for insurance companies Non-dangerous risky investments Manfred - PDF document

for insurance companies Non-dangerous risky investments Manfred Sch?il 6, (e-mail: schael@uni-bonn.de) Angew. Math., Univ. Bonn, D-53115 Bonn, Wegelerstr. Inst. dynamic programming Key words. ruin probability, investment, financial market, stochastic The problem of controlling ruin probabilities is studied in a Cram6r-Lundberg model where the claim process Poisson process with claim size Y n at claim time Ti. is described by a compound The number Ni of claims in (0,t] is a Poisson There is a premium process with claim intensity 1,. (income) rate c which is fixed. invest the capital (surplus/risk in a financial market where 2 The insurance company can reserve) assets can be traded. One of them is called the bond and is described by the interest rate which here is assumed w.l.o.g. to be zero. The other asset by a l-dimensional price process {Sn, n > 1} is called stock. It is described where Sn is the price of one share of the stock at time Tn. The price process will be driven by a jump times compound Poisson process which can by the sequence {Tn, n>l } of market be defined and the sequence of returns {Rn, n21}, where 1 + Rn > 0 and ( l ) S = $ . . ( l + R ) . n n-r n' jumps in (0,t] where {N,} is a Poisson process We write Nt for the number of market with market intensity v. In general, v will be much larger than 1,. Thus the price process is driven by a Lövy process as in the Black-Scholes model. However, a compound Poisson process is chosen in place of a Wiener process. Moreover, only a moment condition is assumed for the distribution of Rn. Thus, the model for the financial market is quite general of the Black-scholes model is the completeness of the and flexible. The main advantage But this property present problem. financial market. is not needed in the control process We define the Poisson {Nr} bf superposition: n ) 1. jump times (2) N, := N, + N, is a Poisson process with parameter l, + v and Trr, We write Kn = 1 if the jump at Tn is caused by the financial market and Kn = 0 if the jump is caused by a claim. Then we make the following assumption: Model Assumption: Zn,= Tr, - Tn_', Yrr, Rrr, The (Zn) are üd and All random variables Krr, n)1, are independent. have an exponential distribution with parameter ?t+v; the (Yn) are iid and positive; the (F.n) are the (Kn).We assume iid as well as P[Rn<0] >0,E[Rn] >0 and E[Rfr] <"", = # = 1 - P [ K n = o ] , P [ K n - 1 ] (3) q ,= [yn] < | for the classical ruin probability qwith start in 0. 3.E

2 In the joint model of insurance and finance, {Tn, n20} are the decision times and the real-valued discrete-time proces. {Xrr, n)0} describes the risk process (surplus process) immediately after a portfolio 0n e R at any time Tn. There 0n represents the time Tn. A dynamic portfolio specifies amount of capital invested in the stock. We have the following law of motion: Kn*, = 1, Xn 2 0. = Xn (4) Xn+1 * c.Zn+l* 0n.Rn+l for for Kn*t = o' Xn 2 o' Xn+l = Xr, * c'zn+l- Yn+l

Xn+l r € For technical reasons we set for Xn < 0. Once in state x e (----oo,Q;, the system moves -"" in the next step. to the absorbing state We write O(x) for the set of all portfolios 0 admissible at x which is assumed to be = [0,^] ,x ) 0. O(x) @(x), x - 0 represents We define @(x) := {0i for x < 0. For 0 e the amount of the capital which is invested in the bond, i.e., which is not invested in the stock. In this model we do not allow for negative amounts thus 0rr, excluding short selling of the stock. A number x < 0 represents a state of ruin. A stationary (investment) plan is a measurable function tp such that q(x) c @(x) for all x. Then q(Xrr) specifies the portfolio 0r, e @(Xrr) for the period (T .T - , ,l. We will sometimes write for the state (risk) process ' n ' n + l r (5) = Xx'Q ' n n Our performance criterion is the ruin probability: (6) V(x,q) '= P [Xä'9 < 0 for some n] At first view, the ruin probability is not a classical performance criterion for control problems. However, one can write the ruin probability as some total cost in an embedded discrete-stage model where one has to pay one unit of cost when entering the ruin state. After this simple observation, results from discrete-time dynamic programming for minimizing costs apply. Lundberg inequalities will be derived for the controlled model which extend the classical inequalities for the uncontrolled model. (2003) where a continuous-time The present paper is related to Gaier, Grandits & Schachennayer control model is studied for a Black-Scholes market by different methods. If ty(x) is the classical ruin probability for an initial reserve x, then V(x) = V(x,qg) where go(x) = 0. If ro > 0 is the classical Lundberg exponent then a classical theorem says 0 < C^.e-ro'^ ( V(*) = V(x,qg) ( e-ro'X for some constant C^ > 0. o o Now let Q be a stationary investment plan such that the decision maker invests a constant fraction time, i.e. 0(x) = T.x, then it was shown by Paulsen y of capital at any (decision) & Gjessing (1997) and Frovola, Kabanov & Pergamenshchikov (2002) that the asymptotic behaviour of the ruin probability is completely different under the investment plan Q. In fact, in the latter case the ruin probability has a polynomial decay (as function of the initial reserve) even if the financial this plan is called market is described by a stock price process with high returns. Therefore, dangerous in Frovola, Kabanov & Pergamenshchikov (2002). However, if Q is an investment plan such that a constant amount A is invested in the stock i.e. Q(x) = A, then one can find some A and some independently of the current risk reserve, exoonenti>r suchthat

< "-t't (7) V(x,,Q) V(x,q) > Ö'"-i '* fo, some (8) 0 < Ö < 1 for every investment plan g and (see A plan with (7) for some Gaier, Grandits & Schachermayer 2003, Schäl 2005). i t ro will be company is poor, call profitable. The plan rQr may however be not admissible when the insurance since rQ(x) = A G @(x) for x < A. In the present paper we study stationary investment plans <p* such that (9) q*(x) = min (x, A) for some A > 0. a plan is admissible. Such a plan q* of the form (9) and some THsoneI\,I There exist r* > rosuch that (9) < "-t*'* . V(x,q*) Then q* is admissible and profitable. rx ( i in (10). Of course in view of (8), we have rx = i if the market intensity THeoneIvI One even can choose v is high. If v is high which is a natural condition, then the model is close to a continuous-time model. 'PARADoxoN' (i) Yn -F.2, i.e. the claims have Assume an Erlang distribution; (ii) the E [Rn] = 0; (iii) v is large. price process is a martingale, i.e. that q(x) = x for small values plan <p Then there exist some stationary such x and = V(x) and = V(x) for small V(x,q) ) V(x,<po) y(x,<p) > V(x,<po) x. Hence, if the system is close to ruin, it may be good to invest all the capital in a martingale. The paper builds on methods from discrete-time control / stochastic dynamic programming / Markov decision processes. References Frovola A G, Kabanov Yu M, Pergamenshchikov S M (2002). Finance and Stochastics 6: 221135 W (2003). Annals of Applied Probability 13: LO54-1016 Gaier J, Grandits P, Schachenneyer HK (1997). Advance in Applied Probability 29:965-985 Paulsen J, Gjessing M (2004). Schäl Scand. Actuarial J. 3: 189-210 M (2005). Math.Meth.Oper.Res. 62: l4l-158 Schäl Schmidli H (2OOZ). Annals of Applied Probability 12: 890-907

N dong,^or^s of a'irhr.1 tnsc.s+rilrnh ün thc insrncrrcr' buslruss h ourhs.l Schä [ ln.ßt. "F"SrJ. t'lath. Uniy. Bonyr

o iis o[ ^uin P^obald Lrf ConhoL by clircnc{e' trrrr., tnocshnGrh qn hecl t1 SchäL f1 4[h, Inrt, An$cto. Bonn LLnr'v. Papr^s tn ecnFi narou{ Htrnc " qS) rrorrn S. (1f s , I IS *1Y,n,U. SchoGlA{ cr'wr.1,, G Grtenclih.P.r (?oo3 I Cl,ooo) l.+t,ppfc.rPLuwr.t'l . (a0o3) Sc.hniielLirhl t*iPP,C,, cllt.+{. ( ZOO4) Sehmi irrj ( tI t? lrt) & 6iess P s ulscn f4ouolcr , Kgb crnov 8, Oof [ 2 ) hov P-gavilensh chi Gf o[, ( r Po'r?} Faninde?

Ca vY\q^hßl owrc\,Jo^h- Ion ltn av\c, Qene^ctl [n (l S ^ - S o stoch p^Lce €.Le t t Lü] Ld, vy p4ocess 1 BLqck- $c,ho[es Ex ' - o - ' U t + ( F - + c r a ) ' t Lr =+ ,r^BS S ; ' - S o ' I { g U r + [ ' t t ) isson wtoolrncl Fo €x 2 eo L L = v t Nrl Poisson wi{h 4crk =+ S; : So'TTH (r + Rr,) con\r4^qencq \Jecrk *t s:t } ( v-'oo) { s; } LLt j -tuur+(F- t zJl B: !o)r] € + J r N 1 1z- n=4 R: \ -t to-ur -+ rl\

ob qondihiovrs Neqessc{zr{ Io4 convezr.gel4c€ r,):. B: := m B + A T i l L - z[: R-l X f + A F E E t4 clectemP. SernL-knq^t=i 3c,t-e- 44 = v' ELBUI'I G..(p-Lcl)'t 1 r1 +. v.Et Rol t n. A - . - L v. Et (R")tl .t -r qz I \ | | a t l_ TxsrnPLe (tq(6) Coaneu Dengb^ Diss. ?aiqer,t* ) | (2oo\) J, Retg.^ T,n, E-unprnical !+ iLIef l Sccr E: : b" ß: E -bu)=1-po P C PC ): p' , v be '�: crz ( e p - 1 ) v ' b ! : f - + t r ? erLso s urttdcte.n+

O c LepL cAu f-aqruf erntryr tR J g(s),: Ef e,- - B(s)l v.[ 1 :- S. \ sl .V.E[Rt] + u.Et Rl oCtJ # I {t 't' A rt F ( ---+) s.F - +- sa g-e r V + s - + J*o* v'[ 1a g e.Cs)] S=a % q^he} ptice of aiste- ( r t