Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Pricing insurance linked securities :interplay between finance and insurance
Arthur Charpentier
http ://perso.univ-rennes1.fr/arthur.charpentier/Atelier Finance & Risque
Université de Nantes, Avril 20081

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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

2

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

survey of literature• 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.

• Bentoglio & Betbeze (2005). L’Etat et l’assurance des risques nouveaux.
La Documentation Française.

3

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

AgendaA short introduction to insurance risks• Catastrophe and (very) large risks• Mortality risks, from short term pandemic to long term risk
Insurance linked securities• Insurance linked securities• Catastrophe or mortality bonds
Financial versus insurance pricing• Insurance : from pure premium to other techniques• Finance : from complete to incomplete markets
Pricing Insurance linked• Distorted premium• Indifference utility
4Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

AgendaA short introduction to insurance risks• Catastrophe and (very) large risks• Mortality risks, from short term pandemic to long term risk
Insurance linked securities• Insurance linked securities• Catastrophe or mortality bonds
Financial versus insurance pricing• Insurance : from pure premium to other techniques• Finance : from complete to incomplete markets
Pricing Insurance linked• Distorted premium• Indifference utility
5Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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 behomogeneous),
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.
6

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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 anconfidence interval (solvency requirement), i.e. if the Xi ’s are independent,


n
X
√


Xi ∈ nE(X) ± 1.96 nVar(X)  with probability 95%.
|
{z
}
i=1

risk based capital needHigh variance, small portfolio, or nonindependence implies more volatility, andtherefore more capital requirement.

7

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, large portfolio (e.g. car insurance)
independent risks, 10,000 insured
●

●

●

●

Fig. 1 – A portfolio of n = 10, 000 insured, p = 1/10.
8

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, large portfolio (e.g. car insurance)
distribution de la charge totale, N(np,, np(1 − p) )independent risks, 10,000 insured, p=1/10

●

0.010

RISK−BASED CAPITAL

0.006

0.008

NEED +7% PREMIUM

0.002

0.004

RUIN
(1% SCENARIO)

0.000

cas indépendant, p=1/10, n=10,000

0.012

●

969

900

950

1000

1050

1100

1150

1200

Fig. 2 – A portfolio of n = 10, 000 insured, p = 1/10.
9

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, large portfolio (e.g. car insurance)
distribution de la charge totale, N(np,, np(1 − p) )independent risks, 10,000 insured, p=1/10

●

0.010

RISK−BASED CAPITAL

0.006

0.008

NEED +7% PREMIUM

0.002

0.004

RUIN
(1% SCENARIO)

0.000

cas indépendant, p=1/10, n=10,000

0.012

●

986

900

950

1000

1050

1100

1150

1200

Fig. 3 – A portfolio of n = 10, 000 insured, p = 1/10.
10Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, small portfolio (e.g. fire insurance)
independent risks, 400 insured
●

●

●

●

Fig. 4 – A portfolio of n = 400 insured, p = 1/10.
11

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, small portfolio (e.g. fire insurance)
distribution de la charge totale, N(np,, np(1 − p) )independent risks, 400 insured, p=1/10

●

0.05
0.03

0.04

RUIN
(1% SCENARIO)

0.02

RISK−BASED CAPITAL

0.01

NEED +35% PREMIUM

0.00

cas indépendant, p=1/10, n=400

0.06

●

39
30

40

50

60

70

Fig. 5 – A portfolio of n = 400 insured, p = 1/10.
12

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

independent risks, small portfolio (e.g. fire insurance)
distribution de la charge totale, N(np,, np(1 − p) )independent risks, 400 insured, p=1/10

●

0.05
0.03

0.04

RUIN
(1% SCENARIO)

0.02

RISK−BASED CAPITAL

0.01

NEED +35% PREMIUM

0.00

cas indépendant, p=1/10, n=400

0.06

●

48
30

40

50

60

70

Fig. 6 – A portfolio of n = 400 insured, p = 1/10.
13

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

nonindependent risks, large portfolio (e.g. earthquake)
independent risks, 10,000 insured
●

●

●

●

Fig. 7 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.
14Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

nonindependent risks, large portfolio (e.g. earthquake)
non−independent risks, 10,000 insured, p=1/10

distribution de la charge totale

●

0.010
0.006

0.008

RUIN
(1% SCENARIO)

0.004

RISK−BASED CAPITAL
NEED +105% PREMIUM

0.002
0.000

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

0.012

●

897

1000

1500

2000

2500

Fig. 8 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.
15

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

nonindependent risks, large portfolio (e.g. earthquake)
non−independent risks, 10,000 insured, p=1/10

distribution de la charge totale

●

0.010
0.006

0.008

RUIN
(1% SCENARIO)

0.004

RISK−BASED CAPITAL

0.002

NEED +105% PREMIUM

2013

0.000

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

0.012

●

1000

1500

2000

2500

Fig. 9 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.
16Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

some stylized facts about natural disasters“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)).

Fig. 10 – Major natural catastrophes (from Munich Re (2006).)
17

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Some stylized facts : natural catastrophes
Includes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,drought, floods...
Date

Loss event

RegionOverall losses

Insured losses

Fatalities

25.8.2005

Hurricane Katrina

USA

125,000

61,0001,322

23.8.1992

Hurricane AndrewUSA

26,50017,00062

17.1.1994

Earthquake NorthridgeUSA

44,00015,30061

21.9.2004

Hurricane Ivan

USA, Caribbean

23,00013,000125

19.10.2005

Hurricane Wilma

Mexico, USA

20,00012,40042

20.9.2005

Hurricane Rita

USA

16,00012,00010

11.8.2004

Hurricane Charley

USA, Caribbean

18,0008,000

3626.9.1991

Typhoon MireilleJapan

10,0007,000

629.9.2004

Hurricane Frances

USA, Caribbean

12,0006,000

3926.12.1999

Winter storm Lothar

Europe11,5005,900

110Tab. 1 – The 10 most expensive natural catastrophes, 1950-2005 (from Munich
Re (2006)).

18

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

basics on 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 agiven threshold, bx
F (x) = P(X ≤ x) = 1 −, where x0 = exp(−a/b)x0Then equivalently log(1 − F (x)) ∼ a + b log x, i.e. for all i = 1, ..., n,log(1 − Fbn (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 (see Zajdenweber (JRI, 1998) or from
Charpentier (BFA, 2005).)
19

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

goodness of fit of the Pareto distribution
Hill estimator of the tail index

1.5
1.0
0.5

Tail index, with 95% confidence interval

!&
!"
!(
!#

=>'?@ slo;e> ! &A"'B=>"'?@ slo;e> !!A%$#!')oga-it01 o3 9u1ulate: ;-o<a<ilites

2.0!

!o#!lo# &areto +lot, -urri0ane losses

!

"

#

$

%

)oga-it01 o3 t0e loss a1ount

&!

0

20

40

60

80

100

Percentage of bservations exceeding the threshold

Fig. 11 – Pareto modeling of hurricanes losses (Pielke & Landsea (WF, 1998)).
20

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

longevity and mortality risksYear

60 years old
40 years old
20 years old

2e−03

5e−04

5e−03

5e−03

2e−02

5e−02

5e−02

Age

5e−05

5e−04

1899
1948
1997

0

20

40

60
Age

80

100

1900

1920

1940

1960

1980

2000

Age

Fig. 12 – Mortality rate surface (function of age and year).
21Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

longevity and mortality risks
Main problem= forecasting future mortality rates, i.e.
m(x, t) =# deaths during calendar year t aged x last birthday
average population during calendar year t aged x last birthday

cf. LifeMetrics (JPMorgan),(1)(2) (2)– Lee & Carter (JASA, 1992), log m(x, t) = βx + βx κt ,(1)(2) (2)(3) (3)– Renshaw & Haberman (IME, 2006), log m(x, t) = βx + βx κt + βx γt−x ,
(1)

(3)

(2)

– Currie (2006), log m(x, t) = βx + κt + γt−x ,Lee Carter model, total France (fit701)

0.036
0.034

Beta2 (Lee Carter)

−3.0
−3.5

Beta1 (Lee Carter)

−4.5

0.030

−4.0

0.032

−2.5

10
5
0
−5
−10
−15

Kappa (Lee Carter)

Lee Carter model, total France (fit701)−2.0

Lee Carter model, total France (fit701)

1950

1960

1970
Year (y)

1980

1990

2000

55

60

65

70
Age (x)

75

80

85

55

60

65

70

75

80

85

Age (x)22

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

AgendaA short introduction to insurance risks• Catastrophe and (very) large risks• Mortality risks, from short term pandemic to long term risk
Insurance linked securities• Insurance linked securities• Catastrophe or mortality bonds
Financial versus insurance pricing• Insurance : from pure premium to other techniques• Finance : from complete to incomplete markets
Pricing Insurance linked• Distorted premium• Indifference utility
23

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

insurance linked securities, from insurance to finance
Finn & Lane (1995) “there are no right price of insurance, there is simply thetransacted market price which is high enough to bring forth sellers and low
enough to induce buyers”.
traditional
reinsurance

traditional
reinsurance

indemnisecuritiza

industryloss
securitization

parametr
securitiza

indemnity
securitizationindustloss
securitiza

parametricsecuritization

ilw
derivatives

ilw
derivatives

24Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

(re)insurance principle

8
6

REINSURANCE COMPANY

4PREMIUM (2)
●

2

●

●

0.0

●

●

●

●

0

Insurance losses

10

12

14

Reinsurance Excess−of−Loss Indemnity

●
●

0.2

●
●

●

●
●

●

●

0.4

0.6

●
●

●
●
●

●

0.8

●

INSURANCE COMPANY

●
●●

●
●

1.0

Time (exposure period)

Fig. 13 – (re)insurance excess-of-loss principle.

25

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

(re)insurance principle

8
6

REINSURANCE COMPANY●

4

INDEMNITY (5)

●

●

2

●

●
●
●

●

●●

●

0.2

PREMIUM (2)

●

●

0

Insurance losses

10

12

14

Reinsurance Excess−of−Loss Indemnity

●
●

0.4

●
● ●

0.6

INSURANCE COMPANY●● ● ●

●

●

●

0.8

Time (exposure period)

Fig. 14 – (re)insurance excess-of-loss principle.

26

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

catastrophe or mortality bonds
14

Parametric cat bond8

CAT BONDNOMINAL (100)6SPECIAL PURPOSE VEHICLE

4

PREMIUM (2)
●

2

●

●

0.0

●

●

●

●

0

Insurance losses

10

12

FINANCIAL INVESTORS

●
●

0.2

●
●

●

●
●

●

●

0.4

0.6

●
●

●
●
●

●

0.8

●

INSURANCE COMPANY

●
●●

●
●

1.0

Time (exposure period)

Fig. 15 – catastrophe or mortality bonds principle.

27

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

catastrophe or mortality bonds
14

Parametric cat bond

NOMINAL + RISKY
COUPON (100+5+2)

8

CAT BONDNOMINAL (100)6SPECIAL PURPOSE VEHICLE

4

PREMIUM (2)
●

2

●

●

0.0

●

●

●

●

0

Insurance losses

10

12

FINANCIAL INVESTORS

●
●

0.2

●
●

●

●
●

●

●

0.4

0.6

●
●

●
●
●

●

0.8

●

INSURANCE COMPANY

●
●●

●
●

1.0

Time (exposure period)

Fig. 16 – catastrophe or mortality bonds principle.

28

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

catastrophe or mortality bonds
14

Parametric cat bond

NOMINAL + RISKY
COUPON (100+5−9)

6

8

CAT BONDNOMINAL (100)SPECIAL PURPOSE VEHICLE●

4

INDEMNITY (9)

●

●

2

●
●
●

●

●●

●

0.2

PREMIUM (2)

●

●
●

0

Insurance losses

10

12

FINANCIAL INVESTORS

●
●

0.4

●
● ●

0.6

INSURANCE COMPANY●● ● ●

●

●

●

0.8

Time (exposure period)

Fig. 17 – catastrophe or mortality bonds principle.

29

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

comparing (re)insurance and financial markets
From AIG (1997),●●

US non−life insurance
and reinsurance capital$200 billion

●

●

●
●
World governementbondmarket, $9,000 billion

Hurricane Andrew$20 billion●●

Hurricane in Florida$100 billion (potential)●●

World Equity market, $15,000 billion●=⇒ capital markets have significantly greater capacity then (re)insurance markets
30Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

insurance linked securities, from insurance to financeMortality/mortality risk :• deviation of the trend (e.g. consequence of obesity)• one off deviation (e.g. avian flu, Spanish influenza)
Natural catastrophe risk :• wind related (e.g. hurricanes in Florida or winter storms in Europe)• flood related (e.g. Paris)• soil related (e.g. earthquakes in California or Japan, or volcanic eruptions)Man-made disasters :• terrorism (e.g. 9/11)• institutional investors : pension funds, (re)insurance ?
• financial markets ?
• governments ?
31

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

AgendaA short introduction to insurance risks• Catastrophe and (very) large risks• Mortality risks, from short term pandemic to long term risk
Insurance linked securities• Insurance linked securities• Catastrophe or mortality bonds
Financial versus insurance pricing• Insurance : from pure premium to other techniques• Finance : from complete to incomplete markets
Pricing Insurance linked• Distorted premium• Indifference utility
32Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

the pure premium as a technical benchmark
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the “produit scalaire des probabilités et des gains”,< p, x >=nXpi xi =i=1n
X

P(X = xi ) · xi = EP (X),i=1

based on the “règle des parties”.
For Quételet, the expected value was, in the context of insurance, the price thatguarantees a financial equilibrium.
From this idea, we consider in insurance the pure premium as EP (X). As in
Cournot (1843), “l’espérance mathématique est donc le juste prix des chances”(or the “fair price” mentioned in Feller (AS, 1953)).
Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. natural
catastrophes)33

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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

1.0

Expected value
●

0.8

●

●

0.6

●

●

0.4

Probability level, P

●

●

0.2

●

●

0.0

●

●

0

2

4

6

8

10

Loss value, X

Fig. 18 – Expected value EP (X) =R

xdFX (x) =R

P(X > x)dx.
34

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

from pure premium to expected utility principleZ
Ru (X) =Zu(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,
1αX
Ru (X) = log EP (e ) ,αi.e. the entropic risk measure.
See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern(PUP, 1994), Rochet (E, 1994)... etc.

35Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Distortion of values versus distortion of probabilities

1.0

Expected utility (power utility function)
●

0.8

●

●

0.6

●

0.4

●

●

0.2

●

●

●

0.0

Probability level, P

●

●

0

2

4

6

8

10

Loss value, X

Fig. 19 – Expected utilityRu(x)dFX (x).

36Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Distortion of values versus distortion of probabilities

1.0

Expected utility (power utility function)
●

0.8

●

●

0.6

●

0.4

●

●

0.2

●

●

●

0.0

Probability level, P

●

●

0

2

4

6

8

10

Loss value, X

Fig. 20 – Expected utilityRu(x)dFX (x).

37Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

from pure premium to distorted premiums (Wang)Z
Rg (X) =Z
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 expectedshortfall), 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.

38

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Distortion of values versus distortion of probabilities

1.0

Distorted premium beta distortion function)
●

0.8

●

●

0.6

●

0.4

●

●

0.2

●

●

●

0.0

Probability level, P

●

●

0

2

4

6

8

10

Loss value, X

Fig. 21 – Distorted probabilities

R

g(P(X > x))dx.

39

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Distortion of values versus distortion of probabilities

1.0

Distorted premium beta distortion function)
●

0.8

●

●

0.6

●

0.4

●

●

0.2

●

●

●

0.0

Probability level, P

●

●

0

2

4

6

8

10

Loss value, X

Fig. 22 – Distorted probabilities

R

g(P(X > x))dx.

40Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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),1log EP (eαX ).
αEsscher’s transform (see Esscher (SAJ, 1936), Bühlmann (AB, 1980)),

i.e. π =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
Z ∞Z ∞E(X) =
F (x)dx and π =Φ(Φ−1 (F (x)) + λ)dx
0

0

41

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

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)u0risk 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}uu0 α (1 + r) + βSd = Cd = max {S0 d − K, 0}

42Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

pricing options in complete markets : the binomial case
The only solution of the system isβ=1
Cu − Cdand α =
S0 u − S0 d
1+r




Cu − Cd
Cu − S0 u.
S0 u − S0 d

C0 is the price at time 0 of that portfolio.
11+r−d(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+rneutral probability measure.
C0 = α + βS0 =43

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

financial versus actuarial pricing, a numerical example
risk-free asset
 1.051→
 1.05risky asset

 110
100 →
 70

contingent claim

 150
??? →
 10

Actuarial pricing : pure premium EP (X) =
probability 75% probability 25% 

1
3
× 150 + × 10 = 115 (since44

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 = 1044Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

financial versus actuarial pricing, a numerical example

145

Comparing binomial risks, from insurance to finance

ESSCHER
TRANSFORM

130

FINANCIAL PRICE(UNDER RISK NEUTRAL MEASURE)

115

120

125

Prices135

140

EXPONENTIALUTILITY

WANG DISTORTED PREMIUMACTUARIAL PURE PREMIUM

●

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Alpha or lambda coefficients

Fig. 23 – Exponential utility, Esscher transform, Wang’s transform...etc.
45Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

risk neutral measure or deflators
The idea of deflators is to consider state-space securities
contingent claim 1

 1
??? →
 0

contingent claim 2

 0
??? →
 1


probability 75% 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 canthen be used to price any contingent claim.
E.g. the final price should be 150 × 0.8333 + 10 × 0.119 = 126.19.

46

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

pricing options in incomplete markets
The valuation of option is based on the idea of replicating portfolios. What if wecannot buy the underlying asset.
risk-free asset
 1.051→
 1.05risky asset

 110
100 →
 70

contingent claim

 150
??? →
 10

Actuarial pricing : pure premium EP (X) =
probability 75% probability 25% 

1
3
× 150 + × 10 = 115.
4
4

Financial pricing : ...
The payoff cannot be replicated =⇒ incomplete market
Remark : this model can be extended in continuous time, with continuous
prices (driven by Browian diffusion).

47

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

pricing options in incomplete markets• the market is not complete, and catastrophe (or mortality risk) cannot be
replicated,• the guarantees are not actively traded, and thus, it is difficult to assume
no-arbitrage• underlying diffusions are not driven by a geometric Brownian motion process48Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

350

50

300

45
40

250

35
30

Munich Re stock price55

60

Impact of WTC 9/11 on stock prices (Munich Re and SCOR)

2001

2002

Fig. 24 – Catastrophe event and stock prices (Munich Re and SCOR).
49

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

AgendaA short introduction to insurance risks• Catastrophe and (very) large risks• Mortality risks, from short term pandemic to long term risk
Insurance linked securities• Insurance linked securities• Catastrophe or mortality bonds
Financial versus insurance pricing• Insurance : from pure premium to other techniques• Finance : from complete to incomplete markets
Pricing Insurance linked• Distorted premium• Indifference utility
50Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

pricing insurance linked securities using distorted premium
Lane (AB, 2000) proposed to fit an econometric model on yield spreads (ER,excess return), as a function of the CEL (conditional expected loss) and the PFL(probability of first loss), as a Cobb-Douglas function,
ER = 0.55 · P F L0.495 · CEL0.574 .
Wang (GPRI, 2004) Assume that X has a normal distribution (where µ and σ
are functions of CEL and PFL), and with λ = 0.45...

51

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

Property Catastrophe Risk Linked Securities, 2001
16

Yield spread (%)

Lane model
Wang model
Empirical

14

12

10

8

6

4

Seismic Ltd

Atlas Re C

Atlas Re B

Atlas Re A

Namazu Re

Gold Eagle B

Gold Eagle A

Kelvin 2nd event

Kelvin 1st event

Residential Re

Juno Re

Concentric Re

Domestic Re

Halyard Re

Mosaic 2B

0

Mosaic 2A2

Fig. 25 – Cat bonds yield spreads, empirical versus models.
52

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

using expected utility principle to price securities
Consider– a risk-free bond,– a risky traded asset, with price Xt ,– a non traded insurance index, with value St
We want to price the option with payoff payoff (ST − K)+ .
In complete market (the index is traded), we price a replicating portfolio.
In incomplete market, we create a portfolio δ = (α, β) of tradable assets whichshould not be too far from the true payoff. See the super-replication price or risk
minimization principle idea.
If X is a random payoff, the classical Expected Utility based premium is obtainby solvingu(ω, X) = U (ω − π) = EP (U (ω − X)).

53

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

using expected utility principle to price securities
Consider an investor selling an option with payoff Xh at time T ,i– either he keeps the option, uδ? (ω, 0) = supδ∈A EP U (ω + (δ · S)T ) ,hi– either he sells the option, uδ? (ω + π, X) = supδ∈A EP U (ω + (δ · S)T − X) .
The price obtained by indifference utility is the minimum price such that the twoquantities are equal, i.e.
π(ω, X) = inf {π ∈ R such that uδ? (ω + π, X) − uδ? (ω, 0) > 0} .
This price is the minimal amount such that it becomes interesting for the seller
to sell the option : under this threshold, the seller has a higher utility keeping theoption, and not selling it.

54

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

conclusion• classical finance, π = EP [(S − d)+ ]• classical insurance, π = e−rT EQ [(ST − K)+ ]• large risks, expected utility approach U (ω − π) = EP (U (ω − [S − d]+ ))• large risks, dual (Yaari) approach ω − π = Eg◦P (ω − S)
• incomplete market, dual (Yaari) approach, i.e. Esscher• incomplete market, expected utility approach, i.e. indifference utility=⇒ a general framework for pricing of both insurance and financial risks.
E.g. capital requirements of Basle II and Solvency II.

55

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.

perspectives• diversification issues
Consider two risks S1 and S2 , π1+2 = EP [S1 + S2 ] = EP [S1 ] + EP [S2 ] = π1 + π2?

What about other premiums ? E.g. Eg◦P [S1 + S2 ] = Eg◦P [S1 ] + Eg◦P [S2 ]For some non subadditive risk measure (e.g. VaR) π1+2 >π1 + π2 .
• econometric and heterogeneity issues
In the case of pure premium, EP (S) = EP

N
X
i=1

!
Yi

= EP (N ) × EP (Y )| {z }| {z }
frequency

average cost

With heterogeneity πi = EP (S|X = xi ) = EP (N |X = xi ) × EP (Y |X = xi )What about other premiums ?

56