__________________________________________________________________________________________________________________________________________________________________________________________________ Optimal Decisions in Finite State Markov Chains: Applications to Personal Finance and Credit Risk Management Mogens Ste¤ensen Linz October 24, 2008 _________________________________________________________________________________________________________________________________________________________________________________________________ 1 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Picture Of A Break, Festival, 1892 G. Laska (19th C. Czech), Oil On Canvas Private Collection, Munich, Germany _________________________________________________________________________________________________________________________________________________________________________________________________ 2 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Based on (all articles joint with Holger Kraft, Goethe-University, Frankfurt) � Optimal Consumption and Insurance: A Continuous-Time Markov Chain Ap- proach ASTIN Bulletin 2008 � The Policyholder’s Static and Dynamic Decision Making of Life Insurance and Pension Payments. Blätter der DGVFM 2008. � Bankruptcy, Counterparty Risk, and Contagion. Review of Finance 2007 � Asset Allocation with Contagion and Explicit Bankruptcy Procedures. Journal of Mathematical Economics 2008 _________________________________________________________________________________________________________________________________________________________________________________________________ 3 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ � Intro � Life Insurance Modelling � Credit Risk Modelling � Life Insurance Decision Making � Optimal Credit Risky Portfolios � Outtro _________________________________________________________________________________________________________________________________________________________________________________________________ 4 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Life insurance Modelling � 0 alive 1 dead ! Figure 1: Survival model N ( t ) = # f s 2 (0 ; t ] ; Z ( s � ) 6 = 0 ; Z ( s ) = 1 g Z t M ( t ) = N ( t ) � 0 I ( s ) � ( s; Y ( s )) ds dB ( t ) = ’death sum’ dN ( t ) + I ( t ) ’pension sum’ d" ( t; n ) � I ( t ) ’premium rate’ dt _________________________________________________________________________________________________________________________________________________________________________________________________ 5 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ � � 0 active/employed 1 disabled/unemployed � � � & . 2 dead Figure 2: Disability/Unemployment model N k ( t ) = # f s 2 (0 ; t ] ; Z ( s � ) 6 = k; Z ( s ) = k g Z t M k ( t ) N k ( t ) � 0 � k ( s; Z ( s ) ; Y ( s )) ds = � � X b Z ( t � ) k dN k ( t ) + � B Z ( t ) ( t ) d" ( t; n ) + b Z ( t ) ( t ) dt dB ( t ) = k : k 6 = Z ( t � ) _________________________________________________________________________________________________________________________________________________________________________________________________ 6 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ 0 husband and wife alive � � . & 1 husband dead, wife alive 2 husband alive, wife dead # � � 0 � 0 & . 3 husband and wife dead Figure 3: Dependent lives ’Weak’ dependence induced by Y ’Medium’ dependence induced by e.g. � 0 > � or � 0 < � ’Strong’ dependence induced by � > 0 _________________________________________________________________________________________________________________________________________________________________________________________________ 7 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Credit Risk Modelling Everybody knows by now what a ’survival model’ looks like so therefore I show you instead a... _________________________________________________________________________________________________________________________________________________________________________________________________ 8 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ ...’Default model’ � 0 solvent 1 default ! Figure 4: Default model N ( t ) = # f s 2 (0 ; t ] ; Z ( s � ) 6 = 0 ; Z ( s ) = 1 g Z t M ( t ) = N ( t ) � 0 I ( s ) � ( s; Y ( s )) ds dB ( t ) = ’recovery payment’ dN ( t ) + I ( t ) ’lump sum coupon’ d" ( t; n ) + I ( t ) ’coupon rate’ dt _________________________________________________________________________________________________________________________________________________________________________________________________ 9 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Everybody knows by now what a ’disability model’ looks like so therefore I show you instead a... _________________________________________________________________________________________________________________________________________________________________________________________________ 10 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ ...’Corporate rating model’: � � 0 rating A 1 rating B � � � & . 2 default Figure 5: Rating model N k ( t ) = # f s 2 (0 ; t ] ; Z ( s � ) 6 = k; Z ( s ) = k g Z t M k ( t ) N k ( t ) � 0 � k ( s; Z ( s ) ; Y ( s )) ds = � � X b Z ( t � ) k dN k ( t ) + � B Z ( t ) ( t ) d" ( t; n ) + b Z ( t ) ( t ) dt dB ( t ) = k : k 6 = Z ( t � ) _________________________________________________________________________________________________________________________________________________________________________________________________ 11 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Everybody knows by now what a ’dependent lives model’ looks like so therefore I show you instead a... _________________________________________________________________________________________________________________________________________________________________________________________________ 12 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ ...’Dependent corporates model’: 0 both corp.s solvent � � . & 1 A default, B solvent 2 A solvent, B default # � � 0 � 0 & . 3 both corp.s default Figure 6: Dependent corporates ’Weak’ dependence induced by Y ’Medium’ dependence induced by e.g. � 0 > � or � 0 < � ’Strong’ dependence induced by � > 0 _________________________________________________________________________________________________________________________________________________________________________________________________ 13 of 25
__________________________________________________________________________________________________________________________________________________________________________________________________ Further comments to (lack of) analogies � Evolutionary …nance (life cycle, fertility, infection, adaptation etc.) � Mortality linked derivatives ( N or Y ) versus credit derivatives ( N or Y ) � Statistical inference versus calibration _________________________________________________________________________________________________________________________________________________________________________________________________ 14 of 25
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