Sharing Risk An Economic Perspective 36th ASTIN Colloquium, Zurich, Andreas Kull, Global Financial Services Risk Management

Transcription

1 Sharng Rsk An Economc Perspectve 36th ASTIN Colloquum, Zurch, Andreas Kull, Global Fnancal Servces Rsk Management q

2 Captal: Shared and competng ssue Assets Captal Labltes Rsk Dmenson Rsk Dmenson Return Dmenson Regulators Captal s key for protecton of polcyholder and for overall fnancal stablty Polcyholders Captal acts as rsk mtgaton Shareholders Expect a return from nvested captal Rsk/Return Dmenso n Management Captal s the key resource for dong busness Rsk transfer strateges should balance rsk and return dmenson by takng nto account 1. Captal constrants (regulators, polcyholders) 2. Economc bottom lne,.e. mpact of rsk transfer on overall return (shareholders, management) 1

3 2 Sharng and Transferrng of Rsk Rsk transfer should balance rsk/return dmensons by takng nto account Captal constrants (regulators, polcyholders) Economc bottom lne, e. mpact of rsk transfer on overall return (shareholders, management) A sound model of rsk transfer and rsk sharng should Allow for relevant constrants (e.g. avalable captal, rsk tolerance) Focus on economc value Include effects of portfolo dversfcaton and dependency ssues Allocate rsk based captal consstently What f rsk exchangng partes employ smlar prncples? Consstently model economcs of rsk transfer (premum prncple) Take nto account portfolo structure of both rsk cedng and recevng party Key ssue: Sharng rsk optmally n ths context

4 What rsks should be shared/transferred? Key parameter Party A - Cost of captal - Portfolo structure - Rsk tolerance level Transfer rsk Key parameter Party B - Cost of captal - Portfolo structure - Rsk tolerance level Rsk A 1 Rsk B 1 Rsk A 2 Rsk A 3 Rsk B 2 Rsk B 3 3 Portfolo A Objectve Party A - Transfer rsk - Meet captal constrant - Maxmze overall return n exchange for premum What rsk for whch premum? Portfolo B Objectve Party B - Take over rsk n exchange for premum - Attan a proftablty hurdle

5 A Smple Model (I) Net underwrtng proft (rsk class ) Economc result (rsk class ) Rsk based captal (rsk class ) Premum for rsk transfer (rsk class ) quanttes wth refer to rsk recevng party Portfolo dstrbuton, Total RBC Objectve and constrant P U Z RBC[ Z = P = U λ RBC[ Z ( L R ) P 1 ] = E[ Z Z FZ ( α)] 1 = (1 λ ) E[ R ] + E[ R R F ( α )] Z Z RBC[ Z] = Z = RBC[ Z ], E[ Z( r)] = max!, RBC[ Z( r)] = const ] P: Premum, L: Loss, P: Rsk transfer premum, R: Loss covered by rsk transfer. RBC: Rsk based captal, λ: cost of captal α: Rsk tolerance level, RBC: Rsk based captal (defned as condtonal expected shortfall relatve to portfolo result) P: Premum for rsk transfer, R: loss covered by rsk transfer, λ: Cost of captal, α: Rsk tolerance level Note addtvty of RBC r: Rsk transfer control parameter(s), eg XL retentons 4

6 A Smple Model (II) Typcal constrant extreme value problem Solved usng Lagrangan multplers κ,.e. by maxmzng expresson r r r φ(r) = E[ Z(r) ] + κ RBC[ Z( )] Ths leads to condton r [( λ λ + κ (1 λ )) E[ R] Rsk recevng portfolo λ (1 κ) E[ R R F ( λ κ) E[ R R F ( α )] + ( α)]] = 0 Other condtons nclude concavty of E[Z(r)] and convexty of κ E[Z(r)]. 1 Z 1 Z! Rsk cedng portfolo 5

7 A Smple Model (IV) r r = 0 s a vector of optmal rsk transfer control parameters (e.g. XL retentons) characterzes optmal rsk transfer per rsk class maxmzes overall economc result on portfolo level for a gven rsk based captal RBC 0. Soluton Man parameters Structure of rsk transferrng and recevng portfolos (Z and Z ) Rsk tolerance levels (α and α ) Captal costs (λ and λ ) avalable rsk based captal (RBC 0, related to Lagrangan parameter κ) Numercal soluton necessary 6

8 Example Illustraton for smple proportonal and non-proportonal rensurance structures Portfolo structures (smple compound Posson processes) Severtty Frequency Type µ σ λ Rsk class A 1 Normal Rsk class A 2 LogNormal Rsk class B 1 Normal Rsk class B 2 LogNormal Rsk tolerance levels and captal costs Portfolo A Portfolo B Rsk Tolerance 10% 10% Cost of Captal 9% 14% 7

9 Example Proportonal Rensurance (I) 8

10 Example Proportonal Rensurance (II) RBC=RBC 0 (const.) Optmal Retentons 9

11 Example Non-Proportonal Rensurance (I) 10

12 Example Non-Proportonal Rensurance (II) RBC=RBC 0 (const.) Optmal Retentons 11

13 Conclusons Sharng rsk and dversfcaton s crucal Workable scheme for companes that have nternal models n place Optmal also for rsk recevng party? Extend the framework to n rsk sharng partes 12

14 Contact Dr. Andreas Kull Ernst & Young Ltd Global Fnancal Servces Rsk Management Brandschenkestrasse 100 CH-8002 Zürch Tel: Emal: 13