Transcript IME 2006 - unipi.gr

Rocco Roberto Cerchiara
University of Calabria – Italy
email: [email protected]
An Overview On Fast Fourier Transform
For Aggregate Losses And Stochastic
Loss Reserves
Introduction
• The development of Aggregate Loss Models is one of the most
important key issues concerning classical actuarial
mathematics and most recently the Solvency II Project.
• In this paper an investigation on Fast Fourier Transform
Method is given to show the possibility to model Non-Life
Insurance Aggregate Losses and so to define Premium and
Loss Reserve estimates under Collective Risk Theory
17 July, 2015
R. R. Cerchiara - IME 2007
2
Collective Risk Theory
~
k
~
~
X   Zi
i 1
 k    1 1

pk  
k

 1  




 

1 



k
k  0,1,2,...;  0;   0
   1   t  1
Pk~ (t )  E t
~
k

Heckman and Meyers [1983] use the “contagion
parameter” c to describe Binomial Negative Distribution:


~
~
~
Var[k ]  E[k ]  1  cE[k ]
17 July, 2015
R. R. Cerchiara - IME 2007
3
Inversion Method
FFT
17 July, 2015
R. R. Cerchiara - IME 2007
4
Inversion Method
• Inversion Method has been used for numerical
evaluation of CDF (see Seal [1971]) or of linked
functions, like aggregate excess of loss premium (see
Heckman and Meyers [1983]), using a transform of
one between Probability Generating Function (PGF),
Moments Generating Function (MGF) or
Characteristic Function (CF).
17 July, 2015
R. R. Cerchiara - IME 2007
5
Inversion Methods of CF
• We can consider three categories:
• Fast Fourier Transform (Robertson [1983], Wang
[1998])
• Direct Numeric Inversion (Heckman and Meyers
[1983]).
• Inversion Method based on Fourier Trasform for
claim size variables which are not identically
distributed (see Ferrara C. et al. [1996]).
17 July, 2015
R. R. Cerchiara - IME 2007
6
FFT
•
Fast Fourier Transform (FFT) is an algorithm based on Fourier Transform,
that is used to inverte the CF to obtain the CDF of discrete random variables
(using Inverse FFT).
•
FFT allows a great computational time saving and it is very common in
several softwares available in Excel, Matlab, R, etc. which have different
maximum points to use for the discrete distribution.
•
See Robertson [1983] and Klugman et al. [1998] for a detailed description of
this technique.
See Bühlmann [1984] for an interesting comparison between FFT and Panjer
recursion formula using the so called “C Criterion”.
Wang S. [1998] showed how FFT can be useful to aggregate Line of Business
results as a possible alternative approach to Copulas.
•
•
 
 X~ (t )  E e
17 July, 2015
~
itX

 Ek~ E e 
~ ~
~
it Z1  Z 2 ... Z k



 k~  k  E ~  ~ (t ) k~  P~  ~ (t ) 
Z
Z
k
k
R. R. Cerchiara - IME 2007
7
Solvency II
QIS
17 July, 2015
R. R. Cerchiara - IME 2007
8
Quantitative Impact Studies
Only one approach for both Premium Risk and
Reserving Risk: QIS3 proposed a combined capital
requirement for these risks (not QIS2).
FFT is useful for Premium Risk and Reserve Risk
analysis and modeling reinsurance too.
In this paper the use of FFT will be showed to model
both Premium and Loss Reserve (with Cape Code
method and Maximum Likelihood Estimation).
17 July, 2015
R. R. Cerchiara - IME 2007
9
Premium Risk
17 July, 2015
R. R. Cerchiara - IME 2007
10
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
•
•
•
•
•
Homer and Clark [2004] has shown a case study for a Non-Life insurance
company which has developed solvency assessment model, based on simulation
and Collective Risk Theory approach, with the goal to calculate the aggregate
losses probability distribution.
The model made the assumption that small and large aggregate losses are
independent. So small and large losses were simulated separately and results
were summed. Typically this assumption is not true. Cerchiara [2007] shows an
example, based on Homer and Clark [2004] results, of integrated use of three
different approaches, where EVT permits the definition of the truncation point
between small and large claims. Two-dimensional FFT allows modeling not only
aggregate losses, but dependence between its basic components too, and Monte
Carlo simulation describes large claims behaviour.
The model works in an efficient way preserving dependencestructure.
Total aggregate loss expected value produced from small and large claims,
E[XSmall,Large], will be calculated, using both example shown in Homer and Clark
[2004] and Danish Fire Insurance database, used in Cerchiara [2006], in relation
to the period 1980-2002, where amounts are expressed in DK.
Calculation algorithms have been developed using Matlab.
17 July, 2015
R. R. Cerchiara - IME 2007
11
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
•
Consider a Small claim size distribution ZSmall which has 6 points (0, 2x106, 4x106, 6x106,
8x106, 10x106) with the following probabilities: 0.0%, 43.8%, 24.6%, 13.8%, 7.8%,
10.0%.
•
This probability distribution can be represented using the bivariate matrix MZ (see Homer
and Clark [2004] and Cerchiara [2006]), which permits the application of two-dimensional
FFT. In particular the first column shows the ZSmall probability distribution with exclusion of
the truncation point 10*106, defined before with EVT and the cell (0,1) the probability that
claim size is above the 10*106 (10%).
Large claim number (kLarge)
Small claim size (ZSmall)
0
1
2
3
4
5
-
0.000
0.100
0.000
0.000
0.000
0.000
2,000,000
0.438
0.000
0.000
0.000
0.000
0.000
4,000,000
0.246
0.000
0.000
0.000
0.000
0.000
6,000,000
0.138
0.000
0.000
0.000
0.000
0.000
8,000000
0.078
0.000
0.000
0.000
0.000
0.000
10,000,000
0.000
0.000
0.000
0.000
0.000
0.000
17 July, 2015
R. R. Cerchiara - IME 2007
This
representation
gives an
important
example how of
two dimensional
FFT is flexible,
because it can
work on variables
measured in
different ways
(number and
amount).
12
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
• In this paper, under dependence assumption, Binomial Negative
•
distribution is used for claim number with mean=10 and
variance=20 (hypothesis, while for the real data considered, the
overall frequency is much higher, however it will be continued with
this simplified assumption for clarity).
So bivariate aggregate distribution will be:
M X  IFFT( PGF( FFT(M Z )))
•
and
PGF  (2  t ) 10
where PGF is Probability Generation Function of Binomial Negative
distribution, FFT is two-dimensional Fast Fourier Transform
procedure and IFFT is two-dimensional Inverse FFT (see Klugman
et al. [1998], Wang [1998], Homer and Clark [2004] and Cerchiara
[2006]).
The matrix MX gives bivariate distribution of large claim number,
kLarge, and small claim aggregate loss, XSmall (discretized with 26
points). Marginal distribution of XSmall is obtained as the sum of
probabilities of each row.
17 July, 2015
R. R. Cerchiara - IME 2007
13
Large claim number (k
X Small
Matrix MX :
Bivariate
Distribution of
XSmall and Klarge
17 July, 2015
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
12,000,000
14,000,000
16,000,000
18,000,000
20,000,000
22,000,000
24,000,000
26,000,000
28,000,000
30,000,000
32,000,000
34,000,000
36,000,000
38,000,000
40,000,000
42,000,000
44,000,000
46,000,000
48,000,000
50,000,000
0
0.37%
0.57%
0.74%
0.91%
1.19%
1.44%
1.71%
1.87%
2.10%
2.23%
2.41%
2.46%
2.55%
2.44%
2.39%
2.31%
2.21%
2.09%
1.95%
1.81%
1.70%
1.66%
1.40%
1.32%
1.16%
1.03%
1
0.15%
0.25%
0.36%
0.50%
0.65%
0.83%
1.03%
1.23%
1.42%
1.78%
1.82%
1.83%
1.90%
1.94%
1.95%
1.93%
1.87%
1.80%
1.71%
1.61%
1.57%
1.55%
1.26%
1.20%
1.04%
0.93%
2
0.03%
0.07%
0.09%
0.14%
0.20%
0.28%
0.36%
0.45%
0.54%
0.62%
0.70%
0.77%
0.82%
0.86%
0.88%
0.89%
0.89%
0.87%
0.84%
0.81%
0.76%
0.72%
0.66%
0.61%
0.56%
0.50%
3
0.01%
0.02%
0.03%
0.04%
0.06%
0.08%
0.10%
0.13%
0.16%
0.19%
0.21%
0.24%
0.26%
0.28%
0.30%
0.31%
0.31%
0.31%
0.31%
0.30%
0.29%
0.28%
0.26%
0.25%
0.23%
0.21%
R. R. Cerchiara - IME 2007
Large
4
0.00%
0.00%
0.00%
0.00%
0.01%
0.01%
0.02%
0.03%
0.03%
0.04%
0.05%
0.06%
0.06%
0.07%
0.08%
0.08%
0.08%
0.09%
0.09%
0.09%
0.08%
0.08%
0.08%
0.08%
0.07%
0.07%
)
5
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.01%
0.01%
0.01%
0.01%
0.01%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
0.02%
6
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.01%
0.01%
0.01%
0.00%
0.00%
0.00%
0.00%
Total

pr X Small  x

0.6%
0.9%
1.2%
1.6%
2.1%
2.6%
3.2%
3.7%
4.3%
4.9%
5.2%
5.4%
5.6%
5.6%
5.6%
5.5%
5.4%
5.2%
4.9%
4.6%
4.4%
4.3%
3.7%
3.5%
3.1%
2.8%
100.0%
14
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
Small
] = 28.0x106.
• So E[ X
• Next, conditional frequency distribution for large claims is
obtained, “rescaling” the matrix MX, where each value is divided by
the corresponding row total value.
• Next Table shows clearly dependence between small and large
claims, observing that in each row there are probabilities to have
ki = 0, 1, 2, . . . , 6 large claims.
• For example, in the column where ki=0, probabilities decrease for
increasing values of XSmall, because if XSmall increases, the
probability of having no large claims has to decrease.
17 July, 2015
R. R. Cerchiara - IME 2007
15
Large
Large claim number (k
X Small
Rescaling of
Matrix MX
17 July, 2015
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
12,000,000
14,000,000
16,000,000
18,000,000
20,000,000
22,000,000
24,000,000
26,000,000
28,000,000
30,000,000
32,000,000
34,000,000
36,000,000
38,000,000
40,000,000
42,000,000
44,000,000
46,000,000
48,000,000
50,000,000
0
64.7%
62.4%
60.3%
57.0%
56.3%
54.5%
53.1%
50.4%
49.3%
45.8%
46.3%
45.8%
45.5%
43.5%
42.5%
41.7%
41.1%
40.3%
39.6%
39.0%
38.4%
38.5%
38.0%
37.9%
37.5%
1
26.2%
27.4%
29.3%
31.3%
30.8%
31.4%
32.0%
33.1%
33.3%
36.5%
35.0%
34.1%
33.9%
34.6%
34.7%
34.8%
34.7%
34.7%
34.7%
34.7%
35.5%
35.9%
34.2%
34.5%
33.7%
2
5.2%
7.2%
7.3%
8.8%
9.5%
10.6%
11.2%
12.1%
12.7%
12.7%
13.5%
14.3%
14.6%
15.3%
15.7%
16.1%
16.5%
16.8%
17.1%
17.4%
17.2%
16.7%
17.9%
17.5%
18.2%
3
1.6%
1.6%
2.0%
2.2%
2.6%
2.8%
2.9%
3.4%
3.6%
3.8%
3.9%
4.4%
4.6%
4.9%
5.2%
5.5%
5.7%
5.9%
6.2%
6.4%
6.4%
6.4%
6.9%
7.1%
7.4%
4
0.8%
0.4%
0.3%
0.2%
0.5%
0.4%
0.6%
0.8%
0.7%
0.8%
1.0%
1.1%
1.1%
1.2%
1.4%
1.4%
1.5%
1.8%
1.9%
1.9%
1.8%
1.9%
2.2%
2.2%
2.4%
)
5
0.8%
0.4%
0.3%
0.2%
0.2%
0.1%
0.1%
0.1%
0.3%
0.3%
0.3%
0.3%
0.3%
0.3%
0.4%
0.4%
0.4%
0.4%
0.5%
0.5%
0.5%
0.5%
0.6%
0.6%
0.6%
6
0.8%
0.4%
0.3%
0.2%
0.2%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.1%
0.2%
Row
Total
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
37.3% 33.8% 18.1% 7.4% 2.5% 0.7% 0.2% 100.0%
R. R. Cerchiara - IME 2007
16
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
• Using the previous matrix it is possible to calculate expected value
of large claim number:
E[k
L arg e
26 7


]    k i  Pr k L arg e  k i X Small  x j  0.871
j 1 i 1
• Next, with 200,000 simulations for ZLarge using Generalized Pareto
Distribution, with parameters
  0.5 and   8.8 (see Cerchiara
[2006]):
E[Z L arg e ]  276.3*106
• Finally, expected value of total aggregate loss is:
E[ X Small ,L arg e ]  E[ X Small ]  E[k L arg e ]  E[Z L arg e ]  268.8x106
17 July, 2015
R. R. Cerchiara - IME 2007
17
FFT, Extreme Value Theory and Simulation to
model Non-Life Insurance claims dependences
•
•
•
Solvency assessment model procedure is more efficient than the
initial independence case and it is based on the following steps:
• a) Defining truncation point with EVT.
• b) Using two-dimensional FFT to calculate marginal distribution of
XSmall.
• c) Calculating expected value of XSmall using the previous marginal
distribution.
• d) Calculating expected value of large claim number E[kLarge] using
conditional distribution to XSmall distribution.
• e) Simulating large claim size in order to obtain E[ZLarge].
Dependence structure has also been preserved.
This example gives a simplified case study of integration between
EVT, FFT and simulations for bivariate random variables which
could be interesting for the Internal Risk Models definition under
the Solvency II European insurance project.
17 July, 2015
R. R. Cerchiara - IME 2007
18
Reserve Risk
17 July, 2015
R. R. Cerchiara - IME 2007
19
QIS3
Loss Reserve
17 July, 2015
R. R. Cerchiara - IME 2007
20
QIS3
Stochastic Loss Reserve
• Use at least two actuarial methods
• Use stochastic models in coherence with actuarial
best practice : in Italian Actuarial Practice Chain
Ladder, Fisher Lange and Taylor Separation
(deterministic) methods are typically used.
• Best Estimate from the “best” (appropriate)
method
• Goodness-of-fit tests
17 July, 2015
R. R. Cerchiara - IME 2007
21
Stochastic Loss Reserve
References
• There is a wide range of books and papers on Stochastic
•
•
•
•
Reserving Models.
It’s worth mentioning books like Loss Reserving Methods
[1981], Claim Reserving Manual [1997], Taylor [2000] and see
also the Casualty Actuarial Society website.
Mack [1993] developed a model to estimate loss reserve and
prediction error without parametric hypothesis.
England and Verrall [2002] presented a wide overview on
models which reproduce Chain Ladder estimate and on
Bornhuetter-Ferguson technique with different approaches like
Generalised Model, Bootstrap, Simulation, etc.
For a more updated (as at June 18, 2007) references list see on:
http://www.math.tudresden.de/sto/schmidt/dsvm/reserve.pdf
17 July, 2015
R. R. Cerchiara - IME 2007
22
Stochastic Loss Reserve
• The application considered is based on Clark [2003] and
•
•
•
•
Meyers [2006], [2007] papers.
Clark [2003] compared Loglogistic and Weibull parametric
models, Cape Code and LDF methods using Overdispersed
Poisson Distribution (ODP) to loss reserve estimation and
Maximum Likelihood Method to estimate parameters.
Meyers [2006], [2007] used FFT and Compound Negative
Binomial (CNB) model to describe reserve distribution and
compared Maximum Likelihood and Bayes’ Theorem to
parameters estimation.
In this paper FFT (Meyers Model) is used to define distribution
and moments of reserve, trying to take into account all typical
elements of underlying risk. Meyers R software has been
used.
Mack and Bootstrap Models comparison results has been
developed.
17 July, 2015
R. R. Cerchiara - IME 2007
23
Meyers [2006] Data - Example 1
Cumulative Claims Paid (€ ,000)
Accident
Devel.
Year
Premium
Year
1
2
3
4
5
6
7
8
9
10
1997
6,929
16,303
23,346
28,721
31,006
34,195
34,346
34,819
34,855
35,492
1998
8,171
18,380
26,770
32,283
38,186
42,811
44,593
44,713
44,795
1999
8,483
19,633
31,899
40,466
46,018
49,218
50,974
51,128
2000
9,262
22,297
32,806
39,527
43,481
46,665
47,647
2001
9,979
24,667
38,834
46,809
50,011
51,746
2002
10,161
22,816
32,954
37,720
41,086
2003
9,136
21,588
30,257
36,571
2004
10,411
23,928
33,345
2005
14,136
31,774
2006
15,164
17 July, 2015
51,157
64,450
67,124
67,843
66,982
69,177
67,676
70,807
86,217
91,484
R. R. Cerchiara - IME 2007
24
From triangular format to tabular format
Premium
Lag
Incremental
loss
1996
51,157
1
6,929
1996
51,157
2
9,374
1996
51,157
3
7,043
1996
51,157
4
5,375
1996
51,157
5
2,285
1996
51,157
6
3,189
1996
51,157
7
151
1996
51,157
8
473
1996
51,157
9
36
1996
51,157
10
637
1997
64,450
1
8,171
1997
64,450
2
10,209
1997
64,450
3
8,390
1997
64,450
4
5,513
1997
64,450
5
5,903
1997
64,450
6
4,625
1997
64,450
7
1,782
1997
64,450
8
120
1997
64,450
9
82
1998
….
…
...
AY
17 July, 2015
The cumulative triangle is actually better
arranged (to apply Maximum Likelihood
estimation) as a table of values
(incremental), rather than in the familiar
triangular format (see Clark [2003]).
R. R. Cerchiara - IME 2007
25
Cape Cod
see Stanard [1985]
•
•
Clark [2003]: The Cape Cod method assumes that there is a known relationship between
the amount of ultimate loss expected in each of the years in the historical period, and
that this relationship is identified by an exposure base. The exposure base is usually
onlevel premium, but can be any other index (such as sales or payroll), which is
reasonably assumed to be proportional to expected loss.
Meyers [2006]: This approach assumes a constant expected loss ratio across the data
time horizon. This hyphotesis can be adjusted using for example a Premium index.
 
~
E X i , j  Pi * ELR * s j
•
•
•
i = 1, 2, ..., 10 is an Accident year index;
•
•
•
Pi is the earned premium for the accident year i;
Note: It’s
possible to
use other
methods like
LDF
j = 1, 2, ..., 10 is an Development year index;
~
X i, j
is the incremental paid loss for the accident year i and the development year j;
ELR is an unknown parameter that represents Expected Loss Ratio;
sj are 10 unknown parameters depending on development year j.
17 July, 2015
R. R. Cerchiara - IME 2007
26
Compound Negative Binomial - Severity
  
F z   1  

z





 1



L.E.V.
   

 ,   1
1  
~
  1   x    
E Z z  
    log  ,
 1



 x  
 
•
The Claim severity distribution is set as a Pareto distribution, with parameters
 = 2 and θ = 10, 25, 50, 75, 100, 125, 150, 150, 150, 150 (values in € ,000),
respectively as the development year runs from 1 to 10 (see Meyers [2006]).
This hypothesis means that more risky claims require a longer time to settle
(see Meyers [2006]).
•
The policy limit is set to € 1,000 thousands.
17 July, 2015
R. R. Cerchiara - IME 2007
27
Compound Negative Binomial - Severity
•
To apply FFT, it’s necessary to discretize Severity distribution on intervals
with length h (step).
•
Meyers [2006] estimated h using premium data. We followed this choice.
•
Mean-Preserving method has been used because this technique permits that
the discretized distribution has the same mean as the original severity
distribution (see Meyers [2006] for technical details and Klugman et al.
[1998] for other techniques).
•
h was chosen so the 214 (16,384) values spanned the probable range of losses
for the insurer (see Meyers [2006]).
17 July, 2015
R. R. Cerchiara - IME 2007
28
Compound Negative Binomial – Claim
Number
 
•
~ , the expected claim number is:
Given E X
i, j
•
In this model,
 ~  ~ 
i , j  E X i , j / E Z j
~
k i , j follows Negative Binomial Distribution, with mean i, j and variance
i, j  ci2, j
•
As a broad assumption, let’s suppose that one estimates that the standard deviation of
the loss ratio (actual loss divided by expected loss) for a line of business can be no
smaller than 20% regardless of the size of the insurer. In Klinker et al. [2003] and IAA
Insurer Solvency Assessment Working Party – App. B [2004] we have:
~
Var[ X ]

P
     
 
~
~2 ~2
~
~
E k Var Z  E Z  cE k E Z
E
 c
~
~ ~
[ k ]
E[k ]E[ Z ]
•
So parameter c is set as 0.04 (Meyers [2006] fit a lower c value from Schedule P
data for Commercial Auto).
•
Note that c describe claim numbers correlation between development years, but
no assumption has been made on inflation (see Clark [2003] and IAA Insurer
Solvency Assessment Working Party – App. B [2004]).
17 July, 2015
R. R. Cerchiara - IME 2007
29
Compound Negative Binomial –
Aggregate Loss
•
Using FFT, it’s possible calculate the entire distribution of a discretized
rounded to the nearest multiple of h:
~
X i, j
~
~
~
~
X i , j  Z j ,1  Z j , 2  ...  Z j ,k~
i, j
•
~
 X i, j
For each accident year Aggregate Loss
will follow a Compound
j
Negative Binomial, with expected claim number:
i ,Tot   i , j
j
•
and Severity distribution:
FZ~i ,Tot 
 i , j  FZ~ j
j
 i , j
j
17 July, 2015
R. R. Cerchiara - IME 2007
30
Compound Negative Binomial –
Aggregate Loss

 
The calculation of df CNB xi , j E X~ i , j , where xi,j are observed data, proceeds in the
following steps:
•
Let pr,j , per r = 0, 1, …, 214-1 e j = 1, …, 10, the probability that severity claim
is equal to hr (remember Mean-Preserving method).
•
Apply FFT to pr,j , to get CF  ~ t  .
Z
j
•
Calculate the expected claim number i,j as shown in previous slide
•
The CF of each X~i , j using  X~
•
Apply Inverse FFT to get f X~  IFFT [ X~i , j (t )] .
•
Finally set r equal to the multiple of h that is nearest to observed data xi,j.
•


1 / c




~
t

1

c



(
t
)

1
i, j
Zj
i, j
So CNB xi , j E X~ i , j  is the rth component of
.
f X~ .
Using FFT, it’s possible to semplify convolution, because FFT of the sum of losses
is equal to the product of FFTs of the summands (see Wang [1998]).
17 July, 2015
R. R. Cerchiara - IME 2007
31
Parameters Estimation
•
Estimation of the 11 parameters ELR and sj (j =1, …, 10) has been based on
Clark [2003] and Meyers [2006] techniques. They used Maximum Likelihood
Method. In front of loss triangle xi , j  , we have the following equation:
Lx     CNB x
10 11i
i, j
•
i 1 j 1
i, j
 
~
E X i, j
Maximizing L function it’s possible estimate ELR and sj .
17 July, 2015
R. R. Cerchiara - IME 2007
32
Parameters Estimation
•
After examing incremental payments empirical paths in the following figure, in
this work Meyers [2006] constraints have been used for sj parameters:
20.000
1
2
15.000
3
4
10.000
s j  s j 1 ,
j  2,3,...,9
5
6
5.000
7
s7 / s8  s8 / s9  s9 / s10
8
1
2
3
4
5
6
7
Anno di sviluppo
Development
Year
•
s1  s2 ,
8
9
10
9
10
10
sj 1
i 1
Meyers [2006]: “The third set of constraints was included to add stability to the
tail estimates. They also reduce the number of free parameters that need to be
estimated from eleven to nine. The last constraint eliminated an overlap with the
ELR parameter and maintained a conventional interpretation of that parameter”.
17 July, 2015
R. R. Cerchiara - IME 2007
33
Parameters Estimation
• The parameters estimation technique for a CNB model requires a long
computational time. Meyers [2006] has developed a procedure that
permits to calculate a good starting set of values ( E X~ i , j ) to input in R
 
optim funcion.
• To get these starting values, Meyers [2006] replaced CNB distribution
with the Overdispersed Poisson Distribution (ODP) given in Clark [2003]
to find the parameters that maximize the ODP likelihood function (see
Meyers [2006] for more details).
17 July, 2015
R. R. Cerchiara - IME 2007
34
Parameters Estimation
•
Nelder Mead algorithm (see Klugman et al. [2004]) has been used to maximize
likelihood function.
•
This recursive procedure permits to define the “good” starting values set (using
also ODP likelihood).
•
Then these starting values are used for CNB likelihood and the Nelder Mead
algorithm is applied again to get the parameters.
•
This procedure requires about 30 minutes to converge (using 1,000 iterations for
optim function).
•
See Meyers [2006] and Meyers [2007] for the Bayes’ Theorem application, as
alternative methodology to Maximum Likelihood Method. These papers show
how to use a priori information (for example from big companies experience) for
a better loss reserve estimation.
•
See Verrall [2007] for another Bayesian methods on loss reserve application.
17 July, 2015
R. R. Cerchiara - IME 2007
35
Results
Example 1
step = h = 50
1
6929
8171
8483
9262
9979
10161
9136
10411
14136
15164
1
2
3
4
5
6
7
8
9
10
2
9374
10209
11150
13035
14688
12655
12452
13517
17638
17143
3
7043
8390
12266
10509
14167
10138
8669
9417
13241
14050
4
5375
5513
8567
6721
7975
4766
6314
7102
8648
9176
sj
1
2
3
4
5
6
7
8
9
10
19.68%
25.64%
21.01%
13.72%
8.65%
6.97%
2.10%
1.18%
0.67%
0.37%
ELR
73.09%
6
3189
4625
3200
3184
1735
3525
3448
3608
4393
4661
7
8
151
1782
1756
982
1028
1062
1039
1087
1324
1404
9
473
120
154
586
579
598
585
612
745
791
Ultimate Loss Ratio
Parameters Estimated
Dev. Year
5
2285
5903
5552
3954
3202
3366
4276
4474
5448
5781
10
Ultimate
Ultimate Loss Ratio
637
35492
69%
177
44972
70%
184
51639
77%
186
48750
72%
184
53863
80%
190
46797
68%
185
46434
69%
194
50767
72%
236
66229
77%
251
68867
75%
TOT.
513809
36
82
327
330
326
337
329
345
420
445
Reserve
IBNR
100%
90%
80%
70%
60%
50%
40%
30%
There is not an
increasing or
decreasing
pattern, then
there could not
be a concern of
bias introduced
in our reserve
estimate (see
Clark [2003]).
20%
10%
0%
17 July, 2015
1997
1998R. 1999
2000 2001- IME
2002
R. Cerchiara
2003
2007
2004
2005
2006
36
Moments and Distribution
Example 1
Values
Percentiles
25%
114,800
Mode = 46.7%
123,200
14,725
Median = 50%
124,400
11.8%
75%
134,600
Skewness
0.25
90%
144,250
Kurtosis
0.11
99%
161,750
99.5%
165,950
Best Estimate (BE)
125,060
Mean
Std Deviation
CV
125,060
Risk Margin_75% / BE
7.63%
Pdf
0 , 16 %
0 , 14 %
0 , 12 %
0 , 10 %
0,08%
0,06%
0,04%
0,02%
0,00%
75000
17 July, 2015
10 0 0 0 0
12 5 0 0 0
R. R. Cerchiara - IME 2007
15 0 0 0 0
17 5 0 0 0
37
Comparison with Mack Model and Bootstrapping
based on Over-Dispersed Poisson Model
Example 1
• FFT Model results have been
compared to the other models
ones:
Acc. Yr
–Mack Model [1993] (software
downloaded on
http://www.casact.org/library/Mack
-Method-handout.xls)
–Bootstrapping based on OverDispersed Poisson Model derived
from England & Verrall [1999]
(software downloaded on
http://www.casact.org/library/Boot
strap-Method-(E&V)-handout.xls),
with 250 simulations.
17 July, 2015
Loss
Reserve
Loss
Reserve
Loss
Reserve
FFT
Mack
Bootstrapp
1997
0
0
0
1998
177
819
819
1999
511
1012
1012
2000
1103
1222
1222
2001
2117
2761
2761
2002
5711
5496
5496
2003
9863
9353
9353
2004
17422
17261
17261
2005
34455
39363
39363
2006
53703
63723
63723
Total
125060
141011
141011
Std Dev
14726
11215
14964
CV
11.8%
8.0%
10.6%
R. R. Cerchiara - IME 2007
-
+
-
38
Example 2
Change in Premiums Level
17 July, 2015
R. R. Cerchiara - IME 2007
39
Example 2: Change in Premiums
Cumulative Claims Paid (€ ,000)
Accident
Devel.
Year
Premium
Year
1
2
3
4
5
6
7
8
9
10
1997
6,929
16,303
23,346
28,721
31,006
34,195
34,346
34,819
34,855
35,492
1998
8,171
18,380
26,770
32,283
38,186
42,811
44,593
44,713
44,795
1999
8,483
19,633
31,899
40,466
46,018
49,218
50,974
51,128
2000
9,262
22,297
32,806
39,527
43,481
46,665
47,647
2001
9,979
24,667
38,834
46,809
50,011
51,746
2002
10,161
22,816
32,954
37,720
41,086
2003
9,136
21,588
30,257
36,571
2004
10,411
23,928
33,345
2005
14,136
31,774
2006
15,164
40,000
43,200
46,656
50,388
54,420
58,773
63,475
68,553
74,037
79,960
•Lower Premium Levels with a costant
premium increase
•Higher Loss Ratios
•No modifications in Claim Data
17 July, 2015
R. R. Cerchiara - IME 2007
40
Results – Example 2
step = h = 40
1
2
3
4
5
6
7
8
9
10
1
6929
8171
8483
9262
9979
10161
9136
10411
14136
15164
2
9374
10209
11150
13035
14688
12655
12452
13517
17638
18535
3
4
7043 5375
8390 5513
12266 8567
10509 6721
14167 7975
10138 4766
8669 6314
9417 8812
14243 9516
15383 10278
sj
1
2
3
4
5
6
7
8
9
10
19.04%
25.05%
20.79%
13.89%
9.01%
7.43%
2.34%
1.31%
0.73%
0.41%
ELR
92,53%
17 July, 2015
6
3189
4625
3200
3184
1735
4041
4364
4713
5090
5497
7
151
1782
1756
982
1180
1274
1376
1486
1605
1734
8
473
120
154
610
659
712
769
830
897
968
9
36
82
316
341
368
398
429
464
501
541
10 Ultimate Ultimate Loss Ratio
637
35492
89%
163
44958
104%
176
51620
111%
190
48788
97%
206
54159
100%
222
47732
81%
240
49042
77%
259
55625
81%
280
70080
95%
302
75069
94%
TOT.
532564
Ultimate Loss Ratio
Parameters Estimated
Dev. Year
5
2285
5903
5552
3954
3202
3366
5293
5716
6174
6667
Reserve
IBNR
110%
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
R. R. Cerchiara - IME 2007
41
Comparison with Mack Model and Bootstrapping
based on Over-Dispersed Poisson Model
Example 2
Percentiles (FFT Model)
Values
Loss
Reserve
Loss
Reserve
Loss
Reserve
FFT
Mack
Bootstrapp
25%
132,320
Mode = 46.8%
141,800
Median = 50%
143,120
1997
0
0
0
75%
154,520
1998
163
819
819
90%
165,360
1999
492
1012
1012
99%
184,880
2000
1141
1222
1222
99.5%
189,560
2001
2413
2761
2761
Best Estimate (BE)
143,816
2002
6646
5496
5496
7.44%
2003
12471
9353
9353
2004
22280
17261
17261
2005
38306
39363
39363
2006
59905
63723
63723
Total
143816
141011
141011
Std Dev
16506
11215
14964
CV
11.5%
8.0%
10.6%
Skewness
0.25
Kurtosis
0.10
Risk Margin_75% / BE
17 July, 2015
Acc. Yr
R. R. Cerchiara - IME 2007
+
-
42
Results
Example 3
17 July, 2015
R. R. Cerchiara - IME 2007
43
Example 3: Example 1 data and c = 0,01
(Meyers [2006] hyphotesis)
h = 50
Acc. Yr
17 July, 2015
Loss
Reserve
Loss
Reserve
Loss
Reserve
FFT
Mack
Bootstrapp
1997
0
0
0
1998
170
819
819
1999
494
1012
1012
2000
1078
1222
1222
2001
2090
2761
2761
2002
5654
5496
5496
2003
9800
9353
9353
2004
17342
17261
17261
2005
34345
39363
39363
2006
53629
63723
63723
Total
124601
141011
141011
Std Dev
9008
11215
14964
CV
7.2%
8.0%
10.6%
Skewness
0.13
Kurtosis
0.03
R. R. Cerchiara - IME 2007
44
Conclusions
17 July, 2015
R. R. Cerchiara - IME 2007
45
Conclusions
•
•
“The FFT method is at its worse when the claim severity distributions
have a high proportion of small claims, but also a long tail, and the
expected claim count is large” (Meyers: Discussion by Glenn Meyers
about Aggregation of Correlated Risk Portfolios by Wang S., Ph.D.
[1999]).
FFT method is better for modelling excess of loss reinsurance, where
the expected number of claims is low: “The FFT method is at its best
in high reinsurance layers where the usual claim severity models are
fairly flat, and the expected claim count is small” (Meyers:
•
Discussion by Glenn Meyers about Aggregation of Correlated Risk
Portfolios by Wang S., Ph.D. [1999]).
As shown in previous slide FFT represents also a good way to
estimate loss reserve distribution.
17 July, 2015
R. R. Cerchiara - IME 2007
46
Conclusions
•
•
•
•
•
In this model is possible to calibrate frequency and severity of predictive
distribution, in presence of variation in liquidation rate, average claim
cost, policy limit, etc.
Cape Code Method permits the definition of loss reserve estimation
using both premium and claim experience. The Cape Code and Maximum
Likelihood estimates are very sensitive to premiums level.
Results are affected by Pareto Distribution hyphotesis. See Meyers
[2006] for some consideration on fitting severity data.
Results comparison with LDF model ones depends also on Loss
Development Factor choices that have an important effect on loss
reserve estimate.
See Clark [2003] for a practical example that includes a demonstration
of the reduction in variability possible from the use of an exposure base
in the Cape Cod reserving method, instead of LDF Model: “The Cape Cod
method may have somewhat higher process variance estimated, but will
usually produce a significantly smaller estimation error.”
17 July, 2015
R. R. Cerchiara - IME 2007
47
Future Improvements
•
Implement LDF model under FFT approach
•
Fitting claim number and severity to real
data
•
Use other parameters estimation method
17 July, 2015
R. R. Cerchiara - IME 2007
48
References
•
•
•
•
•
•
•
•
•
•
•
•
Bühlmann H. [1984]: Numerical Evaluation of the Compound Poisson Distribution:
Recursion or Fast Fourier Transform?, Scandinavian Actuarial Journal, n°1
Casualty Actuarial Society [1996]: Foundations of Casualty Actuarial Science, New
York
Casualty Actuarial Society [2006]: Stochastic Reserving Track Readings from 2006
Spring Meeting
Cerchiara, R.R. [2006]: Simulation Model, Fast Fourier Transform and Extreme Value
Theory to model non life insurance aggregate loss. Ph.D. Thesis, University La
Sapienza, Rome, Italy - Download on (italian language):
http://padis.uniroma1.it/search.py?recid=468
Cerchiara, R.R. [2007]: FFT, Extreme Value Theory and Simulation to Model Non-Life
Insurance Claims Dependences, to appear on Mathematical and Statistical Methods in
Finance, Springer
Clark D. R. [2003]: LDF Curve Fitting and Stochastic Loss Reserving: a Maximum
Likelihood Approach, CAS Forum, pp 41-91
Daykin C. D., Pentikainen T., Pesonen M. [1994]: Practical risk theory for actuaries.
Chapman and Hall, London
Embrechts P., Kluppelbrg C., Mikosch T. [1997]: Modelling extremal events,
Springer Verlag, Berlin
England, P. D., and Verrall, R. J. [1999]: Analytic and bootstrap estimates of
prediction errors in claims reserving, Insurance: Mathematics and Economics, Volume
25, Number 3 , pp. 281-293
England, P. D., and Verrall, R. J. [2002]: Stochastic claims reserving in general
insurance, London, Institute of Actuaries
Homer D.L., Clark D.R. [2004]: Insurance Applications of Bivariate Distributions, PCAS
Forum, pp. 823-852
17 July, 2015
R. R. Cerchiara - IME 2007
49
References
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
IAA Insurer Solvency Assessment Working Party [2004]: A Global Framework for
Insurer Solvency Assessment
Institute of Actuaries [1997]: Claims Reserving Manual, London
Klinker F. L., Lalonde A., Meyers G. [2003], The Aggregation and Correlation of
Insurance Risk, CAS Forum, Summer, 16-82
Klugman S., Panjer H., Willmot G. [1998]: Loss Models - From Data to Decisions, John
Wiley & Sons, New York, First Edition
Klugman S., Panjer H., Willmot G. [2004]: Loss Models - From Data to Decisions, John
Wiley & Sons, New York, Second Edition
Mack T. [1993]: Distribution-Free Calculation of Chain Ladder Reserve Estimates,
ASTIN Bulletin, Vol. 23, No. 2, pp. 213-226
Meyers [1999]: Discussion by Glenn Meyers about Aggregation of Correlated Risk
Portfolios by Wang S., Ph.D. [1998], CAS Proceedings
Meyers G. [2006]: Estimating Predictive Distributions for Loss Reserve Models, CAS
Forum – Arlington
Meyers G. [2007]: Thinking Outside the Triangle, ASTIN Colloquium, Orlando
Nationale Nederlanden [1981]: Loss Reserving Methods, Rotterdam
Robertson J. P. [1983]: The Computation of Aggregate Loss Distributions, PCAS LXXIX
Schmidt K.S. [2007]: A Bibliography on Loss Reserving, download on
http://www.math.tu-dresden.de/sto/schmidt/dsvm/reserve.pdf
Taylor [2000]: Loss Reserving: an actuarial perspective, Kluwer Academic Publishers
Verrall [2007]: Obtaining Predictive Distributions for Reserves Which Incorporate
Expert Opinion, Variance, Volume 01, Issue 01, pp. 53-80
Wang S.S. [1998]: Aggregation of correlated risk portfolios: models and algorithms,
PCAS LXXXV
17 July, 2015
R. R. Cerchiara - IME 2007
50