Risk Modelling in Insurance Hansj org Albrecher Radon Institute, - - PowerPoint PPT Presentation

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Risk Modelling in Insurance

Hansj¨

• rg Albrecher

Radon Institute, Austrian Academy of Sciences and University of Linz, Austria

hansjoerg.albrecher@oeaw.ac.at Tutorial for the Special Semester on Stochastics with Emphasis on Finance, RICAM, Linz September 3, 2008

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Typical Questions in Insurance

◮ Which risks can be insured? ◮ Determination of fair premiums ◮ Stability of insurance activity

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Program

◮ Some Generalities ◮ Aggregate Claim Distributions ◮ Collective Risk Models ◮ Extensions (Dividends, Taxes, Dependence) ◮ Statistical Issues

Collection of ideas and approaches, not exhaustive treatment!

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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)

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Risk Y (random variable) Examples of Measures of Risk

◮ E(Y ), Var (Y ) ◮ CoV(Y ) = √ Var Y E(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

α VaRs ds

(e.g. Swiss Solvency Test)

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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 F ∗(n−1)(x − s)dF1(s) P(X1 + X2 < x) = x 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 2πx3 exp

• − α/2(
• x/µ −
• µ/x)2

, x > 0

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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, ..

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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(X β) < ∞ ⇔ β < α

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

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Aggregate Claims X(t) = N(t)

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

ˆ F(s) := E(e−sXn) = ∞

0 e−sxdF(x),

Qt(z) := E(zN(t)) = ∞

n=0 pn(t)zn 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)ˆ F n(s) = Qt(ˆ F(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.

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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.

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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 Approximations

Under suitable conditions ak (x) = a1(x) + Pk

j=1 Aj f (j)(x),

Improvement in certain parameter ranges.

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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 that limα→α0 κ′

t(α) = ∞.

Var α(St) = tκ′′

t (α)

Pα

St−x q 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 → ∞.

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Asymptotic Approximations of Gt(x) - Superexponential Xn II

1 − Gt(x) = |θ| e−|θ|x Z ∞ e−|θ|v {Mθ(x + v) − Mθ(x)} dv

−σF < θ < 0 with Mθ(x) := P∞

n=0 an F ∗n θ (x) and an := ˆ

F 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 − 1 n « `

b t+b

´α `

t t+b

´n . ˆ F(θ) = 1 + b t ⇒ 1 − Gt(x) ∼ b |t ˆ F

′(θ)|

!α 1 |θ|Γ(α)e−|θ|xxα−1 .

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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 ( inf

0≤t≤T Rt < 0 | R0 = x)

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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

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Solution Methods

◮ Exact Solutions ((P)IDE)

Infinite time horizon: CP c ∂ψ ∂x − λ ψ + λ x ψ(x − y) dF(y) + λ(1 − F(x)) = 0

with lim

x→∞ ψ(x) = 0.

ψ(x) =

• 1 − λµ

c ∞

• n=1

λµ c n (1 − F n∗

I (x)),

with FI(x) = 1

µ

x

0 (1 − FX (y)) dy,

x ≥ 0. Examples:

◮ Xi ∼ Exp(1/µ): ψ(x) = λµ

c e− c−λµ

cµ

x

◮ Xi ∼ Phase-Type

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Solution Methods

◮ Exact Solutions ((P)IDE)

Finite time horizon: CP c ∂ψ ∂x − ∂ψ ∂T − λ ψ + λ x ψ(x − y, T) dF(y) + λ(1 − F(x)) = 0

with lim

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

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Solution Methods

◮ Exact Solutions ((P)IDE)

Finite time horizon with positive interest rate: CP

(c + i x)∂ψ ∂x − ∂ψ ∂T − λ ψ + λ Z x ψ(x − y, T) dF(y) + λ(1 − F(x)) = 0

with lim

x→∞ 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 yn−1e−αy dy (α > 0, n > 0) n ∈ N : Γ(x; α, n) = 1 − e−αx

n−1

X

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)

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◮ Inclusion of (non-linear) dividend barriers

• t

t

• time

t

ruin claims ~ F(y) premiums = ct b dividends dividend barrier b reserve 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.

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Example: bt =

• bm + t

α 1/m

(α, b > 0, m ≥ 1)

(c + i x) ∂φ ∂x + 1 α m bm−1 ∂φ ∂b − λ φ + λ Z x 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 α m bm−1 ∂W ∂b − (δ + λ) W + λ Z x W (x − z, b)dF(z) = 0, with ∂W

∂x

˛ ˛ ˛

x=b = 1.

→ Numerical solution

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Define integral operator

Ag(x, b) =

t∗

Z λe−(λ+δ)t

(c′+x)eδt −c′

Z g (c′ + x)eδt − c′ − z, „ bm + t α «1/m! dF(z)dt +

∞

Z

t∗

λe−(λ+δ)t

“ bm+ t α ”1/m

Z g „ bm + t α «1/m − z, „ bm + t α «1/m! dF(z)dt +

∞

Z

t∗

λe−λt

t

Z

t∗

e−δs B @(c + δ x)eδs − 1 mα “ bm + s

α

”1−1/m 1 C A ds dt, with c′ = c

δ 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 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

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Solution Methods (contd.)

◮ Numerical Techniques (PIDE, Laplace-Transform inversion) ◮ Approximations

◮ Diffusion-/L´

evy-Approximation

◮ Discretizations ◮ Cram´

er-Lundberg-Approximation ∃ R > 0 with E(eR X ) = 1 + R c/λ ⇒

(constant!)

ψ(x) ∼ c − λµ λE(XeRX ) − c e−R x

◮ Heavy-Tail-Approximation: FI(x) ∈ S ⇔ lim

x→∞ 1−F∗2 I (x) 1−FI (x)

= 2, ψ(x) = “ 1 − λµ

c

” P∞

n=1

“ λµ

c

”n (1 − F n∗

I

(x))

⇒ ψ(x) ∼ λµ c − λ µ (1 − FI(x))

convergence rate!

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Solution Methods (contd.)

◮ Inequalities (martingales)

(e.g. Lundberg inequality)

◮ Duality with other models ◮ Simulation

◮ Rare event sampling ◮ Quasi-Monte Carlo techniques

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Discounted penalty function

(Gerber & Shiu (1998))

mδ(x) := E

• w(R(T −

x ), |R(Tx)|) e−δTx 1{Tx<∞}

• w(x1, x2) . . . non-negative function

|R(T )|

−

reserve R(t) time t claims premiums

u

ruin time Tu R(T )

u

u

Special Cases:

◮ w ≡ 1: Laplace transform of Tx

(δ = 0: ruin probability).

◮ w(x1, x2) = 1{x1≤y}1{x2≤z}, δ = 0:

joint distr. of R(T −

x ) and R(Tx).

SLIDE 25

Optimal Control of the Risk Process

◮ Safety Criteria:

Minimize ruin probability by dynamic reinsurance and/or investment

Schmidli (2001), Hipp & Vogt (2003), Browne (1995), Hipp & Plum (2000), Gaier & Grandits (2004)

◮ Profitability Criteria:

Measure insurance portfolio by the value of future profits paid

• ut as dividends to shareholders

de Finetti (1957)

SLIDE 26

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 ≡ b

Paulsen & Gjessing (1997), Gerber, Lin & Yang (2006)

mδ(x, b) = mδ(x) − E(Dx,b) m′

δ(b)

• t

reserve R u b dividends ruin time time

For other claim sizes: Band strategies Extension: L´ evy models, not for renewal models!

SLIDE 27

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) = sup

L

V (u, L) = sup

L

E „Z τ e−βt dLt + Z τ e−βtΛ dt ˛ ˛ ˛ RL

0 = u

«

HJB equation

max  Λ + cV ′(u) + λ Z u V (u − y)dFY (y) − (β + λ)V (u), 1 − V ′(u) ff = 0.

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Optimal solution for exponential claim sizes: 0 ≤ l(u) ≤ M : l∗(u) = 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 10

• 8

10 12 14 16 x0 2 4 6 8 10 x 2 4 6 8 10 12 V x 2 4 6 8 10 x 100 200 300 400 500 EΤx

Λ = 0, 1, 2

SLIDE 29

Alternative approach: Explicit strategies

◮ E.g. Threshold Strategy

• t

claims premiums dividends reserve R time t ruin time dividend threshold b b u

~ F(y) = ct

SLIDE 30

Threshold strategy in the Sparre Andersen model MGF of Du,b . . . M(u, y, b) = M1(u, y, b)I{u<b} + M2(u, y, b)I{u≥b} System of PIDEs:

= @

n

Y

j=1

−c ∂

∂u + λj + δy ∂ ∂y

λj 1 A M1(u, y, b) − (1 − FY (u)) − Z u M1(u − z, y, b) dFY (z), = @

n

Y

j=1

−(c − a) ∂

∂u + (λj − ya) + δy ∂ ∂y

λj 1 A M2(u, y, b) − (1 − FY (u)) − Z u M(u − z, y, b) dFY (z) Boundary conditions are given by lim

b→∞ M1(u, y, b)

= 1, lim

u→∞ M2(u, y, b)

= eya/δ. Moreover, at u = b by continuity (c − a) ∂+ ∂u − δy ∂+ ∂y + ya !j−1 M2 ˛ ˛ ˛ ˛ ˛

u=b

= c ∂− ∂u − δy ∂− ∂y !j−1 M1 ˛ ˛ ˛ ˛ ˛ ˛

u=b

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

SLIDE 31

Wm(u, b) := E(Dm

x,b) :

M(u, y, b) = 1 + ∞

m=1 ym m!Wm(u, b)

System of IDEs:

@

n

Y

j=1

−c ∂

∂u + λj + δ ¯

∆ λj 1 A Wm,1(u, b) − Z u Wm,1(u − z, b) dFY (z) = 0, @

n

Y

j=1

−(c − a) ∂

∂u + (λj − a∆) + δ ¯

∆ λj 1 A Wm,2(u, b) − Z u Wm(u − z, b) dFY (z) = 0, with operators ∆Wm := mWm−1, ¯ ∆Wm := mWm (W0 = 1, W−i = 0 (i ∈ N)). lim

b→∞ Wm,1(u, y, b)

= 0, lim

u→∞ 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).

SLIDE 32

c2W ′′

1,1(u, b) − 2c(δ + λ)W ′ 1,1(u, b) + (δ + λ)2W1,1(u, b) − λ2α e−αu

Z u W1,1(v, b) eαv dv = 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−αu Z u W1(v, b) eαv dv = 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)eR(i) 1 u and W1,2(u, b) = a δ + A(1) 2 (b) eR(1) 2 u

(δ + λ − (c−a) R)2(R + α) − αλ2 = 0. B B B B B B B B B @

1 R(1) 1 +α 1 R(2) 1 +α 1 R(3) 1 +α

−

α R(1) 2 +α

eR(1)

2 b α R(1) 1 +α

eR(1)

1 b α R(2) 1 +α

eR(2)

1 b α R(3) 1 +α

eR(3)

1 b

−eR(1)

2 b

eR(1)

1 b

eR(2)

1 b

eR(3)

1 b

−(c − a)R(1)

2

eR(1)

2 b

cR(1)

1

eR(1)

1 b

cR(2)

1

eR(2)

1 b

cR(3)

1

eR(3)

1 b

1 C C C C C C C C C A 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 = B B B B @

a δ a δ

a 1 C C C C A .

• Analogous solution for ψ(u)

SLIDE 33

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

s ruin tax payments

claims ~ F(y) premia

W1 M1 M2 reserve Rγ(t) W2 σ2 σ1

φγ(s) = 1 − ψγ(s) . . .survival probability with tax rate γ → Simple power relation φγ(u) = (φ0(u))

1 1−γ

SLIDE 34

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 σ2 σ1 W2

time t

◮ Link between φ0(u) and Vmax . . .maximum workload during busy period of M/G/1 queue ◮ Net profit condition c > λ E(Xi) ⇔ traffic intensity ρ < 1 ◮ Cut out excursions from running maximum that “survive”

(u ↔ t)

SLIDE 35

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 W2

time t

◮ Link between φ0(u) and Vmax . . .maximum workload during busy period of M/G/1 queue ◮ Net profit condition c > λ E(Xi) ⇔ traffic intensity ρ < 1 ◮ Cut out excursions from running maximum that “survive”

(u ↔ t)

SLIDE 36

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 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

c P(Vmax ≥ t)

φ0(u) = exp “ − Z ∞

u

α(t) dt ” = exp “ − λ c Z ∞

u

P(Vmax > t) dt ” ⇒ P(Vmax > u) = c λ d du log φ0(u) ∀ u > 0.

SLIDE 37

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) =

λ c (1−γ) P(Vmax ≥ t)

φγ(u) = exp “ − Z ∞

u

αγ(t) dt ” = " exp “ Z ∞

u

α0(t) dt ”#

1 1−γ

= (φ0(u))1/1−γ

◮ Extension to surplus-dependent tax rate ◮ Extension to spectrally negative Levy process

SLIDE 38

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, Diversification

properties Exact Results, Asymptotic Results, Stochastic Ordering

SLIDE 39

Exact solutions for specific dependence structures

Example: Causal dependence Xi ↔ Ti+1 Model:

Xi Xi+1 Ti+1

Di . . . threshold variable

◮

If Xi > Di , then Ti+1 ∼ Exp (λ1)

◮

If Xi ≤ Di , then Ti+1 ∼ Exp (λ2)

◮ net profit condition: µX < c

h

P(Xi >D) λ1

+ P(Xi ≤D)

λ2

i .

◮

lim

x→∞ φi(x) = 1 − ψi(x) = 1

(i = 1, 2) Motivation: earthquakes, volcano eruptions

SLIDE 40

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 condition PM

i=1 πiµX,i < c PM i=1 πiλ−1 i

,

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

SLIDE 41

For Re s ≥ 0 and i = 1, . . . , M: ˜ mi (s) := Z ∞ e−sx mi (x) dx, ˜ bi (s) := Z ∞

x=0

e−sx dBi (x), ˜ ωi (s) := Z ∞

x=0

e−sx Z ∞

x

w(x, y − x) dBi (y) dx.

Linear system of equation: Aδ(s) ˜ mδ(s) = c mδ(0) − Λ P ˜ ω(s) with Aδ(s) := (cs − δ) I − Λ + Λ P ˜ B(s)

I . . .identity matrix, Λ = diag(λ1, . . . , λM), ˜ B(s) = diag(˜ b1(s), . . . , ˜ bM(s)),

• ˜

mδ(s) = ( ˜ mδ,1(s), . . . , ˜ mδ,M(s)),

• ˜

ω(s) = (˜ ω1(s), . . . , ˜ ωM(s)).

SLIDE 42

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 AT

δ (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.

SLIDE 43

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

det A0(s) = 3 − 8s + 4s2 + 6s − 3 1 + s − 4s 3 + s

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

Ruin probabilities:

• ψ(x) =

0.007 0.003

• e−3.161 x +

0.938 0.867

• e−0.065 x.

SLIDE 44

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

det A0(s) = 3 − 8s + 4s2 + 6s − 3 1 + s − 4s 3 + s

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

Density of surplus before ruin:

• f (y1|x) = 1{x≤y1}

„ 9 9 « e−y1 + e−3.161x „ 0.106 0.045 « 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} „ 9.020 8.333 « e−0.0645x−0.935y1 − „ 0.045 0.019 « e−3.161x+2.161y1 !

1 2 3 4 5 6 y1 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 y1 0.1 0.2 0.3 0.4

◮

lim

x→∞ fi(y1|x)/ψi(x)

SLIDE 45

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: lim

x→∞

1 x log ψ(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 < ∞ ∀ n.

M¨ uller & Pflug (2001)

SLIDE 46

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: lim

x→∞

1 x log ψ(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 < ∞ ∀ n.

M¨ uller & Pflug (2001)

SLIDE 47

Some Examples:

• 1. Random Walk (Sparre Andersen model):

n−1 log EeαSn = n−1 log

n

• 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 = −2µ 1 − 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,

SLIDE 48

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

c(t) = (1 + η) PNt

i=1 Xi

t .

(Asmussen (1999))

Use past claim experience to determine premium rate!

Nt

X

i=1

Xi /t − Z t c(s) ds =

Nt

X

j=1

Xj − (1 + η) Z t PNs

j=1 Xj

s ds =

Nt

X

j=1

Xj 1 − (1 + η) log t Ti ! ,

→ Reinterpret as compound Poisson process with time-dependent claims ⇒ κ(α) = λ Z 1 MX ` α ` 1 + (1 + η) log u ´´ du − λ = 0

discrete skeleton {Snh}n≥0

Comparison with constant premium rule: Radap ≥ Rconst

SLIDE 49

• 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

SLIDE 50

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→∞ 1

u log P(maxn Snh > u) = −R

max

t

St ≥ max

n

Snh ≥ max

t

St − c h

lim

u→∞

1 u log ψ(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)

SLIDE 51

Finite time horizon:

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

a.

lim

x→∞

1 x log ψ(x, a x) = −Ra with Ra =

• αa − a κ(αa),

a <

1 κ′(R),

R, a ≥

1 κ′(R).

κ(α) α R θa Ra κ′ = 1/a

◮ Accuracy

SLIDE 52

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.

lim

t→∞

1 vt log E(eαvtSt/at) := κ(α) exists and ∃ increasing function h(t) s.t. g(d) := lim

t→∞

v(a−1(t/d)) h(t) exists ∀d > 0

(plus some technical conditions)

⇒ lim

x→∞

1 h(x) log ψ(x) = const. = − inf

d>0

• g(d) sup

α∈R

(αd − κ(α))

• ⇒ light tails, but possibly long-range dependence

SLIDE 53

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 = 0.5: stationary, but NOT independent increments:

E(Xs Xs+h) = 1

2

“ (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

SLIDE 54

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

lim

x→∞

1 x2−2H log ψ(x) = − inf

d>0

• d−2+2H (d + µ)2/2
• = const.

Weibull-type tail

(cf. also Michna (1998))

SLIDE 55

Heavy-tailed claims

Subexponential class: F ∈ S ⇔ lim

x→∞ 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) =

1 µ

R x

0 (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!

SLIDE 56

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: lim

x→∞

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))

SLIDE 57

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<M1−M0 , {ST1+t − ST1 }0≤t<M2−M1 , {ST2+t − ST1 }0≤t<M3−M2 , . . . are i.i.d.

• P(STk − STk−1 > x) ∼ H(x) with HI ∈ S∗

Under mild additional assumptions:

⇒ ψ(x) ∼ 1 |E(STk − STk−1)| (1 − HI(x))

Examples:

◮ Semi-Markov Model

Long-range dependence: asymptotic behaviour can change

(Mikosch & Samorodnitsky (2000))

SLIDE 58

References I

• H. Albrecher and S. Asmussen.

Ruin probabilities and aggregate claims distributions for shot noise Cox processes.

• Scand. Actuar. J., (2):86–110, 2006.
• H. Albrecher, S. Borst, O. Boxma, and J. Resing.

The tax identity in risk theory - a simple proof and an extension. Insurance Math. Econom., 2008. to appear.

• H. Albrecher and O. J. Boxma.

On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom., 37(3):650–672, 2005.

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Lundberg’s risk process with tax.

• Bl. DGVFM, 28(1):13–28, 2007.
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On a gamma series expansion for the time-dependent probability of collective ruin. Insurance: Mathematics and Economics, 29(3):345–355, 2001.

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On the ruin problem for some adapted premium rules. In Probabilistic Analysis of Rare Events: Theory and Problems of Safety, Insurance and Ruin, pages 1–15. Riga Aviations University, 1999.

SLIDE 59

References II

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Ruin probabilities. World Scientific, Singapore, 2000.

• S. Asmussen, H. Schmidli, and V. Schmidt.

Tail probabilities for non-standard risk and queueing processes with subexponential jumps.

• Adv. in Appl. Probab., 31(2):422–447, 1999.
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Optimal reinsurance and dividend distribution policies in the Cram´ er-Lundberg model.

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Asymptotic expansions for infinite weighted convolutions of rapidly varying subexponential distributions. Prob.Theory Relat. Fields, 141:155–180, 2008.

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Asymptotic expansions of convolutions of regularly varying distributions.

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• M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas.

Actuarial Theory for Dependent Risks. Wiley, Chichester, 2005.

SLIDE 60

References III

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uppelberg, and T. Mikosch. Modelling Extremal Events. Springer, New York, Berlin, Heidelberg, Tokyo, 1997.

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• H. U. Gerber and E. Shiu.

On the time value of ruin. North American Actuarial Journal, 2(1):48–72, 1998.

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Logarithmic asymptotics for steady-state tail probabilities in a single-server queue.

• J. Appl. Probab., 31A:131–156, 1994.

SLIDE 61

References IV

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The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insurance Math. Econom., 33(3):551–566, 2003.

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Ruin probabilities and first passage times for self-similar processes. PhD Thesis, Lund University, 1998.

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Ruin probability with claims modeled by a stationary ergodic stable process.

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The supremum of a negative drift random walk with dependent heavy-tailed steps.

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uller and D. Stoyan. Comparison Methods for Stochastic Models and Risks. Wiley, Chichester, 2002.

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SLIDE 62

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