Transcript Solutions for System-wide Stress Testing

Scenario Generation & Multi-Period Portfolio
Credit Risk Analysis
Dr. Juan M. Licari
Senior Director
[email protected]
Dr. Gustavo Ordoñez-Sanz
Director
[email protected]
Oct-17 2014
1
Multi-period Credit Portfolio Models
3
Multi-period Credit Portfolio Models
Rationale & Drawbacks
» Portfolio life-time analysis: With average maturity (or duration) of the portfolio
assets being longer than a year, any profitability or new deal analysis would
benefit from multi-year modelling capabilities
» Accuracy: Even for 1 year horizon, increasing frequency (e.g. monthly, quarterly)
of calculation improves accuracy
» Time-varying exposures: Exposure today might differ from exposure at default
(or when downgrade occurs)
Practical challenges:
» Complexity: More realism always bring more complexity
» Run-time: Multi-period Monte Carlo models tend to require vast amounts of
processing power
4
Multi-period Credit Portfolio Models
Proposed Analytical Framework
» Accurate: Solutions are “exact” (up to numerical error)
» Fast: Calculations can be performed in seconds (often milliseconds)
(1) Expected Loss is an example of widely used analytical solution for credit
portfolio models. Popular metric because:
» Trivial to compute
𝑁
» Decomposes easily to each issuer/debtor in the portfolio: 𝐸𝐿 =
𝐸𝐿𝑖
𝑖=1
(2) Loss Volatility adds important information to the EL metric
Key Building Blocks:
(A) Scenario Generation + (B) Law of Total Variance & MGF analysis +
(C) Conditional Independence
5
Multi-period Credit Portfolio Models
Multi-period Credit Portfolio Modelling - Key Results
» Analytical Loss Volatility
– Multi-period Stress testing – calculate uncertainty around forecasted ST losses
– Risk Contributions to Loss Volatility – decompose volatility in the same way as EL
– “Instantaneous” risk-based pricing (RAROC) – Fast portfolio profitability analysis
– Optimal portfolio allocation – Find the investment allocation that minimises the loss
volatility for a given level of expected return
» Conditional Expectations
– Tail Risk Contributions (EC Allocation) – contribution of each instrument in portfolio
to any level of loss
– Reverse Stress Testing – Find macroeconomic scenarios (or distribution) that
produce a given level of loss
– Optimal Importance Sampling – Accelerate Monte Carlo simulations to calculate
VaR/EC
2
Macroeconomic Scenario Generation
7
Macroeconomic Scenario Generation
•
Large Scale Macro Models: Laurence Klein (Nobel 1980)
Demand-Supply systems of equations.
Pros: Explicit modeling of industries and macro sectors.
Cons: Not connected to economic theory of consumer behaviour
and production.
•
Dynamic Stochastic General Equilibrium Models (DSGE):
Sargent (Nobel 2011), Prescott (Nobel 2004), Lucas (Nobel 1995)
Modern macro models with micro foundations.
Used widely across central banks and think tanks.
Pros: Feedback effects and optimal responses can be explicitly modeled.
Cons: Limited to key macro series and very math-intensive once extended.
•
VARs and Structural VARs: Sims (Nobel 2011)
Data driven models, work-horse models for short-term forecasting.
Pros: Easier to implement, maintain and simulate. Ideal for satellite models.
Cons: Not connected to economic theory.
8
Macroeconomic Scenario Generation (cont.)
Bayesian Estimation – Prior & Posterior Distributions – Simulations
Parameter
Density
(1)
(2)
(1)
(2)
a
InvGamma
0.10
2.00
Parameter Density
w
Beta
0.50
0.20
b
InvGamma
0.10
2.00
p
Beta
0.50
0.20
g
InvGamma
0.10
2.00
w
Beta
0.50
0.20
i
InvGamma
0.10
2.00

Beta
0.50
0.15
Beta
0.75
0.10
r
InvGamma
0.10
2.00

p
InvGamma
0.10
2.00

Normal
1.25
0.12
w
InvGamma
0.10
2.00
ry
Normal
0.12
0.05
a
Beta
0.50
0.20
ry
Normal
0.12
0.05
b
Beta
0.50
0.20
l
Normal
0.00
2.00
g
Beta
0.50
0.20

Normal
0.40
0.10
i
Beta
0.50
0.20
 ga
Beta
0.50
0.20
r
Beta
0.50
0.20

Gamma
0.62
0.10
0.20
100( 1)
Gamma
0.25
0.10
p
Beta
0.50
1
f  | x  
f  x |   f  
f ( x)
9
.6
Rank Ordering of Macroeconomic Scenarios
Target 1: Max Drop in GDP growth
Scatter over Final Rank Order
0
Max quarterly drop in GDP Growth
Cumulative Proportion
-2
0
2
score_1
4
6
Target 1:
Max Drop in
GDP growth
Scatter over
Marginal
Rank Order
Max quarterly drop in GDP Growth
Cumulative Proportion
.2
Density
.4
Target 1:
Max Drop in
GDP growth
Histogram
across 2.5M
simulations
Max quarterly drop in GDP Growth
10
.15
.1
.05
Max cumulative drop in GDP Growth
0
-5
0
5
10
15
20
score_2
Cumulative Proportion
Density
.2
Target 2: Max
Cumulative Drop in
GDP growth
Histogram across
2.5M simulations
Target 2: Max
Cumulative Drop in
GDP growth
Scatter over Marginal
Rank Order
Max cumulative drop in GDP Growth
Cumulative Proportion
.25
Rank Ordering of Macroeconomic Scenarios (cont.)
Target 2: Max
Cumulative Drop
in GDP growth
Scatter over Final
Rank Order
Max cumulative drop in GDP Growth
11
.1
.15
Rank Ordering of Macroeconomic Scenarios (cont.)
.1
Target 17: Max
Cumulative Drop in
HPI growth
Histogram across
2.5M simulations
.02
.05
Density
.06
.04
Density
.08
Target 12: Max
Cumulative Drop in
Investment growth
Histogram across
2.5M simulations
Max cumulative drop in HPI Growth
0
0
Max cumulative drop in Investment Growth
0
20
score_12
40
60
-10
0
10
score_17
.8
-20
10
5
Density
.4
5
10
15
score_23
20
30
Target 24: Max
Proportional
Increase in
Unemployment
Rate
Histogram
across 2.5M
simulations
.2
Max proportional increase in Unemployment Rate
0
Max Unemployment Rate
0
Density
.6
Target 23: Max
Unemployment Rate
Histogram across
2.5M simulations
20
.9
1
1.1
score_24
1.2
1.3
Cumulative Proportion
8
Rank Ordering of Macroeconomic Scenarios (cont.)
Overall Target
Scatter over Final
Rank Order
2
4
6
Overall Target
Histogram across
2.5M simulations
Final Weighted Score
0
Density
12
.2
.4
score_overall_norm
.6
.8
Final Weighted Score
13
Rank Ordering of Macroeconomic Scenarios (cont.)
GDP q/q Growth Rates, % -- Alternative Severity Points
1
Severity_0_50
Severity_0_99
Severity_0_90
Severity_0_999
Severity_0_95
Severity_0_9999
0.5
0
1
-0.5
-1
-1.5
-2
-2.5
2
3
4
5
6
7
8
9
14
Modelling Market and Credit Risk Drivers
(1) Swaps and Sovereign Curves (term structure models)
Nelson-Siegel approach
(2) Stock Market Returns, Historical and Implied Volatilities
Global Equity Factor (GEF) related to global economic conditions
(3) CDS Spreads by Sector and Rating category
Time series model with Global Credit Factor (GCF)
(4) Credit Migration
Transition matrices for credit portfolios, two stage approach: (i) probit model
combined with (ii) quantile and time-series analysis
Satellite Model 1
1=
1
Satellite Model S
=
=
CORE MODEL
Satellite Model 3
=
Satellite Model 2
=
15
Modelling Market and Credit Risk Drivers (cont.)
Credit Migration
Table 1
Average probabilities (1983M1 - 2007M1)
Aaa
Aa
A
Baa
Ba
B
Caa-c
Def
Aaa
92.10%
7.52%
0.33%
0.00% 0.04% 0.00% 0.00% 0.00%
Aa
0.99% 90.49%
8.07%
0.37% 0.04% 0.03% 0.00% 0.02%
A
0.07%
2.76% 90.65%
5.67% 0.65% 0.15% 0.03% 0.02%
Baa
0.05%
0.24%
5.51% 87.91% 4.75% 1.14% 0.23% 0.17%
Ba
0.01%
0.07%
0.47%
6.35% 82.56% 8.60% 0.60% 1.33%
B
0.01%
0.05%
0.18%
0.52% 5.52% 82.90% 4.74% 6.08%
Caa-c
0.00%
0.02%
0.10%
1.20% 1.19% 7.12% 69.42% 20.96%
Table 2
Average probabilities (2007M6 - 2009M10)
Aaa
Aa
A
Baa
Ba
B
Caa-c
Def
Aaa
78.15% 21.71%
0.04%
0.11% 0.00% 0.00% 0.00% 0.00%
Aa
0.05% 82.65% 16.03%
0.99% 0.11% 0.02% 0.07% 0.09%
A
0.00%
0.88% 89.58%
8.24% 0.44% 0.30% 0.15% 0.41%
Baa
0.01%
0.14%
2.20% 91.95% 4.40% 0.72% 0.20% 0.38%
Ba
0.00%
0.00%
0.04%
5.10% 81.25% 10.46% 1.83% 1.32%
B
0.00%
0.00%
0.07%
0.17% 3.35% 78.31% 13.55% 4.55%
Caa-c
0.00%
0.00%
0.00%
0.14% 0.23% 5.74% 71.19% 22.70%
Figure I: Bi-Modal Nature of Credit Transitions
20
First Mode: Around
Normal/Good Credit
Conditions
10
Second Mode: Around
Stressed Credit Conditions
0
Density
30
40
Bi-Modal Distribution of Baa to Ba Credit Migrations (Bar Chart) vs. a Normal, Symmetric Distribution (Green Solid Line)
0
.02
.04
.06
baa_ba
.08
.1
3
Credit PD Models (conditional on
Macro Scenarios)
US Auto Lending & Mortgage
Markets
17
Credit Models – A Vintage/Cohort Approach
Performance Data – Vintage Segmentation – History and Forecasts
Two examples: First Mortgages & Total Auto, US National Level, Quarterly Vintages
18
Balance (Mil. USD)
100000
80000
60000
40000
10
20
30
0
Age (# of quarters since origination)
10
age
20
30
15
Balance (Mil. USD)
120000
100000
80000
60000
Mid-age Vintage: 2012Q3
Outstanding Balance (Mil.
$ - NSA) over Age
(Quarters since
Origination)
40000
2000000
3000000 auto_num
4000000
Mid-age Vintage: 2012Q3
Number of Accounts over
Age (Quarters since
Origination)
5
10
age
auto_balance
Number
6000000
age
0
Age (# of quarters since origination)
0
Age (# of quarters since origination)
0
5000000
First Vintage: 2007Q1
Outstanding Balance
(Mil. $ - NSA) over Age
(Quarters since
Origination)
20000
auto_balance
Number
First Vintage: 2007Q1
Number of Accounts
over Age (Quarters
since Origination)
0
auto_num
1000000 2000000
3000000 4000000 5000000
US Auto Models – Exposure Lifecycle
Age (# of quarters since origination)
0
5
10
age
15
19
.006
Default Rate (#)
over Vintage,
observed at
different ages
0
Age (# of quarters since origination)
0
10
20
30
.002
.002
.001
Observed PDs
.0025
age
Mid-age Vintage: 2012Q3
Default Rate (#) over Age
(Quarters since
Origination)
.0005
.001 pd
.0015
.003
.004
.005
.005
pd
.01
Observed
First Vintage: 2007Q1
Default Rate (#) over
Age (Quarters since
Origination)
Observed PDs
.015
PDs
US Auto PD Model – Historic Performance
0
Age (# of quarters since origination)
0
5
10
age
15
07Q3
08Q4
340
345
10Q1
11Q2
350
qvintage
355
pd_age_4quarters
pd_age_24quarters
12Q3
360
Vintage
pd_age_12quarters
20
Number
F(
of
48,
=
465
=
726.75
Model
475.617452
48
9.90869691
Prob
=
0.0000
Residual
5.67184058
416
.013634232
R-squared
=
0.9882
Adj
=
0.9869
=
.11677
Total
481.289292
464
Std.
1.03726141
Err.
t
>
obs
416)
F
R-squared
Root
MSE
log_pd
Coef.
P>|t|
[95%
dummy_age_1
1.631347
.0473693
34.44
0.000
1.538234
Conf.
Interval]
1.72446
dummy_age_2
2.217711
.0805545
27.53
0.000
2.059367
2.376056
dummy_age_3
2.041324
.1159124
17.61
0.000
1.813477
2.269171
dummy_age_4
1.656777
.1465386
11.31
0.000
1.368728
1.944825
dummy_age_5
1.279536
.1661623
7.70
0.000
.9529135
1.606158
dummy_age_6
.9403923
.1689269
5.57
0.000
.6083356
1.272449
dummy_age_7
.6509373
.1515249
4.30
0.000
.3530873
.9487873
dummy_age_8
.4172649
.1202897
3.47
0.001
.1808134
.6537164
dummy_age_9
.2342105
.0842995
2.78
0.006
.0685044
.3999166
dummy_age_10
.1064866
.0533284
2.00
0.046
.0016599
_Sage1
.4617167
.0382239
12.08
0.000
.3865806
.5368529
_Sage2
-2.43758
.7499189
-3.25
0.001
-3.911683
-.9634771
_Sage3
3.49146
1.867412
1.87
0.062
-.1792793
_Sage4
.0557114
1.343793
0.04
0.967
-2.585759
2.697182
_Sage5
-2.24524
1.020941
-2.20
0.028
-4.252087
-.2383932
_Sage6
12.29499
5.932834
2.07
0.039
.6329204
23.95706
_Sage7
-73.15822
17.33315
-4.22
0.000
-107.2297
-39.08674
338
.601827
.0729246
8.25
0.000
.4584803
.7451737
339
.5244618
.0729165
7.19
0.000
.3811311
.6677925
340
.4954991
.0730174
6.79
0.000
.35197
.6390282
341
.4845197
.0731541
6.62
0.000
.3407218
.6283175
342
.4445133
.0732872
6.07
0.000
.3004539
.5885728
343
.2103228
.0734188
2.86
0.004
.0660047
.3546408
344
.1467453
.0735917
1.99
0.047
.0020874
.2914032
345
.0346756
.0737311
0.47
0.638
-.1102564
.1796076
346
-.0674085
.0738179
-0.91
0.362
-.2125111
.0776941
347
-.2844975
.0739212
-3.85
0.000
-.4298031
-.1391919
348
-.4057279
.0740411
-5.48
0.000
-.5512692
-.2601866
349
-.2669378
.0741856
-3.60
0.000
-.4127632
-.1211124
350
-.1222775
.0743673
-1.64
0.101
-.26846
.0239049
351
-.2567755
.0745765
-3.44
0.001
-.4033692
-.1101818
352
-.2296637
.0748362
-3.07
0.002
-.376768
-.0825595
353
-.2030854
.0751339
-2.70
0.007
-.3507749
-.0553959
354
-.0445987
.0754846
-0.59
0.555
-.1929775
.1037801
355
-.1113241
.0758966
-1.47
0.143
-.2605128
.0378646
356
-.052321
.0764052
-0.68
0.494
-.2025093
.0978674
357
-.0819127
.0770081
-1.06
0.288
-.2332863
.0694609
358
.1114787
.077755
1.43
0.152
-.041363
.2643204
359
.0163052
.0786526
0.21
0.836
-.1383008
.1709113
360
.0436907
.0797796
0.55
0.584
-.1131306
.200512
361
.0318062
.0812167
0.39
0.696
-.12784
.1914524
362
.1373737
.0831042
1.65
0.099
-.0259829
.3007303
.2113132
7.162199
0
MS
.015
df
pd
SS
.005
Source
.01
PDs
Observed
US Auto PD Model – Fixed-Effects Panel Data Estimation
0
.1125562
.0856832
1.31
0.190
-.0558698
.2809822
364
.0530714
.0894275
0.59
0.553
-.1227147
.2288574
366
-.1760582
.1069851
-1.65
0.101
-.3863569
.0342406
367
-.1396401
.1361635
-1.03
0.306
-.4072943
lbr
.0409895
.0049577
8.27
0.000
.0312443
.0507347
gdp
-.0543942
.0093621
-5.81
0.000
-.0727972
-.0359913
.1280141
_cons
-9.938466
.076754
-129.48
0.000
-10.08934
-9.787592
.01
f_pd
.02
Predicted PDs
1000
1500
Histogram
of Residuals
500
Density
.015
0
363
.005
2000
qvintage
-.002
0
.002
resid_f_pd
.004
Residual
21
.003
+Q3
+Q5
.0025
+Q1
.002
f_pd
2012Q3 Vintage at +Q5
Default Rate (#) over Simulations
(ordered by macro ranking)
+Q4
+Q6
+Q8
.0015
.001
+Q2
0
Age (# of quarters since origination)
0
5
10
15
0
5000
10000
2500
f_pd_sim2_worst200
.0016671
.0014093
10%
.0017151
.0014103
Obs
25%
.0017998
.0014367
Sum
50%
.001903
25000
of
Wgt.
Mean
Largest
Std.
25000
.0019127
Dev.
.0001614
75%
.0020151
.0026969
90%
.0021232
.0027208
Variance
2.60e-08
95%
.0021921
.0027371
Skewness
.3919213
99%
.00234
.002739
Kurtosis
3.339519
0
.0013748
5%
2012Q3 Vintage at
+Q5: Histogram of
Default Rates (#)
2000
Smallest
.0015779
Density
Percentiles
1%
25000
1500
f _ p d PDs - 2012Q3 Vintage at +Q5
Summary Stats for Predicted
20000
Simulation ID - Ranked
1000
pd
f_pd_sim2_best200
15000
sim2
age
500
.004
Predicted PDs
+Q7
+Q9
.002
.003
Predicted PDs
US Auto PD Model – Projections – 2012Q3 Vintage
.0015
.002
.0025
f_pd
.003
Predicted PDs
22
US Auto PD Model – Projections – 2014Q3 Vintage
Predicted PDs
+Q9
+Q7
+Q5
+Q3
.002
.003
.004
Dynamic Forecast – Example of PD Projections for a “Future” Vintage
+Q6
.001
+Q4
0
+Q1
0
+Q2
+Q8
Age (# of quarters since origination)
2
pd
f_pd_sim2_best200
4
age
6
f_pd_sim2_worst200
8
23
1.5e-06
Density
5.0e-07
0
1.5e-04
3000000
345
350
qvintage
11Q2
355
12Q3
360
4000000
f_auto_num
4500000
60000
f_auto_balance
70000
5000000
Number
Vintage
0
340
10Q1
1.0e-04
08Q4
Density
07Q3
3500000
Future
Vintage
2014Q3
observed
at +Q5:
Histogram
for
Outstanding
Balance
5.0e-05
3500000
Future
Vintage
2014Q3
observed
at +Q5:
Histogram
for
Outstanding
Number
1.0e-06
Number of Accounts
Historic Number of
Accounts over Vintage,
observed at age=2
4000000 auto_num
4500000
5000000
5500000
2.0e-06
US Auto Exposure Model – Projections
40000
50000
80000
Balance
24
10
20
.003
.004
.005
0
Age (# of quarters since origination)
0
Default Rate (#)
over Vintage,
observed at
different ages
.002
pd.003
.002
.001
First Vintage: 2007Q1
PD over Age
Observed PDs
.004
.005
Observed PDs
US Mortgage PD Model – Historic Performance
30
.00005
0
Age (# of quarters since origination)
0
5
10
age
15
0
.001
Observed PDs
Mid-age Vintage:
2012Q3
Default Rate (#)
over Age
(Quarters since
Origination)
pd
.0001
.00015
age
07Q3
08Q4
340
345
10Q1
350
qvintage
pd_age_4quarters
pd_age_24quarters
11Q2
355
12Q3
360
Vintage
pd_age_12quarters
25
Predicted PDs
.0012
.0006 f_pd
.0008
.001
+Q7
2012Q3 Vintage at +Q5
Default Rate (#) over Simulations
(ordered by macro ranking)
+Q5
+Q4
.0004
10
15
age (# of quarters since origination)
Summary Stats for Predicted
f _ p d PDs - 2012Q3 Vintage at +Q5
Smallest
.0002616
.0001916
5%
.0002989
.0001948
10%
.0003202
.0002039
Obs
25%
.0003626
.0002069
Sum
50%
.0004174
25000
of
Wgt.
Mean
Largest
Std.
25000
.0004311
Dev.
.0000983
75%
.0004846
.0010698
90%
.0005572
.0010996
Variance
9.65e-09
95%
.0006107
.0012389
Skewness
1.050765
99%
.0007323
.0012793
Kurtosis
5.355223
Density
Percentiles
10000
15000
sim2
f_pd_sim2_worst200
1%
5000
20000
25000
Simulation ID - Ranked
2012Q3 Vintage at
+Q5: Histogram of
Default Rates (#)
0
pd
f_pd_sim2_best200
0
5000
5
4000
0
.0002
0
+Q8 +Q9
3000
.001
+Q3
+Q2
+Q1
2000
.002
+Q6
1000
.004
.003
Predicted PDs
US Mortgage PD Model – Projections – 2012Q3 Vintage
.0002
.0004
.0006
.0008
f_pd
.001
.0012
Predicted PDs
26
US Mortgage PD Model – Projections – 2014Q3 Vintage
Predicted PDs
+Q9
+Q8
.001
.0015
Dynamic Forecast – Example of PD Projections for a “Future” Vintage
+Q7
.0005
+Q6
+Q5
+Q2
+Q3
+Q4
0
+Q1
0
2
pd
f_pd_sim2_best200
6
4
age (# of quarters since origination)
f_pd_sim2_worst200
8
27
4.0e-06
3.0e-06
Density
2.0e-06
1.0e-06
0
1800000
2.5e-05
1500000
2.0e-05
345
350
qvintage
11Q2
355
12Q3
360
Vintage
2200000
f_mortgage_num
2400000
2600000
Number
0
340
10Q1
1.5e-05
08Q4
Density
07Q3
2000000
Future
Vintage
2014Q3
observed at
+Q5: Histogram
for Outstanding
Balance
1.0e-05
1000000
Future
Vintage
2014Q3
observed at
+Q5:
Histogram for
Outstanding
Number
5.0e-06
Number of Accounts
Historic Number of Accounts over
Vintage, observed at age=2
2000000
2500000
3000000
5.0e-06
US Mortgage Exposure Model – Projections
300000
350000
400000
f_mortgage_balance
450000
500000
Balance
4
Multi-period Credit Analytics
Key Results
29
Analytical Solutions – Can we do better than EL?
We can define the loss as:
𝑁
𝑇
𝑍
𝐿=
𝑉𝑖 𝑠′
𝑠𝑡
𝐼𝑖 𝑠 ′
𝑠𝑡
𝑓
𝑑 =
𝑖=1 𝑡=1 𝑠 ′ =1 𝑠=1
𝑤𝑧 𝐿z
𝑧=1
With
𝑉𝑖 𝑠′
𝑠≠ 𝑡|𝑧
𝑉𝑖 𝑠′
= 𝑒𝑖 𝑠 ′
𝑡|𝑧 =
𝑒𝑖 𝑠′
𝑠 𝑡|𝑧
− 𝑒𝑖 𝑠 ′
𝑠 𝑡−1|𝑧
𝑠 𝑡−1|𝑏𝑙
∙ 𝐿𝐺𝐷𝑖 𝑡
And 𝐼𝑖 𝑠′ 𝑠 𝑡
is an indicator function that is 1 if the obligor 𝑖 transitions from credit state 𝑠 ′
to state 𝑠 at time 𝑡 and state of the world and zero otherwise
30
Definitions – Cont.
Note that it is easy to split the losses due to default:
𝑍
𝑁
𝐿=
𝑇
𝑤𝑧
𝑧=1
𝑉𝑖 𝑠′
𝑖=1
𝑡|𝑧 𝐼𝑖 𝑠 ′
𝑡|𝑧
𝑡=1 𝑠 ′ =1
From losses due to downgrades (+ credit spreads / pre-payment / etc.):
𝑍
𝐿=
𝑁
𝑇
−1
𝑤𝑧
𝑧=1
𝑉𝑖 𝑠′
𝑠 𝑡|𝑧 𝐼𝑖 𝑠 ′ 𝑠 𝑡|𝑧
𝑖=1 𝑡=1 𝑠 ′ =1 𝑠=1
And all the results that follow can be split accordingly.
31
Conditional Moment Generating Function
Loss Volatility. Can be computed analytically, assuming conditional independent defaults
(on scenario) and the Law of Total Variance:
𝜎 𝐿 = 𝐸 𝑉𝑎𝑟 𝐿|
+ 𝑉𝑎𝑟 𝐸 𝐿|
Conditional Expectations. Analytical framework that allows to calculate conditional
expectations on a given loss level. Based on Thompson & McLeod (2006): “Analytic
calculation of conditional default and risk contributions using the Ensemble method”.
But extended to:
» More than one period
» Credit migrations
» Credit spreads, pre-payment risk, IR, FX, etc.
Requires the calculation of the MGF:
𝑀𝐺
𝑓 𝑥
𝛼 = 𝐸 𝑒 𝛼𝑓
𝑥
32
US Mortgage – Analytical Loss Volatility
Period
EL
($m)
Analytical
Volatility
($m)
Monte Carlo
Volatility
($m)
Q1
2556
230.2
230.1
2556
Q2
2484
303.0
303.1
5039
Q3
2424
357.5
357.4
Q4
2381
405.7
Q5
2355
Q6
Cumulative EL
($m)
Cumulative
Analytical
Volatility ($m)
Cumulative Monte
Carlo Volatility ($m)
230.2
230.1
485.0
484.8
7464
782.7
782.4
405.6
9845
1120.0
1120.0
450.6
450.3
12200
1906.6
1905.2
2336
491.2
492.0
14536
14958
1495.0
Q7
2318
530.6
530.0
16854
2348.1
2346.0
Q8
2297
567.3
566.2
19150
2819.4
2816.3
Q9
2275
598.6
597.1
21425
3315.1
3310.7
33
US Mortgage – Loss Distribution
0.08
0.06
0.04
0.02
0.00
Probability
0.10
0.12
0.14
Mortgage Cumulative Loss Distribution
20
30
40
Loss ($bn)
50
34
US Auto Lending – Analytical Loss Volatility
Period
EL
($m)
Analytical
Volatility
($m)
Monte Carlo
Volatility
($m)
Cumulative EL
($m)
Cumulative
Analytical Volatility
($m)
Cumulative
Monte Carlo
Volatility ($m)
Q1
1321
80.3
80.3
1321
80.3
80.3
Q2
1322
96.2
96.4
2644
152.4
152.5
Q3
1309
110.0
109.9
3953
235.2
235.5
Q4
1287
123.1
123.2
5239
329.6
329.7
Q5
1265
135.2
135.3
6503
436.0
436.1
Q6
1243
145.5
145.6
7747
552.1
552.3
Q7
1225
155.1
155.2
8972
677.3
677.4
Q8
1210
164.6
164.7
10182
810.6
810.7
Q9
1197
172.1
172.1
11379
950.3
950.4
35
US Auto – Loss Distribution
0.2
0.1
0.0
Probability
0.3
0.4
Autoloans Cumulative Loss Distribution
8
10
12
Loss ($bn)
14
16
36
Corporate bond/CDS portfolio
The examples shown in the following slides are based on a toy portfolio consisting of:
» 95 long bond positions on 95 bonds ($1-25m in each): 36 Aaa, 26 Aa, 34 A
» 5 short bond (5 long CDS) positions ($5-8m in each): 3 Aaa, 1 Aa, 1A
» The calculations are run over 4 years and it is assumed that all bonds redeem after that
date.
» Instruments are marked-to-market at the end of each period taking into account changes
to the credit spreads, IR and FX and any downgrades (or upgrades)
» EDF implied transition matrices
» 100% LGD assumed
» 1m MC runs
» Figures are reported in $m
37
Corporate – Analytical Loss Volatility – Intra-period
Port
Volatility
Analytical
Default EL
Analytical
Transitions
EL
Analytical
Default
Volatility
MC Default
Volatility
Analytical
Transitions
Volatility
MC
Transitions
Volatility
Period 1
0.40
20.23
2.77
2.78
80.66
80.68
Period 2
2.77
16.00
11.60
11.58
73.48
73.49
Period 3
7.33
11.60
22.62
22.72
67.30
67.31
Period 4
11.82
7.27
28.84
28.82
62.06
62.06
Default Loss in period 4
40000
30000
frequency
10000
20000
4e+05
3e+05
2e+05
1e+05
0
0e+00
frequency
5e+05
50000
6e+05
Total Loss in period 4
0
100
200
loss
300
-300
-200
-100
0
100
200
300
400
loss
*Note that both Default and Transition losses incorporate the effect of changing IR, FX and Credit Spreads
38
Corporate – Analytical Loss Volatility – Cumulative
Port
Volatility
Analytical
Default EL
Analytical
Transitions
EL
Analytical
Default
MC Default
Analytical
Transitions
MC
Transitions
Period 1
0.40
20.27
2.77
2.78
80.66
80.68
Period 2
3.16
36.24
13.18
13.18
153.33
153.92
Period 3
10.49
47.83
33.91
33.91
219.66
220.78
Period 4
22.31
55.11
28.84
59.75
280.48
282.07
Cumulative Default Loss at period 4
Cumulative Total Loss at period 4
20000
20000
40000
frequency
60000
frequency
40000
4e+05
3e+05
2e+05
1e+05
0
0
0e+00
frequency
60000
5e+05
80000
6e+05
Cumulative Transition Loss at period 4
-1000
0
200
400
600
800
-1000
-500
0
-500
0
500
1000
1500
500
loss
loss
loss
*Note that both Default and Transition losses incorporate the effect of changing IR, FX and Credit Spreads
39
Risk Contributions to Loss Volatility
We can calculate the contributions to portfolio volatility analytically. So we can manage our
portfolio
𝑁
𝜕𝜎 𝐿
𝑅𝐶𝑖 𝐿 = 𝑛𝑖
𝜕𝑛𝑖
𝜎 𝐿 =
𝑖=1
With
𝑉𝑖 𝑠′
𝑠 𝑡|𝑧
= 𝑛𝑖 ∙ 𝑉𝑖∗𝑠′
𝑅𝐶𝑖 𝐿
𝑠 𝑡|𝑧
40
US Mortgage – Risk Contributions to Loss Volatility
RCs to Cumulative Loss Volatility
RCs to Intra-period Loss Volatility
8
10
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
4
6
RC(%)
6
4
2
2
0
0
RC(%)
8
10
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
0
10
20
Vintage
30
40
0
10
20
Vintage
30
40
41
US Auto Lending – Risk Contributions to Loss Volatility
RCs to Intra-period Loss Volatility
6
8
10
12
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
0
2
4
RC(%)
2
4
6
8
10
12
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
0
RC(%)
RCs to Cumulative Loss Volatility
0
10
20
Vintage
30
40
0
10
20
Vintage
30
40
42
Expectations Conditional on Loss Level
0.04
Probability
0.02
0.02
0.00
0.00
0.01
Probability
0.03
0.06
0.04
0.08
We can find a new probability of the state of the world as a function of only one parameter
𝛼 such that 𝐸𝐿 𝛼 = 𝐿
-6
-6
-4
-2
0
2
4
-4
-2
0
2
4
6
6
Z
Z
Probability of state of the
world 𝑤𝑧 𝛼 =
= 𝑤𝑧
conditional on 𝐸𝐿 𝛼 =
= 𝐸𝐿
Probability of state of the
𝑤𝑧 𝛼 conditional on
𝐸𝐿 𝛼 = 𝑉𝑎𝑅 99.9%
43
US Auto Lending – Tail-Risk Contributions (EC Allocation)
8
Tail Risk Contributions
4
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
0.2
Loss ($bn)
12
Loss ($bn)
14
16
5
10
0
8
TRC (%)
10
0.1
15
Tail Risk Contributions
0.0
Probability
0.3
0
2
0.4
TRC (%)
6
Autoloans Cumulative Loss Distribution
10
15
20
25
Loss ($bn)
30
35
0.05
44
Probability
0.00
0.01
0.02
The solution also give us the distribution for states of the world
0.03
0.04
Analytical Reverse Stress Testing
-6
-4
-2
0
2
4
2
4
6
Probability
0.01
0.08
0.02
0.03
0.04
Z
0.00
-500
0.06
-6
-4
-2
6
Z
y
Probability
abilit
0.04
0.06
500
0.00
-6
-4
-2
0
Z
2
4
6
0.02
0.021000
0.00
Prob
Loss
0.08
0
0.04
0
-6
-4
-2
0
Z
2
4
6
45
US Auto Lending – Uniform distribution of Z
4.0e-05
2.5e-05
3.0e-05
3.5e-05
Probability
4.5e-05
5.0e-05
5.5e-05
L = EL = $11.3bn
0
5000
10000
15000
20000
25000
L = $15.8bn; EL=$11.3bn
4.15e-05
L = $4.4bn; EL=$11.3bn
4.10e-05
4.05e-05
Probability
3.95e-05
4.00e-05
4.0e-05
3.9e-05
3.90e-05
3.8e-05
Probability
4.1e-05
Z
0
5000
10000
15000
20000
25000
0
5000
10000
15000
Z
Z
20000
25000
5
Concluding Remarks and Future
Research Topics
47
Risk Adjusted Pricing (RAROC)
Analytical calculation of the portfolio volatility up to the new deal’s maturity allows us to
instantaneously price.
The Sharpe ratio of the new portfolio with the new deal (or deals) should be larger than
without:
𝜇∗ − 𝑟 𝜇 − 𝑟
>
𝜎 𝐿∗
𝜎 𝐿
BUY
48
Portfolio Optimisation
What is the portfolio composition 𝑛𝑖 that minimises the portfolio loss volatility given a level of
expected loss (and hence return) 𝐸𝐿 = 𝐿?
Optimal Portfolio Allocation for L = EL = $20.63m
15
Using the Lagrange multipliers methodology:
5
Actual; vol = $80.7m;
ELdef = $0.396m ELtrans = $20.233m
Optimal; vol = $71.1m;
ELdef = $0.337m ELtrans = $20.292m
-5
𝑛𝑖 𝐶𝑖 + 𝜆𝐸𝐿∗𝑖 =
0
solving the following system of linear equations:
Balance ($bn)
The efficient frontier can be calculated by
10
Λ 𝑛𝑖 𝜆 = 𝜎 𝐿; 𝑛𝑖 + 𝜆 𝐸𝐿 𝑛𝑖 − 𝐿
𝑁
𝑛𝑖 𝐸𝐿∗𝑖
−𝐿 =
0
20
40
60
80
Vintage
𝑖=1
- Extend the current framework to study DYNAMIC OPTIMISATION (infinite horizon)
Recursive Dynamic Programming (Bellman Equations) and the study of the optimal
solutions to the underlying stochastic difference equations
100
49
Optimal Importance Sampling
Instead of drawing random numbers with unconditional probability from conditional
distribution 𝑤𝑧 draw from conditional 𝑤𝑧 𝛼 . After the simulation reweigh the loss
𝑤
distribution using weight = 𝑧
0.04
Probability
0.02
0.02
0.00
0.01
0.00
-2
0
2
4
-6
6
-4
0.003
0.004
0.005
Total Loss in period 4
probability
-2
0
Z
Z
0.002
-4
0.001
-6
0.000
Probability
0.03
0.06
0.04
0.08
𝑤𝑧 𝛼
-300
-200
-100
0
100
loss
200
300
400
2
4
6
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