HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES

Transcription

1 HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES Workshop on moraliy and longeviy, Hannover, April 20, 2012 Thomas Møller, Chief Analys, Acuarial Innovaion

2 OUTLINE Inroducion Moraliy risk managemen Valuaion Moraliy risk Unsysemaic risk Sysemaic risk Hedging moraliy risk Survivors swaps A modeling framework Illiquid asses Risk-minimizaion Inheren risk and survivor swaps page 2

3 RISK MANAGEMENT IN LIFE INSURANCE Curren siuaion: Marke based valuaion of asses and liabiliies Financial risks: Parly conrolled via invesmens, derivaives ec Moraliy and longeviy risk: An essenial non-hedged and non-hedgeable (?) risk for pension funds Solvency II: Capial requiremens derived from properies for acual producs, inheren risks and invesmens Capial requiremens and risk margins for life annuiies is o be calculaed using a 20 percen reducion in moraliy raes Moraliy derivaives could lead o lower capial requiremens! (e.g. survivor swaps, survivor bonds, oher consrucions) page 3

4 VALUATION AND MORTALITY RISK MANAGEMENT Key issues Expeced fuure moraliy developmen (rend, volailiy) Level of risk premium Naural buyers and sellers of moraliy risk Illusraion of imporance of rend for valuaion Survival probabiliy (iniial age 65) 1 0,8 0,6 0,4 0, Example wih old moraliy: Trend V E 25 p 65 0% % 1% % 2% % 3% % 4% % 0 1% 2% 3% 4% page 4

5 IMPROVEMENT RATES FOR DIFFERENT PERIODS Observed yearly relaive decline, Denmark 5,00% 4,00% 3,00% 2,00% 1,00% 0,00% Age A possible model Difficul o predic fuure changes... Yearly average improvemen raes depend on he period considered. Larges improvemen raes observed in he period Source: PFA Pension & Human moraliy daabase, page 5

6 APPROACH TO LONGEVITY MODELING IN DENMARK Danish FSA has inroduced a moraliy and longeviy benchmark Benchmark for curren moraliy esimaed from daa for insured individuals wih 5 years of daa Pension funds perform yearly saisical ess wheher hey derive from benchmark Benchmark for fuure rend esimaed from oal Danish populaion (20 years of daa) page 6

7 BENCHMARK - OE-RATES MALES , Benchmark deermined using splines from (log) linear regression Special model for high age moraliy - logisic model 0,01 Kanniso model suggesed by Thacher e al, ,001 0, FT benchmark No exponenial increase of moraliy afer age 100 page 7

8 APPROACH TO LONGEVITY MODELING IN DENMARK Benchmark Fuure rend assumpion based on observaions for 20 years Age-dependen yearly decline Differen decline raes for males and females Trend and curren moraliy lead o bes esimae Solvency 2: 20 % reducion of moraliy raes 4,0% 3,0% Females Males Age Males Females 40 84,0 86,5 2,0% 50 83,5 86, ,8 86,6 1,0% 65 84,3 87,0 0,0% ,1 87, ,1 90, ,9 94,7 page 8

9 page 9 TWO TYPES OF MORTALITY RISK

10 TWO FUNDAMENTALLY DIFFERENT TYPES OF MORTALITY RISK Sysemaic moraliy risk: Unexpeced changes in underlying moraliy inensiies and expeced life imes: Same effec on all individuals NOT diversifiable Long-erm risk Risk managemen soluions: new producs, moraliy derivaives Unsysemaic moraliy risk: Randomness of deahs given underlying inensiies: Law of large numbers (Jakob Bernoulli, ) Risk is diversifiable Shor and long erm risk Ineracion is no rivial! page 10

11 LONG TERM SIMULATION OF NUMBER OF SURVIVORS Simulaed number of survivors a age 85, given iniial age 30 0,1 0,08 0,06 0,04 0, ,03 0,025 0,02 0,015 0,01 0, individuals 1000 individuals Expeced number of survivors: 37 Expeced number of survivors: 374 Sd. dev. wihou sysemaic risk: 4.9 Sd. dev. wihou sysemaic risk: 15.4 Sd. dev. wih sysemaic risk: 7.3 Sd. dev. wih sysemaic risk: 57.7 page 11

12 SHORT TERM SIMULATION OF NUMBER OF SURVIVORS - PORTFOLIO OF RETIRED Simulaed number of survivors a age 85, given iniial age 75 0,1 0,08 0,06 0,04 0, ,03 0,025 0,02 0,015 0,01 0, individuals 1000 individuals Expeced number of survivors: 43.5 Expeced number of survivors: 435 Sd. dev. wihou sysemaic risk: 4.9 Sd. dev. wihou sysemaic risk: 15.8 Sd. dev. wih sysemaic risk: 5.2 Sd. dev. wih sysemaic risk: 21.5 page 12

13 page 13 SURVIVOR SWAPS AND MORTALITY RISK MODELING

14 SURVIVOR SWAPS Fix a porfolio of (insured) lives: Pension fund receives difference, if > 0 Expeced number of survivors (fixed leg) Acual number of survivors (variable leg) Pension fund pays difference, if < 0 Possible expiry T Time (years) Expeced number may include a risk premium page 14

15 SURVIVOR SWAP UNDERLYING PORTFOLIO Possibiliies Own porfolio Oher small porfolio Large reference populaion Choice of underlying porfolio: Survivor swap linked o own porfolio may provide perfec hedge bu may be less liquid (no marke) Using a small porfolio on oher lives will ypically no provide a good hedge. Unsysemaic risk will dominae (invesmen insrumen) Large reference populaion gives desired properies if sufficienly long ime horizon (more liquid, possible marke?) page 15

16 SURVIVOR BONDS Y (,x): Number of persons alive a, aged x a ime 0 Survivor index: S(,x) = Y(,x)/Y(0,x) (observable, sochasic process, non-raded) Paymen process: db() = S(,x) d, < T Canno be replicaed wih exising insrumens! Expeced relaive number of survivors: E Q [S(,x)] = p Q x(choice of Q?) Price: E Q T e u r s ds S( u, x) du F( ) T P(, u) E Q S( u, x) F( ) du Discouning facor - ineres rae risk longeviy risk page 16

17 SURVIVOR BONDS: EXAMPLE OF PAYMENTS Acual number of survivors compared o expeced number of survivors (black line) in wo differen sochasic scenarios (red and blue lines). Lef plo shows number of survivors in he insurance porfolio and righ plo shows number of survivors in larger porfolio. (Porfolio sizes and ) page 17

18 SURVIVOR SWAPS Paymen process: db swap () = ( S(,x) p Q x )d, < T Variable paymens Survival probabiliy chosen af ime 0! Pricing: Fixed paymens T P(, u) E Q T Q' S ( u, x) F( ) du P(, u) upx du Valuaion of variable paymens: Similar o a life annuiy Ineres rae dependen Value reflecs moraliy developmen Increases if moraliy decreases Valuaion of fixed paymens: Similar o a cerain annuiy Ineres rae dependen Price does no depend on moraliy developmen For pension fund: Swap par of marke-based accouning Consisen marke values for all asses and liabiliies page 18

19 SURVIVOR SWAPS ACTUARIAL OR FINANCIAL INSTRUMENT? Acuarial inerpreaion own porfolio Mach paymens from life annuiies If more annuians survive, pension fund receives he difference No o be raded? Enered wih reinsurance company Financial inerpreaion Marke value of fuure paymens has similar sensiiviy owards moraliy/longeviy risk as exising liabiliies Value of survivor swap increases afer a longeviy sress (Solvency II) Capial relief Trading possbiliies? 1 0,8 0,6 0,4 0, % 2% 3% 4% page 19

20 MORTALITY MODEL AND PORTFOLIOS Insurance porfolio Populaion Iniial moraliy Developmen process x 2 x Correlaed, imeinhomogeneous CIR x, x, 1 2 Sochasic and ime-dependen Fuure moraliy 0 x, x 1x, 0 x x x, 1 1 2, 2 2 Inuiive and flexible model wih nice analyical properies page 20

21 NUMERICAL EXAMPLES Time-inhomogeneous CIR model known from finance d ( x, ) ( ( x, ) ( x, ) ( x, )) d ( x, ) ( x, ) dw ( ) Parameerizaion Case I Case II ( ~ x, ) ~ ( x, ) ( x, ) ~ e ~ 1 ~ 2 ~ 2 ~ Quaniles, ime horizon 20 years: ~ ~ ~ 5% 25% 50% 75% 95% Case I Case II page 21

22 SIMULATION FOR IMPROVEMENT PROCESS (CASE I) 1,05 0,95 0,85 0, (wih mean-reversion) page 22

23 SIMULATION FOR IMPROVEMENT PROCESS (CASE II) 1,05 0,95 0,85 0, (wihou mean-reversion) page 23

24 page 24 MIXED DYNAMIC AND STATIC HEDGING

25 DYNAMIC RISK-MINIMIZATION FOR PAYMENT STREAMS Idea Minimize condiional expeced (squared) in- or ouflow no generaed by A using a marke measure Q Mehod Marke value decomposiion V *, Q *, Q A A A V 0 ( u) dx ( u) ( u) dy( u) L ( ) 0 0 Risky asses Unhedgeable risk Opimal sraegy o A ( ) ( ) o A ( ) ( ) o ( ) V *, Q ( ) Number of asse X Number of asse Y * A A A ( ) ( ) X ( ) ( ) Y( ) Invesmen in bank accoun Resuls are ypically inuiive! page 25

26 A NAIVE MIXED DYNAMIC AND STATIC RISK-MINIMIZING STRATEGY Se-up wih an illiquid asse X sill raded dynamically Y is an illiquid asse raded a fixed imes i only A naive approach ˆo o ( ) ( ) ˆo o ( ) ( i1) for, i 1 i Invesmen in X unchanged Invesmen in Y fixed a opimal invesmen a beginning of period Does no work if X and Y are o correlaed or rend in () page 26

27 page 27 MIXED DYNAMIC AND STATIC RISK-MINIMIZING STRATEGY Decompose Y wih respec o X Opimal sraegy for dl dx dy Y Y ) ( ˆ ) ( ) ( ) ( ) ( ˆ o o Y o o ) ( ) ( ˆ ) ( ) ( ˆ ) ( ) ( ) ( ˆ * *, Y X A V o o Q o Trick o handle correlaion beween X and Y ) ( ) ( ) ( ) ( ) ( ) ( ˆ i i Y Q i i Y Y A Q o F L E F L u dl u E i i Opimal dynamic sraegy correced by hedgeable par of illiquid asse Risk adjused average of dynamic sraegy on nex inerval i i, 1

28 page 28 CASE STUDY WITH SURVIVOR SWAPS

29 SURVIVOR SWAPS Survivor swap paymen process da swap j d n ~ p d x n N x,, Number of deahs j j j Iniial number of lives x Agreed survival probabiliy Illusraion Pension fund receives difference, if > 0 Expeced number of survivors (fixed leg) Acual number of survivors (variable leg) Pension fund pays difference, if < 0 Possible expiry T Survivor swaps are illiquid page 29

30 SIMULATED INTEREST RATE SCENARIOS Realizaion of he shor rae over a period of 60 years in wo difference sochasic scenarios page 30

31 SIMULATED MORTALITY INTENSITIES Moraliy inensiies for he insurance porfolio (red lines) and he populaion (blue lines) in wo sochasic scenarios (lef plo and righ plo). Deerminisic moraliy inensiies wih a rend of decline (black and grey lines) page 31

32 SIMULATED DEATHS IN THE TWO SCENARIOS Deahs in he insurance porfolio (lef plo) and deahs in he populaion (righ plo) in he wo scenarios page 32

33 SIMULATED SURVIVOR SWAP PRICE PROCESSES The hedging insrumens Inrinsic value processes for survivor swap on he insurance porfolio (lef plo) and survivor swap on he populaion (righ plo) in he wo scenarios page 33

34 SIMULATED INTRINSIC VALUE PROCESS The liabiliy - o be hedged! Inrinsic value processes for he insurance conrac in he wo scenarios. Example: Age 30, Life annuiy saring a age 60; yearly premiums n 1 =100, n 2 =1000, page 34

35 SIMULATED OPTIMAL SURVIVOR SWAPS (OWN PORTFOLIO) Perfec hedge afer reiremen Number of survivor swaps on he insurance porfolio held a ime in he marke (B, P, Z 1 ) page 35

36 SIMULATED OPTIMAL SURVIVOR SWAPS (POPULATION) Number of survivor swaps on he populaion held a ime in he marke (B, P, Z 2 ) Swaps on populaion are no useful here due o shor ime horizon. Main risk is unsysemaic risk page 36

37 ILLIQUID SURVIVOR SWAPS: COMPARISON OF DYNAMIC AND STATIC TRADING STRATEGIES Swaps on insurance porfolio Swaps on populaion porfolio Trading a ime 0 and ime 30 Trading a ime 0 and ime 30 Dynamic rading Trading a ime 0 Dynamic rading Trading a ime 0 Number of rading imes is imporan page 37

38 EFFICIENCY Comparison of iniial inrinsic risk Dynamic rading insurance porfolio Saic hedging ime 0 insurance porfolio Saic hedging ime 0 and 30 insurance / populaion porfolio n1 n2 No swap D1 S1.1 S1.2 S , , ,000 10, , , , , Good resuls even wih few rading imes page 38

39 CONCLUSIONS Survivor swaps: Mach sensiiviy owards moraliy and longeviy risk Opimal number is a weighed assessmen of sysemaic and unsysemaic moraliy risk Size of hedging porfolio maers Small exernal hedging porfolios inroduce new unsysemaic risk and will no be efficien as hedging insrumen Swap on own porfolio or on (oher) oal populaion: Own porfolio mos efficien may be expensive. Less liquid Oher large reference porfolio can work very well. More liquid Compared dynamic o mixed dynamic and saic sraegies Inrinsic risk is almos unaffeced by resricion o saic sraegies page 39

40 RELATED RESEARCH ON MORTALITY MODELING AND MORTALITY RISK MANAGEMENT HEDGING WITH SURVIVOR SWAPS Dahl, Glar, Møller (2011). Mixed dynamic and saic risk-minimizaion wih an applicaion o survivor swaps, European Acuarial Journal MORTALITY RISK AND MORTALITY DERIVATIVES Dahl, Møller (2009). Moraliy derivaives: Longeviy Bonds and survivor swaps, Finans/Inves (In Danish) DYNAMIC HEDGING WITH MORTALITY DERIVATIVES Dahl, Melchior, Møller (2008). On sysemaic moraliy risk and riskminimizaion wih survivor swaps, Scandinavian Acuarial Journal VALUATION AND HEDGING WITH TRADITIONAL BONDS Dahl, Møller (2006). On valuaion and hedging of insurance conracs wih sysemaic moraliy risk, Insurance: Mahemaics and Economics page 40