# 5TH ACTUARIAL AND FINANCIAL MATHEMATICS DAY 5th Actuarial and Financial Mathematics Day PREFACE ...

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KONINKLIJKE VLAAMSE ACADEMIE VAN BELGIE

VOOR WETENSCHAPPEN EN KUNSTEN

5TH ACTUARIAL AND FINANCIAL MATHEMATICS DAY

February 9, 2007

Michèle Vanmaele, Griselda Deelstra, Ann De Schepper,

Jan Dhaene, Huguette Reynaerts, Wim Schoutens & Paul Van Goethem (Eds.)

CONTACTFORUM

KONINKLIJKE VLAAMSE ACADEMIE VAN BELGIE

VOOR WETENSCHAPPEN EN KUNSTEN

5TH ACTUARIAL AND FINANCIAL MATHEMATICS DAY

February 9, 2007

Michèle Vanmaele, Griselda Deelstra, Ann De Schepper,

Jan Dhaene, Huguette Reynaerts, Wim Schoutens & Paul Van Goethem (Eds.)

CONTACTFORUM

Handelingen van het contactforum "5th Actuarial and Financial Mathematics Day" (9 februari 2007, hoofdaanvrager: Prof. M. Vanmaele, UGent) gesteund door de Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten. Afgezien van het afstemmen van het lettertype en de alinea’s op de richtlijnen voor de publicatie van de handelingen heeft de Academie geen andere wijzigingen in de tekst aangebracht. De inhoud, de volgorde en de opbouw van de teksten zijn de verantwoordelijkheid van de hoofdaanvrager (of editors) van het contactforum.

KONINKLIJKE VLAAMSE ACADEMIE VAN BELGIE VOOR WETENSCHAPPEN EN KUNSTEN Paleis der Academiën Hertogsstraat 1 1000 Brussel

© Copyright 2007 KVAB D/2007/0455/00

Niets uit deze uitgave mag worden verveelvoudigd en/of openbaar gemaakt door middel van druk, fotokopie, microfilm of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever. No part of this book may be reproduced in any form, by print, photo print, microfilm or any other means without written permission from the publisher. Printed by Universa Press, 9230 Wetteren, Belgium

KONINKLIJKE VLAAMSE ACADEMIE VAN BELGIE

VOOR WETENSCHAPPEN EN KUNSTEN

5th Actuarial and Financial Mathematics Day PREFACE The fifth edition of the Contactforum “Actuarial and Financial Mathematics Day” attracted many participants, both researchers and practitioners. We welcomed this year several participants from abroad, indicating that this event is becoming to be known internationally. This contactforum aims to bring together young researchers, in particular Ph.D. students and Postdocs, working in the field of Financial and Actuarial Mathematics to discuss recent developments in the theory of mathematical finance and insurance and its application to current issues faced by the industry and to identify the substantive problems confronting academic researchers and finance professionals. We provide a forum for the discussion of advances within this field. In particular, we want to promote the exchange of ideas between practitioners and academics. Renowned practitioners were programmed as main speakers in order to give them a forum to talk about the needs, the problems, the hot topics in their fields. The invited paper about risk measures is included in these transactions. We thank all our speakers and discussants (Jasper Anderluh, Katrien Antonio, Griselda Deelstra, Henrik Jönsson, Nele Vandaele, Maarten Van Wieren, David Vyncke), for their enthusiasm and their interesting contributions which made this day a great success. We are also extremely grateful to our sponsors: the Royal Flemish Academy of Belgium for Science and Arts, and Scientific Research Network “Fundamental Methods and Techniques in Mathematics” of the Fund for Scientific Research - Flanders. They made it possible to spend the day in a very agreeable and inspiring environment. We plan a two day international event for the next meeting in 2008 with the focus on the interplay between finance and insurance.

Griselda Deelstra Ann De Schepper

Jan Dhaene Huguette Reynaerts

Wim Schoutens Paul Van Goethem Michèle Vanmaele

KONINKLIJKE VLAAMSE ACADEMIE VAN BELGIE

VOOR WETENSCHAPPEN EN KUNSTEN

5th Actuarial and Financial Mathematics Day CONTENTS

Invited talk

Capital allocation with risk measures...………………………………………………………. 3 A. Tsanakas Contributed talks

Control variates for callable LIBOR exotics. A preliminary study……...……………...….. 21 J. Buitelaar, R. Lord Hedging under incomplete information. Applications to emmissions markets...…………… 33 U. Cetin, M. Verschuere Hedging guarantees under interest rate and mortality risk….....................................……… 43 A. Chen, A.B. Mahayni Dealing with the volatility smile of Himalayan options...…………………………………... 55 J. Meaney An actuarial approach to short-run monetary equilibrium…… ..…………………….…….. 67 F. Mierzejewski Averaged bond prices in generalized Cox-Ingersoll-Ross model of interest rates..…..……. 77 B. Stehlíková

INVITED TALK

CAPITAL ALLOCATION WITH RISK MEASURES

Andreas Tsanakas

Faculty of Actuarial Science and Insurance, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK. Email: a.tsanakas.1@city.ac.uk

Abstract

This brief review paper covers the use of risk measures for assessing economic capital require- ments and considers the problem of allocating aggregate capital to sub-portfolios.

1. INTRODUCTION

A risk measure is a function that assigns real numbers to random variables representing uncertain pay-offs, e.g. insurance losses. The interpretation of a risk measure’s outcome depends on the context in which it is used. Historically there have been three main areas of application of risk measures:

• As representations of risk aversion in asset pricing models, with a leading paradigm the use of the variance as a risk measure in Markowitz portfolio theory, see Markowitz (1952).

• As tools for the calculation of the insurance price corresponding to a risk. Under this in- terpretation, risk measures are calledpremium calculation principles in the classic actuarial literature, e.g. Goovaerts et al. (1984).

• As quantifiers of the economic capital that the holder of a particular portfolio or risks should safely invest in e.g. Artzner et al. (1999).

This contribution is mainly concerned with the latter interpretation of risk measures. Theeconomic or risk capital held by a (re)insurer corresponds to the level of safely invested

assets used to protect itself against unexpected volatility of its portfolio’s outcome. One has to distinguish economic capital from regulatory capital, which is the minimum required economic capital level as set by the regulator. In fact, much of the impetus for the use of risk measures in the quantification of capital requirements comes from the area of regulating financial institutions.

3

4 A. Tsanakas

Banking supervision (Basel Committee on Banking Supervision) and, increasingly, insurance reg- ulation (European Commission) have been promoting the development of companies’internal models for modelling risk exposures. In that context, the application of a risk measure (most prominentlyValue-at-Risk) on the modelled aggregate risk profile of the insurance company is required.

Economic capital generally exceeds the minimum set by the regulator. Subject to that con- straint, economic capital is determined so as to maximise performance metrics for the insurance company, such as total shareholder return (Exley and Smith 2006). Such maximisation takes into account two conflicting effects of economic capital (Hancock et al. 2001):

• An insurance company’s holding economic capital incurs costs for its shareholders, which can be opportunity or frictional costs.

• Economic capital reduces the probability of default of the company as well as the severity of such default on its policyholders. This enables the insurance company to obtain a better rating of its financial strength and thereby attract more insurance business at higher prices.

Calculation of the optimal level of economic capital using such arguments is quite complicated and depends on factors that are not always easy to quantify, such as frictional capital costs, and on further constraints, such as the ability of an insurance company to raise capital in a particular economic and regulatory environment.

We could however consider that there is a particular calibration of the (regulatory or other) risk measure, which gives for the insurance company’s exposure a level of economic capital that coincides with that actually held by the company. In that sense risk measures can be used to interpret exogenously given economic capital amounts. Such interpretation can be in the context of capital being set to achieve a target rating, often associated with a particular probability of default. Discussion of economic capital in the context of risk measures should therefore be caveated as beingex-post.

Finally, we note that the level of economic capital calculated by a risk measure may be a notional amount, as the company will generally not invest all its surplus in risk-free assets. This can be dealt with by absorbing the volatility of asset returns in the risk capital calculation itself.

2. DEFINITION AND EXAMPLES OF RISK MEASURES

We consider a set of risksX that the insurance company can be exposed to. The elementsX