Minimization of ruin probabilities by investment under transaction costs

Transcription

1 Minimization of ruin probabilities by investment under transaction costs Stefan Thonhauser DSA, HEC, Université de Lausanne 13 th Scientific Day, Bonn,

2 Outline Introduction Risk models and controls Optimal investment in risk theory Investment with transaction costs Statement of optimization problem Characterization of solution Nature of optimal policy Numerical procedure Conclusion 2 / 19

3 Risk models Stochastic model for evolution of surplus process Insurance: classical compound Poisson model (general: spectrally negative Lévy processes) Diffusion risk model (e.g. approximation for large portfolios) Piecewise-deterministic-Markov-processes (Davis (1993)) allows for incorporation of different economic environments includes renewal risk model Determination of risk relevant quantities Theory: classical ruin probabilities, risk measures Practice: fixed time horizon, Value at Risk or Expected Shortfall 3 / 19

4 Control possibilities Classical approach: model is fixed, no further interventions possible need for extensions to control evolution of surplus process For example: Introduce reinsurance possibility Schmidli (22), Hipp & Vogt (23) Allow for investment in financial market ruin probability: exponential decay can turn into power decay! Frolova et al. (22), Kalashnikov & Norberg (22) control of investment crucial Browne (1995), Hipp & Plum (2), Azcue & Muler (29), Th. (213) Additional capital injections (capital issuance) Kulenko & Schmidli (28), Eisenberg & Schmidli (29) 4 / 19

5 Optimal investment Let R = (Rt ) t be surplus process Allow investment in risky asset with price S = (S t ) t If A = (A t ) t denotes invested amount of money surplus with investment T = (T t ) t follows dt t = dr t +A t ds t S t, T = x Accordingly ruin probability is: ψ(x,a) = P x (inf{t T t < } < ) Goal: minimize ruin probability ψ (x) = inf A A ψ(x,a) 5 / 19

6 Comment on model Implicitly assumed: risk-free rate r = If proportion αt invested, then (T t ) t follows ds t dst dt t = dr t +α t T t +(1 α t )T t S t St, T = x since r =, we can use A t = α t T t as control Compare with utility maximization in finance max α E x(u(x T )) ds t dst s.t. dx t = α t X t +(1 α t )X t S t St, X = x Inclusion of insurance risk particular incomplete market 6 / 19

7 Classical results Browne (1995): Rt is Brownian motion with drift A is constant Hipp & Plum (2): Rt is compound Poisson with drift A is function of surplus Asymptotics depend on claims size distribution continuously buying and selling fractions of asset Goal: study problem avoiding continuous interventions - Solutions obtained via dynamic programming method (Schmidli (28)) - (S t) t follows geometric Brownian motion 7 / 19

8 Investment under transaction costs We use a diffusion model: Starting point: dr t = µdt+σdw (1) t, for t...surplus ds t = S t(adt+bdw (2) t ), for t...asset price Insurer invests amount A = A, and leaves it unchanged Surplus T A = (T A t ) t follows: dt A t = dr t +da t, for t, T A = x > da t = A t(adt+bdw (2) t ), for t Associated time of ruin and survival probability: τ A = inf{t T A t }, ϕ (x,a) = P x,a (τ A = ) W (1) = (W (1) ) t,w (2) = (W (2) ) t... independent BMs, (µ, σ, a, b > ) 8 / 19

9 Static situation It turns out that study of ϕ (x,a) is crucial (at one point one needs to decide for a change or not) Observe Mt = ϕ (T A t,a t ) is martingale ϕ (x,a) fulfills: ( ) σ 2 = Lw(x,A) := (µ+aa)w x(x,a)+ 2 + A2 b 2 w xx(x,a) 2 +A 2 b 2 w xa(x,a)+aaw A(x,A)+ A2 b 2 2 waa(x,a) with boundary conditions: { ϕ (,A) =, A, ϕ (x,) = 1 e 2µ σ 2 x, x, lim x ϕ (x,a) = 1, for A, lim A ϕ (x,a) =, for x > (Apply Itô s formula to ϕ (x,a) and use PDE results) 9 / 19

10 Investment not always advantageous Survival probability as function of initial surplus and initial investment x x A A ϕ (x, A) for µ =.3, σ =.8, a =.2, b =.4 (.6) A = 1 (.57) Usage of investment should be controlled! (A is Browne s constant optimal strategy) 1 / 19

11 Introducing transaction costs I Now insurer can modify investment position, but changes are subject to transaction costs: change A A+ A leads to costs K +k A K >... fixed cost, k >... proportional cost factor Observations: continuous adaptions lead to unbounded transaction costs admissible investment policies are impulse controls π = {(θ n,a n )} n N θ i... intervention time, A i... new investment position one needs to know actual investment bivariate modeling needed 11 / 19

12 Introducing transaction costs II Policy π = {(θ n,a n )} n N is admissible, if it fulfills: θ n θ n+1 a.s. for all n N θ n stopping time w.r.t. σ{(w (1) s,w (2) s ), s t, (θ k,a k ), k < n} and θ n τ π for all n N A n is measurable w.r.t. σ{(w (1) s,w (2) s ), s t, (θ k,a k ), k < n} P({lim n θ n T}I {T<τ π }) = for all T > and lim n θ n = τ π a.s. Controlled surplus process T π = (Tt π ) t is: T π t = x+ t dr s + n=1 θn t θ n 1 t A π s(ads+bdw (2) s ) da π s = A π s(ads+bdw (2) s ), for θ n 1 s < θ n A π θ n = A π θ n + A π θ n = A n, A π = A (K + A π θ n )I {θn t} n=1 Value function is ϕ(x,a) = sup π Π ϕ π (x,a) 12 / 19

13 Characterization through iterated optimal stopping Suppose f( ) specifies continuation value, optimally intervene: sup {f(x K k A,A+ A)} =: Mf(x,A) { A D(x,A)} Find optimal intervention time via optimal stopping operator: Characterization: Gf(x,A) := max{ϕ (x,a),supe x,a (Mf(Tθ A,A θ))} θ<τ ϕ n (x,a) := Gϕ n 1 (x,a) ϕ n (x,a) = sup π Πn ϕ π (x,a) (Π n policies with n interventions) lim n ϕ n = ϕ ϕ(x, A) = Gϕ(x, A) (induces approximation procedure) (fixed point property) 13 / 19

14 Analytical characterization Observations: If for (x,a) intervention is optimal ϕ(x,a) = Mϕ(x,A) If around (x,a) it is not optimal to intervene Lϕ(x,A) = (optimal control leads to martingale) Quasi-variational inequalities for this problem are: max{lw,mw w} =, w(,a) =, lim x w(x,a) = 1, lim A w(x,a) = Fixed point property, approximation arguments and Itô s formula show: Characterization: Maximal survival probability ϕ is viscosity solution to QVI 14 / 19

15 Implicit optimal policy Suppose solution to QVI is available, policy π can be defined by: Non-action set: A := {(x,a) ϕ(x,a) > Mϕ(x,A)} A Action set: B = A c stylized intervention and x non-intervention areas Associated intervention times and interventions: θk := inf{t θ k 1 (Tπ t,a π t ) / A} A k := argmax{mϕπn (Tθ π k,aπ θk )} This implicitly given strategy π = {(θ k,a k )} is optimal 15 / 19

16 Comment on numerical implementation Basis of computations is iteration of stopping operator Procedure: ϕ n (x,a) = max{ϕ (x,a),supe x,a (Mϕ n 1 (Tθ A,A θ))} θ<τ Bounded domain [,x] [,A] (x, A sufficiently large for having suitable approximation of ϕ ) Build up strategy π1 via non-action and action sets: A 1 := {(x,a) ϕ (x,a) > Mϕ (x,a)}, B 1 = A c 1 On A1 solve Lw = On B 1 set ϕ π 1 = Mϕ Iteration: based on ϕ π n define A n+1 and B n+1 determines ϕ π n+1 Procedure is a combination of value iteration and policy iteration 16 / 19

17 Numerical examples 3 x x A A 1 x A A 2 1 x.5. Numerical policy and stylized behaviour 17 / 19

18 Effect of optimal investment reduced initial capital for given safety level Define: risk premia p(x) and p(x) implicitly by ϕ(x,) = ϕ(x p(x),a ) and ϕ(x,) = ϕ B (x p(x)) 1..8 Φ 1.5 p x x Different survival probabilities and risk premia x 18 / 19

19 Conclusions We could solve optimal investment problem under transaction costs by use of stochastic control methods Continuous balancing of investment strategy is not that crucial (as long as one can identify bad positions) Optimization techniques can be applied in several fields managing variable annuities (GMWB are combination of investment and consumption) risk management (portfolio optimization under safety constraint) microeconomics of insurance (construction of optimal insurance contracts) 19 / 19