Reserve-Dependent Surrender
RReserve-Dependent Surrender
Kamille Sofie T˚agholt Gad (1), Jeppe Juhl (2), Mogens Steffensen (1)((1) University of Copenhagen, (2) Edlund A/S)
Abstract
We study the modelling and valuation of surrender and other be-havioural options in life insurance and pension. We place ourselvesin between the two extremes of completely arbitrary intervention andoptimal intervention by the policyholder. We present a method thatis based on differential equations and that can be used to approxi-mate contract values when policyholders exhibit optimal behaviour.This presentation includes a specification of sufficient conditions forboth consistency of the model and convergence of the contract values.When not going to the limit in the approximation we obtain a tech-nique for balancing off arbitrary and optimal behaviour in a simple,intuitive way. This leads to our suggestions for intervention modelswhere one single parameter reflects the extent of rationality amongpolicyholders. In a series of numerical examples we illustrate the im-pact of the rationality parameter on the contract values.
Keywords : Behavioural option, Ordinary differential equation, Penaltymethod, Optimal stopping, Solvency II
Modern solvency and accounting rules (Solvency II and IFRS) require thatexpected policyholder behaviour is taken into account. This includes e.g.expected surrender and expected transcription into free policy (paid-up pol-icy). The expectation is supposed to take into account both the economicconditions under which the behaviour takes place as well as the extent towhich intervention is to the benefit of the policyholder. The economic con-ditions and how beneficial it is for the policyholder to intervene may changeover time. Therefore, one should properly speak of dynamic behaviour mod-els when formalizing these effects in the actuarial valuation formulas. Chang-ing economic conditions could e.g. be a changing level of interest rates, andone idea would be to let the intensity or probability of intervention dependon the current (possibly stochastic) level of interest rates. How beneficial anaction is can be formalized by the gain from intervention. Determining thegain may be a delicate issue since both intervening and not intervening opensup for new intervention options in the future that also have to be taken intoaccount. E.g., not to surrender typically opens up for surrendering later,and transcription into free policy changes the effect of the surrender option.This challenge calls for a recursive solution such that the gain is always mea-suring correctly the tendencies of intervening in the future. We disregardthe economic condition by assuming deterministic interest rates and focus1 a r X i v : . [ q -f i n . M F ] D ec n the latter idea of a recursive formula to deal with the benefit of interven-tion. One motivation for this focus is that, perhaps, the external economicconditions are supposed to approximate to the internal benefit.There exists a range of approaches to modelling of behavioural risk. Oneextreme is to say that intervention occurs in a completely arbitrary way, likeinsurance risk. We hereby mean that we model the behaviour as indepen-dent of everything else in our model than the state of the policyholder andthe time measured through calendar time, the policyholder’s age, time sinceinitiation, or time to (deterministic) retirement. Specifically, the behaviourdepends on neither the contract the policyholder holds nor the interest rate.With this approach it is tractable to study various aspects beyond justadding surrender to a survival model. Buchardt et al. [3] studied the for-malistic interaction between semi-Markov modelling of insurance risk andbehavioural risk, including duration dependence of mortality and paymentsin the disability state and recognizing duration dependence of free policypayments. A simpler exposition is found in Buchardt and Møller [2]. Hen-riksen et al. [9] also combine surrender and free policy options and studythe impact on reserving from different simplifying assumptions about thedependence between insurance risk and behaviour risk.Another extreme is to say that intervention occurs in a completely ra-tional, optimal way. We hereby mean that the policyholder, who is assumedto have the same information as the insurance company has, intervenes ac-cording to a strategy that maximizes the value of the insurance contract.This approach was taken in Steffensen [13], who derived general variationalinequalities that characterize the reserve in case of a multi-state Markovmodel for insurance risk and a multi-state model for behavioural risk. Inthe surrender case, this is known as American option pricing of surrenderrisk. Other early references based on this approach to surrender risk areGrosen and Jørgensen [8] and Bacinello [1].In between these extremes exist all different kinds of models where inter-vention is modelled by an intensity, but where the intensity not only dependson time but also some stochastic factors. The dependence on the interestrate appears obvious and is thoroughly examined by De Giovanni [7], whocalculate reserves by solving partial differential equations numerically. Thereexists a large amount of literature examining relevant explanatory variablesbut since these studies appear somewhat marginal to our approach we re-fer to Eling and Kiesenbauer [4] and references therein for a comprehensiveliterature overview.Rather than letting the intensity depend on external factors, one couldlet the intensity depend on internal factors relevant to the specific policy.That could e.g. be to take the difference between the surrender value and(some notion of the) reserve as a measure of how beneficial an interventionis. If the reserve compared with the surrender value does not take futureintervention options into account, the calculation can be split up in two2tandard exercises: First, calculate the reserve without intervention and thenplug this reserve into the intensity for a calculation including surrender. Ifthe reserve compared with the surrender value does take future interventioninto account, the (usually) linear Thiele differential equation characterizingthe reserve becomes in general a non-linear differential equation. The non-linear term comes from the risk premium with respect to the surrender eventthat contains a non-linear function of the reserve itself. The rationale forthis paper is to take a thorough look at this non-linear differential equationin order to motivate it, interpret it, generalize it, and solve it numerically.Last but not least, we present a probabilistic proof of and clarify sufficientconditions for a convergence result that may seem intuitively clear: If thetendency to intervene tends to zero whenever the gain from intervention isnegative and tends to infinity whenever the gain from intervention is positive,we reach in the limit at the reserve based on completely rational behaviour.We establish sufficient convergence of intensities to reach such a conclusion.Thus, our approach to intervention option pricing has two purposes: First,it represents in itself a relevant approach in between the two extremes that,certainly, takes into account the extent to which intervention is to the benefitof the policyholder. Second, for simple parametric forms of the intensity,our calculation approximates the largest possible liability. As such it can beused as a worst-case or stress calculation with respect to surrender risk.The idea of approximating the maximum value by a series of solutionsto differential equation has been known as the penalty method. In compu-tational finance it has been used as an approximation method for Americanoption pricing. In Forsyth and Vetzal [5] the penalty method is comparedwith alternative techniques for pricing of the American put option. In Gadand Pedersen [6] the modelling of non-rational option holder behaviour isstudied in a way similar to what is done here. The contribution of the presentpaper is three-fold: First, we introduce, to the knowledge of the authors, forthe first time the penalty method in intervention option pricing in insurance.Second, we prove sufficient conditions for the convergence to hold. Third,we do not only think of the intensity model as a means of approximatingthe largest value, but as a highly relevant approach to general interventionoption pricing, useful in accounting and solvency. The approach balancesarbitrariness and benefit in a simple form, and in some examples we catchthe notion of rationality in one single parameter. Consider a model with a policyholder who is either alive (active) or dead.We assume the state of the policyholder is governed by a state process witha deterministic, continuous death intensity, µ ( t ), see Figure 1. Let I be theprocess indicating whether the policyholder is alive, and let N be the process3ounting the numbers of deaths of the policyholder. The policyholder isActive Dead (cid:45) µ ( t )Figure 1: Standard survival model.assumed to have the following simple contract. She pays a deterministicpremium with continuous intensity π ( t ) until a terminal time, n , as long asshe is alive. If she is alive at time n she receives a deterministic pension sum b a ( n ), and if she dies before then upon death she gets a deterministic deathsum, b ad ( t ). Thus, the accumulated payments in the time interval [0 , t ] isgiven by the following process of accumulated payments: B ( t ) = B (0) − (cid:90) t π ( u ) I ( u ) du + (cid:90) t b ad ( u ) dN ( u ) + I ( n ) b a ( n )1 ( t ≥ n ) , for t ∈ [0 , n ]. We assume that the market offers a deterministic, contin-uous interest rate, r ( t ). We introduce the reserve corresponding to thepolicyholder being active as the conditional expected present value of futurepayments, V ( t ) = E (cid:20) (cid:90) nt e − (cid:82) ut r ( τ ) dτ dB ( u ) (cid:12)(cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:21) . We then know, e.g. from [11], that the reserve, V , is continuously differen-tiable on [0 , n ) and that it is the solution to Thiele’s differential equation, V (cid:48) ( t ) = r ( t ) V ( t ) + π ( t ) − µ ( t )( b ad ( t ) − V ( t )) , (1)with V ( n − ) = b a ( n ).We now add to our model the possibility that the policyholder surren-ders. That is, we add the possibility that the policyholder terminates hercontract and instead receives a deterministic, continuous surrender value, G ( t ). This can, e.g., be added to the model by assuming that the policy-holder at any time surrenders with some deterministic, continuous intensity, ν ( t ), see Figure 2. We use the term active for when the policy is in force.Surrender (cid:63) ν ( t )Active Dead (cid:45) µ ( t )Figure 2: Standard surrender model.4athematically, the state of surrender is in this model not different fromthe state of death, except that the associated payments are different. Thereserve, V ν , is continuously differentiable and solves the following Thiele’sdifferential equation, see e.g. [11], V (cid:48) ν ( t ) = r ( t ) V ν ( t ) + π ( t ) − µ ( t ) (cid:16) b ad ( t ) − V ν ( t ) (cid:17) − ν ( t ) ( G ( t ) − V ν ( t )) , (2)with V ν ( n − ) = b a ( n ).For V v to be continuously differentiable we need, in general, that ν is continuous as assumed above. However, what is really needed is that ν ( t ) ( G ( t ) − V ν ( t )) is continuous and this can be obtained even when ν isdiscontinuous and properly defined at the point where G ( t ) = V ν ( t ).The surrender value G can be anything exogenously given. In practice itis, typically, a technical value of the same payment stream based on technicalassumptions on interest rates and intensities that we denote by ( r ∗ , µ ∗ ). Inthat case, the surrender value is the technical reserve V ∗ that solves (1) with( r, µ ) replaced by ( r ∗ , µ ∗ ). The forthcoming Solvency II regulations requires that the traditional mod-elling of surrender is revisited. In Article 79 of the Solvency II Directive it isstated that ”Any assumptions made by insurance and reinsurance undertak-ings with respect to the likelihood that policyholders will exercise contractualoptions, including lapses and surrender, shall be realistic and based on cur-rent and credible information. The assumptions shall take account, eitherexplicitly or implicitly, of the impact that future changes in financial andnon-financial conditions may have on the exercise of those options” . Thus,we need to investigate and model what influences the policyholders choiceto surrender and we need to be able to calculate the reserves in the moreadvanced models. In the present section we suggest a way to do this, anddiscuss our method.In a more realistic model of surrender we want to be able to expressboth that surrender is likely influenced by how profitable it is, but alsothat it is still random. On one hand, we wish surrender to be influencedby how profitable it is, because surrender is a decision the policyholdermakes. On the other hand we also have multiple reasons for surrender beingrandom. Randomness is natural because the policyholder most likely lacksinformation to decide what is profitable. Even if she had all the informationthat the pension fund has and were able to use it, then her preferencesmay differ seemingly randomly from the model set up by the pension fundbecause of the policyholders personal preference and economical situation.She might shift her job and get an offer from a new pension fund or shemight need cash for a divorce. 5e can obtain randomness in our model by keeping the surrender mod-elled by an intensity. Further, we model that the policyholders decisiondepends on how profitable it is by letting the surrender intensity dependon how profitable it is for the policyholder to surrender. If she surrendersat time t she gains G ( t ), but she loses the rest of the contract includingher right to exercise later. Hence, she loses V ν ( t ). Therefore, we denoteby G ( t ) − V ν ( t ) her profit from surrendering at time t . We would like thesurrender intensity to be non-negative and increasing in this profit. At firstglance this modelling seems to have a problem that the definition of thesurrender intensity is circular. However, Theorem 3.1 below gives sufficientconditions for this circular definition not to be a problem. Theorem 3.1
For some given non-negative function, h , consider the fol-lowing differential equation in the function U : U (cid:48) ( t ) = r ( t ) U ( t ) + π ( t ) − µ ( t )( b ad ( t ) − U ( t )) − h ( t, U ( t ))( G ( t ) − U ( t )) , (3) with U ( n − ) = ∆ B ( n ) . Suppose (3) has a unique solution, U , and definea surrender intensity by ν ( t ) ≡ h ( t, U ( t )) . Then U is the reserve when thepolicyholder chooses to surrender at time t with intensity ν ( t ) . Proof: The possible problem in this model is the circular definition of thesurrender intensity. However, the existence and uniqueness of the solutionto both (2) and (3) ensures that this does not become a problem.The process ν defined by ν ( t ) ≡ h ( t, U ( t )) is uniquely determined from(3) and the reserve is then uniquely determined from (2). It follows fromthe definition of U that U solves (2), and then from the uniqueness of thesolution to (2) it follows that the reserve is given by U .Once we have decided on a policyholder with a specific policy and afunction h , and thereby also ν , then for this single policyholder, our modeldoes not differ from a model with a deterministic time dependent surrenderintensity as what we had in the classical model of (2). However, when weuse the model for pricing a portfolio of insurance contracts for a group ofpolicyholders, then the model assigns different surrender intensities to eachpolicyholder. Thereby, the reserves in general become higher than if we hadused a constant surrender intensity or a specific time dependent surrenderintensity for the whole portfolio.The relation between the surrender intensity and the profitability maybe chosen in many different ways. Two examples we investigate are: ν ( t ) = h ( t, V ν ( t )) = ψ exp { θ ( G ( t ) − V ν ( t )) } , (4) ν ( t ) = h ( t, V ν ( t )) = θ ( G ( t ) − V ν ( t ) > , (5)where ψ , θ > ψ tells about the overalltendency to surrender, whereas θ tells about how profitability creates devi-ations from this tendency. For equation (5), θ controls both. In both cases6e speak of θ as the rationality parameter. Other intensity functions can bechosen and one should choose a functional form which matches with data.The only mathematical requirement is that the function h has to make itpossible to use Theorem 3.1.One immediate drawback of our model is that we most often do nothave an explicit solution for the differential equation (3). This implies thatwe do not have an explicit expression for the reserve. However, we dohave algorithms available for numerical solutions to ordinary differentialequations. The idea of modelling behaviour by profit dependent intensities may be usedfor other applications as well. Within life insurance the policyholder’s choiceto convert into free policy (paid-up policy) has some resemblance with thesurrender choice. Thereby we may find it reasonable to expand our modelwith the possibility of conversion into free policy in the same way as weadded surrender. Figure 3 displays a simple model where ν af denotes theintensity of conversion into free policy, ν as denotes the intensity of surrenderwhen active and ν fs denotes the intensity of surrender after converted intofree policy. Here the term active is used when the policyholder is payingpremiums. Free Policy (cid:63)(cid:8)(cid:8)(cid:8)(cid:8)(cid:25) (cid:72)(cid:72)(cid:72)(cid:72)(cid:89) (cid:72)(cid:72)(cid:72)(cid:72)(cid:106)(cid:8)(cid:8)(cid:8)(cid:8)(cid:42) ν af ( t ) ν as ( t ) µ ( t ) ν fs ( t ) µ ( t )ActiveSurrender DeadFigure 3: Free policy and surrender model.If all transition intensities are known explicitly, this model is studied in[9]. When a policyholder converts into free policy the payments are reduceddepending of the time of conversion. Let b fd ( t, u ) denote the death sumat time t if converted into free policy at time u , let b f ( n, u ) denote theterminal payment at time n if converted into free policy at time u , and let G f ( t, u ) denote the surrender value at time t when converted into free policyat time u . For the reserves we let V a ( t ) denote the reserve at time t if thepolicyholder is active, and let V f ( t, u ) denote the reserve at time t if thepolicyholder is in the free policy state and converted to free policy at time u . Now, we assume that the intensities are reserve dependent and given in7he form ν as ( t ) = h as ( t, V a ( t )) ,ν af ( t ) = h af ( t, V a ( t )) ,ν fs ( t, u ) = h fs ( t, u, V f ( t, u )) . Then the reserves are given from the following differential equations:dd t V a ( t ) = r ( t ) V a ( t ) + π ( t ) − µ ( t )( b ad ( t ) − V a ( t )) − h as ( t, V a ( t ))( G ( t ) − V a ( t )) − h af ( t, V a ( t ))( V f ( t, t ) − V a ( t )) ,V a ( n − ) = b a ( n ) ,∂∂t V f ( t, u ) = r ( t ) V f ( t, u ) − µ ( t )( b ∗ fd ( t, u ) − V f ( t, u )) − h fs ( t, u, V f ( t, u ))( G f ( t, u ) − V f ( t, u )) ,V f ( n − , u ) = b f ( n, u ) . The only requirement is that the system of differential equations has a uniquesolution. However, the differential equations from above are heavy to workwith, as we need to solve a new differential equation for each value of V f ( t, t ).When modelling the free policy option, this problem is usually overcome byintroducing a scaling function, f , that describes the reduction of paymentsas a result of the conversion to free policy. Thus, b fd ( t, u ) = f ( u ) b ad ( t ), b f ( n, u ) = f ( u ) b a ( n ) and G f ( t, u ) = f ( u ) G f ( t ). Assume the transition in-tensity ν fs does not depend on the time of transition to free policy. Then theprospective reserve, V ∗ f ( t ), from the free policy state based on the payments G f ( t ), b ad ( t ) and b a ( n ) does not depend on this transition time either, andwe get V f ( t, u ) = f ( u ) V ∗ f ( t ) withdd t V ∗ f ( t ) = r ( t ) V ∗ f ( t ) − µ ( t )( b ad ( t ) − V ∗ f ( t )) − ν fs ( t )( G f ( t ) − V ∗ f ( t )) ,V ∗ f ( n − ) = b a ( n ) . This makes V f ( t, u ) a lot easier to calculate. For more on the determinationof the reference payments and scaling function, see [9]. Note however that if ν fs cannot depend on the time of transition to free policy, u , then it cannotdepend on G f ( t, u ) − V f ( t, u ) either and this is a large disadvantage.To get profit dependent choices we may use ν as ( t ) = h as ( t, V a ( t )) = ψ as e θ as ( G ( t ) − V a ( t )) ,ν af ( t ) = h af ( t, V a ( t )) = ψ af e θ ap ( V f ( t,t ) − V a ( t )) ,ν fs ( t, u ) = h fs ( t, u, V f ( t, u )) = ψ fs e θ fs ( G f ( t,u ) − V f ( t,u )) . Approximation of the Worst Case Reserve
In the two previous sections we discussed our model with reserve dependentsurrender and we found it being a reasonable model for predicting the dy-namics of surrender. However, in the following section we discuss how themodel may also be used for determining worst case reserves when the truedynamics of the surrender intensity is not known. This is because our modelis a version of what in the literature is known as the penalty method, and alarge rationality parameter gives us the worst case reserve.Typically the technical reserve is paid out upon surrender (potentiallyminus expenses). In that case, if we take maximum of the technical reserveand the market reserve calculated under the assumption of no surrender,then we get a worst case reserve of either surrendering immediately or neversurrender. However, a surrender strategy somewhere in between the twoextremes may result in a higher market reserve. For determining the worstcase reserve we consider all possible surrender strategies. To do this weconstruct a more general model. We assume that the transition from activeto surrender is governed by a randomized stopping time, τ , with respect tothe state of the policyholder, with randomized stopping times being definedas in [12]. That is, the time of surrender may depend on everything but thefuture time of death and the future interest rate. If the policyholder neversurrenders her contract we let τ = n . The model is illustrated in Figure4. The class of admissible surrender strategies at time t are the variablesin [ t, n ] that are randomized stopping times with respect to the filtrationgenerated from I . We denote this class by T t .Surrender (cid:63) τ Active Dead (cid:45) µ ( t )Figure 4: Optimal surrender model.We hereby disregard the possibility that the policyholder has more in-formation about her future time of death than the insurance company has.We do this despite that such knowledge could influence the policyholdersdecisions.Let V τ denote the prospective reserve if the policyholder surrenders ac-cording to the randomized stopping time τ . Assume G ( n ) = 0, assume G ( n − ) ≤ V ( n − ) and assume G continuous on [0 , n ). Then from [11] it9ollows that V τ is given by: V τ ( t ) = E (cid:20) (cid:90) τt e − (cid:82) ut r ( x ) dx dB ( u ) + e − (cid:82) τt r ( x ) dx G ( τ ) I ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:21) = V ( t ) + E (cid:20) e − (cid:82) τt r ( x ) dx G ( τ ) I ( τ ) − (cid:90) nτ e − (cid:82) ut r ( x ) dx dB ( u ) (cid:12)(cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:21) = V ( t ) + E (cid:104) e − (cid:82) τt r ( x ) dx I ( τ ) G ( τ ) (cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:105) − E (cid:20) e − (cid:82) τt r ( x ) dx I ( τ ) E (cid:20) (cid:90) nτ e − (cid:82) uτ r ( x ) dx dB ( u ) (cid:12)(cid:12)(cid:12)(cid:12) τ, I ( τ ) = 1 (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:21) = V ( t ) + E (cid:104) e − (cid:82) τt r ( x ) dx I ( τ ) ( G ( τ ) − V ( τ )) (cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:105) . Consider the worst case scenario for the pension fund, where the policyholder chooses the surrender strategy as the stopping time, τ , that max-imizes V τ . This is an optimal stopping problem. Any classical stoppingtime from the filtration generated by I must fulfil τ I ( τ ) = t I ( t ) for somedeterministic t ∈ [ t, ∞ ]. The reserve is then given by: V τ ( t ) = V ( t ) + E (cid:104) e − (cid:82) τt r ( x ) dx I ( τ ) ( G ( τ ) − V ( τ )) (cid:12)(cid:12)(cid:12) I ( t ) = 1 (cid:105) = V ( t ) + e − (cid:82) t t r ( x )+ µ ( x ) dx ( G ( t ) − V ( t )) . Thus, for the classical optimal stopping problem, without randomizationallowed, it is optimal to choose t as any time from the set: A t ≡ arg max u ∈ [ t,n ] (cid:16) e − (cid:82) ut r ( x )+ µ ( x ) dx ( G ( u ) − V ( u )) (cid:17) . As the inner part is continuous in u on [0 , n ) and as G ( n − ) − V ( n − ) Theorem 5.1 Suppose that for each θ ≥ we have that h θ is defined in away such that we may use Theorem 3.1 and suppose that the surrender value G is continuous on [0 , n ) with G ( n − ) ≤ V ( n − ) and G ( n ) = 0 . Also assumefor x < : ¯ h θ ( x ) → , θ → ∞ , (6) and for x > : h θ ( x ) → ∞ , θ → ∞ . (7) Then, for every t ∈ [0 , n ] : V θ ( t ) → W ( t ) , θ → ∞ . For a proof, see the appendix. Remark 5.1 Some of the details in the proof has been omitted, but a fullydetailed proof following the same reasoning for a closely related result for anAmerican Put option may be found in [6]. The fact that the penalty methodprovides convergence and the rate of convergence is not new. However, wefind the proof of our article and of [6] interesting. This is because they visu-alize how the error terms may be thought of as probabilities of economicallybad choices of the policyholder times the loss the policyholder faces from herbad choices. In this section we show four examples of how various surrender models im-pact the development of the reserves in four different interest rate situations.In each example we consider a contract with a constant premium intensity of π = 7 , b ad = 1 , , 000 and a pension sum of 2 , , µ ( t ) = 0 . . − . ∗ ( t +35) . This is the death intensity from the Danish life table G82 for females. If thepolicyholder surrenders her contract, she receives a surrender value given bythe technical reserve. The technical reserve is based on the same payments as11he contract, and on a technical interest rate intensity of ˆ r = 0 . 05. Interestrates are chosen high to better visualize the impact of the choice of surrenderintensity. We assume no extra expenses at surrender. Thus, the surrendervalue is given from the differential equation: G (cid:48) ( t ) = ˆ rG ( t ) + π − µ ( t )( b ad − G ( t )) , (8)with G ( n − ) = 2 , , ν a ( t ) = 0 . · exp { . G ( t ) − V θ,ψ ( t )) } Model b : ν b ( t ) = 0 . · ( G ( t ) − V θ ( t )) Model c : ν c ( t ) = 0 . ν d ( t ) = 0Model e : ν e ( t ) = 5 · ( G ( t ) − V θ ( t )) . The first three models are based on a surrender intensity of around 5%.The last model is a model with a rationality parameter θ = 5, which hasbeen found to be high enough for us to approximate the worst case reserve.Additionally, we consider four different developments of the interest rate, r ,used for pricing market reserves. For the two first interest rate situations wecompare the surrender value and the reserves for the five different surrendermodels. For the two last interest rate situations we compare the surrendervalue and the reserves for surrender Model d and Model e. Example 1: Market interest rate is above technical interest rate Assume r = 0 . 12. The reserve developments are displayed in Figure 5.In this situation it is at all time points optimal for the policyholder tosurrender. The worst case reserve corresponds to the surrender value. Thelowest reserve is the market reserve based on no surrender, Model d. Modelswith a chance of surrender has reserves in between. Since there is no riskof surrendering too early, then Model b and the traditional Model c do notdiffer. For Model a we get a slightly higher reserve than the one for Modelb and Model c, because the basic intensity 0 . 05 is slightly increased at alltime points by the exponential factor in the intensity. Example 2: Market interest rate is below technical interest rate Assume r = 0 . 02. The reserve developments are displayed in Figure 6. Inthis situation it is never optimal for the policyholder to surrender. Theworst case reserve corresponds to the market reserve with no surrender.In Model b and Model e the policyholder does not make the mistake ofsurrendering if it is not profitable, and thus, this has an equally high reserve.The surrender value is the lowest value and the reserves of Model a and thetraditional Model c are in between. Model a has a higher reserve than Modelc, because the basic intensity 0 . 05 is slightly increased at all time points bythe exponential factor in the intensity.12 e Figure 5: Example 1. The technical interest rate is ˆ r = 0 . 05. The marketinterest rate is r = 0 . 12. Immediate surrender is always optimal. Example 3: Market interest rate is decreasing Assume r ( t ) = 0 . · ( t ≤ + 0 . · ( t> . The qualitative feature we capture is that the interestrate crosses the guaranteed interest rate downwards. The reserve develop-ments are displayed in Figure 7. In this situation it is optimal to surrenderif the surrender value is higher that the market reserve in Model d withno surrender. Thus, after time t = 20 it is optimal to keep the contractbecause the technical interest rate is higher than the market interest rate.Right before time t = 20 the interest rate of the market is higher than thetechnical interest rate, but this is only for a short time, and thus it is stilloptimal to keep the policy in order to benefit from the technical interestrate later on. At some point before time t = 20 the surrender value andthe market reserve of Model d intersects. Before this time it is optimal tosurrender because the gain from the high market interest rate before time t = 20 is then higher than the future loss from the low market interest rate.All together the worst case reserve is given as the maximum of the surrendervalue and the market reserve with no surrender. Example 4: Market interest rate is increasing Assume r ( t ) = 0 . · ( t ≤ +0 . · ( t> . The qualitative feature we capture is that the interestrate crosses the guaranteed interest rate upwards. The reserve developments13 c de Figure 6: Example 2. The technical interest rate is ˆ r = 0 . 05. The marketinterest rate is r = 0 . 02. Surrender is never optimal.are displayed in Figure 8. In this situation we have that after time t = 20 it isoptimal to surrender. Before time t = 20 it is optimal to plan to surrender attime t = 20. With this strategy the policyholder benefits from both the highmarket interest rate after time t = 20 and the technical interest rate beforetime t = 20 when the market interest rate is low. Thereby, unlike in theprevious three examples, the worst case reserve is no longer the supremumof the surrender value and the market reserve with no surrender. Beforetime t = 20 the worst case reserve is higher than both of the other reserves,because there exists a surrender strategy which is better for the policyholderthan both immediate surrender and no surrender.We recall that the reserves of Model a and Model b converge to the worstcase reserve when the rationality parameter converges to infinity. Thus,if the rationality parameter is sufficiently high and the future increase ininterest rate is sufficiently high, then the reserves of Model a and Model bbecome higher than the maximum of the surrender value and the marketreserve of Model d with no surrender.14 Figure 7: Example 3. The technical interest rate is ˆ r = 0 . 05. The marketinterest rate is r ( t ) = 0 . · ( t ≤ + 0 . · ( t> . Surrender is optimal ifthe surrender value is higher than the market reserve with no surrender. References Figure 8: Example 4. The technical interest rate is ˆ r = 0 . 05. The marketinterest rate is r ( t ) = 0 . · ( t ≤ + 0 . · ( t> . After time t = 20 it isoptimal to surrender. Before time t = 20 it is wise to plan to surrender attime t = 20.[6] Gad KST, Pedersen JL (2014). Profit dependent Exercise of the Ameri-can Put, Preprint, available at arxiv.org/abs/1410.1287[7] De Giovanni D (2010). Lapse rate modeling: a rational expectation ap-proach. Scand Actuar J 2010(1):56–67[8] Grosen A, Jørgensen PL (2000). Fair Valuation of Life Insurance Liabil-ities: The Impact of Interest Rate Guarantees, Surrender Options, andBonus Policies. Insur Math Econ 26(1):37–57[9] Henriksen LFB, Nielsen JW, Steffensen M, Svensson C (2014). Markovchain modeling of policyholder behaviour in life insurance and pension.Eur Actuar J 4(1):1–29. doi: 10.1007/s13385-014-0091-2[10] Kyprianou A. E. (2006) Introductory Lectures on Fluctuations of LevyProcesses with Applications. Springer[11] Møller T, Steffensen M (2007) Market-Valuation Methods in Life andPension Insurance. Cambridge[12] Shiryayev AN (1978) Optimal Stopping Rules. Springer-Verlag1613] Steffensen M. (2002) Intervention options in life insurance. Insur MathEcon 31:71–85 A Proof of Theorem 5.1 The proof is divided in two parts. One part associated with the risk from the ν θ based stopping time surrendering before the optimal time u ∗ and anotherpart associated with the risk from the ν θ based stopping time surrenderingafter the optimal time u ∗ . For this reason we define an intermediate reserve, W θ . The surrender strategy related to W θ resembles the one related to V θ .The only difference is that the strategy related to W θ does not surrender be-fore the optimal time. Mathematically we make the following definition. Letˆ τ θ,t be a stopping time for which the policyholder surrenders at time u withintensity ν θ ( u )1 ( u ≥ u ∗ ( t )) . We may write this stopping time in a convenientway by introducing stopping times, ˆ τ iθ,t , given recursively by ˆ τ θ,t ≡ τ iθ,t for i ∈ N surrenders with intensity ν θ ( u )1 ( u ≥ ˆ τ i − θ,t ) . With these definitions weget: ( I, ˆ τ θ,t ) d = ( I, ∞ (cid:88) i =1 ˆ τ iθ,t (ˆ τ i − θ,t
First we show that for every t ∈ [0 , n ]:lim inf θ →∞ V θ ( t ) ≤ lim inf θ →∞ W θ ( t, t ) . To prove this we use, given t ∈ [0 , n ] and ε > 0, the following notation aboutstopping times, τ : { τ good } = { G ( τ ) − V θ ( τ ) ≥ } , { τ ok } = { G ( τ ) − V θ ( τ ) ∈ [ − ε, } , { τ bad } = { G ( τ ) − V θ ( τ ) < − ε } . Thus, a stopping time, τ , is called good when it is profitable to surrender atthe corresponding time, and it is called bad when the policy holder loses more17han ε on surrendering. In the following, let u ∗ ≡ u ∗ ( t ) and let ˆ τ iθ ≡ ˆ τ iθ,t .By induction one can show that for every m ∈ N : V θ ( t ) = V ( t ) + E t (cid:20) e − (cid:82) ˆ τ θt r ( u )+ µ ( u ) du ( G (ˆ τ θ ) − V (ˆ τ θ )) (cid:21) ≥ V ( t )+ m (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du ( G (ˆ τ iθ ) − V (ˆ τ iθ ))1 (ˆ τ i − θ ≤ u ∗ < ˆ τ iθ , ˆ τ θ ,..., ˆ τ i − θ ok or good ) (cid:21) + E t (cid:20) e − (cid:82) ˆ τm +1 θt r ( u )+ µ ( u ) du ( G (ˆ τ m +1 θ ) − V (ˆ τ m +1 θ ))1 (ˆ τ mθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ mθ ok or good ) (cid:21) + m (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du ( G (ˆ τ iθ ) − V (ˆ τ iθ ))1 (ˆ τ iθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ i − θ ok or good, ˆ τ iθ bad ) (cid:21) − ε m (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du (ˆ τ iθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ i − θ ok or good, ˆ τ iθ ok ) (cid:21) (9)The idea is that the reserve V θ corresponds to the technical reserve, V ,plus the expected gain from surrender. We investigate what happens if thepolicyholder regrets to surrender. The impact if the policy holder regretsto surrender at the observed stopping time, ˆ τ θ , depends on whether thisstopping time was good, ok or bad. If the stopping time is good, then weknow that the value of the gain of surrender is at least as high as waitingfor the next time to surrender, and if the stopping time is ok, then we knowthat the value of the gain of surrender is at most ε worse than waiting forthe next time to surrender. In the above expression we have made thesejudgements for up to m surrender possibilities before the optimal time.The sum in the first line corresponds to the case when one of the first m stopping times reaches beyond the optimal time, u ∗ . The terms of thesecond line correspond to the case when all of the first m stopping timesare before the optimal time, u ∗ , and they have all been ok or good. In thiscase, the value of the gain of surrendering at the first stopping time is nohigher than waiting for the m + 1’th stopping time. The sum in the thirdline corresponds to the case when one of the m first stopping times is badand is before the optimal time, u ∗ . The sum of the fourth line is a correctionof the ε -small loses from ok stopping times.If we display the bound relative to W θ instead of relative to the technical18eserve, V , then we get the following expression: V θ ( t ) ≥ W θ ( t, t ) − ∞ (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du ( G (ˆ τ iθ ) − V (ˆ τ iθ ))1 (ˆ τ i − θ ≤ u ∗ < ˆ τ iθ , ∃ j ∈{ ,...,i − } : ˆ τ jθ bad ) (cid:21) − ∞ (cid:88) i = m +1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du ( G (ˆ τ iθ ) − V (ˆ τ iθ ))1 (ˆ τ i − θ ≤ u ∗ < ˆ τ iθ , ˆ τ θ ,..., ˆ τ i − θ ok or good ) (cid:21) + E t (cid:20) e − (cid:82) ˆ τm +1 θt r ( u )+ µ ( u ) du ( G (ˆ τ m +1 θ ) − V (ˆ τ m +1 θ ))1 (ˆ τ mθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ mθ ok or good ) (cid:21) + m (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du ( G (ˆ τ iθ ) − V (ˆ τ iθ ))1 (ˆ τ iθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ i − θ ok or good, ˆ τ iθ bad ) (cid:21) − ε m (cid:88) i =1 E t (cid:20) e − (cid:82) ˆ τiθt r ( u )+ µ ( u ) du (ˆ τ iθ ≤ u ∗ , ˆ τ θ ,..., ˆ τ i − θ ok or good, ˆ τ iθ ok ) (cid:21) . In the limit of θ , ε and m , then W θ is the only term which does not convergeto 0. To see this, notice that there exists some K > u ∈ [ t, n ]: G ( u ) − V ( u ) , ∈ [ − K, K ] , and G ( u ) − V θ ( u ) ∈ [ − K, K ] . That is, for any stopping time, the adjustment G − V is bounded by K .Thereby we may further bound the value of V θ by replacing each of theseadjustments with − K times an upper bound of the probability of the cor-responding event: V θ ( t ) ≥ W θ ( t, t ) − K P t ( ∃ j ∈ N : ˆ τ jθ bad, ˆ τ jθ ≤ u ∗ ) − K ∞ (cid:88) i = m +1 P t (ˆ τ i − θ ≤ u ∗ < ˆ τ iθ ) − K P t (ˆ τ mθ ≤ u ∗ , ˆ τ θ , . . . , ˆ τ mθ ok or good ) − K P t ( ∃ j ∈ N : ˆ τ jθ bad, ˆ τ jθ ≤ u ∗ ) − ε m (cid:88) i =1 P t (ˆ τ iθ ≤ u ∗ , ˆ τ iθ ok ) ≥ W θ ( t, t ) − K (1 − e ( n − t )¯ h θ ( − ε ) ) − K ∞ (cid:88) i = m +1 P t (ˆ τ i − θ ≤ u ∗ < ˆ τ iθ ) − K P t (ˆ τ mθ ≤ u ∗ ) − K (1 − e ( n − t )¯ h θ ( − ε ) ) − εn. Given θ and ε , then this holds for every n . Thus, the second sum can be madearbitrarily small and so can P t (ˆ τ nθ ≤ u ∗ ), the later follows because given θ ,then the intensity of surrender is bounded on [0 , n ] and thus the distributionof the number of ˆ τ iθ before u ∗ is bounded by a Poisson distribution. Thereby:lim inf θ →∞ V θ ( t ) ≥ lim inf θ →∞ W θ ( t, t ) . 19e find from the calculations above that the lower bound holds becausethe surrender strategy of W θ and V θ only differs by the strategy of W θ ,regretting every surrender before the optimal time. The impact of thisdifference is bounded because the following main reasons: The probabilityof a bad stopping time converges to zero in the limit because of (6). Thenumber of ok or good stopping times occurring before the optimal timeis finite. Regret of a good stopping time decreases the value. Regret ofan ok stopping time has an impact bounded by ε . At last, the technicalcalculations justify that the convergence of ε does not cancel the impact ofthe convergence of (6). Part 2: Consider some arbitrary t ∈ [0 , n ]. We wish to show that: W θ ( t, t ) → W ( t ) , θ → . Let u ∗ ≡ u ∗ ( t ) and ˆ τ θ = ˆ τ θ,t , and notice that since the policyholders relatedto W θ , W and V θ behave similarly before time u ∗ , then convergence at time t corresponds to convergence at time u ∗ . This is seen from: W ( t ) − W θ ( t, t ) = E t (cid:20) e − (cid:82) u ∗ t r ( u )+ µ ( u ) du ( G ( u ∗ ) − V ( u ∗ )) − e − (cid:82) ˆ τθt r ( u )+ µ ( u ) du ( G (ˆ τ θ ) − V (ˆ τ θ )) (cid:105) = e − (cid:82) u ∗ t r ( u )+ µ ( u ) du ( W ( u ∗ ) − W θ ( u ∗ , u ∗ ))= e − (cid:82) u ∗ t r ( u )+ µ ( u ) du ( W ( u ∗ ) − V θ ( u ∗ )) . Thereby, it is sufficient to prove that V θ ( u ∗ ) → W ( u ∗ ) when θ → ∞ . Eitherthis holds, or there is some ε > θ i ) i ∈ N converging toinfinity such that for all i ∈ N : V θ i ( u ∗ ) < W ( u ∗ ) − ε . Thereby V θ i ( u ∗ ) 0, there exists some δ such that( G ( u ∗ ) − V ( u ∗ )) − e − (cid:82) tu ∗ r ( u )+ µ ( u ) du ( G ( t ) − V ( t )) ≤ ε for t ∈ [ u ∗ , u ∗ + δ ].That is, if surrender happens within time δ of the optimal time then theloss of the delay is at most ε .Now, let δ = δ ∧ δ . Then the loss of surrender according to ˆ τ θ i instead20f at the optimal time is bounded in the following way: W ( u ∗ ) − V θ i ( u ∗ )= E (cid:20) ( G ( u ∗ ) − V ( u ∗ )) − ( G (ˆ τ θ i ) − V (ˆ τ θ i )) e − (cid:82) ˆ τθiu ∗ r ( x )+ µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ˆ τ θ i ≤ δ (cid:21) P (ˆ τ θ i ≤ δ )+ E (cid:20) ( G ( u ∗ ) − V ( u ∗ )) − ( G (ˆ τ θ i ) − V (ˆ τ θ i )) e − (cid:82) ˆ τθiu ∗ r ( x )+ µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) (ˆ τ θ i ≤ δ ) c (cid:21) P ((ˆ τ θ i ≤ δ ) c ) ≤ ε + E (cid:20) ( G ( u ∗ ) − V ( u ∗ )) − ( G (ˆ τ θ i ) − V (ˆ τ θ i )) e − (cid:82) ˆ τθiu ∗ r ( x )+ µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) (ˆ τ θ i ≤ δ ) c (cid:21) e − δh θi ( ε ) ≤ ε + 2 Ke − δh θi ( ε ) . Thus V θ i ( u ∗ ) → W ( u ∗ ) as θ → ∞→ ∞