Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies modelled wih a mulisae Markov chain, i is of imporance o consider he ineres and ransiion raes as sochasic processes, and hedging possibiliies of he risks. his is usually done wih an assumpion of independence beween he ineres and ransiion raes. In his paper, i is shown how o valuae life insurance liabiliies using affine processes for modelling dependen ineres and ransiion raes. his approach leads o he inroducion of so-called generalised forward raes. We propose a specific model for surrender modelling, and wihin his model he generalised forward raes are calculaed, and he marke value and he Solvency II capial requiremen are examined for a simple savings conrac. Keywords: Affine Processes; Doubly Sochasic Process; Muli-sae Life Insurance Models; Policyholder Behaviour; Solvency II JL Classificaion: G22 1 Inroducion Life insurance liabiliies are radiionally modelled by a finie sae Markov chain wih deerminisic ineres and ransiion raes. In order o give a marke consisen bes esimae of he presen value of fuure paymens, i has become of increasing ineres o le he ineres and ransiion raes be modelled as sochasic processes. he sochasic modelling is imporan in order o consider hedging possibiliies of he risks. Wih he Solvency II rules, sochasic modelling of he ineres and ransiion raes is also imporan from a risk managemen perspecive. Modelling he ineres and ransiion Deparmen of Mahemaical Sciences, Universiy of Copenhagen, Universiesparken 5, DK- 2100 Copenhagen O and PFA Pension, Sundkrogsgade 4, DK-2100 Copenhagen O, Denmark, hp://mah.ku.dk/~buchard, e-mail: buchard@mah.ku.dk 1
raes as sochasic processes is radiionally done wih an independence assumpion. In his paper, we relax he independence assumpion, and consider basic valuaion wih dependence beween he ineres and one or more ransiion raes. his is done wih coninuous affine processes for he modelling of he dependen raes. he sudy of valuaion of life insurance liabiliies wih dependen raes leads o he definiion of socalled generalised forward raes. hese are naural quaniies ha appear in case of dependence, replacing he usual forward raes, which are no direcly applicable. Using he heory of dependen affine raes, we consider he case of surrender modelling, and propose a specific model for dependen ineres and surrender raes. his is of paricular ineres from a Solvency II poin of view. Wihin his model, a simple savings conrac wih a buy-back opion is considered. We calculae he generalised forward raes, he marke value and he Solvency II capial requiremen. his is done in par wihou hedging, and in par wih a simple saic hedging sraegy. We hen examine he effec of correlaion beween he ineres and surrender rae. he sudy of valuaion of life insurance liabiliies wih sochasic ineres and ransiion raes has received considerable aenion during he las decades. Primarily he ineres and moraliy raes have been modelled as sochasic, which is ofen done wih affine processes. For basic applicaions of affine processes for valuaion of life insurance liabiliies, see [1]. Possibiliies of hedging can be considered, which is imporan for marke consisen valuaion, and for he sudy of valuaion and hedging of life insurance liabiliies wih sochasic ineres and moraliy raes, see [7] and [6]. Anoher approach o modelling sochasic ineres and moraliy is aken in [15], where he ineres and moraliy is modelled wihin a finie sae Markov chain seup. In his paper we exend he sudy of affine ineres and ransiion raes o he case of dependence. We consider how o valuae life insurance liabiliies when he ineres and one or more ransiion raes are modelled as dependen affine processes. his is possible in any decremen/hierarchical Markov chain seup, ha is, in Markov chains where, when he process leaves a sae, i canno reurn. We adop he heory presened in [4], which is reviewed in Secion 2 of his paper. his provides he foundaion for he sudy of mulidimensional affine processes in life insurance mahemaics. he heory presened in [4] is based parly on a resul in [8], and parly on general heory for mulidimensional affine processes presened in [9]. In he financial lieraure, he concep of a forward ineres rae exiss, which is convenien for e.g. represening zero coupon bond prices. his quaniy appears naurally in life insurance mahemaics, when he ineres rae is modelled as a sochasic process. If one also considers a sochasic moraliy, independen of he ineres rae, i becomes naural o define a forward moraliy rae as well. Wih hese forward raes, he expeced presen value of he life insurance liabiliies looks paricularly compelling. However, if one inroduces dependence beween he ineres and moraliy raes, he for- 2
ward raes are no longer applicable. For a general discussion on forward raes, and heir usefulness, see [14], wherein he case of dependence beween he raes is discussed as well. In [11], alernaive forward moraliy raes are defined in order o handle he case of dependence. In his paper, we show ha one of he forward moraliy raes defined in [11] is in general no well defined. Insead, we inroduce he concep of generalised forward raes, which appear naurally and can be used o express he expeced presen value of he life insurance liabiliies in a convenien form. he generalised forward raes indeed generalise he usual definiions of forward raes, in he sense ha when here is independence beween he raes, he generalised forward raes equal he usual forward raes. Modelling policyholder behaviour has become of increasing imporance wih he proposed Solvency II rules, where one is required o consider any dependence beween he economic environmen and policyholder behaviour, see Secion 3.5 in [5]. he sudy of surrender behaviour can eiher be made using a raional approach, where he ouse is, ha he policyholders surrender he conrac if i is raional from some economic viewpoin. his seems a bi exreme, given ha his behaviour is no seen in pracice. Anoher approach is he inensiy approach, where he policyholders surrender randomly, regardless wheher or no i is profiable in he curren economic environmen. his is no a perfec way of modelling eiher, since if he ineres raes decrease a lo, a guaranee given in connecion wih he life insurance conrac moivaes he policyholders o keeping he conrac. For an overview of some of he approaches, see [12]. In [10], an aemp is made on coupling he wo approaches, using wo differen surrender rae models if i is raional or irraional, respecively, o surrender. In his paper, we propose anoher way of coupling he wo approaches. We le he surrender rae be posiively correlaed wih he ineres rae, hus if he ineres rae decreases a lo, he surrender rae also decreases, represening ha he guaranee inheren in he life insurance conrac is of value o he policyholder. he Solvency II capial requiremen is basically, ha he insurance company mus have enough capial, such ha he probabiliy of defaul wihin he nex year is less han 0.5%, represening ha a defaul is a 200-year even. When he insurance company updaes is moraliy ables, or oher ransiion rae ables, his represens a risk ha mus be aken ino accoun when puing up he Solvency II capial requiremen. Mahemaically, his can be done using sochasic raes. For an examinaion of moraliy modelling and he Solvency II capial requiremen, see e.g. [2]. For a basic discussion of he mahemaical formulaion of he Solvency II capial requiremen, see e.g. [3]. In his paper, we deermine he Solvency II capial requiremen for he simple savings conrac, boh in he case of no hedging sraegy, and also in he case of a simple sraegy where ineres rae risk is hedged. he srucure of he paper is as follows. In Secion 2, we presen basic resuls on mulidi- 3
mensional coninuous affine processes, which provides he foundaion for he applicaion of dependen affine processes in life insurance mahemaics. In Secion 3, we presen he general life insurance seup wih sochasic ineres and ransiion raes, and in Secion 4, we propose he definiion of generalised forward raes, which is compared o he usual definiion in Secion 4.1. In Secion 4.2, we discuss oher definiions in he lieraure of forward raes in a dependen seup, and in paricular, we show ha he forward moraliy rae for erm insurances proposed in [11] does no always exis. In Secion 5, we presen a specific model for dependen ineres and surrender raes. he model is inroduced in Secion 5.1. We firs discuss how o find he Solvency II capial requiremen, which is done in Secion 5.3, and a simple hedging sraegy for he ineres rae risk is presened in Secion 5.4. Numerical resuls are presened in Secion 5.5, consising of he generalised forward raes found, and he marke value and Solvency II capial requiremen, presened for differen levels of correlaion. 2 Coninuous Affine Processes he class of affine processes provides a mehod for modelling ineres and ransiion raes, wih he possibiliy of adding dependence. In his secion, we consider general resuls abou coninuous affine processes, which we apply in his paper. For more deails on he heory presened in his secion, see [4]. Le X be a d-dimensional affine process, saisfying he sochasic differenial equaion dx() = (b() + B()X()) d + ρ(, X()) dw (), where W is a d-dimensional Brownian moion. Here, b : R + R d is a vecor funcion, and B : R + R d d is a marix funcion, where we denoe column i by β i (), so ha B() = (β 1 (),..., β d ()). When squared, he volailiy parameer funcion ρ(, x) mus be affine in x, i.e. ρ(, x)ρ(, x) = a() + d α i ()x i, for marix funcions a : R + R d d and α i : R + R d d. Consider now affine ransformaions of X, by defining a vecor funcion c : R + R p and a marix funcion Γ : R + R p d, hereby defining he p-dimensional process, i=1 Y () = c() + Γ()X(). (2.1) We hink of X as socio-economic driving facors, and hen Y is a collecion of he sochasic ineres rae and/or ransiion raes. In his secion, we work in a probabiliy 4
space (Ω, F, F, P ) wih he filraion F = (F()) R+ generaed by he Brownian moion W. For applicaions of Y as ineres and as ransiion raes in finie sae Markov chain models, we presen some essenial relaions. he resuls hold under cerain regulariy condiions, for deails see [4]. Denoe by 1 a vecor wih 1 in all enries, where he dimension is implici. Also, denoe by γ i () he sum of he ih column in Γ(), i.e. γ i () = 11 Γ()e i, where e i is he ih uni vecor, i = 1,..., d. he firs relaion, he basic pricing formula, is for 0 given by 11 Y (s) ds F() = e φ(, )+ψ(, ) X(), (2.2) where φ(, ) is a real funcion, and ψ(, ) is a p-dimensional funcion, given by he sysem of differenial equaions, φ(, ) = 1 2 ψ(, ) a()ψ(, ) b() ψ(, ) + 11 c(), ψ i(, ) = 1 2 ψ(, ) α i ()ψ(, ) β i () ψ(, ) + γ i (), i = 1,..., d, (2.3) wih boundary condiions φ(, ) = 0 and ψ(, ) = 0. For he second relaion, le a vecor κ R p be given, and le u [, ] be some ime poin. hen, 11 Y (s) ds κ ( ) Y (u) F() = e φ(, )+ψ(, ) X() A(,, u) + B(,, u) X(), (2.4) where (φ, ψ) is given by (2.3) as above, A is a real funcion and B is a vecor funcion, given by he sysem of differenial equaions, A(,, u) = ψ(, ) a()b(,, u) b() B(,, u), B i(,, u) = ψ(, ) α i ()B(,, u) β i () B(,, u), i = 1,..., d, (2.5) wih boundary condiions A(u,, u) = κ c(u) and B(u,, u) = κ Γ(u). A paricular example of imporance is κ = e k for some k = 1,..., p, and in his case, we wrie A k and B k o emphasize he dependence on k. his second relaion (2.4) is proven in [8] for u =, and he exension o he case u < is for example given in [4]. he hird relaion is, for anoher ime poin v [, ], and wo inegers k, l = 1,..., p, 5
given by 11 Y (s) ds Y k (u)y l (v) F() = e φ(, )+ψ(, ) X() { ( ) ( ) A k (,, u) + B k (,, u) X() A l (,, v) + B l (,, v) X() } + C kl (,, u, v) + D kl (,, u, v) X(), (2.6) where (φ, ψ) solves (2.3) and (A k, B k ) and (A l, B l ) boh solve (2.5) wih boundary condiions A k (u,, u) = e k c(u), Bk (u,, u) = e k Γ(u) and Al (v,, v) = e l c(v), Bl (v,, v) = e l Γ(v), respecively. he funcions Ckl and D kl are deermined by he following sysem of differenial equaions, Ckl (,, u, v) = B k (,, u) a()b l (,, v) ψ(, ) a()d kl (,, u, v) b() D kl (,, u, v), Dkl i (,, u, v) = B k (,, u) α i ()B l (,, v) ψ(, ) α i ()D kl (,, u, v) β i () D kl (,, u, v), (2.7) for i = 1,..., d, wih boundary condiions 1 C kl (u v,, u, v) = 0 and D kl (u v,, u, v) = 0. his resul is proven in [4]. 3 he Life Insurance Model Consider he usual life insurance seup. Le Z = (Z()) R+ be a Markov process in he finie sae space J, indicaing he sae of he insured. he disribuion of Z is defined via he ransiion raes (µ ij ()) R+, i, j J. Wih (N ij ()) R+, i, j J being he process ha couns he number of jumps for Z from sae i o j, he compensaed process N ij () 0 1 (Z(s )=i) µ ij (s) ds is a maringale. We can allow he ransiion raes (µ ij ) o be sochasic. In his case, he disribuion of Z is defined condiionally on he ransiion raes. We model he ransiion raes as a ime-dependen affine ransformaion of a d-dimensional coninuous affine process X. ha is, for funcions c : R + R p and Γ : R + R p d, le Y be defined as 1 he noaion x y = min{x, y} is used. Y () = c() + Γ()X(). 6
Hence, each of he sochasic ransiion raes are modelled as an elemen in Y. he ineres rae process (r()) R+ is also allowed o be sochasic. his is modelled in he same way, by specifying r as an elemen in Y. By he design of Γ and X, he ineres and ransiion raes can be dependen, independen or deerminisic. Le he filraions F Z = (F Z ()) R+ and F X = (F X ()) R+ be he ones generaed by he processes Z and X, respecively, saisfying he usual hypohesis. We consider he probabiliy space (Ω, F, F, P ), where he filraion F = (F()) R+ is given by F() = F Z () F X (). We consider a life insurance policy, wih paymens specified by he process B = (B()) R+, such ha B() is he oal paymens unil ime. hen we can hink of db() as he paymen a ime, and we can specify B as db() = 1 (Z()=i) b i () d + b ij () dn ij (), i J for deerminisic paymen funcions b i and b ij, i, j J. hen b i () is he paymen while in sae i a ime, and b ij () is he paymen if jumping from sae i o j a ime. he presen value a ime of he fuure paymens associaed wih he life insurance policy is given by P V () = i,j J i j e s r(τ) dτ db(s). For reserving and pricing, one considers he expeced presen value [ V () = e ] s r(τ) dτ db(s) F(), where he expecaion is aken using a marke, risk neural or pricing measure. For acually calculaing V (), he ower propery is applied, ha is, we condiion on F X ( ) o ge [ V X () = e ] s r(τ) dτ db(s) F Z () F X ( ), so ha V () = [ V X () F() ]. Here, V X () is he reserve condiional on he ineres and ransiion raes, hus corresponding o he case of deerminisic raes. When valuaing V X we need he condiional disribuion of Z, and hus B, given he ransiion raes. By consrucion his is known, and well-esablished heory abou life insurance reserves wih deerminisic ineres and ransiion raes (see e.g. [13]) hold. 7
xample 3.1. Consider a surrender model wih 3 saes J = {0, 1, 2}, corresponding o alive, dead and surrendered respecively. he Markov model is shown in Figure 1. Le he ransiion rae from sae alive o sae dead, i.e. he moraliy rae, be deerminisic. We model he ineres rae r and he surrender rae η as dependen affine processes in he form, (r(), η()) = c() + Γ()X(), for a d-dimensional affine process X. Hence, his specificaion is analog o (2.1). By he design of X, he processes X i, i = 1,..., d can be dependen processes, such ha he ineres rae r and he surrender rae η can be dependen processes. Alive 0 1 Dead µ η 2 Surrendered Figure 1: Markov model for he survival-surrender model. Le he paymens be defined by db() = b()1 (Z()=0) d + b d () dn 01 () + U() dn 02 (), where b() is he coninuous paymen rae a ime while alive, b d () is he single paymen if deah occurs a ime, and U() is he paymen upon surrender a ime. he paymen funcions are deerminisic. Condiioning on he inensiies, he expeced presen value V X () is he classic resul, V X () = [ P V () F X (), Z() = 0 ] = e s (r(τ)+µ(τ)+η(τ)) dτ (b(s) + µ(s)b d (s) + η(s)u(s)) ds, see e.g. [13]. Removing he condiion, we find, using heorem (2.2) and (2.4), V () = [ V X () F() ] 8
= = e { s µ(τ) dτ [e ] s (r(τ)+η(τ)) dτ F() (b(s) + µ(s)b d (s)) + [e s (r(τ)+η(τ)) dτ η(s) F() + ] } U(s) ds e ( s µ(τ) dτ+φ(,s)+ψ(,s) X() b(s) + µ(s)b d (s) ) ( A η (, s, s) + B η (, s, s) X() ) U(s) ds. 4 Generalised Forward Raes he form of V () in xample 3.1 moivaes he definiion of quaniies similar o forward raes, ha can be used o express he soluion. In paricular, his leads o a forward ineres rae, bu his is in general no equal he forward rae obained using he usual definiion. Hence, we apply he erm generalised forward raes. Le X, c() and Γ() be given, and le Y be of he form (2.1). Consider firs he moivaing calculaions, 11 Y (s) ds 11 Y ( ) F() = [ e ] 11 Y (s) ds F() = eφ(, )+ψ(, ) X() ( = e φ(, )+ψ(, ) X() ( φ(, ) + X() )) ψ(, ), where we inerchanged inegraion and differeniaion, and applied (2.2). On he oher hand, if we insead apply (2.4) wih κ = 11, we find 11 Y (s) ds 11 Y ( ) F() ( ) = e φ(, )+ψ(, ) X() A(,, ) + X() B(,, ), ( p ) p = e φ(, )+ψ(, ) X() A k (,, ) + X() B k (,, ), k=1 where (A k, B k ), k = 1,..., p are soluions o (2.5) wih boundary condiions given by κ = e k, i.e. A k (,, ) = e k c( ) and Bk (,, ) = e k Γ( ). he las equaliy sign is obained using he relaions p k=1 Ak (,, ) = A(,, ) and p k=1 Bk (,, ) = B(,, ), which hold since (A, B) also solves he linear sysem of differenial equaions k=1 9
(2.5), wih boundary condiions given by κ = 11. Gahering he wo calculaions above, we conclude ha φ(, ) = p A k (,, ), k=1 ψ(, ) = p B k (,, ), k=1 and, in paricular, since φ(, ) = 0 and ψ(, ) = 0, ha φ(, ) = d A k (, s, s) ds, k=1 ψ(, ) = d B k (, s, s) ds. (4.1) k=1 Definiion 4.1. Le X be a d-dimensional coninuous affine process, and le c and Γ be given, such ha Y from (2.1) is defined. Le s and k = {1,..., d}. he generalised forward rae f k (s) for he sochasic rae Y k (s) a ime is hen given by f k (s) = A k (, s, s) + X() B k (, s, s), where (A k, B k ) solves he sysem of differenial equaions (2.5), wih boundary condiions given by κ = e k. Remark 4.2. Using he noaion of he generalised forward raes, we can hen express he relaion (2.3), and for u = also he relaion (2.5), as 11 Y (s) ds F() = e d i=1 f i(s) ds, 11 Y (s) ds Y k ( ) F() = e d (4.2) i=1 f i(s) ds f k ( ). xample 4.3. (xample 3.1 coninued) Using he definiion of he generalised forward raes, we can wrie he expeced presen value as, V () = e s (f r (τ)+µ(τ)+f η (τ)) dτ ( b(s) + µ(s)b d (s) + f η (s)u(s) ) ds. We see ha he expeced presen value is of he same form as he formula ha appears in he case of deerminisic raes, bu wih he ineres and surrender raes exchanged by he corresponding generalised forward raes. Noe ha we used a slighly differen noaion, such ha we wrie f r insead of f 1 and f η insead of f 2. Ofen we wan o consider boh he quaniy 11 Y (s) ds F() = [e ] (r(s)+η(s)) ds F(), 10
4.1 Comparison Wih he Usual Forward Ineres Rae where Y () = (r(), η()), as well as he quaniies arising from he models Y 1 () = (r(), 0) and Y 2 () = (0, η()), 11 Y 1 (s) ds F() = r(s) ds F(), 11 Y 2 (s) ds F() = η(s) ds F(). In such cases, we add a more deailed superscrip o he forward raes f, and specify he model we hink of afer a colon. ha is, we wrie [e ] (r(s)+η(s)) ds F() = e r:(r+η) (f (s)+f η:(r+η) (s)) ds, as well as [e [e r(s) ds η(s) ds ] F() = e ] F() = e f r:r (s) ds, f η:η (s) ds. Noe ha f r:r (s) and f µ:µ (s) are he usual forward raes. 4.1 Comparison Wih he Usual Forward Ineres Rae Le he model Y () = c() + Γ()X() be given, for d > 1, and le r() = Y 1 () be he ineres rae. he forward ineres rae is he funcion g (s) ha saisfies r(s) ds F() = e g (s) ds. his funcion also saisfies, as can be shown by differeniaion, r(s) ds r( ) F() = e g (s) ds g ( ). (4.3) he generalised forward rae for he ineres rae in our model Y, as defined in Definiion 4.1, is denoed f r (s). I saisfies, (r(s)+y 2(s)+...+Y d (s)) ds r( ) F() = e (f r(s)+f 2(s)+...+f d(s)) ds f r ( ), (4.4) where he oher forward raes f i (s) saisfies analogue relaions. We noe, ha while he usual forward ineres rae is defined for any sochasic ineres rae, he generalised forward raes from Definiion 4.1 are only defined for (coninuous) affine sochasic raes. In he case ha r = (r()) R+ is independen of Y 2,..., Y d, he generalised forward rae for he ineres r simplifies o he usual forward ineres rae. his can be seen by 11
4.2 Comparison Wih Oher Dependen Seups wo simple calculaions. Firs, see ha e (f r(s)+f 2(s)+...+f d (s)) ds = = = e [e [e (r(s)+y 2(s)+...+Y d (s)) ds r(s) ds g (s) ds ] F() [e [e ] F() (Y 2(s)+...+Y d (s)) ds (Y 2(s)+...+Y d (s)) ds ] F(). ] F() (4.5) A similar calculaion yields, using (4.4) and (4.3), yields e (f r(s)+f 2(s)+...+f d(s)) ds f r ( ) = e g (s) ds g ( ) [e (Y 2(s)+...+Y d (s)) ds Dividing wih he ideniy (4.5) above, we conclude ha f r ( ) = g ( ), ] F(). which holds for all where <, and we conclude ha he generalised forward rae (on he lef hand side) equals he forward ineres rae (on he righ hand side). he calculaions relied criically on he independence assumpion, and in he general case he generalised forward rae for he ineres is no equal o he forward ineres rae. Inuiively, when he ineres rae appears ogeher wih oher dependen raes, he forward raes need o compensae for his dependence. hus, he generalised forward rae includes a covariance erm, which is no presen in he usual forward ineres rae. 4.2 Comparison Wih Oher Dependen Seups For he case of dependen affine raes, here have been oher proposals for he definiion of forward raes. In [11], he model conains an ineres rae and a moraliy rae which are dependen. his corresponds o he case d = 2, where r() = Y 1 () is he ineres rae and µ() = Y 2 () is he moraliy rae. heir approach is o keep he definiion of he forward ineres rae g : [, ) R + unchanged, and hen find forward moraliy raes ha are compaible wih his definiion. In order o make his idea work, hey define wo differen moraliy raes, one for pure endowmens, and one for erm insurances. We briefly review his approach and compare o he definiion of he generalised forward raes in he previous secion. he forward moraliy rae for pure endowmens, h pe funcion saisfying [e ] (r(s)+µ(s)) ds F() : [, ) R +, is defined as he = e (g (s)+h pe (s)) ds. 12
4.2 Comparison Wih Oher Dependen Seups In erms of he generalised forward raes, f r and f µ, we can use he firs par of (4.2) and wrie he forward moraliy rae for pure endowmen as, h pe (s) = f r (s) + f µ (s) g (s), (4.6) which in paricular shows ha i is well-defined. he forward moraliy rae for pure endowmen can be given an inuiive inerpreaion. Recall ha he generalised forward raes are differen from he usual definiion of forward raes, because he moraliy rae appears ogeher wih anoher dependen rae, hus he generalised forward raes conains a covariance par. he forward moraliy rae for pure endowmens corresponds o moving he covariance from he forward ineres rae ino he forward moraliy rae, insead of having a par in each of he forward raes. In oher words, f r + f µ conains he covariance erms, and subracing g, which does no conain any covariance erms, he covariance erms are conained in h pe. In his way, he original definiion of he forward ineres rae can be kep unalered, bu one can say ha he forward moraliy rae for pure endowmen h pe conains a covariance erm belonging o he ineres rae. he forward moraliy rae for erm insurances, h i funcion saisfying, [ e ] u (r(s)+µ(s)) ds µ(u) du F() = : [, ) R +, is defined as he e u (g (s)+h i (s)) ds h i (u) du. (4.7) o esablish ha h i is well-defined is no as easy as wih he forward moraliy rae for he pure endowmen. Firs, see ha he definiion depends on he choice of. I is naural o make he assumpion ha he forward moraliy rae for erm insurances h i is independen of. his assumpion is impliciy made in he noaion used in [11], and he assumpion is also made for he forward moraliy rae for pure endowmen. Wih his assumpion of independence of, we can differeniae wih respec o, and find he equivalen definiion, (r(s)+µ(s)) ds µ( ) F() = e for. For he res of he paper, his definiion is used. (g (s)+h i (s)) ds h i ( ), (4.8) Comparing o he generalised forward raes, we consider a policy consising of a life annuiy wih a paymen rae b, and a erm insurance wih paymen 1 upon deah. he policy erminaes a ime. he expeced presen value a ime is [ e ] s (r(s)+µ(s)) ds (b + µ(s)) ds F() 13
4.2 Comparison Wih Oher Dependen Seups = = e s r:(r+µ) (f (s)+f µ:(r+µ) e s g(s) ds ( e s hpe (s)) ds ( b + f µ:(r+µ) ) (s) ds (s) ds b + e ) s hi (s) ds h i (s) ds, where we firs wroe i in erms of he generalised forward raes, and aferwards in erms of he forward moraliy raes for pure endowmen and erm insurances, respecively. his illusraes he difference beween he differen ypes of forward raes. he generalised forward rae for moraliy can be used for boh he life annuiy and he erm insurance, whereas wih he oher forward moraliy rae definiions, one need a differen one for a differen produc. If he ineres rae is independen of he moraliy rae, he differen forward moraliy raes simplify and hey all equal he usual usual forward moraliy rae. 4.2.1 Forward Moraliy Rae for erm Insurances no-so-well Defined We have no examined wheher he forward moraliy rae for erm insurances is well defined. I urns ou, ha when here is dependence beween he ineres and moraliy raes, here are cases where he forward moraliy rae for erm insurances defined by (4.8) does no exis. his will in paricular be he case for models wih posiive correllaion. Consider a ineres and moraliy rae model (r(), µ()). his give us a se of generalised forward raes, and a forward moraliy rae for pure endowmen. Assume ha he following assumpions hold. Assumpion 4.4. Le a model for he ineres and moraliy raes r() and µ() be given. he assumpions are, 1. h pe (s) > 0 for all s >. 2. h pe is bounded from below for some imepoin, i.e. here exis ε > 0 and 0 > 0 such ha h pe (s) > ε for all s > 0. 3. he forward ineres rae is greaer han he generalised forward rae for he ineres, g (s) > f r (s), for all s >. I is indeed possible o consruc models where hese assumpions hold, and indeed, hey will hold for mos models when here is a posiive correlaion beween he ineres rae and moraliy rae. he firs wo assumpions sae ha he forward moraliy rae for pure endowmen is posiive and bounded below from some ime, which is saisfied 14
4.2 Comparison Wih Oher Dependen Seups in reasonable models. he hird assumpion assumpion usually holds when here is a posiive correlaion beween he ineres and moraliy rae. he forward moraliy rae for pure endowmen, h pe (s), presen in he assumpions, is no he objec of ineres in his example. In view of (4.6), i can be hough of as a placeholder for f r + f µ g. Proposiion 4.5. Under Assumpion 4.4, here exiss a > 0 such ha he forward moraliy rae for erm insurances h i (s) given by (4.8) does no exis for s >. Proof. Combining (4.8) and (4.2), and hen using (4.6) wice, we ge ha e and by inegraion we find e h i (s) ds h i ( ) = e (f r (s)+f µ (s) g(s)) ds f µ ( ) = e h i (s) ds = 1 e τ h pe (s) ds (h pe ( ) + g ( ) f r ( )), hpe (s) ds (h pe (τ) + g (τ) f r (τ)) dτ. (4.9) Since he lef hand side mus be posiive for any, we conclude ha he condiion e τ hpe (s) ds (h pe (τ) + g (τ) f r (τ)) dτ < 1 (4.10) is necessary for he forward moraliy rae for erm insurances o be well-defined. Under he firs assumpion, he forward moraliy rae for pure endowmen, h pe, defines a disribuion in a wo-sae Markov chain, and we recognise he inegral hpe (s) ds (τ) dτ as a probabiliy: Le Z be a sochasic variable ha denoes he lifeime in a h pe survival model where deah occurs wih rae h pe e τ hpe (s) a ime s. hen (s) ds h pe (τ) dτ = P (Z Z > ). Also, under he second assumpion he probabiliy converges o 1, P (Z Z > ) 1 for. e τ Consider now (4.10). Under he hird assumpion, g (s) > f r (s) for all s >, here exiss ε > 0 and such ha e τ hpe (s) ds (g (τ) f r (τ)) dτ > ε, 15
4.2 Comparison Wih Oher Dependen Seups for all >. his allows us o conclude, for a > large enough, such ha P (Z Z > ) > 1 ε, ha e τ hpe (s) ds (h pe (τ) + g (τ) f r (τ)) dτ > P (Z Z > ) + ε > 1. his conradics (4.10), and he forward moraliy rae for erm insurances does no exis. For compleeness, we specify a model saisfying Assumpion 4.4. Le he 2-dimensional process X saisfy dx 1 () = (1 X 1 ()) d + σ dw 1 (), dx 2 () = (1 X 2 ()) d + σλ dw 1 () + σ 1 λ 2 dw 2 (), wih X(0) = (1, 1). Le he ineres rae and moraliy rae be given by r() = r 0 X 1 (), µ() = µ () + X 2 () 1, wih parameers λ = 0.8, σ = 0.07 and base moraliy µ () = 5 10 4 + 7.5858 10 5 1.09144 50+. ha his model saisfies Assumpion 4.4 is no shown here. 4.2.2 Somewha Generalised Forward Raes A discussion of he concep of forward raes, and generalisaions, should no be underaken wihou a reference o [14]. In he aricle, a scepical view on he concep of forward moraliy raes is adoped, and he fruifulness of he concep is quesioned. In view of his aricle, i is no a all clear ha he concep of forward moraliy raes (and generalised forward raes) is fruiful beyond being a convenien noaion for he quaniies needed for calculaion of cerain life insurance liabilies under a sochasic inensiy assumpion. he aricle [14] also discusses requiremens for more generalised forward raes. he generalised forward raes proposed in his aricle does no mee all he requiremens se up in [14]. In paricular, in life insurance models where one needs o use he relaion (2.6), he generalised forward raes are no applicable. hus, i would probably be more suiing o call hem somewha generalised forward raes. 16
5 Modelling Ineres and Surrender In order o illusrae he mehods proposed, we pu up a specific model for dependen ineres and surrender rae. We model he ineres rae as a sochasic diffusion process r, and he surrender rae by he diffusion process η. he ineres and surrender raes are hen modelled as dependen processes, wihin he affine seup presened in Secion 2. Wihin he Solvency II regime, one is required o model surrender behaviour, and also ake ino consideraion any dependence of he ineres rae (i.e. he economic environmen), see Secion 3.5 in [5]. his model is hus an example of how his can be done. 5.1 Correlaed Ineres and Surrender Model Le η 0 () be a deerminisic surrender rae, corresponding o bes esimae, i.e. he expecaion of he fuure surrender rae. he ineres rae r() and surrender rae η() are hen modelled as an affine ransformaion of X of he form, r() = X 1 (), η() = η 0 ()X 2 (), where X is a 2-dimensional sochasic diffusion process. he process X saisfies he sochasic differenial equaion, ( dx 1 () = (b 1 () β 1 X 1 ()) d + σ 1 1 ρ 2 dw 1 () + ρ ) X 2 () dw 2 (), (5.1) dx 2 () = (b 2 β 2 X 2 ()) d + σ 2 X2 () dw 2 (), where W is a 2-dimensional sandard Brownian moion. he parameers saisfy b 2, β 1, β 2, σ 1, σ 2 R + and ρ [ 1, 1], and he funcion b 1 : R + R is chosen such ha an iniial erm srucure is fied. he process X 2 models relaive deviaions of he surrender rae from he bes esimae, and i says non-negaive, hence he surrender rae η() is non-negaive. he ineres rae process is a mix beween a Hull-Whie Vašíček and a Heson model. he model saisfies our crieria. I is affine, since X is affine and he surrender and ineres rae is an affine ransformaion of X. he surrender rae is non-negaive. Also, choosing no, or lile, mean reversion, sress scenarios produced by he model are close o parallel shifs of he forward raes, which resembles he sress scenarios of he sandard model of Solvency II. 17
5.2 he (Life Insurance) Produc 5.1.1 Correlaion he correlaion beween he ineres rae and he surrender rae is no in general equal o he dependency parameer ρ. If we assume ha [X 2 ()] = 1 for all, we can calculae he correlaion, using sandard mehods 2, Corr [r(), η()] = ρ e(β 1+β 2 ) 1 β 1 + β 2 In he special case where β 1 = β 2, we ge Corr [r(), η()] = ρ. 2β1 e 2β 1 1 2β2 e 2β 2 1. When he parameers are chosen in Secion 5.5.1 below, we see ha indeed [X 2 ()] = 1 and β 1 = β 2 holds. 5.2 he (Life Insurance) Produc Consider a simple savings conrac wih a buy-back opion. he savings conrac consiss of a guaraneed paymen of 1 a reiremen a ime. here is an accoun a he provider wih a guaraneed ineres rae ˆr unil ime. he value a ime of he accoun is hen, U() = e ˆr( ). (5.2) he owner of he savings conrac can hen a any ime before ime surrender he conrac and receive he curren accoun value U(). he accoun value U() is no necessarily idenical o he reserve (marke value) of he savings conrac, hus he savings conrac provider has a risk. In order o bes esimae he value of he accoun, he surrender behaviour should be aken ino accoun. here are differen ways o valuae he surrender opion, see [12] and references herein, and [10]. In his paper we adop he inensiy approach, and assume ha he insured surrenders wih rae η() a ime, i.e. in a shor ime inerval [, + d], he insured surrenders wih probabiliy η() d, given ha surrender has no occured before ime. We adop he life insurance seup of Secion 3, and consider he sae of he insured in he sae space J consising of he wo saes alive and surrendered, corresponding o Figure 2. his savings conrac is a simplified version of he produc considered in xample 3.1 and he Markov model shown in Figure 1. I is sraighforward o include he moraliy modelling done in xample 3.1, bu o keep he noaion simple, we omi i from his example. 2 he quaniies [r()] and [η()] can be found aking expecaion on he Iô represenaion, and solving a differenial equaion. he expecaion [r()η()] can be found analogously, by firs finding a sochasic differenial equaion for he process r()η(). 18
5.3 Solvency II Alive 0 Surrendered 1 η Figure 2: Markov model for he surrender model. he paymens of he conrac consis of a single paymen upon reiremen a ime, and a paymen upon surrender a ime of size U(). ha is, he oal paymens B() a ime is given by db() = U() dn 01 () + 1 (Z()=0) dε (), where ε is he Dirac measure a. Analogously o he calculaions in xample 3.1 and xample 4.3, we find he presen value a ime of he conrac as P V L () = = e s r(τ) dτ db(s) e s r(τ) dτ U(s) dn 01 (s) + e r(τ) dτ 1 (Z()=0), (5.3) and he marke value a ime is, given he savings conrac has no been surrendered, V () = [ P V L () ] F(), Z() = 0 [ = e s (r(τ)+η(τ)) dτ η(s)u(s) ds + e ] (r(τ)+η(τ)) dτ F X () = e s + e r:(r+η) (f (τ)+f η:(r+η) (τ)) dτ f η:(r+η) r:(r+η) (f (τ)+f η:(r+η) (τ)) dτ. (s)u(s) ds Here we used Remark 4.2. he noaion used is inroduced in xample 4.3 above. (5.4) 5.3 Solvency II For Solvency II purposes one wans o conrol he risk of defaul, such ha i is less han 99.5% during he following year. In his secion we specify how o inerpre his in our seup, following he reasoning of Secion 1.1 in [3]. We wan o find he loss afer one year, which is a sochasic variable, and find quaniles in he disribuion of his sochasic variable. Le P V () denoe he presen value a ime of fuure paymens of he insurance company. A ime 0, he Solvency II loss can be wrien as [P V (0) F(1)], 19
5.3 Solvency II where he expecaion is aken using he marke measure, or some reserving measure. For he res of he paper, we refer o his measure as he marke measure. For simpliciy, we ignore he so-called unsysemaic risk during he firs year, ha is, we ake average of he Markov chain Z, condiionally on he underlying inensiies X. Formally, we define he Solvency II loss afer 1 year as L = [ P V (0) F X (1) ]. Boh liabiliies and asses mus be aken ino accoun, so he presen value akes he form P V () = P V L () P V A (), ha is, he presen value of he liabiliies less he asses. For he specific life insurance conrac wih presen value (5.3), we obain, [ P V L (0) F X (1) ] = 1 0 e s 0 (r(τ)+η(τ)) dτ η(s)u(s) ds + e 1 0 (r(s)+η(s)) ds + e 1 1 e s r:(r+η) 1 (f1 (τ)+f η:(r+η) 1 (τ)) dτ f η:(r+η) 1 (s)u(s) ds 0 (r(s)+η(s)) ds e r:(r+η) 1 (f1 (τ)+f η:(r+η) 1 (τ)) dτ. (5.5) he simples possible asse allocaion is o deposi all capial in a savings accoun, earning he risk free ineres rae. In ha case, he presen value of he asses is deerminisic and equals he amoun invesed oday. If he value of he liabiliies is invesed, we have P V A (0) = V (0). For our case, V (0) is given by (5.4). Combining, we ge a Solvency II loss, L = [ P V L (0) P V A (0) F X (1) ] = [ P V L (0) F X (1) ] V (0) which is he difference beween (5.5) and (5.4). Rearranging he erms slighly, we can wrie i on he form, L = 1 0 1 e s 0 (r(τ)+η(τ)) dτ η(s)u(s) ds 0 e s r:(r+η) 0 (f0 (τ)+f η:(r+η) 0 (τ)) dτ f η:(r+η) 0 (s)u(s) ds + e 1 0 (r(s)+η(s)) ds 1 + e 1 1 e s r:(r+η) 1 (f1 (τ)+f η:(r+η) 1 (τ)) dτ f η:(r+η) 1 (s)u(s) ds e s r:(r+η) 0 (f0 (τ)+f η:(r+η) 0 (τ)) dτ f η:(r+η) 0 (s)u(s) ds 0 (r(s)+η(s)) ds e 1 e r:(r+η) 0 (f0 (τ)+f η:(r+η) 0 (τ)) dτ. (f r:(r+η) 1 (τ)+f η:(r+η) 1 (τ)) dτ 20
5.4 Hedging Sraegy wih a Coninuously Paid Coupon Bond he firs wo lines correspond o he losses arising during he firs year because of incorrec expecaions of ineres and surrender behaviour. he las four lines correspond o changed expecaions of he fuure, arising because of informaion received during he firs year. ha is, he hird and fourh line corresponds o changed expecaions of he fuure abou he surrender paymens, and he las wo lines corresponds o changed expecaions of he fuure abou he paymen occuring a reiremen. Inuiively, he informaion received during he firs year allows for an exac discouning during he firs year, and a more precise valuaion of he discouning and surrender behaviour occuring from year 1 an onwards. he loss can be wrien in a simpler form. Using he noaion ha, for s, f r:(r+η) (s) = r(s) and f η:(r+η) (s) = η(s), we can wrie he Solvency II loss as L = 0 e s r:(r+η) 0 (f1 (τ)+f η:(r+η) 1 (τ)) dτ f η:(r+η) 1 (s)u(s) ds 0 e s r:(r+η) 0 (f0 (τ)+f η:(r+η) 0 (τ)) dτ f η:(r+η) 0 (s)u(s) ds + e r:(r+η) 0 (f1 (τ)+f η:(r+η) 1 (τ)) dτ e 0 (f r:(r+η) 0 (τ)+f η:(r+η) 0 (τ)) dτ. (5.6) Recalling ha f r:(r+η) 1 and f η:(r+η) 1 are F X (1) measurable, we can use ha X is a Markov process and see ha f r:(r+η) 1 and f η:(r+η) 1 are r(1) and η(1) measurable. hus, he loss can be found by simulaion of r(s) and η(s) for 0 s 1. he simulaion mus be done under he real world probabiliy measure. his is opposed o he marke, or reserving, measure, ha was used o find he loss. In his paper, we assume for simpliciy ha he wo measures are idenical, and do no adap a change of measure approach, relieving us from discussions of preservaion of he Markov propery during measure changes. 5.4 Hedging Sraegy wih a Coninuously Paid Coupon Bond In pracice, an insurer ries o hedge he ineres rae risk, hereby reducing he loss significanly. We consider a simple saic hedging sraegy, in a bond wih coninuous coupon paymens of he form, c() = e 0 f η:η 0 (τ) dτ f η:η 0 ()U(). (5.7) For more deails, see e.g. [12]. his corresponds o he expeced paymens of he life insurance conrac, condiional on he ineres rae. We can associae a paymen process A wih he bond, given by da() = c() d. he presen value of fuure paymens 21
5.4 Hedging Sraegy wih a Coninuously Paid Coupon Bond associaed wih he bond is hen, P V A () = = e s r(τ) dτ da(s) e s r(τ) dτ e s 0 f η:η 0 (τ) dτ f η:η 0 (s)u(s) ds. his hedging sraegy is he mean-variance opimal saic hedging sraegy when ineres and surrender are independen. If here is a correlaion beween he ineres and surrender rae, his sraegy is no opimal. he mean-variance opimal saic hedging sraegy is in ha case more complicaed. hese consideraions are for simpliciy omied in his paper, and deferred for fuure sudies. We noe ha he sign of he paymens A is opposie of B, where he laer are paymens o he insured, he former are paymens o he insurer. Considering he life insurance conrac and he hedging sraegy ogeher, we obain a Solvency II loss, [ L = e ] s 0 r(τ) dτ ( db(s) da(s)) F X (1) 0 1 = e ( s 0 r(τ) dτ e s 0 η(τ) dτ η(s) e ) s 0 f η:η 0 (s) ds f η:η 0 (s) U(s) ds 0 + e 1 0 (r(s)+η(s)) ds ( + e r:(r+η) 1 (f1 (τ)+f η:(r+η) 1 (τ)) dτ ( e 1 η:η 0 (r(s)+f0 (s)) ds + e 1 1 1 (f r:r η:η 1 (τ)+f0 (τ)) dτ e s r:(r+η) 1 (f1 (τ)+f η:(r+η) 1 (τ)) dτ f η:(r+η) 1 (s)u(s) ds e s 1 ) ) (f r:r 1 (τ)+f η:η 0 (τ)) dτ f η:η 0 (s)u(s) ds = 0 e s r:(r+η) 0 (f1 (τ)+f η:(r+η) 1 (τ)) dτ f η:(r+η) 1 (s)u(s) ds 0 e s 0 (f r:r 1 + e r:(r+η) 0 (f1 (τ)+f η:(r+η) (τ)+f η:η 0 (τ)) dτ f η:η 0 (s)u(s) ds 1 (τ)) dτ e 0 (f r:r η:η 1 (τ)+f0 (τ)) dτ, (5.8) Similar o (5.6), for s, he noaion ha f r:(r+η) η(s) is used for he las equaliy. (s) = f r:r (s) = r(s) and f η:(r+η) (s) = 22
5.5 Numerical Resuls 5.5 Numerical Resuls In his secion we numerically show some consequenses of modelling ineres and surrender as posiively correlaed processes. Firs, he model is specified by choosing a se of parameers, parly inspired by he sress levels in he Solvency II Sandard Formula. Wih his model, we examine he consequenses for he balance shee value of he liabiliies, and he level of he Solvency II capial requiremen, ha is, he liabiliies in 1 years ime. For he Solvency II capial requiremen, in pracice in he indusry, when here is no hedging, mos of he risk is ineres rae risk. Luckily, boh in heory and pracice, a lo of his can be hedged by e.g. buying bonds. For he numerical illusraions of he Solvency II capial requiremen, we consider wo differen sraegies for he asses, corresponding o he wo sraegies considered in Secion 5.3 and Secion 5.4, respecively. Firs, we consider he case where he ineres rae risk is no hedged, and all asses are accumulaed by he risk free ineres rae. Second, we consider he case where he insurer ries o hedge he ineres rae risk, and performs a saic hedge. 5.5.1 Parameers he numerical examples wih he model (5.1) are carried ou for differen level of correlaion, namely ρ {0, 0.3, 0.7}. Also, we consider wo differen guaraneed ineres raes, namely ˆr {1%, 4%}. his corresponds o a low ineres rae, which could be for a newly issued policy, and a high ineres rae, which could be for a policy issued years ago, when he ineres rae level was higher. We noe ha he base deerminisic surrender rae η 0 corresponds o a person aged 40, hus wih = 25, he conrac ends a age 65. he parameers chosen for he ineres and surrender raes are lised in able 1, and in Figure 3 some realisaions of he ineres and surrender raes are shown. he iniial value X 1 (0) and funcion b 1 () are chosen such ha he erm srucure provided by he Danish FSA a Augus 17, 2012 is mached. Le f FSA () denoe he forward rae provided by he Danish FSA. hen he parameers X 1 and b 1 are fied such ha [e 0 r(s) ds] = e 0 f FSA (s) ds, for all 0. he parameers of he model correspond o he measure used for valuaing he marke value of he life insurance liabiliies. hus, wih respec o he ineres rae i is he risk neural measure. For simpliciy, we assume ha his measure equals he real world probabiliy measure. 23
5.5 Numerical Resuls Ineres Rae 0.00 0.02 0.04 0.06 0.08 Simulaion 1 Simulaion 2 Simulaion 3 Forward ineres rae Surrender Rae 0.00 0.02 0.04 0.06 0.08 Simulaion 1 Simulaion 2 Simulaion 3 Base surrender rae 0 5 10 15 20 25 0 5 10 15 20 25 ime ime Figure 3: Illusraive realisaions of he ineres rae (lef) and he surrender rae (righ), wih ρ = 0.7. β 1 0.02 σ 1 0.005 b 2 0.02 β 2 0.02 σ 2 0.15 η 0 () 0.06 0.002 X 2 (0) 1 able 1: Parameers for correlaed ineres and surrender modelling. he iniial value X 1 (0) and he funcion b 1 () are chosen such ha he ineres rae model maches he erm srucure provided by he Danish FSA for valuaing life insurance liabilies, a Augus 17, 2012. 5.5.2 Generalised Forward Raes In Figure 4, he generalised forward raes are shown. hey are calculaed by solving he differenial equaions (2.3) and (2.5) numerically. For he ineres rae, he forward ineres rae supplied by he Danish FSA, f FSA, is shown as well. We see ha for he case ρ = 0 he generalised forward ineres rae f0 r is idenical o he forward rae provided by he Danish FSA. his is as expeced, since in he case ρ = 0 he ineres rae and surrender rae are independen, and in his case he generalised forward raes are equal o he usual forward raes. For a posiive correlaion, he generalised forward raes are smaller. his is because he sochasic variable, e 0 (r(s)+η(s)) ds, which is used o consruc he generalised forward raes, has a heavier ail when he correlaion is sricly posiive, due o he exponenial funcion. Inuiively, here is less diversificaion beween he ineres and surrender rae. For he surrender rae, he basic deerminisic surrender rae η 0 is shown as well as he generalised forward raes. ven hough [η()] = η 0 (), we see ha he generalised 24
5.5 Numerical Resuls ineres rae 0.00 0.01 0.02 0.03 0.04 ρ = 0 ρ = 0.3 ρ = 0.7 Danish FSA surrender rae 0.01 0.02 0.03 0.04 0.05 0.06 ρ = 0 ρ = 0.3 ρ = 0.7 Base surrender rae 0 5 10 15 20 25 0 5 10 15 20 25 ime ime Figure 4: Generalised forward raes. Lef: for he ineres rae, f r:(r+η) 0 (). Righ: for he surrender rae, f η:(r+η) 0 (). he generalised forward raes are shown for differen values of ρ. he forward ineres rae exraced from he Danish FSA a Augus 17, 2012 is also shown, as well as he base deerminisic surrender rae η 0. Higher values of ρ lead o lower values of he forward raes, corresponding o less discouning. forward raes are sysemaically lower han η 0. his is due o Jensens inequaliy, and o see his, consider he case ρ = 0, where we ge, e 0 (f 0 r(s)+f η 0 (s)) ds = = [e > [e [e 0 (r(s)+η(s)) ds] 0 r(s) ds] 0 r(s) ds] e 0 [e 0 η(s) ds] [η(s)] ds = e 0 f r 0 (s) ds e 0 η0 (s) ds, for > 0, using ha he usual forward rae is idenical o he generalised forward rae for ρ = 0. From his inequaliy, we obain, f η 0 () < η0 (), which is wha was observed as he red and black lines in Figure 4. If here is a posiive correlaion, he generalised forward surrender rae, f η:(r+η), is even smaller, similar o he observaion for he ineres raes. 25
5.5 Numerical Resuls 5.5.3 Marke Value he marke value a ime 0, V (0) from (5.4), can be calculaed, solving he inegral numerically. For his, firs use (4.1) o ge V () = e φ(,s)+ψ(,s) X() f η:(r+η) (s)u(s) ds + e φ(, )+ψ(, ) X(), which is easier o handle from a compuaional poin of view, because he funcions φ and ψ are obained in he process of calculaing he generalised forward raes f r:(r+η) and f η:(r+η) when solving (2.3) and (2.5). he marke value V (0), dependen upon he guaraneed ineres rae ˆr and he correlaion ρ, is shown in able 2. he marke values can be compared o he value of he policyholders accoun which is paid ou on surrender. his is given by (5.2), calculaed using he guaraneed ineres rae. he value a ime 0 is presened in able 3. ˆr 4% 1% 0 0.4567 0.6167 ρ 0.3 0.4595 0.6191 0.7 0.4631 0.6222 able 2: Marke value a ime 0, V (0), of he life insurance conrac. he value is shown using hree differen correlaions, corresponding o hree differen ses of generalised forward raes, red, green and blue from Figure 4. wo differen levels of guaraneed ineres rae, ˆr, is used, which leads o differen surrender payous U(). ˆr 4% 1% 0.3679 0.7788 able 3: Iniial value of he policyholders accoun, U(0). For he high guaraneed ineres rae (4%), he value is lower han he marke value from able 2. For he low guaraneed ineres rae (1%), he value is higher han he marke value. he marke value wihou surrender modelling, calculaed seing he surrender rae equal o zero, is 0.5037. I is independen of he guaraneed ineres rae. From able 2 i is seen ha when we include surrender modelling he marke value is somewhere beween he value of he policyholders accoun and he marke value calculaed wihou surrender modelling. For boh cases of guaraneed ineres raes, he marke value increases wih correlaion. When we discussed he generalised forward raes in Secion 5.5.2, we saw ha he 26
5.5 Numerical Resuls generalised forward raes decrease wih increasing correlaion, which is basically due o he convexiy of he exponenial funcion and Jensen s inequaliy. A smaller generalised forward ineres rae leads o an increasing marke value. For he surrender rae, i is more complicaed. For he case of a guaraneed ineres rae of 4%, an increase in he generalised forward surrender rae rae leads o a decrease in he marke value, because he marke value come closer o he value paid ou on surrender. For he case of a guaraneed ineres rae of 1%, he same argumen ells us ha an increase in he generalised forward surrender rae insead leads o an increasing marke value. We see ha he effec of he decreasing generalised forward ineres rae is larges, and in oal, for boh levels of guaraneed ineres rae, he marke value increases when he correlaion increases. 5.5.4 Solvency II We examine he effec on he Solvency II capial requiremen wih wo differen sraegies for he asses. For boh sraegies, he iniial marke value of he asses equals he marke value of he liabiliies. he firs sraegy is no hedging, and he second sraegy is a simple saic hedging sraegy. his corresponds o he wo sraegies discussed in Secion 5.3 and Secion 5.4, respecively. For he firs sraegy, where all asses are invesed in he bank accoun, he Solvency II loss is given by (5.6). For he second sraegy, where he ineres rae risk is hedged saically in a bond wih coninuous paymens, he Solvency II loss is given by (5.8). No Hedge Hedge ˆr ˆr 4% 1% 4% 1% 0 0.069 0.077 0.014 0.025 ρ 0.3 0.072 0.072 0.018 0.030 0.7 0.078 0.060 0.024 0.034 able 4: Simulaed Solvency II loss. Wihou hedging i is given by (5.6) and wih he hedging sraegy i is given by (5.8). Applying an ineres hedging sraegy significanly lowers he Solvency II loss. Also, modelling correlaion beween ineres and surrender has a significan impac on he Solvency II loss. In able 4 he Solvency II loss for he differen cases of hedging sraegy, guaraneed ineres rae risk and correlaion is presened. I is immediaely seen, ha rying o hedge he ineres rae risk by applying he simple hedging sraegy significanly reduces he Solvency II loss. For he case of no hedging sraegy, we see wo differen correlaion effecs. When he guaraneed ineres rae is 4%, a higher correlaion means a higher Solvency II loss, 27
0.010 0.000 0.010 0.06 0.08 0.10 Numerical Resuls 0.04 surrender rae year 1 0.08 0.06 0.04 surrender rae year 1 0.10 5.5 0.020 0.010 ineres rae year 1 0.000 0.010 0.020 ineres rae year 1 0.000 0.010 0.020 ineres rae year 1 0.06 0.08 0.10 0.010 0.04 surrender rae year 1 0.08 0.06 0.04 surrender rae year 1 0.10 Figure 5: Guaraneed ineres rae 4%. Plo of he ineres and surrender rae simulaions afer 1 year in he case wihou any hedging sraegy and correlaion ρ = 0 (lef) and ρ = 0.7 (righ). he color of a mark indicaes he Solvency II loss (5.6), where a darker color is a higher loss, and black colors are losses beyond he 99.5% quanile. 0.010 0.000 0.010 0.020 ineres rae year 1 Figure 6: Guaraneed ineres rae 1%. Plo of he ineres and surrender rae simulaions afer 1 year in he case wihou any hedging sraegy and correlaion ρ = 0 (lef) and ρ = 0.7 (righ). he color of a mark indicaes he Solvency II loss (5.6), where a darker color is a higher loss, and black colors are losses beyond he 99.5% quanile. 28