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CIA Annual MeetingCIA Annual Meeting
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CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Christian-Marc Panneton• e-mail: [email protected]
• Christian-Marc Panneton• e-mail: [email protected]
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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0
0.5
1
1.5
2
2.5
3
50% 100% 150% 200%
0
0.5
1
1.5
2
2.5
3
-40% -20% 0% 20% 40% 60%
• Why do we need stochastic Modeling?• If returns follow a normal distribution %16%,10~ Ny
Distribution of 1-year returnsDistribution of 1-year returns Distribution of value after 1 yearDistribution of value after 1 year
%10yE %94.1112
21
eEeE y
• Prices will follow a log-normal distribution
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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0
5
10
15
-5% 0% 5% 10% 15% 20% 25% 30% 35%
• If a contract pays in one year• Max(Initial deposit, Current value)
Guarantee Pay-off DistributionGuarantee Pay-off Distribution
• How to measure risk associated with such a pay-off?
GuaranteeGuarantee• Alternatively
• Current value + Max(0, Initial deposit – Current value)
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Expected pay-offZero payoffs weights => no tail information
0
5
10
15
-5% 0% 5% 10% 15% 20% 25% 30% 35%
Pay-off DistributionPay-off Distribution
3.41%3.41%
11.42%11.42%
%39.2PayoffE
2.39%2.39%
%41.3%80 V
• Pay-off at a specified probabilityEquivalent to VaR measure
Limited tail information• Expected pay-off with a specified probability
CTE measure
More tail information %42.11%80 CTE
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• CTE calculation• Mathematical definition
Payoff
VaRXXECTE |
nmneSGX ,0max
whenwhen 0VaR
• Involves an Integral
nm
n
eVaRG
S dyyyfGCTE
0
nmn
nmn eVaRGSeSEG |
Not easy to do!Not easy to do!
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Solution: Stochastic Integration
• Simple example: Calculate 5
0
23 dxx
1253 x
125~
• Stochastic integrationGenerate a uniform random number between 0 and 5
Calculate
Repeat n times
Calculate the average of all samples
Multiply by the width of the interval: 5
23x
5,0~U
• From calculus, the exact solution is:
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• More difficult example: Calculate CTE(80%)• 1-year guarantee• At maturity, no lapse, no death, no fees
%4.11~
%16%,10~ Ny
%42.11CTE
• Exact Solution: Hardy, NAAJ April 2001
• Stochastic integrationGenerate a standard normal random number
Calculate Payoff:
Repeat n times
Sort and calculate the average of the 20% highest payoff
sS exp1
1,0~ Ns
11,0max SX
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Stochastic integration is easier to do• No complex integrals to calculate• Need only to simulate market returns and determine pay-off
according to each path
• Drawback: Computer intensive• Only 20% of random paths are used to calculate
CTE(80%)• Aggregation: the worst paths for a specific contract are
not necessarily the same for another contract• Some articles explore these topics
2003 Stochastic Modeling Symposium
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Before doing Stochastic modeling• Need a model
• Log-normal Model 2,~ Nyt• Regime Switching Model with two log-normal regimes
211,~ Nyt If in regime 1 (low volatility regime)If in regime 1 (low volatility regime)
222 ,~ Nyt If in regime 2 (high volatility regime)If in regime 2 (high volatility regime)
• Once a model is selected• How to get model parameters?
• Maximum Likelihood Estimation (MLE)
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Impact on CTE value - Log-Normal Model• CTE(80%), 10-year guarantee, TSX index
18%19%
20%10%
9%
8%
50%
70%
90%
110%
130%
150%
5.30
%80
CTE
To decrease CTE by 10%
Increase from 9.0% to 9.3%
or
Decrease from 19.0% to 18.3%
7.13
%80
CTE
High sensitivity of CTE to parameters
Higher to reflect calibration
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Comparison with Lapse Assumption• CTE(80%), 10-year guarantee, TSX index• No mortality• 8% per year lapse assumption
• Expect high sensitivity because SegFund guarantee is a lapse supported product
Increase from 9.0% to 9.3%
or
Decrease from 19.0% to 18.3%
5.30
%80
CTE
7.13
%80
CTE
To decrease CTE by 10%
9.10%80
lapse
CTEIncrease lapse from 8.0% to 8.9%
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Stability of parameters • Log-normal calibrated model parameters for TSX• From January 1956 to ...
is stable: ± 0.4%
vary more: ±0.85%
CTE(80%) volatile:CTE(80%) volatile:
85% to 147%85% to 147%
Dec 96
Dec 97
Dec 04
Dec 03
Dec 02 Dec 01
Dec 98
Dec 99
Dec 00
18.2%
18.4%
18.6%
18.8%
19.0%
19.2%
8.0% 8.5% 9.0% 9.5% 10.0%
80%90%
100%110%120%130%140%150%160%
96 97 98 99 00 01 02 03 04
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Precision of parameters• Log-normal model calibrated parameters for TSX• From January 1956 to May 2005
MLE
8.61 %
18.71 %
s.e.
2.23 %
0.45 %
CTE(80%)
partial derivative
– 30.3
13.3
Lower CTE(80%)
by 10%
+ 0.1 s.e.
– 1.7 s.e.
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Impact on CTE value - RSLN Model• CTE(80%), 10-year guarantee, TSX index
To decrease CTE by 10%
25%26%
27%-15%
-16%
-17%
80%
90%
100%
110%
120%
11%12%
13%16%
15%
14%
50%
70%
90%
110%
130%
150%
19%20%
21%3%
4%
5%
50%
70%
90%
110%
130%
150%
170%
190%
pp
pp
Increase by 0.5%
or
Decrease by 1.5%
Increase by 1.3%
or
Decrease by 2.3%
Decrease p by 0.2%
or
Increase p by 0.8%
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Stability of parameters• RSLN model parameters for TSX• From January 1956 to ...
: ± 0.7%
p12 : ± 0.5%
CTE(80%) volatile:
86% to 164%
80%90%
100%110%120%130%140%150%160%
96 97 98 99 00 01 02 03 04
Dec 03
Dec 04
Dec 02
Dec 01
Dec 00
Dec 99Dec 98
Dec 97
Dec 96
3.2%
3.4%
3.6%
3.8%
4.0%
4.2%
4.4%
4.6%
14.0% 14.5% 15.0% 15.5% 16.0%
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Precision of parameters• RSLN model parameters for TSX• From January 1956 to May 2005
p12
p21
MLE
15.4 %
11.8 %
-19.1 %
25.2 %
4.3 %
19.4 %
s.e.
2.3 %
0.6 %
11.1 %
0.6 %
1.9 %
6.7 %
CTE(80%)
partial derivative
-20.2
5.9
-7.5
3.6
42.8
-12.2
Lower CTE(80%)
by 10%
+ 0.2 s.e.
– 3.1 s.e.
+ 0.1 s.e.
– 4.9 s.e.
– 0.1 s.e.
+ 0.1 s.e.
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Maximum Likelihood Estimation (MLE)
• Given a particular set of observed data, what set of parameters gives the highest probability of observing the data?
• The likelihood function is proportional to the probability of actually observing the data, given the assumed model and a set of parameters ()
• Maximizing the likelihood function is equivalent to maximizing the probability of observing the data
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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n
iixfLl
1
|loglog
• Maximum Likelihood Estimation (MLE)• The likelihood function, L() is the joint density function of the
observed data (xt) given the parameters in
• If the returns in successive periods are independent, then this density is the product of all the individual density functions
n
iixfL
1
|
• More convenient to work with the log-likelihood
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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1
lnt
tt S
Sy
• Case Study
• Monthly Data (January 1956 to May 2005)• S&P/TSX Total Return Index• S&P 500 Total Return Index• CA-US Exchange Rate• Topix Index (Japan)
• First, convert index values to returns
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #1
• Log-normal model, one variable: S&P/TSX TR• 2 parameters to estimate: and
• Starting values for (monthly) parameters = 1% and = 5%
• Assuming and , find the density associated with each historical return• yFeb, 1956 = 3.84% =>
793.6|%84.3 f
With Excel: With Excel: NormDist(3.84%,1%,5%,False) = 6.793NormDist(3.84%,1%,5%,False) = 6.793
Formula:Formula: 2
2
1
2
1|
iy
i eyf
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #1• Take the log of the density
• Sum all log density• Sum = 994.8
• Use Excel Solver find and which will maximize the log-likelihood value• Constraint:
0
Formula:Formula: 2
21
21 ln2ln|ln
i
i
yyf
• Results:
%762.0
%508.4
8.994l
%14.912
%62.1512
Annual () %92.101
22112 e
%43.17122 1212 ee
Annualized
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #2• LN model, 2 variables: TSX TR and S&P 500 TR
• 5 parameters to estimate:• Same process except, use the joint density
183.4|%72.3%,84.3ln f
,,,, 500500 SPTSXSPTSX
ii
m
yyiY eyf
1'21
21
22
1
-LN(2*PI()) - 0.5 * LN(MDeterm()) - 0.5 * SumProduct(MMult(y; MInverse()); y) = 4.183 = 4.183
With Excel, use matrix functions:With Excel, use matrix functions:
• Then, Maximize with Excel Solver• Constraints:
0 11
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Sensitivity of CTE to Parameters
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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txxtx
tyyty wherewhere ,0~, Ntt
1
1
• Can generate independent random variables
2,0~, INvu tt
10
012I
• Simulation in practice• How to generate correlated random variables?
• Solution: linear transformation
tt u
ttt bvau subject tosubject to
,0~, Ntt
1
1
• Want
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Simulation in practice
• Solve 0tE
0tE
12 tE
12 tE
ttE
a21 b
11 • Solution:
Constraint on Constraint on
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Cases Study #3 and #4
• LN model, 3 var.: TSX, S&P 500 and CA-US• 9 parameters to estimate
• LN model, 4 var.: TSX, S&P 500, CA-US and Topix• 14 parameters to estimate
11
• Practical Issue: Constraints on correlations• Is enough?• Answer is No!• Look at the simulation process
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• With 3 variables• Want:
• Generate 3 independent random numbers: 3,0~,, INwvu ttt
wherewhere
,0~,, Nttt
1
1
1
2313
2312
1312
txxtx
tyyty
tzztz
• Need linear combination: tt uttt vu 2
1212 1
tttt cwbvau
• Solve for 13 ttE 12 tE 23 ttE
212
121323
1
b13a
212
21213232
13 11
c• Solution
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Easier with Matrix notation:
• Mathematically, it means: ,0~ Nt
2,0~ INut
tt u
ttt bvau
t
t
t
t
v
u
ba
01
tt uC
wherewhere
21
C
0 tt uCEE
C
CCCuuCEE Tt
Tt
Tt
Tt
• Square Root of Matrix by Cholesky Decomposition
1...
...1
...1
21
221
121
nn
n
n
nnnn ccc
ccC
...
0...
0...01
21
2221
1
1
21i
kikii cc
1
1
1 j
kjkikij
jjij cc
cc
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Restrictions on correlation values• Correlation matrix must be Semi-Definite Positive
• All eigenvalues are non-negative• The product of the eigenvalues of a matrix equals its determinant
xAxxT ,0
01 2
• 2x2 correlation matrix• Determinant must be non-negative:
• 3x3 correlation matrix:• Determinant must be non-negative: 12 323121
223
231
221
• Determinant of all 2x2 sub-matrices must also be non-negative: 12
21 1231 12
32
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Possible values for a 3x3 Matrix
(1, 1, 1)(1, 1, 1)(–1, 1, 1)(–1, 1, 1)
(–1, 1,–1)(–1, 1,–1)
(–1,–1,–1)(–1,–1,–1) (1,–1,–1)(1,–1,–1)
(1, 1,–1)(1, 1,–1)
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Once one value is set (e.g.: )• Possible values for
21
3231,
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
14.03.0
4.017.0
3.07.01
1
17.0
7.01
3231
32
31
30.031 70.021
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• More and more restrictions as the size is increased• For a 4x4 correlation matrix:
• Determinant must be non-negative
12 434232243
242
232
12 434131243
241
231
12 424121242
241
221
1
222
2222
413242314331422141433221
424332414331414221313221
241
232
242
231
243
221
243
242
241
232
231
221
ρρρρρρρρρρρρ
ρρρρρρρρρρρρ
ρρρρρρρρρρ
12 323121223
231
221
• Determinant of all sub-matrices must be non-negative3x33x3 2x22x2
12 ij
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Possible values for a 4th index
1
14.03.0
4.017.0
3.07.01
434241
43
42
41
41
42
43
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Once one value is set (e.g.: )• Possible values for
41
4342 , 80.042 50.041
43
42
41
15.0
14.03.0
4.017.0
5.03.07.01
4342
43
42
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 142
43
11.08.05.0
1.014.03.0
8.04.017.0
5.03.07.01
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #5• RSLN model, one variable: S&P/TSX TR
• 6 parameters to estimate:
• Initial probabilities to be in regime 1 or 2?• Define the Transition Matrix:
7.3.
2.8.
2221
1211
pp
pp
1,22,12121 ,,,,, pp
• Starting values for (monthly) parameters
%5 %,1, 11
%8 %,1, 22
%202,1 p
%301,2 p
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #5• Initial probabilities to be in regime 1 or 2
• If start in regime 1:
• Regime probabilities for next period %20%807.3.
2.8.%0%100
%0%100
• Regime probabilities in 2 periods %30%707.3.
2.8.%20%80
• The stable distribution of the chain is given by
21212221
1211
pp
pp
• Solve, %40%6021
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #5• MLE
• Calculate densities:When in regime 1
When in regime 2
• With regime switching processStarts in regime 1 and stays in regime 1
793.6| 11956, Febyf
154.4| 21956, Febyf
2608.3|1|,1,1 111,1101 yfpyf
Starts in regime 2 and switch to regime 1
8152.0|2|,2,1 111,2101 yfpyf
Starts in regime 1 and switch to regime 2
Starts in regime 2 and stays in regime 2
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Add densities conditional on regime:
7376.5|,,|2
1
2
11011
a b
ybafyf
• Take the log: 7470.1
• Incorporate the observed return information into regime probabilities
%0.71
7376.5
8152.02608.3
|
|,2,1|,1,1,|1
1
10110111
yf
yfyfyp
• Continue with subsequent historical returns
4.966l• Maximize with Excel Solver
%3.41 %,28.1, 11
%7.26 %,59.1, 22
%3.42,1 p
%4.191,2 p
9.1035l
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Multi-variate RSLN• Each index has its own regime process
• Nice in theory, but not in practice!• Simulate 10 indices
2 and per index => 40
2 regime transition probabilities per index => 20
210 correlation matrices (45 correlations per matrix) => 46,080
• Assuming 40 years of monthly historical dataonly 4,800 data points for the 10 indices!
• The density calculation will involve all possible regime combinations
410 combinations => more than 1 million joint density to value
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Multi-variate RSLN• Global Regime
• Some limitations, but a practical solution!• Simulate 10 indices
2 and per index => 40
2 global regime transition probabilities => 2
2 correlation matrices (45 correlations per matrix) => 90
• The density calculation will involve only 4 regime combinations
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Case Study #6• RSLN model, 2 variables: TSX TR & S&P500 TR
• 12 parameters to estimate
• Parameter drift
22
11
2112 pp
%26.7%59.1
%3.41%28.1
%4.19%3.4
TSX - Uni-variateTSX - Uni-variate
%64.6%79.0
%3.26%32.1
%4.18%5.6
TSX - Multi-variateTSX - Multi-variate
Can be explained by an higher probability to be in the high volatility regime
• Same process except, use the joint density: 1.2363l
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Limitation of Global Regime• Probability of being in either regime is modified
TSX - Uni-variate
High Vol. Regime Prob.
TSX - Change in
High Vol. Regime Prob.
0%
25%
50%
75%
100%
02
-56
02
-61
02
-66
02
-71
02
-76
02
-81
02
-86
02
-91
02
-96
02
-01
-40%
-20%
0%
20%
40%
60%
80%
100%
02
-56
02
-61
02
-66
02
-71
02
-76
02
-81
02
-86
02
-91
02
-96
02
-01
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
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• Limitation of Global Regime• OK if indices are in the same regime at the same time
• If not !• Fix parameters to their univariate estimates for the most significant
exposure (e.g. TSX), then estimate the remaining parameters• Add more global regimes• Add some local regimes
TSX - Uni-variateHigh Vol. Regime Prob.
0%
25%
50%
75%
100%
02
-56
02
-61
02
-66
02
-71
02
-76
02
-81
02
-86
02
-91
02
-96
02
-01
S&P 500 - Uni-variateHigh Vol. Regime Prob.
0%
25%
50%
75%
100%
02
-56
02
-61
02
-66
02
-71
02
-76
02
-81
02
-86
02
-91
02
-96
02
-01
Topix - Uni-variateHigh Vol. Regime Prob.
0%
25%
50%
75%
100%
02
-56
02
-61
02
-66
02
-71
02
-76
02
-81
02
-86
02
-91
02
-96
02
-01
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
• Adding a Third Regime• TSX TR & S&P 500 TR
• TSX TR, S&P 500 TR, CA-US, Topix
# corr. matrices
1
2
1
3
Log-lik
2362.9
2363.1
2366.2
2366.8
CTE(80%)
3.04%
3.34%
3.03%
3.50%
# regimes
2
3
# corr. matrices
1
2
1
3
Log-lik
5197.9
5215.0
5251.1
5253.3
CTE(80%)
0.42%
0.46%
0.59%
0.55%
# regimes
2
3
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
• Copulas• A function that links univariate marginals to
their full multivariate distribution• Sklar theorem:
• Provide a unifying and flexible way to study multivariate distributions
• A lot of interest and research in the context of credit derivatives: tail events
nnn yFyFyFCyyyF ,...,,...,, 221121
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
• Copula test• TSX TR & S&P 500 TR
# corr. matrices
1
2
1
3
Log-lik
2362.9
2363.1
2366.2
2366.8
CTE(80%)
3.04%
3.34%
3.03%
3.50%
# regimes
2
3
GaussianLog-lik
2363.8
2364.0
2372.4
2373.4
CTE(80%)
3.06%
3.27%
3.72%
4.06%
Student
The choice of copula can materially affect the CTE value!
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
• RSLN, 2 regimes, Student copula, 1 correlation matrix
-25%
-20%
-15%
-10% -5%
0% 5%10
%15
%20
%
-25%
-11%
3%
17%
0
5 000
10 000
15 000
20 000
25 000
TSX
SP 500
-15%-10%-5%0%-15%
-10%
-5%
0%
4.0%-5.0%
3.0%-4.0%
2.0%-3.0%
1.0%-2.0%
0.0%-1.0%
CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future
-15%-10%-5%0%-15%
-10%
-5%
0%
4.0%-5.0%
3.0%-4.0%
2.0%-3.0%
1.0%-2.0%
0.0%-1.0%
• RSLN, Student copula
2 regimes, 1 corr. matrix
-15%-10%-5%0%-15%
-10%
-5%
0%
4.0%-5.0%
3.0%-4.0%
2.0%-3.0%
1.0%-2.0%
0.0%-1.0%
3 regimes, 3 corr. matrices
CIA Annual MeetingCIA Annual Meeting
LOOKING BACK…focused on the futureLOOKING BACK…focused on the future