Sensitivity analysis for ruin probabilities: canonical risk model
- PDF / 171,303 Bytes
- 11 Pages / 595 x 842 pts (A4) Page_size
- 38 Downloads / 219 Views
#2001 Operational Research Society Ltd. All rights reserved. 0160-5682/01 $15.00 www.palgrave.com/jors
Sensitivity analysis for ruin probabilities: canonical risk model FJ VaÂzquez-Abad* and P LeQuoc University of Montreal, Canada The surplus process of an insurance portfolio is de®ned as the wealth obtained by the premium payments minus the reimbursements made at the time of claims. When this process becomes negative (if ever), we say that ruin has occurred. The general setting is the Gambler's Ruin Problem. In this paper we address the problem of estimating derivatives (sensitivities) of ruin probabilities with respect to the rate of accidents. Estimating probabilities of rare events is a challenging problem, since naõÈve estimation is not applicable. Solution approaches are very recent, mostly through the use of importance sampling techniques. Sensitivity estimation is an even harder problem for these situations. We shall study three methods for estimating ruin probabilities: one via importance sampling (IS), and two others via indirect simulation: the storage process (SP), which restates the problems in terms of a queuing system, and the convolution formula (CF). To estimate the sensitivities, we apply the Rare Perturbation Analysis (RPA) method to IS, the In®nitesimal Perturbation Analysis (IPA) method to SP and the score function method to CF. Simulation methods are compared in terms of their ef®ciency, a criterion that appropriately weighs precision and CPU time. As well, we indicate how other criteria such as set-up time and prior formulae development may actually be problem-dependent. Keywords: risk; ruin probabilities; sensitivity analysis; rare event simulation; importance sampling; estimation ef®ciency
Introduction: ruin probabilities The canonical model in risk theory assumes that claims due to accidents arrive according to a Poisson process N
t of rate l. The successive claim amounts, denoted fYi g, are i.i.d. random variables with general distribution G and premiums are received at a constant rate c. If the initial endowment is u > 0, the wealth of the insurance company, known as the surplus process is: U
t u ct
N
t P i1
Yi ; t 5 0
1
The event epochs of the process N
t are denoted by fTn ; n 5 0g; and Wn Tn Tn 1 are the Pinterarrival
t times. The cumulative claims process S
t Ni1 Yi is a compound Poisson process. We will often write Un U
TP n to denote the embedded discrete event process and Sn ni1 Yi with an obvious abuse of notation. If we set t minfn: U
Tn < 0g minfn: u cTn 4 Sn g then the ruin probability is c
u; l Pft < 1g and it is a measure of the credit risk of the company. Call b EY1 . If c 4 lb, then c
u; l 1 for all initial endow*Correspondence: Dr FJ VaÂzquez-Abad, Department of Computer Science and Operations Research, University of Montreal, MontreaÂl, Quebec H3C 3J7, Canada. E-mail: [email protected]
ment u.1 As a consequence of this result, it is common to assume that premiums satisfy c > lb, which we shall do. We brie¯y re
Data Loading...
