**5. 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.

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

10 Risk Management – Current Issues and Challenges

Using (15) one obtains

Since *E* 0 

which implies that

 

in the single life case)

,

*ij S*

*i S*

 

 *j S*

*k S*

one gets ( ) *<sup>S</sup> E C*

2 2

 

() ()

*C b IX i*

 

*i i*

 

 

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

*Var C E C X k P X k*

 

*EC X k b*

2 2 ( ) , ( ) ( ), *SS S*

(25)

   

> 

(27)

 

 

 

to get

 

1

   

(29)

 

 

*k S k S*

( 1) ( ) ( ) ( 1) ( ).

. *<sup>j</sup>*

 

(28)

 

1 1 (), *<sup>S</sup>*

 

<sup>2</sup> 22 2 (), *<sup>S</sup>*

(31)

 

*kj k k kj kj*

*va b va p*

*j*

*Y vC v E L S*

 

*k k*

 

 

1

0

*v C vE L X k E L X k v C*

 *C* 

*Var Var C E C E C E C X k P X k*

 

 

*j S EC X k b va b va p*

 

( ). *<sup>S</sup>*

*k S*

( 1) ( ) ( ) ( 1) ,

 

*va b va I X i X j*

<sup>2</sup> <sup>2</sup> ( ) ( ) ( 1) ( ) ( ) ( 1) ( ). *k k kj k k kj kj*

(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

 (30)

*j*

1 1 1

 

*v Y Y v C vE L S E L S*

 

 

 

 and further

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

*ij i i ij*

 

 <sup>2</sup> <sup>2</sup> () ()

*k*

(26)

 

> 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 *E X t t* is fulfilled. We note that the random present value of future cash-flows at time *t* defined by

$$Z\_t = \sum\_{j=0}^{\infty} \upsilon^j \mathbb{C}\_{t+j\prime} \quad \text{ } t = 0, 1, \dots \text{ } \tag{34}$$

coincides with the time- *t* prospective loss defined in (14), that is , 0,1,... *Z Lt t t* . Therefore, the expected value given contract survival equals

$$\mathbf{V}\_t^Z = \mathbb{E}\left[\mathbf{Z}\_t \Big| \mathbf{E}\_t\right] = \sum\_{k \in \mathcal{S}} V\_t^k \cdot P\left(\mathbf{X}\_t = k \Big| \mathbf{E}\_t\right) = \sum\_{k \in \mathcal{S}} V\_t^k \cdot \frac{P\{\mathbf{X}\_t = k\}}{P\{E\_t\}}.\tag{35}$$

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 *Zt* given *<sup>t</sup> E* . Similarly to [1], Section 5.1, the variance formulas (24)-(26) generalize to an arbitrary discrete time 1,2,... *t* . Formula (23) generalizes as

$$L\_t - V\_t \cdot I\left(E\_t\right) = \sum\_{k=t}^{\infty} v^{k-t} \Lambda\_k \,. \tag{36}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 13

*<sup>t</sup> L* time- *it* random prospective loss (14) and time- *it* net premium

obtained from (35). The random present value of future cash-

, 0,1,... .

(40)

(41)

0 1

*i*

(43)

*t tt t t t SR SC V SR SC V* (42)

(39)

(38)

*rs a kr sS k* , is the payment if the contract was in state *r* at time 1 *k*

To the *<sup>i</sup>* -th contract one associates its random future cash-flows ( )*<sup>i</sup> Ck* as defined in (12),

( ) ( )

*i i*

1

( ) <sup>1</sup> , 1,..., , *<sup>i</sup>*

( ) <sup>1</sup> , 1,..., , *<sup>i</sup>*

*CVaR i Z tt t <sup>t</sup> SC CVaR Z E i n V* 

/ , / . *VaR VaR Z CVaR CVaR Z*

To determine these quantities it is necessary to determine the distribution of *Zt* conditional on contract survivals at time *t* , and under the assumption that the remaining lifetimes of all

( ) ( ) 2 ( )

*<sup>n</sup> ii i <sup>Z</sup> t tt t t t*

*ii i*

 

*VaR i Z tt t <sup>t</sup> SC VaR Z E i n V* 

*<sup>n</sup> <sup>Z</sup> Z i t t i V V* 

,( )

. *<sup>i</sup>*

*r rs a k a ka kr s S* of the contract at time

into a benefit & premium part

payment function vector () () () ( ) ( ) ( ), ( ) *i ii i*

*<sup>r</sup> a kr S* , is the payment if the contract at time *k* is in state *r* .

flows of the portfolio is obtained by summing (34) over all contracts and is given by

1 0 0 1

Following Section 2, one defines the *portfolio VaR solvency capital*

contracts are independent of each other. From Theorem 5.1 we have

( ) ()

*i t*

*i ii <sup>n</sup> t tt*

*E C PE EC*

*P E*

<sup>2</sup> ( ) <sup>2</sup> () ()

, 1,..., , , 1,...,

( ) 2

*i ii i*

*i*

, ( )

*E Z E i n V Var Z E i n v Var C*

*n n j j i i t t j t j i j j i Z vC v C t* 

Similarly, summing the individual net premium reserves, one gets the *portfolio reserve*

*k* 0,1,2,... with the two types of payment:

 splitting () () () ( ) ( ) ( ) ( ), , 0,1,2,... *ii i i rr r a k b k k rS k* 

> *i i*

as well as the *portfolio CVaR solvency capital*

and the corresponding solvency capital ratios

2

*v*

0 1

Type 1: () () ( ), *i i*

the corresponding ( )

reserve ,() () ()

*i i i Zi i i t tt V EL E*

Type 2: ( ) ( ) ( ), , 1 *i i*

and is in state state *s* at time *k*

Noting further that , 0,1,... *Z Lt t t* , 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

$$\begin{split} Var\left[\operatorname{Z}\_{t}\big|\operatorname{E}\_{t}\right] &= \sum\_{\tau=0}^{\infty} \upsilon^{2\tau} \cdot Var\left[\operatorname{C}\_{t+\tau}\big|\operatorname{E}\_{t}\right] = \sum\_{\tau=0}^{\infty} \upsilon^{2\tau} \cdot \left(\frac{E\left[\operatorname{C}\_{t+\tau}^{2}\right] \cdot P(E\_{t}) - E\left[\operatorname{C}\_{t+\tau}\right]^{2}}{P(E\_{t})}\right) \\ &E\left[\operatorname{C}\_{t+\tau}^{2}\right] = \sum\_{k\in\mathcal{S}} E\left[\operatorname{C}\_{t+\tau}^{2} \left|X\_{t+\tau} = k\right.\right] \cdot P\left(X\_{t+\tau} = k\right), \quad E\left[\operatorname{C}\_{t+\tau}\right] = \pi^{S}\left(t+\tau\right) = \sum\_{k\in\mathcal{S}} \pi\_{k}^{S}\left(t+\tau\right) \\ &E\left[\operatorname{C}\_{t+\tau}^{2}\left|X\_{t+\tau} = k\right.\right] = \left[b\_{k}\left(t+\tau\right) - \pi\_{k}\left(t+\tau\right)\right]^{2} \\ &+\sum\_{j\in\mathcal{S}} \upsilon a\_{kj}\left(t+\tau+1\right) \cdot \left[b\_{k}\left(t+\tau\right) - \pi\_{k}\left(t+\tau\right) + \upsilon a\_{kj}\left(t+\tau+1\right)\right] \cdot p\_{kj}\left(t+\tau\right). \end{split} \tag{37}$$

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.

#### **5.2. 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 *i n* 1,..., is characterized by the following data elements:


 payment function vector () () () ( ) ( ) ( ), ( ) *i ii i r rs a k a ka kr s S* of the contract at time *k* 0,1,2,... with the two types of payment:

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

Type 2: ( ) ( ) ( ), , 1 *i i rs a kr sS k* , is the payment if the contract was in state *r* at time 1 *k* and is in state state *s* at time *k*

 splitting () () () ( ) ( ) ( ) ( ), , 0,1,2,... *ii i i rr r a k b k k rS k* into a benefit & premium part

To the *<sup>i</sup>* -th contract one associates its random future cash-flows ( )*<sup>i</sup> Ck* as defined in (12), the corresponding ( ) *i i <sup>t</sup> L* time- *it* random prospective loss (14) and time- *it* net premium reserve ,() () () *i i i Zi i i t tt V EL E* obtained from (35). The random present value of future cashflows of the portfolio is obtained by summing (34) over all contracts and is given by

$$Z\_t = \sum\_{i=1}^{n} \sum\_{j=0}^{\infty} v^j \mathbf{C}\_{t\_i + j}^{(i)} = \sum\_{j=0}^{\infty} v^j \left( \sum\_{i=1}^{n} \mathbf{C}\_{t\_i + j}^{(i)} \right), \quad t = 0, 1, \ldots \tag{38}$$

Similarly, summing the individual net premium reserves, one gets the *portfolio reserve*

$$V\_t^Z = \sum\_{i=1}^n V\_{t\_i}^{Z,(i)}.\tag{39}$$

Following Section 2, one defines the *portfolio VaR solvency capital*

$$\text{SC}\_{t}^{\text{VaR}} = \text{VaR}\_{1-x} \left[ Z\_t \middle| E\_{t\_i}^{(i)}, i = 1, \dots, n \right] - V\_t^{Z\_t} \tag{40}$$

as well as the *portfolio CVaR solvency capital*

12 Risk Management – Current Issues and Challenges

2 2

*kj*

contract duration *it* at time *t*

*i <sup>k</sup> <sup>k</sup>*

0,1,2,...

condition for contract survival () ()

Markov chain model of the contract

one-step transition probabilities ( ) () ()

 state space ( )*<sup>i</sup> S* states ( )

*va*

*j S*

 

arbitrary discrete time 1,2,... *t* . Formula (23) generalizes as

variance formulas (conditional version of Theorem 4.2).

*t t t t*

*Var Z E v Var C E v*

*tt k k*

*EC X k b t t*

work.

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

In a first step, we determine the mean and variance of the conditional distribution of *Zt* given *<sup>t</sup> E* . Similarly to [1], Section 5.1, the variance formulas (24)-(26) generalize to an

*L V IE v*

**Theorem 5.1** The conditional variance is determined by the following formulas

2 2

 

()()

associated to this portfolio under the condition that the contracts are still alive.

*i* -th contract *i n* 1,..., is characterized by the following data elements:

*<sup>X</sup>* of the contract over time with values in ( )*<sup>i</sup> <sup>S</sup>*

*i i i i t t E X*

0 0

2 2

 

 *k t tt t k k t*

Noting further that , 0,1,... *Z Lt t t* , one obtains from (36) the following conditional

 

*EC EC X k PX k EC t t*

 

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

 

*t b t t va t p t*

 

 

*t tt t t k k S k S*

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

**5.2. 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

at time *t*

<sup>1</sup> , 0,1,2,... *i ii*

*rs k k p k P X sX r k* , defining the

 

. (36)

2 2

( )

*EC PE EC*

  *t tt*

*P E*

 *t*

2

, () ()

 

, ( )

 

> 

(37)

*S S*

$$\text{SC}\_{t}^{\text{CVaR}} = \text{CVaR}\_{1-\varepsilon} \left[ Z\_{t} \left| \mathbf{E}\_{t\_{i}}^{(i)} \text{, i} = 1, \dots, n \right. \right] - \text{V}\_{t}^{Z} \text{,} \tag{41}$$

and the corresponding solvency capital ratios

$$\text{SSR}\_t^{\text{VaR}} = \text{SC}\_t^{\text{VaR}} / V\_t^Z \text{ \prime} \quad \text{SSR}\_t^{\text{CVaR}} = \text{SC}\_t^{\text{CVaR}} / V\_t^Z \text{ \prime} \tag{42}$$

To determine these quantities it is necessary to determine the distribution of *Zt* 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

$$\begin{aligned} &E\left[Z\_{t}\left|\boldsymbol{E}\_{t\_{i}}^{(i)}\right.\;\mathrm{i}=1,\ldots,n\right]=\boldsymbol{V}\_{t}^{Z}\quad\mathrm{Var}\left[Z\_{t}\left|\boldsymbol{E}\_{t\_{i}}^{(i)}\right.\;\mathrm{i}=1,\ldots,n\right]=\sum\_{\tau=0}^{\alpha}\boldsymbol{v}^{2\tau}\cdot\sum\_{i=1}^{n}\mathrm{Var}\left[\boldsymbol{C}\_{t\_{i}+\tau}^{(i)}\right] \\ &=\sum\_{\tau=0}^{\alpha}\boldsymbol{v}^{2\tau}\cdot\sum\_{i=1}^{n}\frac{E\left[\left(\boldsymbol{C}\_{t\_{i}+\tau}^{(i)}\right)^{2}\right]\cdot P(E\_{t\_{i}}^{(i)})-E\left[\boldsymbol{C}\_{t\_{i}+\tau}^{(i)}\right]^{2}}{P(E\_{t\_{i}}^{(i)})^{2}},\end{aligned}\tag{43}$$

$$\begin{split} &E\left[\mathbf{C}\_{t\_{i}+\tau}^{(i)}\right] = \mathbf{\pi}^{S,(i)}(t\_{i}+\tau) = \sum\_{k \in S^{(i)}} \mathbf{\pi}\_{k}^{S,(i)}(t\_{i}+\tau)\_{\prime} \\ &E\left[\mathbf{C}\_{t\_{i}+\tau}^{(i)}\right]^{2} = \sum\_{k \in S^{(i)}} E\left[\left(\mathbf{C}\_{t\_{i}+\tau}^{(i)}\right)^{2}\Big|\mathbf{X}\_{t\_{i}+\tau}^{(i)} = k\right] \cdot P\left(\mathbf{X}\_{t\_{i}+\tau}^{(i)} = k\right) \\ &E\left[\left(\mathbf{C}\_{t\_{i}+\tau}^{(i)}\right)^{2}\Big|\mathbf{X}\_{t\_{i}+\tau}^{(i)} = k\right] = \left[b\_{k}^{(i)}(t\_{i}+\tau) - \pi\_{k}^{(i)}(t\_{i}+\tau)\right]^{2} \\ &+ \sum\_{j \in S^{(i)}} \text{val}\_{kj}^{(i)}(t\_{i}+\tau+1) \cdot \Big[b\_{k}^{(i)}(t\_{i}+\tau) - \pi\_{k}^{(i)}(t\_{i}+\tau) + \text{val}\_{kj}^{(i)}(t\_{i}+\tau+1)\Big] \cdot p\_{kj}^{(i)}(t\_{i}+\tau). \end{split} \tag{44}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 15

the value of the reserves subject to a

*tt t SCR V V* (47)

*t tt TC SCR RM* (48)

*t tt SR SCR V* (50)

*ij p k* (51)

6%, *CoC <sup>i</sup>* (49)

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

To value the net premium reserves a biometric "best estimate" life table is chosen. In the Markov chain model the life table is replaced by the one-step transition probabilities <sup>1</sup> , 0,1,2,... *ij k k p k P X jX i k* . Given is a single life policy at time of valuation *t*

biometric shock . The one-year *solvency capital requirement* (SCR) for this single policy is

Similarly to the decomposition (7) the *Solvency II target capital* (upper index S2 in quantities)

<sup>2</sup> . *<sup>S</sup>*

where *T* denotes the time horizon, which may depend on the life policy, and *<sup>f</sup> 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*

<sup>2</sup> / . *S Z*

By using a matrix of transition shocks *ij* , we obtain formulas for the Markov chain

, *ij*

which is associated to *ij p k* , for a permanent shift of amount *ij* over all contracts and years 0,1,2,... *k* . In the current specification one has 0.15 *AD* (permanent 15% increase in mortality rates at each age for jumping from the alive state "A" to the dead state "D" for the mortality risk), 0.20 *AD* (permanent 20% decrease in mortality rates at each age for jumping from "A" to "D" for the longevity risk), and 0.35 *AJ* respectively 0.25 *AJ* (increase of 35% in disability rates for the next year, respectively a permanent 25% increase

1

*T t k t CoC f t k k RM i v SCR* 

, . *Z Z*

,

(see Section 6.3). For completeness we briefly recall the QIS5 standard approach.

**6.1. Solvency II standard approach** 

with net premium reserve *<sup>Z</sup> Vt* . Denote by *<sup>Z</sup>*, *Vt*

is understood as the sum of the SCR and a risk margin defined by

under the Solvency II standard approach defined by the quotient

model. Consider the *shifted biometric transition probabilities* defined by

Based on the conditional mean and variance we approximate the distribution function of *Zt* by a gamma distribution as in [1], Section 5. Denote this approximation by ( ) ( ) Pr , 1,..., *<sup>i</sup> i t t <sup>t</sup> F x Z xE i n* . Then, recalling the gamma distribution function, one has

$$F\_t(\mathbf{x}) = \mathbf{G}(\beta\_t \mathbf{x}; \alpha\_t) = \frac{1}{\Gamma(\alpha\_t)} \cdot \int\_0^{\beta\_t \mathbf{x}} t^{\alpha\_t - 1} e^{-t} dt, \quad \alpha\_t = \frac{1}{k\_t^2}, \quad \beta\_t = \frac{1}{k\_t^2 \mu\_t}. \tag{45}$$

where , *t t k* are the conditional mean and coefficient of variation of *Zt* (obtained from (43)- (44)). In this setting, the solvency capital ratio formulas (42) take the forms

$$\rm{SR}\_t^{\rm{VaR}} = \bf{z}\_{1-\varepsilon} \left( k\_t^{-2} \right) \cdot k\_t^2 - 1, \rm{SR}\_t^{\rm{CVaR}} = \bf{z}\_{1-\varepsilon} \left( k\_t^{-2} \right) \cdot k\_t^2 \cdot \frac{\rm{g} \left( \bf{z}\_{1-\varepsilon} \left( k\_t^{-2} \right) \cdot k\_t^{-2} \right)}{\varepsilon}, \tag{46}$$

where <sup>1</sup> <sup>1</sup>*z G* ( ) : (1 ; ) denotes the (1 ) -quantile of the standardized gamma *G x*(; ) and *g*( ; ) '( ; ) *x Gx* 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.
