Slides axa

Arthur CHARPENTIER - AXA Risk College - Modeling and covering catastrophic risks

Modeling and covering catastrophic risks
Arthur Charpentier

AXA Risk College, April 2007

arthur.charpentier@ensae.fr

1


Agenda
Catastrophic risks modelling
• General introduction
• Business interruption and very large claims
• Natural catastrophes and accumulation risk
• Insurance covers against catastrophes, traditional versus alternative
techniques
Risk measures and capital requirements
• Risk measures, an economic introduction
• Calculating risk measures for catastrophic risks
• Diversiﬁcation and capital allocation

2


Agenda
techniques

3


Some stylized facts
“climatic risk in numerous branches of industry is more important than the risk
of interest rates or foreign exchange risk” (AXA 2004, quoted in Ceres (2004)).

Figure 1: Major natural catastrophes (from Munich Re (2006).)

4


Some stylized facts: natural catastrophes
Includes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,
drought, ﬂoods...
Date Loss event Region Overall losses Insured losses Fatalities
25.8.2005 Hurricane Katrina USA 125,000 61,000 1,322
23.8.1992 Hurricane Andrew USA 26,500 17,000 62
17.1.1994 Earthquake Northridge USA 44,000 15,300 61
21.9.2004 Hurricane Ivan USA, Caribbean 23,000 13,000 125
19.10.2005 Hurricane Wilma Mexico, USA 20,000 12,400 42
20.9.2005 Hurricane Rita USA 16,000 12,000 10
11.8.2004 Hurricane Charley USA, Caribbean 18,000 8,000 36
26.9.1991 Typhoon Mireille Japan 10,000 7,000 62
9.9.2004 Hurricane Frances USA, Caribbean 12,000 6,000 39
26.12.1999 Winter storm Lothar Europe 11,500 5,900 110

Table 1: The 10 most expensive natural catastrophes, 1950-2005 (from Munich
Re (2006)).

5


Some stylized facts: man-made catastrophes
Includes industry fire, oil & gas explosions, aviation crashes, shipping and rail
disasters, mining accidents, collapse of building or bridges, terrorism...
Date Location Plant type Event type Loss (property)
23.10.1989 Texas, USA petrochemical∗ vapor cloud explosion 839
04.05.1988 Nevada, USA chemical explosion 383
05.05.1988 Louisiana, USA refinery vapor cloud explosion 368
14.11.1987 Texas, USA petrochemical vapor cloud explosion 282

07.07.1988 North sea platform∗ explosion 1,085
26.08.1992 Gulf of Mexico platform explosion 931
23.08.1991 North sea concrete jacket mechanical damage 474
24.04.1988 Brazil plateform blowout 421

Table 2: Onshore and offshore largest property damage losses (from 1970-1999).

The largest claim is now the 9/11 terrorist attack, with a US$ 21, 379 million
insured loss.
∗
evaluated loss US$ 2, 155 million and explosion on platform piper Alpha, US$ 3, 409 million (Swiss Re (2006)).

6


What is a large claim ?
An academic answer ? Teugels (1982) defined “large claims”,
Answer 1 “large claims are the upper 10% largest claims”,
Answer 2 “large claims are every claim that consumes at least 5% of the
sum of claims, or at least 5% of the net premiums”,
Answer 3 “large claims are every claim for which the actuary has to go and
see one of the chief members of the company”.
Examples Traditional types of catastrophes, natural (hurricanes, typhoons,
earthquakes, floods, tornados...), man-made (fires, explosions, business
interruption...) or new risks (terrorist acts, asteroids, power outages...).
From large claims to catastrophe, the difference is that there is a before the
catastrophe, and an after: something has changed !

7


What is a catastrophe ?
Before Katrina After Katrina

Figure 2: Allstate’s reinsurance strategies, 2005 and 2006.

8


The impact of a catastrophe
• Property damage: houses, cars and commercial structures,
• Human casualties (may not be correlated with economic loss),
• Business interruption
Example
• Natural Catastrophes - USA: succession of natural events that have hit
insurers, reinsurers and the retrocession market
• lack of capacity, strong increase in rate
• Natural Catastrophes - nonUSA: in Asia (earthquakes, typhoons) and
Europe (ﬂood, drought, subsidence)
• sui generis protection programs in some countries

9


The impact of a catastrophe
• Storms - Europe: high speed wind in Europe and US, considered as insurable
• main risk for P&C insurers
• Terrorism, including nuclear, biologic or bacteriologic weapons
• lack of capacity, strong social pressure: private/public partnerships
• Liabilities, third party damage
• growth in indemnities (jurisdictions) yield unsustainable losses
• Transportation (maritime and aircrafts), volatile business, and concentrated
market

10


Probabilistic concepts in risk management
Let X1 , ..., Xn denote some claim size (per policy or per event),
• the survival probability or exceedance probability is

F (x) = P(X > x) = 1 − F (x),

• the pure premium or expected value is
∞ ∞
E(X) = xdF (x) = F (x)dx,
0 0

• the Value-at-Risk or quantile function is
−1
V aR(X, u) = F −1 (u) = F (1 − u) i.e. P(X > V aR(X, u)) = 1 − u,

• the return period is
T (u) = 1/F (x)(u).

11


The density of the exponential distribution The exceedance distribution

0.5

1.0
0.4

0.8
mean = 1
mean = 2
mean = 5 mean = 1
0.3

0.6
mean = 2

Probability
mean = 5
0.2

0.4
0.1

0.2
0.0

0.0
0 2 4 6 8 10 0 2 4 6 8 10

Claim size Claim size

The quantile function of the exponential distribution The return period function
15

12
10
10

8
mean = 1
Claim size

Claim size
mean = 2
mean = 5

6
5

4
mean = 1

2
mean = 2
mean = 5
0

0
0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500

Probability level Time

Figure 3: Probabilistic concepts, case of exponential claims.

12


Modeling catastrophes
• Man-made catastrophes: modeling very large claims,
• extreme value theory (ex: business interruption)
• Natural Catastrophes: modeling very large claims taking into accont
accumulation and global warming
• extreme value theory for losses (ex: hurricanes)
• time series theory for occurrence (ex: hurricanes)
• credit risk models for contagion or accumulation

13


Updating actuarial models
In classical actuarial models (from Cramér and Lundberg), one usually
consider
• a model for the claims occurrence, e.g. a Poisson process,
• a model for the claim size, e.g. a exponential, Weibull, lognormal...
For light tailed risk, Cramér-Lundberg’s theory gives a bound for the ruin
probability, assuming that claim size is not to large. Furthermore, additional
capital to ensure solvency (non-ruin) can be obtained using the central limit
theorem (see e.g. RBC approach). But the variance has to be ﬁnite.
In the case of large risks or catastrophes, claim size has heavy tails (e.g. the
variance is usually inﬁnite), but the Poisson assumption for occurrence is still
relevant.

14


Updating actuarial models
N
Example For business interruption, the total loss is S = Xi where N is
i=1
Poisson, and the Xi ’s are i.i.d. Pareto.
Example In the case of natural catastrophes, claim size is not necessarily huge,
but the is an accumulation of claims, and the Poisson distribution is not relevant.
But if considering events instead of claims, the Poisson model can be relevant.
But the Poisson process is nonhomogeneous.
N
Example For hurricanes or winterstorms, the total loss is S = Xi where N is
i=1
Ni
Poisson, and Xi = Xi,j , where the Xi,j ’s are i.i.d.
j=1

15


Agenda
techniques

16


Some empirical facts about business interruption
Business interruption claims can be very expensive. Zajdenweber (2001)
claimed that it is a noninsurable risk since the pure premium is (theoretically)
inﬁnite.
Remark For the 9/11 terrorist attacks, business interruption represented US$ 11
billion.

17


Some results from Extreme Value Theory
When modeling large claims (industrial ﬁre, business interruption,...): extreme
value theory framework is necessary.
The Pareto distribution appears naturally when modeling observations over a
given threshold,
b
x
F (x) = P(X ≤ x) = 1 − , where x0 = exp(−a/b)
x0

Then equivalently log(1 − F (x)) ∼ a + b log x, i.e. for all i = 1, ..., n,

log(1 − Fn (Xi )) ∼ a + b · log Xi .

Remark: if −b ≥ 1, then EP (X) = ∞, the pure premium is inﬁnite.
The estimation of b is a crucial issue.

18


Cumulative distribution function, with confidence interval
1.0 log−log Pareto plot, with confidence interval

0
logarithm of the survival probabilities

−1
0.8
cumulative probabilities

−2
0.6

−3
0.4

−4
0.2

−5
0.0

0 1 2 3 4 5 0 1 2 3 4 5

logarithm of the losses logarithm of the losses

Figure 4: Pareto modeling for business interruption claims.

19


Why the Pareto distribution ? historical perspective
Vilfredo Pareto observed that 20% of the population owns 80% of the wealth.

80% of the claims 20% of the losses

20% of the claims 80% of the losses

Figure 5: The 80-20 Pareto principle.

Example Over the period 1992-2000 in business interruption claims in France,
0.1% of the claims represent 10% of the total loss. 20% of the claims represent
73% of the losses.

20


Why the Pareto distribution ? historical perspective

Lorenz curve of business interruption claims
1.0
0.8

73% OF
Proportion of claim size

THE LOSSES
0.6
0.4

20% OF
0.2

THE CLAIMS
0.0

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of claims number

Figure 6: The 80-20 Pareto principle.
21


Why the Pareto distribution ? mathematical explanation
We consider here the exceedance distribution, i.e. the distribution of X − u given
that X > u, with survival distribution G(·) deﬁned as

F (x + u)
G(x) = P(X − u > x|X > u) =
F (u)

This is closely related to some regular variation property, and only power
function my appear as limit when u → ∞: G(·) is necessarily a power function.

The Pareto model in actuarial literature
Swiss Re highlighted the importance of the Pareto distribution in two technical
brochures the Pareto model in property reinsurance and estimating property
excess of loss risk premium: The Pareto model.
Actually, we will see that the Pareto model gives much more than only a
premium.

22


Large claims and the Pareto model
The theorem of Pickands-Balkema-de Haan states that if the X1 , ..., Xn are
independent and identically distributed, for u large enough,
 −1/ξ

 1+ξ x
 if ξ = 0,
P(X − u > x|X > u) ∼ Hξ,σ(u) (x) = σ(u)

 exp − x
 if ξ = 0,
σ(u)
for some σ(·). It simply means that large claims can always be modeled using the
(generalized) Pareto distribution.
The practical question which always arises is then “what are large claims”, i.e.
how to chose u ?

23


How to deﬁne large claims ?
• Use of the k largest claims: Hill’s estimator
The intuitive idea is to ﬁt a linear straight line since for the largest claims
i = 1, ..., n, log(1 − Fn (Xi )) ∼ a + blog Xi . Let bk denote the estimator based on
the k largest claims.
Let {Xn−k+1:n , ..., Xn−1:n , Xn:n } denote the set of the k largest claims. Recall
that ξ ∼ −1/b, and then
n
1
ξ= log(Xn−k+i:n ) − log(Xn−k:n ).
k i=1

24


2.5 Hill estimator of the slope Hill estimator of the 95% VaR

10
2.0

8
quantile (95%)
slope (−b)

6
1.5

4
1.0

2
0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200

Figure 7: Pareto modeling for business interruption claims: tail index.

25


• Use of the claims exceeding u: maximum likelihood
A natural idea is to ﬁt a generalized Pareto distribution for claims exceeding u,
for some u large enough.
threshold [1] 3, we chose u = 3
p.less.thresh [1] 0.9271357, i.e. we keep to 8.5% largest claims
n.exceed [1] 87
method [1] “ml”, we use the maximum likelihood technique,
par.ests, we get estimators ξ and σ,
xi sigma
0.6179447 2.0453168
par.ses, with the following standard errors
xi sigma
0.1769205 0.4008392

26


5.0 MLE of the tail index, using Generalized Pareto Model Estimation of VaR and TVaR (95%)

5 e−02
1 e−02
4.5

1−F(x) (on log scale)

95
tail index

2 e−03
4.0

99
5 e−04
3.5

1 e−04
3.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 5 10 20 50 100 200

x (on log scale)

Figure 8: Pareto modeling for business interruption claims: VaR and TVaR.

27


From the statistical model of claims to the pure premium
Consider the following excess-of-loss treaty, with a priority d = 20, and an upper
limit 70.

Historical business interruption claims

140

130

120

110

100

90

80

70

60

50

40

30

20

10

1993 1994 1995 1996 1997 1998 1999 2000 2001

Figure 9: Pricing of a reinsurance layer.

28


The average number of claims per year is 145,
year 1992 1993 1994 1995 1996 1997 1998 1999 2000
frequency 173 152 146 131 158 138 120 156 136

Table 3: Number of business interruption claims.

29


For a claim size x, the reinsurer’s indemnity is I(x) = min{u, max{0, x − d}}.
The average indemnity of the reinsurance can be obtained using the Pareto
model,
∞ u
E(I(X)) = I(x)dF (x) = (x − d)dF (x) + u(1 − F (u)),
0 d

where F is a Pareto distribution.
Here E(I(X)) = 0.145. The empirical estimate (burning cost) is 0.14.
The pure premium of the reinsurance treaty is 20.6.
Example If d = 50 and d = 50, π = 8.9 (12 for burning cost... based on 1 claim).

30


Agenda
techniques

31


Figure 10: Hurricanes from 2001 to 2004.
32


Figure 11: Hurricanes 2005, the record year.
33


Increased value at risk
In 1950, 30% of the world’s population (2.5 billion people) lived in cities. In
2000, 50% of the world’s population (6 billon).
In 1950 the only city with more than 10 million inhabitants was New York.
There were 12 in 1990, and 26 are expected by 2015, including
• Tokyo (29 million),
• New York (18 million),
• Los Angeles (14 million).
• Increasing value at risk (for all risks)
The total value of insured costal exposure in 2004 was
• $1, 937 billion in Florida (18 million),
• $1, 902 billion in New York.

34


Two techniques to model large risks
• The actuarial-statistical technique: modeling historical series,
The actuary models the occurrence process of events, and model the claim size
(of the total event).
This is simple but relies on stability assumptions. If not, one should model
changes in the occurrence process, and should take into account inflation or
increase in value-at-risk.
• The meteorological-engineering technique: modeling natural hazard and
exposure.
This approach needs a lot of data and information so generate scenarios taking
all the policies specificities. Not very flexible to estimate return periods, and
works as a black box. Very hard to assess any confidence levels.

35


The actuarial-statistical approach
• Modeling event occurrence, the problem of global warming.
Global warming has an impact on climate related hazard (droughts, subsidence,
hurricanes, winterstorms, tornados, floods, coastal floods) but not geophysical
(earthquakes).
• Modeling claim size, the problem of increase of value at risk and inflation.
Pielke & Landsea (1998) normalized losses due to hurricanes, using both
population and wealth increases, “with this normalization, the trend of increasing
damage amounts in recent decades disappears”.

36


Impact of global warming on natural hazard

Number of hurricanes, per year 1851−2006
25
Frequency of hurricanes

20
15
10
5
0

1850 1900 1950 2000

Year

Figure 12: Number of hurricanes and major hurricanes per year.

37


More natural hazards with higher value at risk
Consider the example of tornados.

Number of tornados in the US, per month
400
300
Number of tornados

200
100
0

1960 1970 1980 1990 2000

Year

Figure 13: Number of tornadoes (from http://www.spc.noaa.gov/archive/).

38


The number of tornados per year is (linearly) lincreasing.

Distribution of the number of tornados, per year (1960, 1980, 2000)
0.05
0.04
0.03
0.02
0.01
0.00

40 60 80 100 120 140 160

Figure 14: Evolution of the distribution of the number of tornados per year.

39



Return period for tornados: more natural hazard
50
40
Claim size

30

HOMOTHETIC TRANSFORMATION DUE TO
20

MORE NATURAL HASARD PER YEAR
10

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
100 200 300 400 500 600 700 800 900 1000
0

0 20 40 60 80 100

Time (in years)

Figure 15: Impact of global warming on the return period.

40


The most damaging tornadoes in the U.S. (1890-1999), adjusted with wealth, are
the following,
Date Location Adjusted loss
28.05.1896 Saint Louis, IL 2,916
29.09.1927 Saint Louis, IL 1,797
18.04.1925 3 states (MO, IL, IN) 1,392
10.05.1979 Wichita Falls, TX 1,141
09.06.1953 Worcester, MA 1,140
06.05.1975 Omaha, NE 1,127
08.06.1966 Topeka, KS 1,126
06.05.1936 Gainesville, GA 1,111
11.05.1970 Lubbock, TX 1,081
28.06.1924 Lorain-Sandusky, OH 1,023
03.05.1999 Oklahoma City, OK 909
11.05.1953 Waco, TX 899
27.04.1890 Louisville, KY 836

Table 4: Most damaging tornadoes (from Brooks & Doswell (2001)).

41



Return period for tornados: more value at risk
50
40
Claim size

30
20

HOMOTHETIC TRANSFORMATION DUE TO
THE INCREASE OF VALUE AT RISK
10

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
100 200 300 400 500 600 700 800 900 1000
0

0 20 40 60 80 100

Time (in years)

Figure 16: Impact of increase of value at risk on the return period.

42


Cat models: the meteorological-engineering approach
The basic framework is the following,
1. the natural hazard model: generate stochastic climate scenarios, and assess
perils,
2. the engineering model : based on the exposure, the values, the building,
calculate damage,
3. the insurance model: quantify ﬁnancial losses based on deductibles,
reinsurance (or retrocession) treaties.

43


A practical example: Hurricanes in Florida

Figure 17: Florida and Hurricanes risk.

44


perils,

Figure 18: Generating stochastic climate scenarios.

45


perils,

Figure 19: Generating stochastic climate scenarios.

46


perils,

Figure 20: Checking outputs of climate scenarios.

47


perils,

Figure 21: Checking outputs of climate scenarios.

48


calculate damage,

Figure 22: Modeling the vulnerability.

49


calculate damage,

Figure 23: Modeling the vulnerability.

50


Hurricanes in Florida: Rare and extremal events ?
Note that for the probabilities/return periods of hurricanes related to insured
losses in Florida are the following (source: Wharton Risk Center & RMS)

$ 1 bn $ 2 bn $ 5 bn $ 10 bn $ 20 bn $ 50 bn
42.5% 35.9% 24.5% 15.0% 6.9% 1.7%
2 years 3 years 4 years 7 years 14 years 60 years

$ 75 bn $ 100 bn $ 150 bn $ 200 bn $ 250 bn
0.81% 0.41% 0.11% 0.03% 0.005%
123 years 243 years 357 years 909 years 2, 000 years

Table 5: Extremal insured losses (from Wharton Risk Center & RMS).

Recall that historical default (yearly) probabilities are

AAA AA A BBB BB B
0.00% 0.01% 0.05% 0.37% 1.45% 6.59%
- 10, 000 years 2, 000 years 270 years 69 years 15 years

Table 6: Return period of default (from S&P’s (1981-2003)).

51


Are there any safe place to be ?

Figure 24: Looking for a safe place ? going in North-East...?

52


A practical case: North-East Hurricanes in the U.S.

Figure 25: North-East Hurricanes in the U.S.: the 1938 hurricane

53


North-East Hurricanes: the 1938 experience
• Peak Steady Winds - 186 mph at Blue Hill Observatory, MA.
• Lowest Pressure - 946.2 mb at Bellport, NY
• Peak Storm Surge - 17 ft. above normal high tide
• Peak Wave Heights - 50 ft. at Gloucester, MA
• Deaths 700 (600 in New England)
• Homeless 63,000
• Homes, Buildings Destroyed 8,900
• Boats Lost 3,300
• Trees Destroyed - 2 Billion (approx.)
• Cost US$ 300 million (24 billion - 2005 adjusted)

54


North-East Hurricanes: further (recent) experience
1938 New England Hurricane, Cat 5
1954 Carol, Cat 3 (Rhode Island, Connecticut, Massachusetts)
1954 Edna, Cat 3 (North Carolina, Massachusetts, New Hampshire, Maine)
1960 Donna, Cat 5 (New York, Rhode Island, Connecticut, Massachusetts)
1961 Esther, Cat 4 (Massachusetts, New Jersey, New York, New Hampshire)
1985 Gloria, Cat 4 (Virginia, New York, Connecticut)
1991 Bob, Cat 3 (Rhode Island, Massachusetts)
1996 Bertha, Cat 3 (North Carolina)
1999 Floyd, Cat 4 (North Carolina, Virginia, Delaware, Pennsylvania, New Jersey,
New York, Vermont, Maine)
2003 Isabel, Cat 4 (North Carolina, Virginia, Washington D.C., Delaware)
2004 Charley, Cat 4 (Rhode Island, Virginia, North Carolina)

55


North-East Hurricanes: probabilities and return period
According to the United States Landfalling Hurricane Probability Project,
• 21% probability that NY City/Long Island will be hit with a tropical storm
or hurricane in 2007,
• 6% probability that NY City/Long Island will be hit with a major hurricane
(category 3 or more) in 2007,
• 99% probability that NY City/Long Island will be hit with a tropical storm
or hurricane in the next 50 years.
• 26% probability that NY City/Long Island will be hit with a major hurricane
(category 3 or more) in the next 50 years.

56


North-East Hurricanes: potential losses

Figure 26: Coast risk in the U.S. and the nightmare scenario in New Jersey (US$
100 billion).

57


Modelling contagion in credit risk models
cat insurance credit risk
n total number of insured n number of credit issuers
1 if policy i claims 1 if issuers i defaults
Ii = Ii =
0 if not 0 if not
Mi total sum insured Mi nominal
Xi exposure rate 1 − Xi recovery rate

58


Modelling contagion in credit risk models
In CreditMetrics, the idea is to generate random scenario to get the Proﬁt &
Loss distribution of the portfolio.
• the recovery rate is modeled using a beta distribution,
• the exposure rate is modeled using a MBBEFD distribution (see
Bernegger (1999)).
To generate joint defaults, CreditMetrics proposed a probit model.

59


The case of flood

Figure 27: August 2002 floods in Europe, flood damage function, (Munich Re
(2006)).

60


The case of ﬂood

Figure 28: Paris, 1910, the centennial ﬂood.

61


Assessing return period in a changing environment ?

Figure 29: Hydrological scheme of the Seine.
62


Assessing return period in a changing environment ?

Figure 30: Hydrological scheme of the Seine.
63


Comparison of the two approaches
F
F
F
F
F
F
F
F

64


Agenda
• Insurance covers against catastrophes, traditional versus
alternative techniques

65


Risk management solutions ?
• Equity holding: holding in solvency margin
+ easy and basic buffer
− very expensive
• Reinsurance and retrocession: transfer of the large risks to better diversified
companies
+ easy to structure, indemnity based
− business cycle influences capacities, default risk
• Side cars: dedicated reinsurance vehicules, with quota share covers
+ add new capacity, allows for regulatory capital relief
− short maturity, possible adverse selection

66


Risk management solutions ?
• Industry loss warranties (ILW) : index based reinsurance triggers
+ simple to structure, no credit risk
− limited number of capacity providers, noncorrelation risk, shortage of capacity
• Cat bonds: bonds with capital and/or interest at risk when a speciﬁed
trigger is reached
+ large capacities, no credit risk, multi year contracts
− more and more industry/parametric based, structuration costs

67


Insured losses

SELF PRIMARY INSURANCE SIDE CARS
INSURANCE REINSURANCE ILW CAR BONDS
0.04
0.03
Probability density

0.02
0.01
0.00

DEDUCTIBLE

0 20 40 60 80 100

Claim losses

Figure 31: Risk management solutions for diﬀerent types of losses.

68


Additional capital, post−Katrina reinsurance market

2.5 BN$ 27 BN$

4 BN$

ADDITIONAL EQUITY

3.5 BN$

8 BN$ INSURANCE
LINKED
SECURITIES

9 BN$

EXISTING START UP SIDE ILW CAT TOTAL
COMPANIES CARS BONDS

Figure 32: Risk management solutions for diﬀerent types of losses.

69


Retrocession market, 1998−2006

17155

ILW
Retrocession market (including ILW)
Side cars capital
capital markets 12505
Cat bonds issuances

7452
6561

4576
3717
3447
3171
2272

1998 1999 2000 2001 2002 2003 2004 2005 2006

Figure 33: Capital market provide half of the retrocession market.

70


Trigger deﬁnition for peak risk
• indemnity trigger: directly connected to the experienced damage
+ no risk for the cedant, only one considered by some regulator (NAIC)
− time necessity to estimate actual damage, possible adverse selection (audit
needed)
• industry based index trigger: connected to the accumulated loss of the
industry (PCS)
+ simple to use, no moral hazard
− noncorrelation risk

71


Trigger definition for peak risk
• environmental based index trigger: connected to some climate index (rainfall,
windspeed, Richter scale...) measured by national authorities and
meteorological offices
+ simple to use, no moral hazard
− noncorrelation risk, related only to physical features (not financial
consequences)
• parametric trigger: a loss event is given by a cat-software, using climate
inputs, and exposure data
+ few risk for the cedant if the model fits well
− appears as a black-box

72


Figure 34: Actual losses versus payout (cat option).
73


Reinsurance

The insurance approach (XL treaty)

35
30
25
REINSURER

Loss per event

20
15
INSURER

10
INSURED

5
0
0.0 0.2 0.4 0.6 0.8 1.0

Event

Figure 35: The XL reinsurance treaty mechanism.

74


Group net W.P. net W.P. loss ratio total Shareholders’ Funds
(2005) (2004) (2005) (2004)
Munich Re 17.6 20.5 84.66% 24.3 24.4
Swiss Re (1) 16.5 20 85.78% 15.5 16
Berkshire Hathaway Re 7.8 8.2 91.48% 40.9 37.8
Hannover Re 7.1 7.8 85.66% 2.9 3.2
GE Insurance Solutions 5.2 6.3 164.51% 6.4 6.4
Lloyd’s 5.1 4.9 103.2%
XL Re 3.9 3.2 99.72%
Everest Re 3 3.5 93.97% 3.2 2.8
Reinsurance Group of America Inc. 3 2.6 1.9 1.7
PartnerRe 2.8 3 86.97% 2.4 2.6
Transatlantic Holdings Inc. 2.7 2.9 84.99% 1.9 2
Tokio Marine 2.1 2.6 26.9 23.9
Scor 2 2.5 74.08% 1.5 1.4
Odyssey Re 1.7 1.8 90.54% 1.2 1.2
Korean Re 1.5 1.3 69.66% 0.5 0.4
Scottish Re Group Ltd. 1.5 0.4 0.9 0.6
Converium 1.4 2.9 75.31% 1.2 1.3
Sompo Japan Insurance Inc. 1.4 1.6 25.3% 15.3 12.1
Transamerica Re (Aegon) 1.3 0.7 5.5 5.7
Platinum Underwriters Holdings 1.3 1.2 87.64% 1.2 0.8
Mitsui Sumitomo Insurance 1.3 1.5 63.18% 16.3 14.1

Table 7: Top 25 Global Reinsurance Groups in 2005 (from Swiss Re (2006)).

75


Side cars
A hedge fund that wishes to get into the reinsurance business will start a special
purpose vehicle with a reinsurer The hedge fund is able to get into reinsurance
without Hiring underwriters Buying models Getting rated by the rating agencies

76


ILW - Insurance Loss Warranty
Industry loss warranties pay a ﬁxed amount based of the amount of industry loss
(PCS or SIGMA).
Example For example, a $30 million ILW with a $5 billion trigger.

Cat bonds and securitization
Bonds issued to cover catastrophe risk were developed subsequent to Hurricane
Andrew
These bonds are structured so that the investor has a good return if there are no
qualifying events and a poor return if a loss occurs. Losses can be triggered on an
industry index or on an indemnity basis.

77


Cat bonds and securitization

The securitization approach (Cat bond)

35
INVESTORS

30
25
SPV

Loss per event

20
15
INSURER

10
INSURED

5
0
0.0 0.2 0.4 0.6 0.8 1.0

Event

Figure 36: The securitization mechanism, parametric triggered cat bond.

78


Capital structure, Residential Re, 2001

USAA retention of traditional reinsurance USAA
annual
US$ 1.6
0.41% exceedance
billion
probability
Residential Re
Traditional reinsurance
US$ 150 million
US$ 300 million USAA
part of
part of US$ 500 million
US$ 500 million
annual
US$ 1.1
1.12% exceedance
billion
probability

Traditional
reinsurance US$ 360 million USAA
part of US$ 400 million

USAA retention & Florida
hurricane catastrophe fund or
traditional reinsurance

Figure 37: Some cat bonds issued: Residential Re.
79


Capital structure, Redwood Capital I Ltd, 2001

PCS
industry annual
losses exceedence
US$ probability
(billion)

100%
31.5 0.34%
88.9%
30.5 0.37%
77.8%
29.5 0.40%
66.7%
28.5 0.44%
55.6%
27.5 0.48%
44.4%
26.5 0.52%
33.3%
25.5 0.56%
22.2%
24.5 0.61%
11.1%
23.5 0.66%

Figure 38: Some cat bonds issued: Redwood Capital.
80


Capital structure, Atlas Re II, 2001

Traditional retrocession
and retention by SCOR

Atlas Re II retrocessional agreement, US$ 150 million per event
Class A notes, US$ 50 million
annual
0.07% exceedance
probability

Atlas Re II retrocessional agreement, US$ 150 million per event
Class B notes, US$ 100 million

annual
1.33% exceedance
probability

Traditional retrocession
and retention by SCOR

Figure 39: Some cat bonds issued: Redwood Capital.
81


Property Catastrophe Risk Linked Securities, 2001

600

FRENCH WIND

TOKYO EARTHQUAKE
CALIFORNIA EARTHQUAKE

US S.E. WIND

US N.E. WIND
500

SECOND EVENT

EUROPEAN WIND
400

JAPANESE EARTHQUAKE

MONACO EARTHQUAKE
300

MADRID EARTHQUAKE
200

100

0

Figure 40: Distribution of US$ ar risk, per peril.

82


Cat bonds versus (traditional) reinsurance: the price
• A regression model (Lane (2000))
• A regression model (Major & Kreps (2002))

83


Figure 41: Reinsurance (pure premium) versus cat bond prices.

84


Cat bonds versus (traditional) reinsurance: the price
• Using distorted premiums (Wang (2000,2002))
If F (x) = P(X > x) denotes the losses survival distribution, the pure premium is
∞
π(X) = E(X) = 0 F (x)dx. The distorted premium is
∞
πg (X) = g(F (x))dx,
0

where g : [0, 1] → [0, 1] is increasing, with g(0) = 0 and g(1) = 1.
Example The proportional hazards (PH) transform is obtained when g is a
power function.
Wang (2000) proposed the following transformation, g(·) = Φ(Φ−1 (F (·)) + λ),
where Φ is the N (0, 1) cdf, and λ is the “market price of risk”, i.e. the Sharpe
ratio. More generally, consider g(·) = tκ (t−1 (F (·)) + λ), where tκ is the Student t
κ
cdf with κ degrees of freedom.

85


Property Catastrophe Risk Linked Securities, 2001

16 Yield spread (%) Lane model
Wang model
Empirical
14

12

10

8

6

4

2

0
Mosaic 2A

Mosaic 2B

Halyard Re

Domestic Re

Concentric Re

Juno Re

Residential Re

Kelvin 1st event

Kelvin 2nd event

Gold Eagle A

Gold Eagle B

Namazu Re

Atlas Re A

Atlas Re B

Atlas Re C

Seismic Ltd
Figure 42: Cat bonds yield spreads, empirical versus models.

86


Who might buy cat bonds ?
In 2004,
• 40% of the total amount has been bought by mutual funds,
• 33% of the total amount has been bought by cat funds,
• 15% of the total amount has been bought by hedge funds.
Opportunity to diversify asset management (theoretical low correlation with
other asset classes), opportunity to gain Sharpe ratios through cat bonds excess
spread.

87


Insure against natural catastrophes and make money ?

Return On Equity, US P&C insurers
15

KATRINA
RITA
WILMA
10

4 hurricanes

NORTHRIDGE
5

ANDREW
0

9/11

1990 1995 2000 2005

Figure 43: ROE for P&C US insurance companies.
88


Reinsure against natural catastrophes and make money ?

Combined Ratio
Reinsurance vs. P/C Industry

162.4
160

150 9/11

2004/2005
140 ANDREW HURRICANES

129
130
126.5

125.8

124.6
119.2
120
115.8

115.8
114.3
113.6
110.5

110.1

110.1
111
108.8

108.5

107.4
106.9

106.7
110

108
106.5
105.9
104.8
106
105

101.9

100.9
100.8

100.5

98.3
100

90
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Figure 44: Combined Ratio for P&C US companies versus reinsurance.
89


Agenda
techniques

90


Solvency margins when insuring again natural catastrophes
Within an homogeneous portfolios (Xi identically distributed), sufficiently large
X1 + ... + Xn
(n → ∞), → E(X). If the variance is finite, we can also derive a
n
confidence interval (solvency requirement), if the Xi ’s are independent,
 
n
 √ 
Xi ∈  nE(X) ± 2 nVar(X)  with probability 99%.
i=1
premium risk based capital need

Nonindependence implies more volatility and therefore more capital requirement.

91


Implications for risk capital requirements

0.04
0.03
Probability density

99.6% quantile
Risk−based capital need
0.02

99.6% quantile
Risk−based capital need
0.01
0.00

0 20 40 60 80 100

Annual losses

Figure 45: Independent versus non-independent claims, and capital requirements.

92


The premium as a fair price
Pascal and Fermat in the XVIIIth century proposed to evaluate the “produit
scalaire des probabilités et des gains”,
n
< p, x >= pi xi = EP (X),
i=1

based on the “règle des parties”.
For Quételet, the expected value was, in the context of insurance, the price that
guarantees a ﬁnancial equilibrium.

93


What is probability P ?
“my dwelling is insured for $ 250,000. My additional premium for earthquake
insurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more I
look to this, the more it seems that my chances of having a covered loss are about
zero. I’m paying $ 768 for this ? ” (Business Insurance, 2001).
• Estimated annualized proability in Seatle 1/250 = 0.4%,
• Actuarial probability 768/(250, 000 − 43, 750) ∼ 0.37%
The probability for an actuary is 0.37% (closed to the actual estimated
probability), but it is much smaller for anyone else.

94


The short memory puzzle

Percentage of California
Homeowners with Earthquake Insurance

32.9 33 33.2

19.5
17.4 16.8
15.7 15.8
14.6
13.3 13.8

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Figure 46: Trajectory of major hurricanes, in 1999 and 2005.
95


Rational behavior of insurers ?
Between September 2004 and September 2005, the real estate prices (Miami
Dade county) increased of +45%, despite the 4 hurricanes in 2004.

Flyods Hurricane, 1999 The 2005 hurricanes of level 5

64.82
64.82
74.08
83.34
83.34
92.6

92.6

111.12

129.64

166.68

166.68

175.94
185.2
203.72
212.98
203.72
194.46
194.46
212.98
231.5
250.02
250.02
231.5
212.98
194.46
175.94
157.42
166.68
175.94
175.94
148.16

Figure 47: Trajectory of major hurricanes, in 1999 and 2005.

96


von Neumann & Morgenstern: expected utility approach

Ru (X) = u(x)dP = P(u(X) > x))dx

where u : [0, ∞) → [0, ∞) is a utility function.
Example with an exponential utility, u(x) = [1 − e−αx ]/α,
1
Ru (X) = log EP (eαX ) .
α
Musiela & Zariphopoulou (2001) used this premium to price derivatives in
incomplete markets.

97


Yaari: distorted utility approach

Rg (X) = xdg ◦ P = g(P(X > x))dx

where g : [0, 1] → [0, 1] is a distorted function.
Example if g(x) = I(X ≥ α) Rg (X) = V aR(X, α), and if g(x) = min{x/α, 1}
Rg (X) = T V aR(X, α) (also called expected shortfall),
Rg (X) = EP (X|X > V aR(X, α)).

98


Calcul de l’esperance mathématique

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Figure 48: Expected value xdFX (x) = P(X > x)dx.

99


Calcul de l’esperance d’utilité

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Figure 49: Expected utility u(x)dFX (x).

100


Calcul de l’intégrale de Choquet

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Figure 50: Distorted probabilities g(P(X > x))dx.

101


Value-at-Risk and Expected Shortfall
The Value-at-Risk is simply the quantile of a proﬁt & loss distribution,

V aR(X, p) = xp = F −1 (p) = sup{x ∈ R, F (x) ≥ p}.

Remark This notion is closely related to the return period and ruin probabilities.
The Expected Shortfall, or Tail Value-at-Risk, is the expected value above the
VaR,
T V aR(X, p) = E(X|X > V aR(X, p)).

102


Worst-case scenarios
Consider a set of scenarios, i.e. possible probabilities Q. Consider

R(X) = sup {EQ (X)} ,
Q∈Q

the worst case scenarios pure premium.

103


techniques

104


Coherent risk measures
A risk measure is said to be coherent (from Artzner, Delbaen, Eber &
Heath (1999)) if
• R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ),
• R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X),
• R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ,
• R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) + R(Y ).
“subadditivity” can be interpreted as “diversiﬁcation does not increase risk”.
Example: the Expected-Shortfall is coherent, the Value-at-Risk is not.

105


Convex risk measures
A risk measure is said to be convex (from Artzner, Delbaen, Eber & Heath
(1999)) if
• R(·) is convex, i.e. R(λX + (1 − λ)Y ) ≤ λR(X) + (1 − λ)R(Y ), for any
λ ∈ [0, 1].
Hence, if a convex measure satisﬁes the homogeneity condition, it is coherent.

106


Convex risk measures and Expected-Shortfall
All distortion convex risk measure is a mixture of expected shorfalls,
1 1
−1
R(X) = FX (1 − p)dg(p) = ES(X, 1 − p)dµ(p) = E(ES(X, Θ)),
0 0

where Θ is a random variable with values in [0, 1] (Inui & Kijima (2004)).

107


Estimating a Value-at-Risk
A natural idea is to use the Pareto approximation for claims exceeding threshold
u.
n
Let Nu denote the number of claims exceeding u, Nu = I(Xi > u).
i=1
If x > u,

F (x) = P(X > x) = P(X > u)P(X > x|X > u) = F (u)P(X > x|X > u),

where P(X > x|X > u) = G(x)(x − u) is Pareto distributed

G(t) = P(X − u ≤ t|X > u) ∼ Hξ,σ (t).

108


Thus, a natural estimator for F (x) uses a natural estimator for F (u), and the
Pareto approximation for F u (x), i.e.
−1/ξ
Nu x−u
F (x) = 1 − 1+ξ
n σ
for all x > u, and u large enough.
Thus, a natural estimator for V aR(X, p) is
−ξ
σ n
V aRu (X, p) = u + (1 − p) −1 .
ξ Nu

109


Note that Hill’s estimator can also be used
Hill
n −ξn,k
V aRk (X, p) = Xn−k:n (1 − p) ,
k
for some k such that p > 1 − k/n. This estimator can be written
Hill
n −ξn,k
V aRk (X, p) = Xn−k:n + Xn−k:n (1 − p) −1 .
k

110


Estimating an Expected Shortfall
Similarly, the expected shortfall can be estimated simply,

1 σ − ξu
ES u (X, p) = V aRu (X, p) · + .
1−ξ [1 − ξ · V aRu (X, p)]

111


Risk measures for business interruption

Lower CI Estimate Upper CI
u = 0.5 Var(99.9%) 27.50320 46.57878 99.60774
TVar(99.9%) 48.80638 115.46481 205.12015
u=1 Var(99.9%) 29.09757 47.26090 89.10651
TVaR(99.9%) 56.36115 120.26600 205.12015
u=5 Var(99.9%) 25.73079 43.85056 144.09078
TVar(99.9%) 42.10588 238.93939 205.12015

Table 8: Distorted premiums for business interruption claims, using Pareto ap-
proximation (with diﬀerent thresholds u).

112


5 e−02

1 e−02

95

95
2 e−03
5 e−03

99

99
5 e−04
5 e−04

1 e−04
1 e−04

1 2 5 10 20 50 100 200 5 10 20 50 100 200

x (on log scale) x (on log scale)

Figure 51: Estimation of the VaR and the TVaR with levels 99, 9%, where u = 1
on the left, u = 5 on the right.

113


Agenda
techniques

114


Coherent risk measures
A risk measure is said to be coherent (from Artzner, Delbaen, Eber &
Heath (1999)) if
• R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X),
• R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) + R(Y ).
“subadditivity” can be interpreted as “diversiﬁcation does not increase risk”.
Example: the Expected-Shortfall is coherent, the Value-at-Risk is not.

115


Non-subadditivity of Value-at-Risk
In the insurance context, Ewans (2001) pointed out that “probability of ruin may
often be inconsistent with many other reasonable risk management criteria”.
Example: Consider X, Y ∼ LN (0, 1).

116


Possible VaR for the sum of two LN(0,1) risks

40
30
20

VaR(X+Y) > VaR(X)+VaR(Y)
10

VaR(X+Y) < VaR(X)+VaR(Y)
0

0.90 0.92 0.94 0.96 0.98 1.00

Probability levels

Figure 52: Value-at-Risk for the sum of lognormal risks.

117


The non subadditivity puzzle for large risks
Assume that X and Y have tail indices a and b. If min a, b > 1, there exists
p0 ∈ (0, 1) such that for all p ∈ (p0 , 1),

V aR (X + Y, p) < V aR(X, p) + V aR(Y, p).

If min a, b < 1, there exists p0 ∈ (0, 1) such that for all p ∈ (p0 , 1),

V aR (X + Y, p) > V aR(X, p) + V aR(Y, p).

118

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