**1. 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.

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 5

,1 1 0,1,..., 1. *CF D X P j T t t j t jt j t j t j* (1)

, 0,1,..., 1.

(2)

1 1 <sup>1</sup> ( ) , 1,..., 1, *A L ALR t T t t ttt* (3)

(5)

, the *liability VaR solvency criterion* (6) says that at time *t* the

*t t* be a minimum solution to (6), and assume

*tt t TC SC RM* (7)

*tt t ELF V* . Then, the *liability VaR solvency capital*

, 1,2,..., 1 1 ,

*Tt Ft*

Consider the *t j* <sup>1</sup> *F* -measurable discrete time stochastic process *CF j T t t j* , 0,1,..., 1 of

The actuarial liabilities at time *t* , also called time- *t prospective insurance liability*, coincide

Using (1)-(2) and the relationship *D DD tt k tt t t k* , 1 , 1 1, 1 , one obtains the recursive equation <sup>1</sup> 1 1 , 1,..., 1 *t tt t t L L PR X t T* . On the other hand, the random assets over the time horizon 0,*T* satisfy by definition the recursive equation

which implies the following equivalent probabilistic conditions (use that trivially 0 *TL* )

*PA F T t* 0 1,

1 . *t tt PA LF*

initial (deterministic) capital requirement *At* should exceed the random present value of

implies that assets will exceed liabilities with the same probability at each future time over

that the best estimate insurance liabilities at time *t* coincide with the *net premium reserves* (in

 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

. *VaR VaR VaR*

with the random present value of all future insurance cash-flows at time *t* given by

*L D CF t T*

1 , 0

*t tt j t j j*

<sup>1</sup> 1 1 , 1,..., 1 *A A PR X t T t tt t t* . Through subtraction it follows that

 

*VaR A VaR L F <sup>t</sup>* 

sum of the liability VaR solvency capital and the risk margin defined by

*T t*

*PA L t t* 

future cash-flows with a probability of at least 1

the sense defined later in (35), that is let *<sup>Z</sup>*

1 *VaR VaR Z Z t tt tt t SC A V VaR L F V*

Given a default probability 0

the time horizon *T* . Let 1

is called random discount rate.

(4)

(6)

. By (4)-(5) this criterion automatically

predictable stochastic process. The quantity <sup>1</sup> *D R st st* , ,

*future insurance cash-flows* defined by

#### **2. 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 <sup>0</sup> *T <sup>t</sup> <sup>t</sup> <sup>F</sup>* such that *<sup>F</sup>*<sup>0</sup> , and *TF F* . Let *<sup>t</sup> L F* be the space of essentially bounded random variables on ,*F P*, and *B* the space of essentially bounded stochastic processes on ,*F P*, which are adapted to the filtration <sup>0</sup> *T <sup>t</sup> <sup>t</sup> <sup>F</sup>* . The basic discrete time stochastic processes are

, *At t L* : the *assets* and *actuarial liabilities* at time *t*

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:

*<sup>t</sup>* <sup>1</sup> *P* : *loaded premiums* to be paid at time 1 *t* (assumed invested at time 1 *t* )

*Xt* : *insurance costs* to be paid at time *t* (includes insurance benefits, expenses and bonus payments paid during the time period *t t* 1, )

*Rt* : *accumulation factor* for return on investment for the time period *t t* 1, 

We assume that *Xt* is *<sup>t</sup> F* -measurable and *Rt* is *<sup>t</sup>* <sup>1</sup> *F* -measurable. The random *cumulated accumulation factor* for return over the period *st s t T* , ,0 , is denoted by , 1 *t s t j j s R R* . Since *Rt* is *<sup>t</sup>* <sup>1</sup> *F* -measurable *Rs t*, is *<sup>t</sup>* <sup>1</sup> *F* -measurable, and therefore *Rts s t*, , is a predictable stochastic process. The quantity <sup>1</sup> *D R st st* , , is called random discount rate. Consider the *t j* <sup>1</sup> *F* -measurable discrete time stochastic process *CF j T t t j* , 0,1,..., 1 of *future insurance cash-flows* defined by

4 Risk Management – Current Issues and Challenges

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.

which are adapted to the filtration <sup>0</sup>

, *At t L* : the *assets* and *actuarial liabilities* at time *t*

payments paid during the time period *t t* 1,

**2. A general prospective approach to the liability risk solvency capital** 

horizon *T* and a probability space ,*F P*, endowed with a filtration <sup>0</sup>

*T*

*<sup>t</sup>* <sup>1</sup> *P* : *loaded premiums* to be paid at time 1 *t* (assumed invested at time 1 *t* )

*Rt* : *accumulation factor* for return on investment for the time period *t t* 1,

*accumulation factor* for return over the period *st s t T* , ,0 , is denoted by ,

Starting point is a multi-period discrete time stochastic model of insurance. Given is a time

*<sup>F</sup>*<sup>0</sup> , and *TF F* . Let *<sup>t</sup> L F* be the space of essentially bounded random variables on ,*F P*, and *B* the space of essentially bounded stochastic processes on ,*F P*,

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:

*Xt* : *insurance costs* to be paid at time *t* (includes insurance benefits, expenses and bonus

)

We assume that *Xt* is *<sup>t</sup> F* -measurable and *Rt* is *<sup>t</sup>* <sup>1</sup> *F* -measurable. The random *cumulated* 

Since *Rt* is *<sup>t</sup>* <sup>1</sup> *F* -measurable *Rs t*, is *<sup>t</sup>* <sup>1</sup> *F* -measurable, and therefore *Rts s t*, , is a

*<sup>t</sup> <sup>t</sup> <sup>F</sup>* . The basic discrete time stochastic processes are

*T*

*<sup>t</sup> <sup>t</sup> <sup>F</sup>* such that

1

*t s t j j s R R* .

$$CF\_{t+j} = D\_{t+j, t+j+1} \cdot X\_{t+j+1} - P\_{t+j} \quad j = 0, 1, \ldots, T-t-1. \tag{1}$$

The actuarial liabilities at time *t* , also called time- *t prospective insurance liability*, coincide with the random present value of all future insurance cash-flows at time *t* given by

$$L\_t = \sum\_{j=0}^{T-t-1} D\_{t, t+j} \text{CF}\_{t+j}, \quad t = 0, 1, \dots, T-1. \tag{2}$$

Using (1)-(2) and the relationship *D DD tt k tt t t k* , 1 , 1 1, 1 , one obtains the recursive equation <sup>1</sup> 1 1 , 1,..., 1 *t tt t t L L PR X t T* . On the other hand, the random assets over the time horizon 0,*T* satisfy by definition the recursive equation <sup>1</sup> 1 1 , 1,..., 1 *A A PR X t T t tt t t* . Through subtraction it follows that

$$A\_{t+1} - L\_{t+1} = (A\_t - L\_t) \cdot R\_{t+1'} \quad t = 1, \ldots, T - 1,\tag{3}$$

which implies the following equivalent probabilistic conditions (use that trivially 0 *TL* )

$$P\left(A\_T \ge 0 \middle| F\_t\right) \ge 1 - \varepsilon\_\prime \tag{4}$$

$$P\left(A\_{t+\tau} \ge L\_{t+\tau'}\ \tau = 1, 2, \ldots, T - t - 1 \middle| F\_t\right) \ge 1 - \varepsilon,\tag{5}$$

$$P\left(A\_t \ge L\_t \Big| F\_t\right) \ge 1 - \varepsilon. \tag{6}$$

Given a default probability 0 , the *liability VaR solvency criterion* (6) says that at time *t* the initial (deterministic) capital requirement *At* 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 1 *VaR A VaR L F <sup>t</sup> t t* 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 *<sup>Z</sup> tt t ELF V* . Then, the *liability VaR solvency capital* 1 *VaR VaR Z Z t tt tt t SC A V VaR L F V* 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

$$T\mathbb{C}\_t^{VaR} = \mathbb{S}\mathbb{C}\_t^{VaR} + \mathbb{R}M\_t^{VaR}.\tag{7}$$

The cost-of-capital risk margin with cost-of-capital rate 6% *CoC i* is defined by

$$\mathbf{RM}\_t^{VaR} = i\_{\text{CoC}} \cdot \sum\_{k=1}^{T-t} \upsilon\_f^k \cdot \mathbf{SC}\_{t+k}^{VaR} \,\,\,\tag{8}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 7

<sup>1</sup> , 0,1,2,... *ij k k p k P X jX i k* (11)

(13)

(14)

 *actuarial reserve*. In particular, one has 0 *L L* and

(16)

. Inserting (16) into (15) yields

actuarial reserve

, one defines the *time-*

stochastic process 0,1,2,... *<sup>k</sup> <sup>k</sup> <sup>X</sup>* with values in *<sup>S</sup>* . The event *X s <sup>k</sup>* means that the contract at time *k* is in state *s*. We assume that 0,1,2,... *<sup>k</sup> <sup>k</sup> <sup>X</sup>* is a *Markov chain*, which implies that the joint distributions of the random states can be represented in terms of the *one-step transition* 

The set , ( ), , 0,1,2,... *<sup>k</sup> Sak X k* defines the Markov chain model widely discussed in life insurance (Amsler [3]; Hoem [4], [5]; Koller [6]; Milbrodt & Helbig [7]; Wolthuis [8]; etc.). Now, using the indicator function *I*( ) , consider the *random cash-flow* of the GLIFE contract

<sup>1</sup> () () ( 1) . *k ii <sup>k</sup> ij k k*

0

This identifies the insurance loss with the random present value of all future cash-flows.

1 , *<sup>k</sup>*

*k L i vC* 

 

*k L vC* 

. *<sup>k</sup> k*

(12)

*k*

, . *k k*

 

(15)

*C bk k IX i v a k IX i X j*

valued at time *k* 0,1,2,... , which is defined by

*i S i jS*

The *insurance loss* random variable of a GLIFE contract is defined by

Furthermore, for an arbitrary non-negative integer 0,1,...

 

whose (conditional) expected value defines the time-

is called *state- k time-*

*k S V EL X V V ELX k*

*V EL* <sup>0</sup> is the *initial actuarial reserve*, which is not assumed to vanish.

 

**4. 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

<sup>1</sup> *L C vL* . 

 

*probability matrix* , ( ) *ij ij S pk p k* , which is defined by

in year *k k*, 1

*prospective loss* random variable

formula for the random prospective loss

Assume that the contract is in state . *k S* at time

The quantity *<sup>k</sup> V*

where *T* denotes the time horizon, and *<sup>f</sup> 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

$$\mathcal{S} \mathcal{R}\_t^{VaR} = \mathcal{S} \mathcal{C}\_t^{VaR} / V\_t^Z. \tag{9}$$

Alternatively, let 1 1 , *CVaR L F E L L VaR L F F tt t t tt t* be the conditional value-atrisk of the random present value of future cash-flows at the confidence level 1 given the information available at time *t* . The *liability CVaR target capital* 1 *CVaR Z CVaR CVaR CVaR <sup>t</sup> tt t t t t TC CVaR L F V RM SC RM* also meets the insurance risk liabilities and it defines the *CVaR solvency capital ratio* at time *t* :

$$\mathcal{S} \mathcal{R}\_t^{\text{CVaR}} = \mathcal{S} \mathcal{C}\_t^{\text{CVaR}} / V\_t^Z. \tag{10}$$

#### **3. The Markov chain approach to general life contracts**

Consider the Markov chain model of a *general life insurance* (GLIFE) contract with state space *S* and arbitrary payments. The *state space* S is the finite set of states a contract can be during its lifetime. Payments are induced by two kinds of events:

Type 1: payments induced by being in a certain state Type 2: payments induced by a jump of state

The *payment function vector* of a contract at time *k* 0,1,2,... is expressed as a vector *ak a k a k i j S* ( ) ( ), ( ) *i ij* , where the *payment functions* are defined by

Type 1: ( ), *<sup>i</sup> ak i S* , is the payment if the contract at time *k* is in state *i* .

Type 2: ( ), , 1 *ij a k i j Sk* , is the payment if the contract was in state *i* at time 1 *k* and is in state state *j* at time *k* . For convenience set (0) 0 *ij a* for *i jS* and () 0 *ij a k* for *i jS* .

For better interpretation one splits the payment ( ) *<sup>i</sup> a k* into a *benefit part* and a *premium part* such that ( ) ( ) ( ), , 0,1,2,... *iii a k b k k i Sk* , where ( ) 0 *<sup>i</sup> k* denotes the non-negative premium paid at time *k* when the contract is in state *i* . Note that in most applications one has ( ) 0 *<sup>i</sup> k* if the state *i* is different from the state of being "active" (premiums are only paid in this situation). Restricting the attention to biometric risk only, we assume throughout a flat term structure of interest rates with annual interest rate *i* and discount factor *v i* 1 / (1 ) . The state of a GLIFE contract over time is described by the discrete time stochastic process 0,1,2,... *<sup>k</sup> <sup>k</sup> <sup>X</sup>* with values in *<sup>S</sup>* . The event *X s <sup>k</sup>* means that the contract at time *k* is in state *s*. We assume that 0,1,2,... *<sup>k</sup> <sup>k</sup> <sup>X</sup>* is a *Markov chain*, which implies that the joint distributions of the random states can be represented in terms of the *one-step transition probability matrix* , ( ) *ij ij S pk p k* , which is defined by

$$p\_{ij}\left(k\right) = P\left(X\_{k+1} = j \middle| X\_k = i\right), \quad k = 0, 1, 2, \dots \tag{11}$$

The set , ( ), , 0,1,2,... *<sup>k</sup> Sak X k* defines the Markov chain model widely discussed in life insurance (Amsler [3]; Hoem [4], [5]; Koller [6]; Milbrodt & Helbig [7]; Wolthuis [8]; etc.). Now, using the indicator function *I*( ) , consider the *random cash-flow* of the GLIFE contract in year *k k*, 1 valued at time *k* 0,1,2,... , which is defined by

$$\mathbf{C}\_{k} = \sum\_{i \in \mathcal{S}} \left\{ b\_{i}(k) - \pi\_{i}(k) \right\} \cdot I\left(\mathbf{X}\_{k} = i\right) + \boldsymbol{\upsilon} \cdot \sum\_{i \neq j \in \mathcal{S}} a\_{ij}(k+1) \cdot I\left(\mathbf{X}\_{k} = i \wedge \mathbf{X}\_{k+1} = j\right). \tag{12}$$

The *insurance loss* random variable of a GLIFE contract is defined by

6 Risk Management – Current Issues and Challenges

1

The cost-of-capital risk margin with cost-of-capital rate 6% *CoC*

Alternatively, let 1 1 , *CVaR L F E L L VaR L F F* 

*CVaR Z CVaR CVaR CVaR <sup>t</sup> tt t t t t TC CVaR L F V RM SC RM*

liabilities and it defines the *CVaR solvency capital ratio* at time *t* :

**3. The Markov chain approach to general life contracts** 

*ak a k a k i j S* ( ) ( ), ( ) *i ij* , where the *payment functions* are defined by

Type 1: ( ), *<sup>i</sup> ak i S* , is the payment if the contract at time *k* is in state *i* .

its lifetime. Payments are induced by two kinds of events:

Type 1: payments induced by being in a certain state

Type 2: payments induced by a jump of state

such that ( ) ( ) ( ), , 0,1,2,... *iii a k b k k i Sk* 

*i jS* .

has ( ) 0 *<sup>i</sup>* 

1

where *T* denotes the time horizon, and *<sup>f</sup> 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

/ . *VaR VaR Z*

information available at time *t* . The *liability CVaR target capital*

/ . *CVaR CVaR Z*

Consider the Markov chain model of a *general life insurance* (GLIFE) contract with state space *S* and arbitrary payments. The *state space* S is the finite set of states a contract can be during

The *payment function vector* of a contract at time *k* 0,1,2,... is expressed as a vector

Type 2: ( ), , 1 *ij a k i j Sk* , is the payment if the contract was in state *i* at time 1 *k* and is in state state *j* at time *k* . For convenience set (0) 0 *ij a* for *i jS* and () 0 *ij a k* for

For better interpretation one splits the payment ( ) *<sup>i</sup> a k* into a *benefit part* and a *premium part*

premium paid at time *k* when the contract is in state *i* . Note that in most applications one

 *k* if the state *i* is different from the state of being "active" (premiums are only paid in this situation). Restricting the attention to biometric risk only, we assume throughout a flat term structure of interest rates with annual interest rate *i* and discount factor *v i* 1 / (1 ) . The state of a GLIFE contract over time is described by the discrete time

*k* denotes the non-negative

, where ( ) 0 *<sup>i</sup>*

risk of the random present value of future cash-flows at the confidence level 1

 

also meets the insurance risk

*T t VaR k VaR t CoC f t k k RM i v SC* 

*i* is defined by

(8)

*t tt SR SC V* (9)

*t tt SR SC V* (10)

given the

,

*tt t t tt t* be the conditional value-at-

$$L = \sum\_{k=0}^{\infty} v^k \mathbb{C}\_k. \tag{13}$$

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

$$L\_{\tau} = \left(1 + i\right)^{\tau} \cdot \sum\_{k=\tau}^{\infty} v^{k} \mathbb{C}\_{k'} \tag{14}$$

whose (conditional) expected value defines the time-actuarial reserve

$$V\_{\tau} = E\left[L\_{\tau} \middle| \mathbf{X}\_{\tau}\right] = \sum\_{k \in S} V\_{\tau}^{k} \quad V\_{\tau}^{k} = E\left[L\_{\tau} \middle| \mathbf{X}\_{\tau} = k\right]. \tag{15}$$

The quantity *<sup>k</sup> V* is called *state- k time- actuarial reserve*. In particular, one has 0 *L L* and *V EL* <sup>0</sup> is the *initial actuarial reserve*, which is not assumed to vanish.

#### **4. 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

$$L\_{\tau} = C\_{\tau} + \upsilon L\_{\tau+1}.\tag{16}$$

Assume that the contract is in state . *k S* at time . Inserting (16) into (15) yields

$$\mathbf{V}\_{\tau}^{k} = E\left[\mathbf{C}\_{\tau} \, \middle| \, \mathbf{X}\_{\tau} = k\right] + \boldsymbol{\upsilon} \cdot E\left[\, \mathbf{L}\_{\tau+1} \, \middle| \, \mathbf{X}\_{\tau} = k\right]. \tag{17}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 9

1 1 1

 

*vV V b vV v p a V*

 

 

*j S*

 , 1

(21)

 

due to transitions from state *k* to

for a contract in state *k* and the probability

at time

. The

and the

*j S p* 

, one obtains

. Rewrite the latter as

 

for a

of a contract in

**Proof.** Making use of the recursion (18) and the relationship ( ) 1 *kj*

*k j*

state *k* needed to cover the insurance risk in time period

weighted sum of the *sums at risk* ( 1) 1 1 *<sup>j</sup> <sup>k</sup>*

martingale differences 1 *Y Yv* , 0,1,2,...

Through detailed calculation one obtains the following result.

The discrete time stochastic process *Y*

  

*V p va vV b*

 

*j S*

which shows the desired decomposition. ◊

This is the sum of the benefit payment at time

 

state *j* at time 1

Norberg [12], p.10.

this consider the set of 1

sequence of random variables

by the following formulas

The saving premium represents the expected change in actuarial reserve at time

contract in state *k* while the risk premium is the expected value at time

*k k kj kj j S*

*kj kj k k*

*k k k kj kj*

() () ( ) ( ) ( 1)

*S R kk k j*

( ) ( 1) ( ) ( ),

  1

( ) ( ) ( ) ( 1) 1 1. *R k <sup>j</sup>*

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

To evaluate the mean and variance of the random insurance loss (13) of a GLIFE contract, we follow the martingale approach to the *Theorem of Hattendorff* (Bühlmann [13]; Gerber [14]; Gerber et al. [15]; Hattendorff [16]; Kremer [17]; Patatriandafylou & Waters [18]; etc.). For

0 0 *Y E LS* , 1,2,..., *Y EL V* .

*E Cov* 0, , 0, 0 , *LV v* .

**Theorem 4.2** The variance of the random insurance loss of a GLIFE contract is determined

2 0 *Var L v Var C* , 

 

(23)

 ,0,1,...,

(22)

, represent the *discounted one-year* 

is a martingale with respect to *S*

0

(24)

0

contract states *S X*

 

 

 

*insurance losses* and form a sequence of uncorrelated random variables such that

 

*kj a VV* 

*b vp a V V*

 

> 

. The sum at risk is the amount credited to the insured's contract upon a

Using (12) the first expectation in (17) can be rewritten as

$$\begin{split} & \left. v \cdot \sum\_{i \neq j \in S} a\_{ij} (\tau + 1) \cdot P \left( X\_{\tau} = i \wedge X\_{\tau + 1} = j \right| X\_{\tau} = k \right) + \sum\_{i \in S} \left\{ b\_{i}(\tau) - \pi\_{i}(\tau) \right\} \cdot P \left( X\_{\tau} = i \Big| X\_{\tau} = k \right) \\ & = v \cdot \sum\_{j \neq k \in S} a\_{kj}(\tau + 1) \cdot p\_{kj}(\tau) + b\_{k}(\tau) - \pi\_{k}(\tau). \end{split}$$

The second expectation equals

$$E\left[\left.L\_{\tau+1}\right|X\_{\tau}=k\right] = \sum\_{j\in S} E\left[\left.L\_{\tau+1}\right|X\_{\tau+1}=j\right] \cdot P\left(X\_{\tau+1}=j\middle|X\_{\tau}=k\right) = \sum\_{j\in S} V\_{\tau+1}^{j} \cdot p\_{kj}(\tau).$$

Inserting both expressions into (17) and using the made convention ( 1) 0 *kk a* as well as the relationship ( ) 1 *kj j S p* , one obtains the backward recursive reserve formula

$$\boldsymbol{V}\_{\tau}^{k} = \sum\_{j \in \mathcal{S}} p\_{kj}(\tau) \cdot \left(\boldsymbol{\upsilon} \cdot \boldsymbol{V}\_{\tau+1}^{j} + \boldsymbol{\upsilon} \cdot \boldsymbol{a}\_{kj}(\tau+1) + b\_{k}(\tau)\right) - \pi\_{k}(\tau). \tag{18}$$

The actuarial reserve at time given the contract is in state *k S* equals the one-year discounted sum over all possible states of the


which is weighted by the one-step transition probabilities and reduced by the premium paid at time when the contract is in state *k* . The representation (18) is a discrete version of *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].

Let us rearrange (18) in order to obtain the Markov chain analogue of the classical decomposition of the premium into risk premium and saving premium (Gerber [10], [11]), Section 7.5, equation (5.3), and [1], equation (19).

**Theorem 4.1** The premium ( ) *<sup>k</sup>* at time if the contract is in state *k S* is the sum of a *saving premium* ( ) *<sup>S</sup> k* and a *risk premium* ( ) *<sup>R</sup> k* , which are defined as follows:

$$
\pi\_k^S(\tau) = \upsilon \cdot V\_{\tau+1}^k - V\_\tau^k \tag{19}
$$

$$\pi\_k^R(\tau) = b\_k(\tau) - \upsilon \cdot V\_{\tau+1}^k + \upsilon \cdot \sum\_{j \in S} p\_{kj}(\tau) \cdot \left( a\_{kj}(\tau+1) + V\_{\tau+1}^j \right) \tag{20}$$

**Proof.** Making use of the recursion (18) and the relationship ( ) 1 *kj j S p* , one obtains

$$\begin{aligned} \pi\_k^S(\tau) + \pi\_k^R(\tau) &= \upsilon \cdot V\_{\tau+1}^k - V\_{\tau}^k + b\_k(\tau) - \upsilon \cdot V\_{\tau+1}^k + \upsilon \cdot \sum\_{j \in S} p\_{kj}(\tau) \cdot \left( a\_{kj}(\tau+1) + V\_{\tau+1}^j \right) \\ &= -V\_{\tau}^k + \sum\_{j \in S} p\_{kj}(\tau) \cdot \left\{ \upsilon \cdot a\_{kj}(\tau+1) + \upsilon \cdot V\_{\tau+1}^j + b\_k(\tau) \right\} = \pi\_k(\tau), \end{aligned}$$

which shows the desired decomposition. ◊

8 Risk Management – Current Issues and Challenges

*va p b*

*jkS*

The second expectation equals

 

the relationship ( ) 1 *kj j S p* 

The actuarial reserve at time

payments at time 1

payments at time

at time

actuarial reserves at time 1

**Theorem 4.1** The premium ( ) *<sup>k</sup>*

*k* 

*saving premium* ( ) *<sup>S</sup>*

 Using (12) the first expectation in (17) can be rewritten as

( 1) ( ) ( ) ( ).

 

*j S*

,

values in intensity-driven Markov processes, e.g. Norberg [9].

 

> 

discounted sum over all possible states of the

Section 7.5, equation (5.3), and [1], equation (19).

*i jS i S kj kj k k*

 

 

Inserting both expressions into (17) and using the made convention ( 1) 0 *kk a*

*V p vV va b*

 

due to a jump in states,

due if being in a certain state,

at time

 

*k k kj kj*

and a *risk premium* ( ) *<sup>R</sup>*

*ij i i*

<sup>1</sup> . *<sup>k</sup> V EC X k vEL X k*

*v a P X i X jX k b P X iX k*

 

*E L X k E L X j P X jX k V p*

 (17)

> 

> >

> > >

given the contract is in state *k S* equals the one-year

if the contract is in state *k S* is the sum of a

(19)

 

, which are defined as follows:

 

(20)

(18)

 

 

*kj*

as well as

( 1) <sup>1</sup> () ()

<sup>1</sup> 11 1 <sup>1</sup> ( ). *<sup>j</sup>*

 

 

*j S j S*

, one obtains the backward recursive reserve formula

( ) <sup>1</sup> ( 1) ( ) ( ). *<sup>k</sup> <sup>j</sup> kj kj k k*

which is weighted by the one-step transition probabilities and reduced by the premium paid

*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

Let us rearrange (18) in order to obtain the Markov chain analogue of the classical decomposition of the premium into risk premium and saving premium (Gerber [10], [11]),

> <sup>1</sup> ( ) *S kk <sup>k</sup> vV V*

() () 1 1 ( ) ( 1) *R k <sup>j</sup>*

*j S b vV v p a V*

*k* 

when the contract is in state *k* . The representation (18) is a discrete version of

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

$$
\pi\_k^R(\tau) = b\_k(\tau) + \upsilon \cdot \sum\_{j \in S} p\_{kj}(\tau) \cdot \left( a\_{kj}(\tau + 1) + V\_{\tau + 1}^j - V\_{\tau + 1}^k \right). \tag{21}
$$

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* ( 1) 1 1 *<sup>j</sup> <sup>k</sup> kj a VV* due to transitions from state *k* to state *j* at time 1 . 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) of a GLIFE contract, we follow the martingale approach to the *Theorem of Hattendorff* (Bühlmann [13]; Gerber [14]; Gerber et al. [15]; Hattendorff [16]; Kremer [17]; Patatriandafylou & Waters [18]; etc.). For this consider the set of 1 contract states *S X* , 0,1,..., at time and the sequence of random variables

$$Y\_{\tau} = E\left[L\left|\mathbf{S}\_{\tau}\right.\right], \quad \tau = \mathbf{1}, \mathbf{2}, \dots, \quad Y\_0 = E\left[L\right] = V\_0. \tag{22}$$

The discrete time stochastic process *Y* is a martingale with respect to *S* . The martingale differences 1 *Y Yv* , 0,1,2,... , represent the *discounted one-year insurance losses* and form a sequence of uncorrelated random variables such that

$$E\left[\boldsymbol{\Lambda}\_{\eta}\right] = 0,\quad \text{Cov}\left[\boldsymbol{\Lambda}\_{\eta'}, \boldsymbol{\Lambda}\_{\tau}\right] = 0,\quad 0 \le \eta < \tau,\quad L - V\_0 = \sum\_{\tau=0}^{\eta} v^{\tau} \boldsymbol{\Lambda}\_{\tau}.\tag{23}$$

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

$$\operatorname{Var}\left[\boldsymbol{L}\right] = \sum\_{\tau=0}^{\infty} \boldsymbol{v}^{2\tau} \cdot \operatorname{Var}\left[\mathbb{C}\_{\tau}\right]\_{\prime} \tag{24}$$

$$\operatorname{Var}\left[\mathbf{C}\_{\tau}\right] = \sum\_{k \in \mathcal{S}} \operatorname{E}\left[\mathbf{C}\_{\tau}^{2} \middle| \mathbf{X}\_{\tau} = k\right] \cdot \operatorname{P}\left(\mathbf{X}\_{\tau} = k\right) - \pi^{S}(\tau)^{2}, \quad \pi^{S}(\tau) = \sum\_{k \in \mathcal{S}} \pi^{S}\_{k}(\tau), \tag{25}$$

$$\begin{aligned} \operatorname{E}\left[\big\circ\_{\tau}^{2}\middle|X\_{\tau}=k\right] &= \left[b\_{k}(\tau)-\pi\_{k}(\tau)\right]^{2} \\ &+ \sum\_{j\in S} va\_{kj}(\tau+1)\cdot\left[b\_{k}(\tau)-\pi\_{k}(\tau)+va\_{kj}(\tau+1)\right]\cdot p\_{kj}(\tau). \end{aligned} \tag{26}$$

**Proof.** Similarly to Gerber et al. [15], formula (89), one has

$$Y\_{\tau} = \sum\_{j=0}^{\tau - 1} v^j \mathbb{C}\_j + v^{\tau} E \left[ L\_{\tau} \middle| \mathbb{S}\_{\tau} \right]. \tag{27}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 11

 <sup>2</sup> 2 2 1 1 ( 1) () () . *<sup>j</sup> k R*

 

(33)

*kj kj k*

We begin with risk calculations for a single GLIFE contract, and use them to determine the

Given is a single GLIFE contract with random future cash-flows *Ck* defined by (12). We assume that the state space contains a unique distinguished "void" state *Xk*

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

coincides with the time- *t* prospective loss defined in (14), that is , 0,1,... *Z Lt t t* .

. *Zk k <sup>t</sup>*

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

**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

*k S k S t*

*P E*

0

*t tt t t t t*

*V E Z E V P X kE V*

*t tj j Z vC t* 

is fulfilled. We note that the random present value of future cash-flows at

, 0,1,..., *<sup>j</sup>*

(34)

*PX k*

(35)

 

*Var C a V V vp PX k*

**5. The liability VaR & CVaR solvency capital for portfolios of GLIFE** 

*kS jS*

**5.1. Risk calculations for a single GLIFE contract** 

Therefore, the expected value given contract survival equals

value can been chosen as best estimate of the contract liabilities.

liability VaR & CVaR solvency capital for a portfolio of GLIFE contracts.

**contracts** 

*E X t t*

time *t* defined by

Using (15) one obtains

$$\begin{aligned} \boldsymbol{\upsilon}^{\tau}\boldsymbol{\Lambda}\_{\tau} &= \boldsymbol{Y}\_{\tau+1} - \boldsymbol{Y}\_{\tau} = \boldsymbol{\upsilon}^{\tau} \left( \mathbf{C}\_{\tau} + \boldsymbol{\upsilon}E \Big[ \boldsymbol{L}\_{\tau+1} \Big| \mathbf{S}\_{\tau+1} \Big] - E \Big[ \boldsymbol{L}\_{\tau} \Big| \mathbf{S}\_{\tau} \Big] \right) \\ &= \boldsymbol{\upsilon}^{\tau} \Big( \mathbf{C}\_{\tau} + \sum\_{k \in S} \Big[ \boldsymbol{\upsilon}E \Big[ \boldsymbol{L}\_{\tau+1} \Big| \mathbf{X}\_{\tau+1} = k \Big] - E \Big[ \boldsymbol{L}\_{\tau} \Big| \mathbf{X}\_{\tau} = k \Big] \Big) \Big) = \boldsymbol{\upsilon}^{\tau} \Big( \mathbf{C}\_{\tau} + \boldsymbol{\pi}^{S}(\boldsymbol{\tau}) \Big), \end{aligned} \tag{28}$$

$$
\Lambda\_{\tau} = \mathbb{C}\_{\tau} + \pi^S(\tau). \tag{29}
$$

Since *E* 0 one gets ( ) *<sup>S</sup> E C* and further

$$Var\left[\left.\Lambda\_{\tau}\right.\right] = Var\left[\left.\mathcal{C}\_{\tau}\right.\right] = E\left[\left.\mathcal{C}\_{\tau}^{2}\right] - E\left[\left.\mathcal{C}\_{\tau}\right.\right]^{2} = \sum\_{k \in S} E\left[\left.\mathcal{C}\_{\tau}^{2}\right|\mathcal{X}\_{\tau} = k\right] \cdot P\left(\mathcal{X}\_{\tau} = k\right) - \pi^{S}\left(\tau\right)^{2},\tag{30}$$

which is (25). To obtain (26) one uses (12) and the convention ( 1) 0 *ii a* to get

$$\begin{aligned} \mathbf{C}\_{\tau}^{2} &= \sum\_{i \in S} \left[ b\_{i}(\tau) - \pi\_{i}(\tau) \right]^{2} \cdot I\left( \mathbf{X}\_{\tau} = i \right) \\ &+ \sum\_{i, j \in S} v a\_{ij}(\tau + 1) \cdot \left[ b\_{i}(\tau) - \pi\_{i}(\tau) + v a\_{ij}(\tau + 1) \right] \cdot I\left( \mathbf{X}\_{\tau} = i \wedge \mathbf{X}\_{\tau + 1} = j \right) \end{aligned} \tag{31}$$

which implies that

$$E\left[\mathbf{C}\_{\tau}^{2}\Big|\mathbf{X}\_{\tau}=k\right] = \left[b\_{k}(\tau) - \pi\_{k}(\tau)\right]^{2} + \sum\_{j \in S} v a\_{kj}(\tau+1) \cdot \left[b\_{k}(\tau) - \pi\_{k}(\tau) + v a\_{kj}(\tau+1)\right] \cdot p\_{kj}(\tau). \tag{32}$$

**Remark 4.1** In the single life case, the variance formulas in Theorem 4.2 should be compared with the ones for the GLIFE contract with one and multiple causes of decrement in [1], formulas (24)-(26). One can ask if the formula (25) is equivalent to the following one (at least in the single life case)

$$\operatorname{Var}\left[\mathbf{C}\_{\tau}\right] = \sum\_{k \in \mathcal{S}} \left[ \sum\_{j \in \mathcal{S}} \left( a\_{kj} (\tau + 1) + V\_{\tau + 1}^{j} - V\_{\tau + 1}^{k} \right)^{2} v^{2} p\_{kj}(\tau) - \pi\_{k}^{R}(\tau)^{2} \right] \cdot P\left( X\_{\tau} = k \right). \tag{33}$$
