Non-life insurance mathematics

Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Solvency

• Financial control of liabilities under nearly worst-case scenarios

• Target: the reserve– which is the upper percentile of the portfolio liability

• Modelling has been covered (Risk premium calculations)

• The issue now is computation– Monte Carlo is the general tool– Some problems can be handled by simpler,

Gaussian approximations

10.2 Portfolio liabilities by simple approximation

•The portfolio loss for independent risks become Gaussian as J tends to infinity.•Assume that policy risks X1,…,XJ are stochastically independent•Mean and variance for the portfolio total are then

JJE ...)var( and ...)( 11

Introduce ).( and )( and jjjj XsdXE

)...(1

and )...(1

12

1 JJ JJ

which is average expectation and variance. Then

),(1 2

1

NXJ

J

i

d

i

as J tends to infinfity•Note that risk is underestimated for small portfolios and in branches with large claims

Normal approximations

2210

10

and

where

sd( ,E(

become

Tlength of period aover ofdeviation standard andmean the

holders,policy allfor same theare they If losses. individual theof

deviation standard andmean and andintensity claim be Let

zzz

zz

TaTa

JaJa

Χ)Χ)

Χ

The rule of double variance

5

Let X and Y be arbitrary random variables for which

)|var( and )|()( 2 xYxYEx

Then we have the important identities

)}(var{)}(E{) var(Yand )}({)( 2 XXXEYE Rule of double expectation Rule of double variance

Some notions

Examples

Random intensities

Poisson

The rule of double variance

6

)(

)()var(

)()var(

)}|({)}|(var{)var(

22

22

2

2

zz

zz

zz

TJ

E

E

NsdENE

Some notions

Examples

Random intensities

Poisson

slide previous on the formulas in the xand Let Y

Portfolio risk in general insurance

21

2121

)| var( where)(Let

t.independenally stochastic are ,...,, where...

ZZZE

ZZZZZ

Elementary rules for random sums imply

2)|var( and )|( zz NNNNE

The rule of double variance

7

This leads to the true percentile qepsilon being approximated by

JaJaqNO 10

Where phi epsilon is the upper epsilon percentile of the standard normal distribution

Some notions

Examples

Random intensities

Poisson

Fire data from DNB

0 5 10 15

0.0

00

.05

0.1

00

.15

0.2

00

.25

density.default(x = log(nyz))

N = 1751 Bandwidth = 0.3166

De

nsi

ty

Normal approximations in R

z=scan("C:/Users/wenche_adm/Desktop/Nils/uio/Exercises/Branntest.txt");# removes negativesnyz=ifelse(z>1,z,1.001);

mu=0.0065;T=1;ksiZ=mean(nyz);sigmaZ=sd(nyz);a0 = mu*T*ksiZ;a1 = sqrt(mu*T)*sqrt(sigmaZ^2+ksiZ^2);J=5000;qepsNO95=a0*J+a1*qnorm(.95)*sqrt(J);qepsNO99=a0*J+a1*qnorm(.99)*sqrt(J);qepsNO9997=a0*J+a1*qnorm(.9997)*sqrt(J);c(qepsNO95,qepsNO99,qepsNO9997);

The normal power approximationny3hat = 0;n=length(nyz);for (i in 1:n){

ny3hat = ny3hat + (nyz[i]-mean(nyz))**3}ny3hat = ny3hat/n;LargeKsihat=ny3hat/(sigmaZ**3);a2 = (LargeKsihat*sigmaZ**3+3*ksiZ*sigmaZ**2+ksiZ**3)/(sigmaZ^2+ksiZ^2);qepsNP95=a0*J+a1*qnorm(.95)*sqrt(J)+a2*(qnorm(.95)**2-1)/6;qepsNP99=a0*J+a1*qnorm(.99)*sqrt(J)+a2*(qnorm(.99)**2-1)/6;qepsNP9997=a0*J+a1*qnorm(.9997)*sqrt(J)+a2*(qnorm(.9997)**2-1)/6;c(qepsNP95,qepsNP99,qepsNP9997);

Percentile 95 % 99 % 99.97%Normal approximations 19 025 039 22 962 238 29 347 696Normal power approximations 20 408 130 26 540 012 38 086 350

Portfolio liabilities by simulation

• Monte Carlo simulation• Advantages

– More general (no restriction on use)– More versatile (easy to adapt to changing

circumstances)– Better suited for longer time horizons

• Disadvantages– Slow computationally?– Depending on claim size distribution?

An algorithm for liabilities simulation•Assume claim intensities for J policies are stored on file•Assume J different claim size distributions and payment functions H1(z),…,HJ(z) are stored•The program can be organised as follows (Algorithm 10.1)

J ,...,1

*

****

**

j*

***

*

1j

Return 8

)Ulog( and ~ UDraw 7

)( 6

* Zsize claim Draw 5

leRepeat whi 4

)Ulog( and ~ UDraw 3

do ,...,1For 2

0 1

)(),...,( models, size claim ),,..,1( :Input 0

SSUniform

zH

S

SUniform

Jj

zHzHJjT

j

Jj

0

100

200

300

400

500

600

700

Freq

uenc

y

Bin

Fire up to 88 percentile

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

0

10

20

30

40

50

60

70

80

Freq

uenc

y

Bin

Fire above 88th percentile

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

0

2

4

6

8

10

12

1250

000

1500

000

1750

000

2000

000

2250

000

2500

000

2750

000

3000

000

3250

000

3500

000

3750

000

4000

000

4250

000

4500

000

4750

000

5000

000

5250

000

5500

000

5750

000

6000

000

6250

000

6500

000

6750

000

7000

000

7250

000

7500

000

7750

000

8000

000

Mor

e

Freq

uenc

y

Bin

Fire above 95th percentile

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

Searching for the model

16

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching• How is the final model for claim size selected?• Traditional tools: QQ plots and criterion comparisons• Transformations may also be used (see Erik Bølviken’s material)

Descriptive Statistics for Variable Skadeestimat

Number of Observations 185

Number of Observations Used for Estimation 185

Minimum 331206.17

Maximum 9099311.62

Mean 2473661.54

Standard Deviation 1892916.16

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

Results from top 12% modelling

Model Selection Table

Distribution Converged -2 Log Likelihood Selected

Burr Yes 5808 No

Logn Yes 5807 No

Exp Yes 5817 No

Gamma Yes 5799 No

Igauss Yes 5804 No

Pareto Yes 5874 No

Weibull Yes 5799 Yes

Non parametric

Log-normal, Gamma

The Pareto

Extreme value

Searching

Weibull is best in top 12% modelling

Experiments in R

19

7. Mixed distribution 2

5. Mixed distribution 1

4. Weibull

3. Pareto

2. Gamma on log scale

1. Log normal distribution

6. Monte Carlo algorithm for portfolio liabilities

Check out bimodal distributions on wikipedia

Comparison of resultsPercentile 95 % 99 % 99.97%

Normal approximations 19 025 039 22 962 238 29 347 696Normal power approximations 20 408 130 26 540 012 38 086 350Monte Carlo algorithm log normal claims 12 650 847 24 915 297 102 100 605Monte Carlo algorithm gamma model for log claims 88 445 252 401 270 401 6 327 665 905

Monte Carlo algorithm mixed empirical and Weibull 20 238 159 24 017 747 30 940 560Monte Carlo algorithm empirical distribution 19 233 569 24 364 595 32 387 938

Or…check out mixture distributions on wikipedia

Solvency – day 2

Solvency

• Financial control of liabilities under nearly worst-case scenarios

• Target: the reserve– which is the upper percentile of the portfolio liability

• Modelling has been covered (Risk premium calculations)

• The issue now is computation– Monte Carlo is the general tool– Some problems can be handled by simpler,

Gaussian approximations

Structure

• Normal approximation• Monte Carlo Theory• Monte Carlo Practice – an example with

fire data from DNB

Normal approximations

2210

10

and

where

sd( ,E(

become

Tlength of period aover ofdeviation standard andmean the

holders,policy allfor same theare they If losses. individual theof

deviation standard andmean and andintensity claim be Let

zzz

zz

TaTa

JaJa

Χ)Χ)

Χ

Normal approximations

27

This leads to the true percentile qepsilon being approximated by

JaJaqNO 10

Where phi epsilon is the upper epsilon percentile of the standard normal distribution

Some notions

Examples

Random intensities

Poisson

Monte Carlo theory

e

nXXXX

n

nn !)......Pr( 111

Suppose X1, X2,… are independent and exponentially distributed with mean 1.It can then be proved

for all n >= 0 and all lambda > 0. •From (1) we see that the exponential distribution is the distribution that describes time between events in a Poisson process.•In Section 9.3 we learnt that the distribution of X1+…+Xn is gamma distributed with mean n and shape n •The Poisson process is a process in which events occur continuously and independently at a constant average rate•The Poisson probabilities on the right define the density function

which is the central model for claim numbers in property insurance. Mean and standard deviation are E(N)=lambda and sd(N)=sqrt(lambda)

(1)

,...2,1,0,!

)Pr( nen

nNn

Monte Carlo theoryIt is then utilized that Xj=-log(Uj) is exponential if Uj is uniform, and the sum X1+X2+… is monitored until it exceeds lambda, in other words

1Nreturn and stop 5

thenY f 4

)Ulog( and ~ UDraw 3

do ,...2,1For 2

0 1

:Input 0

generator Poisson 2.14 Algorithm

*

*

****

*

n

I

YYUniform

n

Y

Monte Carlo theory

proved. be to wasas

!)!1()!1/()(

and )!1/()( that means This 9.3).(section

n shape andn mean with ddistribute Gamma is But

.)()()|Pr()(

then),( is

functiondensity its If .on ngconditioniby evaluated becan which

)Pr()(

yprobabilit on the based is 2.14 Algorithm ofgenerator Poisson The

....let and 0for )(function density common

t with independenally stochastic be ,...,Let

0

1

0

1)(

1

0

)(

0

1

1

1x-

11

en

dssn

edsnesep

nessf

S

dssfedssfsSXsp

sf

S

SSp

XXSxexf

XX

nnsns

n

snn

n

ns

nnnn

n

n

nnn

nn

n2.14 Algorithmof Proof

(1)

An algorithm for liabilities simulation•Assume claim intensities for J policies are stored on file•Assume J different claim size distributions and payment functions H1(z),…,HJ(z) are stored•The program can be organised as follows (Algorithm 10.1)

J ,...,1

*

****

**

j*

***

*

1j

Return 8

)Ulog( and ~ UDraw 7

)( 6

* Zsize claim Draw 5

leRepeat whi 4

)Ulog( and ~ UDraw 3

do ,...,1For 2

0 1

)(),...,( models, size claim ),,..,1( :Input 0

SSUniform

zH

S

SUniform

Jj

zHzHJjT

j

Jj

Experiments in R

32

7. Mixed distribution 2

5. Mixed distribution 1

4. Weibull

3. Pareto

2. Gamma on log scale

1. Log normal distribution

6. Monte Carlo algorithm for portfolio liabilities

Comparison of results

Percentile 95 % 99 % 99.97%

Normal approximations 19 025 039 22 962 238 29 347 696Normal power approximations 20 408 130 26 540 012 38 086 350Monte Carlo algorithm log normal claims 12 650 847 24 915 297 102 100 605Monte Carlo algorithm gamma model for log claims 88 445 252 401 270 401 6 327 665 905

Monte Carlo algorithm mixed empirical and Weibull 20 238 159 24 017 747 30 940 560Monte Carlo algorithm empirical distribution 19 233 569 24 364 595 32 387 938