@let@token scenario @let@token weights for @let@token ... · constructing scenario weights 1.de ne...
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Scenario Weights for Importance Measurement
An R Package for Sensitivity Analysis
Andreas Tsanakas
joint work with Alberto Bettini, Pietro Millossovich and Silvana Pesenti
https://github.com/spesenti/SWIM
Actuarial Teachers’ and Researchers’ Conference
University of Liverpool, 27-28 June 2019
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Motivation
Complex quantitative models
• Capital modelling and beyond
• Granularity v opaqueness
Questions
• Which parts of the portfolio drive performance?
• Where do model-risk vulnerabilities lie?
Sensitivity analysis
SWIM - An R Package for Sensitivity Analysis 1
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Motivation
Complex quantitative models
• Capital modelling and beyond
• Granularity v opaqueness
Questions
• Which parts of the portfolio drive performance?
• Where do model-risk vulnerabilities lie?
Sensitivity analysis
SWIM - An R Package for Sensitivity Analysis 1
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Motivation
Complex quantitative models
• Capital modelling and beyond
• Granularity v opaqueness
Questions
• Which parts of the portfolio drive performance?
• Where do model-risk vulnerabilities lie?
Sensitivity analysis
• Repeated model runs
• What to do with the results?
SWIM - An R Package for Sensitivity Analysis 1
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Motivation
Complex quantitative models
• Capital modelling and beyond
• Granularity v opaqueness
Questions
• Which parts of the portfolio drive performance?
• Where do model-risk vulnerabilities lie?
Sensitivity analysis
• Repeated model runs Single model run
• What to do with the results? Consistent sensitivity measurement
SWIM - An R Package for Sensitivity Analysis 1
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Overview
Example
Scenario Weights
The SWIM package in R
SWIM - An R Package for Sensitivity Analysis 2
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Example
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A non-linear insurance portfolio
Portfolio consisting of
• Two lines of business
• Same multiplicative factor, e.g. inflation
• Reinsurance layer on the portfolio
• Reinsurance company can default
Input risk factors Output
X1 Claims from 1st LoB Y Portfolio loss
X2 Claims from 2nd LoB
X3 Multiplicative factor
X4 % of RI recovery lost
SWIM - An R Package for Sensitivity Analysis 3
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A non-linear insurance portfolio
Portfolio consisting of
• Two lines of business
• Same multiplicative factor, e.g. inflation
• Reinsurance layer on the portfolio
• Reinsurance company can default
Input risk factors Output
X1 Claims from 1st LoB Y Portfolio loss
X2 Claims from 2nd LoB
X3 Multiplicative factor
X4 % of RI recovery lost
SWIM - An R Package for Sensitivity Analysis 3
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• What if the portfolio VaR was 5% higher
than in the current model?
• How would input factors reflect that
change?
SWIM - An R Package for Sensitivity Analysis 3
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• What if the portfolio VaR was 5% higher
than in the current model?
• How would input factors reflect that
change?
SWIM - An R Package for Sensitivity Analysis 3
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR
SWIM - An R Package for Sensitivity Analysis 4
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR1.05 * VaR
SWIM - An R Package for Sensitivity Analysis 5
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR1.05 * VaR
SWIM - An R Package for Sensitivity Analysis 6
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR1.05 * VaR
SWIM - An R Package for Sensitivity Analysis 7
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR1.05 * VaR
SWIM - An R Package for Sensitivity Analysis 8
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Distribution of portfolio loss
250 300 350 400 450 500 550
0.0
0.2
0.4
0.6
0.8
1.0
Y
VaR1.05 * VaR
SWIM - An R Package for Sensitivity Analysis 9
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Scenario Weights
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Constructing scenario weights
1. Define a stress on a random variable (risk factor or output) as a
change in the value of a risk measure
- Value-at-Risk, Expected Shortfall (TVAR)
- Moments, probabilities, covariances
2. Derive scenario weights (change of measure) such that
- re-weighted output fulfills the required stress
- most plausible / least distorting, by minimising relative entropy
B Typically we have a Monte Carlo sample and work with the empirical
distribution.
SWIM - An R Package for Sensitivity Analysis 10
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Constructing scenario weights
1. Define a stress on a random variable (risk factor or output) as a
change in the value of a risk measure
- Value-at-Risk, Expected Shortfall (TVAR)
- Moments, probabilities, covariances
2. Derive scenario weights (change of measure) such that
- re-weighted output fulfills the required stress
- most plausible / least distorting, by minimising relative entropy
B Typically we have a Monte Carlo sample and work with the empirical
distribution.
SWIM - An R Package for Sensitivity Analysis 10
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Literature
Stressing moments:
• [Csiszar, 1975]
Stressing VaR and ES:
• [Pesenti et al., 2019]
See also:
• [Weber, 2007], [Glasserman & Liu, 2010], [Breuer et al., 2012]
[McNeil & Smith, 2012], [Cambou & Filipovic, 2017]
SWIM - An R Package for Sensitivity Analysis 11
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Scenario weights for a stress on VaR
Prob(Scenario i | high Y )
Prob(Scenario i | low Y )= 4.10
SWIM - An R Package for Sensitivity Analysis 12
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Scenario weights for a stress on VaR and TVaR
Y
scen
ario
wei
ghts
200 250 300 350 400 450 500 550
02
46
810
10% increase in VaR, 13% increase in TVaR
SWIM - An R Package for Sensitivity Analysis 13
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Insurance portfolio - Output
Stress VaR by 10% and TVaR by 13%, at level 0.95
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Y
empi
rical
dis
trib
utio
n
0.0
0.2
0.4
0.6
0.8
1.0
300 400 500
Baseline ModelStressed Model
SWIM - An R Package for Sensitivity Analysis 14
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Insurance portfolio - Inputs
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.05
0.10
0.15
0.20
0.25
50 100 150 200 250 300
X1
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.05
0.10
0.15
0.20
0.25
150 200 250
X2
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.05
0.10
0.15
0.20
0.25
1.00 1.05 1.10
X3
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.2 0.4 0.6 0.8 1.0
X4
empi
rical
dis
tr. fu
nctio
n
diffe
renc
e of
em
piric
al d
istr.
SWIM - An R Package for Sensitivity Analysis 15
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Insurance portfolio - statistics
X1 X2 X3 X4 Y
Mean 150 200 1.05 0.10 362
Mean, stressed 157 202 1.05 0.14 371
Relative increase 5% 1% 0% 44% 3%
Standard deviation 35 20 0.02 0.20 36
Standard deviation, stressed 43 21 0.02 0.26 50
Relative increase 25% 5% 1% 30% 38%
SWIM - An R Package for Sensitivity Analysis 16
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A sensitivity measure
Sensitivity measure for input risk factor Xi
Γi =Estressed(Xi)− E(Xi)
normalised
SWIM - An R Package for Sensitivity Analysis 17
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Real-data example
Distribution of the portfolio loss (blue) and after re-weighting (red).
0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Y
empi
rical
dis
trib
utio
n fu
nctio
ns
SWIM - An R Package for Sensitivity Analysis 18
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Real-data example
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●●
●●
● ● ●
● ●●
● ●●
● ●
● ● ● ●● ● ● ● ● ● ● ●
● ●● ●
●●
● ● ●● ●
● ●● ● ● ●● ●
● ● ●● ● ● ● ●
●
●
● ●● ●
●●
●● ●
●●
● ●
●
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
Input risk factor
Sen
sitiv
ity m
easu
re
1835
3 4712
5938 68 58
4 1634
52 69 1
41 2 61 5 40 6727 44 31 32 24 50 23 66
45 2557 15 8 22 65 64 43 17 21 60 28 13 29 39 20 55 46
9 42 1030 14 7 63 54
51
62
53 7071 36 49
72
1937 48
633 11 56
26
SWIM - An R Package for Sensitivity Analysis 19
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The SWIM package in R
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State of play
Current location of the package:
• https://github.com/spesenti/SWIM
install github("spesenti/SWIM")
Coming soon
• CRAN & vignette
From the editors of the Annals of Actuarial Science
• Special issue on Insurance Data Science
https://www.cambridge.org/core/journals/
annals-of-actuarial-science/information/call-for-papers
SWIM - An R Package for Sensitivity Analysis 20
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State of play
Current location of the package:
• https://github.com/spesenti/SWIM
install github("spesenti/SWIM")
Coming soon
• CRAN & vignette
From the editors of the Annals of Actuarial Science
• Special issue on Insurance Data Science
https://www.cambridge.org/core/journals/
annals-of-actuarial-science/information/call-for-papers
SWIM - An R Package for Sensitivity Analysis 20
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Stress
stress(type = c("VaR", "VaR ES", "mean", "mean
sd", "moment", "prob", "user"), ...)
stress(type = "VaR", x, alpha = c(0.75, 0.9),
q ratio = c(1.2, 1.2), k = 1)
SWIM - An R Package for Sensitivity Analysis 21
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Stress
stress(type = c("VaR", "VaR ES", "mean", "mean
sd", "moment", "prob", "user"), ...)
stress(type = "VaR", x, alpha = c(0.75, 0.9),
q ratio = c(1.2, 1.2), k = 1)
SWIM - An R Package for Sensitivity Analysis 21
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Plotting
plot cdf()
0.00
0.25
0.50
0.75
1.00
−500 0 500 1000 1500
Y
ecdf
stress 1
stress 2
base
SWIM - An R Package for Sensitivity Analysis 22
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Plotting
plot hist()
0
2500
5000
7500
10000
12500
−500 0 500 1000 1500
Y
hist
ogra
m stress 1
stress 2
base
SWIM - An R Package for Sensitivity Analysis 23
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Statistics
summary()
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Sensitivity Measures
sensitivity() importance rank() plot sensitivity()
●
●
●
●
●
●
●
●
●●
0.75
0.77
0.79
0.81
X1 X2 X3 X4 X5
sens
itivi
ty
●
●
stress 1
stress 2
● Gamma
SWIM - An R Package for Sensitivity Analysis 25
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THANK YOU FOR YOUR ATTENTION!
https://github.com/spesenti/SWIM
install github("spesenti/SWIM")
SWIM - An R Package for Sensitivity Analysis 25
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References I
Breuer, T., Jandacka, M., Mencıa, J., & Summer, M. (2012).
A systematic approach to multi-period stress testing of
portfolio credit risk.
Journal of Banking & Finance, 36(2), 332–340.
Cambou, M. & Filipovic, D. (2017).
Model uncertainty and scenario aggregation.
Mathematical Finance, 27(2), 534–567.
Csiszar, I. (1975).
I-divergence geometry of probability distributions and
minimization problems.
The Annals of Probability, 3(1), 146–158.
SWIM - An R Package for Sensitivity Analysis 26
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References II
Glasserman, P. & Liu, Z. (2010).
Sensitivity estimates from characteristic functions.
Operations Research, 58(6), 1611–1623.
McNeil, A. J. & Smith, A. D. (2012).
Multivariate stress scenarios and solvency.
Insurance: Mathematics and Economics, 50(3), 299–308.
Pesenti, S. M., Millossovich, P., & Tsanakas, A. (2019).
Reverse sensitivity testing: What does it take to break the
model?
European Journal of Operational Research, 274(2), 654–670.
SWIM - An R Package for Sensitivity Analysis 27
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References III
Weber, S. (2007).
Distribution-invariant risk measures, entropy, and large
deviations.
Journal of Applied Probability, 44(1), 16–40.
SWIM - An R Package for Sensitivity Analysis 28