Cash flows in life & pension insurance

Transkript

1 Cash flows n lfe & penson nsurance Den Danske Aktuarforenng Krstan Buchardt and Thomas Møller 28. september /84

2 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 2/84

3 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 3/84

4 Den nye regnskabsbekendtgørelse Branchedalog Dalog branchen og med Fnanstlsynet 2015 (og tdlgere): Lvhensættelsesudvalget Forskrng & Penson Fnanstlsynets rådgvende regnskabsudvalg Aktuarforenngens regnskabsudvalg Fælles forståelse Stgende krav tl præcson af modellerng, fx forskrngstageradfærd GY, GY, GY, fuld 7-tlstandsmodel Fortjenstmargen - en mulg model for gennemsntsrente Metodevalg Smulatonsmodeller versus analytske metoder 4/84

5 Balance med gamle regnskabsprncpper Egenkaptal/Basskaptal (BK) Kollektvt bonuspotentale (KB) Bonuspotentale, frpolceydelser (BF) Bonuspotentale, præmer (BP) Garanterede ydelser: GY = BE + RM Rskomargen del af GY Bonus opgjort va resdualopgørelse Garanterede frpolceydelser: GFY = GY + BP Lvsforskrngshensættelse: max(depot, GFY, GY) BK KB BF BP RM BE 5/84

6 Regnskab med nye prncpper ( 66) Lvsforskrngshensættelser (LH) Nutdsværden af bedste skøn af forventede fremtdge betalngsstrømme af ndgåede lvsforskrnger og nvesterngskontrakter Betalngsstrømme/cashflow omfatter Forventede ydelser tl forskrngstagere og parter kontrakter Forventede fremtdge præmer Forventede omkostnnger tl admnstraton Forventede betalnger som følge af forskrngstageroptoner Forventet PAL-skat Derudover ndgår en rskomargen Her, ngen sondrng mellem garanterede/ugaranterede betalnger Bonus ndeholdt lvsforskrngshensættelser 6/84

7 Regnskab med nye prncpper ( 67) Værd af bonus skal være postv: Kan opgøres resdualt, som værden af aktverne, reduceret med: Værd af garanterede betalnger værd af præmer Omkostnnger Rskomargen Nutdsværd af forv. fremtdge fortjeneste (fortjenstmargen) Betnget garanterede betalnger: Som garanterede, alternatvt foretages modellerng Evne og vllghed: Ændrng af grundlag kan nddrages (modelleres) Hvs ydelser genereres drekte ud fra aktver: Værd af cashflow kan opgøres ndrekte ud fra aktvers værd 7/84

8 Regnskab med nye prncpper ( 69, 100, 101) Fortjenstmargen på lvsforskrngskontrakter: Nutdsværden af det forventede fremtdge overskud/fortjeneste de resterende kontraktperoder Fortjenstmargen kke en del af LH, men del af forskrngsmæssge hensættelser (FH) På rentegruppenveau: Garanterede ydelser (nkl. omkostnnger) Indvduelt bonuspotentale og kollektvt bonuspotentale Rskomargen Fortjenstmargen Indvduelle bonuspotentaler (IB): Del af værden af forskrngstagernes bonusret, som er ndeholdt retrospektve hensættelse 8/84

9 Regnskab med nye prncpper Nogle konkrete problemstllnger: Hvordan dekomponeres værd af bonus ndvduelle og kollektve bonuspotentaler (gennemsntsrente)? Hvordan ndvdualseres rskomargen og fortjenstmargen? Hvorfra udsklles rskomargen og fortjenstmargen? Håndterng af markedsrenteprodukter med SUL-dæknnger (Hvad er kontraktperoderne, bundlng/unbundlng?) Notater vedrørende gennemsntsrente: Buchardt, Møller (2015). Opgørelse af lvsforskrngshensættelser efter den nye regnskabsbekendtgørelse, Notat, PFA Penson, august 2015 Frank Rasmussen (2015). Opdelng af forskrngsmæssge hensættelser og noten efter 100, Notat, Pensam 9/84

10 2 repræsentatoner af balancen Fordelng af rentegruppens aktver: BK BK 1. GY + RM hensættes Hvs der kke er aktver dækker BK 2. FM hensættes det omfang der er aktver 3. Resterende aktver er bonuspotentale Bonuspotentale opdeles IB og KB Opdelng af BP ndvduel og kollektv del: Oplagt at opdele GY, RM, FM først. FM KB IB RM GY KB Værdreg. RH FM koll. RM koll. GY koll. FM ndv. IB RM ndv. GY ndv. 10/84

11 Konkret algortme tl opgørelse af IB (1/2) Efter opgørelse af samlet balance: BP opdeles IB og KB Før start: Aktver (nkl. evt. bdrag fra egenkaptalen) deles op 1. Retrospektve hensættelser 2. Kollektve mdler Forplgtelserne (GY, RM, FM) hensættes overordnet af 1. den enkelte polces egen retrospektve hensættelse 2. de kollektve mdler 3. resterende retrospektve hensættelser 11/84

12 Konkret algortme tl opgørelse af IB (2/2) (1) Hensættelser opgøres pr. polce: Den andel der kan hensættes af polcens RH I prorteret rækkefølge: GY, RM, FM (Der kan eksstere evt. overskydende RH (afventer pkt. (3)) (2) Dernæst kollektve hensættelser for resterende GY, RM, FM I prorteret rækkefølge: GY, RM, FM (3) Resterende GY, RM og FM hensættes af resterende retrospektve reserve. BP kan nu opdeles: Resterende kollektve mdler efter (2) er KB Resterende RH efter (1) + (3) er IB 12/84

13 Opgørelse af IB: entydg metode? Opdelng af bonuspotentale er reguleret Regnskabsbkg., Blag 1: 46. Indvduelle bonuspotentaler: Den del af værden af forskrngstagernes bonusret, der er ndeholdt retrospektve hensættelser, jf. pkt. 57. Algortmen bygger på, at RH prmært anvendes som hensættelse tl GY, RM og FM RH prmært anvendes som hensættelse polcens forplgtelser Andre metoder eksempelvs: Rskomargen betragtes (fx Solvens II) som en kollektv størrelse: RM hensættes prmært af kollektve mdler Byg bonus-cashflows: anvend nutdsværd af dsse som IB På kant med defntonen af IB regnskabsbekendtgørelsen Mulgvs blver alt bonuspotentale ndvduelt 13/84

14 Hghlghts of the day Well-known tools: Thele s dfferental equaton for state wse reserves Kolmogorov s forward dfferental equatons for cash flows Runge-Kutta methods for solvng ODE s GLM for statstcal nference n Markov models Tradtonal products (Gennemsntsrentemljøet): Determne value of tax (PAL) Determne value of future profts Include surrender modellng Unt-lnked products: Establsh market consstency between savngs and cash flows Apply same methods as for tradtonal products PAL, future profts, surrender modellng 14/84

15 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 15/84

16 Hghlghts of the classc lfe nsurance setup A model for (the contract of) the polcyholder 0, actve µ a µ a 1, dsabled µ ad µ d 2, dead Calculaton of the reserve (market consstent value): Thele s dfferental equaton Calculaton of expected future payments: Cash flows Kolmogorov s forward dfferental equaton 16/84

17 The classc lfe nsurance setup Fnte state space J = {0, 1,..., J} Markov process Z = (Z(t)) t 0 on J 0, actve µ a µ a 1, dsabled µ ad µ d 2, dead Transton probabltes p j (t, s) = P(Z(s) = j Z(t) = ) Transton rates 1 µ j (t) = lm h 0 h p j(t, t + h), j µ. (t) = j µ j (t) We usually defne the dstrbuton of Z by choosng the transton rates 17/84

18 Payments Payments assocated wth Sojourns n states b (t) Transtons between states b j (t) Jump countng process N j (t) = #{s (0, t] Z(s ) =, Z(s) = j} Payment process db(t) = 1 {Z(t)=} b (t) dt + b j (t) dn j (t) J,j J, j Reference: [Buchardt and Møller, 2015] 18/84

19 Prospectve reserve Present value of payments PV (t) = where r(t) s the (short) nterest rate Defnton (Prospectve reserve) t e s t r(u) du db(s) The (statewse) prospectve reserve at tme t n state s denoted V (t): V (t) = E [PV (t) Z(t) = ] [ = E e ] s t r(u) du db(s) Z(t) = t 19/84

20 Prospectve reserve: Thele s dff. eq. Theorem (Prospectve reserve) The prospectve reserve at tme t n state satsfes V (t) = e ( s t r(u) du p j (t, s) b j (s) + ) µ jl (s)b jl (s) ds. t j J l j Theorem (Thele s dfferental equaton) The prospectve reserve at tme t n state solves the dfferental equaton d dt V (t) = r(t)v (t) b (s) µ j (s) (b j (s) + V j (s) V (s)) j wth boundary condtons V (T ) = 0. 20/84

21 Two useful lemmas To show the formula for the prospectve reserve, two (very useful) results are needed Lemma By the defnton of p j (t, s) E [ 1 {Z(s)=j} Z(t) = ] = p j (t, s) Lemma For any predctable functon F (s) [ s ] E F (s) dn lj (s) Z(t) = = t s (shown usng the predctable compensator lemma) t F (s)p l (t, s)µ lj (s) ds 21/84

22 Cash flow Defnton (Cash flow) The accumulated cash flow at tme t n state assocated wth the payment process (B(t)) t 0 s the functon s A (t, s), A (t, s) = E [B(s) B(t) Z(t) = ] If A (t, ds) has a densty wth respect to the Lebesgue measure A (t, ds) = a (t, s) ds, we refer to a (t, s) as the cash flow. Theorem (Cash flow) The cash flow exsts and a (t, s) = ( p j (t, s) b j (s) + j J l j ) µ jl (s)b jl (s) 22/84

23 Cash flow From the defnton of the prospectve reserve [ V (t) = E e ] s t r(u) du db(s) Z(t) = = = = = t t t t t e s t r(u) du d(e [B(s) Z(t) = ]) e s t r(u) du d(e [B(s) B(t) Z(t) = ]) e s t r(u) du da (t, s) e s t r(u) du a (t, s) ds From the theorem on the prospectve reserve V (t) = e ( s t r(u) du p j (t, s) b j (s) + t j J l j ) µ jl (s)b jl (s) ds 23/84

24 Kolmogorov s dfferental equatons Theorem (Kolmogorov s backward dfferental equaton) The transton probabltes p j (t, s) solve d dt p j(t, s) = µ. (t)p j (t, s) l J,l µ l (t)p lj (t, s) for J, wth boundary condtons p j (s, s) = 1 {=j}. One solve yelds p j (t, s) for all t and for fxed s and j For the cash flow, t and s fxed, and s and j vares... a (t, s) ds = ( p j (t, s) b j (s) + ) µ jk (s)b jk (s) ds j J We need to solve for each s and j k:k j 24/84

25 Kolmogorov s dfferental equatons Theorem (Kolmogorov s forward dfferental equaton) The transton probabltes p j (t, s) solve d ds p j(t, s) = p j (t, s)µ j. (s) + l J,l j wth boundary condtons p j (t, t) = 1 {=j}. p l (t, s)µ lj (s) One solve yelds p j (t, s) for all s and j for fxed t and For the cash flow a (t, s) ds = j J We only need to solve once! ( p j (t, s) b j (s) + k:k j ) µ jk (s)b jk (s) ds 25/84

26 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 26/84

27 Hghlghts of the modellng of polcyholder behavour Incluson of surrenders by straghtforward extenson Requres the surrender payment to be determnstc Incluson of free polcy conversons requres a trck Kolmogorov s ρ-modfed dfferental equaton Sgnfcant mpact on future cash flows 27/84

28 The (Dansh) setup: Wth proft products 1. Polcyholder pays premum(s) 2. Lfe/penson nsurance company guarantees certan benefts Polces are valued wth 2 valuaton bases Techncal bass: Safe-sde Determnes premums and guaranteed payments. Conservatve (low) nterest rate r Safe-sde mortalty rate, dsablty rate, etc Market bass: Best estmate Determnes balance-sheet value of labltes (guaranteed payments). Market nferred forward nterest rate f r. Best estmate mortalty rate, dsablty rate, etc. 28/84

29 Polcyholder behavour 2 polcyholder optons Surrender cancel all future payments, and receves polcy value, accordng to the techncal bass Free polcy (eqv. pad-up polcy) cancel all future premums, and the benefts are reduced, accordng to the techncal bass Optons are based on the techncal bass: Introduces rsk on the market bass (only) Market based valuaton should nclude polcyholder behavour 29/84

30 Polcyholder behavour: Surrender modellng Model polcyholder behavour through random transtons. 3, surrender µ sur 0, actve 1, dsabled 2, dead J Surrender transton: payment V 0 (t) Cash flow a s (t, s) = j J ( p j (t, s) b j (s) + k:k j + p 0 (t, s)µ sur (s)v 0 (s). ) µ jk (s)b jk (s) 30/84

31 State space: Surrender & free polcy 3, surrender µ sur 0, actve 1, dsabled µ free 2, dead J 7, surrender free polcy µ sur 4, actve free polcy 6, dead free polcy 5, dsabled free polcy J f Free polcy at tme t: future payments reduced by factor ρ(t) Introduce duraton U(t): Tme snce free polcy converson: 31/84

32 Free polcy payments Free polcy at tme t: Premums cancelled Future payments reduced by factor ρ(t U(t)) db fs (t) = j J 1 {Z(t)=j} db j (t) + +V0 (t) dn Act,Sur (t) ( + ρ(t U(t)) 1 {Z(t)=j} db + j (t) + j J f j,k J,j k + V,+ 0 (t) dn ActFree,Sur (t) j,k J f,j k ). b jk (t) dn jk (t), b jk (t) + dn jk (t) 32/84

33 Free polcy cash flow Defne ρ-modfed transton probabltes for j J f, s p ρ j (t, s) = p 0 (t, τ)µ free (τ)ρ(τ)p ActFree,j (τ, s) dτ. t Proposton The cash flow s, for J, a fs (t, s) = a s (t, s) + ( (t, s) b j (s) + + ) µ jk (s)b jk (s) + j J f p ρ j + p ρ,actfree (t, s)µ sur(s)v,+ 0 (s) k J f k j Expensve to calculate p ρ j : need a lot of transton probabltes... 33/84

34 p ρ forward dfferental equaton Theorem p ρ j (t, s) satsfy, d ds pρ j (t, s) = 1 {j=actfree}p 0 (t, s) µ free (s)ρ(s) p ρ j (t, s) µ j.(s) + p ρ l (t, s) µ lj(s) p ρ j (t, t) = 0 Compare to Kolmogorov forward dff. eq. l J f l j d ds p j(t, s) = p j (t, s)µ j. (s) + p l (t, s)µ lj (s) l J l j 34/84

35 Numercs: cash flows + polcyholder behavour Example 40 year old male Penson age 65 Lfe annuty, sze Premum 10,000 per year Savngs of 100,000 The techncal bass conssts of Dansh G82M mortalty rate Interest rate r = 1.5% 2-state survval Markov model 35/84

36 Free polcy beneft scalng factor ρ ρ Free polcy factor ρ age 36/84

37 Market bases assumptons Dansh FSA benchmark mortalty Surrender rate: µ sur (x) = (x 40) + Free polcy rate: µ free (x) = 0.05 rate Transton rates Mortalty Surrender Free polcy probablty Transton probabltes Actve Dead Surrendered Free pol., actve Free pol., dead age age 37/84

38 Total cash flow Total cash flow yearly payment age Basc Surrender Sur + free pol 38/84

39 Interest rate senstvty of prospectve reserves Prosp. reserve 0e+00 3e+05 6e+05 Prospectve Reserve Basc Surrender Sur + free pol Techncal reserve Change n nterest rate Basc Surrender Sur. and free pol. Prospectve reserve DV01 Total /84

40 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 40/84

41 Hghlghts of numercal soluton of ordnary dfferental equatons The Euler method s ntutvely appealng Runge-Kutta methods sgnfcantly mproves precson/effcency when solvng ODE s 41/84

42 Numercal soluton of ordnary dfferental equatons Dfferental equaton problem: x (t) = f (t, x(t)), x(a) = x 0. Example wth x(t) = e t, thus f (t, x) = x /84

43 Numercal soluton of ordnary dfferental equatons Euler method wth step sze h x(t + h) = x(t) + h f (t, x(t)). Problem wth x(t) = e x : fnd x(1) from x (t) = x(t) and boundary condton x(0) = x(1) = x(1) = /84

44 Numercal soluton of ordnary dfferental equatons Second order Runge-Kutta x(t + h) = x(t) + h 1 2 (F 1 + F 2 ), F 1 = f (t, x(t)), F 2 = f (t + h, x(t) + hf 1 ). The Runge-Kutta 2 method uses average slope of F 1 and F 2 : F 1 s the Euler -slope F 2 s the slope at the end of the nterval (usng an Euler-predcton for the value of x). x(1) = x(1) = /84

45 Numercal soluton of ordnary dfferental equatons Fourth order Runge-Kutta x(t + h) = x(t) + h 1 6 (F 1 + 2F 2 + 2F 3 + F 4 ), F 1 = f (t, x(t)), F 2 = f (t h, x(t) hf 1), F 3 = f (t h, x(t) hf 2), F 4 = f (t + h, x(t) + hf 3 ). x(1) = x(1) = /84

46 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 46/84

47 Hghlghts of nference n Markov models Statstcal nference s based on occurences and exposures Easly handled wth Generalsed lnear models: The GLM framework s a flexble and wdely-used tool for regresson 47/84

48 Inference n Markov models Portfolo of lfe nsurance contracts: Observed behavour lnked to states n a Markov model 0, actve µ a µ a 1, dsabled µ ad µ d 2, dead How to determne the transton rates from observed behavour? 1 µ j (t) = lm h 0 h p j(t, t + h), j Based on observatons from each person: Exposure: total tme spent n a state Occurrence: count of transtons between two states 48/84

49 Inference n Markov models: How In practce: dependent on covarates: age, gender, portfolo, duraton (requres a sem-markov setup), har-color,... Let κ be the covarate-vector Let K be the set of possble covarates: Example wth gender and age: K = {male, female} {0, 1,..., 120} κ = (g, x) K O j g,x observed jumps j for gender g whle age x Eg,x observed years of sojourns n state for gender g age x Occurence and exposure are measured for each ndvdual and aggregated for each covarate κ. 49/84

50 Inference n Markov models: Posson & GLM Statstcal model O j κ Posson(E κ µ j κ(θ)), κ K, θ Θ. Estmaton of θ based on mnmzaton of the log-lkelhood l(θ) = κ K ( E κ µ j κ(θ) O j κ log(e κµ j κ(θ)) ) Not the correct model: the exposure s stochastc more jumps possble than the observed number of persons The lkelhood s correct! (can be shown) the estmaton of parameters s correct we contnue 50/84

51 Inference n Markov models: Posson & GLM Easy wthn the GLM framework. Assume µ j κ s log-lnear n κ. µ j κ = exp (θ 1 κ θ n κ n ), O j κ Posson(E κ µ j κ(θ)), θ R n. Then apply the GLM framework wth Dstrbuton: Posson Lnk functon: log Lnear predctor: η κ = θ 1 κ θ n κ n Offset: log E κ yeldng the model O j κ Posson log E [ O j κ ] = log E κ + θ 1 κ θ n κ n. (Recall: f X Posson(θ) then EX = θ). 51/84

52 Example of mortalty ft n R Ft Dansh mortalty: Exposure and occurence obtaned from HMD ( Years Ages Ft mortalty ntensty by gender (factor) and age (contnuous) µ g,x = exp (α g + β g x). Extract of dataset oe: Year Age Gender Occ Exp Female Male Female Male Female Male /84

53 Example of mortalty ft n R The model n GLM terms: O g,x Posson ( µ g,x {{ E g,x exp(η g,x ) ) exp(η g,x ) = exp (α g + β g x) In R: use the glm functon loge [O g,x ] = log E g,x + α g + β g x. glm ( Occ Gender Age, famly = posson ( l o g ), o f f s e t = log ( Exp ), data = oe ) The formula determnes the model for the lnear predctor η g,x Gender + Age s an addtve model: η g,x = α g + βx Gender : Age s a cross-effect: η g,x = β g x Gender * Age s both: η g,x = α g + β g x 53/84

54 Example of mortalty ft n R Extract of R output E s t m a t e Std. E r r o r z v a l u e Pr ( > z ) ( I n t e r c e p t ) e <2e 16 GenderMale e <2e 16 Age e <2e 16 GenderMale : Age e <2e 16 The fnal model becomes µ g,x = exp (α g + β g x) α female = α male = β female = β male = /84

55 Example of mortalty ft n R Plot of ftted model wth Occurence-Exposure rates OE g,x = O g,x E g,x, E [OE g,x ] = µ g,x. Intenstet Mænd Kvnder log Intenstet 5e 04 5e 03 5e 02 5e 01 Mænd Kvnder Alder Alder Full R-code: 55/84

56 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 56/84

57 Extensons of the classc Markov setup Stochastc penson age: Model the penson age as a stochastc tme, and scale penson benefts accordngly [Gad and Nelsen, 2015] Sem-Markov: Duraton dependent transton rates and payments [Buchardt et al., 2015] Collectve products: ncludng states for spouses of all ages, wthout assumng knowledge about the state of the polcyholder See followng sldes Causes of dsablty: Includng states for dfferent causes of dsablty, wth dfferent recovery and mortalty rates Wthn the classc Markov setup 57/84

58 Collectve products: The model J obs 0 µ M x+t 1 Sngle γ f 1 σ + µ K 1 γ f N 2 Marred, 1 N + 1 σ + µ K N Marred, N Dead µ M x+t µ M x+t N + 2 Wdow, 1 2N + 1 Wdow, N µ K x 1 +t 2N + 2 Both dead µ K x N +t Cash flow, condtonal on Z(t) = j, da j (t, τ) = p jk (t, τ) db k (τ) + µ kl (τ)b kl (τ)dτ k J l: l k 58/84

59 Cash flow n the collectve model Defne, for j J obs, p 0j(s, t) = P(Z(t) = j Z(t) J obs, Z(s) = 0) = p 0j (s, t) l J obs p 0l(s, t). Cash flow, condtonal on Z(t) J obs : dã(t, τ) = j J obs p 0j(0, t)da j (t, τ). Smplfcaton: Rewrte to p 0k (s, t, τ) = p0j(s, t)p jk (t, τ) j J obs dã(t, τ) = p 0k (0, t, τ) db k (τ) + µ kl (τ)b kl (τ)dτ. k J l: l k 59/84

60 Dfferental equaton for p 0k The probabltes p 0k (s, t, τ) = j J obs p 0j(s, t)p jk (t, τ) solve Kolmogorov s forward dff. eq. (n τ), wth boundary condton at tme t: p 0k (s, t, t) =p0k (s, t), τ p 0k(s, t, τ) = d p dτ 0j(s, t)p jk (t, τ) j J obs = p0j(s, t) p jg (t, τ)µ gk (τ) p jk (t, τ)µ k. (τ) j J obs g:g k = p 0g (0, t, τ)µ gk (τ) p 0k (0, t, τ)µ k. (τ). g:g k 60/84

61 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 61/84

62 Zero coupon rates and forward rates Zero coupon bond prce at tme t (today) exprng at T : P T t Contnuously compoundng zero coupon bond rates: R T t Yearly compoundng zero coupon rates: L T t Instantaneous forward rate f T t at tme t for T : f T t Relaton between dfferent quanttes: P T t = e RT t (T t) = 1 T (1 + L T = e t ft τ dτ t ) T t and f T t = T log PT t Rearrange: log(1 + L T t ) = 1 T ft τ dτ T t t 62/84

63 Dscretely observed zero coupon rates Assume that we only observe the curves L T t, T = t, t + 1, t + 2,..., T For 0 < s < 1, defne Note that Contnuous forward rates f T +s t = L T t +s = (1 s)l T t + sl T t +1 d T +s L t = L T t +1 L T t ds +1 (T + s t)(lt t L T t ) + log( L T +s t L T +s t ) Not defned for s = 0 or s = 1. Use lmts f T + t and f T t 63/84

64 Market reserves and forward rates Payment process db(t) = J 1 {Z(t)=} b (t) dt + b j (t) dn j (t),j J j Use forward rate curve avalable at tme t, f s t State wse market market consstent reserves [ V (t; t ) = E e ] s t f τ t dτ db(s) Z(t) = t = t e s t f τ t dτ j J p j (t, s) b j(s) + k J k j µ jk (s)b jk (s) ds 64/84

65 Thele s dfferental equaton and cash flows Thele wth forward rates d dt V (t; t ) =ft t V (t; t ) b (t) µ j (t) ( b j (t) + V j (t; t ) V (t; t ) ) j J,j Cash flow representaton V (t; t ) = t e s t f τ t dτ da (t, s) da (t, s) = p j (t, s) b j(s) + µ jk (s)b jk (s) ds j J k J k j 65/84

66 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 66/84

67 Expected future reserves Expected reserve at tme u, gven nformaton avalable at t: W (t, u; t ) = j p j (t, u)v j (u; t ) = u e s u f τ t dτ da (t, s) Proof: [ ] E 1 {Z(u)=l} V l (u; t ) Z(t) = = l = = l J u u p l (t, u) u e s u f τ t dτ j J e s u f τ t dτ da l (u, s) p j (t, s) e s u f τ t dτ da (t, s) b j(s) + k J k j µ jk (s)b jk (s) ds 67/84

68 Market values wth tax: Thele Notaton: Assume tax s payable contnuously wth rate φ Market reserve at t for usual payments and tax: V Φ (t; t ) Market consstent return for (τ, τ + dτ): f τ (τ, t ) dτ Introduce state wse tax payment functon: b Φ (t; t ) = φf t t V Φ (t; t ) Thele s dfferental equaton wth tax: d dt V Φ (t; t ) =f t t V Φ (t; t ) b (t) b Φ (t) ( µ j (t) b j (t) + Vj Φ j J,j t V Φ ) (t; t ) V Φ (t; t ) 68/84

69 Market values wth tax: Thele and cash flow representaton Rearrange Thele: d dt V Φ (t; t ) =(1 φ)ft t V Φ (t; t ) b (t) ( ) µ j (t) b j (t) + Vj Φ (t; t ) V Φ (t; t ) Soluton V Φ (t; t ) = W Φ (t, u; t ) = j t j J,j e s t (1 φ)f τ t dτ da (t, s) p j (t, u)v Φ j (u; t ) = u e s u (1 φ)f τ t dτ da (t, s) Classcal result Market value ncludng tax can be obtaned by multplyng forward rate wth (1 φ) Only orgnal cash flow da (t, s) s needed 69/84

70 Cash flows wth tax Cash flow for guaranteed payments and tax: da Φ (t, s; t ) = da(t, s) + j = da(t, s) + j p j (t, s)b Φ j (s; t ) ds p j (t, s)φf s t V j Φ (s; t ) ds = da(t, s) + φf s t W Φ (t, s; t ) ds. Second part of cash flow s term structure dependent Integral equaton for reserve for guaranteed payments and tax: V Φ (t; t ) = t e s t f τ t dτ ( da (t, s) + φf s The tax cash flow s calculated explctly t W Φ ) (t, s; t ) ds 70/84

71 Cash flows wth tax Value of future tax payments V Φ (t; t ) V (t; t ) = = t t (e s t (1 φ)f τ t dτ e s t f τ t dτ ) da (t, s) e s t f τ t dτ φf s t W Φ Algorthm for calculatng the tax cash flow: (t, s; t ) ds Solve Kolmogorov s forward equaton to obtan p j (t, s) Calculate ordnary cash flow da (t, s) Solve Thele s dfferental equaton to obtan V Φ j (s, t ) Calculate term structure dependent tax cash flow from Vj Φ (s, t ), p j (t, s) and ft s Another possblty: Calculate W Φ (t, s, t ) backwards wth the cash flow da (t, s) 71/84

72 Future profts Future profts rate (before tax): γ Future profts rate (after tax): (1 φ)γ Nave defnton: b Γ(t; t ) = (1 φ)γv Φ (t; t ) Here, future profts are lnked to reserve V Φ (t; t ) for guaranteed payments and tax Better defnton: b Γ(t; t ) = (1 φ)γv Φ,Γ (t; t ) Here, future profts are lnked to reserve V Φ,Γ (t; t ) for guaranteed payments, tax and profts Note that we have not yet determned V Φ,Γ (t; t ) We use the second defnton n the followng sldes 72/84

73 Thele s dfferental equaton wth future profts 1/2 Payment functons Contractual payment functon: b (t) Tax payments: b Φ (t; t ) = φf t t V Φ,Γ (t; t ) Future profts: b Γ (t; t ) = (1 φ)γv Φ,Γ (t; t ) Thele equaton d dt V Φ,Γ { (t; t ) = f t t (1 φ)(f t t γ)v Φ,Γ (t;t ) { (t; t ) b Φ (t; t ) b Γ (t; t ) b (t) ( ) µ j (t) b j (t) + V Φ,Γ j (t; t ) V Φ,Γ (t; t ) V Φ,Γ j J,j 73/84

74 Thele s dfferental equaton wth future profts 2/2 Two solutons: V Φ,Γ (t; t ) = Here = t t da Γ (t, s; t ) = j J e s t (1 φ)(f τ t γ)dτ da (t, s; t ) e ( ) s t (1 φ)f τ t dτ da (t, s; t ) + da Γ (t, s, t ) p j (t, s)(1 φ)γv Φ,Γ (t; t ) ds Intepretaton: = (1 φ)γw Φ,Γ (t, s; t ) ds Market value wth tax φ and future profts (1 φ)γ can be calculated from orgnal cashflow and dscount rate (1 φ)(f γ) Alternatvely, we can use term structure dependent future profts cash flow 74/84

75 Future profts: Two approaches Approach 1: Calculate market value ncludng future profts usng modfed forward rate (1 φ)(f γ) Determne value of future profts as resdual t (e s t (1 φ)(f τ t γ)dτ e s t (1 φ)f τ t dτ ) da (t, s; t ) Only the contractual cash s needed Approach 2: Use Thele s dfferental equaton to determne γv Φ,Γ (t; t ) Solve Kolmogorov s dfferental equaton Calculated term structure dependent cash flow da Γ (t, s; t ) 75/84

76 Incluson of surrender modellng Include assumptons about future surrenders Assume market consstent value ( omvalgsbonus ) s pad out upon surrender: Identcal to market consstent value (before (and after) ncluson of surrender) Does not ntroduce rsk on the market bass: Unaffected MCV Sgnfcant mpact on expected cash flows: Beneft payments after retrement reduced Surrender payments before retrement ntroduced Sgnfcant mpact on future profts: Shorter cash flows Less funds under management less profts 76/84

77 Agenda Introducton The classc lfe nsurance setup Polcyholder behavour modellng Numercal soluton of ordnary dfferental equatons Inference n Markov models Extensons of the classc Markov setup Zero coupon rates and forward rates Term structure dependent cash flows for tradtonal products Term structure dependent cash flows for unt-lnked products 77/84

78 Unt-lnked contracts wthout guarantees Standard approach for unguaranteed products Use market consstent smulaton model for savngs (wth return from forward rate curve) Implement polcyholder behavour Construct all cash flows Dscount cash flows to calculate market consstent values Relatvely complex models Our approach Apply same methods as for tradtonal products All accountng quanttes obtaned by solvng sutably parameterzed Thele or Kolmogorov equatons Establsh market consstent payments (constant profle) (assumptons about payment profle can be relaxed) 78/84

79 Cash flows wthout future profts Notaton Savngs at tme t: V,t (free polcy) Unt-payments: b (s) and b j (s) For example, b j (s) = 1 {s n} for some tme n and some state j Tax adjusted forward rate: (1 φ)f τ t Market consstent value of unt-payments: V,Φ (t; t ; f ) Market consstency between savngs and (unguaranteed) payments V (t) = κ(t; t )V,Φ (t; t ; f ) Remarks κ(t; t ) depends on forward rate curve κ(t; t ) s recalculated for each new forward rate curve Term structure dependent, unguaranteed, market consstent cash flows: κ(t; t ) da (t, s) 79/84

80 Cash flows wth future profts Notaton Tax and proft adjusted forward rate: (1 φ)(f τ t γ) Value of unt-payments wth profts: V,Φ (t; t ; f γ) Market consstency: V (t) = Φ,Γ V (t) = κ Γ (t; t )V,Φ (t; t ; f γ) Remarks Cash flow to polcyholder: κ Γ (t; t ) da (t, s) MCV of payments to polcyholder: κ Γ (t; t )V,Φ (t; t ; f ) MCV of future profts: κ Γ (t; t ) (V (t; t ; f γ) V (t; t ; f )) Future profts unt cash flow: da Γ (t, s; t ) =κ Γ (t; t ) da,γ (t, s; t ) =κ Γ (t; t )γ(1 φ)w,φ (t, s; t ; f γ) ds 80/84

81 Cashflows wth future profts and surrender 1/2 Rsk premum n Thele assocated wth surrender s 0, ( ) µ,sur (s) V Φ,Γ (t, s; t Φ,Γ ) V (t, s; t ) = 0 Thus, polcyholder s state wse account unaffected by surrender Thele s dfferental equaton wth tax, profts and surrender d Φ,Γ V (t, s; t ) ds = (1 φ)(ft s Φ,Γ γ) V (t, s; t ) κ Γ (t; t )b (s) ( µ j (s) κ Γ (t; t )bj(s) Φ,Γ + V j (t, s; t ) j J,j µ,sur (s) ( V Φ,Γ (t, s; t ) ) Φ,Γ V (t, s; t ) ) Φ,Γ V (t, s; t ) 81/84

82 Cashflows wth future profts and surrender 2/2 Value of unt-payments wth profts (and surrender): V,Φ (t; t ; f γ; V,Φ (t; t ; f γ)) Unaffected by surrender (snce same forward rate used) Remarks Equvalence not affected MCV of payments to polcyholder and future profts change: κ Γ (t; t )V,Φ (t; t ; f ; V,Φ (t; t ; f γ)) MCV of future profts can be calculated resdually V (t) κ Γ (t; t )V (t; t ; f ; V,Φ (t; t ; f γ)) ( = κ Γ (t; t ) V (t; t ; f γ; V,Φ (t; t ; f γ)) ) V (t; t ; f ; V,Φ (t; t ; f γ)) 82/84

83 Cashflows wth premums, future profts and surrender Assume that the polcyholder also pays premums π(t) Market consstent equvalence between savngs, premums and unguaranteed benefts V (t) = κ Γ (t; t )V,+,Φ π(t)v,,φ (t; t ; f γ; V,+,Φ (t; t ; f γ)) (t; t ; f γ; V,,Φ (t; t ; f γ)) Value of the payments to the polcyholder are obtaned by dscountng the payments wth the unchanged forward rate f κ Γ (t; t )V,+,φ (t; t ; f ; V,+,Φ (t; t ; f γ)) π(t)v,,φ (t; t ; f ; V,,Φ (t; t ; f γ)) Value of the future profts are obtaned resdually as ( ) κ Γ (t; t ) V,+,Φ (t; t ; f γ; V,+,Φ (t; t ; f γ)) V,+,Φ (t; t ; f ; V,+,Φ (t; t ; f γ)) ( ) π(t) V,,Φ (t; t ; f γ; V,,Φ (t; t ; f γ)) V,,Φ (t; t ; f ; V,,Φ (t; t ; f γ)) 83/84

84 References I Buchardt, K. and Møller, T. (2015). Lfe nsurance cash flows wth polcyholder behavor. Rsks, 3: Buchardt, K., Møller, T., and Schmdt, K. B. (2015). Cash flows and polcyholder behavour n the sem-markov lfe nsurance setup. Scandnavan Actuaral Journal, 2015(8): Gad, K. S. T. and Nelsen, J. W. (2015). Reserves and cash flows under stochastc retrement. Scandnavan Actuaral Journal. 84/84