Multistate Models in Health Insurance

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1 Multistate Models in Health Insurance Marcus Christiansen Preprint Series: Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM

2 Multistate models in health insurance Marcus C. Christiansen Institut für Versicherungswissenschaften Universität Ulm D Ulm, Germany

3 Abstract We illustrate how multistate Markov and semi-markov models can be used for the actuarial modeling of health insurance policies, focussing on health insurances that are pursued on a similar technical basis to that of life insurance. On the basis of the general modeling framework of Helwich (2008), we study examples of permanent health insurance, critical illness insurance, long term care insurance, and German private health insurance and discuss the calculation of premiums and reserves on the safe side. In view of the rising popularity of stochastic mortality rate modeling, we present a theoretical foundation for stochastic transition rates and explain why there is a need for future statistical research. Key Words: health insurance; Markov jump process; semi-markov process; prospective reserve; actuarial calculation on the safe side; systematic biometric risk; worst case scenario

4 1 Introduction and motivation Health insurances provide financial protection in case of sickness or injury by covering medical expenses or loss of earnings. Throughout the world there are various types of health insurance products, and there are many different traditions when it comes to their actuarial calculation. All of the calculation techniques are either pursued on a similar technical basis to that of life insurance or pursued on a similar technical basis to that of non-life insurance. The multistate modeling approach that we are presenting here belongs to the first group. That means that we assume that (despite some financial risks) the only source of randomness is a random pattern of (finitely many) states that the policyholder goes through during the contract period. Claim sizes and occurrence times of all health insurance benefits just depend on that random pattern of states. At first let us consider four small examples that illustrate the fundamental ideas of multistate modeling in health insurance. Example 1.1 (disability insurance). A disability insurance or permanent health insurance (PHI) provides an insured with an income if the insured is prevented from working by disability due to sickness or injury. It is usually modeled by a multiple state model with state space S := {a = active/healthy, i = invalid/disabled, d = dead}. a i d Note that in some countries, e.g. Germany, disability insurance is rather categorized as life or pension insurance than health insurance. Example 1.2 (critical illness insurance). A critical illness insurance or dread disease insurance (DD) provides the policyholder with a lump sum if the insured contracts an illness included in a set of diseases specified by the policy conditions. The most commonly diseases are heart attack, coronary artery disease requiring surgery, cancer and stroke. For example, it can be modeled by a multi state structure with state space S := {a = active/healthy, i = ill, d d = dead due to dread disease, d o = dead due to other causes}. a i d o d d 1

5 Example 1.3 (long term care insurance). A long term care insurance (LTC) provides financial support for insured who are in need of nursing or medical care. The need for care due to the frailty of an insured is classified according to the individuals ability to take care of himself by performing activities of daily living such as eating, bathing, moving around, going to the toilet, or dressing. LTC policies are commonly modeled by multistate models, and the state space usually consists of the states active, dead, and the corresponding levels of frailty. For example, in Germany three different levels of frailty are used and, moreover, lapse plays an important role. Thus, we have a state space of S = {a = active/healthy, c I = need for basic care, c II = need for medium care, c III = need for comprehensive care, l = lapsed/canceled, d = dead}. c I a c II d l c III Example 1.4 (German private health insurance). A German private health insurance primarily covers actual medical expenses of the policyholder. In fact, the individual future medical expenses are unknown and, thus, stochastic. However, German private health insurers use deterministic forecasts for individual medical expenses that only depend on the age of the policyholder, so that the state space consists only of S := {a = alive, l = lapsed/canceled, d = dead}. There is no differentiation made between different types and levels of morbidity, and so the only source of randomness is the time of death or lapsation, whichever occurs first. a l d For health insurances pursued on a similar technical basis to that of life insurance (the type of products we are studying here), the contractual guaranteed payments between insurer and policyholder are defined as deterministic functions of time and of the pattern of states of the policyholder. From a mathematical point of view, an insurance contract is just the set of that deterministic payment functions. Beside the specification of the contract terms (the payment functions), one of the main tasks of an actuary is to formulate a stochastic 2

6 model for the random pattern of states of the policyholder. In many cases it is reasonable to assume that the random pattern of states is a Markovian process. The Markovian property reduces complexity and leads to an easily manageable model. However, in some cases we have significant durational effects, that is, the duration of stay in certain states has an effect on the likeliness of future transitions between states. For example, for disabled insured both the probability of recovering and the probability of dying usually significantly decrease with increasing duration of disability (see, for example, Segerer (1993)). In such cases the Markovian assumption clearly oversimplifies matters. The literature shows that then the best compromise between accuracy and feasibility is a semi-markovian model. That is a model where (1) the present state of the policyholder and (2) the actual duration of stay in the present state together form a two-dimensional Markovian process. Nearly all multistate health insurance models presented in the literature fit into that semi-markovian framework. (An interesting exception is the concept of Davis (1984).) By inserting the pattern of states process into the payment functions, we obtain the random future cash flow between insurer and policyholder. The job of the actuary is then to analyze this cash flow with the objective of determining premiums, solvency reserves, portfolio values, profits and losses, and so on. For the interested reader we recommend the monograph of Haberman and Pitacco (1999), which gives a comprehensive survey of actuarial modeling of disability insurance, critical illness cover, and long-term care insurance. A detailed overview of actuarial modeling of private health insurance in Germany is given in Milbrodt (2005). In the insurance literature, the Markovian multistate model first appeared in Amsler (1968) and Hoem (1969). Since then the literature offers a range of papers that study this model, most of them under the topic life insurance. Comprehensive presentations of the Markovian model and lots of further references can be found in the monographs of Wolthuis (1994), Milbrodt and Helbig (1999), and Denuit and Robert (2007). The semi-markovian approach is much less studied in the literature, and we are not aware of a comprehensive monograph here. For the reader who is interested in the mathematical details of the semi-markovian multistate modeling, we recommend Nollau (1980), Janssen and De Dominicis (1984), and Helwich (2008). In the insurance literature the semi-markovian approach first appeared in Janssen (1966) and Hoem (1972). Further references are, for example, Waters (1989), Møller (1993), Segerer (1993), Gatenby and Ward (1994), Möller and Zwiesler (1996), Rickayzen and Walsh (2000), and Wetzel and Zwiesler (2003). 2 Random pattern of states For health insurances pursued on a similar technical basis to that of life insurance, contractual guaranteed payments between insurer and policyholder are defined as deterministic functions of time and of the pattern of states of the policyholder. Therefore, at first we need a model for that pattern of states. Definition 2.1. The random pattern of states is a pure jump process (Ω, F, P, (X t ) t 0 ) with finite state space S and right continuous paths with left-hand limits, where X t represents the state of the policyholder at time t 0. 3

7 We further define the transitions space J := {(i, j) S S i j}, the counting processes the time of the next jump after t the series of the jump times N jk (t) := # { τ (0, t] Xτ = k, X τ = j }, (j, k) J, T (t) := min { τ > t Xτ X t }, S 0 := 0, S n := T (S n 1 ), n N, and a process that gives for each time t the time elapsed since entering the current state, U t := max { τ [0, t] Xu = X t for all u [t τ, t] }. Instead of using a jump process (X t ) t 0, some authors describe the random pattern of states by a chain of jumps. The two concepts are equivalent. 2.1 The semi-markovian approach The random pattern of states (X t ) t 0 is called semi-markovian, if the bivariate process (X t, U t ) t 0 is a Markovian process, which means that for all i S, u 0, and t t n... t 1 0 we have P ( (X t, U t ) = (i, u) ) ( Xtn, U tn,..., X t1, U t1 = P (Xt, U t ) = (i, u) ) Xtn, U tn almost surely. In the following we always assume that the initial state (X 0, U 0 ) is deterministic. (Note that U 0 = 0 by definition.) In practice that means that we know the state of the policyholder when signing the contract. With this assumption and the Markov property of (X t, U t ) t 0 we have that the probability distribution of (X t, U t ) t 0 is already uniquely defined by the transition probability matrix ( ) p(s, t, u, v) := P (X t = k, U t v X s = j, U s = u), 0 u s t <, v 0. (j,k) S 2 Practitioners usually prefer the notation v,t sp jk x+s,u := p jk (s, t, u, v), where x is the age of the policyholder at contract time zero. Alternatively, we can also uniquely define the probability distribution of (X t, U t ) t 0 by specifying the probabilities ( ) p(s, t, u) = p jk (s, t, u) (j,k) S 2, p jk (s, t, u) := P (T (s) t, X T (s) = k X s = j, U s = u), j k, p jj (s, t, u) := P (T (s) t X s = j, U s = u) 4

8 for 0 u s t <. For j = k the corresponding actuarial notation is t sp jj x+s,u := 1 p jj (s, t, u) = P (T (s) > t X s = j, U s = u). A third way to uniquely define the probability distribution of (X t, U t ) t 0 is to specify the cumulative transition intensity matrix ( ) q(s, t) = q jk (s, t), (j,k) S 2 p jk (s, dτ, 0) (2.1) q jk (s, t) := 1 p jj (s, τ, 0), 0 s t <. (s,t] If q(s, t) is differentiable with respect to t, we can also define the so-called transition intensity matrix µ(t, t s) := d ( d dt q(s, t) = p dt jk(s, t, 0) ), 1 p jj (s, t, 0) (j,k) S S which is some form of multi-state hazard rate. The quantity µ jk (t, t s) gives the rate of transitions from state j to state k at time t given that the current duration of stay in j is t s. The corresponding actuarial notation is µ jk x+t,u := µ jk (t, u). If the transition intensity matrix µ exists, it also already uniquely defines the probability distribution of (X t, U t ) t 0. In order to obtain the transition probabilities from the cumulative transition intensities, we can use the Kolmogorov backward integral equation system p ik (s, t, u, v) 1 {i=k} 1 {v u+t s} = p jk (τ, t, 0, v) q ij (s u, dτ) j:j i (s,t] (2.2) + p ik (τ, t, u + τ s, v) q ii (s u, dτ). (s,t] We will base all our actuarial calculations on the cumulative transition intensity matrix q(s, t) and the probabilities p jj (s, t, u), where the latter can be derived from the former by the so-called exponential formula ( ) 1 p ii (s, t, u) = e qc ii (s u,t) qc ii (s u,s) 1 + q ii (s u, τ) q ii (s u, τ ). (2.3) s<τ t Here, q c ii(s u, t) is the continuous part of t q ii (s u, t). Thus, for our actuarial calculations following later on, we have to specify q(s, t), and only q(s, t). The following concepts are frequently used in practice. (a) We estimate the µ jk x+t,u = µ jk (t, u) from statistical data, often by using a parametric method (cf. Haberman and Pitacco (1999), section 4). The cumulative transition intensities q(s, t) are obtained by integrating the transition intensities. In case we are also 5

9 interested in the transition probability matrix p(s, t, u, v), we can obtain it as the unique solution of the Kolmogorov backward differential equation system p(s, t, u, v) = µ(s, u) p(s, t, 0, v) p(s, t, u, v) s u with boundary condition p(t, t, u, v) = 1 {i=k} 1 {v u+t s}. (b) We estimate,t s p jk x+s,u := p jk (s, t, u, ) from statistical data, but only at integer times s, t, u. We generally assume that transitions can only occur at the turn of the years. Then from (2.1) we obtain that q(m, dt) = 0 for non-integer times t, and from (2.2) we get { pjk (n 1, n, n 1 m, ) : j k q jk (m, n) q jk (m, n ) := p jj (n 1, n, n 1 m, ) 1 : j = k for integer times m < n. In practice, the probabilities p jk (n 1, n, n 1 m, ) are often provided by so-called select-and-ultimate tables (cf. Bowers et al. (1997), section 3.8). Such tables generally contain annual rates that depend on the age at selection (e.g. onset of disability) and on the duration since selection. The common actuarial notation is p jk (n 1, n, n 1 m, ) =: p jk [x+m]+n 1 m. (The value in the square brackets is the age of the policyholder at the last transition, and the addend after the brackets gives the duration of stay in the actual state.) Usually, beyond a certain period such as 5 years, the dependence on the time since selection is neglected and the corresponding transition probabilities are combined, resulting in the so-called ultimate table p jk x+n 1 := p jk [x+m]+n 1 m for n 1 m 5. (c) Similarly to (b), we estimate the,t s p jk x+s,u := p jk (s, t, u, ) from statistical data at integer times s, t, u. But differing from (b), we distribute the mass q jk (m, n) q jk (m, n 1) equally on the interval (n 1, n] by defining { pjk (n 1, n, n 1 m, ) : j k, t (n 1, n] µ jk (t, t m) := p jj (n 1, n, n 1 m, ) 1 : j = k, t (n 1, n] for all integer times m. (d) We estimate the,t s p jk x+s,u := p jk (s, t, u, ) from statistical data at integer times s, t, u. For times in between we use linear interpolation. See Helwich (2008, p. 93). If the cumulative transition intensity matrix q(t, s) is regular (the matrix elements q jk (s, ), j k, are monotonic non-decreasing and right-continuous functions that are zero at time s, q jj = k j q jk, q jj (s, t) q jj (s, t ) 1, and in case of q jj (s, t 0 ) q jj (s, t 0 ) = 1 the function q jj (s, ) is constant from t 0 on), then there always exists a corresponding semi-markovian process ((X t, U t )) t 0, which is even strong Markovian (see Helwich (2008), Remark 2.34). In the insurance literature the semi-markov model based on transition intensities was first described by Hoem (1972). The more general model based on cumulative intensities was introduced by Helwich (2008). 6

10 2.2 The Markovian approach For some insurance products we can simplify the semi-markovian model to the special case where (X t ) t 0 on its own is a Markovian process, which means that for all i S and t t n... t 1 0 we have P ( X t = i ) ( Xtn,..., X t1 = P Xt = i ) Xtn almost sure. As a consequence, the transition probabilities p jk (s, t, u, v) and the probabilities p jk (s, t, u) do not depend on u or v anymore, the cumulative transition intensity matrix q(s, t) is constant with respect to its first argument s, and the transition intensity matrix µ(t, u) is constant with respect to the second argument u. Actuaries then use the notation t sp jk x+s := P (X x+t = k X x+s = j) = p jk (s, t, u, ), t sp jj x+s := P (T (s) > t X s = j) = 1 p jj (s, t, u), µ jk x+t := µ jk (t) = µ jk (t, u). In the insurance literature the Markov model based on transition intensities µ jk (t) and a differentiable discounting function was already described by Sverdrup (1965) and Hoem (1969). The more general cumulative intensity approach was introduced by Milbrodt and Stracke (1997). Markovian models are sometimes used to approximate semi-markovian models. The idea is to split all states i with durational effects into a set of states i 1, i 2,..., i n with the meaning i k = insured in state i with a duration between k 1 and k for k = 1,..., n 1, i n = insured in state i with a duration greater than n 1. For example, for disability policies we have the well-known effect that for different durations since disability inception the observed recovery and mortality rates differ. Instead of using the three state model S = {a, i, d} of Example 1.1 in a semi-markovian setup, we could use the multistate model S = {a, i 1,..., i n, d} in a Markovian setup. a i 1 d i 2. We have to specify q(t) in such a way that the transitions (i 1, i 2 ),..., (i n 1, i n ) occur with probability one at the turn of the years. Durational effects are taken into account by using different recovery and mortality rates for the states i 1,..., i n. This differentiation of disability states has been originally proposed by Amsler (1968). 7

11 2.3 Other approaches More realistic and sophisticated models can be built by considering for example the dependence on the age x at policy issue, the dependence on the total time spent in some states since policy issue, stressing the health history of the insured. Both approaches complicate the actuarial models. The first one can be included by estimating v,t s p jk x+s,u, t s p jj x+s,u, or µ jk x+t,u separately for each starting age x. Another interesting approach is given by Davis (1984). However, reasonable models can be defined by considering just the semi-markovian case of above. 3 Payment functions As already mentioned in the introduction, premiums and benefits are paid according to the pattern of states of the policyholder. Definition 3.1 (contractual payments). Payments between insurer and policyholder are of two types: (a) Lump sums are payable upon transitions between two states and are specified by deterministic nonnegative functions b jk (t, u) with bounded variation on compacts. The amount b jk (t, u) is payable if the policy jumps from state j to state k at time t and the duration of stay in state j was u. In order to distinguish between payments from insurer to insured and vice versa, benefit payments get a positive sign and premium payments get a negative sign. (b) Annuity payments fall due during sojourns in a state and are defined by deterministic functions B j (s, t), j S. Given that the last transition occurred at time s, B j (s, t) is the total amount paid in [s, t] during a sojourn in state j. We assume that the B j (s, ) are right-continuous and of bounded variation on compacts. Example 3.2 (disability insurance). Suppose we have a contract that pays a continuous disability annuity with rate r > 0 as long as the insured is in state disabled, and a constant (lump sum) premium p > 0 has to be paid yearly in advance as long as the policyholder is in state active. Following a modeling framework of Pitacco (1995), let (n 1, n 2 ) denote the insured period, meaning that benefits are payable if the disability inception time belongs to this interval, let f denote the deferred period (from disability inception), let m be the maximum number of years of annuity payment (from disability inception), and let T ( n 2 + f) be the stopping time of annuity payment (from policy issue at time zero). 8

12 Then the contractual payment functions are B a (s, t) = n 2 1 k=0 B i (s, t) = 1 (n1,n 2 )(s) B d (s, t) = 0 p 1 [k, ) (t), min{t,s+f+m,t } min{t,s+f} r du, for all 0 s t. The transitions payments b jk (t, u) are all constantly zero. Example 3.3 (critical illness insurance). Let the state space be S := {a = active/healthy, i = ill, d = dead}, and denote by T the policy term. Suppose that a constant (lump sum) premium p > 0 has to be paid yearly in advance as long as the policyholder is in state active. A lump sum is payed upon death or a dread disease diagnosis, whichever occurs first. The death benefit is l d. The dread disease benefit is l i, but only if the policyholder outlives a deferred period f, otherwise the sum l d is paid. We assume that a recovery from state i to state a is impossible. The contractual payment functions are T 1 B a (s, t) = p 1 [k, ) (t), B i (s, t) = l i 1 (0,T f] (s) 1 [s+f, ) (t), B d (s, t) = 0, k=0 b ad (t, u) = l d 1 (0,T ] (t), b ai (t, u) = 0, b id (t, u) = l d 1 (0,T ] (t) 1 [0,f) (u), for all 0 s t and u 0. Without the deferred period (f = 0), we can alternatively write T 1 B a (s, t) = p 1 [k, ) (t), B i (s, t) = 0, B d (s, t) = 0, k=0 b ad (t, u) = l d 1 (0,T ] (t), b ai (t, u) = l i 1 (0,T ] (t), b id (t, u) = 0. Example 3.4 (long-term care insurance). Suppose we have a German policy with state space S = {a = active/healthy, c I = need for basic care, c II = need for medium care, c III = need for comprehensive care, l = lapsed/canceled, d = dead}. Continuous annuities are paid with rates r I, r II, r III as long as the insured is in state c I, c II, c III, respectively. A constant (lump sum) premium p > 0 has to be paid yearly in advance till retirement at contract time R, lapse, or death, whichever occurs first. The contractual payment functions are R 1 B a (s, t) = p 1 [k, ) (t), k=0 B c I/II/III(s, t) = (t min{t, s}) r I/II/III, B l (s, t) = 0, B d (s, t) = 0 for all 0 s t. The transitions payments b jk (t, u) are all constantly zero. Interestingly, the benefits B i (s, dt) at time t s do not really depend on s. Hence, we can simplify the 9

13 payment functions by skipping the first argument, R 1 B a (t) = p 1 [k, ) (t), k=0 B c I/II/III(t) = t r I/II/III, B l (t) = 0, B d (t) = 0. Example 3.5 (German private health insurance). As already mentioned above, German health insurers calculate on the basis of deterministic benefit forecasts that only depend on the age of the policyholder. This deterministic approach is justified by the argument that, though the individual medical expenses are far from being deterministic, the portfolio average of medical expenses is nearly deterministic in case of a large portfolio of independent insured because of the law of large numbers. The law of large numbers does not work for systematic changes of medical expenses with progression of calendar time, but German health insurers have the exceptional right to adapt their deterministic forecasts of the medical expenses if calculated values and real values differ significantly. (The special right of German and Austrian health insurers to change the payments functions after(!) signing of the contract is unique in private health insurance.) Let the average yearly medical expenses (Kopfschäden) of a policyholder with age x + k be a deterministic quantity K x+k, where x is the age of the policyholder at inception of the contract. Recall that the state space is S := {a = alive, l = lapsed/canceled, d = dead}. A constant (lump sum) premium p > 0 has to be paid yearly in advance till death or lapsation. The contractual payment functions are B a (s, t) = ( Kx+k p ) 1 [k, ) (t), k=0 B l (s, t) = 0, B d (s, t) = 0 for all 0 s t. The transitions payments b jk (t, u) are all constantly zero. All payments are independent of the duration of stay. Hence, we can simplify the payment functions by skipping the first argument, 4 Reserves B a (t) = ( Kx+k p ) 1 [k, ) (t), k=0 B l (t) = 0, B d (t) = 0. By statute the insurer must at any time maintain a reserve in order to meet future liabilities in respect of the contract. This reserve bears interest with some rate ϕ(t). On the basis of this interest rate we define a discounting function, v(s, t) := e t s ϕ(r)dr. (4.1) 10

14 We can interpret v(s, t) as the value at time s of a unit payable at time t s. At next, we study the present value of future payments between insurer and policyholder, that is, the discounted sum of all future benefit and premium payments, A(t) := j S n=0 + (j,k) J v(t, τ) 1 {Xτ =j} 1 {Sn τ<s n+1 } B j (S n, dτ) v(t, τ) b jk (τ, U τ ) dn jk (τ). (4.2) A(t) gives the amount that an insurer needs at time t in order to meet all future obligations in respect of the contract, but, because of its randomness, we do not know its value. However, if we have a homogeneous portfolio of stochastically independent policies, then the law of large numbers yields that the average present value per policy is close to its mean. This motivates the following definition. Definition 4.1. The prospective reserve at time t in state (i, u) is defined by given that the expectation exists. V i (t, u) := E ( A(t) (Xt, U t ) = (i, u) ), V i (t, u) is the amount that the insurer needs on portfolio average in order to meet all future obligations. At present time t we do not only know the value of (X t, U t ) but also the complete history of (X τ, U τ ) τ 0 up to time t. But since we assumed that (X τ, U τ ) τ 0 is Markovian, in the above definition we may just condition on (X t, U t ). Theorem 4.2. The prospective reserves V i (t, u), i S, 0 u t, equal almost sure the unique solution of the integral equation system V i (t, u) = v(t, τ) (1 p ii (t, τ, u)) B i (t u, dτ) + ( ) v(t, τ) (1 p ii (t, τ 0, u)) b ij (τ, τ t + u) + V j (τ, 0) q ij (t u, dτ). j:j i (4.3) A proof can be found in Helwich (2008). The integral equation system (4.3) is also denoted as Thiele integral equations (of type 1). The theorem provides a way for the calculation of V i (t, u). Generally, we can use the following method. Algorithm 4.3. (1) At first solve the integral equation system (4.3) for u = 0, V i (t, 0) = v(t, τ) (1 p ii (t, τ, 0)) B i (t, dτ) + ( ) v(t, τ) (1 p ii (t, τ 0, 0)) b ij (τ, τ t) + V j (τ, 0) q ij (t, dτ). j:j i 11

15 If there is no closed form solution, we have to use numerical methods in order to obtain at least approximations on a fine time grid. In practice we will always have some time T < such that V i (t, 0) = 0 for all t T and i S. We start from that T and make small backward steps till we arrive at t = 0. (2) On the basis of the solution from (1), we can now calculate the integrals in (4.3), at least approximately by using numerical integration. Recall that the probability p ii (t, τ, 0) has the explicit representation (2.3). We now demonstrate this algorithm for some examples. We will see that in many practical examples the calculation method can be considerably simplified. Example 4.4 (disability insurance). We continue with Example 3.2. Suppose that the transition intensity matrix µ(t, u) exists. As there are no payments in state d = dead, and since the transitions (d, a) and (d, i) are not possible, we obtain V d (t, u) = 0 for all 0 u t. Furthermore, we have V a (t, u) = 0 and V i (t, u) = 0 for all t T, because there are no payments after stopping time T. Thus, we get V a (t, u) = v(t, τ) (1 p aa (t, τ, u)) B a (t u, dτ) + V i (t, u) = + (t,t ] T t (t,t ] T t v(t, τ) (1 p aa (t, τ, u)) V i (τ, 0) µ ai (τ, τ t + u) dτ, v(t, τ) (1 p ii (t, τ, u)) B i (t u, dτ) v(t, τ) (1 p ii (t, τ, u)) V a (τ, 0) µ ia (τ, τ t + u) dτ. Inserting the definitions of B a and B i, for u = 0 we get V a (t, 0) = + V i (t, 0) = n 2 1 k=0,k>t T + t v(t, k) (1 p aa (t, k, 0)) ( p) v(t, τ) (1 p aa (t, τ, 0)) V i (τ, 0) µ ai (τ, τ t) dτ, (t,t ] (n 1,n 2 ) (t+f,t+f+m) T t v(t, τ) (1 p ii (t, τ, 0)) r dτ v(t, τ) (1 p ii (t, τ, 0)) V a (τ, 0) µ ia (τ, τ t) dτ. (4.4) Since the mapping t (V a (t, 0), V i (t, 0)) is continuous at non-integer times t, we can approximate this mapping in between integer times by a piecewise constant function on a fine time grid. At integer times t = n we have V a (n, 0) = V a (n+, 0) v(t, n) (1 p aa (t, n, 0)) p and V i (n, 0) = V i (n+, 0). We start from (V a (T, 0), V i (T, 0)) = (0, 0) and use a backward approximation scheme with sufficiently small steps. When reaching t = 0, we have an approximation of t (V a (t, 0), V i (t, 0)) for all t 0, and in a second step we can now approximate the V a (t, u) and V i (t, u) by using numerical integration in formula (4.4). Sometimes actuaries 12

16 further simplify the model by assuming that there is no reactivation (i, a) possible. In this case, V i (t, u) can be directly calculated from V i (t, u) = v(t, τ) (1 p ii (t, τ, u)) B i (t u, dτ), (t,t ] and then V a (t, u) can be directly calculated from (4.4), if necessary by using numerical integration. Example 4.5 (critical illness insurance). We continue with Example 3.3. Again suppose that the transition intensity matrix µ(t, u) exists. Generally, we can use Algorithm 4.3 similarly to Example 4.4. In case the deferred period f is zero, we have V i (t, u) = 0 and V d (t, u) = 0 for all 0 u t, and it suffices to solely look at V a (t, u), V a (t, u) = T 1 k=0,k>t T + + t T t v(t, k) (1 p aa (t, k, u)) ( p) v(t, τ) (1 p aa (t, τ, u)) b ai (τ, τ t + u) µ ai (τ, τ t + u) dτ v(t, τ) (1 p aa (t, τ, u)) b ad (τ, τ t + u) µ ad (τ, τ t + u) dτ. (4.5) Since the right hand side does not contain any prospective reserves anymore, we just have to (numerically) integrate the right hand side in order to obtain V a (t, u). Example 4.6 (long-term care insurance). We continue with Example 3.4. Our model with state space S = {a = active/healthy, c I = need for basic care, c II = need for medium care, c III = need for comprehensive care, l = lapsed/canceled, d = dead} is a so called hierarchical model since the transition probability matrix p(s, t) has a triangular form. As a consequence, for each state i in the ordered set [a, c I, c II, c III, l, d] formula (4.3) depends only on prospective reserves of states that follow state i in [a, c I, c II, c III, l, d]. We start with state d for which we get V d (t, u) = 0 for all 0 u t. Then we continue with state l for which we also have V d (t, u) = 0 for all 0 u t. At next we calculate V c III(t, u), V c III(t, u) = v(t, τ) (1 p ii (t, τ, u)) r III dτ. Then we go on with c II and calculate V c II(t, u) on the basis of V d (t, 0), V l (t, 0), and V c III(t, 0). The last steps are the calculations of V c I(t, u) and V a (t, u). While for the general method according to Algorithm 4.3 we have to solve an integral equation system, we can here simply use consecutive (numerical) integration in order to obtain the prospective reserves. Example 4.7 (German private health insurance). We continue with Example 3.5. As there are no payments in state l and state d, and since we cannot jump back to state a, we have V l (t, u) = 0 and V d (t, u) = 0 for all 0 u t, and it suffices to solely look at V a (t, u), V a (t, u) = k=0,k>t v(t, k) (1 p aa (t, k, u)) (K x+k p). 13

17 According to the definition of B a (s, t) in Example 3.5, all medical expenses are paid as lump sums yearly in advance. This simplification is motivated by the discrete time approach (b) of section 2.1, where state changes may only occur at the turn of the years. German private health insurers usually use such a discrete time approach and additionally assume that the random pattern of states is a Markovian process according to section 2.2. This Markov assumption is controversial because statistical data shows that withdrawal probabilities decrease with the time elapsed since inception of the contract. This effect can be explained by the fact that policyholders partly lose their ageing provision (the prospective reserve) if they switch insurance companies. The aging provision rises with increasing contract time and with it the loss due to lapse. Nevertheless, German private health insurers use the Markovian approach, and for integer times t and k with t < k we get 1 p aa (t, k, u) = P (T (t) > k X t = a, U t = u) = P (T (t) > k X t = a) k 1 = P (T (j) > j + 1 X j = a) = j=t k 1 j=t 1p aa x+j. The probabilities 1 p aa x+j = 1 1p al x+j 1 p ad x+j =: 1 ω x+j q x+j are estimated from statistical data. Referring to approach (b) in section 2.1, we define the cumulative transition intensity matrix q(s, t) by q al (m, n) q al (m, n ) := ω x+n 1, q ad (m, n) q ad (m, n ) := q x+n 1, and q(m, n ) q(m, n 1) = 0 for all integer times n 1. Remark 4.8 (Discrete time approach). Following approach (b) in section 2.1, let transitions between different states only happen at integer times. Then q(s, dτ) is zero at non-integer times. Suppose that also the B i (s, dτ) are zero at non-integer times, which means that payments during sojourns in a state are paid either yearly in advance or yearly in arrears. As transitions shall only occur at the turn of the years, that does not mean a loss of generality. Let T < be the contract term. Then we can rewrite (4.3) to V i (n, m) = T τ=n+1 + j:j i v(n, τ) (1 p ii (n, τ, m)) ( B i (τ, n m) B i (τ, n m) ) T τ=n+1 ( ) v(n, τ) (1 p ii (n, τ 0, m)) b ij (τ, τ n + m) + V j (τ, 0) ( q ij (n m, τ) q ij (n m, τ ) ) for all integer times 0 m n. We obtained a recursion formula that can be easily solved backwards starting from some n = T and making time steps of 1 till we arrive at t = 0. Recall that the probabilities p ii (n, τ, m) can be easily calculated with the help of (2.3). Remark 4.9 (Absolute continuity approach). If not only the cumulative transition intensities are differentiable with respect to their second argument (the transition intensity matrix 14

20 According to Helwich (2008, chapter 4.D), the prospective reserves V i (t, u) can also be seen as the unique solution of the following Thiele integral equation system (of type 2) V i (t, u) = B i (t u, dτ) V i (t, u + τ t) ϕ(τ) dτ + (6.1) R ij (τ, u + τ t) q ij (t u, dτ), j:j i where R ij (τ, v) is the so-called sum at risk associated with a possible transition from state i to state j at time τ, R ij (τ, v) := b ij (τ, v) + V j (t, 0) + B j (τ, τ) V i (τ, v) (B i (τ v, τ) B i (τ v, τ )). (We use the convention B i (τ, τ ) := 0.) Now suppose that we have another set of actuarial assumptions ϕ (t) and q (s, t) with corresponding prospective reserves Vi (t, u) and sums at risk Rij(t, u). Let W i (t, u) := Vi (t, u) V i (t, u). From (6.1) we get W i (t, u) = W i (τ, u + τ t) ϕ(τ) dτ Vi (τ, u + τ t) (ϕ (τ) ϕ(τ)) dτ + ( Wj (τ, 0) W i (τ, u + τ t) ) q ij (t u, dτ) j:j i + Rij(τ, u + τ t) (qij q ij )(t u, dτ). j:j i For the sum of the second and fourth integral we write Vi (τ, u + τ t) (ϕ (τ) ϕ(τ)) dτ (t,r] + R ij (τ, u + τ t) (qij q ij )(t u, dτ) j:j i (t,r] =: C i (t u, dτ) = C i (t u, r) C i (t u, t). (t,r] Suppose that the cumulative transition intensities qij and q ij have representations of the form qij(s, t) = λ ij(τ, τ s) dλ ij (τ), (s,t] q ij (s, t) = λ ij (τ, τ s) dλ ij (τ). (s,t] For example, if the transition intensity matrices µ and µ exist, then let the Λ ij be Lebesgue- Borel measures and define λ ij := µ ij and λ ij := µ ij. In the discrete time model according 17

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