In the insurance business risky investments are dangerous

In the insrance bsiness risy investments are dangeros Anna Frolova 1, Yri Kabanov 2, Sergei Pergamenshchiov 3.

1 Noname manscript No. (will be inserted by the editor) In the insrance bsiness risy investments are dangeros Anna Frolova 1, Yri Kabanov 2, Sergei Pergamenshchiov 3 1 Alfaban, Moscow, Rssia 2 Laboratoire de Mathématiqes, Université de Franche-Comté, Besançon, France and Central Economics and Mathematics Institte, Moscow, Rssia 3 Toms State University, Rssia Dedicated to the memory of Vladimir Kalashniov. Abstract We find an exact asymptotics of the rin probability Ψ() when the capital of insrance company is invested in a risy asset whose price follows a geometric Brownian motion with mean retrn a and volatility σ >. In contrast to the classical case of non-risy investments where the rin probability decays exponentially as the initial endowment tends to infinity, in this model we have, if ρ := 2a/σ 2 > 1, that Ψ() K 1 ρ for some K >. If ρ < 1, then Ψ() = 1. Key words: ris process, geometric Brownian motion, rin probabilities JEL Classification Nmbers: G22, G23 Mathematics Sbject Classification (2): 62P5, 6J25 1 Introdction It is well-nown that the prosperity of an insrance company is de not only to earnings in its principal bsiness bt also to intelligent investments of the money at its disposal. This is the reason why the modern trend in actarial mathematics is toward incorporating an economic environment into models, see, e.g., [9], [11], [6], and many others. Apparently, risy investment can be dangeros: disasters may arrive at the period when the maret vale of assets is low and the company will not be able to cover losses by selling these assets jst becase of price flctations. Reglators are rather attentive to this isse and impose stringent constraints on company portfolios. Typically, jn bonds are prohibited, a prescribed (large) part of the portfolio shold contain non-risy assets (e.g., Treasry bonds) while in the remaining part only risy assets with good ratings are allowed.

2 2 Anna Frolova et al. The common idea that investments in an asset with stochastic interest rate may be too risy for an insrance company can be jstified mathematically. In [4] it is noticed that in the classical Lndberg Cramér model the rin probability may decrease not as an exponential bt a power fnction if the wealth is invested in the stoc whose price follows a geometric Brownian motion. In the setting of [4] the ris process is Marov; the eqation for the exit (rin) probability can be redced to a differential eqation which belongs to a well-stdied class. In May 1999 the second athor had the pleasre of visiting the University of Copenhagen and discssing with Vladimir Kalashniov the topic of interest. Vladimir indly provided him with the manscript [5] containing pper and lower bonds allowing s to complete the stdy initiated in [4]. In the present wor we consider in detail the model of [4] and find an exact asymptotics for the rin probability. The conclsion: independently of the safety loading, the investments in an asset with large volatility lead to the banrptcy with probability one while for the small volatility the rin probability decreases as a power fnction. Kalashniov s bonds (developed frther in his joint wor with Ragnar Norberg [6]) play an important role in or stdy. Or techniqes is elementary. More profond and general reslts can be fond in [1], [7], and [8]. 2 The model We are given a stochastic basis with a Wiener process w independent of the integervaled random measre p(dt, dx) with the compensator p(dt, dx). Let s consider a process X = X of the form t t t X t = + a X s ds + σ X s dw s + ct xp(ds, dx), (1) where a and σ are arbitrary constants and c. We shall assme that p(dt, dx) = αdtf (dx) where F (dx) is a probability distribtion on ], [. In this case the integral with respect to the jmp measre is simply a compond Poisson process. It can be written as N t i=1 ξ i where N is a Poisson process with intensity α and ξ i are random variables with common distribtion F ; w, N, ξ i, i N, are independent. In or main reslt (Theorem 1) we assme that F is an exponential distribtion. Let τ := inf{t : Xt } (the date of rin), Ψ() := P (τ < ) (the rin probability), and Φ() := 1 Ψ() (the non-rin probability). The parameter vales a =, σ =, correspond to the Lndberg Cramér model for which the ris process is sally written as X t = +ct N t i=1 ξ i. In the considered version (of non-life insrance) the capital evolves de to continosly incoming cash flow with rate c and otgoing random payoffs ξ i at times forming an independent Poisson process N with intensity α. For the model with positive safety loading and F having a non-heavy tail, the Lndberg ineqality provides an encoraging information: the rin probability decreases exponentially as the initial capital tends to infinity. Moreover, for the exponentially distribted claims the rin probability admits an explicit expression, see [1] or [2].

3 In the insrance bsiness risy investments are dangeros 3 The more realistic case a >, σ =, corresponding to non-risy investments, does not pose any problem. We stdy here the case σ >. Now the eqation (1) describes the evoltion of the capital of an insrance company which is continosly reinvested into an asset with the price following a geometric Brownian motion (i.e. the relative price increments are adt + σdw t ) It is well-nown (see, e.g., the discssion in [9] for more general insrance models) that for the Marov process given by (1) the non-exit probability Φ() satisfies the following eqation: 1 2 σ2 2 Φ () + (a + c)φ () αφ() + α Φ( y)df (y) =. (2) With σ >, this eqation is of the second order and, hence, reqires two bondary conditions in contrast to the classical case (a =, σ = ) where it degenerates to an eqation of the first order reqiring a single bondary condition, see [2]. Theorem 1 Let F (x) = 1 e x/µ, x >. Assme that σ >. (i) If ρ := 2a/σ 2 > 1, then for some K > Ψ() = K 1 ρ (1 + o(1)),. (3) (ii) If ρ < 1, then Ψ() = 1 for all. The same model serves well in the sitation where only a fixed part γ ], 1] of the capital is invested in the risy asset (one shold only replace the parameters a and σ in (1) by aγ and σγ). The proofs will be given in Sections 5 and 4, respectively. Section 3 contains, in a certain sense, preliminary reslts which happen to be sefl to accomplish an analysis of soltions to the differential eqation for rin probability and obtain its exact asymptotics in the model of interest. For this reason we do not try to loo here for more delicate formlations and penetrate, e.g., into a specific strctre of coefficients to get rid of the logarithm in Proposition 1. In Section 4 we provide simple argments revealing the fact that for ρ < 1 the imbedded process is ergodic with the invariant measre charging the negative axes and, hence, leaves the positive half-axes with probability one. 3 Kalashniov s bonds Here we establish a reslt for generally distribted claims. Proposition 1 Let ρ := 2a/σ 2 > 1. (i) If Eξ ρ 1 1 <, then there exists a constant C sch that Ψ() C 1 ρ (ln ) 1 (ρ 1), 2. (4) ii) If P (ξ 1 > x) > for all x, then there are constants b, B, > sch that Ψ() b B,. (5)

4 4 Anna Frolova et al. Let τ n be the instant of the n-th jmp of N and let θ n := τ n τ n 1 with τ :=. We define the discrete-time process S = S with S n := X τn. Since the rin may occr only when X jmps downwards, Ψ() = P (T < ) where T := inf{n 1 : Sn }. Pt κ := a σ 2 /2 and wt n := w t+τn 1 w τn 1. Let s introdce the notations λ n := exp{σw n θ n + κθ n }, η n := c θn exp{σ(w n θ n w n ) + κ(θ n )} d. Solving the linear stochastic eqation we get that S n = λ n S n 1 + η n ξ n. (6) Ptting E n := Π n λ, we may se also the representation S n = E n + E n n Notice that λ n are i.i.d. random variables and Eλ ν 1 = E 1 (η ξ ). (7) α α + (1 ρ ν)νσ 2 /2. (8) We dedce Proposition 1 from reslts on the general discrete-time process given by (6) where (λ n, η n ) is a seqence of (two-dimensional) i.i.d. random variables, λ n >, and each ξ n is independent from the σ-algebra generated by the family {λ, η, ξ m, N, m N \ {n}}. In particlar, the assertion (i) follows immediately from (8) and Proposition 2 Let η n. Assme that Eλ β 1 = q β where q β < 1 if β ], β [ and q β = 1. If Eξ β 1 <, then there is a constant C sch that P (T < ) C β (ln ) 1 β, 2. (9) Proof. It is easily seen from the formla (7) that P (T < ) P (ζ > ) where ζ n := n ξ. Applying Lemma 1 below we get the reslt. E 1 Lemma 1 Let ζ n := n χ where χ and Eχ β l βqβ with q β < 1 if β ], β [ and q β = 1. Then there is a constant C sch that P (ζ > ) C β (ln ) 1 β, 2. Proof. Let M be a positive integer. Let β 1. Tae arbitrary β ], β [. Using the Chebyshev ineqality and taing into accont that x+y r x r + y r, r 1, we infer that ( ) β 2 M ( β 2 P (ζ M > /2) Eχ β ) l β M

5 In the insrance bsiness risy investments are dangeros 5 and, similarly, P (ζ ζ M > /2) ( ) β 2 ( ) β Eχ β 2 qβ M l β. 1 q β =M+1 Choosing M = M β as the integer part of (ln q β ) 1 ln β β, we get the reslt. Let β > 1. The first line above can be modified as follows: ( ) ( β M ) β 2 ( ) β 2 M ( β E χ M β 1 Eχ β 2 ) l β M β. Using the bond for the tail of the series with β = 1 and ptting M = M 1, we obtain the desired ineqality. The assertion (ii) is a corollary of the following general reslt. Proposition 3 Assme that the following conditions hold: (a) there exists a constant l < 1 sch that P (λ 1 l) > ; (b) P (ξ 1 > x) > for any x. Then there are b, B > sch that for all sfficiently large P (T < ) b B. (1) Proof. The assmption (a) implies that for some constants K > and p 1 > P (λ 1 l, η 1 ξ 1 K) = p 1. The assmption (b) and the independence of ξ 1 and (λ 1, η 1 ) imply that there are constants L > and p 2 > for which P (λ 1 L, η 1 ξ 1 2LK/(1 l)) = p 2. Let M := 1 + [(ln K(1 l) ln )/ ln l] where [.] denotes the integer part. Obviosly, l M K/(1 l). Define the sets and On the set A M A M := M {λ l, η ξ K}, D M+1 := {λ M+1 L, η M+1 ξ M+1 2LK/(1 l)}. S M = E M + E M M E 1 (η ξ ) l M + This implies that on the set A M D M+1 Ths, M l M K l M + S M+1 LS M 2LK/(1 l) L(l M K/(1 l)). P (T < ) P (A M D M+1 ) p 2 p M 1 K 1 l. 1+(ln K(1 l) ln )/ ln l p 2 p1 and we get the desired reslt with b = (ln p 1 )/ ln l. Remar. The exit probability for the soltion of the difference eqation (6) with random coefficients was stdied in [5] and [6]. The reslts of this section, althogh slightly different in formlations and proofs, are strongly inspired by these wors.

6 6 Anna Frolova et al. 4 Large volatility: the rin is imminent We show that the investments in a stoc with large volatility, namely, when ρ < 1, lead to a rin with probability one whatever is the initial capital. Clearly, it is sfficient to consider the case where a >. Inspecting the formla (8) we infer that Eλ ν 1 < 1 for certain ν ], 1[ and the reqired assertion follows from the general reslt below on the exit probability for the linear eqation (6). Proposition 4 Assme that the following conditions hold: (a) there is a constant ν ], 1[ sch that Eλ ν 1 = q < 1 and E η n ξ n ν < ; (b) P (ξ 1 > x) > for any x. Then P (T < ) = 1 for every. Proof. Pt E n := E n/e, S n (p) := n =n p+1 E n (η ξ ), and n (p) := S n S n (p), p N, p n. Then ( ) n p n (p) = En p n E n p + E n p (η ξ ) = En ps n n p. Since S n ν λ ν n S n 1 ν + η n ξ n ν and λ n and S n 1 are independent, we obtain from (a) that E S n ν < C and E n (p) ν < Cq p for some constant C. Let A n := {S n > }. For any ε > the set i m A pi is a sbset of ({S pi (p) > ε} { pi (p) > ε}) { pi (p) > ε} {S pi (p) > ε}. i m i m i m Since S pi (p), i = 1, 2,..., is a seqence of i.i.d. random variables, it follows that P (T = ) P ( i m A pi ) mcε ν q p + (P {S p (p) > ε}) m. (11) Notice that the distribtion of S p (p) coincides with the distribtion of p ϑ p := E 1 (η ξ ). As ϑ p is a partial sm of a series absoltely convergent in L ν, the seqence ϑ p converges a.s. to a finite random variable ϑ which taes negative vales with positive probability (becase ξ 1 is independent of all other random variables and satisfies (b)). Ths, taing the limit in p, we get that P (T = ) P (ϑ > ε) m. Choosing ε small enogh to ensre that P (ϑ > ε) < 1 and letting m tend to infinity, we obtain the reslt. Remar. One can extend the above argments and show that S is a Harris-recrrent, hence, ergodic process. The distribtion of ϑ is its invariant measre.

7 In the insrance bsiness risy investments are dangeros 7 5 Small volatility: decay of the rin probability Assme that the claims are exponentially distribted, i.e. F (x) = 1 e x/µ. Similarly to the classical case, this assmption allows s to obtain for the rin probability an ordinary differential eqation (bt of a higher order). Indeed, now the eqation (2) is 1 2 σ2 2 Φ () + (a + c)φ () αφ() + α µ Notice that d d Φ( y)e y/µ dy = Φ() 1 µ Φ( y)e y/µ dy =. (12) Φ( y)e y/µ dy. Differentiating (12) and exclding the integral term we arrive to a third order differential eqation. The good news is that it does not contain the fnction itself. In other words, we obtain a second order differential eqation for G = Φ which can be written as G + p()g + q()g =, (13) where p() := 1 ( µ a ) 1 σ 2 + 2c 1 σ 2 q() := 2a µσ ( a α + c µ 2, ) 2 1 σ 2 2. The sbstittion G() = R()Z(/(2µ)) with { R() := exp 1 2 eliminates the first derivative and yields the eqation where Z (1 + Q )Z = 1 } p(s)ds Q := 2 (1 a ) 1 4 σ A i i with certain constants A i which are of no importance. Notice that Q 2 is integrable at infinity and hence, according to [3], pp , the eqation has a fndamental soltion { ( Z ± () = exp ± + 1 )} Q r dr (1 + o(1)) = e ± ±(1 a/σ2) (1 + o(1)) 2 1 i=2

8 8 Anna Frolova et al. as. Since R() = e 1 2µ (1+a/σ2) f(), where f is a decreasing fnction on [1, [ bonded away from zero, f(1) = e 1 2µ, we obtain that (13) admits, as soltions, fnctions with the following asymptotics: G + () = 2a/σ2 (1 + o(1)), G () = 2 e 1 µ (1 + o(1)),. The differential eqation of the third order for Φ has the soltions Φ () = 1 and Φ + () = Φ () = r 2a/σ2 (1 + β 1 (r)) dr, r 2 e 1 µ r (1 + β 2 (r)) dr, where β i (r) as r. The rin probability Ψ := 1 Φ is the linear combination of these fnctions, i.e. Ψ() = C + C 1 Φ + () + C 2 Φ (). For the case ρ > 1 we now from Proposition 1 (i) that Ψ( ) =. Ths, Ψ() = C 1 r ρ (1 + β 1 (r)) dr + C 2 r 2 e 1 µ r (1 + β 2 (r)) dr. The first integral decreases at infinity as the power fnction 1 ρ /(1 ρ) while the second is exponentially decreasing. Bt Proposition 1 (ii) asserts that Ψ behaves at infinity as a power fnction. This implies that C 1 and we obtain the assertion (i) of Theorem 1. References 1. Asmssen S. Rin Probabilities. World Scientific, Singapore, Grandell I. Aspects of Ris theory. Springer, Berlin, Fedory M.V. Asymptotic analysis: linear ordinary differential eqations. Springer, Berlin, Frolova A.G. Some mathematical models of ris theory. All-Rssian School- Colloqim on Stoch. Methods in Geometry and Analysis. Abstracts, 1994, Kalashniov V. Rin probability nder random interest rate. Manscript, Kalashniov V., Norberg R. Power tailed rin probabilities in the presence of risy investments. Preprint. Laboratory of Actarial Math., Univ. of Copenhagen, Nyrhinen H. On the rin probabilities in a general economic environment. Stoch. Proc. Appl., 83 (1999), Nyrhinen H. Finite and infinite time rin probabilities in a stochastic economic environment. Stoch. Proc. Appl., 92 (21), Palsen J. Stochastic Calcls with Applications to Ris Theory. Lectre Notes, Univ. of Bergen and Univ. of Copenhagen, Palsen J. Sharp conditions for certain rin in a ris process with stochastic retrn on investments. Stoch. Proc. Appl., 75 (1998), Palsen J., Gjessing H. K. Rin theory with stochastic retrn on investments. Adv. Appl. Probab., 29 (1997), 4, Shiryaev A.N. Essentials of Stochastic Finance. World Scientific, Singapore, 1999.