Computation of bonus in multi-state life insurance
CComputation of bonus in multi-state life insurance
Jamaal Ahmad (cid:63) , Kristian Buchardt , and Christian Furrer Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100Copenhagen Ø, Denmark. PFA Pension, Sundkrogsgade 4, DK-2100 Copenhagen Ø, Denmark. (cid:63)
Corresponding author. E-mail: [email protected].
July 9, 2020
Abstract
We consider computation of market values of bonus payments in multi-state with-profit life insur-ance. The bonus scheme consists of additional benefits bought according to a dividend strategythat depends on the past realization of financial risk, the current individual insurance risk, thenumber of additional benefits currently held, and so-called portfolio-wide means describing theshape of the insurance business. We formulate numerical procedures that efficiently combine si-mulation of financial risk with more analytical methods for the outstanding insurance risk. Spe-cial attention is given to the case where the number of additional benefits bought only dependson the financial risk.
Keywords:
Market consistent valuation; With-profit life insurance; Participating life insurance;Economic scenarios; Portfolio-wide means
JEL Classification:
G22; C63
The potential of systematic surplus in multi-state with-profit life insurance (sometimesreferred to as participating life insurance) leads to bonus payments that depend on thedevelopment of the financial market and the states of the insured. This dependence istypically non-linear and involves the whole paths of the processes governing the financialmarket and the states of the insured. Consequently, the computation of market values of1 a r X i v : . [ q -f i n . R M ] J u l INTRODUCTIONbonus payments lies outside the scope of classic backward and forward methods. In thispaper, we present computational schemes for a selection of these more involved marketvalues using a combined approach in which we simulate the financial risk while retainingmore analytical methods for the outstanding insurance risk.In Denmark, the investment strategy and dividend strategy are to a great extent con-trolled by the insurer, and practitioners have traditionally determined the market valueof bonus payments residually by imposing the equivalence principle on the market basis,cf. [14, Chapter 2]. In reality, this valuation method is only applicable if – among otherthings – one includes payments to and from the equity, since such payments appear na-turally in the context of e.g. cost of capital and other expenses. Thus a decomposition ofthe total market value that specifically displays the market value of bonus payments, asrequired by the Solvency II and IFRS 17 regulative frameworks, cf. [10, 11, 1], cannot bederived residually unless the market value of payments to and from the equity is easy todetermine. Since the latter generally is not the case, more sophisticated computationalmethods are required. The provision of these kinds of methods constitutes the maincontribution of this paper.The study of systematic surplus and bonus payments in multi-state with-profit life in-surance goes back to [19, 17, 18], where one finds careful definitions of various conceptsof surplus, discussions of general principles for its redistribution, and the introduction offorecasting techniques in a so-called Markov chain interest model, see also [16]. In [20],partial differential equations for market values of so-called predetermined payments andbonus payments are derived in a Black-Scholes model.The projection of bonus payments in multi-state life insurance and the computation of as-sociated market values has recently received renewed attention, see [12, 13, 2, 6]. In [13],the focus is on projection of bonus payments conditionally on the insured sojourning ina specific state; this approach targets e.g. product design and bonus prognosis from theperspective of the insured rather than market valuation. Conversely, the paper [12] alsodeals with projection of bonus payments but on a portfolio level, which ensures computa-tional feasibility but does not shed light on the full complexity of multi-state with-profitlife insurance. Although with-profit life insurance focuses on the collective and althoughdecisions by the insurer (so-called future management actions), including possible deter-mination of dividend yields, often depend mainly on the performance of the collective,one ought to take into account that bonus payments are individual in nature. This isthe starting point in [2], where the focus is on deriving differential equations for relevantretrospective reserves given a dividend strategy (used to buy additional benefits) thatdepends in an affine manner on the reserves themselves. The process governing the stateof the insured is assumed Markovian. In [6], the results of [2] are extended to allowfor policyholder behavior, namely the options of surrender and free policy conversion.In [2, 6], the dependence of the dividend strategy on the performance of the collective,2 INTRODUCTIONencapsulated in what we shall term the shape of the insurance business, and the practicaland computational challenges arising from this are not highlighted.In this paper, we derive methods for the computation of market values of bonus paymentsin a Markovian multi-state model for a financial market consisting of one risky asset inaddition to a bank account governed by a potentially stochastic interest rate. Theinsurance risk and financial risk are assumed independent. We include the policyholderoptions surrender and free policy conversion following [8, 3, 4] and focus on the bonusscheme known as additional benefits , where dividends are used to buy extra benefits;this bonus scheme is common in practice and is e.g. the focal point of [14, Chapter 2].In practice, the dividend strategy depends on product design, regulatory frameworks,and decisions made by the insurer. In this paper, we assume that the dividend strategyis explicitly computable based on the following information: the past realization offinancial risk, the current individual insurance risk (state of insured and time sincefree policy conversion), the current shape of the insurance business, and the number ofadditional benefits currently held. Furthermore, the dividend strategy must be affinein the number of additional benefits. The shape of the insurance business consists ofso-called portfolio-wide means, cf. [14, Chapter 6], which reflect on a portfolio levelthe current financial state of the insurance business. Consequently, the shape of theinsurance business depends on the dividend strategy, which again depends on the shapeof the insurance business.Using classic techniques, we derive a system of differential and integral equations forthe computation of the expected accumulated bonus cash flows conditionally on therealization of financial risk. This allows us to formulate a procedure for the computationof the market value of bonus payments which efficiently combines simulation of financialrisk with classic methods for the remaining insurance risk. We identify the specialcase where the number of additional benefits depend only on financial risk – the stateindependent case – and show how this significantly simplifies the numerical procedure.It is our impression that the state independent model is aligned to current actuarialpractice, where it might e.g. serve as an approximation for valuation on a portfolio level.We should like to stress that while our results are subject to important technical reg-ularity conditions, it is the general methodology and conceptual ideas that constitutethe main contributions of this paper. Furthermore, our concepts, methods, and resultsare targeted academics and actuarial practitioners alike, and, consequently, we aim atkeeping the presentation at a reasonable technical level.The paper is structured as follows. In Section 2, we present the setup. The general resultsand general numerical procedure are given in Section 3, while the state independent caseis the subject of Section 4. Finally, Section 5 concludes with a comparison with recentadvances in the literature and a discussion of possible extensions.3 SETUP
In the following, we describe the mathematical framework. Subsections 2.1–2.3 intro-duce the processes governing the financial market, the state of the insured, and the insu-rance payments, and we discuss the valuation of so-called predetermined payments. Thedividend and bonus scheme is described in Subsection 2.4, which leads to a specificationof the total payment stream as a sum of predetermined payments and bonus payments.Contrary to the predetermined payments, the bonus payments depend on the develop-ment of the financial market, which adds an extra layer of complexity to the valuationproblem. The focal point of this paper is to establish explicit methods for the compu-tation of the market value of the bonus payments; a precise description of this problemis given in Subsection 2.5. In the remainder of the paper, the problem is studied for aspecific class of dividend processes specified in Subsection 2.6.A background probability space (Ω , F , P) is taken as given. Unless explicitly stated orevident from the specific context, all statements are in an almost sure sense w.r.t. P. Theprobability measure P relates to market valuation and therefore corresponds to some riskneutral probability measure. Due to the presence of insurance risk, the market is notcomplete, which implies that the risk neutral probability measure is not unique. Sincewe shall assume financial risk and insurance risk to be independent, one can think of theprobability measure P as the product measure of some risk neutral probability measurefor financial risk and some probability measure for insurance risk.
The state of the insured is governed by a non-explosive jump process Z = { Z ( t ) } t ≥ on a finite state space J with deterministic initial state Z (0) ≡ z ∈ J . Denote by N the corresponding multivariate counting process with components N jk = { N jk ( t ) } t ≥ for j, k ∈ J , k (cid:54) = j given by N jk ( t ) = { s ∈ (0 , t ] : Z ( s − ) = j, Z ( s ) = k } . Let S = { S ( t ) } t ≥ be the price process for some risky asset (diffusion process, inparticular continuous) and let r = { r ( t ) } t ≥ be a suitably regular short rate processwith corresponding bank account S ( t ) = S (0) exp (cid:16)(cid:82) t r ( v ) d v (cid:17) , S (0) ≡ s >
0, andsuitably regular forward interest rates f ( t, · ), t ≥
0, satisfyingE (cid:104) e − (cid:82) Tt r ( s ) d s (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) = e − (cid:82) Tt f ( t,s ) d s for all 0 ≤ t < T as well as f ( t, t ) = r ( t ) for all t ≥
0; here F S is the natural filtrationgenerated by S := ( S , S ), which exactly represents available market information. The4 SETUPavailable insurance information is represented by the filtration F Z naturally generatedby Z , and the total information available is represented by the filtration F = F S ∨ F Z naturally generated by ( S, Z ).To allow for free policy behavior and surrender, we suppose the state space J can bedecomposed as J = J p ∪ J f , with J p := { , . . . , J } and J f := { J + 1 , . . . , J + 1 } for some J ∈ N . Here J p containsthe premium paying states, while J f contains the free policy states, and transition to { J } and { J + 1 } corresponds to surrender as premium paying and free policy, respectively,cf. [3, 4]. We suppose that J f is absorbing and can only be reached via a transition from { } to { J + 1 } , { J } and { J + 1 } are absorbing, and that { J } and { J + 1 } can onlybe reached from { } and { J + 1 } , respectively. The setup is depicted in Figure 1. · · · iJ − J p J Surrender J + 1 · · · J + 1 + i J J f J + 1Surrenderas freepolicy Figure 1:
General finite state space extended with a surrender state { J } and free policy states J f . The states J p \ { J } contain the biometric states of the insured, e.g. active, disabled, anddead. The states J f are a copy of J p , and a transition from { } to { J + 1 } corresponds to a freepolicy conversion. A transition to { J } or { J + 1 } corresponds to a surrender of the policy. The life insurance contract is described by a payment stream B = { B ( t ) } t ≥ giving accumulated benefits less premiums. It consists of predetermined payments B ◦ = { B ◦ ( t ) } ≤ t ≤ n , stipulated from the beginning of the contract, and additional bonuspayments determined when market and insurance information are realized during thecourse of the contract; details regarding the latter are given in later subsections.We specify the predetermined payments as in [3, 4]. For simplicity, we suppose thatthe predetermined payments regarding the classic states J p consist of suitably regulardeterministic sojourn payment rates b j and transition payments b jk ; in particular, sur-render results in a deterministic payment. In the free policy states, no premiums arepaid and the benefit payments are reduced by a factor ρ ∈ [0 ,
1] depending on the timeof free policy conversion. In rigorous terms, we haved B ◦ ( t ) = d B ◦ , p ( t ) + ρ ( τ ) d B ◦ , f ( t ) , B ◦ (0) = 0 , d B ◦ , p ( t ) = (cid:88) j ∈J p ( Z ( t − )= j ) (cid:18) b j ( t ) d t + (cid:88) k ∈J p k (cid:54) = j b jk ( t ) d N jk ( t ) (cid:19) , B ◦ , p (0) = 0 , d B ◦ , f ( t ) = (cid:88) j ∈J f ( Z ( t − )= j ) (cid:18) b + j (cid:48) ( t ) d t + (cid:88) k ∈J f k (cid:54) = j b + j (cid:48) k (cid:48) ( t ) d N jk ( t ) (cid:19) , B ◦ , f (0) = 0 , with J f (cid:51) j (cid:55)→ j (cid:48) := j − ( J + 1) and x + := max { , x } , and where τ is the time of freepolicy conversion given by τ = inf { t ∈ [0 , ∞ ) : Z ( t ) ∈ J f } . We have τ = 0 if and only if z ∈ J f ; in this case, the policy is initially a free policy.Without loss of generality we thus let ρ (0) = 1. Furthermore, we suppose there are nosojourn payments in the surrender states, i.e. b J ≡ B ◦ into benefit and pre-mium parts. We add the superscript ± to denote the benefit and premium part, respec-tively. Then we have B ◦ , − ( t ) = B ◦ , p , − ( t ) ,B ◦ , + ( t ) = B ◦ , p , + ( t ) + ρ ( τ ) B ◦ , f ( t ) , d B ◦ , p , ± ( t ) = (cid:88) j ∈J p ( Z ( t − )= j ) (cid:18) b ± j ( t ) d t + (cid:88) k ∈J p k (cid:54) = j b ± jk ( t ) d N jk ( t ) (cid:19) , B ◦ , p , ± (0) = 0 . In the following, we assume the existence of a maximal contract time n ∈ (0 , ∞ ) in thesense that all sojourn payment rates and transition payments, including those of theunit bonus payment stream, cf. Subsection 2.4, are zero for t > n .6 SETUP The life insurance contract is written on the technical basis , also called the first orderbasis, which is at least originally designed to consist of prudent assumptions on financialrisk and insurance risk. The technical basis is modeled via another probability measureP (cid:63) under which the short rate process r (cid:63) is deterministic and suitably regular, while Z is independent of S and Markovian with suitably regular transition rates µ (cid:63) . Theassumptions regarding absorption, as illustrated in Figure 1, are retained under P (cid:63) .Policyholder behavior is not included on the technical basis, which entails the followingconstraints on the transition rates, surrender payments, and free policy factor, see [3, 4]: µ (cid:63)jk = µ (cid:63)j (cid:48) k (cid:48) , j, k ∈ J f , k (cid:54) = j,b J = (cid:101) V (cid:63) , (0 , ∞ ) (cid:51) t (cid:55)→ ρ ( t ) = (cid:101) V (cid:63) ( t ) (cid:101) V (cid:63), +0 ( t ) , where for j ∈ J p \ { J } the state-wise technical reserve (cid:101) V (cid:63)j of predetermined paymentsand the corresponding valuation of benefits only (cid:101) V (cid:63), + are given by (cid:101) V (cid:63)j ( t ) = E (cid:63) (cid:20) (cid:90) nt e − (cid:82) st r (cid:63) ( v ) d v d B ◦ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) Z ( t ) = j (cid:21) , (2.1) (cid:101) V (cid:63), + j ( t ) = E (cid:63) (cid:20) (cid:90) nt e − (cid:82) st r (cid:63) ( v ) d v d B ◦ , + ( s ) (cid:12)(cid:12)(cid:12)(cid:12) Z ( t ) = j (cid:21) , (2.2)with E (cid:63) denoting integration w.r.t. P (cid:63) . It it possible to show that the state-wise technicalreserves of predetermined payments satisfy the following differential equations of Thieletype:dd t (cid:101) V (cid:63)j ( t ) = r (cid:63) ( t ) (cid:101) V (cid:63)j ( t ) − b j ( t ) − (cid:88) k ∈J p \{ J } k (cid:54) = j (cid:0) b jk ( t ) + (cid:101) V (cid:63)k ( t ) − (cid:101) V (cid:63)j ( t ) (cid:1) µ (cid:63)jk ( t ) , (cid:101) V (cid:63)j ( n ) = 0 , (2.3)for j ∈ J p \ { J } . By adding +’s as superscripts, one finds an identical system ofdifferential equations concerning the valuation of benefits only.We are now ready to define the technical reserve of predetermined payments denoted V (cid:63), ◦ . First, for the purpose of bonus allocation, the definitions of state-wise reserves ofpredetermined payments are naturally extended from j ∈ J p \ { J } to j ∈ J via V (cid:63), ◦ j ( t ) = (cid:101) V (cid:63)j ( t ) if j ∈ J p \ { J } ,ρ ( τ ) (cid:101) V (cid:63), + j (cid:48) ( t ) if j ∈ J f \ { J + 1 } , j ∈ { J, J + 1 } . (2.4)7 SETUPThe technical reserve of predetermined payments V (cid:63), ◦ is then defined according to V (cid:63), ◦ ( t ) = V (cid:63), ◦ Z ( t ) ( t ). Note that V (cid:63), ◦ j depends on τ in the free policy states, thus beingstochastic, while it is deterministic in the premium paying states.We now turn our attention to valuation under the market basis modeled via P. Here weassume that Z and S are independent and that Z is Markovian with suitably regulartransition rates µ . The market reserve V ◦ of predetermined payments is then given by V ◦ ( t ) = E (cid:20) (cid:90) nt e − (cid:82) st r ( u ) d u d B ◦ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) (cid:21) = (cid:90) nt e − (cid:82) st f ( t,u ) d u A ◦ ( t, d s ) , (2.5)with A ◦ the so-called expected accumulated predetermined cash flows given by A ◦ ( t, s ) = E (cid:2) B ◦ ( s ) − B ◦ ( t ) | F Z ( t ) (cid:3) . (2.6)Denote with p the transition probabilities of Z under P. Following [3, 4], on ( Z ( t ) ∈ J f ), A ◦ ( t, d s ) = ρ ( τ ) (cid:88) j ∈J f p Z ( t ) j ( t, s ) (cid:18) b + j (cid:48) ( s ) + (cid:88) k ∈J f k (cid:54) = j b + j (cid:48) k (cid:48) ( s ) µ jk ( s ) (cid:19) d s, (2.7)while on ( Z ( t ) ∈ J p ), A ◦ ( t, d s ) = (cid:88) j ∈J p p Z ( t ) j ( t, s ) (cid:18) b j ( s ) + (cid:88) k ∈J p k (cid:54) = j b jk ( s ) µ jk ( s ) (cid:19) d s + (cid:88) j ∈J f p ρZ ( t ) j ( t, s ) (cid:18) b + j (cid:48) ( s ) + (cid:88) k ∈J f k (cid:54) = j b + j (cid:48) k (cid:48) ( s ) µ jk ( s ) (cid:19) d s (2.8)where the so-called ρ -modified transition probabilities p ρjk , j ∈ J p and k ∈ J , are definedby p ρjk ( t, s ) = E[ ( Z ( s )= k ) ρ ( τ ) ( τ ≤ s ) | Z ( t ) = j ] and satisfy for k ∈ J f so-called ρ -modifiedversions of Kolmogorov’s forward differential equations:dd s p ρjk ( t, s ) = (cid:88) (cid:96) ∈J f (cid:96) (cid:54) = k p ρj(cid:96) ( t, s ) µ (cid:96)k ( s ) + ( k = J +1) p j ( t, s ) µ k ( s ) ρ ( s ) − p ρjk ( t, s ) µ k • ( s ) ,p ρjk ( t, t ) = 0 , (2.9)while p ρjk ( t, s ) = p jk ( t, s ) for k ∈ J p . With premiums determined by the principle of equivalence based on the prudent techni-cal basis, the portfolio creates a systematic surplus if everything goes well. This surplus8 SETUPmainly belongs to the insured and is to be paid back in the form of dividends. Following[17, 18], we let D = { D ( t ) } t ≥ denote the accumulated dividends, and we suppose itonly consists of absolutely continuous dividend yields:d D ( t ) = δ ( t ) d t, D (0) = 0 , where δ = { δ ( t ) } t ≥ is suitably regular and F -adapted. In Subsection 2.6, we specifythe dividend strategy further.We suppose that the dividends are used as a premium to buy additional benefits on thetechnical basis corresponding to a so-called unit bonus payment stream B † that onlyconsists of benefits and thus is unaffected by the free policy option. It is given byd B † ( t ) = (cid:88) j ∈J ( Z ( t − )= j ) (cid:18) b † j ( t ) d t + (cid:88) k ∈J k (cid:54) = j b † jk ( t ) d N jk ( t ) (cid:19) , B † (0) = 0 , where the payment functions in the premium paying states J p , b † j and b † jk , are suitablyregular non-negative deterministic functions with b † J ≡
0, while b † j = b † j (cid:48) and b † jk = b † j (cid:48) k (cid:48) , j, k ∈ J f , k (cid:54) = j,b † J = (cid:101) V (cid:63), † , where for j ∈ J p \ { J } we denote by (cid:101) V (cid:63), † j the state-wise technical unit reserves of B † given as (2.1) with B ◦ replaced by B † . Again, these state-wise technical reserves satisfydifferential equations of Thiele type, namely (2.3) with added superscripts † .For the purpose of bonus allocation, the state-wise technical unit reserves are naturallyextended from j ∈ J p \ { J } to j ∈ J via V (cid:63), † j ( t ) = (cid:101) V (cid:63), † j ( t ) if j ∈ J p \ { J } , (cid:101) V (cid:63), † j (cid:48) ( t ) if j ∈ J f \ { J + 1 } , j ∈ { J, J + 1 } , (2.10)when the technical value of the additional benefits V (cid:63), † reads V (cid:63), † ( t ) = V (cid:63), † Z ( t ) ( t ).The expected accumulated unit bonus cash flows A † of B † on the market basis can befound analogously to A ◦ and read A † ( t, d s ) = a † ( t, s ) d s, (2.11) a † ( t, s ) = (cid:88) j ∈J p Z ( t ) j ( t, s ) (cid:18) b † j ( s ) + (cid:88) k ∈J k (cid:54) = j b † jk ( s ) µ jk ( s ) (cid:19) . (2.12)9 SETUPThe state-wise counterparts are denoted A † i and a † i , i ∈ J . They satisfy A † Z ( t ) ( t, d s ) = a † Z ( t ) ( t, s ) d s = a † ( t, s ) d s = A † ( t, d s ) by taking the form A † i ( t, d s ) = a † i ( t, s ) d s, (2.13) a † i ( t, s ) = (cid:88) j ∈J p ij ( t, s ) (cid:18) b † j ( s ) + (cid:88) k ∈J k (cid:54) = j b † jk ( s ) µ jk ( s ) (cid:19) . (2.14)Let Q ( t ) denote the number of additional benefits held at time t . Since δ is used as apremium to buy B † on the technical basis, we have thatd Q ( t ) = d D ( t ) V (cid:63), † Z ( t ) ( t ) = δ ( t ) V (cid:63), † Z ( t ) ( t ) d t, Q (0) = 0 . (2.15)Imposing this bonus mechanism, the total payment stream consisting of both predeter-mined payments and bonus payments is given byd B ( t ) = d B ◦ ( t ) + Q ( t ) d B † ( t ) , B (0) = 0 . (2.16)In this paper, we implicitly think of Q as weakly increasing, although this is not amathematical requirement. This way of thinking is reflected in the terminology. Alongthese lines, we define the payment process B g by B g ( t, d s ) = d B ◦ ( s ) + Q ( t ) d B † ( s ) , B g ( t, t ) = B ( t ) , (2.17)and refer to it as the payments guaranteed at time t ≥
0, while the remaining payments( Q ( s ) − Q ( t )) d B † ( s )are referred to as bonus (payments) .In the remainder of the paper, we focus on valuation of the payment stream (2.16), inparticular the bonus payments. We assume that Q exists and is suitably regular, so thatthe technical arguments in the remainder of the paper are legitimate. This is an implicitcondition that must be checked for any specific model. Thinking of time zero as now, the present life insurance liabilities of the insurer aredescribed by the market value of the total payment stream B evaluated at time zero: V (0) = E (cid:20)(cid:90) n e − (cid:82) t r ( v ) d v d B ( t ) (cid:21) .
10 SETUPBy (2.16), this amounts to market valuation of the predetermined payments and bonuspayments. Thus V (0) = V ◦ (0) + V b (0) where V ◦ (0) is given by (2.5) and V b (0) = E (cid:20)(cid:90) n e − (cid:82) t r ( v ) d v Q ( t ) d B † ( t ) (cid:21) . (2.18)is the time zero market value of bonus payments. Remark . By setting Q (0) = 0, we think of time zero as the time of initialization of theinsurance contract. To determine the market value of bonus payments after initializationof the contract, one could extend the filtration F to include additional information attime zero and consider a general F (0)-adapted Q (0). This extension is straightforwardand achieved by focusing on Q ( · ) − Q (0) rather than Q ( · ), and thus the requirement Q (0) = 0 is only really made for notational convenience. (cid:52) There exists well-established methods to calculate V ◦ (0) explicitly using the expectedaccumulated cash flows of predetermined payments on the market basis from (2.7)–(2.8);in particular, this computation does not depend on the dividend strategy δ nor furtherrealizations of the financial market (only the forward rate curve f (0 , · ) is required). Onthe contrary, the time zero market value of bonus payments V b (0) does depend on thestrategy δ . Due to possibly non-linear path dependencies regarding both the financialand biometric/behavioral scenarios, this implies that classic computational methods via( ρ -modified) Kolmogorov’s forward differential equations are not applicable.The focal point of the paper is to establish methods to calculate the market value ofbonus payments V b (0). We consider an approach that combines simulations of thefinancial market with more analytical methods for calculations involving the state ofthe insured. Everything else being equal, this approach should be numerically superiorto a pure simulation approach for which one would simulate both the financial marketand the state of the insured. To formalize the main idea, we define what we shall term Q -modified transition probabilities (at time 0) for j ∈ J by p Qz j (0 , t ) = E (cid:2) Q ( t ) ( Z ( t )= j ) (cid:12)(cid:12) F S ( t ) (cid:3) (2.19)for all t ≥
0. We immediately have the following result:
Proposition 2.2.
Under suitable regularity conditions the time zero market value of thebonus payments is given by V b (0) = E (cid:20)(cid:90) n e − (cid:82) t r ( v ) d v A b (0 , d t ) (cid:21) , (2.20) A b (0 , d t ) = a b (0 , t ) d t, (2.21) a b (0 , t ) := (cid:88) j ∈J p Qz j (0 , t ) (cid:16) b † j ( t ) + (cid:88) k ∈J k (cid:54) = j b † jk ( t ) µ jk ( t ) (cid:17) . (2.22)11 SETUP Furthermore, if Q is adapted to F S , then p Qz j (0 , t ) = Q ( t ) p z j (0 , t ) , (2.23) a b (0 , t ) = Q ( t ) a † (0 , t ) . (2.24) Proof.
Since { Q ( t ) } t ≥ is continuous and adapted, it is predictable. Using martingaletechniques, we find that V b (0) = E (cid:34) (cid:90) n e − (cid:82) t r ( v ) d v (cid:88) j ∈J Q ( t ) ( Z ( t − )= j ) (cid:18) b † j ( t ) + (cid:88) k ∈J k (cid:54) = j b † jk ( t ) µ jk ( t ) (cid:19) d t (cid:35) . Due to continuity assumptions, we might replace ( Z ( t − )= j ) by ( Z ( t )= j ) . Using the lawof iterated expectations and Fubini’s theorem, we conclude that V b (0) = E (cid:34) (cid:90) n e − (cid:82) t r ( v ) d v (cid:88) j ∈J E (cid:2) ( Z ( t )= j ) Q ( t ) (cid:12)(cid:12) F S ( t ) (cid:3) (cid:18) b † j ( t ) + (cid:88) k ∈J k (cid:54) = j b † jk ( t ) µ jk ( t ) (cid:19) d t (cid:35) = E (cid:34) (cid:90) n e − (cid:82) t r ( v ) d v (cid:88) j ∈J p Qz j (0 , t ) (cid:18) b † j ( t ) + (cid:88) k ∈J k (cid:54) = j b † jk ( t ) µ jk ( t ) (cid:19) d t (cid:35) = E (cid:34) (cid:90) n e − (cid:82) t r ( v ) d v a b (0 , t ) d t (cid:35) . Furthermore, if Q is F S -adapted, then the Q -modified transition probabilities satisfy p Qz j (0 , t ) = E (cid:2) ( Z ( t )= j ) Q ( t ) (cid:12)(cid:12) F S ( t ) (cid:3) = Q ( t ) p z j (0 , t ) , and thus a b (0 , t ) = Q ( t ) a † (0 , t ), cf. (2.12).Since the so-called expected accumulated bonus cash flow A b (0 , · ) is F S -adapted, theresult provides a representation of V b (0) motivating a computational scheme based onsimulation of the financial market. For each simulated financial scenario, we shouldcompute A b (0 , · ) explicitly in each scenario, which in general requires computation ofof p Qz j (0 , · ) for all j ∈ J ; this we study in Section 3. In the special case where Q is F S -adapted, it holds that p Qz j (0 , · ) = Q ( · ) p z j (0 , · ), and the problem simplifies to adirect calculation of Q that does not involve the biometric/behavioral states, and canessentially be solved by a classic computation of the expected accumulated cash flow A † (0 , · ) via Kolmogorov’s forward differential equations; this is studied in Section 4.As mentioned above, the computation of the expected accumulated bonus cash flowdepends on the actual specification of the dividend strategy δ during the course of thecontract, and in practice, this strategy is a control variable that depends on what we12 SETUPrefer to as the shape of the insurance business. In the following subsection, we formalizethe shape of the insurance business and its corresponding controls, which leads to aspecification of a class of dividend strategies. We now introduce the shape of the insurance business consisting of key quantities on aportfolio level that the insurer needs at future time points to determine the controls, i.e.the dividend strategy and the investment strategy. We only introduce a few key financialindicators, but we believe that our general methodology allows for the implementationof additional shape variables.To describe the shape of the insurance business, we first consider the liabilities, specif-ically the technical value and the market value of guaranteed payments on a portfoliolevel. Recall that the payments B g ( t, · ) guaranteed at time t ≥ V g is thus given by V g ( t ) = E (cid:20) (cid:90) nt e − (cid:82) st r ( v ) d v B g ( t, d s ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) (cid:21) = (cid:90) nt e − (cid:82) st f ( t,v ) d v A g ( t, d s ) , (2.25)with A g denoting the expected accumulated guaranteed cash flows, A g ( t, d s ) = A ◦ ( t, d s ) + Q ( t ) A † Z ( t ) ( t, d s ) . (2.26)Similary, the technical reserve of guaranteed payments is given by V (cid:63) ( t ) = V (cid:63), ◦ ( t ) + Q ( t ) V (cid:63), † Z ( t ) ( t ) . (2.27)The so-called portfolio-wide means of V (cid:63) and V g are now obtained by averaging outthe unsystematic insurance risk by applying the law of large numbers w.r.t. a collectionof independent and comparable insured in the portfolio, see e.g. the discussions in [14,Chapter 6] and [15]. The portfolio-wide means take the form¯ V g ( t ) = E (cid:2) V g ( t ) | F S ( t ) (cid:3) and ¯ V (cid:63) ( t ) = E (cid:2) V (cid:63) ( t ) | F S ( t ) (cid:3) for t ≥
0. The portfolio-wide means represent values of liabilities under the assumptionthat the insurance portfolio is of such a size that unsystematic insurance risk can bedisregarded. It corresponds to what is often referred to as mean-field approximations inthe literature. In Subsection 3.1, we show how to compute these.We now turn our attention to the assets. They are described by a portfolio of S whichis self-financed by the premium less benefits that the portfolio of insured pays to theinsurer. We denote the value process by U = { U ( t ) } t ≥ . We think of this process as theassets for the whole portfolio, but in our presentation the payments involved are only13 SETUPthe contributions of a single insured. Since an individual insured pays − d B ( t ) to theinsurer, this contribution to the total payments of the portfolio can be represented bythe expected cash flow − (cid:0) A ◦ (0 , d t ) + A b (0 , d t ) (cid:1) . Thus we let U take the formd U ( t ) = θ ( t ) d S ( t ) + η ( t ) d S ( t ) − (cid:0) A ◦ (0 , d t ) + A b (0 , d t ) (cid:1) , U (0) ≡ u , where ( θ, η ) = ( θ ( t ) , η ( t )) t ≥ is a suitably regular F S -adapted investment strategy. Wethink of η as a control variable for the insurer, since the number of units invested intothe bank account is determined residually by θ ( t ) = ( U ( t ) − η ( t ) S ( t )) /S ( t ). This givesd U ( t ) = r ( t )( U ( t ) − η ( t ) S ( t ))d t + η ( t ) d S ( t ) − (cid:0) A ◦ (0 , d t ) + A b (0 , d t ) (cid:1) . (2.28)In this paper, we only consider a single insured and the portfolio-wide mean reserves re-present the contribution of this insured to the shape of the insurance business. To includethis observation into the setting, one can consider Z (0) as stochastic with distributioncorresponding to the empirical distribution of initial states in the portfolio. The lattercan be described by weights w j with the j ’th weigth giving the proportion of insuredthat are initially in state j ∈ J . The corresponding portfolio-wide means would in thiscase read (cid:88) j ∈J w j E j (cid:2) V g ( t ) | F S ( t ) (cid:3) and (cid:88) j ∈J w j E j (cid:2) V (cid:63) ( t ) | F S ( t ) (cid:3) , where E j corresponds to expectation under the assumption that Z (0) ≡ j . Additionally,the insured typically belong to different cohorts implying that e.g. the transition ratesand payment processes differ among insured. This is handled in a similar way. Also, thesame considerations apply to the payments affecting the value process U . We considerthese kinds of extensions from a single insured to a whole portfolio straightforward anddo not give them further attention in the remainder of the paper.Let S ( ·∧ t ) = { S ( u ) } ≤ u ≤ t . We can now make the concepts of shape and controls precise. Definition 2.3.
The shape of the insurance business I is the triplet I = (cid:0) U ( t ) , ¯ V g ( t ) , ¯ V (cid:63) ( t ) (cid:1) t ≥ , while the controls are the pair ( δ ( t ) , η ( t )) t ≥ . Assumption 2.4.
We suppose that ( δ, η ) are chosen such that the setting is well-specifiedin the sense that Q exists and is suitably regular. Furthermore, we assume that η takesthe form η ( t ) = η ( t, S ( · ∧ t ) , I ( t )) (2.29)14 SETUP for some explicitly computable and suitably regular deterministic mapping η , and weassume that δ takes the form δ ( t ) = δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t ))+ δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) ρ ( τ ) ( τ ≤ t ) + δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) Q ( t ) , (2.30) for some suitably regular deterministic mappings δ , δ and δ that we are able to computeexplicitly.Remark . In Remark 2.1 we discussed the extension to general Q (0) and the idea offocusing on Q ( · ) − Q (0). By rewriting (2.30) in the following manner, δ ( t ) = δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) + δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) Q (0)+ δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) ρ ( τ ) ( τ ≤ t ) + δ ( t, S ( · ∧ t ) , Z ( t ) , I ( t )) ( Q ( t ) − Q (0)) , we see how this idea would manifest itself in relation to Assumption 2.4. (cid:52) In the following, we also use the shorthand notations t (cid:55)→ δ i ( t, Z ( t )), i = 0 , ,
2, whichonly highlights F Z -measurable quantities.The assumption that the controls depend only on portfolio-wide means rather thanactual realizations of the balance sheet and the assets is the key choice of this paper.The risk we hereby account for is only the systematic risk, i.e. the risk that affects allinsured.Note that it is the assumption of δ being dependent on U that makes η a process thataffects the payments to the insured, thus justifying it as a control. Note also that weallow δ to depend on Z , τ , and Q , while this is not the case for η . This is since thedividends are allocated to the individual insured while the assets are a portfolio levelquantity. The specific affine structure on δ mirrors that of B , cf. (2.16). This is importantfor practical applications, as the following example highlights. Example 2.6 (Second order interest rate) . Dividends may arise by accumulating thetechnical reserve V (cid:63) from (2.27) with a second order interest rate r δ that is determinedbased on the shape of the insurance business. This is obtained by letting δ ( t ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) V (cid:63) ( t ) ,r δ ( t ) = Φ( t, S ( · ∧ t ) , I ( t )) , for some explicitly computable and suitably regular mapping Φ. This corresponds to15 SCENARIO-BASED PROJECTION MODELsetting δ ( t, j ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) ( j ∈J p \{ J } ) (cid:101) V (cid:63)j ( t ) ,δ ( t, j ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) ( j ∈J f \{ J +1 } ) (cid:101) V (cid:63), + j (cid:48) ( t ) δ ( t, j ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) V (cid:63), † j ( t ) , for all j ∈ J . ◦ The aim of this paper is to develop methods to compute the market value of bonuspayments V b (0). Recall from Proposition 2.2 that this can be done via the computationof the expected accumulated bonus cash flow A b (0 , · ), which depends on the financialmarket through Q . To achieve this within the setup of Assumption 2.4, we adopt asimulation approach. It follows from (2.15) that for a simulated financial scenario,i.e. a realization of the whole path of S , we need the shape of the insurance business I ( t ) = ( U ( t ) , ¯ V (cid:63) ( t ) , ¯ V g ( t )) and corresponding controls ( δ ( t ) , η ( t )) for all time points t ≥
0. In other words, starting today from time zero, we must project the shape of theinsurance business and the controls into future time points for each simulated financialscenario.In the following sections, we formulate our scenario-based projection models demonstrat-ing how to project the shape of the insurance business in a specific financial scenario,and how to apply these projections to calculate the expected accumulated bonus cashflow A b (0 , · ). Section 3 concerns the general case where Q is allowed to be F Z ∨ F S -adapted and where we apply (2.21)–(2.22). In the subsequent Section 4 we specializeto Q being state independent (of Z ), i.e. F S -adapted, where we instead can apply thesimpler formula (2.24). This section contains the main contributions of the paper and provides the founda-tion for the special case in Section 4. In Subsection 3.1, we formulate our generalscenario-based projection model demonstrating how to project the shape of the insur-ance business into future time points in a given financial scenario. The projections arethen in Subsection 3.2 used to calculate the Q -modified transition probabilities p Qz j (0 , · )and corresponding expected accumulated bonus cash flow A b (0 , · ). Based on this, wepresent in Subsection 3.3 a procedure for the computation of V b (0) via an applicationof Proposition 2.2.As noted in Proposition 2.2, we are able to simplify calculations of A b (0 , · ) to what wecoin state independent calculations of Q and p if Q is assumed F S -adapted. This special16 SCENARIO-BASED PROJECTION MODELcase leads to a notion of a state independent scenario-based projection model , which isstudied in more details in Section 4. We now turn our attention to projection of the shape of the insurance business. Thisconsists of computation of I = ( U, ¯ V g , ¯ V (cid:63) ) for realizations of S , where each realizationexactly represents a simulated financial scenario.The method for computation of U for a realization of S follows immediately from thedynamics of the assets according to (2.28). The computational issue reduces to that ofcomputing p Qz j (0 , · ), cf. (2.21)–(2.22) and (2.28). Thus we focus on the projection of theportfolio-wide means ¯ V g and ¯ V (cid:63) .First, we consider the portfolio-wide mean of the market value of guaranteed payments,¯ V g . From (2.25), calculation of ¯ V g is a matter of calculating the portfolio-wide means¯ A g of the expected accumulated guaranteed cash flows A g defined by¯ A g ( t, s ) = E (cid:2) A g ( t, s ) | F S ( t ) (cid:3) for 0 ≤ t ≤ s < ∞ . Proposition 3.1.
The portfolio-wide means ¯ A g of the expected accumulated guaranteedcash flows A g read ¯ A g ( t, d s ) = A ◦ (0 , d s ) + (cid:88) j ∈J p Qz j (0 , t ) A † j ( t, d s ) for all t ≥ .Proof. By (2.26), (2.19), and due to the assumed independence between Z and S , weimmediately find that¯ A g ( t, s ) = E (cid:2) A ◦ ( t, s ) | F S ( t ) (cid:3) + (cid:88) j ∈J E (cid:104) ( Z ( t )= j ) Q ( t ) A † Z ( t ) ( t, s ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) = E[ A ◦ ( t, s )] + (cid:88) j ∈J p Qz j (0 , t ) A † j ( t, s ) . By (2.6) and the iterated law of expectations,E[ A ◦ ( t, s )] = E[ B ◦ ( s ) − B ◦ ( t )]= A ◦ (0 , s ) − E[ B ◦ ( t ) − B ◦ (0)] .
17 SCENARIO-BASED PROJECTION MODELSince the latter term does not depend on s , we find that¯ A g ( t, d s ) = A ◦ (0 , d s ) + (cid:88) j ∈J p Qz j (0 , t ) A † j ( t, d s )as desired.Calculation of ¯ V g ( t ) now proceeds by discounting ¯ A g ( t, · ) with the forward rate curveavailable at time t according to the following expression:¯ V g ( t ) = (cid:90) nt e − (cid:82) st f ( t,v ) d v ¯ A g ( t, d s ) . (3.1)Consequently, given A ◦ and A † the computational issue has been reduced to that ofcomputing the Q -modified transition probabilities p Qz j (0 , · ).Next we consider the portfolio-wide mean of the technical reserve of guaranteed pay-ments, ¯ V (cid:63) . We could follow the same approach above and calculate the technical reservesvia expected (accumulated) cash flows, however, since the technical interest rate is de-terministic, a range of technical reserves, including V (cid:63), † , (cid:101) V (cid:63) , and (cid:101) V (cid:63), + , can be computedmore efficiently by solving the differential equations of Thiele type derived from (2.3),cf. Subsection 2.3 and Subsection 2.4.Denote by ¯ V (cid:63), ◦ the portfolio-wide mean technical reserves of predetermined paymentsgiven by ¯ V (cid:63), ◦ ( t ) = E (cid:2) V (cid:63), ◦ ( t ) | F S ( t ) (cid:3) for t ≥
0. Since Z and S are assumed independent, we could replace the conditionalexpectation by an ordinary expectation. Proposition 3.2.
The portfolio-wide mean technical reserve of guaranteed paymentsreads ¯ V (cid:63) ( t ) = ¯ V (cid:63), ◦ ( t ) + (cid:88) j ∈J p Qz j (0 , t ) V (cid:63), † j ( t ) , while the portfolio-wide mean technical reserve of predetermined payments reads ¯ V (cid:63), ◦ ( t ) = (cid:88) j ∈J p j (cid:54) = J p z j (0 , t ) (cid:101) V (cid:63)j ( t ) + (cid:88) j ∈J f j (cid:54) =2 J +1 p ρz j (0 , t ) (cid:101) V (cid:63), + j (cid:48) ( t ) . (3.2) Proof.
By (2.27) and (2.19), direct calculations yield¯ V (cid:63) ( t ) = E (cid:2) V (cid:63), ◦ ( t ) | F S ( t ) (cid:3) + (cid:88) j ∈J E (cid:104) ( Z ( t )= j ) Q ( t ) V (cid:63), † Z ( t ) ( t ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) = ¯ V (cid:63), ◦ ( t ) + (cid:88) j ∈J p Qz j (0 , t ) V (cid:63), † j ( t ) .
18 SCENARIO-BASED PROJECTION MODELTo obtain (3.2), we split V (cid:63), ◦ according to the events of Z ( t ) being in J p \ { J } , J f \{ J + 1 } , and { J, J + 1 } . According to (2.4), we then have¯ V (cid:63), ◦ ( t ) = E (cid:104) ( Z ( t ) ∈J p \{ J } ) (cid:101) V (cid:63)Z ( t ) ( t ) + ( Z ( t ) ∈J f \{ J +1 } ) ρ ( τ ) (cid:101) V (cid:63), + Z ( t ) (cid:48) ( t ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) = E (cid:34) (cid:88) j ∈J p j (cid:54) = J ( Z ( t )= j ) (cid:101) V (cid:63)j ( t ) + (cid:88) j ∈J f j (cid:54) =2 J +1 ( Z ( t )= j ) ρ ( τ ) (cid:101) V (cid:63), + j (cid:48) ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F S ( t ) (cid:35) = (cid:88) j ∈J p j (cid:54) = J p z j (0 , t ) (cid:101) V (cid:63)j ( t ) + (cid:88) j ∈J f j (cid:54) =2 J +1 p ρz j (0 , t ) (cid:101) V (cid:63), + j (cid:48) ( t ) , as desired.As already mentioned, the technical reserves V (cid:63), † , (cid:101) V (cid:63) , and (cid:101) V (cid:63), + can be computed effi-ciently using differential equations of Thiele type, while the ρ -modified transition prob-abilities are simply computed according to (2.9). Thus Proposition 3.2 reduces the com-putational complexity to that of computing Q -modified transition probabilities p Qz j (0 , · ).This computation is studied in details in the next subsection. Q -modified transition probabilities We are now ready to present a system of differential equations for the Q -modified transi-tion probabilities p Qz j (0 , · ); here p ρz j (0 , · ) := p z j (0 , · ) for z ∈ J f , which is in accordancewith τ = 0 for z ∈ J f and the assumption ρ (0) = 1. Theorem 3.3.
The Q -modified transition probabilities p Qz j (0 , · ) satisfy for j ∈ J thedifferential equations dd t p Qz j (0 , t ) = p z j (0 , t ) δ ( t, j ) + p ρz j (0 , t ) δ ( t, j ) + p Qz j (0 , t ) δ ( t, j ) V (cid:63), † j ( t ) − p Qz j (0 , t ) µ j • ( t ) + (cid:88) k ∈J k (cid:54) = j p Qz k (0 , t ) µ kj ( t ) , p Qz j (0 ,
0) = 0 . (3.3) Proof.
The boundary conditions follows by the assumption that Q (0) = 0. Referringto (2.19) and (2.15), we have p Qz j (0 , t ) = E (cid:2) ( Z ( t )= j ) Q ( t ) (cid:12)(cid:12) F S ( t ) (cid:3) = E ( Z ( t )= j ) (cid:90) t δ ( u ) V (cid:63), † Z ( u ) ( u ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F S ( t ) with δ ( t ) = δ ( t, Z ( t )) + δ ( t, Z ( t )) ρ ( τ ) ( τ ≤ t ) + δ ( t, Z ( t )) Q ( t ) .
19 SCENARIO-BASED PROJECTION MODELNote that for 0 ≤ u ≤ t and k ∈ J ,E (cid:34) ( Z ( u )= k ) p Qz k (0 , u ) p z k (0 , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F S ( t ) (cid:35) = E (cid:2) ( Z ( u )= k ) Q ( u ) (cid:12)(cid:12) F S ( t ) (cid:3) , E (cid:34) ( Z ( u )= k ) p ρz k (0 , u ) p z k (0 , u ) (cid:35) = E (cid:2) ( Z ( u )= k ) ρ ( τ ) ( τ ≤ u ) (cid:3) . Thus by Markovianity of Z and independence between Z and S , p Qz j (0 , t ) = E (cid:34) ( Z ( t )= j ) (cid:90) t (cid:88) k ∈J ( Z ( u )= k ) b Qk ( u ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F S ( t ) (cid:35) (3.4)with b Qk , k ∈ J , given by b Qk ( u ) = δ ( u, k ) + δ ( u, k ) p ρz k (0 ,u ) p z k (0 ,u ) + δ ( u, k ) p Qz k (0 ,u ) p z k (0 ,u ) V (cid:63), † k ( u ) (3.5)for all u ≥
0. The assumption of independence between Z and S , Markovianity of Z ,and Fubini’s theorem finally yield p Qz j (0 , t ) = (cid:90) t (cid:88) k ∈J p z k (0 , u ) p kj ( u, t ) b Qk ( u ) d u. (3.6)The statement of the theorem is now established by differentiation as follows. Leibniz’integration rule givesdd t p Qz j (0 , t ) = (cid:88) k ∈J ( k = j ) p z k (0 , t ) b Qk ( t ) + (cid:90) t (cid:88) k ∈J p z k (0 , u ) (cid:18) dd t p kj ( u, t ) (cid:19) b Qk ( u ) d u = δ ( t, j ) p z j (0 , t ) + δ ( t, j ) p ρz j (0 , t ) + δ ( t, j ) p Qz j (0 , t ) V (cid:63), † j ( t )+ (cid:90) t (cid:88) k ∈J p z k (0 , u ) (cid:18) dd t p kj ( u, t ) (cid:19) b Qk ( u ) d u. Applying Kolmogorov’s forward differential equations and (3.6) to the last line of theequation we find thatdd t p Qz j (0 , t ) = δ ( t, j ) p z j (0 , t ) + δ ( t, j ) p ρz j (0 , t ) + δ ( t, j ) p Qz j (0 , t ) V (cid:63), † j ( t ) − p Qz j (0 , t ) µ j • ( t ) + (cid:88) (cid:96) ∈J (cid:96) (cid:54) = j p Qz (cid:96) (0 , t ) µ (cid:96)j ( t )as desired. 20 SCENARIO-BASED PROJECTION MODEL Remark . There exists a clear link between Q -modified transition probabilities andso-called state-wise retrospective reserves. Referring to (3.4) and (3.5), we see that fora fixed financial scenario, W j ( · ) := p Qz j (0 , · ) p z j (0 , · )corresponds to the state-wise retrospective reserve of [15] (in the presence of information G ( t ) = F S ( t ) ∨ σ ( Z ( t )), cf. [15] Subsection 5.B) with payments − (cid:88) j ∈J ( Z ( t )= j ) b Qj ( t ) d t and interest rate zero. Contrary to the primary setup of [15], the payments consideredhere are functions of the state-wise retrospective reserves W j ( · ). (cid:52) The system of differential equations for p Qz j (0 , · ) from Theorem 3.3 involves the shapeof the insurance business I through the mappings δ , δ , and δ . Together with theresults of the previous subsection, Theorem 3.3 allows us formulate a procedure for thecalculation of V b (0). The procedure is presented in the next subsection. Based on the results of the previous subsections, we demonstrate a procedure for thescenario-based projection model. In what follows, we suppose we are given mappings( δ, η ) serving as controls. They are assumed to satisfy Assumption 2.4.Besides the financial scenarios, the input consists of the following quantities which canbe precalculated independently of the financial scenarios:(1) The expected accumulated cash flow of predetermined payments A ◦ (0 , s ) for s ≥ V (cid:63), ◦ ( t ) forall t ≥ t ≥
0, state-wise expected accumulated unit bonus cash flows A † j ( t, s ) forall s ≥ t and j ∈ J as in (2.13)–(2.14).(4) State-wise technical unit reserves V (cid:63), † j ( t ) for all t ≥ j ∈ J as in (2.10).(5) Transition probabilities p z j (0 , t ) for all t ≥ j ∈ J .21 SCENARIO-BASED PROJECTION MODELAs discussed previously, this input can be calculated using classic methods for solvingdifferential equations of Thiele type as well as ( ρ -modified) Kolmogorov forward dif-ferential equations.The financial scenarios are N realizations { S k ( t ) } t ≥ , k = 1 , . . . , N , of { S ( t ) } t ≥ withcorresponding short rate r k and forward rate curves f k . We consider them as output ofan economic scenario generator.The procedure essentially consists of computing p Qz j (0 , · ), j ∈ J , and U ( · ) in eachfinancial scenario by solving a system of (stochastic) differential equations. The involvedpart is to evaluate the differentials. The procedure looks as follows. For each financialscenario k = 1 , . . . , N : ◦ Initialize with p Q,kz j (0 ,
0) = 0 for all j ∈ J and U k (0) = u . ◦ Apply a numerical algorithm to solve the coupled (stochastic) differential equationsystems for p Q,kz j (0 , · ), j ∈ J , and U k ( · ) from Theorem 3.3 and (2.28), respectively. – Evaluating the differentials at time t involves the mappings ( δ , δ , δ , η )from (2.29)–(2.30). By inspection of the differentials and these mappings,we see that we require the shape of the insurance business I k ( t ) = (cid:16) U k ( t ) , ¯ V g,k ( t ) , ¯ V (cid:63),k ( t ) (cid:17) , the expected bonus cash flow a b,k (0 , t ), as well as the input. Computation of¯ V g,k ( t ), ¯ V (cid:63),k ( t ), and a b,k (0 , t ) is achieved via Proposition 3.1, Proposition 3.2,and (2.22). ◦ We emphasize that as part of evaluating the differentials we computed the expectedbonus cash flow a b,k (0 , · ).The procedure completes by computing the market value of bonus payments V b (0) via V b (0) ≈ N N (cid:88) k =1 (cid:90) n e − (cid:82) t r k ( v ) d v a b,k (0 , t ) d t using an algorithm for numerical integration.Note that we require the input (3), which are the state-wise expected accumulated unitbonus cash flows A † j ( · , · ) evaluated on the two-dimensional time grid { ( t, s ) ∈ [0 , ∞ ) : t ≤ s } . To precompute this input, one must solve Kolmogorov’s forward differentialequations many times, once for every t ≥ j ∈ J . This significantly impacts thenumerical efficiency of the procedure. Furthermore, the algorithm itself depends on themarket basis for the specific insured through the transition rates µ . In practice, where22 STATE INDEPENDENT SCENARIO-BASED PROJECTION MODELthe algorithm must be executed for many insured, one must view the specific transitionrates for a single insured as input.In the following section, we present the simpler state independent scenario-based projec-tion model, where we require that the dividend strategy be specified (or approximated)such that Q is F S -adapted. By presenting a numerical procedure for the model, we showhow this requirement on the dividend strategies leads to a numerical speedup. This section concerns the formulation of the state independent scenario-based projectionmodel. The model is a special case of the projection model from Section 3 which relieson ensuring Q to be an F S -adapted process such that the simplified case of Proposi-tion 2.2 applies. In Subsection 4.1, we provide sufficient conditions on δ such that Q is F S -adapted. Next, Subsection 4.2 revisits the projection of the shape under this sim-plification. Finally, in Subsection 4.3 we present a procedure for the computation of themarket value of bonus payments in the state independent projection model. Recall from (2.15) and (2.30) that Q is the solution to the differential/integral equationd Q ( t ) = δ ( t, Z ( t )) + δ ( t, Z ( t )) ρ ( τ ) ( τ ≤ t ) + δ ( t, Z ( t )) Q ( t ) V (cid:63), † Z ( t ) ( t ) d t, Q (0) = 0 . To ensure that Q is an F S -adapted process, it suffices to require that δ , δ and δ areon the form δ i ( t, Z ( t )) = (cid:101) δ i ( t ) V (cid:63), † Z ( t ) ( t ) , i = 0 , , (4.1) δ ( t, Z ( t )) = 0 , (4.2)where we have used the shorthand notation (cid:101) δ i ( t ) = (cid:101) δ i ( t, S ( · ∧ t ) , I ( t )) for suitably regulardeterministic mappings (cid:101) δ i , i = 0 ,
2. This is a consequence of the following observation.When (4.1)–(4.2) hold, then simplyd Q ( t ) = (cid:0) (cid:101) δ ( t ) + (cid:101) δ ( t ) Q ( t ) (cid:1) d t, Q (0) = 0 . (4.3)This implies p Qz j (0 , t ) = Q ( t ) p z j (0 , t ), cf. (2.23). Remark . Since the class of dividend strategies presented here builds on Assump-tion 2.4, affinity in Q is more or less implicitly assumed. The simplifications we obtainin the following Subsections 4.2–4.3 build on Q being F S -adapted rather than the divi-dend strategy being affine in Q . The results are therefore trivially extendable to dividendstrategies that are non-affine in the number of additional benefits held. (cid:52)
23 STATE INDEPENDENT SCENARIO-BASED PROJECTION MODEL
For the portfolio-wide means ¯ A g we observe a simplification in the part that concernsfuture bonus payments similar to what we previously saw concerning the predeterminedpayments: Corollary 4.2.
Assume that the dividend strategy δ is on the form (4.1) – (4.2) . Theportfolio-wide means ¯ A g of the expected accumulated guaranteed cash flows A g then read ¯ A g ( t, ds) = A ◦ (0 , d s ) + Q ( t ) A † (0 , d s ) . Proof.
From Proposition 3.1 and its proof, we have¯ A g ( t, s ) = A ◦ (0 , s ) − E [ B ◦ ( t ) − B ◦ (0)] + E (cid:104) Q ( t ) A † ( t, s ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) . Since by assumption Q is F S -adapted and Z and S are independent, referring to (2.5)with superscript ◦ replaced by † and applying the law of iterated expectations yieldsE (cid:104) Q ( t ) A † ( t, s ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) = Q ( t ) E (cid:104) B † ( s ) − B † ( t ) (cid:105) = Q ( t ) A † (0 , s ) − Q ( t ) E (cid:104) B † ( t ) − B † (0) (cid:105) Consequently, ¯ A g ( t, d s ) = A ◦ (0 , d s ) + Q ( t ) A † (0 , d s )as desired.For the technical reserve, the result is similar. Before we present the result, let theportfolio-wide mean technical unit bonus reserve ¯ V (cid:63), † be given by¯ V (cid:63), † ( t ) = E (cid:104) V (cid:63), † Z ( t ) ( t ) (cid:12)(cid:12)(cid:12) F S ( t ) (cid:105) for t ≥
0. Since Z and S are assumed independent, we could replace the conditionalexpectation by an ordinary expectation. It is then a trivial observation that¯ V (cid:63), † ( t ) = (cid:88) j ∈J p z j (0 , t ) V (cid:63), † j ( t ) . (4.4) Corollary 4.3.
Assume that the dividend strategy δ is on the form (4.1) – (4.2) . Theportfolio-wide mean technical reserve of guaranteed payments then reads ¯ V (cid:63) ( t ) = ¯ V (cid:63), ◦ ( t ) + Q ( t ) ¯ V (cid:63), † ( t ) . Proof.
Since by assumption, Q is F S -adapted and Z and S are independent, the resultfollows immediately from (2.23), Proposition 3.2, and (4.4).24 STATE INDEPENDENT SCENARIO-BASED PROJECTION MODELThe following example is a continuation of Example 2.6 regarding the accumulation ofthe technical reserve with a second order interest rate. Example 4.4 (Second order interest rate continued) . The dividend strategy from Ex-ample 2.6 regarding accumulation of the technical reserve V (cid:63) with a second order interestrate r δ does not satisfy the requirements on δ from (4.1)–(4.2). Instead, the strategy δ ( t ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) ¯ V (cid:63) ( t )¯ V (cid:63), † ( t ) V (cid:63), † Z ( t ) ( t ) , (4.5)satisfies (4.1)–(4.2) with (cid:101) δ ( t ) = ( r δ ( t ) − r (cid:63) ( t )) ¯ V (cid:63), ◦ ( t )¯ V (cid:63), † ( t ) and (cid:101) δ ( t ) = ( r δ ( t ) − r (cid:63) ( t )) . One may think of this strategy as an accumulation of the portfolio-wide mean technicalreserve ¯ V (cid:63) with r δ instead, since by (4.3),¯ V (cid:63), † ( t ) d Q ( t ) = (cid:0) r δ ( t ) − r (cid:63) ( t ) (cid:1) ¯ V (cid:63) ( t ) d t. By multiplying the strategy (4.5) with V (cid:63) ( t )¯ V (cid:63) ( t ) and ¯ V (cid:63), † ( t ) V (cid:63), † Z ( t ) ( t )one arrives at strategy of Example 2.6. If the two ratios are close to one, the strategy(4.5) approximates the strategy of Example 2.6. Note that E (cid:2) V (cid:63) ( t ) / ¯ V (cid:63) ( t ) (cid:12)(cid:12) F S ( t ) (cid:3) = 1,i.e. the portfolio-wide mean of the first ratio is equal to one. For the latter ratio, this isnot necessarily the case since it is non-linear in V (cid:63), † Z ( t ) ( t ). ◦ Based on the results of the previous subsections, we demonstrate a procedure for thestate independent scenario-based projection model. In what follows, we suppose we aregiven mappings ( δ, η ) serving as controls. They are assumed to satisfy Assumption 2.4with δ on the form (4.1)–(4.2).Besides the financial scenarios, the input consists of the following quantities which canbe precalculated independently of the financial scenarios:(1) The expected accumulated cash flow of predetermined payments A ◦ (0 , s ) for all s ≥ V (cid:63), ◦ ( t ) forall t ≥ a † (0 , s ) for all s ≥ V (cid:63), † ( t ) for all t ≥ ρ -modified) Kolmogorov forward dif-ferential equations.The financial scenarios are N realizations { S k ( t ) } t ≥ , k = 1 , . . . , N , of { S ( t ) } t ≥ withcorresponding short rate r k and forward rate curves f k . We consider them as output ofan economic scenario generator.The procedure essentially consists of computing Q ( · ) and U ( · ) in each financial sce-nario by solving a system of (stochastic) differential equations. The involved part is toevaluate the differentials. The procedure looks as follows. For each financial scenario k = 1 , . . . , N : ◦ Initialize with Q k (0) = 0 and U k (0) = u . ◦ Apply a numerical algorithm to solve the coupled (stochastic) differential equationsystems for Q k ( · ) and U k ( · ) from (4.3) and (2.28), respectively. – Evaluating the differentials at time t involves the mappings ( (cid:101) δ , (cid:101) δ , η ) from(2.29) and (4.1). By inspection of the differentials and these mappings, wesee that we require the shape of the insurance business I k ( t ) = (cid:16) U k ( t ) , ¯ V g,k ( t ) , ¯ V (cid:63),k ( t ) (cid:17) , the expected bonus cash flow a b,k (0 , t ) = Q k ( t ) a † (0 , t ), cf. (2.24), as well asthe input. Computation of ¯ V g,k ( t ) and ¯ V (cid:63),k ( t ) is achieved via Corollary 4.2and Corollary 4.3. ◦ We emphasize that as part of evaluating the differentials we computed the expectedbonus cash flow a b,k (0 , · ).The procedure completes by computing the market value of bonus payments V b (0) via V b (0) ≈ N N (cid:88) k =1 (cid:90) n e − (cid:82) t r k ( v ) d v a b,k (0 , t ) d t using an algorithm for numerical integration.Note that in comparison with the procedure of Subsection 3.3, the expected unit bonuscash flows a † j ( t, · ), j ∈ J , have only to be precomputed for j = z and t = 0. This leads26 OUTLOOKto a speedup. Additionally, the procedure itself does not depend on the market basis forthe specific insured (except potentially through the mappings (cid:101) δ , (cid:101) δ , and η ). These arethe primary practical advantages that are gained by strengthening the requirements onthe dividend strategy to (4.1)–(4.2). In this section, we compare our methodology and results with recent advances in theliterature and discuss possible extension in demand by practitioners. Subsection 5.1 con-tains comparisons with [2, 6, 12], while the inclusion of both duration effects (so-calledsemi-Markovianity) and the bonus scheme consolidation is the focal point of Subsec-tion 5.2.
In [2] and the follow-up paper [6], where the methods and results of the former aregeneralized to allow for surrender and free policy conversion, primary attention is givento the derivation of differential equations for quantities such asE (cid:2) ( Z ( t )= j ) V (cid:63) ( t ) (cid:12)(cid:12) F S ( t ) (cid:3) . Since V (cid:63) = V (cid:63), ◦ + Q · V (cid:63), † , we find that t (cid:55)→ ( Z ( t )= j ) V (cid:63) ( t ) is an affine function of t (cid:55)→ ( Z ( t )= j ) Q ( t ). Thus disregarding free policy conversion, we see a direct link betweenthe differential equations derived in [2, 6] and those of Theorem 3.3. For these resultssuitable affinity of the dividend strategy is a key assumption.The inclusion of the policyholder option of free policy conversion adds an additionallayer of complexity. We assumed the unit bonus payment stream B † to be unaffected bythe free policy option, which leads to the total payment stream given by (2.16). No suchassumption is made in [6], which leads to more involved payment streams, although bysetting B † = B ◦ , + , our payment stream equals that of [6, Subsection 4.2, cf. (11)–(12)].We consider some key concepts and provide practical insights that are not within thescope of [2, 6]. We explicitly include financial risk, which serves as a good starting pointfor the extension to doubly stochastic models with dependence between the financialmarket and the stochastic transition rates. Moreover, we identify and discuss the theo-retical and practical challenges arising from the fact that the dividend strategy dependson the shape of the insurance business. Furthermore, we provide ready-to-implementnumerical schemes for the computation of the market value of bonus payments. Fi-nally, we discuss potential simplifications arising when the number of additional benefits27 OUTLOOKis (approximated to be) F S -adapted – the state independent case, which might be ofparticular interest to practitioners.The projection model described in [12, Section 4] appears to be conceptually very close toexactly our state independent model. As an example, additional benefits are in [12, seep. 196] bought according to the portfolio-wide mean ¯ V (cid:63), † of the technical reserve ratherthan the actual technical reserve V (cid:63), † Z ( · ) ; this is exactly in the spirit of our Example 4.4.Consequently, we believe that our presentation among other things serves to forma-lize and generalize the pragmatic approach found in [12] and, correspondingly, aims atbridging the gap between the methods and results found in [2, 6] and [12]. In both theory and practice, the generalization to so-called semi-Markovian models in-troducing duration dependence in the transition rates and payments is popular andimpactful, cf. [9, 7, 5, 4]. We believe that the methods we use here can easily be adaptedto semi-Markovian models.The increase in numerical speed from the general case to the state independent case is in-creasing in the complexity of the intertemporal dependence structure, which can be seenas follows. Referring to Subsection 3.3 and Subsection 4.3, the general projection modelrequires as input the expected unit bonus cash flows evaluated on a two-dimensional timegrid, while evaluation on a one-dimensional time grid suffices for the state independentmodel. When including duration effects, the complexity increases, which ought to entaila four-dimensional time/duration grid for the expected unit bonus cash flows in generalprojections and a two-dimensional time/duration grid in state independent projections.The gain in numerical speed by assuming the state independent special case is thus fargreater in the semi-Markovian model compared to the Markovian model.In Denmark, the bonus scheme known simply as consolidation (in Danish: styrkelse ) seeswidespread use in practice, cf. [12, Subsection 4.1]. Consolidation involves two technicalbases: a low (more prudent) basis and a high (less prudent) basis. At the onset of thecontract, the predetermined payments, i.e. the payments guaranteed at time zero, sat-isfy an equivalence principle for which some payments are valuated on the high technicalbasis and the remaining payments are valuated on the low technical basis. Dividendsare then used to shift these payments from the high to the low basis while upholdingthe relevant equivalence principle. Typically consolidation is combined with the bonusscheme additional benefits in the following manner. When all predetermined paymentshave been shifted to the low technical basis, future dividends are used to buy additionalbenefits. This ruins a key affinity assumption, which increases the complexity signifi-cantly. In particular, an extension of Theorem 3.3 appears to require more sophisticated28EFERENCESmethods. In the state independent case, the assumption of affinity is not required, cf.Remark 4.1. Consequently, we believe that it is straightforward to extend the stateindependent projection model to include consolidation in combination with additionalbenefits.
Acknowledgments and declarations of interest
Our work was at first motivated by unpublished notes by Thomas Møller. We thankThomas Møller for fruitful discussions and for sharing his insights with us.Christian Furrer’s research is partly funded by the Innovation Fund Denmark (IFD)under File No. 7038-00007. The authors declare no competing interests.
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