multistate models in health insurance

AStA Adv Stat Anal (2012) 96:155186DOI 10.1007/s10182-012-0189-2O R I G I NA L PA P E R

Multistate models in health insurance

Marcus C. Christiansen

Received: 13 December 2010 / Accepted: 14 March 2012 / Published online: 3 April 2012 Springer-Verlag 2012

Abstract We illustrate how multistate Markov and semi-Markov models can be usedfor the actuarial modeling of health insurance policies, focusing on health insurancesthat are pursued on a similar technical basis to that of life insurance. In the first part,we give an overview of the basic modeling frameworks that are commonly used andexplain the calculation of prospective reserves and net premiums. In the second part,we discuss the biometric insurance risk, focusing on the calculation of implicit safetymargins. We present new results on implicit margins in the semi-Markov model andon biometric estimation risk in the Markov model, and we explain why there is a needfor future research concerning the systematic biometric risk.

1 Introduction

Health insurances provide financial protection in case of sickness or injury by cov-ering medical expenses or loss of earnings. Throughout the world there are varioustypes of health insurance products, and there are many different traditions when itcomes 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 firstgroup. That means that we assume that (despite some financial risks) the only sourceof randomness is a random pattern of (finitely many) states that the policyholder goesthrough during the contract period. Claim sizes and occurrence times of all healthinsurance benefits just depend on that random pattern of states.

M.C. Christiansen ()Institute of Insurance Science, University of Ulm, 89069 Ulm, Germanye-mail: [email protected]

156 M.C. Christiansen

For health insurances pursued on a similar technical basis to that of life insurance,the contractual guaranteed payments between insurer and policyholder are defined asdeterministic functions of time and of the pattern of states of the policyholder. Froma mathematical point of view, an insurance contract is just the set of that determin-istic payment functions. Beside the specification of the contract terms (the paymentfunctions), one of the main tasks of an actuary is to formulate a stochastic modelfor the random pattern of states of the policyholder. In many cases it is reasonableto assume that the random pattern of states is a Markovian process. The Markovianproperty reduces complexity and leads to an easily manageable model. However, insome cases we have significant durational effects; that is, the duration of stay in cer-tain states has an effect on the likeliness of future transitions between states. Forexample, for disabled insured both the probability of recovering and the probabilityof dying usually significantly decrease with increasing duration of disability (see, forexample, Segerer 1993). In such cases, the literature shows that a good compromisebetween 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 thepresent state together form a two-dimensional Markovian process. Nearly all multi-state health insurance models presented in the literature fit into that semi-Markovianframework. (An interesting exception is the concept of Davis 1984.) However, Am-sler (1968) proposed a method that approximates the semi-Markovian model by anaugmented Markov model, which is also widely accepted as a reasonable compro-mise.

By inserting the pattern of states process into the payment functions, we obtainthe random future cash flow between insurer and policyholder. In health insurancepursued on a similar technical basis to that of life insurance, the risk implied bythe uncertainty of the pattern of states process is usually met with safety marginson the transition probabilities. The focus and the main contribution of the presentpaper is on the calculation and estimation of those safety margins. A natural ideais to define the safety margins as differences of best estimates and upper or lowerconfidence bounds. However, an insurer is not really interested in confidence inter-vals for the transition rates, but in confidence limits for the capital reserves that heshould hold in order to be able to meet all insurance liabilities. The capital reservesare functionals of the transition rates, and the proper confidence condition is there-fore a functional confidence condition. Unfortunately, the latter condition has manysolutions, and we have to impose a further condition in order to uniquely identify rea-sonable safety margins. In the present paper we add a risk-proportionality principle.The functional confidence condition and the risk-proportionality principle togetherform a nonstandard confidence estimation problem, which is (partially) solved in thepresent paper.

Sections 2 and 3 give an overview of commonly used multistate models forhealth insurance policies. We present a number of examples including permanenthealth insurance, critical illness insurance, long-term care insurance, and Germanprivate health insurance. We base our presentation on the modeling framework ofHelwich (2008), which is the most general framework for semi-Markovian mul-tistate health insurance models. For the interested reader we also recommend themonograph of Haberman and Pitacco (1999), which gives a comprehensive survey

Multistate models in health insurance 157

of actuarial modeling of disability insurance, critical illness cover, and long-termcare insurance. A detailed overview of actuarial modeling of private health insur-ance in Germany can be found in Milbrodt (2005). In the insurance literature, theMarkovian multistate model first appeared in Amsler (1968) and Hoem (1969). Sincethen the literature has offered a range of papers that study this model, most ofthem under the topic life insurance. Comprehensive presentations of the Marko-vian model and numerous further references can be found in the monographs ofWolthuis (2003), Milbrodt and Helbig (1999), and Denuit and Robert (2007). Thesemi-Markovian approach is much less studied in the literature, and we are not awareof a comprehensive monograph here. For the reader who is interested in the math-ematical details of the semi-Markovian multistate modeling, we recommend Nol-lau (1980), Janssen and De Dominicis (1984), and Helwich (2008). In the insur-ance literature the semi-Markovian approach first appeared in Janssen (1966) andHoem (1972). Waters (1989) provided one of the first examples of semi-Markovianmodels to practical actuarial work in the field of disability annuities. Further refer-ences are, for example, Mller (1993), Segerer (1993), Gatenby and Ward (1994),Mller and Zwiesler (1996), Rickayzen and Walsh (2000), and Wetzel and Zwiesler(2003).

In Sect. 4 we discuss the risk implied by the uncertainty of the random patternof states of the policyholder, here denoted as biometric insurance risk. All maincontributions of the paper are found in Sect. 4. The focus is on the definition andcalculation of implicit biometric safety margins, which are margins on the transi-tion probabilities with the aim to let premiums and reserves be on the safe sidefrom the perspective of the insurer. First we extend a safe-side condition, that iswell known for the Markovian case, to the semi-Markovian case. In three subsec-tions we separately analyze the following risk parts: (1) unsystematic risk, (2) es-timation risk, and (3) systematic risk. Concerning the unsystematic risk, we brieflyrecapitulate a recent result of Christiansen (2011) that is based on the delta method.Previous papers, e.g., Pannenberg (1997), were not able to truly deal with the mul-tistate case. We then show that the ideas of Christiansen (2011) can be further ex-tended such that they can also be used to deal with the estimation risk. A numericalexample is given that illustrates the new method. In contrast to the existing liter-ature (for some references, see Haberman and Pitacco 1999), our method is non-parametric and offers a truly risk-oriented distribution of the total safety marginto implicit safety margins with respect to type and time of transition. Finally, wediscuss the systematic risk and explain why there is a need for future research inspite of the contributions of Renshaw and Haberman (2000) and Christiansen et al.(2012).

2 The random pattern of states of the policyholder

For health insurances pursued on a similar technical basis to that of life insurance,contractual guaranteed payments between insurer and policyholder are defined asdeterministic functions of time and of the pattern of states of the policyholder. Beforewe introduce a general modeling framework for that pattern of states, we give fourexamples of customary health insurance contracts.


Example 2.1 (disability insurance) A disability insurance or permanent health insur-ance (PHI) provides an insured with an income if the insured is prevented from work-ing by disability due to sickness or injury. It is usually modeled by a multiple statemodel with state space S := {a = active/healthy, i = invalid/disabled, d = dead}.

Note that in some countries, e.g., Germany, disability insurance is categorized as lifeor pension insurance rather than health insurance.

Example 2.2 (critical illness insurance) A critical illness insurance or dread dis-ease insurance (DD) provides the policyholder with a lump sum if the insured con-tracts an illness included in a set of diseases specified by the policy conditions. Themost common diseases are heart attack, coronary artery disease requiring surgery,cancer, and stroke. For example, it can be modeled by a multistate structure withstate space S := {a = active/healthy, i = ill, dd = dead due to dread disease , do =dead due to other causes}.

Example 2.3 (long-term care insurance) A long-term care insurance (LTC) pro-vides financial support for insureds who are in need of nursing or medical care.The need for care due to the frailty of an insured is classified according tothe individuals ability to take care of himself by performing activities of dailyliving such as eating, bathing, moving around, going to the toilet, or dress-ing. LTC policies are commonly modeled by multistate models, and the statespace usually consists of the states active, dead, and the corresponding levels offrailty. 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, cI = need for basic care, cII = need for medium care, cIII =


need for comprehensive care, l = lapsed/canceled, d = dead}.

Example 2.4 (German private health insurance) A German private health insuranceprimarily covers actual medical expenses of the policyholder. In fact, the individualfuture medical expenses are unknown and, thus, stochastic. However, German pri-vate health insurers use deterministic forecasts for individual medical expenses thatonly depend on the age of the policyholder, so that the state space consists only ofS := {a = alive, l = lapsed/canceled, d = dead}. No differentiation is made betweendifferent types and levels of morbidity, and so the only source of randomness is thetime of death or lapsation, whichever occurs first.

Definition 2.5 The random pattern of states is a pure jump process(,F,P , (Xt )t0) with finite state space S and right continuous paths with left-handlimits, where Xt represents the state of the policyholder at time t 0.

We further define the transitions space J := {(i, j) S S|i = j}, the countingprocesses

Njk(t) := #{ (0, t] |X = k,X = j

}, (j, k) J,

the time of the next jump after tT (t) := min{ > t |X = Xt },

the series of the jump timesS0 := 0, Sn := T (Sn1), n N,

and a process that gives for each time t the time elapsed since entering the currentstate,

Ut := max{ [0, t] |Xu = Xt for all u [t , t]

}.


Instead of using a jump process (Xt )t0, some authors describe the random patternof states by a chain of jumps. The two concepts are equivalent.

2.1 The semi-Markovian approach

The random pattern of states (Xt )t0 is called semi-Markovian, if the bivariate pro-cess (Xt ,Ut )t0 is a Markovian process, which means that for all i S , u 0, andt tn t1 0 we have

P((Xt ,Ut ) = (i, u)|Xtn,Utn, . . . ,Xt1,Ut1

) = P ((Xt ,Ut ) = (i, u)|Xtn,Utn)

almost surely. In the following we always assume that the initial state (X0,U0)is deterministic. (Note that U0 = 0 by definition.) In practice that means that weknow the state of the policyholder when signing the contract. With this assumptionand the Markov property of (Xt ,Ut )t0 we have that the probability distribution of(Xt ,Ut )t0 is already uniquely defined by the transition probability matrix

p(s, t, u, v) := (P(Xt = k,Ut v |Xs = j,Us = u))(j,k)S 2,

0 u s t < , v 0.Alternatively, we can also uniquely define the probability distribution of (Xt ,Ut )t0by specifying the probabilities

p(s, t, u) = (pjk(s, t, u))(j,k)S 2,

pjk(s, t, u) := P(T (s) t,XT (s) = k |Xs = j,Us = u

), j = k,

pjj (s, t, u) := P(T (s) t |Xs = j,Us = u

).

A third way to uniquely define the probability distribution of (Xt ,Ut )t0 is to specifythe cumulative transition intensity matrix

q(s, t) = (qjk(s, t))(j,k)S 2,

qjk(s, t) :=

(s,t]pjk(s,d,0)

1 pjj (s, ,0), 0 s t < .

(2.1)

If q(s, t) is differentiable with respect to t , we can also define the transition intensitymatrix

(t, t s) := ddt

q(s, t) =( d

dt pjk(s, t,0)1 pjj (s, t,0)

)

(j,k)SS,

which is some form of multistate hazard rate. The quantity jk(t, t s) gives therate of transitions from state j to state k at time t given that the current duration ofstay in j is t s. If the transition intensity matrix exists, it also already uniquelydefines the probability distribution of (Xt ,Ut )t0. In order to obtain the transition


probabilities from the cumulative transition intensities, we can use the Kolmogorovbackward integral equation system

pik(s, t, u, v) 1{i=k} 1{vu+ts} =

j :j =i

(s,t]pjk(, t,0, v) qij (s u,d)

+

(s,t]pik(, t, u + s, v) qii(s u,d).

(2.2)

If the cumulative transition intensity matrix q(t, s) is regular (the matrix elementsqjk(s, ), j = k, are monotonic nondecreasing and right continuous functions thatare zero at time s, qjj = k =j qjk , qjj (s, t) qjj (s, t) 1, and in case ofqjj (s, t0) qjj (s, t0) = 1 the function qjj (s, ) is constant from t0 on), then therealways exists a corresponding semi-Markovian process ((Xt ,Ut ))t0, which is evenstrong Markovian (see Helwich 2008, Remark 2.34).

In the insurance literature the semi-Markov model based on transition intensitieswas first described by Hoem (1972). The more general model based on cumulativeintensities was introduced by Helwich (2008).

2.2 The Markovian approach

If durational effects are negligible, we can simplify the semi-Markovian model to thespecial case where (Xt )t0 on its own is a Markovian process, which means that forall i S and t tn t1 0 we have

P(Xt = i|Xtn, . . . ,Xt1) = P(Xt = i|Xtn)almost surely. As a consequence, the transition probabilities pjk(s, t, u, v) and theprobabilities pjk(s, t, u) do not depend on u or v anymore, the cumulative transitionintensity matrix q(s, t) is constant with respect to its first argument s, and the transi-tion intensity matrix (t, u) is constant with respect to the second argument u. TheKolmogorov backward integral equation system reduces to

pik(s, t) 1{i=k} =

j

(s,t]pjk(, t) qij (d). (2.3)

In the insurance literature the Markov model based on transition intensities jk(t)and a differentiable discounting function was described by Sverdrup (1965) andHoem (1969). The more general cumulative intensity approach was introduced byMilbrodt and Stracke (1997).

Compared to the semi-Markovian approach, the Markovian approach reduces di-mensionality and is technically much easier to handle. In order to make up for the lossof durational effects, Amsler (1968) proposed a method that allows one to reintegratedurational effects. The idea is to split all states i with durational effects into a set ofstates i1, i2, . . . , in with the meanings


ik = insured in state i with a duration between k 1 and k for k = 1, . . . , n 1, in = insured in state i with a duration greater than n 1.For example, for disability policies we have the well-known effect that for differ-ent durations since disability inception the observed recovery and mortality ratesdiffer. Instead of using the three-state model S = {a, i, d} of Example 2.1 in asemi-Markovian setup, we could use the multistate model S = {a, i1, . . . , in, d} ina Markovian setup.

We have to specify q(t) in such a way that the transitions (i1, i2), . . . , (in1, in) oc-cur with probability one at the turn of the years. Durational effects are taken intoaccount by using different recovery and mortality rates for the states i1, . . . , in. Forexample, in the Netherlands this approach is commonly used with parameter n = 6,where i6 contains all policyholders that have been disabled for more than five years;see Gregorius (1993).

In this paper we will base all our actuarial calculations on the cumulative transitionintensity matrix q(t) and the probabilities pjj (s, t), where the latter can be derivedfrom the former by the exponential formula

1 pii(s, t) = eqcii (t)qcii (s)

s


(b) We estimate pjk(s, t) from statistical data, but only at integer times s, t . We gen-erally assume that transitions can only occur at the turn of the years. Then from(2.1) we obtain that q(dt) = 0 for non-integer times t , and from (2.3) we get

qjk(n) qjk(n) :={pjk(n 1, n): j = kpjj (n 1, n) 1: j = k

for integer times n.In order to consider durational effects in the way that Amsler (1968) proposed,

the probabilities pjk(n 1, n) are sometimes provided by select-and-ultimate ta-bles (cf. Bowers et al. 1997, Sect. 3.8). These tables generally contain annual ratesthat depend on the age at selection (e.g. onset of disability) and on the durationsince selection. Usually, beyond a certain period such as five years, the depen-dence on the time since selection is neglected and the corresponding transitionprobabilities are combined, resulting in the ultimate table.

(c) Similarly to (b), we estimate the pjk(s, t) from statistical data at integer timess, t . But differing from (b), we distribute the mass qjk(n) qjk(n 1) equallyon the interval (n 1, n] by defining

jk(t) :={pjk(n 1, n): j = k, t (n 1, n]pjj (n 1, n) 1: j = k, t (n 1, n]

for integer times n.(d) We estimate the pjk(s, t) from statistical data at integer times s, t . For times in

between we use linear interpolation. See Helwich (2008, p. 93).

2.3 Other approaches

More realistic and sophisticated models can be built by considering, for example,

the total number of jumps into a certain state, the dependence on the total time spent in some states since policy issue, stressing

the health history of the insured.

Both approaches make the actuarial models technically much more demanding. An-other interesting approach was given by Davis (1984). However, reasonable modelscan be defined by considering just the Markovian approach in combination with themethod of Amsler (1968).

3 The health insurance contract

As already mentioned in the introduction, premiums and benefits are paid accordingto the pattern of states of the policyholder.

Definition 3.1 (contractual payments) Payments between insurer and policyholderare of two types:


(a) Lump sums are payable upon transitions between two states and are specifiedby deterministic nonnegative functions bjk(t, u) with bounded variation on com-pacts. The amount bjk(t, u) is payable if the policy jumps from state j to state kat time t and the duration of stay in state j was u. In the Markovian approach theparameter u plays no role, and we write bjk(t, u) = bjk(t).

In order to distinguish between payments from insurer to insured and viceversa, benefit payments get a positive sign and premium payments get a negativesign.

(b) Annuity payments fall due during sojourns in a state and are defined by deter-ministic functions Bj (s, t), j S . Given that the last transition occurred at times, Bj (s, t) is the total amount paid in [s, t] during a sojourn in state j . We as-sume that the Bj (s, ) are right continuous and of bounded variation on com-pacts. In the Markovian approach the parameter s plays no role, and we writeBj (s, t) = Bj (t).

Example 3.2 (disability insurance) Suppose we have a contract that pays a continuousdisability annuity with rate r > 0 as long as the insured is in state disabled, anda constant regular premium p > 0 has to be paid yearly in advance as long as thepolicyholder is in state active. Following a modeling framework of Pitacco (1995), let (n1, n2) 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 incep-

tion), and let T ( n2 + f ) be the stopping time of annuity payment (from policy issue

at time zero).Then the contractual payment functions are

Ba(s, t) =n21

k=0p 1[k,)(t),

Bi(s, t) = 1(n1,n2)(s) min{t,s+f+m,T }

min{t,s+f }r du,

Bd(s, t) = 0

for all 0 s t . The transitions payments bjk(t, u) are all constantly zero. Bi(s, t) isthe only payment function that depends on the duration parameter s. In order to putthe contract into a Markovian framework according to the ideas of Amsler (1968),we split the state disabled i into a finite series of states i0, i1, i2, . . . that representthe number of time periods (e.g., years) that the policyholder has already been instate disabled. The payment functions Bil (t), l = 0,1,2, . . . are either constantlyzero or equal to Bil (t) = r min{t, T } depending on whether t l is between n1 andn2, whether l is greater than f , and whether l is smaller than f + m.


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 con-stant regular premium p > 0 has to be paid yearly in advance as long as the policy-holder is in state active. A lump sum is payed upon death or a dread disease diagnosis,whichever occurs first. The death benefit is ld . The dread disease benefit is li , but onlyif the policyholder outlives a deferred period f , otherwise the sum ld is paid. We as-sume that a recovery from state i to state a is impossible. The contractual paymentfunctions are

Ba(s, t) =T 1

k=0p 1[k,)(t), Bi(s, t) = li 1(0,T f ](s)1[s+f,)(t),

Bd(s, t) = 0,bad(t, u) = ld 1(0,T ](t), bai(t, u) = 0, bid(t, u) = ld 1(0,T ](t)1[0,f )(u),

for all 0 s t and u 0. In order to put the contract into a Markovian frameworkaccording to the ideas of Amsler (1968), we have to split the state ill i into two statesof illness that distinguish between policyholders that are in the deferred period andpolicyholders that have already outlived the deferred period.

Example 3.4 (long-term care insurance) Suppose we have a German policy with statespace S = {a = active/healthy, cI = need for basic care, cII = need for medium care,cIII = need for comprehensive care, l = lapsed/canceled, d = dead}. Continuous an-nuities are paid with rates rI , rII , rIII as long as the insured is in state cI , cII , cIII ,respectively. A constant regular premium p > 0 has to be paid yearly in advance untilretirement at contract time R, lapse, or death, whichever occurs first. The contractualpayment functions do not depend on duration of stay and have the form

Ba(t) =R1

k=0p 1[k,)(t),

BcI/II/III (t) = t rI/II/III ,Bl(t) = 0, Bd(t) = 0

for all t 0. The transitions payments bjk(t) are all constantly zero.

Example 3.5 (German private health insurance) As already mentioned above, Ger-man health insurers calculate on the basis of deterministic benefit forecasts that onlydepend on the age of the policyholder. This deterministic approach is justified by theargument that, though the individual medical expenses are far from being determin-istic, the portfolio average of medical expenses is nearly deterministic in the case ofa large portfolio of independent insureds because of the law of large numbers. Thelaw of large numbers does not work for systematic changes of medical expenses withprogression of calendar time, but German health insurers have the exceptional rightto adapt their deterministic forecasts of the medical expenses if calculated values and


real values differ significantly. (The special right of German and Austrian health in-surers to change the payments functions after signing of the contract is unique inprivate health insurance.) Let the average yearly medical expenses (Kopfschden)of a policyholder with age x + k be a deterministic quantity Kx+k , where x is theage of the policyholder at inception of the contract. Recall that the state space isS := {a = alive, l = lapsed/canceled, d = dead}. A constant regular premium p > 0has to be paid yearly in advance till death or lapsation. The contractual paymentfunctions do not depend on duration of stay and are of the form

Ba(t) =

k=0(Kx+k p)1[k,)(t),

Bl(t) = 0, Bd(t) = 0for all t 0. The transitions payments bjk(t) are all constantly zero.

3.1 Reserves

By statute the insurer must at any time maintain a reserve in order to meet futureliabilities in respect of the contract. This reserve bears interest with some rate (t).On the basis of this interest rate we define a discounting function,

v(s, t) := e ts (r)dr . (3.1)

We can interpret v(s, t) as the value at time s of a unit payable at time t s. Next,we study the present value of future payments between insurer and policyholder, thatis, the discounted sum of all future benefit and premium payments,

A(t) :=

jS

n=0

(t,)v(t, )1{X=j} 1{Sn


that cause tremendous capital costs that make the insurance business unprofitable.Therefore, an insurer only reserves an amount of money that is sufficient to coverthe future liabilities with a probability close to 1 but not equal to 1. (In practice theshareholders of the insurer also allocate capital, helping to achieve a satisfactory levelof safety and allowing further reduction of the reserves.) Mathematically, that meansthat we are looking for deterministic quantities V i (t, u) that satisfy

P(A(t) V i (t, u)|(Xt ,Ut ) = (i, u)

) (3.4)

for some solvency level (0,1) that is close to one. At time t the insurer knowswhich values the process (Xs,Us)s assumes on [0, t], and he makes use of this in-formation. However, since (Xs,Us)s satisfies the Markov property, only the state of(Xs,Us)s at time t is relevant. In the case of the Markovian approach where theduration of stay plays no role, (3.4) reduces to

P(A(t) V i (t)|Xt = i

) . (3.5)

If the probability distribution of A(t) is completely known, and if we have a homoge-neous portfolio of stochastically independent policies, then the law of large numbersyields that the average present value per policy is close to its mean. (An illustrationof this fact can be found in Pitacco and Olivieri 2012 for the two-state model withstates active and dead.) This motivates the following definition.

Definition 3.6 The prospective reserve at time t in state (i, u) is defined by

Vi(t, u) := E(A(t)|(Xt ,Ut ) = (i, u)

),

given that the expectation exists.

As already mentioned after (3.4), at time t we do not only know the value of(Xt ,Ut ) but also the complete history of (X ,U )0 up to time t . But since weassumed that (X ,U )0 is Markovian, in the above definition we just condition on(Xt ,Ut ). In the case of the Markovian approach where the duration of stay plays norole, we define

Vi(t) := E(A(t)|Xt = i

).

Proposition 3.7 The prospective reserves Vi(t, u), i S , 0 u t , equal almostsurely the unique solution of the integral equation system

Vi(t, u) =

(t,)v(t, )

(1 pii(t, , u)

)Bi(t u,d)

+

j :j =i

(t,)v(t, )

(1 pii(t, 0, u)

)(bij (, t + u) + Vj (,0)

)

qij (t u,d). (3.6)


If the transition rates and the actuarial payments do not depend on duration of stay,the integral equation system can be reduced to

Vi(t) =

(t,)v(t, )

(1 pii(t, )

)Bi(d)

+

j :j =i

(t,)v(t, )

(1 pii(t, 0)

)(bij ( ) + Vj ()

)qij (d). (3.7)

The proposition provides a method for the calculation of Vi(t, u). A proof can befound in Helwich (2008). The integral equation system (3.6) is also denoted as Thieleintegral equations (of type 1). From the proposition we see that the prospective re-serve is uniquely defined by the payment functions, the discounting function, and thecumulative transition intensities. Recall that the probabilities pii(t, , u) are alreadyuniquely defined by the cumulative transition intensities. Later on we will see theprospective reserve as a functional of the cumulative transition intensity matrix q anduse the intuitive notation

Vi(t, u) = Vi(t, u;q). (3.8)

Remark 3.8 (discrete time approach) Similarly to approach (b) in Sect. 2.2, let tran-sitions between different states only happen at integer times. Then q(s,d) is zeroat non-integer times. Suppose that also the Bi(s,d) are zero at non-integer times,which means that payments during sojourns in a state are paid either yearly in ad-vance or yearly in arrears. As transitions shall only occur at the turn of the years, thatdoes not mean a loss of generality. Let T < be the contract term. Then we canrewrite (3.6) as

Vi(n,m) =T

=n+1v(n, )

(1 pii(n, ,m)

)(Bi(, n m) Bi(, n m)

)

+

j :j =i

T

=n+1v(n, )

(1 pii(n, 0,m)

)

(bij (, n + m) + Vj (,0))(

qij (n m,) qij (n m,))

for all integer times 0 m n. We obtained a recursion formula that can be easilysolved backwards starting from some n = T and making time steps of 1 until wearrive at t = 0. Recall that the probabilities pii(n, ,m) can be easily calculated withthe help of (2.4).

Remark 3.9 (absolute continuity approach) If not only the cumulative transition in-tensities are differentiable with respect to their second argument (the transition inten-sity matrix exists), but also the Bj (s, t) are differentiable in the second argumentwith derivatives bj (t, t s), then we can alternatively obtain the prospective reserves


Vi(t, u) by solving the system of partial differential equations

tVi(t, u) =

uVi(t, u) bi(t, u) + (t)Vi(t, u)

j :j =i

(bij (t, u) + Vj (t,0) Vi(t, u)

)ij (t, u). (3.9)

See Helwich (2008, Theorem 4.11). In the special case of (3.7), (3.9) reduces to asystem of ordinary differential equations,

ddt

Vi(t) = bi(t) + (t)Vi(t)

j :j =i

(bij (t) + Vj (t) Vi(t)

)ij (t). (3.10)

Definition 3.10 (equivalence premium) The equivalence principle states that premi-ums should be chosen in such a way that at time zero (the beginning of the contract)the expected present value of future premiums should equal the expected presentvalue of future benefits. In mathematical terms that is just the case if and only if

BX0(0,0) + VX0(0,0) = 0.(Recall that we assumed X0 to be deterministic, and that VX0(0,0) just contains thepayments strictly after time zero and, thus, we have to add BX0(0,0).)

The typical way to calculate the equivalence premium is as follows. First wechoose the benefits. Second, we choose a premium scheme such as a single premiumat the beginning of the contract, a yearly constant premium, or a regular premium thatincreases each year by some specified factor. The last step is then to find a scalingfactor for the premiums such that the equivalence principle is met.

4 Biometric insurance risk

Recall that at time t the present value A(t) is the true sum that an insurer needs atthat time in order to meet all obligations in respect of the contract. However, A(t) is afunctional of the random pattern of states of the policyholder and depends on randomevents in the future. In order to meet the future liabilities A(t), the insurer is obligedto hold a reserve V i (t, u) defined according to (3.4), given that (Xt ,Ut ) = (i, u).The fact that the reserve V i (t, u) may still be insufficient with a probability of up to1 , implies a risk. Unsystematic biometric risk. The probability distribution of the random pattern

of states of the policyholder is perfectly known, but payments between insurer andpolicyholder may deviate from expected values. (Pitacco and Olivieri 2012 use theterm process risk.)

However, in practice the probability distribution of the random pattern of states of thepolicyholders is usually not perfectly known.


Biometric estimation risk. The probability distribution of the random pattern ofstates of the policyholder is estimated from data, but because of the finite samplesize the estimated probabilities may deviate from the true probabilities.

Systematic biometric risk. The probability distribution of the random pattern ofstates of the policyholder is perfectly known for the past and the present, but theremay be systematic demographic changes that are not anticipated today.

For each of those three risks safety margins are added on reserves and premiums. Inlife insurance and health insurance pursued on a similar basis to life insurance, thecommon way to add safety margins is not to add them directly on reserves

V i (t, u) = Vi(t, u) + safety margin(and in turn on the equivalence premium), but to add implicit safety margins on topof q such that Vi(t, u;q) is a conservative reserve,

V i (t, u) = Vi(t, u;q) = Vi(t, u;q + safety margin).

In the actuarial literature, the reserve Vi(t, u) is commonly denoted as best estimatereserve, and the conservative reserve V i (t, u) is accordingly defined as the sum ofbest estimate reserve and safety margin.

Definition 4.1 (implicit safety margin) Let q be a best estimate for the cumulativetransition intensity matrix of the random pattern of states of the policyholder, andlet q be the cumulative transition intensity matrix that is used for reserving andpremium calculation. Then qij (u,dt) qij (u,dt) is called the implicit safety marginfor transition i to j at time t and current duration of stay t u.

In general it is not clear whether the safety margins should be positive or negative,that is, whether overestimating or underestimating of the transition rates is conserva-tive. In the following proposition we give a sufficient safe-side condition. It is wellknown in the literature in the case of the Markovian framework, but here it is pre-sented for the first time in the more general semi-Markovian framework.

Proposition 4.2 Suppose that the cumulative transition intensities qij and qij haverepresentations of the form

qij (s, t) =

(s,t]ij (, s)dij ( ),

qij (s, t) =

(s,t]ij (, s)dij ( ).

(4.1)

We define the sum at risk for a transition from state i to state j at time and currentsojourn time v by

Rij (, l) := bij (, l) + Vj (t,0) + Bj (, ) Vi(, l) (Bi( l, )

Bi( l, )),


where we use the convention Bi(, ) := 0. IfRjk(, l) 0 jk(, l) jk(, l) (4.2)

for all j, k S , j = k, 0 l , then we haveVi

(t, u;q) Vi(t, u;q), i S, 0 u t.

The proof is given in the Appendix. The sum at risk Rij (, l) gives the mean coststhat a transition of a policyholder from state i to state j at time t and current so-journ time l causes for the insurer. Though the proposition tells us the sign that theimplicit safety margins should have, we still do not know how to choose their abso-lute values. A possible solution is to define q by confidence estimations. However,there are infinitely many solutions, and the confidence level that we choose for q isnot necessarily similar to the confidence levels that result for premiums and reserves.Moreover, a simple confidence estimation for q without considering the risk struc-ture of the insurance contract may lead to a distribution of the total safety margin toimplicit safety margins that does not necessarily reflect the significance of the differ-ent risk sources. In the following we discuss that problem and possible solutions inmore detail.

Decomposition of risk For mathematical technical reasons, we generally limit our-selves to the Markovian approach according to Sect. 2.2, recalling that the methodof Amsler (1968) still allows us to consider durational effects. Further, let T > 0be a finite time horizon after which there are no payments anymore. Suppose wehave a homogeneous portfolio of n N policyholders with corresponding countingprocesses Npij (t), p {1, . . . , n}, that are all distributed according to q . The NelsonAalen estimator for qij is defined as

Q(n)ij (t) :=

(0,t]1

np=1 1Xpu=i

d(

n

p=1N

pij (u)

)

; (4.3)

see Andersen et al. (1991, Sect. IV.1). According to Christiansen (2011), we have

A(n)(t) :=n

p=1Ap(t) =

n

p=1VXpt

(t;Q(n)).

Recall that A(n)(t) is the true sum that the insurer needs at time t in order to meetall obligations in respect of the portfolio. However, Q(n) depends on future eventsand is not known today. Let q0 be the cumulative transition intensity matrix of aformer generation of m N policyholders whose transitions happened in the pastand are known today. Let q(m) be the NelsonAalen estimator for q0. Then we candecompose Q(n) to

Q(n) = (Q(n) q) + (q q0) + (q0 q(m)) + q(m).


Simply transferring our past experience to the future, the last addend q(m) on theright-hand side can be seen as a crude estimation for Q(n). The difference Q(n) q(m)is the biometric insurance risk, which we decompose to unsystematic biometric riskQ(n) q , systematic biometric risk q q0, and biometric estimation risk q0 q(m).In the same way we can decompose A(n)(t) to

A(n)(t) =n

p=1VXpt

(t;Q(n))

=n

p=1

(VXpt

(t;Q(n)) VXpt (t;q)

) +n

p=1

(VXpt

(t;q) VXpt(t;q0))

+n

p=1

(VXpt

(t;q0) VXpt

(t;q(m))) +

n

p=1VXpt

(t;q(m)),

where the four addends on the right-hand side are interpreted as unsystematic biomet-ric risk, systematic biometric risk, biometric estimation risk, and crude estimation forA(n)(t), respectively. The actuarial task is now to add implicit safety margins on thecrude estimation q(m) such that the resulting q leads to reserves and premiums thatcover the true liabilities A(n)(t) with a probability close to one.

As the safety margins lead to premiums that are exaggerated and not seen as fairanymore, a common practice is to compare the conservative premiums with the pre-miums that would have been adequate in retrospect and to partly refund the part ofthe premium that was overpaid. The sum that was overpaid is called surplus, andthe part of the surplus that is refunded is called bonus. This refunding does not onlyhappen at the end but also during the contract period. As q implicitly defines thesurplus, bonus payments once made cannot be called back, and premiums cannot beraised if the surplus of the insurer turns to a loss, a risk adequate choice of q is avery important. An exception is the German private health insurances, where the reg-ulatory regime allows one to adapt premiums under specified circumstances. Thesecircumstances are given if the average yearly medical expenses or the mortality rateschange significantly. However, a German health insurer still needs carefully chosensafety margins, since the regulatory regime does not allow premium adaptations atthe discretion of the insurer but imposes sharp restrictions. Only if the implicit safetymargins reflect the true risks of the contract, can we be sure that the calculation ofsurplus is reasonable.

Remark 4.3 The total safety margin for a policyholder in state i S at time t isVi(t;q)Vi(t;q(m)). By using a generalized Taylor expansion, Christiansen (2011)showed that the total safety margin is asymptotically equal to

Vi(t;q) Vi

(t;q(m)) =

(j,k)S

(0,T ]v(0, )p(m)ij (0, )R

(m)jk ( )d

(qjk q(m)jk

)( )

+ o(q q(m)), (4.4)


where the p(m)ij (0, ) are the transition probabilities that correspond to q(m),

R(m)jk ( ) := bjk( ) + Vk

( ;q(m)) + (Bk() Bk()

)

Vj( ;q(m)) (Bj () Bj ()

) (4.5)

is the sum at risk according to Proposition 4.2 but in the Markovian framework, and Tis the time horizon of the insurance contracts. That means that the total safety margincan be asymptotically decomposed to a weighted sum of the different implicit safetymargins d(qjk q(m)jk )( ) with weights v(0, )p(m)ij (0, )R(m)jk ( ). The sign of theweights is consistent with condition (4.2).

In order to keep the notation simple, in the following we say that the present isalways time zero. General results for arbitrary times t 0 can be easily derived bydoing a time shift and by using the (at time t available) information about the actualstates of the policyholders at t .

4.1 Implicit safety margins for the unsystematic risk

We now solely focus on the risk that the true transitions Q(n) deviate from the the-oretical rate q , where the latter shall have an intensity . What amount of implicitsafety margin dq dq shall we add on q in order to cover the risk

n

p=1

(VXp0

(0;Q(n)) VXp0 (0;q)

)?

Given that the intensity of q exists, from the Delta Method we get

n(Q

(n)jk qjk

) d Ujk, (4.6)

where the Ujk are independent standard Gaussian processes; see Andersen et al.(1991, Example IV.1.9). That means that the differences dQ(n)jk ( ) dqjk( ), whichdescribe the uncertainty in the number of transitions from j to k at time , are asymp-totically independent and normally distributed. By using the asymptotic expansion(4.4) and applying the Delta Method again, Christiansen (2011) showed that

1n

n

p=1

(VXp0

(0;Q(n)) VXp0 (0;q)

)

d

(j,k)S

n

p=1

(0,T ]v(0, )pXp0 j (0, )Rjk( )dUjk( ), (4.7)

where the transition probabilities pXp0 j (0, ) and the sum at risks Rjk( ) (see (4.5))are calculated on the basis of q . The (uncountably many) addends on the right-hand


side,n

p=1v(0, )pXp0 j (0, )Rjk( )dUjk( ), (j, k) J, (0, T ], (4.8)

can be interpreted as the contributions that the uncertainties in the transitions (j, k) attimes make to the total unsystematic biometric risk. Note that they are all stochas-tically independent since the increments dUjk( ) for different transitions (j, k) anddifferent times are all stochastically independent. If we measure the unsystematicrisk by using the variance as risk measure, which makes sense here since we are closeto normality and the variance contains all risk information in the case of a normal dis-tribution, the independence property of the addends in (4.8) leads to an asymptoticallyadditive decomposition of the unsystematic risk with respect to different transitionsat different times,

Var

(1n

n

p=1

(VXp0

(0;Q(n)) VXp0 (0;q)

))

n

(j,k)S

(0,T ]Var

(n

p=1v(0, )pXp0 j (0, )Rjk( )dUjk( )

)

;

see Christiansen (2011). We now group the portfolio to subportfolios where the poli-cyholders have the same initial states Xp0 and focus on just one of the subportfolios.Without loss of generality let {1, . . . , n1}, n1 n, be the policyholders that have theinitial state X10. The total unsystematic biometric risk of that subportfolio has theform

n1

(VX10

(0;Q(n1)) VX10 (0;q)

)

and is asymptotically normally distributed with zero mean and variance

2X10

:=

(j,k)S

(0,T ]pX10j

(0, ) (v(0, )Rjk( ))2 dqjk( ). (4.9)

In order to let the distribution of the total safety margin to implicit risk marginsbe risk-oriented, Christiansen (2011) suggested that the contribution of the implicitsafety margin dqjk( ) dqjk( ) to the total safety margin should be proportionalto the contribution of the uncertainty in dQ(n1)jk ( ) dqjk( ) to the total variance,at least asymptotically. Then the asymptotic variance decomposition of above, theasymptotic expansion (4.4), and the confidence condition

P

(n1

p=1

(VXp0

(0;Q(n1)) VXp0 (0;q)

) n1

p=1

(VXp0

(0;q) VXp0 (0;q)

))

=

P (VX10(0;Q(n1)) VX10

(0;q)) =


imply that we asymptotically should define q by (cf. Christiansen 2011)

dqjk(t) =(

1 + u 2X10

1n1

v(0, t)Rjk(t))

dqjk(t), (j, k) J, t (0, T ],

(4.10)where u is the -quantile of the standard normal distribution. The (asymptotic) con-fidence estimation (4.10) is consistent with condition (4.2); moreover, not only doesit give the proper sign of the implicit safety margins, but it also gives an adequateabsolute value.

4.2 Implicit safety margins for the estimation risk

We now focus on the risk that the estimator q(m) deviates from its theoretical value q0,where the latter shall have an intensity 0. We see q(m) as the theoretical estimatorthat is random and write q(m) for a realization of q(m) for some given data. As q0is unknown, we have to calculate on the basis of q(m), but we want to add a safetymargin of q on q(m) in order to cover the risk

n

p=1

(VXp0

(0;q0) VXp0

(0;q(m))). (4.11)

The insurance literature offers various statistical methods for the estimation of (past)transition rates from data. For the interested reader we suggest Haberman and Pitacco(1999, Sect. 4) and Milbrodt and Helbig (1999, Sect. 3.F). The common way to defineimplicit safety margins for the estimation risk is to estimate confidence bands ljk andujk for the empirical rates q(m)jk ,

P(dljk(t) dq(m)jk (t) dujk(t), t 0, (j, k) J

) , (4.12)

and then to choose q such that dqjk(t) = dq(m)jk (t) + dqjk(t) is equal to the loweror upper bound, whatever is to the disadvantage of the insurer (cf. Proposition 4.2).Two problems arise with this method. First, the confidence level in (4.12) generallydiffers from the confidence level that we finally obtain for premiums and reserves.Second, there are infinitely many confidence estimates that satisfy (4.12) but that leadto different surpluses and bonuses. We now present a new method for the calculationof implicit safety margins for biometric estimation risk that is able to cope with thosetwo problems. The two main ideas are (1) to use a top-down approach of the form

P(VX0

(0;q0) VX0

(0;q(m) + q)) = (4.13)

instead of (4.12) and (2) to apply the asymptotic risk decomposition of Christiansen(2011) according to the previous section.

Because of (4.4) the total biometric estimation risk (4.11) for m to infinity can beasymptotically decomposed to a (uncountable) sum with addends

nv(0, t)p0X0j (0, t)R0jk(t)d(q0jk q(m)jk

)(t), t 0, (j, k) J,


that give the contribution that the uncertainty in transition (j, k) at time t makes tothe total randomness. (The superscript 0 on p0X0j (0, t) and R0jk(t) signifies that wecalculate on the basis of q0.) An implicit safety margin of q on the estimation q(m)leads to a total safety margin of

n(VX0

(0; q(m) + q) VX0

(0; q(m)))

that can also be asymptotically decomposed with the help of (4.4). The addends areof the form

nv(0, t) p(m)X0j (0, t) R(m)jk (t)dq

(t), t 0, (j, k) J, (4.14)

where p(m)X0j (0, t) and R(m)jk (t) are calculated on the basis of q(m). In order to let

the implicit safety margins be risk adequate, we use the asymptotic variance propor-tionality principle of the previous section, which means here that the contribution ofdqjk(t) to the total safety margin shall be asymptotically proportional to the contri-bution that the uncertainty in d(q0jk q(m)jk )(t) makes to the total variance, i.e.,

nv(0, t)p(m)X0j (0, t) R(m)jk (t)dq

(t) = nCmm

p0X0j (0, t)(v(0, t)R0jk(t)

)2 dq0(t)(4.15)

with proportionality factor Cm > 0. (The left-hand side follows from (4.14), and theright-hand side is a consequence of (4.9).) However, in practice q0 is unknown, andp0X0j (0, t) and R0jk(t), which are calculated on the basis of q0, are unknown aswell.

Proposition 4.4 For all X0 S , (j, k) J , and T > 0 we have

(0,T ]p

(m)X0j

(0, t)(v(0, t)R(m)jk (t))2 dq(m)(t)

P

(0,T ]p0X0j (0, t)

(v(0, t)R0jk(t)

)2 dq0(t).

The proof is given in the Appendix. Because of Proposition 4.4 we may rewrite(4.15) as

nv(0, t) p(m)X0j (0, t) R(m)jk (t)dq

(t) = n Cmm

p(m)X0j

(0, t) (v(0, t) R(m)jk (t))2 dq(m)(t)

without violating the asymptotic proportionality principle. Rearranging the equation,we arrive at

dq(t) = Cmm

v(0, t) R(m)jk (t)dq(m)(t).

The proportionality factor Cm shall be chosen in such a way that the confidencecondition (4.13) is met, at least asymptotically. The implicit safety margin q was


constructed in such a way that the corresponding total safety margin asymptoticallyequals Cm times the variance of (4.11). Since (4.11) is asymptotically normally dis-tributed, (4.13) is asymptotically met if the factor Cm makes the total safety margin(asymptotically) equal to u times the standard deviation of (4.11), where u is the-quantile of the standard normal distribution. Hence, we set Cm = u (mX0)1with X0 given by (4.9) but calculated on the basis of q0. As q0 is unknown in prac-tice, we apply Proposition 4.4 and substitute X0 by the asymptotically equivalentquantity

(m)X0

:=(

(j,k)S

(0,T ]p

(m)X0j

(0, ) (v(0, ) R(m)jk ( ))2 dq(m)jk ( )

)1/2.

All in all, we obtain that

dq(t) = um

(m)X0

v(0, t) R(m)jk (t)dq(m)(t) (4.16)

defines an implicit safety margin that asymptotically satisfies confidence condition(4.13) and the asymptotic variance proportionality principle.

Example 4.5 (critical illness insurance) Consider a critical illness insurance that paysa lump sum either in case of death or the first time the policyholder is diagnosed witha disease specified by the policy conditions, whichever occurs first. The male policy-holder starts at age 30 from state a = active, and the contract ends at age 65. In ourstate space S = {a, i, d} only the transitions (a, i) and (a, d) are of interest becausethe lump sum benefit is paid not more than once. Suppose that transition benefits ofbai(t) = 10 000 and bad(t) = 5 000 are paid for transitions (a, i) and (a, d). Let q0be given by

0ai(t) = 0.0025 + 0.000004 100.065 t ,0ad(t) = 0.0005 + 0.000075858 100.038 t .

Let the interest intensity be (t) = ln(1.0225), which corresponds to a yearly interestrate of 2.25 %. The premium shall be paid continuously with intensity ddt Ba(t) =128.28, which was chosen such that the equivalence principle holds at age 30 un-der q0. We simulated a homogeneous portfolio of m = 1 000 independent policy-holders distributed according to q0. Figures 1 and 2 show the theoretical cumula-tive transition intensities q0ad and q

0ai and their NelsonAalen estimates q

(1000)ad and

q(1000)ai . The estimates are quite close to the theoretical values, but we also see clear

differences which indicate that we might have a significant biometric estimationrisk. Figure 3 shows the theoretical prospective reserve Va(t;q0) and the empiri-cal prospective reserve Va(t; q(1000)). The empirical reserve is the amount that theinsurer actually holds if calculating on the basis of q(1000). The theoretical reserveis the amount that the insurer should hold but that is unknown in practice. In or-der to be on the safe side, an implicit safety margin q shall be added on q(1000)such that Va(30; q(1000) + q) is not smaller than Va(30;q0) with a probability of


Fig. 1 Theoretical cumulativemortality transition intensityq0ad

(line) and its NelsonAalenestimate q(1000)

ad(dots)

Fig. 2 Theoretical cumulativeillness transition intensity q0

ai(line) and its NelsonAalenestimate q(1000)

ai(dots)

= 99 %. (Note that t = 30 is the beginning of the contract.) We can achieve thatlevel of safety (at least asymptotically) by defining q by (4.16). Figure 5 shows therelative difference of q(1000) + q and q(1000). Figure 4 shows the theoretical reserveVa(t;q0) and the empirical safe-side reserve Va(t; q(1000) + q). We see that the


Fig. 3 Theoretical andempirical prospective reservesVa(t;q0) (line) andVa(t; q(1000)) (dots)

Fig. 4 Theoretical andsafe-side prospective reservesVa(t;q0) (line) andVa(t; q(1000) + q) (dots)

estimated safe-side reserve Va(t; q(1000) + q) is significantly above the empiricalreserve Va(t; q(1000)) without safety margin, cf. Fig. 3.

The estimate (4.16) is based on two important assumptions. First, we assumedthat the left-hand side of Proposition 4.4 yields a good approximation for the right-hand side. In our example, the theoretical standard deviation a with respect to q0 is


Fig. 5 Relative differences(q

(1000)ad

+ q

ad

)/q(1000)ad

(lowerline) and(q

(1000)ai

+ q

ai

)/q(1000)ai

(upperline)

Fig. 6 Quantile-quantile plot ofthe distribution of Va(0;q(20))(y-axis) and its asymptoticapproximation according to(4.7) (x-axis)

3266.135 and its empirical estimate (1000)a is 3315.592, which means that the erroris just 1.5 % here. The second important assumption for (4.16) was that the left-handside of (4.7) is close to the normal distribution of the right-hand side of (4.7). Figure 6shows the distributions of the left- and right-hand sides of (4.7) in a quantile-quantileplot for a portfolio size of just m = 20. (The distribution of the left-hand side was


simulated with a sample size of 100 000.) We see that already for this very smallportfolio of m = 20 policyholders the approximation in (4.7) is astonishingly close.

4.3 Implicit safety margins for the systematic biometric risk

We now discuss the risk that the cumulative transition intensity matrix q of the ran-dom pattern of states of the policyholders differs from the cumulative transition in-tensity matrix q0 of a previous generation. History showed that the difference q q0can be significant. For example, the world life expectancy more than doubled over thepast two centuries (see Oeppen and Vaupel 2002), and disability and recovery ratesare affected by the rapid progress in medical treatment and fast-changing require-ments in the world of work. Since many health insurance policies have rather longcontract periods, these changes can imply a considerable difference between Vi(t;q)and Vi(t;q0), exposing the insurer to a systematic biometric risk.

Among actuaries there is an increasing awareness of the systematic biometric riskand a strong trend to model q q0 stochastically. Some examples are the time-dependent Gompertz approach of Milevsky and Promislow (2001), the stochasticPerks model of Cairns et al. (2006), the extension of the Lee-Carter model of Del-warde et al. (2007), or the forward model of Bauer et al. (2008). Further referencescan be found in Pitacco et al. (2009) and Pitacco and Olivieri (2012). Apart from thetwo-state model, where the only transition is from active to dead, very few studiesinvestigated time trends in transition rates for multistate actuarial models. Noticeableexceptions are Renshaw and Haberman (2000) and Christiansen et al. (2012). Ren-shaw and Haberman (2000) considered the sickness recovery and inception transitionrates, together with the mortality rates when sick, which form the basis of the UKcontinuous mortality investigation Bureaus multistate model. These authors identi-fied the underlying time trends from the observation period 19751994 using separatePoisson GLM regression models for each transition. Christiansen et al. (2012) con-sider a three-state disability model, using a multivariate Lee-Carter type model thatis fitted as described in Hyndman and Ullah (2007), that is, by means of a functionaldata approach.

However, systematic changes of transition rates are not very well understood sofar. On top of that, health insurance policies often have contract terms of severaldecades, and so q q0 has to be modeled quite far into the future. For the unsystem-atic risk and the estimation risk, we used a central limit theorem in order to approxi-mate the probability distributions of the risks. That works quite well as the sizes n andm of the present portfolio and the previous generation portfolio are relatively large inpractice and the independence assumption is quite reasonable. Using a nonparametricapproach, the model risk was negligible. For the systematic risk, the typical approachis to model the difference qq0 as a functional of yearly risk contribution factors thatrepresent the uncertainty in systematic changes from one year to another. The yearlyrisk contribution factors are assumed to be independent and identically distributed,which makes it possible to define estimators for their probability distribution (or justparameters of it). However, the sample size is limited by the number of years forwhich the insurance companies have reliable data. In practice the available samplesizes are often even smaller than the number of years that the forecasts reach into


the future. As a consequence, parametric approaches are frequently used. However,those parametric models are often not justified by medical or biological insight butare chosen according to mathematical tractability. This implies a huge model risk,and indeed we observe that the different models that can be found in the literatureproduce very different forecasts. Thus, there is a vast need for future research.

In order to gain more robustness with respect to model risk, a recent idea is touse worst-case techniques in combination with confidence sets. Instead of trying todescribe the stochastic process t q(t) in detail, which usually is based on manya priori assumptions that can hardly be verified, we just estimate a confidence bandM for q . This estimation may be based on a detailed model for t q(t), but theresulting M is less dependent on the a priori assumptions. The next step is then tofind the worst case in M ,

qworst := arg maxqM

VX0(0;q),

and to define implicit safety margins by qworst q0; see for example Christiansen(2010). The confidence level of M is a lower bound for the confidence level ofVX0(0;q) VX0(0;q0). The disadvantages here are that we cannot directly controlthe confidence level that the resulting premiums and reserves finally have and that theestimation of the confidence set M still comes with some model risk.

5 Conclusion

For many different types of health insurances, reasonable actuarial models can bebuilt by describing the medical history of the policyholder by a multistate jumpprocess. Usually the random pattern of states of the policyholder is assumed to beMarkovian or semi-Markovian. We have given an overview of common multistatehealth insurance models in the literature, basing our presentation on the comprehen-sive modeling framework of Helwich (2008).

The uncertainty in the future random pattern of states leads to unsystematic bio-metric risk, biometric estimation risk, and systematic biometric risk. In order to beable to meet future liabilities with a probability close to one, health insurers usuallyadd implicit safety margins on the transition probabilities. We showed which sign theimplicit safety margins should have in the general semi-Markovian model of Helwich(2008) by extending a classical result for the Markovian model. Within the Markovianframework, we presented a new method for the definition of implicit safety marginsfor biometric estimation risk and illustrated the concept by simulating an example.The idea is based on a concept of Christiansen (2011) for the unsystematic biometricrisk. Finally, we discussed the systematic biometric risk and pointed out unsolvedproblems that call for future research.

Acknowledgements The author thanks Hartmut Milbrodt from the University of Rostock, Hans-Joachim Zwiesler from the University of Ulm, and two anonymous referees for comments that have beenvery helpful for revising a previous version of the present work.


Appendix

Proof of Proposition 4.2 According to Helwich (2008, Chap. 4.D), the prospectivereserves Vi(t, u) can also be seen as the unique solution of the following Thiele inte-gral equation system (of type 2):

Vi(t, u) =

(t,)Bi(t u,d)

(t,)Vi(t, u + t) ( )d

+

j :j =i

(t,)Rij (, u + t) qij (t u,d). (A.1)

Let Wi(t, u) := V i (t, u) Vi(t, u). From (A.1) we get

Wi(t, u) =

(t,)Wi(, u + t) ( )d

+

j :j =i

(t,)(Wj(,0) Wi(,u + t)

)qij (t u,d)

+

j :j =i

(t,)Rij (, u + t)

(qij qij

)(t u,d).

For the third integral we write

j :j =i

(t,r]Rij (, u + t)

(qij qij

)(t u,d) =:

(t,r]Ci(t u,d).

By interpreting the Ci as synthetic annuity payments, we can see the integral equa-tion system for the Wi(t, u) as a Thiele integral equation system of the form (A.1)of a policy with cumulative annuity payments Ci , zero transition benefits, and a cu-mulative transition intensity matrix q. It has the unique solution (see Helwich 2008,Chap. 4)

Wi(t, u) =

(j,k)J

(t,)

[0,)v(t, )Rjk(, l)

(jk(, l) jk(, l)

)

pij (t, , u,dl)djk( ).

From this equation we can directly derive the result of the proposition.

Proof of Proposition 4.4 According to Andersen et al. (1991, Theorem IV.1.1), theNelsonAalen estimator q(m) is uniformly consistent on compact intervals,

q(m) q0 :=

(j,k)Jsup

0tT

q(m)jk (t) q0jk(t)

p 0.


We now want to show that the functional

F : q

(0,T ]pX0j (0, t;q)

(v(0, t)Rjk(t;q)

)2 dqjk(t)

is continuous with respect to the supremum norm, which will allow us to transfer theconsistency of the NelsonAalen estimator to the consistency statement of Proposi-tion 4.4. The transition probability matrix p(0, t;q) has a representation as a productintegral of the cumulative transition intensity matrix q; see Andersen et al. (1991).According to Gill (1994), the difference of two product integrals that correspond tointegrators q and q is bounded by Kq q for any q and q of uniformly boundedvariation on [0, T ]. The constant K > 0 depends on the uniform variation bound. Ascumulative transition intensities are monotone, the uniformly bounded variation forq and q is in our case equivalent to uniform boundedness in supremum norm. Thus,for all q and q with supremum norm bounded by some arbitrary but fixed M > 0, wehave

sup0tT

p(s, t;q) p(s, t; q KM q q, 0 s < t T .

Using that Vi(t;q) has a representation of the form

Vi(t;q) =

jS

(t,T ]v(t, s)pij (t, s;q)dBj (s)

+

(j,k)J

(t,T ]v(t, s) bjk(s)pij (t, s;q)dqjk(s),

see Helwich (2008), we can show that

sup0


of Rjk(t;q) and the bounded variation of bjk we get thatsup

0 )

= limmP

(F(q(m)

) F(q) > ,q(m) q0 > )

limmP

(F(q(m)

) F (q0) > ,q(m) q0 )

0 + limmP

(K

q(m) q0 > )

= 0,

where the constant K is chosen as K M with M = + q0 q(m).

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Multistate models in health insuranceAbstractIntroductionThe random pattern of states of the policyholderThe semi-Markovian approachThe Markovian approachOther approaches

The health insurance contractReserves

Biometric insurance riskDecomposition of riskImplicit safety margins for the unsystematic riskImplicit safety margins for the estimation riskImplicit safety margins for the systematic biometric risk

ConclusionAcknowledgementsAppendixReferences

multistate models in health insurance

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