Risk Modelling in Insurance - Ricam | Welcome · 2008. 9. 8. · Risk Modelling in Insurance Hansj¨org Albrecher Radon Institute, Austrian Academy of Sciences and University of Linz,

Risk Modelling in Insurance

Hansjörg Albrecher

Radon Institute, Austrian Academy of Sciences

and University of Linz, Austria

[email protected]

Tutorial for the Special Semester onStochastics with Emphasis on Finance, RICAM, Linz

September 3, 2008

Typical Questions in Insurance

◮ Which risks can be insured?

◮ Determination of fair premiums

◮ Stability of insurance activity

Program

◮ Some Generalities

◮ Aggregate Claim Distributions

◮ Collective Risk Models

◮ Extensions (Dividends, Taxes, Dependence)

◮ Statistical Issues

Collection of ideas and approaches, not exhaustive treatment!

Insurance Portfolio

Premium Income P(t)

Claim Payments X1 + X2 + · · · + XN(t)

Initial Capital x

Reserve at time t:

R(t) = x + P(t) −

N(t)∑

i=1

Xi

−→ Diversification (in the collective, over time,..)

Quantitative Approach: Risk Theory(Modelling, Measuring Risk, Control)

Risk Y (random variable)

Examples of Measures of Risk

◮ E(Y ), Var (Y )

◮ CoV(Y ) =√

Var YE(Y ) , skewness S(Y ) =

E((Y−E(Y ))3)(Var(Y ))3/2

◮ Value at Risk: VaRα(Y ) = qα = inf{y |FY (y) ≥ α}−→ Capital requirements (Basel II, Solvency II)

(normal distribution)

◮ Coherent Risk Measures (Axioms)◮ Expected Shortfall:

ESα(Y ) = E[Y |Y > qα] =1

1−α

∫ 1

αVaRs ds

(e.g. Swiss Solvency Test)

Crucial: Distribution of Aggregate Claims X (t) =∑N(t)

n=1 Xn

Individual Model∑n

i=1 Xi :

◮ Single payments Xi ≥ 0 with Fi (x) = P(Xi ≤ x) (policy i)

◮ Xi independent, distribution identical or volume-dependent

◮ Convolutions!

P(X1 + X2 + ..+ Xn < x) := F∗n(x) =

∫ x

0

F ∗(n−1)(x − s)dF1(s)

P(X1 + X2 < x) =

∫ x

0

F2(x − s)dF1(s)

Explicit e.g. for

◮ Gamma distribution:

f (x) =αα

µαΓ(α)xα−1 exp(−xα/µ), x > 0

◮ Inverse Gauss distribution:

f (x) =

√

µα

2πx3exp

(

− α/2(√

x/µ−√

µ/x)2)

, x > 0

Aggregate portfolio usually inhomogeneous!What matters: aggregate claim distribution

◮ Collective Model∑N(t)

i=1 Xi ,

N(t).. claim number,Xi iid (d.f. F , “mixed distribution”, reliable estimate)

Modelling N(1): e.g.

◮ Poisson distribution:

P(N(1) = n) = e−λλn/n!, n = 0, 1, ..

◮ Mixed Poisson, e.g. negative binomial:

P(N(1) = n) =

(

α+ n − 1

n

)

pα(1 − p)n, n = 0, 1, ..

Models for Claim Sizes:Heavy Tails: “Few claims determine aggregate claim size”

Candidates: e.g.

◮ (Heavy-Tailed) Weibull:

f (x) = bxb−1 exp(−xb), x > 0, 0 < b < 1.

All Moments exist

◮ Lognormal: log Xi ∼ N(µ, σ2)

f (x) = x−1(2πσ2)−1/2 exp(−(log(x)−µ)2/(2σ2)), x > 0, µ ∈ R, σ2 > 0.

All Moments exist

◮ Pareto:

F (x) = 1 −„

b

x

«α

, x ≥ b > 0, α > 0

resp. Shifted Pareto E (Xβ) < ∞ ⇔ β < α

Subexponential Class F ∈ S : P(X1 + X2 + . . .+ Xn > x) ∼ n(1 − F (x))∀ n ≥ 2 as x → ∞.

Aggregate Claims X (t) =∑N(t)

n=1 Xn, Xi iid, independent of N(t)

F̂ (s) := E(e−sXn) =∫ ∞0 e

−sxdF (x),

Qt(z) := E(zN(t)) =

∑∞n=0 pn(t)z

n with pn(t) = P(N(t) = n)

Gt(x) := P{X (t) ≤ x} =∑∞

n=0 pn(t)F∗n(x)

E [e−sX (t)] =∞

∑

n=0

pn(t)F̂n(s) = Qt(F̂ (s))

→ Moments of X (t): E (X (t)) = E (N)E (Xn),Var(X (t)) =

Var(N(t))E 2(Xn) + E (N(t))Var(Xn), etc.

Approximations for X (t)

◮ Moment matching

◮ Normal Approximation Gt(x) ≈ Φ„

x−E [X (t)]√Var [X (t)]

«

◮ Shifted Gamma, etc.

◮ Edgeworth approximation, orthogonal polynomials

◮ Discretization of claim sizes→ recursive methods (Panjer recursions, FFT etc.)

◮ Asymptotic approximations◮ Subexponential Claims◮ Superexponential Claims

Also useful in modelling credit risk, operational risk etc.

Asymptotic Approximations of Gt(x) - Subexponential Xn

If E (zN(t)) < ∞ for some z > 1:

Gt(x) = P(X1 + . . .+ XN(t) > x) =∞X

n=0

P(N(t) = n)F ∗n(x)

∼∞X

n=0

P(N(t) = n)n F (x) = E(N(t)) F (x)

◮ Simple! Useful for relevant range of x?

F (x) = x−3.3, N ∼Poisson(2),

10 15 20 25

-4.0

-3.5

-3.0

-2.5

-2.0

a3

a2

a1

lower bound

upper bound

10 15 20 25

0.2

0.4

0.6

0.8

1.0

a3

a2

a1

lower bound

upper bound

Higher-order ApproximationsUnder suitable conditions ak (x) = a1(x) +

Pkj=1 Aj f

(j)(x),

Improvement in certain parameter ranges.

Asymptotic Approximations of Gt(x) - Superexponential Xn

Saddlepoint approximations

t κt(α) := log E(eαSt )

Consider the tilted probability measure

Pα(St ∈ dx) = E(eαSt−tκt(α) 1{St∈dx}),

Choice of α = α(x): Eα(St ) = tκ′t (α) = x ,

As x → ∞, α approaches right abscissa of convergence α0 = sup{α : κt (α) < ∞}, given thatlimα→α0 κ

′t (α) = ∞.

Var α(St ) = tκ′′t (α)

Pα

St−xq

t κ′′t (α)< y

!

→ Φ(y)

=⇒ 1 − Gt(x) ∼e−α(x)x+κt (α(x))

α(x)√

2π κ′′t (α(x))as x → ∞.

Asymptotic Approximations of Gt(x) - Superexponential Xn II

1−Gt(x) = |θ| e−|θ|xZ ∞

0

e−|θ|v {Mθ(x + v) − Mθ(x)} dv

−σF < θ < 0

with Mθ(x) :=P∞

n=0 an F∗nθ (x) and an := F̂

n(θ) pn(t)

s

1

If a(x) := a[x] ∈ R as x ↑ ∞:

1 − Gt(x) ∼ e−|θ|xa (x , θ)

Appropriate choice of θ!

Example: Pascal process: pn(t) =

„

α+ n − 1n

«

`

bt+b

´α ` tt+b

´n.

F̂ (θ) = 1 +b

t⇒ 1 − Gt(x) ∼

b

|tF̂ ′(θ)|

!α1

|θ|Γ(α)e−|θ|x

xα−1 .

A “Robust” View: Classical Collective Risk Model

t

reserve R

premiums

ruin

time

t

claims ~ F(y)

x

Rt = x + c t −

N(t)∑

n=1

Xn

N(t). . . homogeneous Poisson process (λ)Xn. . . iid random variables (d.f. F )c . . . premium density

Ruin Probability ψ(x ,T ) = P ( inf0≤t≤T

Rt < 0 |R0 = x)

A “Robust” View: Classical Collective Risk Model

t

reserve R

premiums

ruin

time

claims ~ F(y)

u

t

Rt = u + c t −

N(t)∑

n=1

Xn

Generalizations:

◮ more general point processes

◮ inflation, interest on the surplus, dividend and tax payments

◮ investment in financial market, reinsurance

◮ delay in claim settlement, dependency

Solution Methods

◮ Exact Solutions ((P)IDE)

Infinite time horizon: CP

c∂ψ

∂x− λψ + λ

∫ x

0

ψ(x − y) dF (y) + λ(1 − F (x)) = 0

with limx→∞

ψ(x) = 0.

ψ(x) =

(

1 −λµ

c

) ∞∑

n=1

(

λµ

c

)n

(1 − F n∗I (x)),

with FI (x) =1µ

∫ x

0 (1 − FX (y)) dy , x ≥ 0.

Examples:

◮ Xi ∼ Exp(1/µ): ψ(x) =λµc

e−c−λµ

cµ x

◮ Xi ∼ Phase-Type

Solution Methods


Finite time horizon: CP

c∂ψ

∂x−∂ψ

∂T− λψ + λ

∫ x

0

ψ(x − y ,T ) dF (y) + λ(1 − F (x)) = 0

with limx→∞

ψ(x,T ) = 0 (T ≥ 0) and ψ(x, 0) = 0 (x ≥ 0)

Solution Methods


Finite time horizon with positive interest rate: CP

(c + i x)∂ψ

∂x− ∂ψ∂T

− λψ + λZ x

0

ψ(x − y ,T ) dF (y) + λ(1 − F (x)) = 0

with limx→∞

ψ(x,T ) = 0 (T ≥ 0) and ψ(x, 0) = 0 (x ≥ 0) i ..real interest force

E.g. λ = k i , X ∼ Exp(α): Gamma Series Expansion ψ(x, t) = a0(t) +Pk

n=1 an (t)Γ(x ;α, n)

with Γ(x ;α, n) =αn

Γ(n)

Z

u

0yn−1

e−αy

dy (α > 0, n > 0)

n ∈ N : Γ(x ;α, n) = 1 − e−αx

n−1X

j=0

(αx)j

j!,

∂Γ(x ;α, n)

∂x= α

“

Γ(x ;α, n − 1) − Γ(x ;α, n)”

Recurrence equation

an+1(t) =1

αc

“

(λ + αc − i n)an(t) + a′n(t) + (i (n − 1) − λ) an−1(t)

”

a1(t) =1

αc

`

a′0(t) + λ a0(t)´

, a0(t) = U(0, t)

◮ Inclusion of (non-linear) dividend barriers

��

�� t

t

��

��

time

t

ruin

claims ~ F(y)premiums = ct

b

dividends dividend barrier breserve R

u

Integro-differential equation approach: Markovian property!bt = f (b, t) . . . monotone increasing in t and satisfying

f (b, t) = f(

f (b, t1), t − t1

)

∀ b > 0 and ∀ t > t1 > 0.

⇒ Functional equation!Translation equation: among functions that are mon. incr. in b and t and continuous in b, general solution:

f (b, t) = h(

h−1(b) + t)

,

where h(t) = f (b0, t) is a given initial function.

Example:

bt =(

bm +t

α

)1/m

(α, b > 0,m ≥ 1)

(c + i x)∂φ

∂x+

1

αm bm−1∂φ

∂b− λφ+ λ

Z x

0

φ(x − z , b)dF (z) = 0,

∂φ

∂x

˛

˛

˛

x=b= 0, lim

b→∞φ(x , b) = φ(x).

W (x, b).. expected present value of future dividend payments

(c + δ x)∂W

∂x+

1

αm bm−1∂W

∂b− (δ + λ)W + λ

Z x

0

W (x − z , b)dF (z) = 0,

with ∂W∂x

˛

˛

˛

x=b= 1.

→ Numerical solution

Define integral operator

Ag(x, b) =

t∗Z

0

λe−(λ+δ)t

(c′+x)eδt−c′Z

0

g

(c′

+ x)eδt

− c′− z,

„

bm

+t

α

«1/m!

dF (z)dt

+

∞Z

t∗

λe−(λ+δ)t

“

bm+ tα

”1/m

Z

0

g

„

bm

+t

α

«1/m

− z,

„

bm

+t

α

«1/m!

dF (z)dt

+

∞Z

t∗

λe−λt

tZ

t∗

e−δs

0

B

@(c + δ x)e

δs−

1

mα“

bm + sα

”1−1/m

1

C

Ads dt,

with c′ = cδ

and (c′ + x)eδt∗

− c′ =“

bm + t∗

α

”1/m.

⇒ W (x, b) is a fixed point of A

For g1, g2 ∈ L∞(µ) and ∀ 0 ≤ x ≤ b < ∞

||Ag1(x, b) − Ag2(x, b)|| ≤ ||g1 − g2||

∞Z

0

λe−(λ+δ)t

dt

≤λ

λ + δ||g1 − g2||

contracting operator ⇒ fixed point of A is unique by Banach’s theorem.

Iterate operator −→ high-dimensional integral! Quasi-Monte Carlo methods

Solution Methods (contd.)

◮ Numerical Techniques (PIDE, Laplace-Transform inversion)

◮ Approximations

◮ Diffusion-/Lévy-Approximation

◮ Discretizations

◮ Cramér-Lundberg-Approximation

∃R > 0 with E(eR X ) = 1 + R c/λ⇒ (constant!)

ψ(x) ∼c − λµ

λE(XeRX ) − ce−R x

◮ Heavy-Tail-Approximation: FI (x) ∈ S ⇔ limx→∞1−F∗2

I(x)

1−FI (x)= 2,

ψ(x) =“

1 − λµc

”

P∞n=1

“

λµc

”n(1 − Fn∗I (x))

⇒ ψ(x) ∼λµ

c − λµ(1 − FI (x)) convergence rate!

Solution Methods (contd.)

◮ Inequalities (martingales)(e.g. Lundberg inequality)

◮ Duality with other models

◮ Simulation◮ Rare event sampling◮ Quasi-Monte Carlo techniques

Discounted penalty function(Gerber & Shiu (1998))

mδ(x) := E(

w(R(T−x ), |R(Tx)|) e−δTx 1{Tx

Optimal Control of the Risk Process

◮ Safety Criteria:Minimize ruin probability by dynamic reinsurance and/orinvestmentSchmidli (2001), Hipp & Vogt (2003), Browne (1995), Hipp & Plum (2000), Gaier & Grandits (2004)

◮ Profitability Criteria:Measure insurance portfolio by the value of future profits paidout as dividends to shareholdersde Finetti (1957)

Dividend Strategies

◮ Optimality results: max E(Dx) (Dx ..discounted dividends)

HJB approach

◮ Compound Poisson: Gerber (1969), Azcue & Muler (2005), Schmidli (2008)

Optimal solution for exponential claims:

Horizontal barrier bt ≡ bPaulsen & Gjessing (1997),

Gerber, Lin & Yang (2006)

mδ(x , b) = mδ(x) − E(Dx,b) m′δ(b) ��

��

��

��

��

��

��

��

treserve R

u

b

dividends

ruin time

time

For other claim sizes: Band strategies

Extension: Lévy models, not for renewal models!

Problem: Under horizontal barrier strategy: ψ(u) = 1!

Compound Poisson model:

◮ Ruin constraints◮ Stochastic control problem very difficult!

◮ Consideration of the time value of ruin

V (u) = supL

V (u, L) = supL

E

„Z τ

0

e−βt

dLt +

Z τ

0

e−βtΛ dt

˛

˛

˛R

L0 = u

«

HJB equation

max

Λ + cV ′(u) + λ

Z u

0

V (u − y)dFY (y) − (β + λ)V (u), 1 − V ′(u)ff

= 0.

Optimal solution for exponential claim sizes:

0 ≤ l(u) ≤ M : l∗(u) =

{

0 u < u0,M u ≥ u0.

Threshold/Barrier type

−→ nonlinear equation for u0

Xi ∼Exp(2), λ = 3, β = 0.03, c = 1.75

2 4 6 8 10L

8

10

12

14

16

x0HLL

2 4 6 8 10x

2

4

6

8

10

12

V *HxL

2 4 6 8 10x

100

200

300

400

500

EHΤx L

Λ = 0, 1, 2

Alternative approach: Explicit strategies

◮ E.g. Threshold Strategy

��

��

��

��

��

��

t

claimspremiums

dividendsreserve R

time t

ruin time

dividend threshold b

b

u

~ F(y)= ct

Threshold strategy in the Sparre Andersen model

MGF of Du,b . . .M(u, y , b) = M1(u, y , b)I{u

Wm(u, b) := E(Dmx ,b) :

M(u, y , b) = 1 +∑∞

m=1ym

m!Wm(u, b)

System of IDEs:0

@

nY

j=1

−c ∂∂u

+ λj + δ∆̄

λj

1

A Wm,1(u, b) −

Z

u

0Wm,1(u − z, b) dFY (z) = 0,

0

@

nY

j=1

−(c − a) ∂∂u

+ (λj − a∆) + δ∆̄

λj

1

A Wm,2(u, b) −

Z

u

0Wm(u − z, b) dFY (z) = 0,

with operators ∆Wm := mWm−1, ∆̄Wm := mWm (W0 = 1,W−i = 0 (i ∈ N)).

limb→∞

Wm,1(u, y, b) = 0,

limu→∞

Wm,2(u, y, b) =

„

a

δ

«m

.

By continuity

c∂−

∂u

!j−1

Wm,1

˛

˛

˛

˛

˛

˛

u=b

=

(c − a)∂+

∂u+ a∆

!j−1

Wm,2

˛

˛

˛

˛

˛

u=b

, (j = 1, . . . , n).

c2W

′′1,1(u, b) − 2c(δ + λ)W

′1,1(u, b) + (δ + λ)

2W1,1(u, b) − λ

2α e

−αuZ

u

0W1,1(v, b) e

αvdv = 0

and

(c − a)2W

′′1,2(u, b) − 2(c − a)(δ + λ)W

′1,2(u, b) + (δ + λ)

2W1,2(u, b) − a(2λ + δ)

− λ2α e

−αuZ

u

0W1(v, b) e

αvdv = 0

together with W1,1(b−, b) = W1,2(b+, b) and c∂−W1,1

∂u(u, b)

˛

˛

˛

u=b= (c − a)

∂+W1,2∂u

(u, b) + a˛

˛

˛

u=b.

W1,1(u, b) =P3

i=1 A(i)1 (b)e

R(i)1

uand W1,2(u, b) =

aδ

+ A(1)2 (b) e

R(1)2

u

(δ + λ− (c−a) R)2(R + α) − αλ2 = 0.

0

B

B

B

B

B

B

B

B

B

@

0 1

R(1)1

+α

1

R(2)1

+α

1

R(3)1

+α

− α

R(1)2

+αeR

(1)2

b α

R(1)1

+αeR

(1)1

b α

R(2)1

+αeR

(2)1

b α

R(3)1

+αeR

(3)1

b

−eR

(1)2

beR

(1)1

beR

(2)1

beR

(3)1

b

−(c − a)R(1)2 e

R(1)2

bcR

(1)1 e

R(1)1

bcR

(2)1 e

R(2)1

bcR

(3)1 e

R(3)1

b

1

C

C

C

C

C

C

C

C

C

A

0

B

B

B

B

B

B

@

A(1)2

A(1)1

A(2)1

A(3)1

1

C

C

C

C

C

C

A

=

0

B

B

B

B

@

0

aδaδ

a

1

C

C

C

C

A

.

- Analogous solution for ψ(u)

How do tax payments change the ruin probability?

Definition of “profit” of insurance company?

(equalization reserves, claims reserves (IBNR, RBNS,...)) In practice: Tax privileges were reduced recently!

Model: Tax rate 0 ≤ γ ≤ 1 in “profitable” times:

��

��

��

��

��

��

time t

sruin

tax payments

claims ~ F(y)

premia

W1

M1

M2

reserve Rγ (t)

W2σ2σ1

φγ(s) = 1 − ψγ(s) . . .survival probability with tax rate γ

→ Simple power relation φγ(u) = (φ0(u))1

1−γ

A Simple Proof via a Queueing Approach

Rescale time: R∗t = u + t −∑N∗t

i=1 Xi (with N∗t ...hom. Poisson process (λ/c))

time t

claims ~ F(y)

premia

u

W1

reserve R0(t)

M1

M2

σ2σ1

W2

time t

◮ Link between φ0(u) and Vmax . . .maximum workload during busy period of M/G/1 queue

◮ Net profit condition c > λE(Xi) ⇔ traffic intensity ρ < 1◮ Cut out excursions from running maximum that “survive” (u ↔ t)




time t

claims ~ F(y)

premia

u

W1

reserve R0(t)

M1

M2

W2

time t

◮ Link between φ0(u) and Vmax . . .maximum workload during busy period of M/G/1 queue

◮ Net profit condition c > λE(Xi) ⇔ traffic intensity ρ < 1◮ Cut out excursions from running maximum that “survive” (u ↔ t)




time t

claims ~ F(y)

premia

u

W1

reserve R0(t)

M1

M2

W2

time t

Interpretation: φ0(u) = P(no events during ”time” interval [u,∞) of an inhom.Poisson process with time-dependent rate α(t) = λ

cP(Vmax ≥ t)

φ0(u) = exp“

−Z ∞

u

α(t) dt”

= exp“

− λc

Z ∞

u

P(Vmax > t) dt”

⇒ P(Vmax > u) = cλ

d

dulog φ0(u) ∀ u > 0.

time t

claims ~ F(y)

premia

u

W1

reserve R0(t)

M1

M2

W2

Interpretation: φγ(u) = P(no events during ”time” interval [u,∞) of an inhom.Poisson process with time-dependent rate αγ(t) =

λc (1−γ)

P(Vmax ≥ t)

φγ(u) = exp“

−Z ∞

u

αγ(t) dt”

=

"

exp“

Z ∞

u

α0(t) dt”

# 11−γ

= (φ0(u))1/1−γ

◮ Extension to surplus-dependent tax rate

◮ Extension to spectrally negative Levy process

Challenges

◮ Dependence between and within portfolios

◮ How to detect dependence in data?

◮ How to model dependence?

◮ How to deal with dependence in risk management?

◮ Classical insurance principles are not necessarily valid anymore!

◮ Law of large numbers, Central limit theorem, Diversificationproperties

Exact Results, Asymptotic Results, Stochastic Ordering

Exact solutions for specific dependence structures

Example: Causal dependence Xi ↔ Ti+1

Model:

Xi Xi+1

T i+1

Di . . . threshold variable

◮ If Xi > Di , then Ti+1 ∼ Exp (λ1)

◮ If Xi ≤ Di , then Ti+1 ∼ Exp (λ2)

◮ net profit condition: µX < ch

P(Xi>D)λ1

+ P(Xi≤D)λ2

i

.

◮ limx→∞

φi(x) = 1 − ψi(x) = 1 (i = 1, 2)

Motivation: earthquakes, volcano eruptions

Generalization: A Semi-Markov Model

- {Zn, n ≥ 0} . . . irreducible Markov chain on

{1, . . . ,M},

- P = ((pij ), 1 ≤ i, j ≤ M)

- claim Fj

- intensity λj

t

reserve R

premiums

ruin

time

t

claims ~ F(y)

x

Janssen & Reinhard (1985), Adan & Kulkarni (2003)

P(Wn+1 ≤ x,Xn+1 ≤ y, Zn+1 = j |Zn = i, (Wr ,Xr , Zr ), 0 ≤ r ≤ n)

= P(W1 ≤ x,X1 ≤ y, Z1 = j |Z0 = i) = (1 − e−λi x )pij Bj (y),

Causal dependence model from before is embedded here (M = 2):

◮ If Xi > Di , then Ti+1 ∼ Exp (λ1)

◮ If Xi ≤ Di , then Ti+1 ∼ Exp (λ2)

◮ p11 = p21 = P(X > D), p12 = p22 = P(X < D)

◮ F1 = X |X > D, F2 = X |X < D

net profit conditionPM

i=1 πiµX ,i < cPM

i=1 πiλ−1i ,

(π1, . . . , πM ) . . . stationary distribution of {Zn}, µX,i = E(Xi ).

For Re s ≥ 0 and i = 1, . . . ,M:

m̃i (s) :=

Z ∞

0e−sx

mi (x) dx,

b̃i (s) :=

Z ∞

x=0e−sx

dBi (x),

ω̃i (s) :=

Z

∞

x=0e−sx

Z

∞

xw(x, y − x) dBi (y) dx.

Linear system of equation:

Aδ(s) ~̃mδ(s) = c ~mδ(0) − Λ P ~̃ω(s)

withAδ(s) := (cs − δ) I − Λ + Λ P B̃(s)

I . . .identity matrix,

Λ = diag(λ1, . . . , λM),

B̃(s) = diag(b̃1(s), . . . , b̃M(s)),

~̃mδ(s) = (m̃δ,1(s), . . . , m̃δ,M(s)),

~̃ω(s) = (ω̃1(s), . . . , ω̃M(s)).

Aδ(s) ~̃mδ(s) = c ~mδ(0) − Λ P ~̃ω(s)

det(Aδ(s)) = 0 . . . generalized Lundberg fundamental equation

◮ M zeroes s1, . . . , sM with Re(si ) > 0 for δ > 0

Determine (non-trivial) ~ki with AT (si )~ki = ~0.

0 = ~̃mδ(si )T

ATδ (si )~ki = (c ~mδ(0) − Λ P ~̃ω(si ))T ~ki ,

⇒ M linear equations for mδ,1(0), . . . ,mδ,M(0).

◮ Explicit solution for ~mδ(0)

◮ For any fixed δ, zeroes s1, . . . , sM can be obtained numerically.

◮ For claim size distributions with rational Laplace transform, mδ(x)can then be derived explicitly.

Illustration: D ∼ Exp(2),F ∼ Exp(1), c = 2, λ1 = 3, λ2 = 1.

detA0(s) = 3 − 8s + 4s2 +6s − 31 + s

− 4s3 + s

(s1 = −3.161, s2 = −0.065, s3 = 0, s4 = 1.226)

Ruin probabilities:

~ψ(x) =

(

0.0070.003

)

e−3.161 x +

(

0.9380.867

)

e−0.065 x .

Illustration: D ∼ Exp(2),F ∼ Exp(1), c = 2, λ1 = 3, λ2 = 1.

detA0(s) = 3 − 8s + 4s2 +6s − 31 + s

− 4s3 + s

(s1 = −3.161, s2 = −0.065, s3 = 0, s4 = 1.226)

Density of surplus before ruin:

~f (y1|x) = 1{x≤y1}

„

99

«

e−y1 + e

−3.161x

„

0.1060.045

«

e−2.226y1 +

„

0.061−0.026

«

e−y1

!

+ e−0.065x

„

−0.574−0.531

«

e−2.226y1 +

„

−8.446−7.802

«

e−y1

!

+ 1{x≤y1}e1.226x−2.226y1

„

1.476−0.186

«

+ 1{x≥y1}

„

9.0208.333

«

e−0.0645x−0.935y1 −

„

0.0450.019

«

e−3.161x+2.161y1

!

1 2 3 4 5 6y1

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6y1

0.1

0.2

0.3

0.4

◮ limx→∞

fi (y1|x)/ψi (x)

Asymptotic Results: A general criterion

Ui = Xi − c Ti , Sn =∑n

i=1 Un with E(Ui ) < 0

N

SN

n

Sn Ruin

x

Light-Tail Claims:

limx→∞

1

xlogψ(x) = −R

if ∃κ and R , ε > 0 with

◮ n−1 log EeαSn → κ(α) for |α− R | < ε, Glynn & Whitt (1994)

◮ κ(R) = 0, κ′(R) > 0, Nyrhinen (1998)

◮ EeRSn

Asymptotic Results: A general criterion

Ui = Xi − c Ti , Sn =∑n

i=1 Un with E(Ui ) < 0

N

SN

n

Sn Ruin

x

R

κ(α)

α

Light-Tail Claims:

limx→∞

1

xlogψ(x) = −R

if ∃κ and R , ε > 0 with

◮ n−1 log EeαSn → κ(α) for |α− R | < ε, Glynn & Whitt (1994)

◮ κ(R) = 0, κ′(R) > 0, Nyrhinen (1998)

◮ EeRSn

Some Examples:

1. Random Walk (Sparre Andersen model):

n−1 log EeαSn = n−1 logn

∏

i=1

EeαUi = log EeαUi = κ(α)

(Lundberg equation)

2. Ornstein-Uhlenbeck processes: Ui jointly normal distributed,E(Ui ) = µ < 0, Var(Ui )=1, Cov(Ui ,Uj) = e

−α |i−j|

⇒ R = −2µ1 − e−α

1 + e−α

Alternative Interpretation: Ui yearly increment!

3. Autoregressive AR(1) processes:Ui+1 = a Ui + Zi+1, i ≥ 0, Zi ∼ Z (iid), |a| < 1

⇒ R = RZ · (1 − a)

Extension: ARMA(p,q) processes:

Ui = a1 Ui−1 + . . .+ ap Ui−p + Zi + b1 Zi−1 + . . .+ bq Zi−q,

4. Adaptive Premium Rule: Fix security loading η and consider

c(t) = (1 + η)

PNti=1 Xi

t.

(Asmussen (1999))

Use past claim experience to determine premium rate!

NtX

i=1

Xi/t −

Z

t

0c(s) ds =

NtX

j=1

Xj − (1 + η)

Z

t

0

PNsj=1

Xj

sds =

NtX

j=1

Xj

1 − (1 + η) logt

Ti

!

,

→ Reinterpret as compound Poisson process with time-dependent claims

⇒ κ(α) = λZ 1

0

MX`

α`

1 + (1 + η) log u´´

du − λ = 0

discrete skeleton {Snh}n≥0

Comparison with constant premium rule: Radap ≥ Rconst

5. A shot-noise intensity process A. & Asmussen (2006)

1 2 3 4 5

0.5

1

1.5

2

2.5

Catastrophies, External Events

Stochastic process λt = λ+∑

n∈N h(t − Wn,Yn)

{Wn}n∈N . . . epochs of a hom. Poisson process (ρ), {Yn}n∈N i.i.d. (ind. of PP),

h(t, x) ≥ 0 with h(t, x) = 0 for t < 0.

◮ Specific case: h(t, x) = g(t) x g(t) = e−δt : PDMP

◮ Model is also used for delayed claim settlement itself

Light Tail Asymptotics:

discrete skeleton {Snh}n∈N: κnh(α)/n → h κ(α)

κ(α) = λ(MX (α) − 1) − α c + ρ(

EY (e(MX (α)−1)H(∞,Y )) − 1

)

.

κ(R) = 0, ∃ α > 0 : MX (α) < ∞, E(exp(αH(∞,Y ))) < ∞:

=⇒ limu→∞1u

log P(maxn Snh > u) = −R

maxt

St ≥ maxn

Snh ≥ maxt

St − c h

limu→∞

1

ulogψ(u) = −R .

1 2 3 4 5

0.5

1

1.5

2

2.5

R

κ(α)

α

Strengthening: C1e−Ru ≤ ψ(u) ≤ e−Ru (C1 > 0)

Finite time horizon:

κ(α) convex, α ∈ R+ and αa solution of κ′(α) = 1

a.

limx→∞

1

xlogψ(x , a x) = −Ra

with Ra =

{

αa − a κ(αa), a <1

κ′(R) ,

R , a ≥ 1κ′(R) .

κ(α)

α

R θaRa

κ′ = 1/a

◮ Accuracy

Up to now: Crucial criterion: n−1 log EeαSn → κ(α)

◮ light claim tails

◮ short-range dependence

Generalization: Duffield/O’Connell (1995)If ∃ functions at , vt : R

+ → R+ with at , vt ր ∞ s.t.

limt→∞

1

vtlog E(eαvtSt/at ) := κ(α)

exists and ∃ increasing function h(t) s.t.

g(d) := limt→∞

v(a−1(t/d))h(t)

exists ∀d > 0

(plus some technical conditions)

⇒ limx→∞

1

h(x)logψ(x) = const. = − inf

d>0

[

g(d) supα∈R

(αd − κ(α))]

⇒ light tails, but possibly long-range dependence

Example: Fractional Brownian Motion (Index 0 < H < 1)

Definition: Stochastic Process X : R → R with

◮ Xt continuous, X0 = 0 a.s.

◮ Increment Xt+h − Xt ∼ N(0, h2H ) ∀t ≥ 0, h > 0

Properties:

◮ H=0.5: Brownian Motion

◮ H 6= 0.5: stationary, but NOT independent increments:E(Xs Xs+h) =

12

“

(s + h)2H + s2H − h2H”

◮ H > 0.5: long-range dependence, i.e.P∞

n=1 Cov(X1,Xn+1 − Xn) = ∞◮ Covariance between future and past increments positive for H > 0.5

and negative for H < 0.5

◮ Self-Similarity: Xt ∼ γ−HXγt ∀γ > 0

Sample Path of FBM (H=0.7) Sample Path of FBM (H=0.3)

limx→∞

1

x2−2Hlogψ(x) = − inf

d>0

[

d−2+2H (d + µ)2/2]

= const.

Weibull-type tail (cf. also Michna (1998))

Heavy-tailed claims

Subexponential class: F ∈ S ⇔ limx→∞

1−F∗2(x)1−F (x) = 2,

i.e. P(X1 + . . . + Xn > x) ∼ P(max(X1, . . . ,Xn) > x).

Renewal Model (independence!): FI (x) ∈ S:

ψ(x) ∼µ

cE(T ) − µ(1 − FI (x)),

where FI (x) =1µ

R x0 (1 − FX (y)) dy, x ≥ 0

Note: Only E(T ) matters, not its distribution! (in the geom. bounded case)

Heuristic: ruin is caused by one large claim ⇒

Expect: above formula insensitive to “weak” dependence in the tail!

General Criteria for Insensitivity I

Dependency among interclaim times

Vn = T1 + · · · + Tn (time of n-th claim occurrence)

Xi are i.i.d., E(Ti ) = λ−1

FI ∈ S and for all ε > 0 sufficiently small:

limx→∞

P(supn≥1{n(λ−1 − ε) − Vn} ≥ x)

1 − FI (x)= 0,

ψ(x) ∼µ

cE(T ) − µ(1 − FI (x))

Examples:

◮ Super-position of renewal processes

◮ all T1, . . . ,Tn s.t. P(supn≥1{n(λ−1 − ε) − Vn} ≥ x) is exponentially bounded(cf. LD-criterion!)

e.g. stationary autoregressive, Markov modulation etc. (Asmussen, Schmidli & Schmidt (1999))

General Criteria for Insensitivity II

Regenerative processes:

claim surplus process St s.t. ∃ renewal process with epochs M0 = 0 ≤ M1 < M2 · · · such that

{ST0+t− ST0

}0≤t

References I

H. Albrecher and S. Asmussen.

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References IV

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Risk Modelling in Insurance - Ricam | Welcome · 2008. 9. 8. · Risk Modelling in Insurance Hansj¨org Albrecher Radon Institute, Austrian Academy of Sciences and University of Linz,