Optimal Risk Classification and Underwriting Risk for Substandard Annuities

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Erlage-Nürberg Gudru Hoerma, Muich Hato Schmeiser,

1 1 Optimal Risk Classificatio ad Uderwritig Risk for Substadard Auities Nadie Gatzert, Uiversity of Erlage-Nürberg Gudru Hoerma, Muich Hato Schmeiser, Istitute of Isurace Ecoomics, Uiversity of St. Galle

2 2 Ageda Itroductio The model framework Basic model Optimal risk classificatio Optimal risk classificatio ad costs of uderwritig risk Market etry barriers ad advatages Summary

3 3 Itroductio Motivatio Substadard auities offer icreased pesio paymets for idividuals with below-average life expectacy - Surprisigly rare except for i the U.K. market - Risk classificatio geerally icreases profitability (see Doherty, 1981), e.g., o-life - Private pesios for perso's with impaired health - Reluctace of isurace compaies to offer substadard auity products

4 4 Itroductio Motivatio Sellig substadard auities is a challegig task: - Establish classificatio system based o isured's life expectacy - Adequate uderwritig guidelies are ecessary Uderwritig criteria: medical coditios or lifestyle factors - Iclude classificatio costs whe pricig the cotract - Demad for product is determied by auity amout

5 5 Itroductio Motivatio Aim of this paper: - Comprehesive aalysis with respect to substadard auities - Combie two strads of literature: substadard auities ad risk classificatio - Develop a model to determie optimal profit-maximizig risk classificatio system for substadard auities - Crucial: accout for classificatio costs ad uderwritig risk - For (prospective) providers ad stadard isurers

6 6 Ageda Itroductio The model framework Basic model Optimal risk classificatio Optimal risk classificatio ad costs of uderwritig risk Market etry barriers ad advatages Summary

7 7 The Model Framework Basic Model Geeral populatio of potetial risks with average populatio mortality Mortality heterogeeity i the geeral populatio cosidered by meas of a frailty model Idividual probabilities of death by applicatio of a stochastic frailty factor to the populatio mortality table d q x, d qx < 1 qx d x x d qx x, otherwise ( ) = 1, = mi % {, K, ω} : % 1 for {, K, ω}

8 8 The Model Framework Basic Model Frailty factor specifies idividual's state of health Frailty distributio represets distributio of differet states of health (differet life expectacies) i the geeral populatio No-egative Cotiuous Flat at, right-skewed Expected value of 1

9 9 The Model Framework Basic Model Subpopulatio Frailty Distributio tio la u p o l P ra e e f G o e g ta e rc e P 1 H Low Mortality Subpopulatio h High Mortality (Frailty Factor)?? (Frailty Factor) H differet subpopulatios with differig mortality level

10 1 The Model Framework Basic Model Characteristics of subpopulatio h, h=1,,h Number of risks N h Price-demad fuctio f h () R P h P Cost Fuctio Price-Demad Fuctio fh ( ) Mootoously decreasig Reservatio price P h R decreases with icreasig h: P1 > P2 > L > P H f h (N h ) = R R R N h

11 11 The Model Framework Basic Model Cost fuctio g h () = P h A Actuarial premium for coverig the cost of (oe uit of) auity isurace for the average potetial isured i subpopulatio h) Depedig o the average frailty factor d h R P h A P h P g ( ) Cost Fuctio h Price-Demad Fuctio fh N h ( )

12 12 The Model Framework Optimal Risk Classificatio Subpopulatio Frailty Distributio tio la u p o l P ra e e f G o e g ta e rc e P 1 H Low Mortality Subpopulatio h High Mortality (Frailty Factor)?? I m risk classes i=1,, I m i classificatio system m M set of all possible classificatio systems m (Frailty Factor)

13 13 The Model Framework Optimal Risk Classificatio P R P1 Aggregate cost ad price-demad fuctio i risk class i cosistig of 2 subpopulatios Cost Fuctio g ( ) Price-Demad Fuctio 1 f1 ( ) P Cost Fuctio g2 ( ) Price-Demad Fuctio f2 ( ) R P1 P Cost Fuctio g Price-Demad Fuctio i ( ) fi ( ) R P2 R P2 A P1 A P2 A P1 N 1 N 2 N + N 1 2

14 14 The Model Framework Optimal Risk Classificatio Aggregate cost ad price-demad fuctio i risk class i, geeral formulas ν 1 1 fi ( ) = fi ( fi ( Pi) ) = fi fs ( Pi) s= 1 ν ν ν A i ( ) = s ( i ( )) s( i ( )) = s ( i) s( i) = s s s= 1 s= 1 s= 1 g f f g f f P g P P if, for = 1,..., 1 ν+ 1 R R ( ν ) ( ν+ 1) ν 1 1 fs P fs P ν Si s= 1 s= 1 Iν = ν 1 R ( ν ) f, for ν = i. s P N S i s= 1

15 15 The Model Framework Optimal Risk Classificatio Profit i risk class i Classificatio costs Total profit from classificatio system m Optimizatio problem ( ) ( ) ( ) ( ) ( ) Π = E C = f g, = 1,..., N i i i i i i k( I m 1) Im ( 1,..., I ) i ( i) k( Im 1) Π = Π m {( 1,..., )} i = 1 (,..., ) I max max I m Π 1 m M m

16 16 The Model Framework Optimal Risk Classificatio Maximizatio i recurret steps: Fid optimal price-demad combiatios for each risk class i withi each classificatio system m M = argmax Π ν ( ) ν * i i i ν I ν, ν = 1, K, i S i R P i P ( ) Cost Fuctio gi Price-Demad Fuctio fi ( ) * m M ( 1 ) m = argmax Π,..., I m * P i A P i Π i * ( i ) * i N i

17 17 The Model Framework Optimal Risk Classificatio & Costs of Uderwritig Risk Uderwritig risk is oe of the mai reasos why isurers are reluctat to egage i risk classificatio Ex.: Effect of uderwritig errors for two risk classes Uderestimate true cost of isurace R P i * P i * P j A P i P Π i * ( i ) Π i ( ij) ( ) Cost Fuctio gi Price-Demad Fuctio fi ( ) R P j * P j A P j P ( ) Cost Fuctio g j Price-Demad Fuctio f j ( ) * i ij N i * j N j

18 18 The Model Framework Optimal Risk Classificatio & Costs of Uderwritig Risk Expected profit i risk class i, give probability p ij of wrogly classifyig (risk class j istead of i ) Error probability depeds o classificatio system: Larger umber of risk classes => smaller differeces betwee risk classes => higher error probabilities for adjacet classes, but smaller effects of wrog classificatio Exteded optimizatio problem: ( ) % p Π i ij i ij j i Π = m M ( ) 1 ** m = argmax Π %,..., I m

19 19 Ageda Itroductio The model framework Basic model Optimal risk classificatio Optimal risk classificatio ad costs of uderwritig risk Market etry barriers ad advatages Summary

20 2 Market etry barriers ad advatages Market etry barriers Classificatio costs Uderwritig risk Market very competitive Competitio by other fiacial products Low market awareess requires strog distributio system Attractive ad iovative product desig eeded Impact o existig portfolios

21 21 Market etry barriers ad advatages Advatages Huge market potetial Attractive for ew market players Possibility to reach broader populatio or iche markets Early ivolvemet prevets defesive reactios Beefit of competitive advatage Avoidace of adverse selectio problems

22 22 Ageda Itroductio The model framework Basic model Optimal risk classificatio Optimal risk classificatio ad costs of uderwritig risk Market etry barriers ad advatages Summary

23 23 Summary Propose a model for a risk classificatio system i a mortality heterogeeous geeral populatio Solve for optimal umber ad size of risk classes Profit-maximizig price-demad combiatio i each risk class Extesio: accout for costs of uderwritig risk i optimizatio Discuss market etry barriers ad advatages I summary: Risk classificatio i auity markets ot oly icreases profitability of isurace compaies but also beefits society at large, sice formerly uisurable persos gai access to private pesios

24 24 Optimal Risk Classificatio ad Uderwritig Risk for Substadard Auities Thak you very much for your attetio! Nadie Gatzert, Uiversity of Erlage-Nuremberg Gudru Hoerma, Muich Hato Schmeiser, Istitute of Isurace Ecoomics, Uiversity of St. Galle

Subject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.

Subject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries. Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical