Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach

© 2012 Hürlimann, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach


Introduction
The main theoretical goal of the present exposé is to extend the results presented in Hürlimann [1] to the Markov chain model of life insurance, which enables modeling all single life/multiple life traditional contracts subject to biometric risk with multiple causes of decrement. In particular, a complete risk modeling of single-life insurance products with mortality and disability risks requires the specification of a Markov model with three states. As novel illustration we offer to the interested practitioner an in-depth treatment of endowment contracts with waiver of premium by disability.
The present investigation is restricted to biometric risks encountered in traditional insurance contracts within a discrete time Markov chain model. The current standard requirements for the Solvency II life risk module have been specified in QIS5 [2], pp.147-163. QIS5 prescribes a solvency capital requirement (SCR), which only depends on the time of valuation (=time at which solvency is ascertained) but not on the portfolio size (=number of policies). It accounts explicitly for the uncertainty in both trends (=systematic risk) and parameters (=parameter risk) but not for the random fluctuations around frequency and severity of claims (=process risk). In fact, the process risk has been disregarded as not significant enough, and, in order to simplify the standard formula, it has been included in the systematic/parameter risk component. For the purpose of internal models and improved risk management, it appears important to capture separately or simultaneously all risk components of biometric risks. A more detailed account of our contribution follows.
As starting point, we recall in Section 2 the general solvency rule for the prospective liability risk derived in [1], Section 2, which has resulted in two simple liability VaR & CVaR target capital requirements. In both stochastic models, the target capital can be decomposed into a solvency capital component (liability risk of the current period) and a risk margin component (liability risk of future periods), where the latter must be included (besides the best estimate liabilities) in the technical provisions. This general decomposition is in agreement with the current QIS5 specification. The proposed approach is then applied to determine the biometric solvency risk capital for a portfolio of general traditional life contracts within the Markov chain model of life insurance. For this, we assume that the best estimate liabilities of a general life contract coincide with the so-called "net premium reserves". After introduction of the Markov chain approach to life insurance in Section 3, we recall in Section 4 the ubiquitous backward recursive actuarial reserve formula and the theorem of Hattendorff. Based on this we determine in Section 5 the conditional mean and variance of a portfolio's prospective liability risk (=random present value of future cashflows at a given time of valuation) and use a gamma distribution approximation to obtain the liability VaR & CVaR solvency capital as well as corresponding solvency capital ratios. These first formulas include only the process risk and not the systematic risk. To include the latter risk in solvency investigations we propose either to shift the biometric transition probabilities, as done in Section 6.2, or apply a stochastic model, which allows for random biometric transition probabilities, as explained in Section 6.3. Section 7 illustrates numerically and graphically the considered VaR & CVaR solvency capital models for a cohort of endowment contracts with waiver of premium by disability and compares them with the current Solvency II standard approach. Finally, Section 8 summarizes, concludes and provides an outlook for possible alternatives and extensions.

A general prospective approach to the liability risk solvency capital
Starting point is a multi-period discrete time stochastic model of insurance. Given is a time horizon T and a probability space   , , F P  endowed with a filtration   0 In a total balance sheet approach, their values depend upon the stochastic processes in B  , which describe the random cash-in and cash-out flows of any type of insurance business: 1 t P  : loaded premiums to be paid at time 1 t  (assumed invested at time 1 t  ) t X : insurance costs to be paid at time t (includes insurance benefits, expenses and bonus payments paid during the time period  1, t t   ) t R : accumulation factor for return on investment for the time period  1, We assume that t X is t F -measurable and t R is 1 t F  -measurable. The random cumulated accumulation factor for return over the period  , , 0 The actuarial liabilities at time t , also called timet prospective insurance liability, coincide with the random present value of all future insurance cash-flows at time t given by Using (1)-(2) and the relationship On the other hand, the random assets over which implies the following equivalent probabilistic conditions (use that trivially 0 Given a default probability 0   , the liability VaR solvency criterion (6) says that at time t the initial (deterministic) capital requirement t A should exceed the random present value of future cash-flows with a probability of at least 1   . By (4)-(5) this criterion automatically implies that assets will exceed liabilities with the same probability at each future time over the time horizon T . Let  be a minimum solution to (6), and assume that the best estimate insurance liabilities at time t coincide with the net premium reserves (in the sense defined later in (35), that is let represents the capital available at time t to meet the insurance risk liabilities with high probability. A risk margin is added to this capital requirement (recall that in Solvency II the sum of the best estimate insurance liabilities and the risk margin determines the Technical Provisions). The liability VaR target capital is the sum of the liability VaR solvency capital and the risk margin defined by .

VaR
VaR The cost-of-capital risk margin with cost-of-capital rate 6% where T denotes the time horizon, and f v is the risk-free discount rate. For comparison with other solvency rules, one considers the VaR solvency capital ratio at time t defined by / .
Alternatively, let The insurance loss random variable of a GLIFE contract is defined by This identifies the insurance loss with the random present value of all future cash-flows. Furthermore, for an arbitrary non-negative integer 0,1,...

 
, one defines the time- prospective loss random variable whose (conditional) expected value defines the time- actuarial reserve , .
The quantity k V  is called state-k time- actuarial reserve. In particular, one has 0 L L  and 0 V E L      is the initial actuarial reserve, which is not assumed to vanish.

Backward recursive reserve formula and the theorem of Hattendorff
In a first step, we derive a recursion formula for the actuarial reserves. Recall the recursion formula for the random prospective loss 1 .
Assume that the contract is in state . k S  at time  . Inserting (16) into (15) yields Using (12) the first expectation in (17) can be rewritten as The second expectation equals   The actuarial reserve at time  given the contract is in state k S Thiele's differential equation. Thiele's differential equation is a simple example of a Kolmogorov backward equation, which is a basic tool for determining conditional expected values in intensity-driven Markov processes, e.g. Norberg [9].
Proof. Making use of the recursion (18) and the relationship ( ) 1 The saving premium represents the expected change in actuarial reserve at time  for a contract in state k while the risk premium is the expected value at time  of a contract in state k needed to cover the insurance risk in time period  , 1 Rewrite the latter as This is the sum of the benefit payment at time  for a contract in state k and the probability weighted sum of the sums at risk   The sum at risk is the amount credited to the insured's contract upon a transition, namely the lump sum payable immediately plus the adjustment of the actuarial reserve. The obtained results constitute a discrete time version of those mentioned in Norberg [12], p.10.
To evaluate the mean and variance of the random insurance loss (13) The discrete time stochastic process   Y  is a martingale with respect to   S  . The martingale differences , represent the discounted one-year insurance losses and form a sequence of uncorrelated random variables such that Through detailed calculation one obtains the following result.

Theorem 4.2
The variance of the random insurance loss of a GLIFE contract is determined by the following formulas Proof. Similarly to Gerber et al. [15], formula (89), one has Using (15) one obtains which is (25). To obtain (26) one uses (12) and the convention (

The liability VaR & CVaR solvency capital for portfolios of GLIFE contracts
We begin with risk calculations for a single GLIFE contract, and use them to determine the liability VaR & CVaR solvency capital for a portfolio of GLIFE contracts.

Risk calculations for a single GLIFE contract
Given is a single GLIFE contract with random future cash-flows   k C defined by (12). We assume that the state space contains a unique distinguished "void" state   k X   meaning that the contract has terminated at time k . We assume contract survival, i.e. a contract is still alive at time of valuation t , which implies that the conditional event We note that the random present value of future cash-flows at time t defined by coincides with the timet prospective loss defined in (14), that is , . Therefore, the expected value given contract survival equals In contrast to (15) the reserve defined in (35) is state independent and called net premium reserve, see Bowers et al. [19], Chap.17.7, p. 500, for a special case. Following Section 2, this value can been chosen as best estimate of the contract liabilities.

Remarks 5.1 (i)
The motivation for state-independent reserves is second-to-die life insurance, where during lifetime the insurer may not be informed about the first death. An endowment with waiver of premium during disability, which is our illustration in Section 7, seems to contradict this concept because it cannot be argued that the insurer is unaware of the state occupied while the premium is being waived. However, at a given arbitrary time of valuation (including starting dates of contracts) future states of contracts are unknown, and therefore it is reasonable in a first step to assume state independent reserves for the design of a general method. Later refinement might be necessary to cover all possible cases.
(ii) State independent reserves have been introduced by Frasier [20] for the last-survivor status, see also The Actuary [21] and Margus [22]. The choice between state independent and state dependent reserves depends upon loss recognition in the balance sheet (recognition or not of a status change). With state independent reserves, the insurance company administers the contract as if it had no knowledge of any decrements, as long as the contract is not terminated. Only the latter situation is considered in the present work.
In a first step, we determine the mean and variance of the conditional distribution of t Z given t E . Similarly to [1], Section 5.1, the variance formulas (24)-(26) generalize to an arbitrary discrete time Noting further that , , one obtains from (36) the following conditional variance formulas (conditional version of Theorem 4.2).

Theorem 5.1
The conditional variance is determined by the following formulas As shown in the next Subsection, these formulas can be used to determine the target capital and solvency capital ratio of a portfolio of GLIFE contracts using appropriate approximations for the distribution of the random present value of future cash-flows associated to this portfolio under the condition that the contracts are still alive.

Solvency capital and solvency capital ratio for a portfolio of GLIFE contracts
Towards the ultimate goal of solvency evaluation for an arbitrary life insurance portfolio, we consider now a set of n policyholders alive at time t . From Section 3 one knows that the i -th contract is characterized by the following data elements: To the i -th contract one associates its random future cash-flows   (12), the corresponding ( ) i i t L timei t random prospective loss (14) and timei t net premium reserve The random present value of future cashflows of the portfolio is obtained by summing (34) over all contracts and is given by Similarly, summing the individual net premium reserves, one gets the portfolio reserve Following Section 2, one defines the portfolio VaR solvency capital as well as the portfolio CVaR solvency capital and the corresponding solvency capital ratios To determine these quantities it is necessary to determine the distribution of t Z conditional on contract survivals at time t , and under the assumption that the remaining lifetimes of all contracts are independent of each other. From Theorem 5.1 we have ( ) 2 0 1 , 1,..., , , 1,..., Based on the conditional mean and variance we approximate the distribution function of t Z by a gamma distribution as in [1], Section 5. Denote this approximation by Then, recalling the gamma distribution function, one has are the conditional mean and coefficient of variation of t Z (obtained from (43)-(44)). In this setting, the solvency capital ratio formulas (42) take the forms    denotes its probability density. The limiting results for a portfolio of infinitely growing size are similar to those in [1], Remark 5.1. If the coefficients of variation tend to zero, the gamma distributions converge to normal distributions and the solvency capital ratios converge to zero. This holds under the following assumption. Whenever insured contracts are independent and identically distributed, and if the portfolio size is large enough, then the ratio of observed state transitions to portfolio size is close to the given rates of transition with high probability. This assumption is related to the process risk, which describes the random fluctuations in the biometric transition probability matrix. However, if the ratio of observed state transitions to portfolio size is not close to the given rates of transition, even for large portfolio sizes, systematic risk exists, e.g. Olivieri & Pitacco [23], Section 2.1. In this situation, the rates of transition are uncertain and assumed to be random, and we consider stochastic models that include the process and systematic risk components. This is the subject of Section 6.3.

Comparing the standard approach with variants of the stochastic approach
Since the present Section has some overlap with [1], Section 6, it is treated more briefly, but can be read independently. Facts peculiar to the Markov chain approach are added whenever felt necessary. Recall that biometric risks in QIS5 accounts for the uncertainty in trends and parameters, the so-called systematic/parameter risk, but not for the process risk. We note that the solvency capital models of Section 5.2 only apply to the process risk. For full coverage of the process and systematic risk components, these solvency models are revised and extended. For this, we either shift the biometric transition probability matrix (see Section 6.2) or apply a stochastic biometric model with random biometric rates of transition (see Section 6.3). For completeness we briefly recall the QIS5 standard approach.

Solvency II standard approach
To value the net premium reserves a biometric "best estimate" life Similarly to the decomposition (7) the Solvency II target capital (upper index S2 in quantities) is understood as the sum of the SCR and a risk margin defined by where T denotes the time horizon, which may depend on the life policy, and f v is the riskfree discount rate. Since Solvency II uses a total balance sheet approach, the defined single policy quantities must be aggregated on a portfolio and/or line of business level. For comparison with internal models it is useful to consider the solvency capital ratio at time t under the Solvency II standard approach defined by the quotient 2 / .

Stochastic approach: Shifting the biometric transition probability matrix
Following the Sections 5.2 and 6.1, we consider the "shifted" random present value t Z  of future cash-flows of the portfolio at time t with conditional mean and variance The distribution of t Z  conditional on contract survivals at time t is again approximated by a gamma distribution denoted by 2 2 ( ) ( ; ), where the conditional mean and coefficient of variation , t t k    of t Z  are obtained from the formulas (55)-(56). Making use of (46) and (47) one sees that the portfolio VaR & CVaR solvency capitals under the shifted biometric transition probability matrix are given by the expressions 2 2 ( ), , , , 1,..., 1 , ; .
The observations in [1], Section 6.2, hold for the Markov chain model. By small coefficients of variation the gamma distributions converge to normal distributions, and the corresponding solvency capitals converge to those of normal distributions such that Asymptotically, the solvency capital ratios tend to the following minimum values , , By vanishing coefficients of variation the VaR & CVaR solvency capital ratios converge to the Solvency II solvency capital ratio. In this situation, the process risk has been fully diversified away, and, as expected, only the parameter/systematic risks remain.

Stochastic approach: Poisson-gamma model of biometric transition
For simplicity let us fix the states , i j of the transition probabilities   , 0,1,2,... ij p k k  . In case the ratio of observed state transitions to portfolio size is not close to the given rates of transition, even for large portfolio sizes, systematic risk exists. In this situation, the transition rates are uncertain and assumed to be random. This situation is modelled similarly to [1], Section 6.3. We assume a Bayesian Poisson-Gamma model such that the number of transitions is conditional Poisson distributed with a Gamma distributed random transition probability, which results in a negative binomial distribution for the unconditional distribution of the number of transitions. Then, we consider a Poisson-Gamma model with time-dependence of the type introduced in Olivieri & Pitacco [23], which up-dates its parameters to experience. Given is a fixed time t and biometric transition probabilities   , 0,1,2,... ij p k k  , for the given fixed states, which is based on an initial cohort of size t  at time t . Let It follows that the unconditional distribution of the number of transitions in the first time period is negative binomially distributed such that In contrast to the expected number of transitions ( ) t ij p t  predicted by the biometric transition probability matrix, one has ( ).
To model a systematic deviation from the expectation, one assumes that the quotient /   is different from one, for example greater than one for transitions produced by the mortality and disability risks and less than one for those produced by the longevity risk. Suppose that at time 1 t  , the number of transitions t d observed in the cohort over the first time period is available, and let , which implies a coefficient of variation for s t D  equal to 10%. One shows that the choice (72) with (1 )     implies that the transition probabilities (71) coincide with the corresponding shifted entries in the biometric life table. In this special case, we observe that the stochastic model of Section 6.3 provides the same results as the shift method of Section 6.2. In general, the stochastic model of Section 6.3 is more satisfactory and flexible because it allows the use of effective observed numbers of transitions as time elapses.

The endowment contract with waiver of premium by disability
For a clear and simple Markov chain illustration we restrict the attention to a single cohort of identical n -year endowment contracts with waiver of premium in the event of disability and fixed one-unit of sum insured payable upon death or survival at maturity date. The treatment of other similarly complex disability contracts is left to future research. For some further possibilities consult Example 2.1 in Christiansen et al. [24].

Markov model for mortality and disability risks
A complete risk model for single-life insurance products with mortality and disability risks requires the specification of a Markov model with three states. A policyholder aged x at contract issue changes state at time 0 t  according to the following diagram For a n -year endowment contract with waiver of premium by disability without recovery from disability, one has 0, 1,2,..., 1 x k r k n     . This simplifying assumption is sometimes made in practice and justified in economic environments with a small number of disabled persons, for which the probability of recovery can be neglected. For example, the Swiss Federal Insurance Pension applies such a model and uses a biometric life table called "EVK Table", where EVK is the abbreviation for "Eidgenössische VersicherungsKasse", e.g. Koller [6], p.129, or Chuard [25] for a detailed historical background.

State dependent actuarial reserves and net level premiums
The net level premium of the n -year endowment with waiver of premium and one unit of sum insured for a life in the active state at age x is denoted by ( : ) needs the survival probabilities of staying in the active or disabled state. Denote by  the maximum attainable age. Then, the active survival probabilities (probability a life in the active state at age x will attain age x k  in the active state without disablement) are given by Similarly, the disabled survival probabilities (probability a life in the disabled state at age x will attain age x k  in the disabled state without recovery) are given by   Corresponding to these survival probabilities one associates n -year life annuities for a life aged x being in the active or disabled state whose actuarial present values are defined by The actuarial present value (APV) of future benefits for the n -year endowment with waiver of premium and one unit of sum insured for a life in the active (respectively disabled) state at age x is denoted by ( : ) a A x n (respectively ( : ) i A x n ). Using the backward recursive formulas for the state dependent actuarial reserves let us determine formulas for the evaluation of the introduced APVs. In particular, an explicit formula for the net level premium is derived. The backward recursive reserve formulas are given by     One has 2 0 0 V  because the life is in the state "a" at contract issue, 2 1 0 V  because the life can only be in the state "i" after at least one year and then no actuarial reserve is available, and 3 0, 0,1,..., , because no actuarial reserve is required in case the insured life has died. Since actuarial reserves represent differences between APVs of future benefits and future premiums one has further the relationships

Numerical illustration
The Markov chain parameterization of the present contract type has been given at the beginning of Section 7.2. We assume that all the policyholders are aged x at time 0 t  . Our construction of the biometric life table with mortality and disability risk factors is based on the classical textbook Saxer [26], Section 2.5. Besides the one-year probabilities introduced in Section 7.1, one considers further the partial or independent rates of decrement, see Saxer [26], Section 2.4, or Bowers et al. [19], Section 9.5, denoted by * a x t q  : one-year independent rate of active mortality at time t * x t i  : one-year independent rate of disability at time t The independent rates of decrement are linked to the probabilities of active mortality and disability through the relationship, e.g. Saxer [26], formulas (2.5.1) and (2.5.2), For the purpose of illustration only and by lack of another reference, we base our calculations on Table 1, which is obtained by combining the Tables 4 and 5  . We use the shifted biometric life table with Solvency II standard like specifications, namely at each age 20% decrease for the probability to die as active (longevity risk) respectively 15% increase for the probability to die as disabled (mortality risk), 35% increase for the first year probability to disable and then 25% increase at each future age (disability risk). The interest rate and the risk-free interest rate is 3%. Table 2 displays shifted coefficients of variation under varying cohort sizes. The values are sufficiently small so that the normal approximation to the gamma distribution can be applied. Table 3, which is based on (60), displays the cohort size dependent solvency capital ratios and their limiting values (61) for a portfolio of infinitely growing size. The chosen confidence level is 99.5% for VaR and 99% for CVaR (the accepted level, which corresponds to a 99.5% Solvency II calibration). Table 1. One-step transition probabilities for the mortality and disability Markov chain In the present case study, we observe that for all cohort sizes and contract times, the current standard approach prescribes almost negligible solvency capital ratios. For small cohort sizes and early contract times, the discrepancies between the stochastic and standard approach increase with age and contract duration attaining solvency capital ratios above 200% for small cohort sizes with 100 insured lives. In fact, as already explained, the current QIS5 specification neglects the process risk. Moreover, we note that the chosen results for the normal distribution are only approximate, especially for small cohort sizes. In this respect, we think that the displayed figures are most likely lower bounds due to the fact that often a normal approximation rather underestimates than overestimates risk. A more detailed analysis of this point is left as open issue for further investigation (however, the use of the gamma approximation makes no big difference). On the other hand, solvency capital ratios of cohort sizes exceeding 10'000 policyholders and late contract times tend more and more to the lower limiting bound as expected from the central limit theorem. Fig. 2 visualizes these findings. In virtue of the made confidence level calibration, the VaR & CVaR solvency capital ratios are of the same order of magnitude. Finally, the considered example points out to another difficulty. Though almost negligible in absolute value, we note that the standard solvency capital ratios change their signs repeatedly over the time axis. In this respect, one can ask whether fixed transition shifts are the "crucial scenarios". As a response to this "biometric worst-and best-case scenarios" are proposed in Christiansen [27], [28].

Conclusions and outlook
Let us summarize the present work. We have derived a general solvency rule for the prospective liability, which has resulted in two simple liability VaR & CVaR target capital requirements. The proposed approach has been applied to determine the biometric solvency risk capital for a portfolio of general traditional life contracts within the Markov chain model of life insurance. Our main actuarial tools have been the backward recursive actuarial reserve formula and the theorem of Hattendorff. Based on this we have determined the conditional mean and variance of a portfolio's prospective liability risk and have used a gamma approximation to obtain the liability VaR & CVaR solvency capital. Since our first formulas include only the process risk and do not take into account the possibility of systematic risk, we have proposed either to shift the biometric transition probabilities, or apply a stochastic model, which allows for random biometric transition probabilities.
Similarly to [1], Section 8, the adopted general methodology is in agreement with several known facts as (i) the process risk is negligible for portfolios with increasing size and has a small impact on medium to large insurers (ii) all else equal, process risk will increase (decrease) with higher (lower) coefficients of variation (aggregated effect of both decrement rates and sums at risk). Another interesting observation has been made at the end of Section 6.3 that the model with shifted biometric transitions can be Moreover, a detailed analysis for a single cohort of identical endowment contracts with waiver of premium by disability has been undertaken in Section 7. Besides a complete Markov chain specification, which seems to be missing in the literature, the numerical illustration has shown, as expected, that the cohort size is a main driving factor of process risk. Due to the statistical law of large numbers, the larger the cohort size the less solvency capital is actually required. In contrast to the life annuity "longevity risk" study in [1], the stochastic approach penalizes almost all insurers (except the very large ones) because the current standard approach prescribes almost negligible solvency capital ratios and does not measure explicitly the process risk effects.
The interested actuary might challenge the proposed approach with alternatives from other regulatory environments than Solvency II. Moreover, it is important to point out that a lot of technical issues remain to be settled properly. They are not only regulatory specific but also related to the complex mathematics of related software products and go beyond the Markov chain model. Today's life insurance contracts include many embedded options and are henceforth even more complex. A challenging issue is the definition of capital requirements for unit-linked contracts without and with guarantee and variable annuities with guaranteed minimum benefits (so-called variable GMXB annuities).