Killing the Law of Large Numbers:

Transcription

1 Kllng he Law of Large umbers: Fnancal Valuaon of Moraly Rsk va he Insananeous Sharpe Rao By: Moshe A. Mlevsky Schulch School of Busness York Unversy, Torono, Canada Jon work wh: Davd Promslow (York U.) & Vrgna Young (U. of Mchgan) Longevy Rsk Symposum

2 Oulne & Agenda Basc revew of how he LL breaks down under aggregae moraly rsk and why hs mpacs rsk managemen. Dscuss he exsng academc leraure. Inroduce our Sharpe Rao based approach for prcng longevy-lnked nsrumens n dscree me. Bref overvew of how o model hs n connuous-me (me permng) Longevy Rsk Symposum

3 Longevy Insurance Payoff: Pay: (+L) p 50% Chance (-p) 50% Chance Pays $ Alve Pays $0 Dead Longevy Rsk Symposum

4 Back o Classcal Bascs: w 0 Pr ( p) Pr ( p) E[ w p var[ w p( p) + ( p)(0 p) 4p( p) Longevy Rsk Symposum

5 Back o Classcal Bascs: w 0 Pr Pr E [ w var[ w SD [ w Longevy Rsk Symposum

6 Longevy Rsk Symposum Addng-up he longevy bes: Wha s oal payou when sellng polces? ) ( 4 var var[ [ p p w W p w E E W w W Term s larger when rsks are dependen

7 Refresher: Law of Large umbers: Under..d. payous lm w E[ w Whn he conex of nsurance hs mples ha evenually here s no uncerany (.e. rsk) n wha you wll be payng ou per polcy and he payou rsk s no compensaed n equlbrum. In he language of porfolo heory, s dversfable. Longevy Rsk Symposum

8 The dosyncrac Rsk whch s he oal sandard devaon per polcy, goes o zero p( p) SD[ W 0 You expec each polcy o generae a payou of $ wh zero uncerany (rsk), as long as you sell enough polces. Longevy Rsk Symposum

9 I can also use he Cenral Lm Theorem (CLT) o compue probables Pr[ W C W E[ W Pr SD[ W C E[ W SD[ W Pr Z C p 4p( p) Or I can go hru he brue-force way usng he Bnomal dsrbuon. More on hs laer Longevy Rsk Symposum

10 umercal Example: Sell 0,000 Longevy Insurance Polces Toal Payou Ousde he Range of: ($5, $5,000) ($7, $,500) ($9, $,000) ($9, $0,500) ($9, $0,300) Even Probably: 0.000% 0.000% 0.000% 0.000% 0.7% oe: Sandard devaon of payou s only $00, snce p 0.5. Longevy Rsk Symposum

11 Wha happens under sochasc hazard parameers The frs (classcal) example assumed we know he parameer value p wh perfec cerany. Wha f we are unsure of p? Are we enled o smply compue he expecaon E[p and use ha esmae for prcng & rsk managemen formulas? o! In fac, he oal payou dsrbuon becomes a mxure of Bnomals, he Law of Large umbers (LL) -- nvoked for prcng nsurance -- breaks down and some rsk can never be elmnaed. Longevy Rsk Symposum

12 Longevy Insurance Payoff: Sochasc Hazard Parameers ~ Pays $ Alve Pay: (+L) ~ Pays $0 Dead Longevy Rsk Symposum

13 Longevy Rsk Symposum Modern Bascs (II): Sochasc Hazard Parameers [ ~ ) ( Pr ) ( Pr ~ ~ Pr 0 ~ Pr p p E p p w +

14 Modern Bascs (II): Example Sochasc Hazard Parameers w 0 Pr Pr ~ ~ ~ Pr Pr E[ ~ 0.5 -> Whch was he assumed survval probably n he radonal case Longevy Rsk Symposum

15 Longevy Rsk Symposum Addng-up he Longevy bes under Sochasc Hazard Parameers [ var[ [var[ var[ [ ~ [ E W W E W E w E E W w W +

16 Longevy Rsk Symposum Addng-up he Longevy bes under Sochasc Hazard Parameers [ var[ [var[ var[ [ ~ [ E W W E W E w E E W w W +

17 Decomposng he Payou Varance E[var[ W 4 ( ) var[ E[ W 4 p ( p )( ) The oal (payou) varance s he sum of hese wo componens. oe ha he second poron would be zero under he classcal approach. Longevy Rsk Symposum

18 Decomposng he Payou Varance: A umercal Example E[var[ W 4 (0.6)(0.4) 0.96 var[ E[ W 4 (0.5)(0.5)(0.) 0.04 var[ W Longevy Rsk Symposum oce ha when, he varance equals $ whch s he same as classcal case. A sngle polcy s no rsker! Bu, when ges large we have a problem

19 In hs case he Rsk per polcy does no go o zero whch s our man problem and movaon lm SD[ W ( ) p( p) 0 When 0.5, whch mples no parameer uncerany, he expresson does go o zero. The same dea apples when p or p equal zero. The classcal model s a specal case. Longevy Rsk Symposum

20 How Fas Does Rsk Declne? Toal Sandard Devaon per Polcy umber of Polces Sold: ,000 Infny Deermnsc Hazard Parameer p 0.5 $.000 $0.707 $0.447 $0.00 $0.03 $0.00 $0.000 Sochasc Hazard Parameer E[ 0.5 $.000 $0.7 $0.48 $0.3 $0.0 $0.00 $0.00 Longevy Rsk Symposum 0.6 or 0.4 wh even odds

21 Longevy Rsk Symposum The Mxure of Bnomal Dsrbuons: Anoher way o ge he same resuls k k p p p B B p B p k W p B k W + ) ( : ), ( ), ( ) ( ), ( Pr[ ), ( Pr[ 0 0 Where:

22 You sold 00 Longevy Insurance polces: Comparng he wo cases: Toal Payou Larger Than: $0 $0 $0 $30 Pr[ W > C Dversfable Pr[ W > C on-dversfable Longevy Rsk Symposum oe: Survvor ges $ E[ (0.6)(0.5)+(0.4)(0.5) p 0.5

23 A Quck Break: Revew Prevous & Curren Leraure Lee & Carer (99), Olver (00) Mlevsky & Promslow (00), Dahl (004), Bffs (005), Schrager (006), Balloa & Haberman (006), Bffs & Mllossovch (006). Denu & Dhaene (006). Carns, Blake & Dowd (005), Cox & Ln (004), Webb & Fredberg (006). Longevy Rsk Symposum

24 Refresher on he Sharpe Rao: Common fnancal language used n hnkng abou compensaon for marke rsk. E [ X R α : SD [ X E R [ X α 0.5 SD [ X 0.0 Roughly n-lne wh hsorcal reurns for equy markes. Longevy Rsk Symposum

25 Our conrbuon: (Thnk abou) prcng va he Sharpe Rao How much would you charge for he longevy nsurance polcy f you waned o be compensaed for hs (non dversfable) rsk n proporon o he Sharpe Rao? As a crude example, f your sandard devaon per polcy s convergng o $0.0 and you demand a Sharpe Rao of α 0.5, hen you would charge a loadng of L 5% above he rsk-free rae. Yes. Ths s jus anoher premum prncple. Longevy Rsk Symposum

26 Ok. How does hs model work n connuous me? d λ a( λ, ) d + b( )( λ λ ) dw λ dr µ ( r, ) d + σ ( r, ) dw The varable λ denoes he Insananeous Force of Moraly (IFM) whch s akn o he sochasc hazard parameer n he dscree case. I s beng drven by a Brownan moon (.e. a random walk). Longevy Rsk Symposum

27 50.0 Dffuson Hazard Raes lambda Curren Age 50, Realze30 yrs Curren Age 50, Expec 30 yrs Curren Age Longevy Rsk Symposum Age and Tme Sac 005 Table (w/o projecon) Hazard for 955 cohor (expeced) Hazard for 955 cohor (realzed)

28 Fnancal Perspecve on Moraly: Do o Confuse The Two E Q [ λ Fnancal Economc E P [ λ Bo-sascal Longevy Rsk Symposum

29 Remember ha f a moraly rsk premum exss E Q [ λ E P [ λ Longevy Rsk Symposum

30 One Possble Hazard Rae Process: Mlevsky & Promslow (00) λ λ exp{ g + ξy } 0 dy κ Y + db Mean Reverng Brownan-drven Gomperz (MRBG) model. Relaed o he Black-Derman-Toy Model for neres raes. Longevy Rsk Symposum

31 Calbrang Moraly Dffusons Usng Annuy Payous Pc\Age A. Longevy Rsk Symposum Source: CAEX Fnancal 00 (unsex) Monhly ncome per $00,000 premum

32 Longevy Rsk Symposum The Evoluon of Impled Hazard Raes Age 65: Jan-85 Jul-85 Jan-86 Jul-86 Jan-87 Jul-87 Jan-88 Jul-88 Jan-89 Jul-89 Jan-90 Jul-90 Jan-9 Jul-9 Jan-9 Jul-9 Jan-93 Jul-93 Jan-94 Jul-94 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00

33 Back o our sory: Hedgng Porfolo for a Pure Endowmen Polcy Π P( r, λ, ) + F( r, ) Sell he pure endowmen (longevy nsurance) for P and hen hedge he rsk usng a porfolo of rsk-free bonds denoed by F. In hs se-up he varable denoes he hedge rao. Longevy Rsk Symposum

34 Local Sandard Devaon (Insananeous Sharpe Rao) of he Hedgng Porfolo lm h 0 h var( Π + h F ) b ( )( λ λ ) Pλ ( r, λ, ) + λp ( r, λ, ) We value (prce) he pure endowmen by seng he drf of he hedgng porfolo equal o he shor rae r, mes porfolo value plus α mes he local sandard devaon. Longevy Rsk Symposum

35 Longevy Rsk Symposum Value (prce) of he clam solves: ),, ( ) ( ) ( ) ( T P r P P b P r P b ap P P P rr r Q λ λ λ λ α λ λ λ σ µ λ λλ λ Our paper proves ha P sasfes a number of appealng properes: () I s sub-addve n he number of conracs sold. () The prcng survval probably s greaer han he physcal probably

36 Praccal Concluson Mos researchers (and praconers) now agree ha here s a moraly rsk premum. A a mnmum hs means we as nsrucors -- should change he way we each (lfe, annuy) nsurance prcng o sudens. We also beleve he concep of Sharpe Raos can be helpful n undersandng he rsk & reward radeoff, as well as lnkng hese deas o fnancal economcs. Sldes & updaed paper a Longevy Rsk Symposum