Slides saopaulo-catastrophe (1)

Arthur CHARPENTIER - Modeling and covering catastrophes

Modeling and covering catastrophes
Arthur Charpentier
Sao Paulo, April 2009

arthur.charpentier@univ-rennes1.fr

http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/

1


Agenda
Catastrophic risks products and models
• General introduction
• Modeling very large claims
• Natural catastrophes and accumulation risk
• Insurance covers against catastrophes, traditional versus alternative techniques
Risk measures and pricing covers
• Pricing insurance linked securities
• Risk measures, an economic introduction
• Calculating risk measures for catastrophic risks
• Pricing cat bonds : the Winterthur example
• Pricing cat bonds : the Mexican Earthquake

2


Agenda

3


Source : Swiss Re (2007).

4


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)).

Fig. 1 – Major natural catastrophes (Source : Munich Re (2006)).

5


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

Tab. 1 – The 10 most expensive natural catastrophes, 1950-2005 (Source : Munich
Re (2006)).

6


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

Tab. 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)).

7


Some stylized facts : ... mortality risk
“there seems to be broad agreement that there exists
a market price for systematic mortality risk. Howe-
ver, there seems to be no agreement on the structure
and level of this price, and how it should be incorpo-
rated when valuating insurance products or mortality
derivatives” Bauer & Russ (2006).

“securitization of longevity risk is not only a good
method for risk diversifying, but also provides low
beta investment assets to the capital market” Liao,
Yang & Huang (2007).

8


longevity and mortality risks

Yea
r
Age

5e−02
5e−02

2e−02
60 years old
40 years old
20 years old
5e−03

5e−03
2e−03
5e−04

1899
1948
1997

5e−04
5e−05

0 20 40 60 80 100 1900 1920 1940 1960 1980 2000

Age Age

Fig. 2 – Mortality rate surface (function of age and year).

9


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 !

10


Source : Swiss Re (2008).

11


Source : AXA (2008).

12


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

13


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

14


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 −1
V aR(X, u) = F (u) = F (1 − u) i.e. P(X > V aR(X, u)) = 1 − u,

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

15


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
• time series theory for occurrence
• credit risk models for contagion or accumulation

16


Updating actuarial models
In classical actuarial models (from Cramer 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, Cram´r-Lundberg’s theory gives a bound for the ruin
e
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.

17


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

18


Agenda

19


Example : 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.

20


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.

21


Cumulative distribution function, with confidence interval
1.0 lo#!lo# %areto *lot, ,it. /onfiden/e inter3al

0
lo)arit.m of t.e sur5i5al 6ro7a7ilities

!1
0.8
cumulative probabilities

!#
0.6

!$
0.4

!%
0.2

!5
0.0

0 1 2 3 4 5 0 1 # $ % 5

logarithm of the losses lo)arit.m of t.e losses

Fig. 3 – Pareto modeling for business interruption claims.

22


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

Fig. 4 – 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.

23


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

Fig. 5 – The 80-20 Pareto principle.
24


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.

25


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 ?

26


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

27


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

Fig. 6 – Pareto modeling for business interruption claims : tail index.

28


Extreme value distributions...
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

29


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)

Fig. 7 – Pareto modeling for business interruption claims : VaR and TVaR.

30


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

Fig. 8 – Pricing of a reinsurance layer.

31


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

Tab. 3 – Number of business interruption claims.

32


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 u = ∞, π = 8.9 (12 for burning cost... based on 1 claim).

33


Agenda
Catastrophic risks modelling
• Business interruption and very large claims
Risk measures and capital requirements
• Diversiﬁcation and capital allocation

34


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.

35


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.

36


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”.

37


Impact of global warming on natural hazard

!u#$er o) *urricanes, per 2ear 3853!6008
25
Frequency of hurricanes

20
15
10
5
0

1850 1900 1950 2000

Year

Fig. 9 – Number of hurricanes and major hurricanes per year.

38


More natural hazards with higher value at risk
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

Tab. 4 – Most damaging tornadoes (from Brooks & Doswell (2001)).

39



40


Cat models : the meteorological-engineering approach
The basic framework is the following,

• the natural hazard model : generate stochastic climate scenarios, and assess
perils,
• the engineering model : based on the exposure, the values, the building,
calculate damage,
• the insurance model : quantify ﬁnancial losses based on deductibles,
reinsurance (or retrocession) treaties.

41


Source : GIEC (2008).

42



43



44



45



46


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

Tab. 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

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

47


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

48


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.

49


Agenda
• Insurance covers against catastrophes, traditional versus alternative
techniques
Risk measures and capital requirements
• Diversiﬁcation and capital allocation

50


Insurance versus credit, an historical background
The Babylonians developed a system which was re-
corded in the famous Code of Hammurabi (1750 BC)
and practiced by early Mediterranean sailing mer-
chants. If a merchant received a loan to fund his
shipment, he would pay the lender an additional sum
in exchange for the lender’s guarantee to cancel the
loan should the shipment be stolen.

cf. cat bonds.

51


Why a reinsurance market ?
“reinsurance is the transfer of part of the hazards of risks that a direct insurer
assumes by way of reinsurance contracts or legal provision on behalf of an
insured, to a second insurancce carrier, the reinsurer, who has no direct
contractual relationship with the insured” (Swiss Re, introduction to reinsurance)
Reinsurance allwo (primary) insurers to increase the maximum amount they can
insure for a given loss : they can optimize their underwriting capacity without
burdening their need to cover their solvency margin.
The law of large number can be used by insurance companies to assess their
probable annual loss... but under strong assumptions of identical distribution
(hence past event can be used to estimate future one) and independence.

52


Which reinsurance treaty is optimal ?
In a proportional agreement, the cedent and the reinsurer will agree on a
contractually deﬁned ratio to share (identically) the premiums and the losses
In a non-proportional reinsurance treaty, the amount up to which the insurer will
keep (entierely) the loss is deﬁned. The reinsurance company will pay the loss
above the deductible (up to a certain limit).
The Excess-of-Loss (XL) trearty, as the basis for non-proportional reinsurance,
with
• a risk XL : any individual claim can trigger the cover
• an event (or cat) XL : only a loss event involving several individual claims are
covered by the treaty
• a stop-loss, or excess-of-loss ratio : the deductible and the limit og liability are
expressed as annnual aggregate amounts (usually as percentage of annual
premium).

53


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

54


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

55


Source : Guy Carpenter (2008).

56


57


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

58


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

59


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

Fig. 10 – The XL reinsurance treaty mechanism.

60


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

Tab. 7 – Top 25 Global Reinsurance Groups in 2005 (from Swiss Re (2006)).

61


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, nor getting rated by the rating agencies

62


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.

63



64



65



66


Source : Banks (2005).

67


Cat Bonds and securitization
Secutizations in capital markets were intiated with
mortgage-backed securities (MBS)
collaterized mortgage obligations (CMO)
asset-backed securities (ABS)
collaterized loan obligations (CLO)
collaterized bond obligations (CBO)
collaterized debt obligations (CDO)

68



69


Insurance Linked Securities
indemnity trigger
index trigger
parametric trigger

70



71



72



73



74


Mortality bonds


75



76


Source : Goldman Sachs (2006).

77


USAA’s hurricane bond(s) : Residential Re
USAA, mutually owned insurance company (auto, householders, dwelling,
personal libability for US military personal, and family).
Hurricane Andrew (1992) : USD 620 million
Early 1996, work with AIR and Merrill Lynch (and later Goldman Sachs and
Lehman Brothers) to transfer a part of their portfolio
Bond structured to give the insurer cover of the Excess-of-Loss layer above USD
1 billon, to a maximum of USD 500 million, at an 80% rate (i.e. 20% coinsured),
provided by an insurance vehicule Residential Re, established as a Cayman SPR.
The SPR issued notes to investors, in 2 classes of 3 tranches,
class A-1, rated AAA, featuring a USD 77 million tranche of principal
protected notes, and USD 87 million of principal variable notes,
class A-2, rated BB, featuring a USD 313 million of principal variable notes,
Trigger is the single occurrence of a class 3-5 hurricane, with ultimate net loss as
deﬁned under USAA’s portfolio parameters (indemnity trigger)

78


class A-1, rated AAA, hurricane bond


79


class A-2, rated BB, hurricane bond


80


81


Source : Lane (2006).

82



83



84


Agenda

85


survey of literature on pricing
• Fundamental asset pricing theorem, in ﬁnance, Cox & Ross (JFE, 1976),
Harrison & Kreps (JET, 1979), Harrison & Pliska (SPA, 1981, 1983).
Recent general survey
– Dana & Jeanblanc-Picque (1998). March´s ﬁnanciers en temps continu :
´ e
´
valorisation et ´quilibre. Economica.
e
– Duffie (2001). Dynamic Asset Pricing Theory. Princeton University Press.
– Bingham & Kiesel (2004). Risk neutral valuation. Springer Verlag
• Premium calculation, in insurance.
– Buhlmann (1970) Mathematical Methods in Risk Theory. Springer Verlag.
¨
– Goovaerts, de Vylder & Haezendonck (1984). Premium Calculation in
Insurance. Springer Verlag.
– Denuit & Charpentier (2004). Math´matiques de l’assurance non-vie, tome
e
´
1. Economica.

86


survey of literature on pricing
• Price of uncertain quantities, in economics of uncertainty, von Neumann
& Morgenstern (1944), Yaari (E, 1987). Recent general survey
– Quiggin (1993). Generalized expected utility theory : the rank-dependent
model. Kluwer Academic Publishers.
– Gollier (2001). The Economics of Risk and Time. MIT Press.

87


from mass risk to large risks
insurance is “the contribution of the many to the misfortune of the few”.
1. judicially, an insurance contract can be valid only if claim occurrence satisfy
some randomness property,
2. the “game rule” (using the expression from Berliner (Prentice-Hall, 1982),
i.e. legal framework) should remain stable in time,
3. the possible maximum loss should not be huge, with respect to the insurer’s
solvency,
4. the average cost should be identiﬁable and quantiﬁable,
5. risks could be pooled so that the law of large numbers can be used
(independent and identically distributed, i.e. the portfolio should be
homogeneous),
6. there should be no moral hazard, and no adverse selection,
7. there must exist an insurance market, in the sense that demand and supply
should meet, and a price (equilibrium price) should arise.

88


risk premium and regulatory capital (points 4 and 5)
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), i.e. if the Xi ’s are independent,
 
n
√
Xi ∈ nE(X) ± 1.96 nVar(X)  with probability 95%.
 
i=1
risk based capital need

High variance, small portfolio, or nonindependence implies more volatility, and
therefore more capital requirement.

89


independent risks, large portfolio (e.g. car insurance)

independent risks, 10,000 insured

q q

q q

Fig. 11 – A portfolio of n = 10, 000 insured, p = 1/10.

90



independent risks, 10,000 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) )
,

q

0.012
cas indépendant, p=1/10, n=10,000

0.010
RISK−BASED CAPITAL
NEED +7% PREMIUM

0.008
0.006
RUIN
(1% SCENARIO)

0.004
0.002
0.000 969
q

900 950 1000 1050 1100 1150 1200


91



independent risks, 10,000 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) )
,

q

0.012
cas indépendant, p=1/10, n=10,000

0.010
NEED +7% PREMIUM

0.008
0.006
RUIN
(1% SCENARIO)

0.004
0.002
0.000 986
q

900 950 1000 1050 1100 1150 1200


92


independent risks, small portfolio (e.g. ﬁre insurance)

independent risks, 400 insured

q q

q q

Fig. 14 – A portfolio of n = 400 insured, p = 1/10.

93



independent risks, 400 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) )
,

q

0.06
cas indépendant, p=1/10, n=400

0.05
RUIN

0.04
(1% SCENARIO)

0.03

0.02
NEED +35% PREMIUM

0.01
0.00
q 39

30 40 50 60 70


94



independent risks, 400 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) )
,

q

0.06
cas indépendant, p=1/10, n=400

0.05
RUIN

0.04
(1% SCENARIO)

0.03

0.02
NEED +35% PREMIUM

0.01
0.00
q 48

30 40 50 60 70


95


nonindependent risks, large portfolio (e.g. earthquake)

independent risks, 10,000 insured

q q

q q

Fig. 17 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.

96



non−independent risks, 10,000 insured, p=1/10 distribution de la charge totale

q

0.012
nonindependant case, p=1/10, n=10,000

0.010
RUIN
(1% SCENARIO)

0.008
0.006

0.004
NEED +105% PREMIUM

0.002
0.000 897
q

1000 1500 2000 2500


97



non−independent risks, 10,000 insured, p=1/10 distribution de la charge totale

q

0.012
nonindependant case, p=1/10, n=10,000

0.010
RUIN
(1% SCENARIO)

0.008
0.006

0.004
NEED +105% PREMIUM

0.002
0.000 2013
q

1000 1500 2000 2500


98


the pure premium as a technical benchmark
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the “produit scalaire des probabilit´s et des gains”,
e
n n
< p, x >= pi xi = P(X = xi ) · xi = EP (X),
i=1 i=1

based on the “r`gle des parties”.
e
For Qu´telet, the expected value was, in the context of insurance, the price that
e
guarantees a ﬁnancial equilibrium.
From this idea, we consider in insurance the pure premium as EP (X). As in
Cournot (1843), “l’esp´rance math´matique est donc le juste prix des chances”
e e
(or the “fair price” mentioned in Feller (AS, 1953)).
Problem : Saint Peterburg’s paradox, i.e. inﬁnite mean risks (cf. natural
catastrophes)

99


the pure premium as a technical benchmark
∞
For a positive random variable X, recall that EP (X) = P(X > x)dx.
0

Expected value
1.0

q

q
0.8

q

q
Probability level, P

0.6

q

q
0.4

q

q
0.2

q

q
0.0

q

0 2 4 6 8 10

Loss value, X

Fig. 20 – Expected value EP (X) = xdFX (x) = P(X > x)dx.

100


from pure premium to expected utility principle

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
i.e. the entropic risk measure.
See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern
(PUP, 1944), ... etc.

101


Distortion of values versus distortion of probabilities

Expected utility (power utility function)

1.0 q

q
0.8

q

q

0.6

q

q
0.4

q

q
0.2

q

q
0.0

q

0 2 4 6 8 10

Loss value, X

Fig. 21 – Expected utility u(x)dFX (x).

102



Expected utility (power utility function)

1.0 q

q
0.8

q

q

0.6

q

q
0.4

q

q
0.2

q

q
0.0

q

0 2 4 6 8 10

Loss value, X

Fig. 22 – Expected utility u(x)dFX (x).

103


from pure premium to distorted premiums (Wang)

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 ≥ 1 − α) Rg (X) = V aR(X, α),
• if g(x) = min{x/(1 − α), 1} Rg (X) = T V aR(X, α) (also called expected
shortfall), Rg (X) = EP (X|X > V aR(X, α)).
See D’Alembert (1754), Schmeidler (PAMS, 1986, E, 1989), Yaari (E, 1987),
Denneberg (KAP, 1994)... etc.
Remark : Rg (X) will be denoted Eg◦P . But it is not an expected value since
Q = g ◦ P is not a probability measure.

104



Distorted premium beta distortion function)

1.0 q

q
0.8

q

q

0.6

q

q
0.4

q

q
0.2

q

q
0.0

q

0 2 4 6 8 10

Loss value, X

Fig. 23 – Distorted probabilities g(P(X > x))dx.

105



Distorted premium beta distortion function)

1.0 q

q
0.8

q

q

0.6

q

q
0.4

q

q
0.2

q

q
0.0

q

0 2 4 6 8 10

Loss value, X

Fig. 24 – Distorted probabilities g(P(X > x))dx.

106


some particular cases a classical premiums
The exponential premium or entropy measure : obtained when the agent
as an exponential utility function, i.e.
π such that U (ω − π) = EP (U (ω − S)), U (x) = − exp(−αx),
1
i.e. π = log EP (eαX ).
α
Esscher’s transform (see Esscher (SAJ, 1936), B¨hlmann (AB, 1980)),
u
EP (X · eαX )
π = EQ (X) = ,
EP (eαX )
for some α > 0, i.e.
dQ eαX
= αX )
.
dP EP (e
Wang’s premium (see Wang (JRI, 2000)), extending the Sharp ratio concept
∞ ∞
E(X) = F (x)dx and π = Φ(Φ−1 (F (x)) + λ)dx
0 0

107


pricing options in complete markets : the binomial case
In complete and arbitrage free markets, the price of an option is derived using
the portfolio replication principle : two assets with the same payoff (in all
possible state in the world) have necessarily the same price.
Consider a one-period world,

 S = S u( increase, d > 1)
u 0
risk free asset 1 → (1+r), and risky asset S0 → S1 =
 Sd = S0 d( decrease, u < 1)

The price C0 of a contingent asset, at time 0, with payoff either Cu or Cd at time
1 is the same as any asset with the same payoff. Let us consider a replicating
portfolio, i.e. 
 α (1 + r) + βS = C = max {S u − K, 0}
u u 0
 α (1 + r) + βSd = Cd = max {S0 d − K, 0}

108


pricing options in complete markets : the binomial case
The only solution of the system is
Cu − Cd 1 Cu − Cd
β= and α = Cu − S0 u .
S0 u − S0 d 1+r S0 u − S0 d

C0 is the price at time 0 of that portfolio.
1 1+r−d
C0 = α + βS0 = (πCu + (1 − π) Cd ) where π = (∈ [0, 1]).
1+r u−d

C1
Hence C0 = EQ where Q is the probability measure (π, 1 − π), called risk
1+r
neutral probability measure.

109


ﬁnancial versus actuarial pricing, a numerical example
risk-free asset risky asset contingent claim
   
 1.05  110  150 probability 75% 
1→ 100 → ??? →
 1.05  70  10 probability 25% 
3 1
Actuarial pricing : pure premium EP (X) = × 150 + × 10 = 115 (since
4 4
p = 75%).
1
Financial pricing : EQ (X) = 126.19 (since π = 87.5%).
1+r
The payoﬀ can be replicated as follows,

 −223.81 · 1.05 + 3.5 · 110 = 150
and thus −223.81 · 1 + 3.5 · 100 = 126.19.
 −223.81 · 1.05 + 3.5 · 70 = 10

110


ﬁnancial versus actuarial pricing, a numerical example

Comparing binomial risks, from insurance to finance

145

EXPONENTIAL
UTILITY ESSCHER
TRANSFORM
140
135
Prices

130

FINANCIAL PRICE
125

(UNDER RISK NEUTRAL MEASURE)
120

WANG DISTORTED PREMIUM

ACTUARIAL PURE PREMIUM
115

q

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Alpha or lambda coefficients

Fig. 25 – Exponential utility, Esscher transform, Wang’s transform...etc.

111


risk neutral measure or deflators
The idea of deflators is to consider state-space securities

contingent claim 1 contingent claim 2
  
 1  0 probability 75% 
??? → ??? →
 0  1 probability 25% 

Then it is possible to replicate those contingent claims
 
 −1.667 · 1.05 + 0.025 · 110 = 1  2.619 · 1.05 + −0.02 · 110 = 0
 −1.667 · 1.05 + 0.025 · 70 = 0  2.619 · 1.05 + −0.02 · 70 = 1

The market prices of the two assets are then 0.8333 and 0.119. Those prices can
then be used to price any contingent claim.
E.g. the final price should be 150 × 0.8333 + 10 × 0.119 = 126.19.

112


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

113



114



115



116



117



118


119


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.

120

0
2
4
6
8
10
12
14
16

Mosaic 2A

Mosaic 2B

Halyard Re
Yield spread (%)

Domestic Re

Concentric Re

Juno Re

Residential Re

Kelvin 1st event

Kelvin 2nd event

Gold Eagle A

Gold Eagle B

Namazu Re
Empirical
Lane model

Atlas Re A
Wang model

Atlas Re B

Atlas Re C

Seismic Ltd
Property Catastrophe Risk Linked Securities, 2001

121


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.

122


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

Fig. 26 – ROE for P&C US insurance companies.
123


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

110

106.7

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

Fig. 27 – Combined Ratio for P&C US companies versus reinsurance.
124


Source : Morton Lane (2008).

125



126



127



128



129


Agenda

130


Description of th WinCat Cat Bond structure

Source : Cr´dit Suisse (1997).
e

131


Description of th WinCat Cat Bond structure

Source : Cr´dit Suisse (1997).
e

132


Pricing

Source : Schmock (1999).

133


Claim number of past events


134


Modeling the number of event
Consider major storm of hail storm events, causing more than 1000 claims. Let
N denote the number of events per year. Assume that N ∼ P(λ) with parameter
λ > 0. The empirical estimate of λ is λ = 17 (17 events in 10 years).
10


135


Going further on the Poisson model
Several models have been considered in Schmock (1999), such as
• a linear trend of parameter λ


136


• a log-linear trend of parameter λ


137


• a (modiﬁed) log-linear trend of parameter λ


138


• a modiﬁed linear trend of parameter λ


139


Distribution for the knock-out event
Since events can be extreme, it might be natural to consider a Pareto
distribution,  α
 1− θ
, for x ≥ θ,

P(X ≤ x) = x

 0 , for x < θ,
A natural estimator for θ is 1000.
Then classical estimators can be considered for α.
• maximum likelihood,
Since likelihood can be writen
n
αθα
log L(x1 , · · · , xn , α, θ) = log
i=1
xα+1
i
n
= n log α + αn log θ − (α + 1) log xi ,
i=1

140


the maximum (in α) is obtained as
n −1
1 xi
α= log
n i=1 θ

n
Sinece E(α) = α, set
n−1
n −1
n−1 1 xi
α1 = α= log .
n n−1 i=1 θ
Then
α2
E(α1 ) = α and V ar(α1 ) = .
n−2
• method of moments,
For the Pareto distribution
αθ θ2 α
E(X) = and V ar(X) = 2 (α − 2)
, pour α > 2.
α−1 (α − 1)

141


Thus
X
α2 = .
X −θ
Asymptotical properties are then
α(α − 1)2
E(α2 ) → α et V ar(α2 ) → .
n(α − 2)
• OLS regression,
If the logarithm of survival probabilities log[1 − F (x)] are linear in log x, i.e.
log[1 − F (x)] = log F (x) = β0 + β1 log x,
we obtain a Pareto distribution. In that case
Yi = log[1 − F (Xi )] = log F (Xi ) = β0 + β1 log Xi + εi .

The OLS estimator for β = (β0 , β1 ) is then
n n n
−n i=1 log Xi · log F (Xi ) + i=1 log Xi · i=1 log F (Xi )
β1 = −α3 = n n 2
n 2 −
i=1 [log Xi ] [ i=1 log Xi ]
142


Since F is unknown, it should be replaced by an empirical version,
n
1
F (x) = 1(Xi > x)
n i=1

such that F (Xj ) is equal to 1 − rank(Xj )/n.
• empirical quantiles,
For instance, if we expect quantiles of order 10% and 90% to be equal,
 α
 0.10 = 1 − θ
q10%
α
 0.90 = 1 − θ
q90%

where q10% and q10% are the empirical quantiles
α
q90%
Since = 0.9/0.1 = 9, a natural estimator is then
q10%
log 9
α4 = .
log q90% − log q10%

143


• Gini’s index,
In the case of a Pareto distribution
1
1−1/α 1
G=1−2 1 − (1 − u) du =
0 2α − 1

Thus, if γ denotes the empirical version of Gini’s index
1+γ
α5 = .
2γ

144


In that case, p6000 = P(X > 6000) ∼ 6−1.37 ∼ 8.57%. It is also possible to derive
bounds for this probability,

p6000 ∈ [4.5%; 16.2%] with 68% chance.


145


To go further, it is also possible to use a Generalized Pareto Distribution
−α
 1− 1+ x

, for x ≥ 0,
P(X ≤ x) = ασ
 0 , for x < θ,

Maximum likelihood estimator are here α = 1.3806 and σ = 660.7.


146



Thus, p6000 = P(X > 6000) ∼ 7.575%.

147



148


Agenda

149


Source : Cabrera (2006).

150


Natural Catastrophes in Mexico
For Fonden, a ”catastrophe” is a claim larger than annual average catastrophe,
plus the standard deviation,

1996 1997 1998 1999 2000 2001 2002 2003 2004
amount 109 310 330 739 511 120 261 216 32

the average is 292 and standard deviation is 48, thus a catastrophe is obtained
when the loss exceed 512.
The AIR annual expected loss probabilities

Annual expected loss probabilities
00zone 100 00zone 200 00zone 500
0.63% 0.96% 0.30%

151



152



153



154



155


The Mexican Cat Bond
The cat bond was issued by a SPV Cayman Islands Cat-Mex Ltd, structured by
Swiss Reinsurance Company (SRC) and Deutsche Bank.
The 160 million cat bond pays a tranche equal to the LIBOR + 235 basis points.
The cat bond is a part of a total coverage of 450 million provided by the reinsurer
for three years against earthquakes risk, with a total premium of 26 millions.
Payment of losses is conditional upon conﬁrmation by AIR, which modeled the
seismic risk. Only 3 zones (out of 9) are insured in the transaction : zones 1, 2
and 5, with coverage of 150 million (in each case).
The cat bond payment would be triggered if there is an event, i.e. an earthquake
higher (or equal) than 8M w hitting zone 1 or zone 2, or an eathquake higher or
equal than 7.5M w hitting zone 5.

156


Insurance market intensity λ1
Consider an homogeneous Poisson process with intensity λ1 . Under the
non-artbitrage framework, the compounded discount actuarial fair insurance
price at time t = 0, in the reinsurance market is
3 3
H = E 450 · 1 (τ < 3) e−rτ τ = 450 e−rt t fτ (t)dt = 450 e−rt t λ1 e−λ1 t dt
0 0

i.e. the insurance premium is equal to the value of the expected discounted loss
from earthquake.
With constant interest rate, rt = log(1.0541). Thus
3
26 = 450 e− log(1.0541)t λ1 e−λ1 t dt, where 1 − e−λ1 t is the probability of
0
occurence of an event over period [0, t]. Hence, we get an intensity rate from the
reinsurance market λ1 ≈ 0.0214.
The probability of having (at least) one event in three years is 0.0624, i.e. 2.15
events in one hundred years.

157


Capital market intensity λ2
The cat bond pays to the investors the principal P equal to 160 million at
maturity T = 3 years, and gives coupon C every 3 months during the bond’s life
in case of no event. The coupon bonds pays a ﬁxed spread rate z = 235 basis
points over LIBOR.
Hence the annual interest rate is r = 5.1439%, and thus
r+z 5.1439% + 2.35%
C= P = · 160 = 3.1055
4 4

158


Let G be the random variable representing the investor’s gain,
1
P = E τ
G
(1 + r)
12
t 1 1
= E C · 1(τ > ) + P · 1(τ > 3)
t=1
4 (1 + r)t/4 (1 + r)3
12
1 1
= Ce−λ2 t/4 + P e−3λ2
t=1
(1 + r)t/4 (1 + r)3

Substituting the values of the principal P = 160 million and the coupons
12
e−λ2 t/4 e−3λ2
C = 3.1055 million, i.e. 160 = 3.06 t/4
+ 160 3
. From this
t=1
(1.0541) (1.0541)
expression, the capital market intensity can be obtained λ2 ≈ 0.0241. The
probability of having (at least) one event in three years is 0.0699, i.e. 2.4 events
in one hundred years.

159


Historical intensity λ3
Finaly, the historical intensity rate that describes the eathquake process λ3 can
be obtained.
Over 104 years, there were 192 earthquakes higher than 6.5M w, but only half
occured in the insured zones (mainly zone 2).

zone frequency (%)
1 30 16%
2 42 22%
5 18 9%
other 102 53%

The probability of occurence of the trigger event is p = 3/192.
Hence
0.005140
λ= = 1.8504
360
160


Consequently the annual historical intensity is λ3 = λp = 0.0289.

161


References (internet)
Artemis (Alternative Risk Transfer Internet Portal) http ://www.artemis.bm/
Guy Carpenter http ://www.guycarp.com/
Lane Financial LLC http ://www.lanefinancialllc.com/
Munich Re http ://www.munichre.com/
AON Bneﬁeld http ://www.benfieldgroup.com/
Swiss Re http ://www.swissre.com/

162


References
Banks, E. (2004). Alternative Risk Transfer : Integrated Risk Management
through Insurance, Reinsurance, and the Capital Markets. Wiley.
Banks, E. (2005). Catastrophic Risk. Wiley.
Cabrera, B., B. (2006). Pricing catastrophic bonds for earthquakes in Mexico.
Master thesis, Humboldt-Universit¨t zu Berlin.
a
Cardenas, V. & Mechler, R. (2005). The Mexico cat bonds - a solution for
country at risk ? Munich Re Foundation Symposium.
Cox, S.H. & Pedersen, H.W. (2000). Catastrophe risk bonds. North American
Actuarial Journal, 4, 4, 56-..
Kreps, R. (2005). Riskiness leverage models. CAS Forum.
Lane, M. & Mahul, O. (2008). Catastrophe Risk Pricing : an empirical analysis.
World Bank, WPS 4765.
Mata, A.J. (2004) Catastrophe Excess of Loss. in Encyclopedia of Actuarial
Sciences.
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Meyers, G. (2004) Catastrophe Models and Catastrophe Loads. in Encyclopedia of
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Muerlann, A. (2004) Catastrophe Derivatives. in Encyclopedia of Actuarial
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