
 Lower default dependency => lower credit risk

Homogeneous portfolio
 Theory development
 Simplicity
 Highly analytical tractability
 Analytical approximation to a real debt
portfolio
 Similar debts are managed under the same
portfolio
 Around 5% to 10% model error

Unified maturity
 RM is artificially set to one year
 The lender invests in many debts with maturity longer
than one year or without fixed maturity will review and
control the credit risk at the end of the following one year
 The lender invests in many debts with maturity shorter
than one year will invest the proceeds at maturity in
similar debts up to one year
 The debts with maturity short than one year accounts for
the minority of the homogeneous portfolio (< 10%)

Outline
 Gaussian copula
 Appendices

Independent homogeneous portfolio
 Portfolio EAD
 Shared equally among all borrowers
 LGD
 Same for all debts
 PD
 Same for all borrowers
 NOB
 > 30
 Borrowers totally independent of one another

Credit risk factors
– Independent homogeneous portfolio
Credit risk
Default loss
Portfolio
exposure
at default
Default chance
Loss given
default
Probability
of default
Diversification
effect
No. of
Borrowers (-)

Combination
 The number of ways to place k objects in N
slots where the order of the k objects does
NOT matter
   
   
     
N k
N × N - 1 × N - 2 × ... × 3 × 2 × 1
C =
k × k - 1 × k - 2 × ... × 3 × 2 × 1 ×
N - k × N - k - 1 × N - k - 2 × ... × 3 × 2 × 1
  
  

Combination
 A and B in five slots
 [AB***]
 [A*B**]
 [A**B*]
 [A***B]
 [*AB**]
 [*A*B*]
 [*A**B]
 [**AB*]
 [**A*B]
 [***AB]
 B and A in five slots
 [BA***]
 [B*A**]
 [B**A*]
 [B***A]
 [*BA**]
 [*B*A*]
 [*B**A]
 [**BA*]
 [**B*A]
 [***BA]

Default status of five borrowers
 One default (1) among five
borrowers
 [10000]
 [01000]
 [00100]
 [00010]
 [00001]
 Two defaults (1,1) among
five borrowers
 [11000]
 [10100]
 [10010]
 [10001]
 [01100]
 [01010]
 [01001]
 [00110]
 [00101]
 [00011]

Binomial distribution
 Probability mass function
 Cumulative probability distribution function
 Average = PD × NOB
 
 
 
 
NOB-kk
NOB k
M
NOB-kk
NOB k
k=0
Probability k defaults out of NOB borrowers
= C × PD × 1 - PD
Probability Up to M defaults out of NOB borrowers
= C × PD × 1 - PD 
 
Example 3.3

Extreme case default rate
 Extreme case no. of defaults
 In Microsoft Excel
 Extreme case default rate
 
 
Q
NOB-kk
NOB k
k=0
C × PD × 1 - PD = 99.9%
Q = Critbinom NOB, PD, 99.9%
Q
XCDR =
NOB

Example 3.4

Cumulative probability of up to
k defaults out of NOB borrowers

Portfolio credit risk measure
 Extreme case loss
XCL = Portfolio EAD × LGD × XCDR

to extreme case loss
 For fixed portfolio EAD, LGD and PD
 Lower concentration of borrowers
 Larger NOB
 Smaller XCDR
 Smaller XCL
 Higher concentration of borrowers
 Smaller NOB
 Larger XCDR
 Larger XCL

Outline
 Gaussian copula
 Appendices

Bernoulli random variable
 A random no. B
 Either 1 with probability PD
 Or 0 with probability 1 - PD
 If B = 1, then the borrower defaults

Standard uniform random variable
 A random no. u between 0 and 1
 If u < PD, then the borrower defaults

Standard normal random variable
 A real random no. z
 Mapped to a standard uniform random
variable u
 
2
z
-
1 τ
u = exp - dτ = Normsdist z
22π
0 u 1

 
 
 
 

Example 3.5

Standard normal random variable
 If y and z are
independent standard
normal random
variables
 then x is also a
standard normal
random variable
 
   
 
       
 
   
2 2
x = y ρ + z 1 - ρ
E x = E y ρ + z 1 - ρ
= E y ρ + E z 1 - ρ
= 0 ρ + 0 1 - ρ
= 0
Var x = Var y ρ + z 1 - ρ
= Var y ρ + Var z 1 - ρ
= 1 ρ + 1 1 - ρ
= 1
SD x = Var x =1
 
 
 
 
 
 
 
 

Modelling one borrower
 A systematic standard normal random
variable y
 A specific standard normal random variable z
 u = Φ y ρ + z 1 - ρ
Example 3.6

Correlated standard normal
random variables
 If y, z1 and z2 are
independent standard
normal random
variables
 then x1 and x2 are
standard normal
random variables with
copula correlation
coefficient ρ
 
 
 
 
 
 
1 1
2 2
1 2 1 2
1
2
1 2
1
1 2
x = y ρ + z 1 - ρ
x = y ρ + z 1 - ρ
Cov x ,x = Cov y ρ + z 1 - ρ, y ρ + z ρ
= Cov y, y ρ ρ
+ Cov y,z ρ 1 - ρ
+ Cov y,z ρ 1 - ρ
+ Cov z ,z 1 - ρ 1 - ρ
= 1 ρ + 0 + 0 + 0
= ρ
Cov x ,x
Corr x ,x =
 
 
 
 
 
 

 
   
2
1 2
ρ
= = ρ = CCC
SD x SD x 1 1 

Modelling two borrowers
with same PD
 A systematic standard normal random variable y
 Two specific standard normal random variables z1 and z2
 Mapped to standard uniform random variables u1 and u2
 If u1 < PD, then borrower 1 defaults
 If u2 < PD, then borrower 2 defaults
 The larger the CCC, the higher the default dependency
between the two borrowers
 
 
1 1
2 2
u = Φ y CCC + z 1 - CCC
u = Φ y CCC + z 1 - CCC
Example 3.7

Homogeneous borrowers
 NOB different borrowers
 Same PD
 Same CCC between any two borrowers

Modelling NOB
homogeneous borrowers
 A systematic standard normal random variable y
 NOB specific standard normal random variables z1,
z2, z3, … zNOB
 Mapped to standard uniform random variables u1, u2,
u3, … uNOB
 If uk < PD, then borrower k defaults
 The larger the CCC, the higher the default
dependency among the NOB borrowers
 k ku = Φ y CCC + z 1 - CCC k=1,2,3,...NOB
Example 3.8

CCC for retail exposures
 Residential mortgage
 Qualifying revolving retail exposure
 Other retail exposure
 
CCC = 0.15
CCC = 0.04
CCC = 0.03 + 0.13exp -35PD

CCC for institution exposures
 Institution exposures
 Small and medium enterprise
 Annual revenue (S) between EUR 5 mn and 50 mn
 Large financial institution
 Total assets > USD 100 bn
 
 
 
CCC = 0.12 1 + exp -50PD
CCC = 0.12 1 + exp -50PD
CCC =
S - 50
+
1125
0 1 + ex.15 p -50PD
  
  
  

CCC under Basel III

Outline
 Gaussian copula
 Appendices

Finite homogeneous portfolio
 Portfolio EAD
 LGD
 PD
 NOB
 > 30
 CCC
 Same between any two borrowers

Credit risk factors
– Finite homogeneous portfolio
Credit risk
Default loss
Portfolio
exposure
at default
Default chance
Loss given
default
Probability
of default
Diversification
effect
No. of
Borrowers (-)
Copula
correlation
coefficient

 Concentration of debts
 Measured by the NOB
 Approaching one when fully concentrated
 Approaching infinity when fully granular
 Default dependency
 Quantified by the CCC

Monte Carlo simulation
 Generate a systematic standard normal random no. y
 For each borrower k (k = 1 to NOB)
 Generate a specific standard normal random no. zk
 Map to standard uniform random no. uk
 If uk < PD, then borrower k defaults
 Register the no. of borrowers in default
 Repeat the above steps for 1,000,000 time
 k ku = Φ y CCC + z 1 - CCC
Example 3.12
Example 3.13

 
Extreme case no. of defaults
= Percentile No. of defaults, 99.9%
Extreme case no. of defaults
XCDR =
NOB

XCL of finite homogeneous portfolio
CCC NOB
PD
XCDR
LGD
Portfolio
EAD
XCL
(-)
(+)
(+)
(+)
(+)
(+)

Properties of the XCL
 Smaller XCL for
 Smaller portfolio EAD and LGD – less loss upon default
 Smaller PD – higher credit quality
 Larger NOB – lower concentration
 Smaller CCC – lower default dependency
 Lower risk for
 Larger portfolio EAD and LGD – more loss upon default
 Larger PD – lower credit quality
 Smaller NOB – higher concentration
 Larger CCC – higher default dependency
 XCL is a good quantitative measure of credit risk for finite
homogeneous portfolio
 Having taken into account the diversification effect

Outline
 Gaussian copula
 Appendices

Infinite homogeneous portfolio
 Portfolio EAD
 LGD
 PD
 NOB
 → Infinity
 CCC
 Same between any two borrowers
 Default rate (“DR”)
 The percentage of borrowers in default

Credit risk factors
– Infinite homogeneous portfolio
Credit risk
Default loss
Portfolio
exposure
at default
Default chance
Loss given
default
Probability
of default
Diversification
effect
Copula
correlation
coefficient

Vasicek default rate distribution
 
     
 
     
   
22 -1 -1-1
2
-1 -1-1
DR
0
-1 -1
1 - CCC Φ DR - Φ PDΦ DR1 - CCC
f DR = exp -
CCC 2 2CCC
Φ PD - 1 - CCC Φ τΦ τ1 - CCC
F DR = exp - dτ
CCC 2 2CCC
1 - CCC Φ DR - Φ PD
= Φ
CCC
       
 
 
 
    
 
 
 
 
 
  

 Probability density function
 Cumulative probability distribution function

Vasicek default rate distribution
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0% 20% 40% 60% 80% 100%
Default rate
Defaultratedensity.

Vasicek default rate model
 Mean
   -1 -1
Mean = PD
Φ PD + CCC × Φ 99.9%
XCDR = Φ
1 - CCC
 
 
  
Example 3.9

XCDR vs PD and CCC

Example 3.10

to extreme case loss
 For fixed portfolio EAD, LGD and PD
 Lower default dependency among borrowers
 Smaller CCC
 Smaller XCDR
 Smaller XCL
 Higher default dependency among borrowers
 Larger CCC
 Larger XCDR
 Larger XCL

Application of
infinite homogeneous portfolio
 To approximate a real debt portfolio with similar
debts lent to many similar but different borrowers
 Similar debts
 Similar EAD
 Similar LGD
 Similar PD
 RM unified to one year
 Similar borrowers – borrowers with
 Similar credit quality
 Similar default dependency between any two borrowers

Model validity of
infinite homogeneous portfolio
Risk factor Criteria
EAD Coefficient of variation < 10%
LGD Coefficient of variation < 10%
PD Same credit rating or FICO score category
RM
Longer/Non-fixed term debts subject to review and control
Short term debts subject to re-investment
Short term debts < 10%
NOB > 300
CCC Same CCC formula
Example 3.11

Loss distribution of a single debt

Loss distribution of a debt portfolio

Outline
 Gaussian copula
 Appendices

Debt basket
 A collection of debts lent to a smaller number
of borrowers from the same lender
1 1
2 2
3 3
NOB NOB
EL 1-year EL
EL 1-year EL
= EL = 1-year EL
EL 1-year EL
   
   
   
   
   
   
      
Basket EL Basket 1- year EL

 
 
 
 
k
-12
NOB k
NOB-k-
-1
-12
NOB k
Probability k defaults out of NOB borrowers
Φ PD - τ CCCτ
exp - Φ
2 1 - CCCC
= dτ
2π
Φ PD - τ CCC
1 - Φ
1 - CCC
Φ PD - τ CCCτ
exp - Φ
2 1 - CCCC
2π


     
   
      
   
  
    
 
 
  


 
k
5
NOB-k-5
-1
dτ
Φ PD - τ CCC
1 - Φ
1 - CCC
  
 
   
   
  
    

Example 3.14

 
 
 
k
-1
M
NOB k
NOB-k-
-1k=0
Probability Up to M defaults out of NOB borrowers
Φ PD - τ CCCτ
exp - Φ
2 1-CCCC
dτ
2π
Φ PD - τ CCC
1 - Φ
1-CCC
Ave


                     
       
    
       
 
rage = PD NOB

 
 
k
-12
Q
NOB k
NOB-k-
-1k=0
Φ PD - τ CCCτ
exp - Φ
2 1 - CCCC
dτ = 99.9%
2π
Φ PD - τ CCC
1 - Φ
1 - CCC
Q
XCDR =
NOB


                      
       
    
       
 
Example 3.15

03.2 homogeneous debt portfolios

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