Optimal dividend distribution under a ruin constraintbrixen07/slides/hipp.pdf · The optimal dividend strategy D is similar; it has, however, a dynamic argument ﬁ; and the functions

Christian HippFinance, Banking and Insurance

Optimal dividend distribution under a ruin constraint

Christian HippInstitute for Finance, Banking, and Insurance

Universität Karlsruhe (TH) (1/3) http://insurance.fbv.uni-karlsruhe.de

[email protected]

Wolfgang Runggaldier‘s 65th birthdayBressanone/Brixen July 17, 2007


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Giovanni Andreatta, Università di Padova Alain Bensoussan, University of Texas, Dallas Francesca Biagini, Universität München Michele Bonollo, Banco Popolare di Verona e Novara Carl Chiarella, University of Technology, Sydney Carlo Alberto Clarotti, ENEA, Roma Giovanni Battista Di Masi, Università di Padova Ernst Eberlein, Universität Freiburg Robert J. Elliott, University of Calgary Gino Favero, Università Bocconi, Milano Lorenzo Finesso, ISIB-CNR, Padova Rüdiger Frey, Universität Leipzig Marco Frittelli, Università degli Studi, Milano Martino Grasselli, Università di Padova Stefan Jaschke, Munich Re, Munich Yuri Kabanov, Universitè de Franche-Comtè, Besancon Juerg Kohlas, University of Freiburg Robert Liptser, Tel Aviv University Fabio Mercurio, Banca IMI, Milano Sanjoy Mitter, Massachusetts Institute of Technology Hideo Nagai, University of Osaka Fulvio Ortu, Università Bocconi, Milano Sara Pasquali, IMATI-CNR, Milano Giorgio Picci, Università di Padova Eckhard Platen, University of Technology, Sydney Maurizio Pratelli, Università di Pisa Giorgio Romanin Jacur, Università di Padova Martin Schweizer, ETH, Zurich Dieter Sondermann, University of Bonn Fabio Spizzichino, Università di Roma - La Sapienza Peter J.C. Spreij, University of Amsterdam Michael Taksar, University of Missouri Karl Thomaseth, ISIB-CNR, Padova Marco Tolotti, Università Bocconi, Milano Nizar Touzi, Ecole Polytechnique, Paris Omar Zane, ABN-AMRO, London

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1 Summary

Dynamic dividend optimization

• started with de Finetti (1957, Act. Congress NY)• is connected with one of the major research topics of Wolfgang

(discrete approximation of continuous control problems, papers in1994/5/6/9 and 2001/2)

• is a challenge under constraints• is on my desk since 2002 (paper H(2003))

Here and in H(2003): infinite time horizon, path dependent constraintruin probability.

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Solving a control problem under a constraint

• Pareto optimal solution to a two objective problem.• with an extra state variable• this state variable is at the same time a control variable• modification of dynamic equations• stopping times in the continuous case• numerical computation via discrete approximations

We consider two objectives: ruin probability and expected discounteddividends.

Ruin probability: objective function for policy holders or for supervision.

Expected discounted dividends: objective function for the stock holder.

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Optimal dividend distribution – without a ruin constraint – leads tocertain ruin which is not acceptable for the policy holders. Minimizingruin probability leads to no dividend payment which is not acceptablefor stock holders.

A ruin probability less than one is possible only for a risk process whichtends to infinity.

Optimal dividend payment with ruin constraint also leads to a reservetending to infinity, but later.

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Continuous time model

The company’s value process modelled by Brownian motion with drift

X(t) = x + µt + σW (t), t ≥ 0,

µ, σ > 0, and W (t), t ≥ 0 standard Brownian motion.Maximize the accumulated discounted expected dividends

E[∫ τD

0

e−ρtdD(t)] (1)

under the ruin constraint

ψD(x) = P{τD < ∞} ≤ α. (2)

τ ruin time of the controlled processρ positive interest for discounting of future payments.Maximum is taken over all non-decreasing adapted processes D(t), t ≥ 0.

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The infinite horizon ruin probability without paying dividends,

ψ0(x) = P{X(t) > 0 for all t ≥ 0} = exp(−2µσ2

x).

Solve the problem without ruin constraint: via a variational inequality

0 = max{−ρu(x) + µu′(x) + 12σ2u′′(x), 1− u′(x)}. (3)

In the range u′(x) > 1 we have a linear differential equation (LDG) withconstant coefficients. The characteristic equation reads

0 = −ρ + µz + 12σ2z2

with solutions z1 < 0 and z2 > 0. The general solution for (LDG) is

u(x) = C1 exp(z1x) + C2 exp(z2x).

Let M be the unique value with u′(M) = 1. Using the initial valueu(0) = 0 and the smooth paste conditions u′(M) = 1, u′′(M) = 0, we

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arrive at C1 = −C2 = C, where C and M are defined via

M =log(z21)− log(z22)

z2 − z1 ,

C = 1/(z1 exp(z1M)− z2 exp(z2M)).u(x, 1) = u(x) = C(exp(z1x)− exp(z2x)).

With ruin constraint (2):

u(x, α) = sup

{E[

∫ τD

0

e−ρtdD(t)] : ψD(x) ≤ α}

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Mw

u(w)

Optimal dividend payment:pay all above M, pay nothing below M.Certain ruin, bounded wealth.

Dividends: continuous case

Characterization of u(x, α):

0 = supδ,f{−ρu(x, α) + (µ− δ)ux(x, α) + 12σ

2uxx(x, α) (4)

+σ2fux,α(x, α) +12σ2f2uαα(x, α)}.

or

0 = max{−ρu(x, α) + µux(x, α) + 12σ

2uxx(x, α) (5)

−12σ2

ux,α(x, α)2

uαα(x, α), 1− ux(x, α)

}

These equations do not help.

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Discrete approximation (time/state space)

For initial surplus x ≥ 0 (integer), discount factor 0 < v < 1 and dividendpayment strategy D = (d0, d1, ...) with integers dt ≥ 0 : define

XD(t) = x + X1 + ... + Xt − d0 − ...− dt−1, t ≥ 0,

P{Xt = 1} = 1− P{Xt = −1} = p > 1/2Choose D such that

E

τD−1∑t=1

vtdt

= max!

under the constraint

ψD(x) = P{τD < ∞} ≤ α

τD ruin time.u(x, α) value function.

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Dynamic equation for u(x) = u(x, 1) (without constraint):

u(x) = d + maxδ

v [pu(x + 1− δ) + (1− p)u(x− 1− δ)]= max [1 + u(x− 1), v(pu(x + 1) + (1− p)u(x− 1))]

Leads to a barrier strategy.

Dynamic equation for u(x, α) :

u(x, α) = supδ,β

[δ + v(pu(x + 1− δ, β) + (1− p)u(x− 1− δ, β))] (6)

δ ∈ {0, 1, ..., x}, (7)ψ0(x + 1− δ) ≤ β ≤ 1 (8)pβ + (1− p)β = α (9)ψ0(x− 1− δ) ≤ β ≤ 1 (10)

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Alternative:

u(x, α) = max

(sup

β

[v(pu(x + 1, β) + (1− p)u(x− 1, β))] , 1 + u(x− 1, α)

)

ψ0(x + 1) ≤ β ≤ 1pβ + (1− p)β = αψ0(x− 1) ≤ β ≤ 1

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'$

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'$

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'$

x, α´

´´

´́3

x− 1, β

QQ

QQQs

x + 1, βp

1− p

pβ + (1− p)β = α

Solves the problem: solution of equation exists, verification argumentworks, numerical algorithm follows.

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Algorithm:

un+1(x, α) = max

(sup

β

[v(pun(x + 1, β) + (1− p)un(x− 1, β))

], 1 + un(x− 1, α)

)

Problems:

• discretization for the range of β• β not in the grid• truncation of x• slow convergence• needs huge main memory• slow approximation

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Continuous problem:An alternative dynamic equation for u(x, α) using stopping times:

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'$

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'$

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'$

x, α´

´´

´́3

s, β

QQ

QQQs

M,βp(x)

1− p(x)

p(x)β + (1− p(x))β = α

t = 0 t = τ

s < x < M, τ = inf{t : XD(t) /∈ (s,M)}, p(x) = P{Xτ = M}

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Resulting equation for u(x, α) in the region ux(x, α) > 1 :

u(x, α) = sups,M,β

[p(x)G(x)u(M,β) + (1− p(x))H(x)u(s, β)]

G(x) = E[exp(−ρτ)1(X(τ)=M)]H(x) = E[exp(−ρτ)1(X(τ)=s)]

α = p(x)β + (1− p(x))βψ0(M) ≤ β ≤ 1ψ0(s) ≤ β ≤ 1

p(x) solves

0 = µf ′(x) +12f ′′(x)

G(x),H(x) solve

0 = −ρf(x) + µf ′(x) + 12f ′′(x)

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Algorithm:

un+1(x, α) = sups,M,β

[p(x)G(x)un(M,β) + (1− p(x))H(x)un(s, β)

]

Too complex for numerical calculation.

Experiments with MAPLE or Mathematica show a considerable improve-ment in each iteration; I had to stop after 10 iterations (too many recur-sions).

The following numerical results are derived via discrete approximations.

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Approximation:

n → ∞∆ = σ/

√n step size

p = 1/2 +µ

2σ√

n

v = exp(−ρ/n)

In the numerical example we have n = 10.000, µ = σ = 1, ρ = 0.1.The figures show results after 2.000 iterations.At the end, the improvement in each iteration was still significant:

sup |uk+1(x, α)− uk(x, α)| = 0.0157.

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Results

Define s0(α) through ψ0(s0(α))) = α. A not optimal dividend strategysatisfying the ruin constraint is given by the following rule: stop payingdividends for ever if you touch s0(α). Pay dividends above M0(α) =M + s0(α), where M is the optimal barrier in the dividend problemwithout constraint.

The optimal dividend strategy D is similar; it has, however, a dynamicargument α, and the functions M(α) and s(α) are different.It is a function of XD(t) and b(t), the time t allowed ruin probability.b(t) is a mean α martingale with dynamics

db(t) = β(XD(t), b(t))dW (t).

Dividend is paid above a barrier M(b(t)). After hitting s(b(t)) paymentof dividends is stopped for ever. So the process (XD(t), b(t)) stays be-tween the curves s(α) and M(α). A path of XD(t) going to ruin hasb(t) → 1. The functions s0, s,M0,M satisfy

s(α) < s0(α) < M(α) < M0(α).

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M0(α)

s0(α)

s(α)M(α)

Simulations

Optimal dividend strategy in the discrete model.Parameters as above.Based on the functions β(x, α), β(x, α),M(α).Reserve process R(t) and allowed ruin probability process b(t) definedrecursively:

b(0) = α, R(0) = xb(t + 1) = β(R(t− 1) + 1, b(t− 1)) if Xt = +1b(t + 1) = β(R(t− 1)− 1, b(t− 1)) if Xt = −1

R(t) = min[R(t− 1) + Xt,M(b(t))].

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zweite.pdfBrixenHipp5.pdfBrixenHipp4.pdfBrixenHipp3.pdfBrixen2007.pdfBricen2007.pptOptimal dividend distribution under a ruin constraint

Letzte.pdf

Simulationen.ppt

Documents

Optimal dividend distribution under a ruin constraintbrixen07/slides/hipp.pdf · The optimal dividend strategy D is similar; it has, however, a dynamic argument ﬁ; and the functions