Life insurance cash flows with policyholder behaviour


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1 Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension, Sundkrogsgade 4, DK2100 Copenhagen Ø, Denmark November 3, 2013 Absrac The problem of valuaion of life insurance paymens wih policyholder behaviour is sudied. Firs a simple survival model is considered, and i is shown how cash flows wihou policyholder behaviour can be modified o include surrender and free policy behaviour by calculaion of simple inegrals. In he second par, a more general disabiliy model wih recovery is sudied. Here, cash flows are deermined by solving a modified Kolmogorov differenial equaion. This mehod has been suggesed recenly in Buchard e al. [2]. We conclude he paper wih numerical examples illusraing he impac of modelling policyholder behaviour. Keywords: Kolmogorov differenial equaions; surrender; free policy; Solvency 2 1 Inroducion In a classic mulisae life insurance seup, we consider how o include he modelling of policyholder behaviour when calculaing he expeced cash flows. In his paper, he policyholder behaviour consiss of wo policyholder opions. Firs, he surrender opion, where he policyholder may surrender he conrac cancelling all fuure paymens and insead receiving a single paymen corresponding o he value of he conrac on a echnical basis. Second, he free policy opion 2, where he policyholder may cancel he fuure premiums, and have he benefis reduced according o he echnical 1 Corresponding auhor, 2 he free policy opion is someimes referred o as a paidup policy in he lieraure. 1
2 basis. Policyholder modelling has a significan influence on fuure cash flows. If he echnical basis differs considerably from he marke basis, policyholder behaviour can also have a subsanial impac on he marke value of he conrac. The policyholder behaviour is modelled as random ransiions in a Markov model, and raionaliy behind surrender and free policy modelling is hus disregarded. In conras, one can consider surrender and free policy exercises as raional, where hey purely occur if i is benificial for he policyholder wih some objecive measure. For an inroducion o policyholder modelling, see [7] and references herein. Aemps o couple he wo approaches have been made for surrender behaviour, where surrender occurs randomly, bu where he probabiliy is somewha conrolled by raional facors, e.g. [4] and [1]. From a Solvency II poin of view, he modelling of policyholder behaviour is required, see Secion 3.5 in [3]. In he firs par of he paper a simple survival model is considered. We calculae cash flows wihou policyholder behaviour as inegral expressions. Then we exend he model by including firs surrender behaviour and hen boh surrender and free policy behaviour. We see ha hese exensions can be obained via simple modificaions of he cash flows wihou policyholder behaviour. This can be viewed as a formula for exending curren cash flows wihou policyholder behaviour. However, his modificaion of he cash flows is only correc for he survival model, and no for e.g. a disabiliy model. If he mehod is applied o cash flows from a disabiliy model, i could be viewed as an approximaion o a more correc way of modelling policyholder behaviour. Also, we show ha he cash flows wih policyholder behaviour can be derived from cash flows wih surrender behaviour. This mehod can be used in he case where one has access o cash flows wih surrender behaviour bu no free policy behaviour. In pracice, many life insurance companies work wih cash flows wihou policyholder behaviour, hence, he proposed mehod may be viewed as a simple alernaive o full, combined modelling of policyholder behaviour and insurance risk. The qualiy of hese formulae as an approximaion is no assessed in his paper; This issue is sudied numerically in [5], where hey examine ways o simplify he calculaions when modelling policyholder behaviour. In he second par, we consider he more correc way of modelling policyholder behaviour in a mulisae life insurance seup. This model is presened in [2] for he general semi Markov seup, and here we presen he special case of a Markov process for he disabiliy model wih recovery. Wihin his seup, he ransiion probabiliies are firs calculaed using Kolmogorov s differenial equaions, and hen he cash flow can be deermined. When including policyholder behaviour, duraion dependence is inroduced since he fuure paymens are affeced by he ime of he free policy conversion. This complicaes calculaions significanly. We presen he main resul from [2] ha allow us o effecively dismiss he duraion dependence and calculae cash flows wih policyholder behaviour by simply calculaing a slighly modified Kolmogorov differenial equaion. The complexiy 2
3 of he calculaions is herefore no increased significanly by inclusion of policyholder behaviour. In he hird par of he paper a numerical example is sudied, which illusraes he imporance of including policyholder modelling when valuaing cash flows. We see ha he srucure of he cash flows changes significanly in our example, and he dollar duraion measuring ineres rae risk is reduced by more han 50%. For hedging of ineres rae risk, i is hus essenial o consider policyholder behaviour. 2 Life insurance seup The general seup is he classic mulisae seup in life insurance, consising of a Markov process, Z, in a finie sae space J {0, 1,..., J} indicaing he sae of he insured. We associae paymens wih sojourns in saes and ransiions beween saes, and his specifies he life insurance conrac. We go hrough he seup and basic resuls; for more deails, see e.g. [8], [6] or [7]. Assume ha Z is a Markov process in J, and ha Z0) 0. The ransiion probabiliies are defined by p ij s, ) P Z) j Zs) i), for i, j J and s. Define he ransiion raes, for i j, 1 µ ij ) lim h 0 h p ij, + h), µ i. ) j J j i µ ij ). We assume ha hese quaniies exis. Define also he couning processes N ij ), for i, j J, i j, couning he ransiions beween sae i and j. They are defined by N ij ) # {s 0, ] Zs) j, Zs ) i}, where we have used he noaion f ) lim h 0 f h). The paymens consis of coninuous paymen raes during sojourns in saes, and single paymens upon ransiions beween saes. Denoe by b i ) he paymen rae a ime if Z) i, and le b ij ) be he paymen upon ransiion from sae i o j a ime. Then, he accumulaed paymens a ime are denoed B), and are given by db) 1 {Z)i} b i ) d + b ij ) dn ij ). 2.1) i J i,j J i j 3
4 Posiive values of he paymen funcions b i ) and b ij ) correspond o benefis, while negaive values corresponds o premiums. I is also possible o include single paymens during sojourns in saes, bu ha is for noaional simpliciy omied here. We assume ha he ineres rae r) is deerminisic. Then, he presen value a ime of all fuure paymens is denoed P V ), and i is given by P V ) rτ) dτ dbs). The formula is inerpreed as he sum over all fuure paymens, dbs), which are discouned by rτ) dτ. For an acual valuaion, we ake he expecaion condiional on he curren sae, E [P V ) Z) i]. This expeced presen value is called he prospecive saewise) reserve. Definiion 2.1. The prospecive reserve a ime for sae i J is denoed V i ), and given as [ V i ) E e ] s rτ) dτ dbs) Z) i. The prospecive reserve can be calculaed using he following classic resuls. Proposiion 2.2. The prospecive reserve a ime given Z) i, i J, saisfies, V i ) rτ) dτ j J p ij, s) b js) + k J k j µ jk s)b jk s) ds. Proposiion 2.3. The prospecive reserve a ime given Z) i, i J, saisfies Thiele s differenial equaion, d d V i) r)v i ) b i ) j J,j i wih boundary condiions V i ) 0, for i J. µ ij ) b ij ) + V j ) V i )), Remark 2.4. If a imepoin T 0 exiss such ha b i ) b ij ) 0 for > T and all i, j J, hen he boundary condiions V i T ) 0 for i J are used wih Thiele s differenial equaion. I can be convenien o calculae no only he expeced presen value he prospecive reserve), bu also he expeced cash flow. From here on, we simply refer o he expeced cash flow as he cash flow, and i is a funcion giving he expeced paymens a any fuure ime s. The cash flow is, in his seup, independen of he ineres rae, and hus he cash flow can be useful for hedging and for an assessmen of he ineres rae risk associaed wih he life insurance liabiliies. 4
5 Definiion 2.5. The cash flow a ime associaed wih he paymen process B)) 0, condiional on Z) i, i J, is he funcion s A i, s), given by for s [, ). A i, s) E [Bs) B) Z) i], A formal calculaion yields an expression for he cash flow: From Definiion 2.1, we noe ha V i ) rτ) dτ de [Bs) Z) i]) rτ) dτ de [Bs) B) Z) i]) rτ) dτ da i, s), where we have used ha B) is a consan and doesn change he dynamics in s. We sae he resul in a proposiion. Proposiion 2.6. The cash flow A i, s) saisfies, V i ) rτ) dτ da i, s), da i, s) p ij, s) b js) + µ jk s)b jk s) ds. j J k J k j The second resul in Proposiion 2.6 follows from he firs resul and from Proposiion 2.2. In order o acually calculae he cash flow, one mus firs calculae he ransiion probabiliies p ij s, ). In sufficienly simple models, socalled hierarchical models, where you can no reurn o a sae afer you lef i, he ransiion probabiliies can be calculaed using only inegrals and known funcions. These kind of models are considered in Secion 3. In general Markov models, closed form expressions for he ransiion probabiliies ypically do no exis. Insead, he ransiion probabiliies can be found numerically by solving Kolmogorov s forward and backward differenial equaions. Proposiion 2.7. The ransiion probabiliies p ij, s), for i, j J, are unique soluions o Kolmogorov s backward differenial equaion, d d p ij, s) µ i. )p ij, s) µ ik )p kj, s), k J k i 5
6 2.1 Technical basis and marke basis wih boundary condiions p ij s, s) 1 {ij}, and Kolmogorov s forward differenial equaion, d ds p ij, s) p ij, s)µ j. s) + p ik, s)µ kj s), k J k j wih boundary condiions p ij, ) 1 {ij}. Using Kolmogorov s differenial equaions, he ransiion probabiliies needed in order o calculae he cash flow from Proposiion 2.6 can be found. I is worh noing, ha for calculaing he cash flow, he forward differenial equaions are he easies way o obain he desired ransiion probabiliies. 2.1 Technical basis and marke basis In pracice and in our examples, we disinguish beween calculaions on he socalled echnical basis, used o sele premiums, and he marke basis, used he calculae he marke consisen value of he life insurance liabiliies, referred o as he marke value. A basis is a se of assumpions used for he calculaions of life insurance liabiliies, and i ypically consiss of an ineres rae r) and a se of ransiion raes µ ij )) i,j J. There can also be differen adminisraion coss associaed wih differen bases, however adminisraion coss are no considered in his paper. The Markov model can also be differen in differen bases, and he policyholder behaviour modelling of his paper is an example of his. Here, policyholder behaviour is no included in he echnical basis, bu is included in he marke basis, so he Markov models differ by he surrender and free policy saes. Throughou he paper we le ˆr) and ˆµ ij ) be he firs order ineres and ransiion raes, respecively, i.e. he ineres and ransiion raes associaed wih he echnical basis. We le r) and µ ij ) be he ineres and ransiion raes, respecively, for he marke basis. In general, values marked wih aˆare associaed wih he echnical basis. Thus, V ) is he prospecive reserve for he marke basis, and ˆV ) is he prospecive reserve for he echnical basis. 2.2 The policyholder opions We sudy life insurance conracs wih wo opions for he policyholder. She can surrender he conrac a any ime or she can sop he premium paymens and conver he policy ino a socalled free policy. If he policyholder surrenders he conrac a ime, all fuure paymens are cancelled, and insead he policyholder receives a compensaion for he premiums she has paid 6
7 2.2 The policyholder opions so far. Usually, he prospecive reserve calculaed on he echnical basis, ˆV i ), is paid ou, bu he formula allows i o be any deerminisic value. In his paper, we allow for a deducible, and say ha he paymen upon surrender is 1 κ) ˆV i ). Since any deerminisic value can be chosen, in paricular, we can choose κ o be ime dependen. If he policyholder sops he premium paymens, i.e. exercises he free policy opion, all fuure premiums are cancelled, and he size of he benefis are decreased o accoun for he missing fuure premium paymens. If he free policy opion is exercised a ime, all fuure benefis are decreased by a facor ρ). In order o handle his, we spli he paymen process in posiive and negaive paymens, corresponding o benefis and premiums, respecively. The benefi and premium cash flows are denoed by A + and A, respecively, and are given by da + i, s) p ij, s) b js) + + µ jk s)b jk s) + ds, j J k J k j da i, s) p ij, s) b js) + µ jk s)b jk s) ds, j J k J k j where he noaion fx) + maxfx), 0) and fx) max fx), 0) for a funcion fx) is used. The prospecive reserve can hen be decomposed as well, and we have V + i ) V i ) rτ) dτ da + i, s), rτ) dτ da i, s), and V i ) V i + ) Vi ). The relaions also hold on he echnical basis, hus ˆV i ) ˆV i + ) ˆV i ), where ˆV i + ) and ˆV i ) are he values of he fuure benefis and premiums, respecively, valuaed on he echnical basis. If he free policy is exercised a ime, hen a fuure ime s, he paymen rae while in sae i is ρ)b i s) + and he paymen if a ransiion from sae i o j occurs, is ρ)b ij s) +. Hence, he prospecive reserve on he echnical basis a ime s in sae i, given he free policy opion is exercised a ime s, is s e u s ˆrτ) dτ ρ) dâ+ i + s, u) ρ) ˆV i s), where dâ+ i s, u) is he cash flow calculaed wih he firs order ransiion probabiliies and raes, deermined by ˆµ ij ). 7
8 The facor ρ) should be deerminisic and is usually chosen according o he equivalence principle on he echnical basis: The prospecive reserve for he echnical basis should no change as a consequence of he exercise of he free policy opion. We assume in his paper ha he free policy conversion can only occur from sae 0. Thus, if he free policy opion is exercised a ime, he prospecive reserve on he echnical basis before he free policy opion is exercised, ˆV ), should be equal o he prospecive reserve afer he exercise, ρ) ˆV + ). Thus, we require ˆV ) ρ) ˆV + ), yielding ρ) ˆV ) ˆV + ). Here, we omied he subscrip 0 from ˆV 0 ), and we do ha in general he res of he paper when here is no ambiguiy. We see ha ρ is he value on he echnical basis of benefis less premiums, divided by he value on he echnical basis of he benefis only. We refer o ρ) as he free policy facor. 3 The survival model We consider he survival model and exend i gradually o include policyholder behaviour. Firs, we include he surrender opion, and aferwards, we include he free policy opion as well. The survival model consiss of wo saes, 0 alive) and 1 dead), corresponding o Figure 1. 0, alive 1, dead Figure 1: Survival Markov model Assume he insured is x years old a ime 0. The paymens consis of a benefi rae b) and a premium rae π), and a paymen b ad ) upon deah a ime. Referring o he general seup, we have b 0 ) b) π), b 01 ) b ad ). and also, we denoe he moraliy inensiy µ 01 ) µ ad ). The prospecive reserve on he echnical basis a ime in sae 0 is given by Proposiion 2.2, and we ge ˆV ) ˆru) du s ˆp x+ bs) πs) + ˆµ ad s)b ad s)) ds, 8
9 3.1 Survival model wih surrender modelling We have used he acuarial noaion for he survival probabiliy, s ˆp x+ ˆp 00, s), and i is given by, ˆp x e 0 ˆµ adx+u) du. Thus, ˆp x is he survival probabiliy of an xyear old reaching age x +, calculaed on he echnical basis. The marke values of benefis and premiums, respecively, are hen given by V + ) V ) ru) du s p x+ bs) + µ ad s)b ad s)) ds, ru) du s p x+ πs) ds, and he associaed cash flows, condiioning on being alive a ime, are da +, s) s p x+ bs) + µ ad s)b ad s)) ds, da, s) s p x+ πs) ds, 3.1) wih V ) V + ) V ) and da, s) da +, s) da, s) being he oal prospecive reserve and cash flow, respecively. Here, we have omied he subscrip 0 from he noaion dâ0, s). The free policy facor is deermined by ρ) ˆV ) ˆV + ), where ˆV + ) is he value on he echnical basis of he benefis only. If he free policy opion is exercised immediaely, he marke value is ρ)v + ) ˆV ) ˆV + ) V + ), and in Denmark, his is ofen referred o as he marke value of he guaraneed free policy benefis. 3.1 Survival model wih surrender modelling We coninue he example from above and deermine he marke value including valuaion of he surrender opion. The Markov model is exended o include a surrender sae, corresponding o Figure 2. The surrender modelling is only included in he marke basis, and he valuaion on he echnical basis does no change. 9
10 3.1 Survival model wih surrender modelling 2, surrender 0, alive 1, dead Figure 2: Survival Markov model wih surrender. On he marke basis, we denoe he surrender rae by µ as ). We inroduce a quaniy p s x which is he probabiliy ha an xyear old does no die nor surrender before ime x +. I is hus he probabiliy of saying in sae 0, and is given by, s p s x+ : p 00, s) µ adτ)+µ asτ)) dτ s p x+ µasτ) dτ. Here, he ransiion raes µ ad and µ as are for an xyear old a ime 0, which for simpliciy is suppressed in he noaion. The paymen upon surrender a ime s is 1 κ) ˆV s), and he cash flow valuaed a ime is, by Proposiion 2.6, da s, s) s p s x+ bs) πs) + µ ad s)b ad s) + µ as s)1 κ) ˆV ) s) ds. 3.2) We decompose he cash flow in all paymens excluding he surrender paymens, and he surrender paymens, da s1, s) s p s x+ bs) πs) + µ ad s)b ad s)) ds µasτ) dτ da, s), da s2, s) s p s x+µ as s)1 κ) ˆV s) ds. Here, da, s) is he cash flow from he model in Figure 1, as defined by 3.1). The marke value calculaed on he marke basis including surrender is denoed V s ), and is given by V s ) + rτ) dτ da s1, s) + da s2, s) ) rτ) dτ µasτ) dτ da, s) rτ) dτ s p s x+µ as s)1 κ) ˆV s) ds. 3.3) We see ha he cash flow and marke value including surrender modelling are found using he original cash flow wihou surrender modelling, da, s), and muliplying he probabiliy of no surrender µasτ) dτ. Thus, finding he cash flow and he marke value in he survival model wih surrender is paricularly simple when he exising cash flow is known. 10
11 3.2 Survival model wih surrender and free policy modelling 3.2 Survival model wih surrender and free policy modelling We exend he model o include free policy modelling on he marke basis, and he Markov model is exended in Figure 3 o include free policy saes. The moraliy and surrender ransiion raes in he free policy saes are idenical o hose in he premium paying saes, µ ad and µ as. 2, surrender 0, alive 1, dead 5, surrender free policy 3, alive free policy 4, dead free policy Figure 3: Survival Markov model wih surrender and free policy. We inroduce a free policy rae µ af ), which is he ransiion rae of becoming a free policy a ime. We inroduce he noaion s p fs x+ µ adτ)+µ asτ)+µ af τ)) dτ µasτ)+µ afτ)) dτ s p x+ µ afτ) dτ s p s x+, which is he probabiliy of saying in sae 0, i.e. no becoming a free policy, surrendering nor dying. If he free policy ransiion occurs a ime, he fuure benefis are reduced by a facor ρ), and he fuure premiums are cancelled. Thus, in he free policy sae a a laer ime s, he paymen rae is ρ)bs), and he paymen upon deah is ρ)b ad s). The surrender paymen, if surrender occurs as a free policy, is ρ)1 κ) ˆV + s), where ρ) ˆV + s) is he prospecive reserve on he echnical basis. The paymen process is dependen on he exac ime of he free policy ransiion, i.e. he paymens are dependen on he duraion since he free policy ransiion. I can be shown ha he cash flow valuaed a ime is given by da f, s) s p fs bs) πs) + µ ad s)b ad s) + µ as s)1 κ) ˆV ) s) ds + x+ s τ p fs x+µ af τ) s τ p s x+τ ρτ)bs) + µ ad s)ρτ)b ad s) + µ as s)ρτ)1 κ) ˆV + s) ) dτ ds. 3.4) The resul can be obained as a special case of Proposiion 4.1 below, where he disabiliy rae is se o 0, bu for compleeness, a separae proof is given in Appendix A. The firs 11
12 3.2 Survival model wih surrender and free policy modelling line is he paymens in sae 0 and he paymens upon deah and surrender. The second and hird lines conain he paymens as a free policy. This expression can be inerpreed as he probabiliy of saying in sae 0 unil ime τ, hen becoming a free policy a ime τ, and hen neiher dying nor surrendering from ime τ o ime s. This is muliplied wih he paymens as a free policy a ime s, given he free policy occured a ime τ. Finally, we inegrae over all possible free policy ransiion imes from s o. The cash flow is decomposed ino four pars. Firs, he benefis and premiums, excluding surrender paymens, while alive and no a free policy, da f1, s) s p fs x+ bs) πs) + µ ad s)b ad s)) ds µasu)+µ afu)) du da +, s) da, s) ). 3.5) Then, he surrender paymens, if he free policy ransiion has no occured, da f2, s) s p fs x+µ as s)1 κ) ˆV s) ds µ afu) du s p s x+µ as s)1 κ) ˆV s) ds. 3.6) Noe ha hese cash flows correspond o he cash flows in he surrender model, bu reduced wih he probabiliy of he free policy ransiion no happening. The hird cash flow is he benefis while a free policy da f3, s) s s τ p fs x+µ af τ)ρτ) s τ p s x+τ dτ bs) + µ ad s)b ad s)) ds 3.7) τ p fs x+µ af τ)ρτ) τ µasu) du da + τ, s) dτ, and he fourh cash flow is he surrender paymens while a free policy, da f4, s) s τ p fs x+µ af τ)ρτ) s τ p s x+τ dτ µ as s)1 κ) ˆV + s) ds. 3.8) The hird cash flow 3.7) seems complicaed, since he cash flows a ime s evaluaed a ime τ, da + τ, s), is needed for any τ, s) and all s. However, a sraighforward calculaion yields, τ p fs x+µ af τ)ρτ) τ µasu) du da + τ, s) e τ µasu)+µ afu)) du µ af τ)ρτ) τ µasu) du τ p x+ da + τ, s) e τ µasu)+µ afu)) du µ af τ)ρτ) τ µasu) du da +, s), which simplifies hings, and inserion of his ino da f3 yields, s da f3, s) e τ µasu)+µ afu)) du µ af τ)ρτ)e ) s τ µasu) du dτ da +, s). 12
13 3.2 Survival model wih surrender and free policy modelling Define he quaniy and noe ha r ρ, s) s e τ µ afu) du µ af τ)ρτ) dτ, 3.9) da f3, s) r ρ, s) µasu) du da +, s), da f4, s) r ρ, s) s p s x+µ as s)1 κ) ˆV + s) ds. The marke value including surrender and free policy modelling is denoed V f ), and i may finally be wrien as V f ) e ) s rτ) dτ da f1, s) + da f2, s) + da f3, s) + da f4, s) rτ) dτ µasu)+µ afu)) du da +, s) da, s) ) rτ) dτ s p fs x+µ as s)1 κ) ˆV s) ds rτ) dτ r ρ, s) µasu) du da +, s) rτ) dτ r ρ, s) s p s x+µ as s)1 κ) ˆV + s) ds. The las four lines in 3.10) have he following inerpreaion. 3.10) The firs line is he value of he original cash flow 3.1) wihou policyholder behaviour, reduced by he probabiliy of no surrendering and no becoming a free policy. The second line is he value of he surrender paymens, when no a free policy. The hird line is he benefi paymens as a free policy, i.e. he posiive paymens reduced wih he free policy facor ρτ) a he ime τ of he free policy ransiion. The fourh line is he surrender paymens if surrender occurs afer he free policy ransiion. The formula gives he marke value of fuure guaraneed paymens, including valuaion of he surrender and free policy opions. In order o calculae he value, he following quaniies are needed The original cash flows da +, s) and da, s), 13
14 3.3 Free policy modelling when surrender is already modelled The prospecive reserve on he echnical basis ˆV + s) and ˆV s), for all fuure ime poins s, which allow us o deermine he surrender paymens and he free policy facor ρs). The facor r ρ, s), which is a simple inegral of he free policy ransiion rae. 3.3 Free policy modelling when surrender is already modelled In he previous secion, we found he marke value including surrender and free policy modelling based on cash flows wihou any policyholder behaviour modelling. I is also possible o find his marke value based on cash flows including surrender modelling. This could be relevan if he exising cash flows already include surrender modelling, and one wishes o modify hese cash flows o include free policy modelling. Thus, we assume ha he cash flow including surrender behaviour modelling, 3.2), are available, and ha i is spli in a cash flow associaed wih he benefis and a cash flow associaed wih premiums, i.e. da s,+, s) s p s x+ bs) + µ ad s)b ad s) + µ as s)1 κ) ˆV ) + s) ds, da s,, s) s p s x+ πs) + µ as s)1 κ) ˆV ) 3.11) s) ds. Noe ha he paymen upon surrender is spli beween he wo cash flows, hrough he decomposiion ˆV ) ˆV + ) ˆV ), i.e. he value of he fuure benefis less he value of he fuure premiums. The marke value wih surrender modelling, bu no free policy modelling, V s ) from 3.3), is hen given by V s ) rτ) dτ da s,+, s) da s,, s) ). We find he cash flow including free policy modelling by modifying he exising cash flows ino wo cash flows: One, which is reduced by he probabiliy of no becoming a free policy, and a special free policy cash flow. Wih a few calculaions using 3.5), 3.6) and 3.11), we see ha da f1, s) + da f2, s) s p fs x+ and also, by 3.7), 3.8), 3.9) and 3.11), da f3, s) + da f4, s) s τ p fs x+µ af τ)ρτ) s τ p s x+τ dτ r ρ, s) da s,+, s). bs) πs) + µ ad s)b ad s) + µ as s)1 κ) ˆV ) s) ds µ afu) du da s,+, s) da s,, s) ), bs) + µ ad s)b ad s) + µ as s)1 κ) ˆV ) + s) ds 14
15 The oal cash flow is hen given as da f, s) µ afu) du da s,+, s) da s,, s) ) + r ρ, s) da s,+, s). This cash flow can be inerpreed as a weighed average beween he original cash flow, reduced wih he probabiliy of no becoming a free policy, and he paymens as a free policy. The paymens as a free policy are he posiive paymens muliplied wih r ρ, s). The quaniy r ρ, s) is inerpreed as he probabiliy of becoming a free policy muliplied wih he free policy facor ρτ) a he ime τ of he free policy ransiion. The marke value from before, V f ), can hen be calculaed as V f ) + e ) s rτ) dτ da f1, s) + da f2, s) + da f3, s) + da f4, s) rτ) dτ µ afu) du da s,+, s) da s,, s) ) rτ) dτ r ρ, s) da s,+, s). If we only include surrender modelling, he needed exra quaniies are simple inegrals of he surrender rae µ as ). If we in addiion include free policy modelling, he free policy facor ρ) mus also be found, which requires access o fuure prospecive reserves on he echnical basis. When hese are found, he marke value is relaively simple o calculae. An essenial assumpion for hese calculaions is ha here are no paymens afer leaving he acive sae, i.e. ha he prospecive reserve is 0 in he dead and surrender saes. Tha is, afer he paymen upon deah or surrender, here are no fuure paymens. If one adds a disabiliy sae, similar simple resuls can only be obained if he prospecive reserve is 0 in he disabiliy sae. This is ypically no saisfied, and as such he mehods of modifying he cash flows presened here are no applicable. However, he mehod may be used as an approximaion o resuls obained wih more sophisicaed policyholder behaviour in a more general model, e.g. a disabiliy model. 4 A general disabiliy Markov model In his secion we consider he survival model exended wih a disabiliy sae, from which i is possible o recover. We exend he model furher by including saes for surrender and free policy, and end up wih an 8sae model, see Figure 4. By solving cerain ordinary differenial equaions for he relevan ransiion probabiliies and a special free policy quaniy, similar o r ρ from 3.9), he cash flow and prospecive reserve can be found. 15
16 3, surrender 0, acive 1, disabled 2, dead J 7, surrender free policy 4, acive free policy 5, disabled free policy 6, dead free policy J f Figure 4: The 8sae Markov model, wih disabiliy, surrender and free policy. The ransiion raes beween saes 0, 1 and 2 are idenical o he ransiion raes beween saes 4, 5 and 6. The wo surrender saes can be considered one sae, and hen his model is known as he socalled 7sae model. The resuls can easily be exended o more general Markov models han he disabiliy model, as long as free policy conversion only occurs from he acive sae 0. A more general seup is sudied in [2], which is here specialised o he case of he survivaldisabiliy model. For valuaion on he echnical basis, he survivaldisabiliy Markov model, consising of saes 0, 1 and 2, are used. In his secion, he paymens are labelled by he sae hey correspond o insead of he labels used previously. Thus, he paymen rae in sae 0, acive, is b 0 ) and in sae 1, disabled, i is b 1 ). Upon disabiliy here is a paymen b 01 ), upon deah as acive here is a paymen b 02 ), and upon deah as disabled here is a paymen b 12 ). The paymen funcion in sae 0, b 0 ), is decomposed in posiive paymens b 0 ) +, which are benefis, and negaive paymens, b 0 ), which are premiums. Thus, b 0 ) b 0 ) + b 0 ). We assume ha all oher paymens funcions are posiive. The noaion corresponds o he noaion used in 2.1) for he paymen funcions b 0 ), b 1 ), b 01 ), b 02 ) and b 12 ), and all oher paymen funcions b i and b ij are zero. The ransiion raes are also labelled by numbers, e.g. he ransiion rae from sae i o j is µ ij ). 16
17 Using Proposiion 2.6, he cash flow for sae 0 under he echnical basis is, dâ, s) ˆp 00, s) b 0 s) + ˆµ 02 s)b 02 ) + ˆµ 01 s)b 01 s)) + ˆp 01, s) b 1 s) + ˆµ 12 s)b 12 s)), where he noaion ˆp and ˆµ refers o he ransiion probabiliies and raes on he echnical basis. The firs line conains paymens while in sae 0, acive, and paymens during ransiions ou of sae 0. The paymens on he second line are paymens in sae 1, disabled, and paymens during ransiions ou of sae 1. We decompose he cash flow in posiive and negaive paymens, and define, dâ+, s) ˆp 00, s) b 0 s) + + ˆµ 02 s)b 02 s) + ˆµ 01 s)b 01 s) ) ds + ˆp 01, s) b 1 s) + ˆµ 12 s)b 12 s)) ds, dâ, s) ˆp 00, s)b 0 s) ds, such ha dâ, s) dâ+, s) dâ, s). The prospecive reserve on he echnical basis ˆV ) is also decomposed, ˆV + ) ˆV ) ˆru) du dâ+, s), ˆru) du dâ, s), and we have ˆV ) ˆV + ) ˆV ). Here we again omi he noaion 0 for he sae in he reserves and cash flows. For valuaion on he marke basis, we consider he exended Markov model in Figure 4. We define a duraion, U), which is he ime since he free policy opion was exercised or since surrender), U) inf {s 0 Z s) {0, 1, 2}}. If he free policy opion is exercised, and he curren ime is, he ime of he free policy ransiion is hen U). Upon ransiion o a free policy, he benefis are reduced by he facor ρ U)), and he premiums are cancelled. The paymens in he free policy saes a hus duraion dependen, and a ime hey are, b 4, U)) ρ U))b 0 ) +, b 5, U)) ρ U))b 1 ), b 45, U)) ρ U))b 01 ), b 46, U)) ρ U))b 02 ), b 56, U)) ρ U))b 12 ). 17
18 Upon surrender from sae 0, an amoun 1 κ) ˆV ) is paid ou, where ˆV ) is he prospecive reserve on he echnical basis. If he free policy opion is exercised and surrender occurs from sae 4, he prospecive reserve on he echnical basis is he value of he fuure benefis, reduced by he free policy facor ρ U)). Thus, he paymen upon surrender as a free policy is 1 κ)ρ U)) ˆV + ). The parameer κ is a surrender srain and is usually 0. We have, b 03 ) 1 κ) ˆV ), b 47, U)) 1 κ)ρ U)) ˆV + ). The oal paymen process is hen given by, db) 1 {Z)0} b 0 ) + 1 {Z)1} b 1 ) ) d + b 01 ) dn 01 ) + b 02 ) dn 02 ) + b 12 ) dn 12 ) + 1 κ) ˆV ) dn 03 ) 1{Z)4} + ρ U)){ b 0 ) {Z)5} b 1 ) ) d 4.1) + b 01 ) dn 45 ) + b 02 ) dn 46 ) + b 12 ) dn 56 ) + 1 κ) ˆV } + ) dn 47 ). The firs wo lines conain he benefis and premiums in he saes 0, alive, 1, disabled and 2, dead. Line hree conains he paymen upon surrender as a premium paying policy, and line six conains he paymen upon surrender as a free policy. Lines four and five conain he paymens as a free policy. We find he cash flow, and o his end i is convenien o define he quaniy p ρ ij, s) E [ 1 {Zs)j} ρs Us)) Z) i ], for i {0, 1, 2}, j {4, 5, 6}, and s. Then, i holds ha s p ρ ij, s) p i0, τ)µ 04 τ)p 4j τ, s)ρτ) dτ. 4.2) For a proof of 4.2), see Appendix B. For ρ) 1, his quaniy is simply he ransiion probabiliy from sae i o j: I is he probabiliy of going form sae i o 0 a ime τ, and hen ransiioning o sae 4 a ime τ, and finally going from sae 4 o sae j from ime τ o s. Since a ransiion from a sae i {0, 1, 2} o a sae j {4, 5, 6} can only occur hrough a ransiion from sae 0 o 4, his gives he ransiion probabiliy. When ρ) 1, he quaniy corresponds o he ransiion probabiliy muliplied by ρ) a he ime of ransiion o a free policy. We now sae he cash flow. The proof is found in Appendix C. 18
19 Proposiion 4.1. The cash flow in sae 0, da f, s), for paymens a ime s valued a ime, is given by da f, s) p 00, s) b 0 s) + µ 01 s)b 01 s) + µ 02 s)b 02 s) + µ 03 s)1 κ) ˆV ) s) ds + p 01, s) b 1 s) + µ 12 s)b 12 s)) ds + p ρ 04 b, s) 0 s) + + µ 45 s)b 01 s) + µ 46 s)b 02 s) + µ 47 s)1 κ) ˆV ) + s) ds + p ρ 05, s) b 1s) + µ 56 s)b 12 s)) ds. Calculaion of he cash flow requires p ρ ij, s) o be calculaed, and wih 4.2), his requires he ransiion probabiliies p 4j τ, s) for all s and τ saisfying τ s. However, i urns ou ha his is no necessary, since since here exiss a differenial equaion for p ρ ij, s) similar o Kolmogorov s forward differenial equaion. Using his, one can calculae all he usual ransiion probabiliies and he p ρ ij, s) quaniies ogeher. This eliminaes he need o calculae p 4j τ, s) for all τ and s saisfying τ s. Proposiion 4.2. The quanies p ρ ij, s) saisfy he forward differenial equaions, for i {0, 1, 2} and j {4, 5, 6}, d ds pρ ij, s) 1 {j4}p i0, s)µ 04 s)ρs) p ρ ij, s)µ j.s) + wih boundary condiions p ρ ij, ) 0. l {4,5,6} l j p ρ il, s)µ ljs), A more general version of his resul is presened in Theorem 4.2 in [2] for he general semimarkov case, and can also be found for he general Markov case as equaion 4.8) in [2]. For compleeness, a sraighforward proof is given in Appendix D. For he proposiion, we recall ha µ j. s) is he sum of all he ransiion raes ou of sae j. Noe in paricular, ha if j 4, he las sum is simply he one erm p ρ i5, s)µ 54s), and if j 5, he las erm is p ρ i4, s)µ 45s). The marke value including surrender and free policy modelling is denoed V f ) and is given by, V f ) rτ) dτ da f, s) 19
20 rτ) dτ p 00, s) b 0 s) + µ 01 s)b 01 s) + µ 02 s)b 02 s)) ds rτ) dτ p 01, s) b 1 s) + µ 12 s)b 12 s)) ds rτ) dτ p 00, s)µ 03 s)1 κ) ˆV s) ds rτ) dτ p ρ 04, s) b 0 s) + + µ 45 s)b 01 s) + µ 46 s)b 02 s) ) ds rτ) dτ p ρ 05, s) b 1s) + µ 56 s)b 12 s)) ds rτ) dτ p ρ 04, s)µ 47s)1 κ) ˆV + s) ds. The firs hree lines are he paymens when he free policy opion is no exercised, and he las hree lines are paymens as a free policy. The firs wo lines are he paymens wihou policyholder behaviour which is similar o he firs line in 3.10). The hird line is he surrender paymens when he free policy opion is no exercised and his is similar o line wo in 3.10). The fourh and fifh line are he paymens as a free policy, wihou he surrender paymen, corresponding o he hird line in 3.10). The las line is he surrender paymens as a free policy which corresponds o he fourh line in 3.10). If he disabiliy sae is removed, he formula simplifies o 3.10). 5 Numerical Example We presen a numerical example which illusraes ha he modelling of policyholder behaviour may have a considerable effec on he srucure of he cash flows. In paricular, he ineres rae sensiiviy duraion) of he cash flow is significanly reduced, which is of imporance if one applies duraion maching echniques in order o hedge he ineres rae risk. The example presened is similar o he numerical example in [2]. For simpliciy, we omi disabiliy and only consider a survival model. In he formulae, his can obained by seing he disabiliy ransiion rae equal o 0. We consider a 40 year old male wih a life annuiy a reiremen, age 65. A premium is paid of 10,000 annually, and he curren savings consis of 100,000. The echnical basis consiss of 2sae survival Markovmodel, as given in Figure 1. Ineres rae of 1.5% Moraliy rae, he Danish G82M able, µ x) x. On he echnical basis, he equivalence principle gives a life annuiy of size 41,
21 We consider hree differen marke bases, wihou policyholder behaviour, wih surrender modelling, and wih boh surrender and free policy modelling. This corresponds o he Markov models in Figures 1, 2 and 3, respecively. Common for he hree marke bases are Ineres rae from he Danish FSA of 8 May Moraliy rae from he Danish FSA benchmark For ages less han 65, he surrender and free policy ransiion raes are given by µ as x) x 40) +, µ af x) 0.05, respecively, where x is he age, and for x 65, he raes are 0. The ransiion raes loosely resemble he ones used in pracice by a large Danish pension fund in he compeiive marke. The ransiion raes are shown in Figure 5 ogeher wih he ransiion probabiliies, which have been calculaed using Kolmogorov s forward differenial equaions, Proposiion 2.7. The probabiliy of surrender and free policy are significan, and already a age 47 he probabiliy of having surrendered or made a free policy conversion is greaer han he probabiliy of sill being acive. We also noe ha he ransiion probabiliies are smooh excep a age 65 where he surrender and free policy ransiion raes drop o 0. The lef par of Figure 6 conains he premium cash flows. We see ha boh surrender and free policy modelling grealy reduces he premium cash flow, which is as expeced, since fuure premiums are cancelled upon eiher surrender or free policy conversion. The surrender and benefi cash flow are also shown in Figure 6 righ): The paymens before age 65 are he paymens from surrender, and in paricular we see ha in he basic model where surrender is no modelled, here are no paymens before age 65. Afer age 65, i is no longer possible o surrender, and here is only he life annuiy. We see ha he inroducion of surrender modelling reduces he life annuiy par of he cash flow, which is replaced by a significan amoun of surrender paymens. Wih free policy modelling, boh he benefis and he surrender paymens are reduced furher, since here is less premiums, and hus smaller surrender paymens and benefis. In Figure 7, he oal cash flows are shown. They are calculaed as he sum of he premium, benefi and surrender cash flows. Boh he posiive and negaive pars are significanly reduced wih policyholder behaviour, bu since i affecs boh pars, he change in he marke value as measured by he prospecive reserve is no of he same magniude. However, he change in he srucure has a significan effec on he ineres rae sensiiviy, which is seen in he righ par of Figure 7. Wihou policyholder behaviour, he prospecive reserve is 21
22 rae Transiion raes Moraliy Surrender Free policy probabiliy Transiion probabiliies Acive Dead Surrendered Free pol., acive Free pol., dead age age Figure 5: Transiion raes lef) and ransiion probabiliies righ) in he Markov model wih surrender and free policy modelling. The surrender rae decreases linearlly and he free policy rae is consan. Policyholder behaviour is seen o be quie significan, and already a age 47 is he probabiliy of eiher surrender or free policy greaer han he probabiliy of sill being acive. yearly paymen Premium cash flow Basic Surrender Sur + free pol yearly paymen Benefis and surrender cash flow age age Figure 6: Cash flows of premiums lef) and benefis righ). The premiums are significanly reduced when including surrender and free policy, as expeced. The benefis are also significanly reduced, and we see ha he surrender paymens appear before age
23 yearly paymen Toal cash flow Basic Surrender Sur + free pol Prosp. reserve 0e+00 2e+05 4e+05 6e+05 Prospecive Reserve Basic Surrender Sur + free pol Technical reserve age Change in ineres rae Figure 7: Toal cash flow lef) and prospecive reserve ploed agains parallel shifs in he marke ineres rae srucure righ). The oal cash flows are numerically significanly smaller wih policyholder behaviour. To he righ he effec of his change is seen on he ineres rae sensiiviy, which is significanly less wih policyholder behaviour. more sensiive o changes in he ineres rae, and his sensiiviy is significanly lowered wih policyholder behaviour. We see, ha a lile above he curren marke ineres rae level, he hree lines inersec. Thus, a his level of marke ineres raes, he cash flow modelling does no have a grea effec on he prospecive reserve. Basic Surrender Sur. and free pol. Prospecive reserve 129, , ,734 DV01 Toal 93,284 46,346 38,087 DV01 Pos. paymens 115,550 62,044 46,625 DV01 Premiums 22,266 15,697 8,538 Table 1: Prospecive reserves and dollar duraions DV01), wih and wihou policyholder behaviour. The duraion is grealy reduced when policyholder behaviour is included, boh for he oal cash flows and also for he posiive paymens and premiums separaely. In Table 1, he prospecive reserve is shown ogeher wih he dollar duraion DV01), which measures he change in he prospecive reserve for a 100 basis poin change in he ineres rae srucure. The prospecive reserve is reduced wih policyholder behaviour, as could also be seen from Figure 7. In he example, surrender modelling reduces he dollar duraion by approximaely 50%, and free policy modelling reduces i by anoher 18%. Thus, if in pracice one applies duraion maching echniques in order o hedge he life insurance liabiliies, i is essenial o ake ino accoun boh surrender and free 23
24 REFERENCES policy modelling. Acknowledgemens We are graeful o Krisian Bjerre Schmid for general commens and assisance wih he numerical examples. References [1] Krisian Buchard. Dependen ineres and ransiion raes in life insurance. Preprin), Deparmen of Mahemaical Sciences, Universiy of Copenhagen, hp://mah.ku.dk/~buchard, [2] Krisian Buchard, Krisian Bjerre Schmid, and Thomas Møller. Cash flows and policyholder behaviour in he semimarkov life insurance seup. Preprin), Deparmen of Mahemaical Sciences, Universiy of Copenhagen and PFA Pension, [3] CEIOPS. CEIOPS advice for level 2 implemening measures on Solvency II: Technical provisions, Aricle 86 a, Acuarial and saisical mehodologies o calculae he bes esimae. hps://eiopa.europa.eu/publicaions/siifinall2advice/ index.hml, [4] Domenico De Giovanni. Lapse rae modeling: A raional expecaion approach. Scandinavian Acuarial Journal, 1:56 67, [5] Lars Frederik Brand Henriksen, Jeppe Woemann Nielsen, Mogens Seffensen, and Chrisian Svensson. Markov chain modeling of policy holder behavior in life insurance and pension. Available a SSRN: hp://ssrn.com/absrac , June [6] Michael Koller. Sochasic models in life insurance. European Acuarial Academy Series. Springer, [7] Thomas Møller and Mogens Seffensen. Markevaluaion mehods in life and pension insurance. Inernaional Series on Acuarial Science. Cambridge Universiy Press, [8] Ragnar Norberg. Reserves in life and pension insurance. Scandinavian Acuarial Journal, 1991:1 22,
25 A Cash flow for Secion 3.2 We here prove he formula 3.4) for he model presened in Figure 3. Define firs he duraion U) since enering he free policy sae, ha is U) inf {s 0 Z s) {0, 1, 2}}. Now, he paymens in he seup lead o he paymen process db) 1 {Z)0} b) π)) d + b ad ) dn ad ) + 1 κ) ˆV ) dn as ) + ρ U)) 1 {Z)3} b) d + b ad ) dn af,df ) + 1 κ) ˆV ) + ) dn af,sf ). Here, N ad is he couning process ha couns he number of jumps from sae acive o sae dead. Similarly, N as, N af,df and N af,sf couns he number of jumps from sae acive o surrender, from sae acive, free policy o dead, free policy, and from sae acive, free policy o surrender, free policy, respecively. The cash flow is hen given as T da f, s) [ T ] E dbs) Z) 0 [ T ] E 1 {Zs)0} bs) πs)) ds Z) 0 [ T ] + E b ad s) dn ad s) + 1 κ) ˆV s) dn as s) Z) 0 [ T ] + E ρs Us))1 {Zs)3} bs) ds Z) 0 [ T + E ρs Us)) b ad s) dn af,df s) + 1 κ) ˆV + s) dn af,sf s)) ] Z) 0 T T s p fs x+bs) πs)) ds + s p fs x+ b ad s)µ ad s) + 1 κ) ˆV ) s)µ as s) ds [ T ] + E ρs Us))1 {Zs)3} bs) ds Z) 0 [ T + E ρs Us)) b ad s) dn af,df s) + 1 κ) ˆV + s) dn af,sf s)) ] Z) 0. 25
26 For he firs expecaion, we used ha he expecaion of an indicaor funcion is a probabiliy. For he second expecaion, we recall ha he couning process here can be replaced by he predicable compensaor. For he hird expecaion, we condiion on he sochasic variable s Us), which is he ime of ransiion. Then, condiional on Z) 0, and wih he indicaor funcion 1 {Zs)3}, we know ha a ransiion has occured, hus s Us), s), which deermine he inegral limis. Also, he densiy of he ime of he ransiion from sae 0 o sae 3 is τ τ p fs x+µ af τ). Using hese observaions, we calculae T E [ ρs Us))1 {Zs)3} bs) Z) 0 ] ds T s E [ ρτ)1 {Zs)3} bs) Z) 0, s Us) τ ] dp s Us) τ Z) 0) ds T s T s T s ρτ) E [ 1 {Zs)3} Zτ) 3 ] bs) τ p fs x+µ af τ) dτ ds ρτ) s τ p s x+τ bs) τ p fs x+µ af τ) dτ ds τ p fs x+µ af τ)ρτ) s τ p s x+τ dτbs) ds. A.1) For he fourh expecaion, we only consider he firs par, since he second par is analogous. Since Us) is coninuous whenever N af,df s) and N af,sf s)) increase in value, we can replace Us) by Us ). Using ha ρs Us )) is predicable, we can inegrae wih respec o he compensaor of he couning process N af,df s) insead, so we ge [ T ] E ρs Us))b ad s) dn af,df s) Z) 0 [ T ] E ρs Us ))b ad s) dn af,df s) Z) 0 [ T ] E ρs Us ))b ad s)1 {Zs )3} µ ad s) ds Z) 0 T T s E [ ρs Us ))1 {Zs )3} Z) 0 ] b ad s)µ ad s) ds τ p fs x+µ af τ)ρτ) s τ p s x+τ dτb ad s)µ ad s) ds. Since in he second las line, he expression is analogous o he hird expecaion, he las line was obained using he same calculaions as A.1). Gahering he resuls, he cash flow da f, s) is obained. 26
27 B Proof of equaion 4.2) Condiioning on he ime of ransiion from sae 0 o 4, s Us), and using ha he densiy for he ransiion ime is p i0, τ)µ 04 τ), we find p ρ ij, s) E [ 1 {Zs)j} ρs Us)) Z) i ] s s s s E [ 1 {Zs)j} ρs Us)) Z) i, s Us) τ ] dp s Us) τ Z) i) E [ 1 {Zs)j} Z) i, s Us) τ ] ρτ)p i0, τ)µ 04 τ) dτ p i0, τ)µ 04 τ) E [ 1 {Zs)j} Zτ) 4 ] ρτ) dτ p i0, τ)µ 04 τ)p 4j τ, s)ρτ) dτ. A line five we used ha if we know ha s Us) τ, hen in paricular, we know ha Zτ) 4. Since Z is Markov, we can hen drop he condiion ha Z) i and s Us) τ. Noe, ha for he proof, i is essenial ha i {0, 1, 2} and j {4, 5, 6}, since we a he condiioning on line hree use ha s Us), s), i.e. a ransiion o he free policy saes occurs in he ime inerval, s). We can do ha, since i mus hold if Z) i {0, 1, 2} and Zs) j {4, 5, 6}. C Proof of Proposiion 4.1 Proof. The cash flow is given as T [ T ] da f, s) E dbs) Z) 0, where B) is given in 4.1). Insering B) yields, T T + da f, s) T [ 1{Zs)0} E b 0 s) + 1 {Zs)1} b 1 s) ) ] Z) 0 ds [ E ρs Us)) 1 {Zs)4} b 0 s) {Zs)5} b 1 s) ) ] Z) 0 ds C.1) C.2) [ T + E b 01 s) dn 01 s) + b 02 s) dn 02 s) + b 12 s) dn 12 s) + 1 κ) ˆV s) dn 03 s)) ] C.3) Z) 0 27
28 [ T + E ρs Us)) b 01 s) dn 45 s) + b 02 s) dn 46 s) + b 12 s) dn 56 s) + 1 κ) ˆV + s) dn 47 s)) ] C.4) Z) 0 The four expecaions C.1) C.4) are calculaed separaely. The firs expecaion C.1) is he expecaion of indicaor funcions, and we replace by he ransiion probabiliies, T p 00, s)b 0 ) + p 01, s)b 1 )) ds. In he second expecaion C.2), he same calculaions as in Secion B can be performed o obain, T p ρ 04, s)b 0s) + + p ρ 05, s)b 1s) ) ds. In he hird expecaion C.3), we inegrae deerminisic funcions wih respec o a couning process. Taking he expecaion, we can insead inegrae wih respec o he predicable compensaor, and we ge, T [ E 1 {Zs )0} b 01 s)µ 01 s) + b 02 s)µ 02 s)) + 1 {Zs )1} b 12 s)µ 12 s) + 1 {Zs )0} 1 κ) ˆV ] s)µ 03 s) ds Z) 0 ds T p 00, s) b 01 s)µ 01 s) + b 02 s)µ 02 s)) + p 01, s)b 12 s)µ 12 s) + p 00, s)1 κ) ˆV ) s)µ 03 s) ds. For he fourh expecaion C.4), we sar by replacing Us) wih Us ), since whenever any of N 45 s), N 46 s), N 56 s) or N 47 s) are increasing, hen Us) is coninuous. Thus, we inegrae a predicable process wih respec o a couning process, and we can inegrae wih respec o he predicable compensaor insead, [ T E ρs Us )) b 01 s) dn 45 s) + b 02 s) dn 46 s) + b 12 s) dn 56 s) + 1 κ) ˆV + s) dn 47 s)) ] Z) 0 T [ E ρs Us )) 1 {Zs )4} b 01 s)µ 45 s) + b 02 s)µ 46 s)) + 1 {Zs )5} b 12 s)µ 56 s) + 1 {Zs )4} 1 κ) ˆV ) ] + Z) s)µ 47 s) 0 ds T p ρ 04, s) b 01s)µ 45 s) + b 02 s)µ 46 s)) + p ρ 05, s)b 12s)µ 56 s) + p ρ 04, s)1 κ) ˆV ) + s)µ 47 s) ds. 28
29 For he las equaliy, we again used he calculaions from Secion B. Gahering he four expecaions, he resul is obained. D Proof of Proposiion 4.2 Proof. We differeniae p ρ ij, s) for i {0, 1, 2} and j {4, 5, 6}, d ds pρ ij, s) d ds s p i0, τ)µ 04 τ)p 4j τ, s)ρτ) dτ p i0, s)µ 04 s)p 4j s, s)ρs) + 1 {j4} p i0, s)µ 04 s)ρs) + s s p i0, τ)µ 04 τ) p 4jτ, s)µ j. s) + p i0, τ)µ 04 τ) d ds p 4jτ, s)ρτ) dτ l {4,5,6} l j 1 {j4} p i0, s)µ 04 s)ρs) p ρ ij, s)µ j.s) + l {4,5,6} l j p 4l τ, s)µ lj s) ρτ) dτ p ρ il, s)µ ljs). For he hird equaliy sign, we used ha p 4l τ, s)µ lj s) 0 for l {4, 5, 6}. 29
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