Estimating Credit Exposure and
Economic Capital Using Monte Carlo
Simulation
Ronald Lagnado
Vice President, MKIRisk
IPAM Conference on Financial Mathematics
January 11, 2001
Monte Carlo Simulation for
Integrated Market/Credit Risk

Random sampling generates potential
future paths of market/credit risk sources

Provides time profile of credit exposure
and distribution of losses

Facilitates effective management of credit
limits and optimal allocation of capital
Benefits of Monte Carlo
Simulation for Credit Risk Analysis
Efficient Capital Allocation
Avoid overstating credit exposure by correctly
aggregating across master agreements, time, and
market scenarios
Account for netting, collateral, less-than-perfect
correlation, mean reversion, etc.
Prudent Capital Allocation
Account for default correlation, risky collateral,
margin call lags, correlation instability, etc.
MKI Integrated Risk
Management Solution
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• Trades/deals
• Static Data
• Prices, Curves, ...
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Systems
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consistent, complete,
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systems
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Sources
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Optional
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Systems
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systems
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A
P
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Consolidation
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s
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Analytics
RV CARMA
Irregularity
notifications
Reports
Monte Carlo Simulation
Value
Begin With Current Mark-to-Market
Base
MarktoMarket
Time
Nodes
1
2
3
4
5
6
7
8
9
Time (Nodes)
Monte Carlo Simulation
Value
Advance to a Future Date
Base
MarktoMarket
Time
Nodes
1
2
3
4
5
6
7
8
9
Monte Carlo Simulation
Value
EVOLVE
RISK
DRIVERS
Base
MarktoMarket
Time
Nodes
1
2
3
4
5
6
7
8
9
Monte Carlo Simulation
Value
EVOLVE
RISK
DRIVERS
VALUE
EVERY
DEAL
Base
MarktoMarket
Time
Nodes
1
2
3
4
5
6
7
8
9
Monte Carlo Simulation
Value
EVOLVE
RISK
DRIVERS
VALUE
EVERY
DEAL
Base
MarktoMarket
Time
Nodes
ASSIGN TO
PORTFOLIOS
1
2
3
4
5
6
7
8
9
Monte Carlo Simulation
Value
NEW
MARKET
DATA
VALUE
EVERY
DEAL
Base
MarktoMarket
ASSIGN TO
PORTFOLIOS
APPLY
NETTING,
COLLATERAL,
ETC.
Time
Nodes
1
2
3
4
5
6
7
8
9
Time (Nodes)
Monte Carlo Simulation
Value
Base
MarktoMarket
Time
Nodes
Repeat for Successive Time Nodes
1
2
3
4
5
6
7
8
9
Time (Nodes)
Monte Carlo Simulation
Distribution of
Portfolio Values,
Exposures, etc.
Value
Base
MarktoMarket
Time
Nodes
Runs
1
2
3
4
5
6
7
8
9
Time (Nodes)
Credit Exposure Profiles
Portfolio Exposure Dynamics
Exposure
Max Exposure
Future
Potential
Exposure
1 Std Dev
‘Y’ Std Dev
Mean
Current
Exposure
01
T
Future Simulation Dates
Credit Relationships
Counterparty C
- Guaranteed
or not B
Counterparty
- Guaranteed
or not A
Counterparty
- Guaranteed or not
Master Agreement A2
Master Agreement A1
CSA A12
CSA A11
Trade 10003
Collateral
Trade 10002
Trade 10001
Counterparty Exposure (Netting)
Net credit exposure to Counterparty i:

NEi    Vijt 
j 1  t 1
N MA,i
N T ,ij
N CSA,ij N T ,ijk
 V
ijkt
k 1
t 1

N CSA,ij
C
usable
ijk
k 1
N T ,ijk



usable
Cijk  max 0, min  Cijk , Vijkt 
t 1








Market Risk Drivers
Interest Rates

Base Term Structures

Spread Term Structures
Exchange Rates
Equities

Indexes

Individual Stocks
Commodities

Spot Prices

Forward Prices
Implied Volatility Surfaces
Example: Interest Rate Process
d ln r (t )  A[μ  ln r (t )]dt  ΣdZ(t )
r
vector of interest rates drivers

vector of mean reversion levels
A
matrix of mean reversion speeds

instantaneous covariance matrix
Z
vector of independent Brownian motions
Example: Interest Rate Process
Integrate over time step: discrete VAR(1) process
x (t )  ln r (t )
x (t  t )  ( I  e  tA )   e  tA x (t )  e  tA e tA
t  t
A
e
 dZ( )
t
x (t  t )  w ( t )   1 ( t )x (t )  
 ~ N (0,  )
  e
 tA
t
  e  e
0
A
T A T
d  e
 tA T
USD Libor Rates (1991-2000)
10
9
8
6
5
4
3
Jan-98
Jan-97
Jan-96
Jan-95
Jan-94
Jan-93
0
Jan-92
1
Jan-00
2
Jan-99
1-month rate
1-year rate
10-year rate
Jan-91
Rate (%)
7
Parameter Estimates: USD Libor
rates:
1m
3m
6m
1y
2y
3y
5y
7y
10y
speed:
0.51
0.37
0.42
0.51
0.50
0.64
0.78
0.80
0.78
volatility:
0.23
0.19
0.20
0.20
0.16
0.16
0.15
0.14
0.13
correlation:
1.
0.39
1.
0.34
0.48
1.
0.24
0.35
0.53
1.
0.23
0.35
0.40
0.51
1.
0.22
0.33
0.38
0.49
0.97
1.
0.20
0.31
0.36
0.46
0.93
0.95 1.
0.19
0.29
0.34
0.44
0.88
0.91 0.96
1.
0.17
0.27
0.31
0.42
0.83
0.87 0.93
0.96 1.
Interest Rate Drivers
initial rates
long-term reversion levels
10
9
8
Rate (%)
7
6
5
4
3
2
1
0.0833333
0.25
0.5
1
2
Maturity (Years)
3
5
7
10
Exposure Profile
5-Year Swap
mean (GBM)
99% (GBM)
mean (MRH)
99% (MRH)
14
Exposure (% Notional)
12
10
8
6
4
2
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time (Years)
3.5
4.0
4.5
5.0
Exposure Profile
5-Year Swap
mean (MRH)
99% (MRH)
mean (MR0)
99% (MR0)
14
Exposure (% Notional)
12
10
8
6
4
2
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time (Years)
3.5
4.0
4.5
5.0
Option Exposure: Comparison of
Exact Results with Monte Carlo
Equity Index Call Option
expiration: 2 years
implied volatility: 20%
initially at-the-money
Underlying Stochastic Parameters
drift: 15%
volatility: 20%
Monte Carlo Simulation: Weekly Time-Steps
Exact Results: Obtained with Gauss-Hermite Quadrature
Exposure Profile
2-Year Call Option
mean
mean + 2.33 std
99%
Exposure (USD Millions)
160
140
120
100
80
60
40
20
0
0
0.25
0.5
0.75
1
Time (Years)
1.25
1.5
1.75
2
Exposure Profile
2-Year Call Option
99% (exact)
99% (1000 paths)
99% (10000 paths)
Exposure (USD Millions)
160
140
120
100
80
60
40
20
0
0
0.25
0.5
0.75
1
Time (Years)
1.25
1.5
1.75
2
Exposure Profile
2-Year Call Option
99% (cv)
99% (sv: vol = 1.5, corr = 0)
99% (sv: vol = 1.5, corr = -0.7)
Exposure (USD Millions)
160
140
120
100
80
60
40
20
0
0
0.25
0.5
0.75
1
Time (Years)
1.25
1.5
1.75
2
Simulation of Dynamic Collateral
and Margin Call Lags
Example:
Single Counterparty
Single Transaction: 2-year equity call option
Margin Call Parameters
Threshold: $30 Million
Margin Call Lag: 4 weeks
Delivery Lag: 1 week
Excess Collateral Returned Immediately
Monte Carlo Simulation: 10000 paths
Option Exposure Profile
Margin Call Lag = 4w, Delivery Lag = 1w
80
99% Exposure (USD millions)
70
60
50
40
30
time steps: daily
20
time steps: 3 months
10
time steps: 1.5 months
0
1
51
101
151
201
251
Day
301
351
401
451
501
Option Exposure Profile
Margin Call Lag = 4w, Delivery Lag = 1w
80
99% Exposure (USD millions)
70
60
50
40
30
time steps: daily
20
time steps: 11w,1w,...
10
time steps: 7w,5w,...
0
1
51
101
151
201
251
Day
301
351
401
451
501
Option Exposure Profile
Margin Call Lag = 4w, Delivery Lag = 1w
80
99% Exposure (USD Millions)
70
60
50
40
30
20
time steps: daily
10
time steps: 7w,4w,1w,...
0
1
51
101
151
201
251
Day
301
351
401
451
501
Losses and Capital Calculation
Model Requirements

Exposure Profiles

Credit Quality Migration and Default (Correlated)

Stochastic Recovery
Benefits

Loss Reserves and Economic Capital

Capital Allocation across Business Units

Performance Measures (RAROC)

Incremental Capital and Capital-Based Pricing
The Losses Distribution
Distribution of Losses
(Integrated
Market/Credit Risk Simulation)
Losses
PDF
0
PV(Losses))
The Losses Distribution
Distribution of Losses
(Integrated
Losses
PDF
Market/Credit Risk Simulation)
Expected Losses
0
PV(Losses))
The Losses Distribution
Distribution of Losses
(Integrated
Losses
PDF
Market/Credit Risk Simulation)
Expected Losses
Unexpected Losses
0
PV(Losses))
The Losses Distribution
Distribution of Losses
(Integrated
Losses
PDF
Market/Credit Risk Simulation)
Expected Losses
(Reserves)
Unexpected Losses
(Economic Capital)
0
PV(Losses))
Credit Migration Model
Markov chain with transition probability matrix:
P(t , T )  [ pij (t , T )]
pij (t , T )  probability of migrating from rating i to
rating
j
during the time interval [t , T ]
P(T1, T3 )  P(T1, T2 )P(T2 , T3 )
(T1  T2  T3 )
Credit Migration Model
Time Inhomogeneous:
P(T1, T1  T )  P(T2 , T2  T )
Time Homogeneous:
P(t , T )  exp[( T  t )G]
Typical Transition Matrix (1-Year)
Initial
Rating
Year-End Rating
AAA
AA
A
BBB
BB
B
CCC
D
AAA
90.81
8.33
0.68
0.06
0.12
0
0
0
AA
0.70
90.65
7.79
0.64
0.06
0.14
0.02
0
A
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
BBB
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
BB
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
B
0
0.11
0.24
0.43
6.48
83.46
4.07
5.20
CCC
0.22
0
0.22
1.30
2.38
11.24 64.86 19.79
Credit Quality Migration and
Default Correlation
Factor Model for Asset Value Return
For each counterparty (i
 1,2,..., N c )
Nf
RAi   w R f j  w Z i
j 1
i
j
RAi ~ N (0,1)
i
s
Credit Migration Quantiles
BBB
BB
A
B
AA
CCC
AAA
D
0
% Change in Firm Value (Normalized)
Relating Asset Returns to Default
Correlation
Asset-Return Correlation:
Nf
Nf
 A,ij   w wl corr ( R f , R f )
i
k
k 1 l 1
j
k
Default Correlation:
 D ,ij 

l

N 2 N ( pi ), N ( p j ),  A,ij  pi p j
1
1
pi (1  pi ) p j (1  p j )
Default Correlation vs. Asset Correlation
Default correlation
(Identical Default Probabilities = 0.02)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4
Asset correlation
0.6
0.8
1
Losses
discrete time nodes:
t  0,1,..., T
r0 , r1,..., rT 
idiosyncratic credit driver path: z1 , z 2 ,..., zT 
market risk driver path:
default stopping time:
T
i
Nc
L[0,T ]    (t )Vi (t; r0 ,..., rt )di (t; r0 ,..., rt ; z1,..., zt )
t 1 i 1
1  i ( t 1,t ]
d i (t;...)  {
0  i ( t 1,t ]
Loss Statistics (Simplified Case)
Single-period; Independent exposure and default
Nc
E ( L)  Vi pi
i 1
Nc
Nc
var( L)   (Vi ) pi (1  pi )    i pi
2
i 1
Nc
2
i 1
i 1
 2  ij i j pi p j
i 1 j 1
Nc
i 1
 2  (ViV j  ij i j ) D ,ij
i 1 j 1
pi (1  pi ) p j (1  p j )
Loss Statistics (Simplified Case)
Single-period
Constant and identical exposures
Identical default probabilities and correlations
E ( L)  N c p
var( L)  N c p(1  p)[1  ( N c  1)  D ]
lim var( L / N c )  p(1  p) D
N c 
Loss distributions:
500 counterparties, constant exposures, p = 0.05
rhoa = 0, rhod = 0
rhoa = 0.05, rhod = 0.012
400
600
300
400
200
200
100
0
10
20
30
40
50
0
0
rhoa = 0.1, rhod = 0.026
50
100
rhoa = 0.25, rhod = 0.077
800
1500
600
1000
400
500
200
0
0
50
100
150
0
0
100
200
300
Tolerance Intervals
Ordered sample of losses from Monte Carlo simulation:
L(1)  L( 2 )    L( n )
Estimated 100 p % quantile:
Qˆ p  L([ np]1)
Distribution of order statistics:
n
P{L( r )  x}   Cn ,i [ FL ( x )] [1  FL ( x )]
i
i r
n i
Tolerance Intervals
Construct non-parametric
for estimated quantile:
100 % confidence interval
s 1
P{L( r )  Q p  L( s ) }   Cn ,i p (1  p )
i
i r
n i

Convergence of Unexpected
Losses
500 counterparties, 550 deals, 1 year horizon
Runs
1000
10000
30000
99th
Percentile
4,516,000
(10)
4,818,000
(100)
4,971,322
(300)
90%
Tolerance
5,324,000
(6)
5,225,000
(87)
5,225,000
(278)
95%
Tolerance
5,768,010
(5)
5,361,000
(84)
5,290,000
(272)
99%
Tolerance
7,041,449
(3)
5,494,000
(78)
5,394,000
(261)
Summary
Monte Carlo simulation is preferred approach for
integrated market/credit risk analysis

Reveals distributions of future credit exposure and
losses to default

Facilitates efficient capital allocation by correctly
aggregating exposure across time and market
scenarios

Leads to prudent capital allocation by accounting for
market complexities