Chapter 7 homogeneous debt portfolios

Copyright © 2016 CapitaLogic Limited
This presentation file is prepared in accordance with
Chapter 7 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 7
Homogeneous
Debt Portfolios

Copyright © 2016 CapitaLogic Limited 2
Declaration
Copyright © 2016 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 (林日辉林日辉林日辉林日辉),
Principal, Structured Products Analytics, 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 one-year EL
A debt portfolio comprises N different debts
( ){ }
[ ]( )
k
N
k
k=1
N
RM
k k k k
k=1
N
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 one-year EL
Failed to incorporate diversification effect
More borrowers with smaller EADs => lower risk
Lower default dependency => lower risk
Not a valid credit risk measure for a debt
portfolio
Example 7.1
Example 7.2

– Debt portfolio
Credit risk
Default loss
Exposure
at default
Default
likelihood
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
High 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
Assume that a lender invests in many debts
with maturity
longer than one year, will review and control the
credit risk at the end of the following one year
shorter than one year, will invest the proceeds at
maturity in similar debts up to one year

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

Concentration of debts
Measured by no. of borrowers
Approaching one when fully concentrated
Approaching infinity when fully granular
Default dependency
Quantified by copula correlation coefficient

Outline
Gaussian copula
Appendices

Portfolio EAD
EAD same for all debts
EAD × NOB
LGD
Same for all debts
PD
Same for all borrowers
RM
Unified to one year
NOB
> 10
Borrowers totally independent of one another

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 mass function
Average = PD × NOB
( )
( )
( )
( )
NOB-kk
NOB k
M
NOB-kk
NOB k
k=0
Prob k defaults out of NOB borrowers
= C × PD × 1 - PD
Confidence level Up to M defaults out of NOB borrowers
= C × PD × 1 - PD 
 ∑
Example 7.3

Worst case default rate
Worst case no. of defaults
In Microsoft Excel
( )
( )
Q
NOB-kk
NOB k
k=0
C × PD × 1-PD = 99.9%
Q = Critbinom NOB, PD, 99.9%
Q
WCDR =
NOB
∑
Example 7.4

Confidence level of up to
k defaults out of NOB borrowers

Portfolio credit risk measure
Worst case loss
WCL = Portfolio EAD × LGD × WCDR

to worst case loss
For fixed portfolio EAD, LGD and PD
Lower concentration of borrowers
Larger NOB
Smaller WCDR
Smaller WCL
Higher concentration of borrowers
Smaller NOB
Larger WCDR
Larger WCL

Outline
Gaussian copula
Appendices

Bernoulli random variable
A random no. B
Either 1 with likelihood PD
Or 0 with likelihood 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. x
Mapped to a standard uniform random
variable u
( )
2
x
-
1 t
u = exp - dt = Normsdist x
22π
0 u 1
∞
 
 
 
≤ ≤
∫
Example 7.5

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 = Normsdist y ρ + z 1 - ρ
Example 7.6

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 different 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 = Normsdist y CCC + z 1 - CCC
u = Normsdist y CCC + z 1 - CCC
Example 7.7

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

Modelling NOB homogeneous borrowers
A systematic standard normal random variable y
N 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 = Normsdist y CCC + z 1 - CCC k=1,2,3,...NOB
Example 7.8

CCC for retail exposures
Residential mortgage
Qualifying revolving retail exposure
Other retail exposure
( )
( )
CCC = 0.15
CCC = 0.04
1 - exp -35PD
CCC = 0.16 - 0.13
1 - exp -35
 
 
  

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
( )
( )
( )
( )
( )
( )
1 - exp -50PD
CCC = 0.24 - 0.12
1 - exp -50
1 - exp -50PD
CCC = 0.24 - 0.12
1 - exp -50
1 - exp -50PD
CCC = 0.24 - 0.12
1 - exp -50
S - 50
+
1125
1.25
 
 
  
 
 
  
   
  
    

CCC under Basel III

Outline
Gaussian copula
Appendices

Portfolio EAD
EAD × NOB
LGD
Same for all debts
PD
RM
Unified to one year
NOB
> 10
CCC
Same between any two borrowers

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 < PDk, then borrower k defaults
Count the no. of borrowers in default
Repeat the above steps for 10,000 time
( )k k k ku = Normsdist y CCC + z 1 - CCC
Example 7.12

Worst case loss
( )
= Percentile No. of defaults, 99.9%
WCDR =
NOB

Worst case loss of
finite homogeneous portfolio
CCC NOB
PD
WCDR
LGD
Portfolio
EAD
WCL
(-)
(+)
(+)
(+)
(+)
(+)

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

Outline
Gaussian copula
Appendices

Portfolio EAD
EAD × NOB
PD
RM
Unified to one year
NOB
→ Infinity
CCC
Same between any two borrowers
Default rate (“DR”)
The percentage of borrowers in default

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 xΦ x1 - CCC
F DR = exp - dx
CCC 2 2CCC
1 - CCC Φ DR - Φ PD
= Φ
CCC
  ×     
 
 
 
  ×  
 
 
 
 ×
 
  
∫
Default rate density function
Cumulative default rate 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%
WCDR = Φ
1 - CCC
 
 
  
Example 7.9

WCDR vs PD and CCC

Worst case loss
Example 7.10

to worst case loss
For fixed portfolio EAD, LGD and PD
Lower default dependency among borrowers
Smaller CCC
Smaller WCDR
Smaller WCL
Higher default dependency among borrowers
Larger CCC
Larger WCDR
Larger WCL

Application of
infinite homogeneous portfolio
To approximate a real debt portfolio with similar
debts lent to many similar (but different) borrowers
Similar debts – Debts with
Similar EAD
Similar LGD
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 < 5%
LGD Coefficient of variation < 5%
PD Same credit rating or group of FICO score
RM Reasonably unified to1 year
NOB > 300
CCC Same CCC formula
Example 7.11

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 manageable
number of borrowers from the same lender
1 1
2 2
3 3
N N
EL 1-year EL
EL 1-year EL
= EL = 1-year EL
EL 1-year EL
   
   
   
   
   
   
      
Basket EL Basket 1- year EL
⋮ ⋮

t is a standard normal random no.
( )
( )
( )
k
-12
NOB k
NOB-k-
-1
-12
NOB k
Φ PD - t CCCt
exp - Φ
2 1 - CCCk defaults out of C
Prob = dt
NOB borrowers 2π Φ PD - t CCC
1 - Φ
1 - CCC
Φ PD - tt
exp - Φ
2C
2π
∞
∞
                   
         
   
     
 
 
 
≈
∫
( )
k
5
NOB-k-5
-1
CCC
1 - CCC
dt
Φ PD - t CCC
1 - Φ
1 - CCC
             
     
   
     
∫
Example 7.13

( )
( )
k
-12
NOB k
NOB-k-
-1
Up to M defaults
Confidence level
out of NOB borrowers
Φ PD - t CCCt
exp - Φ
2 1 - CCCC
= dt
2π Φ PD - t CCC
1 - Φ
1 - CCC
∞
∞
 
 
 
                     
       
    
       
∫
M
k=0
Average = PD NOB






×
∑

( )
( )
2
k
-1Q
NOB k
-
k=0
NOB-k
-1
t
exp -
2
Φ PD - t CCCC
Φ dt = 99.9%
2π 1 - CCC
Φ PD - t CCC
1 - Φ
1 - CCC
Q
WCDR =
NOB
∞
∞
  
   
      
                 
                 
∑ ∫
Example 7.14

Chapter 7 homogeneous debt portfolios

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