cia annual meeting

CIA Annual MeetingCIA Annual Meeting

LOOKING BACK…focused on the futureLOOKING BACK…focused on the future

CIA Annual MeetingCIA Annual MeetingSession 2403: Stochastic Modeling Session 2403: Stochastic Modeling


• Christian-Marc Panneton• e-mail: [email protected]

• Christian-Marc Panneton• e-mail: [email protected]



• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Copulas



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



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)



• 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



• 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!



• 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:



• 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



• 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



• 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)



• 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



• 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


9.10%80

lapse

CTEIncrease lapse from 8.0% to 8.9%



• 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



• 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.



• Impact on CTE value - RSLN Model• CTE(80%), 10-year guarantee, TSX index


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%



• 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%



• 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.



• 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



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



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



• 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



• 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



• 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



• Agenda• Why we need Stochastic Modeling• Importance of Parameters• LN Model and Parameters• Simulation - Correlation• RSLN Model and Parameters• Sensitivity of CTE to Parameters



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



• Simulation in practice

• Solve 0tE

0tE

12 tE

12 tE

ttE

a21 b

11 • Solution:

Constraint on Constraint on



• 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



• 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



• 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



• 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



• 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)



• 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



• 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



• Possible values for a 4th index

1

14.03.0

4.017.0

3.07.01

434241

43

42

41

41

42

43



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• 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!



• 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%



-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

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