Download - Non-life insurance mathematics
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
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*
n
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n
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Monte Carlo theory
proved. be to wasas
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and )!1/()( that means This 9.3).(section
n shape andn mean with ddistribute Gamma is But
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yprobabilit on the based is 2.14 Algorithm ofgenerator Poisson The
....let and 0for )(function density common
t with independenally stochastic be ,...,Let
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1
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)(
0
1
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11
en
dssn
edsnesep
nessf
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dssfedssfsSXsp
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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