FAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004

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1 FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004

2 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES : THE BINOMIAL and THE BLACK/ SCHOLES CASES 4. FAIR VALUE OF LIFE INSURANCE PARTICIPATING CONTRACTS 5. FAIR VALUE OF VARIABLE ANNUITIES 6. CONCLUSION Pierre Devolder 09/004

3 . Inroducion o FAIR VALUE Inernaional norms IAS / IFRS for all financial insiuions in Europe soon ( ) A lo of discussions linked o accouning principles and arificial volailiy General principle of Fair valuaion of elemens for asses as well as for liabiliies: marke values insead of hisorical values Pierre Devolder 09/004 3

4 . Inroducion o FAIR VALUE Basic principle : from an hisorical or sauory accouning poin of view o fair value bases Fair value : price a which an insrumen would be raded if a liquid marke exised for his insrumen ASSETS : marke values LIABILITIES :??? If no marke value : principle of esimaion of fuure cash flows properly discouned and aking ino accoun he differen kinds of risk Pierre Devolder 09/004 4

5 . Inroducion o FAIR VALUE Need o develop good models of valuaion especially for acuarial liabiliies where here is no marke price Consisency beween modern financial pricing heory and classical acuarial models Imporan example : guaranees in life insurance and pension:one of he mos challenging risk nowaday Even if for compeiion reasons mehods of pricing could remain very classical, fair valuaion will require new insighs aking ino accoun modern finance Pierre Devolder 09/004 5

6 . Risk neural pricing and Deflaors Purpose : - inroducion o he modern financial paradigm of risk neural pricing - link wih deflaor mehodology - link wih classical acuarial principle of discouning - developmen in a simple discree marke model Pierre Devolder 09/004 6

7 . Risk neural pricing and Deflaors Paradigm of risk neural pricing: Purpose: o compue he presen price of a fuure sochasic cash flow correlaed wih financial marke in an uncerain environmen Wha we could expec : price = discouned expeced value of he fuure cash flow: L(0) = E(L(T)) NO!!! T ( + i) Pierre Devolder 09/004 7

8 . Risk neural pricing and Deflaors You have o change eiher he probabiliy measure, eiher he discouning facor Change of probabiliy measure : risk neural mehod: You can say wih a classical discouning facor bu he expecaion of he cash flows mus be done using anoher probabiliy measure han he real one Change of discouning facor : deflaors mehod: You can use he real probabiliy measure bu he discouning facor has o be changed and becomes sochasic Pierre Devolder 09/004 8

9 . Risk neural pricing and Deflaors Risk neural pricing : L(0) = ( + r) T E Q (L(T)) Deflaor mehod : Q= risk neural measure L (0) = E (D(T) L(T)) D(T) =deflaor = sochasic discouning Pierre Devolder 09/004 9

10 . Risk neural pricing and Deflaors Single period marke model : one riskless asse : S 0( ) = S0(0)( + r) wih r = riskfree rae d risky asses defined on a probabiliy space : Ω = S { ω, ω,..., ωn } wih p = P({ ω }) =,..., N i( ) = ( Si(, ω ), Si(, ω ),..., Si(, ωn )) i =,..., d classical assumpion of absence of arbirage opporuniies Pierre Devolder 09/004 0

11 . Risk neural pricing and Deflaors STATE PRICE : Basic propery : if he marke is wihou arbirage here exiss a random variable ψ such ha for any asse : S i ( 0) Ψ S (, ω ) wih Ψ = Ψ( ω ) 0 = N = i If he marke is complee, he sae price is unique. Consequence : for asse i =0 ( risk free asse) r Ψ = N Ψ = = + Pierre Devolder 09/004

12 . Risk neural pricing and Deflaors RISK NEUTRAL MEASURE Definiion : arificial probabiliy measure given by: q Ψ = Q( ω = ω) = = ( + r) Properies : ) 0 q Ψ N and q = = ) for each asse: mean reurn= risk free rae S (0) Ψ N i = q S i(, ω) ( + r) = Pierre Devolder 09/004

13 . Risk neural pricing and Deflaors DEFLATOR : random variable D defined by : Properies : D(ω )= D N i. p D = E(D) = = + r ( expeced value of he deflaor = classical discoun ) = Ψ p ii. S (0) = N i p D S i(, ω) = Pierre Devolder 09/004 3

14 . Risk neural pricing and Deflaors Generalizaion : if X is a financial insrumen on his marke (replicable by he underlying asses) and giving for scenario a cash flow X(,) hen he iniial value of his insrumen can be wrien as : i. Sae price ii.risk neural vision : vision: X(0) = N = Ψ N X(0) = q + r = X(, ) X(, ) = + r E Q (X()) iii.deflaor vision:x(0) N = = p D X(, ) = E(D X()) Pierre Devolder 09/004 4

15 . Risk neural pricing and Deflaors Muliple periods model : discree ime model ( =0,,, T) Riskfree asse: S0( ) = S0(0)( + r) wih r = riskfree rae Risky asses : S i ( ) = ( Si(, ω ), Si(, ω),..., Si(, ωn )) i =,..., d STATE PRICE : S i ( 0) Ψ ( ) S (, ω ) wih Ψ ( ) = Ψ( ω, ) 0 = N = i Pierre Devolder 09/004 5

16 . Risk neural pricing and Deflaors DEFLATOR : Ψ ( ) D ( ) = = discoun facor from o 0if p scenario Pricing : if X is a financial replicable insrumen on his marke generaing successive sochasic cash flows : { C(, ω); =,..., T; ω Ω} Then he iniial price of X can be wrien alernaively : X ( 0) C(, ω ) Ψ = T N = = ( ) Pierre Devolder 09/004 6

17 . Risk neural pricing and Deflaors Or wih risk neural measure: X(0) ( + r) = T N = = q C(, ω ) Or wih deflaors : X(0) = T N = = p C(, ω)d () = E(D()C()) T = Pierre Devolder 09/004 7

18 3.. THE BINOMIAL CASE Single period model: Risky asse : S( ) = S(0) u = S(0) d wih probabiliy p wih probabiliy -p Absence of arbirage opporuniies if: 0 < d < + r < Oher form of he risky asse : u u = + r + λ + µ d = + r + λ µ Pierre Devolder 09/004 8

19 3.. THE BINOMIAL CASE Wih condiion : 0 < λ < µ λ = risk premium µ = Equaions of he STATE PRICE : volailiy For i=0: ( + r) Ψ + ( + r) Ψ = For i=: uψ d + Ψ = Pierre Devolder 09/004 9

20 3.. THE BINOMIAL CASE Soluion for he STATE PRICE: up Ψ = + r d ( + r)( u d) = µ λ µ ( + r) down Ψ = u ( + r) ( + r)( u d) = µ + λ µ ( + r) Safey principle : Ψ = Ψ if λ = Ψ Ψ if λ > 0( normal case) 0 Pierre Devolder 09/004 0

21 3.. THE BINOMIAL CASE Fair value in a binomial environmen single period : If X is a financial insrumen on his marke wih fuure sochasic cash flows given respecively by : X (, ω = X ) = X and X (, ω) Then he iniial fair value of X is given by : X ( 0) = X Ψ + X Ψ Or : X(0) = (X + X) + (X X ) ( + r) λ µ ( + r) Pierre Devolder 09/004

22 3.. THE BINOMIAL CASE Muliple periods model: Risky asse u u u d d d Pierre Devolder 09/004

23 3.. THE BINOMIAL CASE Srucure of STATE PRICES in muliple periods: Assumpion: financial produc having successive cash flows depending only on he curren siuaion of he marke (no pah dependan). Sae price Ψ Ψ Ψ Ψ = sae price a ime if scenario Ψ Ψ 3 Pierre Devolder 09/004 3

24 3.. THE BINOMIAL CASE Value of he STATE PRICES: Ψ = C Ψ Ψ Where - = number of up (=,..,+) -- = number of down And: C is he number of pahs in he ree wih - up in periods Pierre Devolder 09/004 4

25 3.. THE BINOMIAL CASE Fair valuaion in muliple periods binomial : If X is a financial insrumen having successive cash flows in he ree given by : X = cash flowa ime if Then he iniial fair value of X is given by : scenario X (0) = T = + = X Ψ Pierre Devolder 09/004 5

26 3.. THE BLACK / SCHOLES CASE Coninuous exension : Black and Scholes model Riskless asse : ds0 () = rs0() Risky asse : d ds() = δs()d + σs() dw() Where:- w is a sandard brownian moion - δ is he mean reurn of he asse (δ>r) - σ is he volailiy of his reurn Pierre Devolder 09/004 6

27 3.. THE BLACK / SCHOLES CASE Risk neural measure : ds () = rs = rs ()d ()d + S + σs ()(( δ r)d ()dw *() + σdw()) wih : w *() = w() + δ σ r Q = risk neural measure ( Girsanov Theorem) Under Q he process w*is a sandard brownian moion Pierre Devolder 09/004 7

28 3.. THE BLACK / SCHOLES CASE Deflaors: sochasic process D such ha for i= (risk free asse) and for i= ( risky asse): Soluion: Si (0) = E (D()S i()) D() = e r e δ r δ r w() σ σ Safey principle : Lower values of «w» give higher values for deflaor Pierre Devolder 09/004 8

29 4. Fair value of paricipaing conracs 4.. Liabiliy side: Life insurance conrac wih profi : guaraneed ineres rae + paricipaion 4.. Asse side : Sraegic asse allocaion: Cash + Bonds + Socks 4.3. Valuaion of he conrac : Fair value and equilibrium condiion Pierre Devolder 09/004 9

30 4. Fair value of paricipaing conracs Need for a consisen ALM approach: Double link beween asse and liabiliy in his kind of produc : Liabiliy Asse : Invesmen sraegy mus ake ino accoun he specificiies of he underlying liabiliy Asse Liabiliy : Paricipaion liabiliy linked wih invesmen resuls Pierre Devolder 09/004 30

31 4.. Liabiliy side Pure Endowmen conrac : - iniial age a =0 : x - mauriy : N - Benefi if alive a ime =N : - Benefi in case of deah before N : 0 - Conrac wih single or periodical premiums (pure premium) - Technical parameers : -moraliy able: { l x } - guaraneed echnical rae : i - paricipaion rae on surplus: η Pierre Devolder 09/004 3

32 4.. Liabiliy side Bonus sysems : general definiion: percenage of he surplus (Asses Liabiliies) Three possible schemes : Terminal bonus : bonus compued a mauriy Reversionary bonus : bonus compued each year and fully inegraed in he conrac as addiional premium Cash bonus : bonus compued each year and paid direcly o he clien Pierre Devolder 09/004 3

33 4.. Asse side Cash Bonds- Socks model ( CBS model): The underlying porfolio of he insurer is supposed o be invesed in hree big classes of asse: - Cash : shor erm posiion ( money accoun) - Bonds : zero coupon bonds wih a mauriy no necessarily mached wih he duraion of he conrac - Socks : sock index Pierre Devolder 09/004 33

34 4.. Asse side Cash model : Money marke accoun : dβ() = r() β()d wih r() = risk free rae Risk free rae :Ornsein-Uhlenbeck process dr() = a(b r())d + σr dw() wih w = s an dard brownian moion Pierre Devolder 09/004 34

35 4.. Asse side Bond model P(,T)= price a ime of a zero coupon wih mauriy T General evoluion equaion : dp(,t) = P(,T) µ (,T) d P(,T) σ(,t) dw() Paricular case : VASICEK Model µ (,T) = r() + λσ B(T ) σ(,t) = σ wih : B(s) B(T ( e Pierre Devolder 09/ r = a r ) as )

36 4.. Asse side Socks model : S()= value a ime of a sock index ds() = S()( µ d + σ S ( ρdw () + ρ dw ())) wih ρ= correlaion socks / in eres raes w = s an dard brownian moion independan of w Pierre Devolder 09/004 36

37 4.. Asse side Porfolio : α α α C B S = proporion = proporion = proporion in bonds socks αc + αb + αs = main assumpions : - proporions remain consan ( coninuous rebalancing) - self financed sraegy in in cash Pierre Devolder 09/004 37

38 Bond sraegy : 4.. Asse side Assumpion : a each ime zero coupon bonds of only one mauriy can be held bu he mauriy has no necessarily o mach he duraion of he liabiliy ( price of mismachinglong duraion of life insurance conrac) Sraegy : mached sraegy : T=N Sraegy : shorer mauriy of he bond and successive reinvesmens ill end of he conrac Pierre Devolder 09/004 38

39 4.. Asse side Bond sraegy s=0 3 n N s = 0,,...,n : number of reinvesmens 0 = 0,,,..., n : ins an s of reinvesmen Paricular case : mached sraegy : n=0 Pierre Devolder 09/004 39

40 4.. Asse side Evoluion of he porfolio : V () = marke value of Evoluion equaion : dv() V() = α S ds() S() he underlying porfolio + α B dp(, i) P(, ) i + α C dβ() β() (for i < < i ) Pierre Devolder 09/004 40

41 4.. Asse side Value a mauriy of he asses : ln V(N) ln V(0) = wih each n s= 0 erm (ln V( (ln V( s+ s+ ) ln V( ) ln V( µ (N) = E(ln V(N) ln V(0)) σ (N) = var(ln V(N) ln V(0)) s s )) )) = normallydisribued Pierre Devolder 09/004 4

42 Pierre Devolder 09/ Asse side Explici form : σ ρ σ α α σ + α σ α σ ρ α σ α + ρ σ α + λσ + α + α α µ + α = s s s s s s s s s r S B S s B B S S s r B S S S S s r B C B S s s u))du B( u) ( B ( (u) u))dw B( ( (u) dw u))du B( )r(u) ( ( ) ln V( ) ln V(

43 4.3. Valuaion of he conrac Example of a conrac wih single premium and erminal bonus : Fair value a mauriy = pay off of he conrac FV(N) = ( + i) N +η max(v(n) ( + i) N ; 0) Iniial fair value : (? In line wih he single premium) FV(0) = N N px (( + i) P(0, N) + ηcall(v; N;( + i) N )) Pierre Devolder 09/004 43

44 4.3. Valuaion of he conrac Equilibrium condiion : ( + i) N P(0, N) + η call(v; N;( + i) N ) = Consequences : ) i< R(0, N) ( wih P(0, N) = ( + R(0, N)) ) depends on he invesmen sraegy hrough he value of V(N) N ) Pierre Devolder 09/004 44

45 4.3. Valuaion of he conrac 3 ) implici relaion for he echnical rae i ; 4 ) explici relaion for he paricipaion rae η: η = N ( + i) P(0, N) N call(v; N;( + i) ) An explici formula of he call can be obained in he CBS model presened before. Pierre Devolder 09/004 45

46 4.3. Valuaion of he conrac Risk forward neural measure mehod : call = where P(0, N)E Q N Q N (max((v(n) ( + i) );0) = forward risk neural measure In he CBS model we have : call = Φ(D (i)) ( + i) wih : D ± + Nln ( + i) ln P(0,T) ± / (i) = υ N T P(0, N) Φ(D (i)) υ Pierre Devolder 09/004 46

47 υ 4.3. Valuaion of he conrac Wih for insance for he mached sraegy : wih = α : S σ S N + ( α N B (N) = ( e a a N B(N) = + ( e 3 a a B ) an ) σ an r ) B 3 a (N) + α ( e an S ( α ) B ) σ S σ ρb (N) r Pierre Devolder 09/004 47

48 5. FAIR VALUE OF VARIABLE ANNUITIES Purposes : How o valuae pension annuiies no in erms of echnical basis bu in erms of marke fair values; Influence of reversionary bonus ( variable annuiies) on he level of provision; Sensiiviy of he provision wih respec o financial parameers; How o fix he echnical ineres rae. Pierre Devolder 09/004 48

49 5.. Liabiliy side - Immediae lifeime annuiy for an affiliae o a pension fund - x : iniial age a ime =0 - Liabiliy o pay: cases : ) fixed annuiy : L ) variable annuiy : amoun o pay L, = a ime for scenario ( possibiliy o increase yearly he pension depending on he financial performances asse side) - Paymen a he end of he year ill deah or during a fixed period of n years Pierre Devolder 09/004 49

50 5.. Liabiliy side Acuarial firs order bases : i =echnical discoun rae p x = survival probabiliy a ime Technical provision for a consan pension ( case ): n L, = V x L x n n = L a = L p x = ( + i) Pierre Devolder 09/004 50

51 5.. Asse side Binomial model : mixed financial sraegy of he pension fund beween riskless asse (r= riskfree rae) and risky asse (binomial model u / d) γ γ : par invesed in he risky asse ( 0 γ : par invesed in he riskless asse ) Pierre Devolder 09/004 5

52 5.3. Bonus scheme Definiion of he reversionary bonus for variable < annuiies Used rule of bonus : comparison each year beween he effecive reurn of he asses and he riskfree rae; a par of his surplus is given back o he affiliae: 0 β : paricipaion rae Pierre Devolder 09/004 5

53 5.3. Bonus scheme Yearly rae of increase of he pension : If he risky asse is up : γ u + ( γ )( + r) + k = + β ( ) ( + r) λ + µ or + k = + βγ ( ) + r If he risky asse is down : γ d + ( γ )( + r) + + l = + β ( ) = ( + r) Pierre Devolder 09/

54 5.3. Bonus scheme Final form of he liabiliies of he variable annuiy: L, = L + ( + k) Where -+ is he number of imes of up permiing o give a bonus. As expeced THE LIABILITY DEPENDS ON TIME AND IS STOCHASTIC Pierre Devolder 09/004 54

55 5.4. Valuaion of he conrac Compuaion of he fair value of he liabiliies : (fixed annuiy) FV = x = = ( ) n + L p L Ψ x, n = L n + p x = = x = + Ψ n = L p ( ) r r = L a x n Pierre Devolder 09/004 55

56 Pierre Devolder 09/ Valuaion of he conrac Compuaion of he fair value of he liabiliies (variable annuiy) Acuarial valuaion : no so simple: liabiliies no deerminisic Fair valuaion : general formula of valuaion : ( ) Ψ = + = = n x n x k L p L FV,, ( ) ( ) [ ] n x k p L + + Ψ Ψ = = ( ) ( ) + Ψ Ψ = + = + = n x k C p L

57 5.4. Valuaion of he conrac Compuaion of he fair value of he liabiliies (variable annuiy) FV ( L k ) x, n n = L p + βγ µ λ x = + r µ ( + r) n = L p ( ) i * x = + i a = L * x n Pierre Devolder 09/004 57

58 5.4. Valuaion of he conrac Equilibrium relaion : i* = equilibrium consan discoun rae given by : i * = ( r µ λ βγ ) µ ( + r) / ( + µ λ βγ ) µ ( + r) if β = 0 or γ = 0 : i * = r if β > 0 and γ > 0 : i * < r Pierre Devolder 09/004 58

59 5.5. Numerical illusraion Cenral scenario: u=. d=0.99 r=0.03 Risk premium : λ =0.0 Volailiy : µ = 0.06 i=0.05 Moraliy: GRM 95 n=5 Pierre Devolder 09/004 59

60 5.5. Numerical illusraion %socks , 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 40%socks Fixed annuiy 0% <,5%socks 0%sock 40%socks 60%socks 80%socks 00%socks Pierre Devolder 09/004 60

61 5.5. Numerical illusraion Volailiy sensiiviy analysis : ( 60% in risky asse) ,0 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0, "Volailiy facor" Acuarial valuaio n Deflaor valuaio n B= 0,5 B= 0,3 B= 0,5 B= 0,65 B= 0,8 B= Pierre Devolder 09/004 6

62 5.5. Numerical illusraion Value of he equilibrium discoun rae : cenral scenario for β = 0.5 and γ =0.6 : i*=.% for β = and γ =0.6 : i*=.4% 3 for β = 0.9 and γ =0.4 : i*=.05% 4 for β = and γ = : i*= 0.4% λ = µ = % 6% r=3% i=.5% Pierre Devolder 09/004 6

63 5.5. Numerical illusraion Value of he equilibrium discoun rae : oher scenario ( less volaile) for β = 0.5 and γ =0.6 : i*=.60% versus.% for β = and γ =0.6 : i*=.% versus.4% 3 for β = 0.9 and γ =0.4 : i*=.5% versus.05% 4 for β = and γ = : i*=.68% versus 0.4% λ = % µ = 3% r=3% i=.5% Pierre Devolder 09/004 63

64 5.5. Numerical illusraion Equilibrium echnical rae 0,03 0,05 0,0 0,05 0,0 0, %socks 0, 0,3 0,5 Paricipaion rae 0,7 0,9 00%socks 00%socks 80%socks 60%socks 40%socks 0%socks 0%socks Pierre Devolder 09/004 64

65 5.5. Numerical illusraion EQUILIBRIUM TECHNICAL RATE(II) 0,035 0,03 0,05 0,0 0,05 0,0 0, ,0 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0, "Volailiy facor" B= B=0.8 B=0.65 B=0.5 B=0.3 B=0.5 B=0 Pierre Devolder 09/004 65

66 6.CONCLUSION Sae prices, risk neural measures and deflaors are easy ools in order o valuae sochasic fuure cash flows correlaed o fuure financial markes. This is exacly he siuaion of life insurance producs One of he mos imporan resul is he way o valuae bonus and o define properly a echnical ineres rae in a complee ALM framework Pierre Devolder 09/004 66

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