Stochastic Processes and their Applications 98 (2002) 211 – 228
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Power tailed ruin probabilities in the presence
of risky investments
Vladimir Kalashnikova;1 , Ragnar Norbergb;∗; 2
a Laboratory
of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5,
DK-2100 Copenhagen I, Denmark
b Department of Statistics, London School of Economics, Houghton Street, WC2A 2AE London, UK
Received 8 March 2000; received in revised form 23 October 2001; accepted 9 November 2001
Abstract
The present paper addresses the situation where the reserve of an insurance business is currently invested in an asset that may yield negative interest. Upper and lower bounds for the
probability of ruin are obtained in the case where the cash 0ow of premiums less claims and
the logarithm of the asset price are both L2evy processes. These bounds are in general power
c 2001 Elsevier Science B.V. All rights reserved.
functions of the initial reserve. MSC: 60K99; 60G70
Keywords: Probability of ruin; Stochastic interest; L2evy processes; Two-sided bounds
1. Introduction
1.1. Background and scope of the present study
The aim of this paper is to show that the probability of ultimate ruin decreases slowly
(not faster than a power function) if the reserve is currently invested in a risky asset,
that is, an asset that may bear negative interest. The result pertains to the frequently
asked question “which is more important, insurance risk or :nancial risk?” and states
∗
Corresponding author. Fax: +44-207955-7416.
E-mail address: [email protected] (R. Norberg).
1 This work was partly supported by the Russian Foundation for Basic Research (grant 98-01-00855) and
INTAS (grant 98-1625).
2 This work was partly supported by the Mathematical Finance Network under the Danish Social Science
Research Council, Grant No. 9800335.
c 2001 Elsevier Science B.V. All rights reserved.
0304-4149/01/$ - see front matter PII: S 0 3 0 4 - 4 1 4 9 ( 0 1 ) 0 0 1 4 8 - X
212 V. Kalashnikov, R. Norberg / Stochastic Processes and their Applications 98 (2002) 211 – 228
that risky investments may impair the insurer’s solvency just as severely as do large
claims, roughly speaking.
A breakthrough in the studies of ruin problems with compounding assets was made
with the seminal paper of Harrison (1977). Working with :xed, positive interest, he
observed that the discounted value of the cash 0ow is a well de:ned random variable
and discovered a fundamental relationship between its distribution and the probability
of ruin, leading to entirely new techniques for determining the latter. Closed form
expressions were obtained in the special case with Brownian motion driven cash 0ow.
The case is discussed in Norberg (1999); with no interest the probability of ruin
is exponential, whereas with positive interest (which can only help, of course) it is
super-exponential. Paulsen (1993,1998) extended the theory to situations with stochastic
interest and showed that possible losses on investments may radically alter the picture,
see also Norberg (1999); in the case where the cash 0ow and the log accumulation
factor are independent Brownian motions with drift the probability of ruin is explicitly
determined and behaves essentially like a power function. The same feature presents
itself in a recent paper by Nyrhinen (1999), who employs large deviations techniques
in a discrete time set-up and determines the asymptotic form of the probability of ruin
by increasing initial reserve. The previously announced results of the present paper
provide a general explanation for these :nds.
1.2. Outline of the paper
We set out in Section 2 by describing the framework model, a bivariate L2evy process scenario, and de:ne the probability of ruin. In Section 3 we derive, by a direct
probabilistic argument, a power function lower bound for the probability of ruin. The
bound is coarse, but valid for all values of the initial reserve above a certain known
minimum. In Section 4 we obtain two-sided bounds for the ruin probability by use of
Cram2er techniques in conjunction with the previous result.
2. Framework model and ruin probabilities
2.1. The bivariate L7evy driven risk process
A real-valued continuous time stochastic process is said to be L2evy if it commences
at 0 at time 0, has independent stationary increments, and is continuous in probability.
The archetype L2evy processes are the trivial straight line through (0,0), Brownian
motion, compound Poisson processes, and a vast family of pure jump processes with
in:nitely many jumps in every :nite time interval (e.g. the gamma process). Any
:nite linear combination of independent L2evy processes is itself L2evy. For an account
of L2evy processes, see Bertoin (1996).
We consider an insurance business commencing at time 0 (say) with an initial reserve
x ¿ 0, and we are interested in its future development. The cash 0ow of premiums less
claims is described by a L2evy process, P. In particular, it could be the classical risk
V. Kalashnikov, R. Norberg / Stochastic Processes and their Applications 98 (2002) 211 – 228 227
is :nite for suTciently large s ¿ 0. Then (4.27) becomes
E
∞
1
P
(R bR (1)V ) j e−R V −1rV =
= 1;
j!
P + 1r + R (1 − bR (1))
j=0
which reduces to
1r + R (1 − bR (1)) = 0:
A unique positive solution exists if and only if BR (0−) ¿ 0 and
R
d
(1 − bR (1))|1=0 = R EYR ¿ − r:
d1
(4.42)
For instance, if BR is an atomic distribution assigning probabilities ;k ¿ 0 to points
<k , k = 1; : : : ; m, then (4.42) becomes
r + R
m
<k ; k ¿ 0
k=1
and BR (0−) ¿ 0 holds if and only if <k ¡ 0 for at least one k. Assume that the claim
size YP is not identically 0 and EYP1 ¡ ∞. Under these assumptions, the relations
(4.32), (4.38), and (4.40) hold as all other conditions mentioned in Theorems 2 and
3 are automatically satis:ed.
Acknowledgements
We thank Angelos Dassios, Yurij Kabanov, Deimante Rusaityte, and particularly an
anonymous referee for useful discussions and remarks that helped to improve the text
and correct some inaccuracies.
References
Bertoin, J., 1996. L2evy Processes. Cambridge University Press, Cambridge.
Embrechts, P., KlZuppelberg, C., Mikosch, T., 1997. Modelling Extremal Events. Springer, Berlin.
Embrechts, P., Veraverbeke, N., 1982. Estimates for the probability of ruin with special emphasis on the
possibility of large claims. Insurance Math. Econom. 1, 55–72.
Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.
Goldie, C.M., 1991. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab.
1, 126–166.
Grandell, J., 1991. Aspects of Risk Theory. Springer, Berlin.
Grandell, J., 1998. Simple approximations of ruin probabilities. Preprint of Department of Mathematics,
Royal Institute of Technology, Trita-Mat-1998-01.
Harrison, J.M., 1977. Ruin problems with compounding assets. J. Stochastic Proc. Appl. 5, 67–79.
Kalashnikov, V., 1996. Two-sided bounds of ruin probabilities. Scand. Actuar. J. 1996, 1–18.
Kalashnikov, V., 1997. Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic
Publishers, Dordrecht.
228 V. Kalashnikov, R. Norberg / Stochastic Processes and their Applications 98 (2002) 211 – 228
Kesten, H., 1973. Random diSerence equations and renewal theory for products of random matrices. Acta
Math. 131, 207–248.
Norberg, R., 1999. Ruin problems with assets and liabilities of diSusion type. J. Stochastic Proc. Appl. 81,
255–269.
Nyrhinen, H., 1999. On the ruin probabilities in a general economic environment. J. Stochastic Proc. Appl.
83, 319–330.
Paulsen, J., 1993. Risk theory in a stochastic environment. J. Stochastic Proc. Appl. 21, 327–361.
Paulsen, J., 1998. Ruin theory with compounding assets—a survey. Insurance Math. Econom. 22, 3–16.