Insurance: Mathematics and Economics 26 2000 59–73
Ruin theory with risk proportional to the free reserve and securitization
Thomas Siegl
a, ∗
, Robert F. Tichy
b
a
Arthur Andersen, Risikomanagement Beratung, Mergenthalerallee 10-12, D-65760 Eschborn, Germany
b
Institut für Mathematik, TU-Graz, Steyrergasse 30II, A-8010 Graz, Austria Received January 1999; received in revised form August 1999
Abstract
A model is proposed for addressing investment risk of the free reserve, in the form of credit or currency risk. This risk is expressed by a constant factor α that represents the recovery rate of a bond or a devaluation factor. Securitization e.g.
with a CAT-bond like product yields a constant amount K upon such an event. The model equation is an integro-differential equation with deviating arguments. We compute the analytical solution for the probability of survival and also show results
of simulations using quasi-Monte Carlo methods. ©2000 Published by Elsevier Science B.V. All rights reserved.
MSC: 62P05; 34K10
Keywords: Ruin theory; Credit risk; Currency risk; Securitization; Deviating arguments; Quasi-Monte Carlo method
1. Introduction
We extend the classical ruin model with the two main components of premium income and claims payment. The third component reflects a risk proportional to the free reserve such as a currency devaluation or credit event. In the
later case we assume that the free reserve x is invested in bonds with credit risk. Upon a credit event the value of the free reserve drops according to the recovery rate α from x to αx. Then the free reserve is reinvested in bonds
with similar risk.
Another application lies in modeling currency devaluation. We assume that the free reserve x
A
of the insurance is invested in currency A and that claims and premiums are paid in currency B. Let x denote the equivalent to x
A
in currency B. If currency A is devaluated with respect to currency B, the value of the free reserve x
A
with respect to the ability to pay claims changes from x in currency B to αx in currency B, or x
→ αx for short. Furthermore, we assume that the insurer will hedge against these events by e.g. a securitization bond where the
notional is at risk. We model this hedge by a constant payment K the bond notional that is added to the free reserve immediately after the event x
→ αx + K. The premium for this embedded option is paid for by a continuous payment of a part
ˆc of the total premium income ˜c in the form of a spread over the applicable interest rate a larger
∗
Corresponding author. E-mail addresses:
thomas.sieglt-online.de T. Siegl, tichyweyl.math.tu-graz.ac.at R.F. Tichy. 0167-668700 – see front matter ©2000 Published by Elsevier Science B.V. All rights reserved.
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60 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
regular coupon. The bond is re-emitted after the event. The proposed model may help in choosing the optimal terms of such a risk exchange.
In the following we will present the model for arbitrary claim size distributions F and prove existence and uniqueness for the general case, while an analytic solution is found for the special case F y
= 1 − exp−y. Furthermore, we compare the performance of hybrid and pseudo-Monte Carlo sequences for this case. We will
show that a significant improvement by a factor of 2–3 is possible as far as the required sample size for a given error is concerned.
2. Model with upper barrier
