Slides axa

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

Modeling and covering catastrophic risks

Arthur Charpentier

AXA Risk College, April 2007

[email protected]

1


Agenda

Catastrophic risks modelling

• General introduction

• Business interruption and very large claims

• Natural catastrophes and accumulation risk

• Insurance covers against catastrophes, traditional versus alternativetechniques

Risk measures and capital requirements

• Risk measures, an economic introduction

• Calculating risk measures for catastrophic risks

• Diversification and capital allocation

2


Agenda










3


Some stylized facts“climatic risk in numerous branches of industry is more important than the riskof 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 MunichRe (2006)).

5


Some stylized facts: man-made catastrophes

Includes industry fire, oil & gas explosions, aviation crashes, shipping and raildisasters, 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 millioninsured 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 thesum of claims, or at least 5% of the net premiums”,

Answer 3 “ large claims are every claim for which the actuary has to go andsee one of the chief members of the company”.

Examples Traditional types of catastrophes, natural (hurricanes, typhoons,earthquakes, floods, tornados...), man-made (fires, explosions, businessinterruption...) or new risks (terrorist acts, asteroids, power outages...).

From large claims to catastrophe, the difference is that there is a before thecatastrophe, 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 hitinsurers, reinsurers and the retrocession market

• lack of capacity, strong increase in rate

• Natural Catastrophes - nonUSA: in Asia (earthquakes, typhoons) andEurope (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 concentratedmarket

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) =∫ ∞

0

xdF (x) =∫ ∞

0

F (x)dx,

• the Value-at-Risk or quantile function is

V aR(X,u) = F−1(u) = F−1

(1− u) i.e. P(X > V aR(X, u)) = 1− u,

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

11


0 2 4 6 8 10

0.00.1

0.20.3

0.40.5

The density of the exponential distribution

Claim size

mean = 1mean = 2 mean = 5

0 2 4 6 8 10

0.00.2

0.40.6

0.81.0

The exceedance distribution

Claim size

Proba

bility


0.0 0.2 0.4 0.6 0.8 1.0

05

1015

The quantile function of the exponential distribution

Probability level

Claim

size mean = 1

mean = 2 mean = 5

0 100 200 300 400 500

02

46

810

12

The return period function

Time

Claim

size


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 accontaccumulation 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 usuallyconsider

• 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 ruinprobability, assuming that claim size is not to large. Furthermore, additionalcapital to ensure solvency (non-ruin) can be obtained using the central limittheorem (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. thevariance is usually infinite), but the Poisson assumption for occurrence is stillrelevant.

14


Updating actuarial models

Example For business interruption, the total loss is S =N∑

i=1

Xi where N is

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.

Example For hurricanes or winterstorms, the total loss is S =N∑

i=1

Xi where N is

Poisson, and Xi =Ni∑

j=1

Xi,j , where the Xi,j ’s are i.i.d.

15


Agenda










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$ 11billion.

17


Some results from Extreme Value Theory

When modeling large claims (industrial fire, business interruption,...): extremevalue theory framework is necessary.

The Pareto distribution appears naturally when modeling observations over agiven threshold,

F (x) = P(X ≤ x) = 1−(

x

x0

)b

, where x0 = exp(−a/b)

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


0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Cumulative distribution function, with confidence interval

logarithm of the losses

cum

ula

tive p

robabili

ties

0 1 2 3 4 5

−5

−4

−3

−2

−1

0

log−log Pareto plot, with confidence interval

logarithm of the losseslo

ga

rith

m o

f th

e s

urv

iva

l pro

ba

bili

ties

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.

20% of the claims

80% of the claims 20% of the losses

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 represent73% of the losses.

20


Why the Pareto distribution ? historical perspective

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lorenz curve of business interruption claims

Proportion of claims number

Prop

ortio

n of

claim

size

20% OFTHE CLAIMS

73% OFTHE LOSSES

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 giventhat X > u, with survival distribution G(·) defined as

G(x) = P(X − u > x|X > u) =F (x + u)

F (u)

This is closely related to some regular variation property, and only powerfunction 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 technicalbrochures the Pareto model in property reinsurance and estimating propertyexcess of loss risk premium: The Pareto model.

Actually, we will see that the Pareto model gives much more than only apremium.

22


Large claims and the Pareto model

The theorem of Pickands-Balkema-de Haan states that if the X1, ..., Xn areindependent and identically distributed, for u large enough,

P(X − u > x|X > u) ∼ Hξ,σ(u)(x) =

(1 + ξ

x

σ(u)

)−1/ξ

if ξ 6= 0,

exp(− x

σ(u)

)if ξ = 0,

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 claimsi = 1, ..., n, log(1− Fn(Xi)) ∼ a + blog Xi. Let bk denote the estimator based onthe k largest claims.

Let {Xn−k+1:n, ..., Xn−1:n, Xn:n} denote the set of the k largest claims. Recallthat ξ ∼ −1/b, and then

ξ =

(1k

n∑

i=1

log(Xn−k+i:n)

)− log(Xn−k:n).

24


0 200 400 600 800 1000 1200

1.0

1.5

2.0

2.5

Hill estimator of the slope

slo

pe

(−

b)

0 200 400 600 800 1000 1200

24

68

10

Hill estimator of the 95% VaR

quantile

(95%

)

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


0.5 1.0 1.5 2.0 2.5 3.0 3.5

3.0

3.5

4.0

4.5

5.0

MLE of the tail index, using Generalized Pareto Model

tail

index

5 10 20 50 100 200

1 e

−0

45

e

−0

42

e

−0

31

e

−0

25

e

−0

2x (on log scale)

1−

F(x

) (o

n lo

g s

ca

le)

99

95

99

95

Estimation of VaR and TVaR (95%)

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 upperlimit 70.

Historical business interruption claims

1993 1994 1995 1996 1997 1998 1999 2000 2001

10

20

30

40

50

60

70

80

90

100

110

120

130

140

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 Paretomodel,

E(I(X)) =∫ ∞

0

I(x)dF (x) =∫ u

d

(x− d)dF (x) + u(1− F (u)),

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










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. In2000, 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 modelchanges in the occurrence process, and should take into account inflation orincrease in value-at-risk.

• The meteorological-engineering technique: modeling natural hazard andexposure.

This approach needs a lot of data and information so generate scenarios takingall the policies specificities. Not very flexible to estimate return periods, andworks 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 bothpopulation and wealth increases, “with this normalization, the trend of increasingdamage amounts in recent decades disappears”.

36


Impact of global warming on natural hazard

1850 1900 1950 2000

05

1015

2025

Number of hurricanes, per year 1851−2006

Year

Frequ

ency

of hu

rrican

es

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

37


More natural hazards with higher value at risk

Consider the example of tornados.

1960 1970 1980 1990 2000

010

020

030

040

0

Number of tornados in the US, per month

Year

Numb

er of

torna

dos

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

38



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

40 60 80 100 120 140 160

0.00

0.01

0.02

0.03

0.04

0.05

Distribution of the number of tornados, per year (1960, 1980, 2000)

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

39



0 20 40 60 80 100

010

2030

4050

Return period for tornados: more natural hazard

Time (in years)

Claim

size

100100

200200

300300

400400

500500

600600

700700

800800

900900

10001000 1100 1200 1300 1400

HOMOTHETIC TRANSFORMATION DUE TOMORE NATURAL HASARD PER YEAR

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

40



The most damaging tornadoes in the U.S. (1890-1999), adjusted with wealth, arethe 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



0 20 40 60 80 100

010

2030

4050

Return period for tornados: more value at risk

Time (in years)

Claim

size

100100

200200

300300

400400

500500

600600

700700

800800

900900

10001000 1100 1200 1300 1400

HOMOTHETIC TRANSFORMATION DUE TOTHE INCREASE OF VALUE AT RISK

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 assessperils,

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


A practical example: Hurricanes in Florida1. the natural hazard model: generate stochastic climate scenarios, and assess

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


A practical example: Hurricanes in Florida2. the engineering model : based on the exposure, the values, the building,

calculate damage,

Figure 22: Modeling the vulnerability.

49


A practical example: Hurricanes in Florida2. the engineering model : based on the exposure, the values, the building,

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 insuredlosses 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) experience1938 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 stormor 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 stormor 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

Ii =1 if policy i claims

0 if notIi =

1 if issuers i defaults

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 (seeBernegger (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 versusalternative 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 diversifiedcompanies

+ 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 specifiedtrigger is reached

+ large capacities, no credit risk, multi year contracts

− more and more industry/parametric based, structuration costs

67


0 20 40 60 80 100

0.00

0.01

0.02

0.03

0.04

Insured losses

Claim losses

Prob

abilit

y den

sity

SELFINSURANCE

PRIMARY INSURANCE REINSURANCESIDE CARS

ILW CAR BONDS

DEDUCTIBLE

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

68


Additional capital, post−Katrina reinsurance market

EXISTING

COMPANIES

9 BN$

START UP

8 BN$

SIDE

CARS

3.5 BN$

ILW

4 BN$

CAT

BONDS

2.5 BN$

TOTAL

27 BN$

ADDITIONAL EQUITY

INSURANCELINKED

SECURITIES

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

69


Retrocession market, 1998−2006

1998

2272

1999

3171

2000

3447

2001

3717

2002

4576

2003

7452

2004

6561

2005

12505

2006

17155

Cat bonds issuances

Side cars capital

Retrocession market (including ILW)

ILW

capital markets

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 (auditneeded)

• industry based index trigger: connected to the accumulated loss of theindustry (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 andmeteorological offices

+ simple to use, no moral hazard

− noncorrelation risk, related only to physical features (not financialconsequences)

• parametric trigger: a loss event is given by a cat-software, using climateinputs, 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

INSURED

INSURER

REINSURER

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

35

The insurance approach (XL treaty)

EventLoss

per

eve

nt

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 specialpurpose vehicle with a reinsurer The hedge fund is able to get into reinsurancewithout 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 HurricaneAndrew

These bonds are structured so that the investor has a good return if there are noqualifying events and a poor return if a loss occurs. Losses can be triggered on anindustry index or on an indemnity basis.

77


Cat bonds and securitization

INSURED

INSURER

SPV

INVESTORS

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

35

The securitization approach (Cat bond)

EventLoss

per

eve

nt

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

78


Capital structure, Residential Re, 2001

US$ 1.1billion

US$ 1.6billion

1.12%

0.41%annualexceedanceprobability

annualexceedanceprobability

USAA retention & Florida

hurricane catastrophe fund or

traditional reinsurance

USAA

USAA

USAA

Traditional

reinsurance US$ 360 million

part of US$ 400 million

Residential Re

US$ 150 million

part of

US$ 500 million

Traditional reinsurance

US$ 300 million

part of US$ 500 million

USAA retention of traditional reinsurance

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


Capital structure, Redwood Capital I Ltd, 2001

PCSindustry

lossesUS$

(billion)

annualexceedenceprobability

23.5 0.66%

24.5 0.61%

25.5 0.56%

26.5 0.52%

27.5 0.48%

28.5 0.44%

29.5 0.40%

30.5 0.37%

31.5 0.34%

11.1%

22.2%

33.3%

44.4%

55.6%

66.7%

77.8%

88.9%

100%

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


Capital structure, Atlas Re II, 2001

1.33%

0.07%



Traditional retrocessionand retention by SCOR

Traditional retrocessionand retention by SCOR

Atlas Re II retrocessional agreement, US$ 150 million per event

Class A notes, US$ 50 millionClass A notes, US$ 50 million

Atlas Re II retrocessional agreement, US$ 150 million per event

Class B notes, US$ 100 million

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


Property Catastrophe Risk Linked Securities, 2001

0

100

200

300

400

500

600

CALIF

ORNI

A EA

RTHQ

UAKE

FREN

CH W

IND

US S

.E. W

IND

US N

.E. W

IND

TOKY

O EA

RTHQ

UAKE

SECO

ND E

VENT

EURO

PEAN

WIN

D

JAPA

NESE

EAR

THQU

AKE

MONA

CO E

ARTH

QUAK

E

MADR

ID E

ARTH

QUAK

E

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 isπ(X) = E(X) =

∫∞0

F (x)dx. The distorted premium is

πg(X) =∫ ∞

0

g(F (x))dx,

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 apower 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 Sharperatio. More generally, consider g(·) = tκ(t−1

κ (F (·)) + λ), where tκ is the Student t

cdf with κ degrees of freedom.

85


Property Catastrophe Risk Linked Securities, 2001

0

2

4

6

8

10

12

14

16 Yield spread (%)

Mosa

ic 2A

Mosa

ic 2B

Halya

rd R

e

Dome

stic R

e

Conc

entric

Re

Juno

Re

Resid

entia

l Re

Kelvi

n 1st

even

t

Kelvi

n 2nd

even

t

Gold

Eagle

A

Gold

Eagle

B

Nama

zu R

e

Atlas

Re A

Atlas

Re B

Atlas

Re C

Seism

ic Ltd

Lane model

Wang model

Empirical

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 withother asset classes), opportunity to gain Sharpe ratios through cat bonds excessspread.

87


Insure against natural catastrophes and make money ?

1990 1995 2000 2005

05

1015

Return On Equity, US P&C insurers

ANDREW

NORTHRIDGE

9/11

4 hurricanes

KATRINA

RITA

WILMA

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

88


Reinsure against natural catastrophes and make money ?

Combined Ratio Reinsurance vs. P/C Industry

110.5

108.8

1991

126.5

115.8

1992

105 10

6.9

1993

113.6

108.5

1994

119.2

106.7

1995

104.8 10

6

199610

0.8 101.9

1997

100.5

105.9

1998

114.3

108

1999

106.5

110.1

2000

162.4

115.8

2001

125.8

107.4

2002

111

110.1

2003

124.6

98.3

2004

129

100.9

200590

100

110

120

130

140

150

160

ANDREW

9/11

2004/2005HURRICANES

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

89


Agenda










90


Solvency margins when insuring again natural catastrophes

Within an homogeneous portfolios (Xi identically distributed), sufficiently large

(n →∞),X1 + ... + Xn

n→ E(X). If the variance is finite, we can also derive a

confidence interval (solvency requirement), if the Xi’s are independent,

n∑

i=1

Xi ∈

nE(X)︸︷︷︸premium

± 2√

nVar(X)︸︷︷︸risk based capital need

with probability 99%.

Nonindependence implies more volatility and therefore more capital requirement.

91


0 20 40 60 80 100

0.00

0.01

0.02

0.03

0.04

Implications for risk capital requirements

Annual losses

Prob

abilit

y de

nsity

99.6% quantile

99.6% quantile

Risk−based capital need

Risk−based capital need

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 “produitscalaire des probabilités et des gains”,

< p, x >=n∑

i=1

pixi = EP(X),

based on the “règle des parties”.

For Quételet, the expected value was, in the context of insurance, the price thatguarantees a financial equilibrium.

93


What is probability P ?

“my dwelling is insured for $ 250,000. My additional premium for earthquakeinsurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more Ilook to this, the more it seems that my chances of having a covered loss are aboutzero. 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 estimatedprobability), but it is much smaller for anyone else.

94


The short memory puzzle

Percentage of California Homeowners with Earthquake Insurance

32.9

1994

33

1995

33.2

1996

19.5

1997

17.4

1998

16.8

1999

15.7

2000

15.8

2001

14.6

2002

13.3

2003

13.8

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 (MiamiDade county) increased of +45%, despite the 4 hurricanes in 2004.

148.16175.94

175.94166.68157.42175.94194.46212.98231.5250.02250.02231.5212.98194.46

194.46203.72

212.98

203.72

185.2

175.94

166.68

166.68

129.64

111.12

92.6

92.6

83.3483.34

74.0864.82

64.82

250.02

194.46

Flyods Hurricane, 1999 The 2005 hurricanes of level 5

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,

Ru(X) =1α

log(EP(eαX)

).

Musiela & Zariphopoulou (2001) used this premium to price derivatives inincomplete 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) = TV aR(X, α) (also called expected shortfall),Rg(X) = EP(X|X > V aR(X,α)).

98


0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’esperance mathématique

Figure 48: Expected value∫

xdFX(x) =∫P(X > x)dx.

99


0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’esperance d’utilité

Figure 49: Expected utility∫

u(x)dFX(x).

100


0 1 2 3 4 5 6

0.00.2

0.40.6

0.81.0

Calcul de l’intégrale de Choquet

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 theVaR,

TV 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) = supQ∈Q

{EQ(X)} ,

the worst case scenarios pure premium.

103











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 anyλ ∈ [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,

R(X) =∫ 1

0

F−1X (1− p)dg(p) =

∫ 1

0

ES(X, 1− p)dµ(p) = E(ES(X, Θ)),

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 thresholdu.

Let Nu denote the number of claims exceeding u, Nu =n∑

i=1

I(Xi > u).

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 thePareto approximation for Fu(x), i.e.

F (x) = 1− Nu

n

(1 + ξ

x− u

σ

)−1/bξ

for all x > u, and u large enough.

Thus, a natural estimator for V aR(X, p) is

V aRu(X, p) = u +σ

ξ

((n

Nu(1− p)

)−bξ− 1

).

109



Note that Hill’s estimator can also be used

V aRk(X, p) = Xn−k:n

(n

k(1− p)

)−ξHilln,k

,

for some k such that p > 1− k/n. This estimator can be written

V aRk(X, p) = Xn−k:n + Xn−k:n

((n

k(1− p)

)−ξHilln,k − 1

).

110


Estimating an Expected Shortfall

Similarly, the expected shortfall can be estimated simply,

ESu(X, p) = V aRu(X, p) ·(

1

1− ξ+

σ − ξu

[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


1 2 5 10 20 50 100 200

1

e−

04

5

e−

04

5

e−

03

5

e−

02

x (on log scale)

1−

F(x

) (o

n lo

g s

cale

)

99

95

99

95

5 10 20 50 100 200

1

e−

04

5

e−

04

2

e−

03

1

e−

02

x (on log scale)1

−F

(x)

(on

log

sca

le)

99

95

99

95

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

113


Agenda










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 mayoften be inconsistent with many other reasonable risk management criteria”.

Example: Consider X,Y ∼ LN(0, 1).

116


0.90 0.92 0.94 0.96 0.98 1.00

010

2030

40

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

Probability levels

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

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

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 existsp0 ∈ (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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