slides nantes

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

Universite de Nantes, Avril 2008

1


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-Picque (1998). Marches financiers en temps continu :valorisation et equilibre. Economica.

– 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). Mathematiques de l’assurance non-vie, tome

1. Economica.

2


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 Francaise.

3


Agenda

A 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


Agenda









5


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 satisfysome 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’ssolvency,

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 supplyshould meet, and a price (equilibrium price) should arise.

6


risk premium and regulatory capital (points 4 and 5)

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), i.e. if the Xi’s are independent,

n∑i=1

Xi ∈

nE(X)± 1.96√nVar(X)︸︷︷︸

risk based capital need

with probability 95%.

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

7


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



●

●

independent risks, 10,000 insured, p=1/10

900 950 1000 1050 1100 1150 1200

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

cas

ind

ép

en

da

nt,

p=

1/1

0,

n=

10

,00

0

distribution de la charge totale, N((np,, np((1 −− p))))

RUIN(1% SCENARIO)

RISK−BASED CAPITALNEED +7% PREMIUM

969


9



●

●

independent risks, 10,000 insured, p=1/10

900 950 1000 1050 1100 1150 1200

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

cas

ind

ép

en

da

nt,

p=

1/1

0,

n=

10

,00

0


RUIN(1% SCENARIO)


986


10


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

●

●

independent risks, 400 insured

●

●

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

11



●

●

independent risks, 400 insured, p=1/10

30 40 50 60 70

0.0

00

.01

0.0

20

.03

0.0

40

.05

0.0

6

cas

ind

ép

en

da

nt,

p=

1/1

0,

n=

40

0


RUIN(1% SCENARIO)


39


12



●

●

independent risks, 400 insured, p=1/10

30 40 50 60 70

0.0

00

.01

0.0

20

.03

0.0

40

.05

0.0

6

cas

ind

ép

en

da

nt,

p=

1/1

0,

n=

40

0


RUIN(1% SCENARIO)


48


13


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



●

●

non−independent risks, 10,000 insured, p=1/10

1000 1500 2000 2500

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

distribution de la charge totale

no

nin

de

pe

nd

an

t ca

se,

p=

1/1

0,

n=

10

,00

0

RUIN(1% SCENARIO)


897


15



●

●

non−independent risks, 10,000 insured, p=1/10

1000 1500 2000 2500

0.0

00

0.0

02

0.0

04

0.0

06

0.0

08

0.0

10

0.0

12

distribution de la charge totale

no

nin

de

pe

nd

an

t ca

se,

p=

1/1

0,

n=

10

,00

0

RUIN(1% SCENARIO)


2013


16


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


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 (from Munich

Re (2006)).

18


basics on 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 · logXi.

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 fromCharpentier (BFA, 2005).)

19


goodness of fit of the Pareto distribution

! " # $ % &!

!'

!#

!(

!"

!&

!

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

)oga-it01 o3 t0e loss a1ount

)oga

-it01

o3 9

u1ul

ate:

;-o

<a<i

lites

=>'?@ slo;e> ! &A"'B=>"'?@ slo;e> !!A%$#

0 20 40 60 80 1000.

51.

01.

52.

0

Hill estimator of the tail index

Percentage of bservations exceeding the threshold

Tai

l ind

ex, w

ith 9

5% c

onfid

ence

inte

rval

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

20


longevity and mortality risks

Year

Age

0 20 40 60 80 100

5e−0

55e

−04

5e−0

35e

−02

Age

189919481997

1900 1920 1940 1960 1980 20005e

−04

2e−0

35e

−03

2e−0

25e

−02

Age

60 years old40 years old20 years old

Fig. 12 – Mortality rate surface (function of age and year).

21


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),– Lee & Carter (JASA, 1992), logm(x, t) = β

(1)x + β

(2)x κ

(2)t ,

– Renshaw & Haberman (IME, 2006), logm(x, t) = β(1)x + β

(2)x κ

(2)t + β

(3)x γ

(3)t−x,

– Currie (2006), logm(x, t) = β(1)x + κ

(2)t + γ

(3)t−x,

1950 1960 1970 1980 1990 2000

−15

−10

−50

510

Lee Carter model, total France (fit701)

Year (y)

Kapp

a (L

ee C

arte

r)

55 60 65 70 75 80 85

−4.5

−4.0

−3.5

−3.0

−2.5

−2.0


Age (x)

Beta

1 (L

ee C

arte

r)

55 60 65 70 75 80 85

0.03

00.

032

0.03

40.

036


Age (x)

Beta

2 (L

ee C

arte

r)

22


Agenda









23


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 lowenough to induce buyers”.

traditional indemni industry parametr ilwreinsurance securitiza loss securitiza derivatives

securitization

traditional indemnity indust parametric ilwreinsurance securitization loss securitization derivatives

securitiza

24


(re)insurance principle

● ●

●

●

●● ●

●

●

●●

●

●●●

●●

●

● ●

●●

●

● ● ●

0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

1214

Reinsurance Excess−of−Loss Indemnity

Time (exposure period)

Insu

ranc

e lo

sses

●

●

INSURANCE COMPANY

REINSURANCE COMPANY

PREMIUM (2)

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

25


(re)insurance principle

●

●

●● ●

●

●

●

●

●

●

●

●

●

●●

●● ●

● ●● ●

0.2 0.4 0.6 0.8

02

46

810

1214

Reinsurance Excess−of−Loss Indemnity


Insu

ranc

e lo

sses

●

●

INSURANCE COMPANY

REINSURANCE COMPANY

PREMIUM (2)INDEMNITY (5)

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

26


catastrophe or mortality bonds

● ●

●

●

●● ●

●

●

●●

●

●●●

●●

●

● ●

●●

●

● ● ●

0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

1214

Parametric cat bond


Insu

ranc

e lo

sses

●

●

INSURANCE COMPANY

SPECIAL PURPOSE VEHICLE

PREMIUM (2)

FINANCIAL INVESTORS

CAT BONDNOMINAL (100)

Fig. 15 – catastrophe or mortality bonds principle.

27



● ●

●

●

●● ●

●

●

●●

●

●●●

●●

●

● ●

●●

●

● ● ●

0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

1214

Parametric cat bond


Insu

ranc

e lo

sses

●

●

INSURANCE COMPANY


PREMIUM (2)

FINANCIAL INVESTORS


NOMINAL + RISKYCOUPON (100+5+2)


28



●

●

●● ●

●

●

●

●

●

●

●

●

●

●●

●● ●

● ●● ●

0.2 0.4 0.6 0.8

02

46

810

1214

Parametric cat bond


Insu

ranc

e lo

sses

●

●

INSURANCE COMPANY


PREMIUM (2)

FINANCIAL INVESTORS


NOMINAL + RISKYCOUPON (100+5−9)

INDEMNITY (9)


29


comparing (re)insurance and financial markets

From AIG (1997),

●

●

●

●

●

●

●

●

●

●

●

World Equity market, $15,000 billion

World governement bondmarket, $9,000 billion

US non−life insuranceand reinsurance capital

$200 billion

Hurricane Andrew$20 billion

Hurricane in Florida$100 billion (potential)

=⇒ capital markets have significantly greater capacity then (re)insurance markets

30


insurance linked securities, from insurance to finance

Mortality/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


Agenda









32


the pure premium as a technical benchmark

Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth centuryproposed to evaluate the “produit scalaire des probabilites et des gains”,

< p,x >=n∑i=1

pixi =n∑i=1

P(X = xi) · xi = EP(X),

based on the “regle des parties”.

For Quetelet, 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 inCournot (1843), “l’esperance mathematique 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. naturalcatastrophes)

33


the pure premium as a technical benchmark

For a positive random variable X, recall that EP(X) =∫ ∞

0

P(X > x)dx.

●

●

●

●

●

●

●

●

●

●

●

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Expected value

Loss value, X

Pro

babi

lity

leve

l, P

Fig. 18 – Expected value EP(X) =∫xdFX(x) =

∫P(X > x)dx.

34


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,

Ru(X) =1α

log(EP(eαX)

),

i.e. the entropic risk measure.

See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern

(PUP, 1994), Rochet (E, 1994)... etc.

35


Distortion of values versus distortion of probabilities

●

●

●

●

●

●

●

●

●

●

●

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Expected utility (power utility function)

Loss value, X

Pro

babi

lity

leve

l, P

Fig. 19 – Expected utility∫u(x)dFX(x).

36



●

●

●

●

●

●

●

●

●

●

●

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Expected utility (power utility function)

Loss value, X

Pro

babi

lity

leve

l, P

Fig. 20 – Expected utility∫u(x)dFX(x).

37


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) = TV 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 sinceQ = g ◦ P is not a probability measure.

38



●

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●

●

●

●

●

●

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Distorted premium beta distortion function)

Loss value, X

Pro

babi

lity

leve

l, P

Fig. 21 – Distorted probabilities∫g(P(X > x))dx.

39



●

●

●

●

●

●

●

●

●

●

●

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Distorted premium beta distortion function)

Loss value, X

Pro

babi

lity

leve

l, P

Fig. 22 – Distorted probabilities∫g(P(X > x))dx.

40


some particular cases a classical premiums

The exponential premium or entropy measure : obtained when the agentas an exponential utility function, i.e.

π such that U(ω − π) = EP(U(ω − S)), U(x) = − exp(−αx),

i.e. π =1α

log EP(eαX).

Esscher’s transform (see Esscher (SAJ, 1936), Buhlmann (AB, 1980)),

π = EQ(X) =EP(X · eαX)

EP(eαX),

for some α > 0, i.e.dQdP

=eαX

EP(eαX).

Wang’s premium (see Wang (JRI, 2000)), extending the Sharp ratio concept

E(X) =∫ ∞

0

F (x)dx and π =∫ ∞

0

Φ(Φ−1(F (x)) + λ)dx

41


pricing options in complete markets : the binomial case

In complete and arbitrage free markets, the price of an option is derived usingthe portfolio replication principle : two assets with the same payoff (in allpossible state in the world) have necessarily the same price.

Consider a one-period world,

risk free asset 1→ (1+r), and risky asset S0 → S1 =

Su = S0u( increase, d > 1)

Sd = S0d( decrease, u < 1)

The price C0 of a contingent asset, at time 0, with payoff either Cu or Cd at time1 is the same as any asset with the same payoff. Let us consider a replicatingportfolio, i.e. α (1 + r) + βSu = Cu = max {S0u−K, 0}

α (1 + r) + βSd = Cd = max {S0d−K, 0}

42


pricing options in complete markets : the binomial case

The only solution of the system is

β =Cu − CdS0u− S0d

and α =1

1 + r

(Cu − S0u

Cu − CdS0u− S0d

).

C0 is the price at time 0 of that portfolio.

C0 = α+ βS0 =1

1 + r(πCu + (1− π)Cd) where π =

1 + r − du− d

(∈ [0, 1]).

Hence C0 = EQ

(C1

1 + r

)where Q is the probability measure (π, 1− π), called risk

neutral probability measure.

43


financial versus actuarial pricing, a numerical example

risk-free asset risky asset contingent claim

1→

1.05

1.05100→

110

70???→

150

10

probability 75%

probability 25%

Actuarial pricing : pure premium EP(X) =

34× 150 +

14× 10 = 115 (since

p = 75%).

Financial pricing :1

1 + rEQ(X) = 126.19 (since π = 87.5%).

The payoff can be replicated as follows, −223.81 · 1.05 + 3.5 · 110 = 150

−223.81 · 1.05 + 3.5 · 70 = 10and thus −223.81 · 1 + 3.5 · 100 = 126.19.

44


financial versus actuarial pricing, a numerical example

0.00 0.01 0.02 0.03 0.04 0.05 0.06

115

120

125

130

135

140

145

Comparing binomial risks, from insurance to finance

Alpha or lambda coefficients

Pric

es

ACTUARIAL PURE PREMIUM

FINANCIAL PRICE(UNDER RISK NEUTRAL MEASURE)

WANG DISTORTED PREMIUM

ESSCHERTRANSFORM

EXPONENTIALUTILITY

●

Fig. 23 – Exponential utility, Esscher transform, Wang’s transform...etc.

45


risk neutral measure or deflators

The idea of deflators is to consider state-space securities

contingent claim 1 contingent claim 2

???→

1

0???→

0

1

probability 75%

probability 25%

Then it is possible to replicate those contingent claims −1.667 · 1.05 + 0.025 · 110 = 1

−1.667 · 1.05 + 0.025 · 70 = 0

2.619 · 1.05 +−0.02 · 110 = 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


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 risky asset contingent claim

1→

1.05

1.05100→

110

70???→

150

10

probability 75%

probability 25%

Actuarial pricing : pure premium EP(X) =

34× 150 +

14× 10 = 115.

Financial pricing : ...

The payoff cannot be replicated =⇒ incomplete market

Remark : this model can be extended in continuous time, with continuousprices (driven by Browian diffusion).

47


pricing options in incomplete markets

• the market is not complete, and catastrophe (or mortality risk) cannot bereplicated,

• the guarantees are not actively traded, and thus, it is difficult to assumeno-arbitrage

• underlying diffusions are not driven by a geometric Brownian motion process

48


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

2001 2002

3035

4045

5055

60

250

300

350

Mun

ich

Re

stoc

k pr

ice

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

49


Agenda









50


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 · PFL0.495 · 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


Property Catastrophe Risk Linked Securities, 2001

0

2

4

6

8

10

12

14

16 Yield spread (%)

Mosai

c 2A

Mosai

c 2B

Halya

rd Re

Dome

stic Re

Conce

ntric R

e

Juno R

e

Resid

ential

Re

Kelvin

1st ev

ent

Kelvin

2nd e

vent

Gold E

agle A

Gold E

agle B

Nama

zu Re

Atlas R

e A

Atlas R

e B

Atlas R

e C

Seism

ic Ltd

Lane model

Wang model

Empirical

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

52


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 StWe 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 riskminimization principle idea.

If X is a random payoff, the classical Expected Utility based premium is obtainby solving

u(ω,X) = U(ω − π) = EP(U(ω −X)).

53


using expected utility principle to price securities

Consider an investor selling an option with payoff X at time T ,– either he keeps the option, uδ?(ω, 0) = supδ∈AEP

[U(ω + (δ · S)T )

],

– 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 sellerto sell the option : under this threshold, the seller has a higher utility keeping theoption, and not selling it.

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conclusion

• classical finance, π = EP[(S − d)+]• classical insurance, π = e−rTEQ[(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


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∑i=1

Yi

)= EP(N)︸︷︷︸

frequency

× EP(Y )︸︷︷︸average cost

With heterogeneity πi = EP(S|X = xi) = EP(N |X = xi)× EP(Y |X = xi)

What about other premiums ?

56

slides nantes

Economy & Finance