Slides General Insurance Mathematics

Transcription

Dumaria R. Tampubolon, Ph.D
Statistics Research Division
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung, Bandung, Indonesia
SEAMS School – Universitas Sanata Dharma – Yogyakarta
August 2016
Outline
 What is “Actuarial Science”?
 What is “General Insurance” and what is its nature?
 What are the major problems in General Insurance?
 What is a “runoff triangle”?
 What is “outstanding claims liability”?
 Using “Leverage” as a tool to measure the sensitivity of an
estimate of outstanding claims liability due to small changes in
the incremental payments (paid claims).
 Determining a probability model for the moment magnitudes of
earthquake mainshocks where the source of earthquake is in the
region of Megathrust Mid 2 Sumatera
 Determining the average recurrence interval of certain moment
magnitudes.
8-9 August 2016
D. R. Tampubolon: SEAMS School at Universitas
Sanata Dharma
2
Actuarial Science
A
branch of knowledge which applies mathematics,
probability and statistics, economics, and finance in
assessing risks of financial losses due to measurable
non-prevented events. Such events are those which
will occur at a certain time in the future with
probability greater than zero but the time of
occurrence of the event and the corresponding
amount of the financial losses cannot be determined
with certainty at the time of valuation.
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Definition
 Insurance:
A vehicle to transfer pure economic (or financial) risks
or
liability to compensate for loss or damage arising
from specified contingencies such as natural disaster,
fire, theft, negligence, injury, death, etc
 General Insurance:
Relates to the insurance of property and liability; may
also relate to the insurance of the person which is not
covered by life insurance.
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Nature of General Insurance
 Severity (amount of claims) and Frequency (number of
claims )can not be determined with certainty at the time of
valuation
 Time of disastrous event which lead to financial
compensation can not be determined with certainty at the
time of valuation
 Claims are not usually paid as soon as they occur. Hence,
there is a delay: between occurrence of event and reporting
of a claim; and between the time of reporting and
settlement of the claim.
 A closed claim might be reopened and additional money
need to be paid
→Short-tailed and Long-tailed business
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Major Problems in General Insurance
 How to determine the premium?
Risk (Adjusted) Premium = Pure Premium +
Loadings
 Pure Premium is the Expected of Claims, that is
the mean of the distribution of claims.
 How to determine the loading factor?
 How to determine the outstanding claims reserve?
OR
How to determine the outstanding claims liability?
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Outstanding Claims Liability
 The present values of expected future payments which include
Incurred But Not Reported (IBNR) claims, the evaluation of future
payments on claims already notified and the management
expenses of future claims payments of claims incurred as at the
balance date.
(Hart, D. G., Buchanan, R.A., and Howe, B. A. (1996). The Actuarial Practice of General
Insurance, page 26)
 The provision for the outstanding claims is usually the largest
component of a general insurance company’s liabilities; hence,
changes in the outstanding claims liability have a direct, and
possibly large, impact on the company’s profits and tax liabilities.
 (General) Insurance companies need to reserve enough of their
premium income to cover future claim payments from past and
current policies
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Sensitivity of the Estimate
 Gain insights on the forecasting methodology
used:
→ very or moderately or not sensitive?
 Gain insights on the data:
→ absolute and relative importance
 Gain insights on the uncertainty of the
estimate of the outstanding claims liability
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Measurement of Sensitivity
Measurement of the sensitivity of the estimate
of the outstanding claims liability to small
perturbation in an incremental payment:
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D. R. Tampubolon: SEAMS School at
Universitas Sanata Dharma
9
EXAMPLE
Estimate of the outstanding claims
liability of AFG Data using Hertig’s model
 The data used as an example is the Automatic
Facultative General (AFG) Liability, excluding Asbestos
and Environmental, from the Historical Loss
Development study, which was also considered by
Mack (1994b). Following Mack, the runoff triangle of
the incurred payments of the AFG data is represented
in thousands ($’000).
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EXAMPLE
Estimate of the outstanding claims
liability of AFG Data using Hertig’s model
 The estimate of the outstanding claims liability for
the AFG data, using Hertig’s Model (Hertig, 1985) is
$86.889 million or approximately $87million.
 What happen if there is an additional $1,000 in the
incremental payment on a cell of the runoff triangle?
Or, what happen if there is a delay of payment (of a
small amount) of $1,000?
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Hertig’s Model Leverage
(1 unit increase)
0
1
2
3
4
5
6
7
8
9
0
-1.292
-161.585
-1.352
-0.659
-7.935
-3.322
-13.908
-3.344
2.265
22.815
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1
-1.311
1.03
-0.629
-0.469
0.318
0.037
0.367
1.177
2.309
2
-0.513
-1.596
-0.034
0.025
0.254
0.671
1.51
1.664
3
-0.11
0.762
0.257
0.47
0.626
0.842
1.405
4
0.48
0.877
0.643
0.626
0.804
1.454
5
1.201
1.323
1.142
0.996
0.996
6
2.116
2.073
1.677
1.528
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3.237
3.707
2.678
8
5.489
6.455
9
12.161
12
(1 unit increase)
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 Given the Hertig’s model, the leverage is
calculated as follows. Let us say that there is a
$1000 increase in cell (1,0) of the runoff
triangle of the incremental payments (an
increase of $1000 is small enough since
increases of $500 and $1 in the incremental
payments also result in the same leverage
values).
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 The resulting leverage is -161.585. This means
that there is a decrease in the estimate of the
outstanding claims liability of almost 162 times
the increase in the cell. In other words, had the
claims paid in accident year 1 and development
year 0 been $107,000 instead of $106,000, then
the resulting Hertig’s model estimate of the
outstanding claims liability will be approximately
$162,000 lower than the original estimate of
$86,889,000.
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 In another example, for accident year 0, let us say
that there is an increase of $1000 in the paid
claims at the final development year (at the tail).
Then the change in estimated total outstanding is
approximately 12 times as much. This means
that, had the $1000 claims been paid later, the
resulting Hertig’s model estimate of the
outstanding claims liability will be approximately
$12,000 higher than the original estimate.
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 What happens if claim payments are delayed?
For a particular accident year:
Pay early → a “decrease” in outstanding claims liability estimate
Pay later → an “increase” in outstanding claims liability estimate
 What happens where there are very few observations available
to do forecasting?
Large leverage in the last accident year and at the tails
 There is an extremely large leverage in cell (1,0). It turned out
that the incremental payment in that cell is “unusual” compare
to the other incremental payments in development year 0. The
leverage values of the Hertig’s model indicate “unusual
observations” in the incremental payment data.
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Earthquake Insurance
 How to determine the premium to cover loss due to
earthquake hazard?
 How to determine the outstanding claims liability?
 There is an expression used in seismology and
geophysics ; for example:
“10% PE in 50 years given the return period of
475 years “.
What does that mean?
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Determining Premium using
Catastrophe Model for Earthquake
 Let say someone asks an insurance company to provide
protections against earthquakes; that is a coverage on
damages to a property or on business’ interruption
caused by an earthquake. What is the premium need
to be charged by the insurance company?
 To determine the premium, one need to estimate the
financial loss caused by the earthquake; or one need to
determine the “probable maximum loss”; or one need
to determine the probability of the loss to exceed a
certain amount.
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Peta Tim Revisi Gempa 2010
2010 Indonesia Hazard Map
(Open a Different File)
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Significant earthquakes in Indonesia
2004 – 2011 (Source: USGS website)
26 December 2004 Sumatera-Andaman Islands
(Aceh): Mw 9.1 (tsunami); 227,898 fatalities
 28 March 2005 Northern Sumatera (Nias region):
Mw 8.6; 1,313 fatalities
 27 May 2006 Java (Yogyakarta): Mw 6.3;
5,749 fatalities
 12 September 2007 Southern Sumatera (Bengkulu):
Mw 8.5; 25 fatalities

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Significant earthquakes in Indonesia
2004 – 2011 (Source: USGS website)




16 November 2008 North Sulawesi (Gorontalo):
Mw 7.4; 6 fatalities
30 September 2009 Southern Sumatera (Padang):
Mw 7.6; 1,117 fatalities
25 October 2010 Kepulauan Mentawai :
Mw 7.7 (tsunami); 670 fatalities
4 April 2011 South Java: Mw 6.7
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Catastrophe Model
-Earthquake
(Moment)
Magnitude
-Distance between
the Site and the
Source of Earthquake
-Soil Condition
HAZARD
VULNERABILITY
LOSS
INVENTORY
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Case Study
Megathrust Mid 2 Sumatera
97.298°E – 101.947°E and -5.418°S – 0.128°N
West Sumatera Province, Indonesia
(Irsyam et al., 2010)
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Modeling Hazard: Moment Magnitude
 Gutenberg-Richter Law
Gutenberg-Richter (1941; 1944) describe the
relationship between the frequency and earthquake
magnitude through the equation
log10 𝑁 𝑚 = 𝑎 − 𝑏𝑚
where N(m) is the number of earthquakes with
magnitudes greater than or equal to m; a and b are
parameters which indicate the characteristics of
seismic activities at a particular site.
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 In practice, it is of interest to examine earthquake
magnitudes which are greater than or equal to a
particular value mt . Hence, the Gutenberg-Richter
equation becomes
log10 𝑁 𝑚 = 𝑎𝑡 − 𝑏 𝑚 − 𝑚𝑡
or
𝑁 𝑚 = 10𝑎 𝑡 −𝑏
𝑚 −𝑚 𝑡
where m is greater than or equal to mt ; and at is the
logarithm of the number of earthquakes with
magnitude greater than or equal to mt .
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 In this research, we will use the moment magnitude
scale instead of the Richter scale to measure the
earthquake magnitude. Let Mw be a random variable
which denote the moment magnitude of an
earthquake.
 Let Z be a random variable which denote the scalar
seismic moment in Newton-meter or Nm. The
relationship between earthquake moment magnitude
and the scalar seismic moment is
2
𝑀𝑤 = 3 log 𝑍 − 6
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 Hence, the number of earthquakes with Mw greater
than or equal to M is given by the equation
𝑁 𝑀 = 10𝑎 𝑡 −𝑏
or
𝑁 𝑀 = 10
𝑎 𝑡 −𝑏
𝑀−𝑀𝑡
2
2
log 𝑍− log 𝑍𝑡
3
3
 Since at = log10 N(Mt) then it can be shown that the
equation above is equivalent to:
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𝑁 𝑀 = 𝑁 𝑀𝑡
where
𝑍𝑡
𝑍
𝛽
2
𝛽= 𝑏
3
 Hence,
𝑍𝑡
Pr 𝑀𝑤 > 𝑀 𝑀𝑤 ≥ 𝑀𝑡 =
𝑍
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𝛽
29
 So, given Mw greater than or equal to Mt , the
equation above is the survival function of a Pareto
distribution with parameters β and Zt . That is, given
the moment magnitude is greater than or equal to a
threshold Mt , the seismic moment Z follows a Pareto
distribution with parameters β and Zt .
 This result leads to a hypothesis that the moment
magnitude of earthquake mainshocks might follow a
Generalized Pareto distribution.
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 The data used in this research is the earthquake data
from the National Earthquake Information Centre –
United States Geological Survey (USGS, 2012)
earthquake catalog, from January 1973 to December
2011.
 The selected earthquake data are those of which
centre are in the area of Megathrust Mid 2 Sumatera.
We use the report by “Tim Revisi Peta Gempa
Indonesia tahun 2010” (Irsyam et al, 2010) in defining
the area of Megathrust Mid 2 Sumatera.
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 The earthquake mainshocks are separated from the
earthquake foreshocks and aftershocks. The process
is called seismicity declustering. The Gardner –
Knopoff algorithm (1974) and the program written by
Stiphout et al (2012) are used to decluster the
earthquakes data.
 After declustering, 6.82% or 137 earthquakes are
categorized as earthquake mainshocks. The
descriptive statistics of the earthquake mainschoks
data are as follows:
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N
Mean
Variance
Standard Deviation
Skewness
Kurtosis
Lower Quartile
Median
Upper Quartile
Range
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137
5.6814
0.2006
0.4479
3.0609
14.4983
5.4146
5.5727
5.8155
3.4066
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 We fit a Generalized Pareto distribution to the
moment magnitudes of the earthquake mainshocks
data.
 The distribution function of a Generalized Pareto
distribution with parameters ξ and θ, and threshold u
is:
𝑥−𝑢
1− 1+𝜉
𝐹𝑋 𝑥 =
𝜃
1−𝑒
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−
𝑥−𝑢
𝜃
1
−
𝜉
,
,
if 𝜉 ≠ 0 , 𝜃 > 0
if 𝜉 = 0 , 𝜃 > 0
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 To estimate the parameters, the Maximum Likelihood
Estimation method is applied.
 At 5% significance level, the Cramér-von Mises test
statistics showed that, given a threshold of moment
magnitude Mt = 5.4, the moment magnitudes of the
earthquake mainshocks (with the source of earthquakes
in the area of Megathrust Mid 2 Sumatera) follows a
Generalized Pareto distribution with parameters
ξ = 0.14447 and θ = 0.30891.
 With the parameters obtained, the distribution of the
moment magnitudes has mean 5.76107 and standard
deviation 0.42820.
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Modeling Hazard:
Average Recurrence Interval
𝑀𝑤 be a random variable which denote the moment magnitudes of earthquakes (mainshocks)
𝑉 be a random variable which denote the number of years needed until an earthquake with
moment magnitude at least a certain value, 𝑀𝑤 ≥ 𝑀𝑡 , occurs for the first time.
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Modeling Hazard:
 V follows a Geometric distribution with parameter
𝑝 = Pr 𝑀𝑤 ≥ 𝑀𝑡
 The expected number of years needed until an
earthquake with Mw at least Mt occurs for the first
time is
1
𝐸𝑉 =
Pr 𝑀𝑤 ≥ 𝑀𝑡
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Modeling Hazard:
 The “average recurrence interval“ (some literature
used the term “return period”) is defined as the
expected number of years until an earthquake with
Mw at least M, given a moment magnitude threshold
Mt , occurs for the first time in a region.
Average Recurrence Interval = 𝜏 = Pr 𝑀
1
𝑤 >𝑀 𝑀𝑤 >𝑀𝑡
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Modeling Hazard:
1
The value is called the “average recurrence rate”.
𝜏
 Let N be the random variable which denote the
number of earthquakes with Mw at least M, given a
moment magnitude threshold Mt , occurring in t
years in a region.
 It is assumed that the earthquake (mainshock) is
independent of time and independent of past
earthquakes (mainshocks).
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Modeling Hazard:
 The random variable N may be modeled by a
Poisson distribution, that is
Pr 𝑁 = 𝑛 =
𝑡
−
𝑒 𝜏
𝑛!
𝑡
𝜏
𝑛
for n = 0,1,2,…
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Modeling Hazard:
 The probability of at least one earthquake with Mw
at least M, given a moment magnitude threshold Mt ,
occurring in t years in a region is:
Pr 𝑁 ≥ 1 = 1 − 𝑒𝑥𝑝
𝑡
𝜏
 The above equation can be used to calculate the
seismic risk expressed as: “x% PE in t years” (x%
Probability of Exceedance in t years) for a given
recurrence interval of earthquakes with a certain
moment magnitude or greater.
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Modeling Hazard:
 Example:
Let t = 50 years and let the probability of exceedance
0.1
Then
0.1 = 1 − 𝑒𝑥𝑝 −
50
𝜏
or the average recurrence interval is approximately
475 years
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Modeling Hazard:
Averange Recurrence Interval
 For a threshold Mt = 5.4, let Mw follows a Generalized
Pareto distribution with parameters
𝜉 = 0.14447
and
 Let
𝜃 = 0.30891
Pr 𝑀𝑤 > 𝑀 = 𝑆𝑀𝑤
1
𝑀 =
475
Then M is approximately 8.475
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Modeling Hazard:
 This means that “the probability of at least one
earthquake with moment magnitude at least 8.475
occurring in 50 years in the region, given the average
recurrence interval of 475 years, is 10%”.
 Using the expression usually used by seismologists:
“10% PE in 50 years given the average recurrence
interval of 475 years with moment magnitude at least
8.5 “.
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Modeling Hazard:
 Another example:
The probability of at least one earthquake with
moment magnitude at least 7.935 occurring in 50
years in the region, given the average recurrence
interval of 224 years, is 20%.
Using the expression usually used by seismologists:
“20% PE in 50 years given the average recurrence
interval of 224 years with moment magnitude at least
7.9 “
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Still Need to be Done!!!
 Modeling the distribution of Peak Ground Acceleration
(PGA) utilizing the probability distribution of moment
magnitudes (Hazard Module)
 Determining the Modified Mercalli Intensity (MMI)
 Determining the Damage Curve (Vulnerability Module)
 Determining the “Probable Maximum Loss”; or
determining the Distribution of Loss (Loss Module).
 Determining the Premium
 Determining the Outstanding Claims Liability
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References
 [1]
 [2}
 [3]
 [4]
Andaria, R. (2013). Penentuan Average Recurrence
Interval, Peak Ground Acceleration dan Modified
Mercalli Intensity: Sumber Gempa Wilayah Megathrust Mid 2
Sumatera, Tesis Program Studi Magister Matematika
(Supervisor: Tampubolon, D. R.), FMIPA, Institut Teknologi
Bandung.
Chen, W. and Scawthorn, C. (2003). Earthquake Engineering
Handbook. CRC Press.
Choulakian, V. and Stephens, M. (2001). “Goodness-of-Fit
Tests
for
the
Generalized
Pareto
Distribution”.
Technometrics, American Statistical Association and
American Society for Quality Control, 43, 4, 478-484
Hertig, J. (1985), "A Statistical Approach to IBNR-Reserves
in Marine Reinsurance”, ASTIN Bulletin, 15, 2, 171-183.
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Reference
 [5] Irsyam, M., Sengara, I. W., Aldiamar, F., Widiyantoro, S.,
Triyoso, W., Natawidjaja, D. H., Kertapati, E., Meilano, I.,
Suhardjono, Asrurifak, M., and Ridwan, M. (2010).
Ringkasan Hasil Studi Tim Revisi Peta Gempa Indonesia
2010, Technical Report, Departemen Pekerjaan Umum,
Indonesia.
 [6] Kagan, Y. (2002). “Seisimic Moment Distribution
Revisited: I. Statistical Results”, Geophysical Journal
International, 148, 520-541
 [7] Klugman, S., Panjer, H., Willmot, G. (2004). Loss
Models: From Data to Decisions, 2nd edition, New York:
Wiley.
8-9 August 2016
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48
Reference
 [8]
Mack, T. (1994b), "Measuring the Variability of Chain Ladder
Reserve Estimates”, Casualty Actuarial Society Forum, Spring,
101-182.
 [9] Pisarenko, V., Sornette, A., Sorenette, D., and Rodkin, M.
(2008). “Characterization of the Tail of the Distributions of
Earthquake Magnitudes by Combining the GEV and GPD
Descriptions of Extreme Value Theory”,
http://arxiv.org/ftp/arxiv/papers/0805/0805.1635.pdf
 [10] Pradana, A. A. (2013). Pemodelan Magnitudo Gempa
Bumi
Menggunakan Distribusi Peluang Generalized Pareto: Studi
Kasus Megathrust Mid 2 Sumatera dan Megathrust Jawa,
Laporan Tugas Akhir Program Studi Sarjana Matematika
(Supervisor: Tampubolon, D. R.), FMIPA, Institut Teknologi
Bandung.
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Reference
 [11] Putra, R. R., Kiyono, J., Ono, Y., dan Parajuli, H. R. (2012).
“Seismic Hazard Analysis for Indonesia”, Journal of
Natural Disaster Science, 33, 2, 59-70.
 [12] Stiphout, T., Zhuang, J., and Marsan, D. (2012).
“Seismicity Declustering”, Community Online Resource
for Statistical Seismicity Analysis.
 [13] Tampubolon, D. R. (2008). Uncertainties in the
Estimation of Outstanding Claims Liability in General
Insurance, PhD Thesis, Macquarie University, Australia
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Reference
 [14] Tse, Y. K. (2009). Non-life Actuarial Models: Theory,
Methods and Evaluation, New York: Cambridge University
Press.
 [15] United States Geological Survey (USGS) website.
Earthquake search.
http://earthquake.usgs.gov/earthquakes/eqarchives
 [16] Wang, Z. (2007). “Seismic hazard and risk assessment
in the intraplate environment: The New Madrid seismic
zone of the central United States”, The Geological
Society of America. Special Paper 425, 363-373.
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Thank You
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52

Slides General Insurance Mathematics

Transcription

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