**6. 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.

#### **6.1. Solvency II standard approach**

14 Risk Management – Current Issues and Challenges

( )

*j S*

where , *t t* 

*G x*(; ) 

**approach** 

where <sup>1</sup>

 

<sup>1</sup>*z G* ( ) : (1 ; )

 and *g*( ; ) '( ; ) *x Gx* 

*i*

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

*i i*

( )

*k S*

 

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

( ) ( )

*i i kj i k i*

*va t b t*

 

 2 2 <sup>1</sup> 1, *VaR t tt SR z k k* 

denotes the (1 )

 

components. This is the subject of Section 6.3.

( 1) ( )

*EC X k b t t*

( ) ,( ) ,( )

*i S i S i t i k i*

*EC t t*

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

2 2

*gz k k*


(46)

;

*k k*

    (44)

*t va t p t*

*ii i k i kj i kj i*

 

 

( )

() () ,

 

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

*t t <sup>t</sup> F x Z xE i n* . Then, recalling the gamma distribution function, one has

0

*t tt t t*

*t t x*

1

*t*

*k* are the conditional mean and coefficient of variation of *Zt* (obtained from (43)-

2 2

*SR z k k*

1

*t tt*

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

**6. Comparing the standard approach with variants of the stochastic** 

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

*t t tt*

(45)

<sup>1</sup> 2 2

denotes its probability density. The limiting results for a

, *t t CVaR*

<sup>1</sup> 1 1 () ( ; ) ,, , ( )

( ) ( ),

( ) 2 2 () () ( )

*E C E C X k PX k*

*k S i ii i t tt t*

 

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

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

( ) () ()

*Fx G x t e dt*

(44)). In this setting, the solvency capital ratio formulas (42) take the forms

*i i i i t t ki ki*

> 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* with net premium reserve *<sup>Z</sup> Vt* . Denote by *<sup>Z</sup>*, *Vt* the value of the reserves subject to a biometric shock . The one-year *solvency capital requirement* (SCR) for this single policy is

$$\text{SCR}\_{\text{t}} = V\_{\text{t}}^{Z,\Lambda} - V\_{\text{t}}^{Z}. \tag{47}$$

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

$$T\mathcal{C}\_t^{S2} = \mathcal{SCR}\_t + \mathcal{RM}\_t.\tag{48}$$

$$\mathbf{RM}\_t = \mathbf{i}\_{\text{CoC}} \cdot \sum\_{k=1}^{T-t} \mathbf{v}\_f^k \cdot \mathbf{SCR}\_{t+k'} \quad \mathbf{i}\_{\text{CoC}} = \mathbf{6\%} \mathbf{o}\_t \tag{49}$$

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* under the Solvency II standard approach defined by the quotient

$$\text{SSR}^{S2}\_t = \text{SCR}\_t / V^Z\_t. \tag{50}$$

By using a matrix of transition shocks *ij* , we obtain formulas for the Markov chain model. Consider the *shifted biometric transition probabilities* defined by

$$p\_{ij}^{\Lambda\_{\overline{ij}}} \left( k \right)\_{\prime} \tag{51}$$

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

in disability rates at each age in following years for jumping from "A" to the disability state "J" for the disability risk). To calculate the portfolio reserve *<sup>Z</sup> Vt* in (39) and the corresponding shifted value *<sup>Z</sup>*, *Vt* under a matrix of transition shocks *ij* , we use (34) and the backward recursion formula (18) to get

$$\begin{aligned} V\_t^Z &= \sum\_{i=1}^n V\_{t\_i}^{Z,(i)} = \sum\_{i=1}^n \sum\_{k \in S} V\_{t\_i}^{k,(i)} \cdot P\left(X\_{t\_i}^{(i)} = k \middle| E\_{t\_i}^{(i)}\right) \\ V\_{t\_i}^{k,(i)} &= \sum\_{j \in S} p\_{kj}(t\_i) \cdot \left\{ \upsilon \cdot V\_{t\_i+1}^{j,(i)} + \upsilon \cdot a\_{kj}^{(i)}(t\_i+1) + b\_k^{(i)}(t\_i) \right\} - \pi\_k^{(i)}(t\_i) \end{aligned} \tag{52}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 17

 

(57)

*<sup>k</sup>* of *Zt*

 

2 2

;

*t t*

 

are obtained from the

.

(59)

*Z*

1

(61)

 

2 2 ( ) ( ; ), ( ) , ( ) / , *t tt t t t t t Fx G x k k*

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

2 2 , ( ), , ,

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

, 1,...,

*t tt t*

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

, 1 ,, , (1 ) (1 ) , . *VaR <sup>Z</sup> CVaR <sup>Z</sup> <sup>t</sup> <sup>t</sup> tt t <sup>t</sup> t t SC SCR k V SC SCR k V*

*SCR z k k V*

corresponding solvency capitals converge to those of normal distributions such that

Asymptotically, the solvency capital ratios tend to the following minimum values

0 0

diversified away, and, as expected, only the parameter/systematic risks remain.

**6.3. Stochastic approach: Poisson-gamma model of biometric transition** 

*t t*

, ,

*t t <sup>Z</sup> k k <sup>t</sup> SCR SR SR V* 

lim lim .

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

For simplicity let us fix the states *i j* , of the transition probabilities , 0,1,2,... *ij pkk* . 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

*VaR CVaR t*

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

<sup>1</sup> 2 2 ,

(60)

*gz k k*

(58)

, , ( ),

*SC CVaR Z E i n V*

*i CVaR i Z tt t t*

1

1

where the conditional mean and coefficient of variation , *t t*

expressions

$$\begin{split} \boldsymbol{V}\_{t}^{\boldsymbol{Z},\boldsymbol{\Lambda}} &= \sum\_{i=1}^{n} \boldsymbol{V}\_{t\_{i}}^{\boldsymbol{Z},\boldsymbol{\Lambda}(i),\boldsymbol{\Lambda}} = \sum\_{i=1}^{n} \sum\_{\boldsymbol{t} \in \mathcal{S}} \boldsymbol{V}\_{t\_{i}}^{k,(i),\boldsymbol{\Lambda}} \cdot \boldsymbol{P} \Big(\mathbf{X}\_{t\_{i}}^{(i),\boldsymbol{\Lambda}} = \boldsymbol{k} \Big| \boldsymbol{F}\_{t\_{i}}^{(i),\boldsymbol{\Lambda}} \Big), \\ \boldsymbol{V}\_{t\_{i}}^{k,(i),\boldsymbol{\Lambda}} &= \sum\_{j \in \mathcal{S}} p\_{kj}^{\Lambda\_{ij}} \{\boldsymbol{t}\_{i}\} \cdot \Big{(}\boldsymbol{v} \cdot \boldsymbol{V}\_{t\_{i}+1}^{j,(i),\boldsymbol{\Lambda}} + \boldsymbol{v} \cdot \boldsymbol{a}\_{kj}^{(i)} \{\boldsymbol{t}\_{i}+1\} + \boldsymbol{b}\_{k}^{(i)} \{\boldsymbol{t}\_{i}\} \Big{)} - \pi\_{k}^{(i),\boldsymbol{\Lambda}} \{\boldsymbol{t}\_{i}\}. \end{split} \tag{53}$$

Similarly to (47)-(50) and using (52)-(53) we obtain the risk capital formulas

$$\begin{aligned} \text{SCR}\_{t} &= V\_{t}^{Z,\Lambda} - V\_{t}^{Z} \quad \text{TC}\_{t}^{S2} = \text{SCR}\_{t} + \text{RM}\_{t'}\\ \text{RM}\_{t} &= i\_{\text{CoC}} \quad \sum\_{k=1}^{\max\{\alpha - \{x\_{i} + t\_{i}\}\}} v\_{f}^{k} \cdot \text{SCR}\_{t+k'} \quad \text{SR}\_{t}^{S2} = \frac{\text{SCR}\_{t}}{V\_{t}^{Z}} . \end{aligned} \tag{54}$$

#### **6.2. Stochastic approach: Shifting the biometric transition probability matrix**

Following the Sections 5.2 and 6.1, we consider the "shifted" random present value *Zt* of future cash-flows of the portfolio at time *t* with conditional mean and variance

( ), , 2 ( ), ( ), 0 1 <sup>2</sup> ( ), <sup>2</sup> ( ), ( ), 2 ( ), 2 0 1 , 1,..., , , 1,..., ()() , ( ) *i ii i ii i <sup>n</sup> i ii <sup>Z</sup> t t tt t t i i ii <sup>n</sup> t tt i i t E Z E i n V Var Z E i n v Var C E C PE EC v P E* (55) ( ) ( ) ( ), ,( ), ,( ), ( ), 2 2 ( ), ( ), ( ), <sup>2</sup> ( ), ( ), <sup>2</sup> ( ) ( ), ( ) ( ) ( ), () () , ( ) () () ( *<sup>i</sup> <sup>i</sup> <sup>i</sup> i i <sup>i</sup> <sup>i</sup> i i i S i S i t i ki k S i ii i t tt t k S i i i i t t ki k i i kj E C t t E C E C X k PX k EC X k b t t va t* ( ) ( ), ( ) ( ), ( ) 1) ( ) ( ) ( 1) ( ). *kj i i ii i i ki k i kj i kj i j S b t t va t p t* (56)

The distribution of *Zt* conditional on contract survivals at time *t* is again approximated by a gamma distribution denoted by

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 17

$$F\_t^\Lambda(\mathbf{x}) = \mathbf{G}(\mathcal{J}\_t^\Lambda \mathbf{x}; \alpha\_t^\Lambda), \quad \alpha\_t^\Lambda = (k\_t^\Lambda)^{-2}, \; \mathcal{J}\_t^\Lambda = (k\_t^\Lambda)^{-2} / \,\mu\_t^\Lambda,\tag{57}$$

where the conditional mean and coefficient of variation , *t t <sup>k</sup>* of *Zt* 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

16 Risk Management – Current Issues and Challenges

and the backward recursion formula (18) to get

1 1

*i i kS*

*i i*

1 1

*i i kS*

*kj i i*

*j S*

*j S*

corresponding shifted value *<sup>Z</sup>*, *Vt*

2

*v*

0 1

( )

*j S*

The distribution of *Zt*

*i*

( )

*i kj*

*va t*

a gamma distribution denoted by

*i i*

(

in disability rates at each age in following years for jumping from "A" to the disability state "J" for the disability risk). To calculate the portfolio reserve *<sup>Z</sup> Vt* in (39) and the

,( ) ,( ) ( ) ( )

*i i ii*

, ,( ), ,( ), ( ), ( ),

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

*V V V P X kE*

Similarly to (47)-(50) and using (52)-(53) we obtain the risk capital formulas

*ZZS*

 , 2 max ( )

*i i*

*SCR V V TC SCR RM*

*tt t t t t x t*

**6.2. Stochastic approach: Shifting the biometric transition probability matrix** 

Following the Sections 5.2 and 6.1, we consider the "shifted" random present value *Zt*

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

1

future cash-flows of the portfolio at time *t* with conditional mean and variance

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

, ( )

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

( ), 2

( )

( ) ( ),

*i ii i*

*i*

( ), 2 2 ( ), ( ), ( ),

*E C E C X k PX k*

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

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

( ) () ()

*i i i i t t ki k i*

*EC X k b t t*

*b t*

() () ,

*k S i ii i t tt t*

 

()()

( ), ,( ), ,( ),

*i S i S i t i ki*

*E C t t*

*i t*

( )

*k S*

 

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

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

*E C PE EC*

*P E*

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

*V V V P X kE*

,( ) ,( ) ( ) () () 1

*k i j i i ii t t kj i kj i k i k i*

*V p t vV va t b t t*

,( ), ,( ), ( ) ( ) ( ), 1

*t CoC f tk t Z*

( ), , 2 ( ), ( ),

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

*i ii*

 

> 

 

 

*i ii i i ki k i kj i kj i*

 

( ), ( ) ( ), ( ) 1) ( ) ( ) ( 1) ( ). *kj*

conditional on contract survivals at time *t* is again approximated by

*SCR RM i v SCR SR*

*V p t vV va t b t t*

*i i ii*

*k i j i i ii t t kj i kj i k i k i*

( ) ( 1) ( ) ( ),

, ,

*k t*

( ) ( 1) ( ) ( ).

under a matrix of transition shocks *ij* , we use (34)

,

2

(54)

*k S t*

, .

*V*

0 1

 *t va t p t*

*i*

(52)

(53)

of

(55)

(56)

,

2 2 , ( ), , , 1 1 , 1,..., 1 , *<sup>i</sup> VaR i Z Z t t <sup>t</sup> t t tt t SC VaR Z E i n V SCR z k k V* (58) , , ( ), 1 2 2 <sup>1</sup> 2 2 , 1 , 1,..., ; . *i CVaR i Z tt t t t t Z t tt t SC CVaR Z E i n V gz k k SCR z k k V* (59)

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

$$\mathbf{SC}\_{t}^{\Lambda, \text{VaR}} = \mathbf{SCR}\_{t} + \boldsymbol{\Phi}^{-1}(\mathbf{1} - \boldsymbol{\varepsilon})k\_{t}^{\Lambda}V\_{t}^{Z,\Lambda}, \quad \mathbf{SC}\_{t}^{\Lambda, \text{CVaR}} = \mathbf{SCR}\_{t} + \frac{\boldsymbol{\varphi}\Big[\boldsymbol{\Phi}^{-1}(\mathbf{1} - \boldsymbol{\varepsilon})\Big]}{\boldsymbol{\varepsilon}}k\_{t}^{\Lambda}V\_{t}^{Z,\Lambda}.\tag{60}$$

Asymptotically, the solvency capital ratios tend to the following minimum values

$$\lim\_{k\_t^{\Lambda} \to 0} \mathcal{S} \mathcal{R}\_t^{\Lambda, VaR} = \lim\_{k\_t^{\Lambda} \to 0} \mathcal{S} \mathcal{R}\_t^{\Lambda, CVaR} = \frac{\mathcal{S} \mathcal{R} \mathcal{R}\_t}{V\_t^Z}. \tag{61}$$

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.

#### **6.3. 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 pkk* . 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 pkk* , for the given fixed states, which is based on an initial cohort of size *<sup>t</sup>* at time *t* . Let *Dt* 1 denote the random number of transitions produced by the cohort in the time period *t t* 1, , 1,2,... . For the first time period 1 , we assume that there is no experience available and that the random number of transitions is conditional Poisson distributed such that

$$D\_t \sim \text{Po}\left(\ell\_t \mathbf{Q}\_t\right), \quad \mathbf{Q}\_t \sim \text{Gamma}\left(\alpha, \frac{\beta}{p\_{\dot{\eta}}(t)}\right). \tag{62}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 19

2 1

 

0 1

*k t ij*

 

1

*t k*

*pt k*

1

*t k*

*pt k*

( 1)

1

*d*

( ) ( ), ( ) ( ), 1,2,...,

 

1

*t k ij*

( 1)

, ,..., ~ , , . <sup>1</sup> ( 1)

1

1

*k*

 

, ,..., ( 1)

*d*

1

*t k ij*

The conditional expected number of transitions compares with the given expected number

11 2 1 1

*t tt t t ij*

*ED d d d p t*

*k*

( ) ( 1).

1

If biometric experience is consistent with what is expected, the quotient of both expected values remains constant over time. On the other hand, if experience is better (worse) than

In practice one proceeds as follows. Given a fixed time *t* , consider for each pair of fixed states the Poisson-Gamma transition probabilities obtained from (64) and (70) defined by

*k*

1 0

*ij ij ij ij*

*p t pt p t p t*

*PG PG PG PG k ij k ij ij ij*

*pt pt pt k p*

1

( ) ( ) 1 ( ) , (0) 1, 0,1,...

Replacing everywhere in the formulas (55)-(59) the superscripts *ij* by the superscript *PG* for the relevant transition probabilities and using (71) in calculations, we obtain portfolio VaR & CVaR solvency capital formulas under the Poisson-Gamma model of biometric transition similar to (58) and (59). Similar limiting results apply. An implementation requires a detailed specification. To be consistent with the Solvency II standard approach, one can assume that future transitions deviate systematically from the biometric life table according to the

( )

1

1

 

 

 

(71)

*k*

is obtained from the recursion

*t k ij*

*p t*

(68)

 1, 2 ,

, the up-dated cohort

( 1)

(69)

*pt k*

 

(70)

*t ij*

*p t*

. The corresponding predictive negative binomial

*t ij*

2 1

 

2

*p t D d NB d*

Iterating the above Bayesian scheme, one generalizes as follows. At time *t*

 1, 

~ ,, . <sup>1</sup> ( )

 

1 2

*t t t*

having observed the annual number of transitions 1 2 , , ..., *tt t dd d*

size for the next time period *t t*

distribution of the number of transitions is then given by

*t tt t t k*

*D d d d NB d*

 

1 1

*E D p tp t*

*t t ij ij*

 

expected, the same quotient will increase (decrease) over time.

*PG PG k*

prescribed shock *ij* for the given fixed states, such that

 

<sup>122</sup> , *tk tk tk d* 2,3,..., *k*

as follows:

11 2

It follows that the unconditional distribution of the number of transitions in the first time period is negative binomially distributed such that

$$D\_t \sim \text{NB}\left(\alpha, \frac{\theta\_1}{\theta\_1 + 1}\right), \quad \theta\_1 = \frac{\beta}{p\_{ij}(t)}.\tag{63}$$

In contrast to the expected number of transitions ( ) *t ij p t* predicted by the biometric transition probability matrix, one has

$$E\left[\left.D\_t\right.\right] = \frac{\alpha}{\beta} \cdot \ell\_t p\_{\vec{\eta}}(t) \,. \,\tag{64}$$

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 *<sup>t</sup> d* observed in the cohort over the first time period is available, and let *t tt* <sup>1</sup> *d* be the up-dated cohort size. A calculation shows that the posterior distribution of *Qt* conditional on the information *D d t t* is Gamma distributed

$$\mathcal{Q}\_t \Big| d\_t \sim \text{Gamma}\left(\alpha + d\_{t'} \frac{\beta + \ell\_{i\bar{j}} p\_{i\bar{j}}(t)}{p\_{i\bar{j}}(t)}\right),\tag{65}$$

which shows that the initial structural systematic risk parameters are up-dated as follows:

$$\left(\left(a,\beta\right)\quad\rightarrow\quad\left(a+d\_{t'}\beta+\ell\_{t}p\_{ij}(t)\right).\tag{66}$$

Passing to the second time period *t t* 1, 2 , we assume similarly to the first period that

$$D\_{t+1} \Big| d\_t \sim \text{Po}\left(\ell\_{t+1} Q\_{t+1}\right), \quad Q\_{t+1} \Big| d\_t \sim \text{Gamma}\left(\alpha + d\_t, \frac{\beta + \ell\_t p\_{\dot{\eta}}(t)}{p\_{\dot{\eta}}(t+1)}\right). \tag{67}$$

This implies a so-called predictive distribution of negative binomial type

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 19

$$D\_{t+1} \left| d\_t \sim \text{NB} \left( \alpha + d\_{t'} \frac{\theta\_2}{\theta\_2 + 1} \right) , \quad \theta\_2 = \frac{\beta + \ell\_i p\_{ij}(t)}{\ell\_{t+1} p\_{ij}(t)}. \tag{68}$$

Iterating the above Bayesian scheme, one generalizes as follows. At time *t* 1, 2 , having observed the annual number of transitions 1 2 , , ..., *tt t dd d* , the up-dated cohort size for the next time period *t t* 1, is obtained from the recursion <sup>122</sup> , *tk tk tk d* 2,3,..., *k* . The corresponding predictive negative binomial distribution of the number of transitions is then given by

18 Risk Management – Current Issues and Challenges

initial cohort of size *<sup>t</sup>* at time *t* . Let *Dt*

produced by the cohort in the time period *t t*

transitions is conditional Poisson distributed such that

period is negative binomially distributed such that

transition probability matrix, one has

Gamma distributed

transition probabilities , 0,1,2,... *ij pkk* , for the given fixed states, which is based on an

1 , we assume that there is no experience available and that the random number of

~ ,~ , . ( ) *t tt t*

It follows that the unconditional distribution of the number of transitions in the first time

1

In contrast to the expected number of transitions ( ) *t ij p t* predicted by the biometric

( ). *<sup>t</sup> t ij ED p t* 

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 *<sup>t</sup> d* observed in the cohort over the first time period is available, and let *t tt* <sup>1</sup> *d* be the up-dated cohort size. A calculation shows that the posterior distribution of *Qt* conditional on the information *D d t t* is

~ ,, ( )

 

~ ,~ , . ( 1)

 

*p t Q d Gamma d p t*

which shows that the initial structural systematic risk parameters are up-dated as follows:

 

*p t D d Po Q Q d Gamma d p t*

*t t t*

*t t tt t t t*

This implies a so-called predictive distribution of negative binomial type

 

1 11 1

Passing to the second time period *t t* 1, 2

To model a systematic deviation from the expectation, one assumes that the quotient /

 

1 ~,, . 1 () *<sup>t</sup>*

*D NB*

1

 

*ij*

*p t*

( )

, *d pt t t ij* , (). (66)

, we assume similarly to the first period that

*ij*

 

(67)

( )

*t ij*

*t ij*

*ij*

*D Po Q Q Gamma p t*

 1, , 1,2,...

*ij*

 

(62)

. (64)

 

1 denote the random number of transitions

. For the first time period

(63)

(65)

 

$$D\_{t+\tau-1} \left| d\_{t'}, d\_{t+1'}, \dots, d\_{t+\tau-2} \sim \text{NB} \left( a + \sum\_{k=0}^{\tau-2} d\_{t+k'} \frac{\theta\_{\tau}}{\theta\_{\tau} + 1} \right), \quad \theta\_{\tau} = \frac{\beta + \sum\_{k=1}^{\tau-1} \ell\_{t+k-1} p\_{\dot{\gamma}} (t+k-1)}{\ell\_{t+\tau-1} p\_{\dot{\gamma}} (t+\tau-1)}. \tag{69}$$

The conditional expected number of transitions compares with the given expected number as follows:

$$\begin{split} &E\left[D\_{t+\tau-1}\big|d\_{t},d\_{t+1},...,d\_{t+\tau-2}\right] = \frac{\alpha + \sum\_{k=1}^{\tau-1} d\_{t+k-1}}{\beta + \sum\_{k=1}^{\tau-1} \ell\_{t+k-1} p\_{ij}(t+k-1)} \cdot \ell\_{t+\tau-1} p\_{ij}(t+\tau-1) \\ &\overset{\leq}{\geq} \sum\_{k} \left[D\_{t+\tau-1}\right] = \ell\_{t} \cdot \underset{\tau-1}{\cdot} p\_{ij}(t) p\_{ij}(t+\tau-1). \end{split} \tag{70}$$

If biometric experience is consistent with what is expected, the quotient of both expected values remains constant over time. On the other hand, if experience is better (worse) than expected, the same quotient will increase (decrease) over time.

In practice one proceeds as follows. Given a fixed time *t* , consider for each pair of fixed states the Poisson-Gamma transition probabilities obtained from (64) and (70) defined by

$$\begin{split} p\_{ij}^{\rm PC}(t) &= \frac{\alpha}{\beta} \cdot p\_{ij}(t), \quad p\_{ij}^{\rm PC}(t+\tau) = \frac{\alpha + \sum\_{k=1}^{\tau} d\_{t+k-1}}{\beta + \sum\_{k=1}^{\tau} \ell\_{t+k-1} p\_{ij}(t+k-1)} \cdot p\_{ij}(t+\tau), \quad \tau = 1, 2, \ldots, \\\ p\_{ik} p\_{ij}^{\rm PC}(t+\tau) &= \,\_{k-1} p\_{ij}^{\rm PC}(t+\tau) \cdot \Big(1 - p\_{ij}^{\rm PC}(t+\tau+k-)\big)\_{i \ge 0} p\_{ij}^{\rm PC}(0) = 1, \quad \tau = 0, 1, \ldots \end{split} \tag{71}$$

Replacing everywhere in the formulas (55)-(59) the superscripts *ij* by the superscript *PG* for the relevant transition probabilities and using (71) in calculations, we obtain portfolio VaR & CVaR solvency capital formulas under the Poisson-Gamma model of biometric transition similar to (58) and (59). Similar limiting results apply. An implementation requires a detailed specification. To be consistent with the Solvency II standard approach, one can assume that future transitions deviate systematically from the biometric life table according to the prescribed shock *ij* for the given fixed states, such that

$$\ell \cdot \ell\_{s+t+k} = \ell\_{s+t+k-1} - d\_{s+t+k-1}, \quad d\_{s+t+k-1} = (1 - \Delta) \cdot \ell\_{s+t+k-1} q\_{s+t+k-1}, \quad k = 1, 2, \dots \tag{72}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 21

*ij k* , 1,2,3 , 0,1,2,... (73)

1

Type 1:

Type 2:

<sup>1</sup>*a k*( ) 

1,

<sup>12</sup> *a k*( )

*k n*

1,

( ) 1 ,

*iq i q*

*ij x k xk xk xk*

*pk r r q q*

[6], p.129, or Chuard [25] for a detailed historical background.

**7.2. State dependent actuarial reserves and net level premiums** 

state. In our notations the *payment functions* of this contract are defined by

<sup>2</sup> , () *a k*

, 0,1,..., 1

0, 1,2,..., 1

will attain age *x k* in the disabled state without recovery) are given by

*k n*

*<sup>a</sup> Pk n*

*k n*

0 01

 

*a a xk xk xk xk*

*i i*

For a *n* -year endowment contract with waiver of premium by disability without recovery from disability, one has 0, 1,2,..., 1 *x k rk 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 "**E**idgenössische **V**ersicherungs**K**asse", e.g. Koller

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 (:) *a a P P xn* , where the upper index indicates that the premium is only due if the contract remains in the active life

To describe the state and time dependent actuarial reserves , 1,2,3 , 0,1,..., *<sup>i</sup> Vi k n <sup>k</sup>* , one

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

> <sup>1</sup> <sup>1</sup> 1 10 , 1, 1, ..., . *a aa <sup>a</sup> k x k x xk xk x ppq i p k x*

Similarly, the *disabled survival probabilities* (probability a life in the disabled state at age *x*

<sup>1</sup> 1 , 1, 1, ..., 1 0 *i ii i k x k x xk x ppq p k x*

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

needs the survival probabilities of staying in the active or disabled state. Denote by

1,

<sup>13</sup> , *ak k n* ( ) 1, 1,2,..., , <sup>21</sup> *a k*( ) 0.

*k n*

0, 1,2,..., 1

<sup>3</sup> , *a k*() 0

the

(74)

(75)

*k n*

This choice is consistent with the expected number of transitions in the first period *st st ED d* if in (64) one sets (1 ) . Assume further that 100 , which implies a coefficient of variation for *Ds t* 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.

### **7. 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].

#### **7.1. 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

**Figure 1.** Markov chain states and their jump probabilities

The possible state changes occur with the following probabilities


The states are *S* 1,2,3 , , *aid* . The one-step transition probabilities are given by

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 21

$$p\_{ij}(k) = \begin{pmatrix} 1 - i\_{\mathbf{x}+k} - q\_{\mathbf{x}+k}^a & i\_{\mathbf{x}+k} & q\_{\mathbf{x}+k}^a \\ r\_{\mathbf{x}+k} & 1 - r\_{\mathbf{x}+k} - q\_{\mathbf{x}+k}^i & q\_{\mathbf{x}+k}^i \\ 0 & 0 & 1 \end{pmatrix}, \ i, j \in \{1, 2, 3\}, k = 0, 1, 2, \dots \tag{73}$$

For a *n* -year endowment contract with waiver of premium by disability without recovery from disability, one has 0, 1,2,..., 1 *x k rk 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 "**E**idgenössische **V**ersicherungs**K**asse", e.g. Koller [6], p.129, or Chuard [25] for a detailed historical background.

#### **7.2. 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 (:) *a a P P xn* , where the upper index indicates that the premium is only due if the contract remains in the active life state. In our notations the *payment functions* of this contract are defined by

Type 1:

20 Risk Management – Current Issues and Challenges

*st st ED d* if in (64) one sets

(1 )

*x t*

*a*

*i*

11 1 1 1 , (1 ) , 1,2,... . *stk stk stk stk stk stk d d q k* (72)

implies that the transition probabilities (71) coincide with the corresponding

. Assume further that 100

*<sup>i</sup> q <sup>x</sup> <sup>t</sup>*

, which implies a

This choice is consistent with the expected number of transitions in the first period

coefficient of variation for *Ds t* equal to 10%. One shows that the choice (72) with

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

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

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

active disabled

*<sup>x</sup> <sup>t</sup> i*

*<sup>x</sup> <sup>t</sup> r*

dead

**7. The endowment contract with waiver of premium by disability** 

(1 )

of effective observed numbers of transitions as time elapses.

further possibilities consult Example 2.1 in Christiansen et al. [24].

contract issue changes state at time 0 *t* according to the following diagram

a i

d

**7.1. Markov model for mortality and disability risks** 

**Figure 1.** Markov chain states and their jump probabilities

*x t q* : one-year probability of active mortality at time t

*x t q* : one-year probability of disabled mortality at time t

*i* : one-year probability of disability at time t

*a <sup>x</sup> <sup>t</sup> q*

*x t r* : one-year probability of recovery at time t

The possible state changes occur with the following probabilities

The states are *S* 1,2,3 , , *aid* . The one-step transition probabilities are given by

$$a\_1(k) = \begin{cases} \ -P^t, \ k = 0, 1, \dots, n-1 \\\ 1, \ k = n \end{cases}, a\_2(k) = \begin{cases} \ 0, k = 1, 2, \dots, n-1 \\\ \ 1, k = n \end{cases}, a\_3(k) = 0$$

Type 2:

$$a\_{12}(k) = \begin{cases} \quad 0, k = 1, 2, \dots, n - 1 \\\quad 1, k = n \end{cases}, a\_{13}(k) = 1, k = 1, 2, \dots, n, \quad a\_{21}(k) = 0.1$$

To describe the state and time dependent actuarial reserves , 1,2,3 , 0,1,..., *<sup>i</sup> Vi k n <sup>k</sup>* , one 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

$$\mathbf{p}\_{k}p\_{\mathbf{x}}^{a} = \prescript{}{k-1}{p}\_{\mathbf{x}}^{a} \cdot \left(\mathbf{1} - \mathbf{q}\_{\mathbf{x}+k-1}^{a} - \mathbf{i}\_{\mathbf{x}+k-1}\right) \quad \prescript{}{0}{p}\_{\mathbf{x}}^{a} = \mathbf{1}, \quad k = \mathbf{1}, \dots, a \mathbf{o} - \mathbf{x}. \tag{74}$$

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

$$p\_k p\_x^i = \_{k-1} p\_x^i \cdot \left(1 - q\_{x+k-1}^i\right)\_{\prime} \quad \_0 p\_x^i = 1, \quad k = 1, \ldots, \alpha - \infty \tag{75}$$

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

$$a^a(\mathbf{x}:n) = \sum\_{k=0}^{n-1} v^k \cdot \, \_k p^a\_{\mathbf{x}'} \text{ for the } \, n-year \text{ active life annuity} \tag{76}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 23

21 1 2 1

( : ) ( : ) / ( : ). *a aa P xn A xn a xn* (85)

(84)

(86)

(88)

(87)

<sup>0</sup> *V* 0 . Using (79) one obtains the explicit formula (use (76), (84))

Besides the state dependent actuarial

1 1

, 1,2,..., 1, 0,

2

( )

*EC PE t k Var Z E v t n P E*

(:) (:) , 1,..., 2,

() ( )

1 1

(83)

1 11

Using these results and proceeding through backward induction, one obtains the following explicit formula for the evaluation of the APV of future benefits for a life in the active state

1 12 2

*j x xj k j xj xk j x xj n j xj*

*k x xk n x xn*

*n k <sup>n</sup> k a i i na i*

*v pi p q v pi p*

The net level premium is determined by the actuarial equivalence principle, which states

We determine the conditional mean and variance given survival of the time-*t* prospective

reserves (79), we consider the net premium reserve (35), which coincides with the

*PX a PX i V p Vi p V V t n*

1 1 <sup>0</sup>

( ) , 1,2,..., 1, ( ) 0.

Furthermore, a calculation based on Theorem 5.1 yields the following conditional variances

( ): , 0,1,..., 1,

*<sup>S</sup> n k*

*x a a aa i i k xk k x x k x xk*

*EC P x n vv P x n q p vi p q k n*

<sup>1</sup> 1 1 21 1 ( : ) ( : ) (1 ) , *<sup>a</sup> <sup>a</sup> a a i i xn n x x n x xn P x n vv P x n p p vi p q*

*k k*

2 2 <sup>1</sup>

 

, 0,1,2,..., 1,

*PX i i p t n PX i PE p i p t n PE*

1 2

, 0,1,..., 1, () ()

1 1 0

*Ax n kk v p q v p k n*

( :) , 2,3,..., 1.

*j xnk xnkj k xnk*

1

*i j i i ki*

1 2 3

(:) (1 ) *<sup>n</sup> <sup>a</sup> k aa n a a*

2 0 0

**7.3. Conditional mean and variance of the prospective insurance loss** 

, 0,1,..., 1.

1 2

*t tt t t t t t t*

*V E Z E V P X aE V P X iE*

*a*

*PX a p t n*

2 2 2

0

*Z k tk k*

(:) (:) ,

*a a a*

2 2 <sup>2</sup>

*t t a i t t tx xt x*

1 2 1 1

*PE PE pi p*

*i*

 

*a i t t t tx t xt x*

*k j j*

*A xn v p q v p q*

*k*

*j*

1

1

1

0

2 2

*EC P x n vv P x n q*

2 2

*t t t*

0

*n*

*E C*

*n t <sup>k</sup> t t k k*

*Z vC t n*

*t tx*

*t xt x a i t tx xt x*

*k*

that at contract issue <sup>1</sup>

insurance loss

conditional mean

:

*Z*

1

$$a^i(\mathbf{x}:n) = \sum\_{k=0}^{n-1} v^k \cdot \, \_k p^i\_{\mathbf{x}'} \text{ for the } \, n-year \text{ divided life annuity} \tag{77}$$

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 (:) *<sup>a</sup> A x n* (respectively (:) *<sup>i</sup> 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

$$\begin{aligned} V\_k^1 &= \upsilon \cdot \left( q\_{x+k}^a + p\_{x+k}^a V\_{k+1}^1 + i\_{x+k} V\_{k+1}^2 \right) - P^a, \quad k = 0, 1, \ldots, n-1, \quad V\_n^1 = 1, \\ V\_k^2 &= \upsilon \cdot \left( q\_{x+k}^i + p\_{x+k}^i V\_{k+1}^2 \right), \quad k = 2, 3, \ldots, n-1, \quad V\_n^2 = 1. \end{aligned} \tag{78}$$

One has <sup>2</sup> <sup>0</sup> *<sup>V</sup>* 0 because the life is in the state "a" at contract issue, <sup>2</sup> <sup>1</sup> *V* 0 because the life can only be in the state "i" after at least one year and then no actuarial reserve is available, and <sup>3</sup> 0, 0,1,..., *Vk n <sup>k</sup>* , 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

$$\begin{aligned} V\_k^1 &= A^a(\mathbf{x} + k : n - k) - P^a(\mathbf{x} : n) \cdot a^a(\mathbf{x} + k : n - k), \quad k = 0, 1, \ldots, n - 1, \\ V\_k^2 &= A^i(\mathbf{x} + k : n - k), \quad k = 2, 3, \ldots, n - 1. \end{aligned} \tag{79}$$

On the other hand the APVs of the active life annuities in (79) satisfy the recursions

$$a^a(\mathbf{x} + k \mathbf{:} \, n - k) = \mathbf{1} + \boldsymbol{\upsilon} \cdot \boldsymbol{p}\_{\mathbf{x} + k}^a \cdot a^a(\mathbf{x} + k + \mathbf{1} \, \mathbf{:} \, n - k - \mathbf{1}), \quad k = 0, 1, \ldots, n - 1. \tag{80}$$

Inserting (79) and (80) into (78) one obtains the backward recursions for APVs

$$A^{i}(\mathbf{x}+n-k:k) = \upsilon \cdot \begin{pmatrix} q\_{\mathbf{x}+n-k}^{a} + p\_{\mathbf{x}+n-k}^{a}A^{a}(\mathbf{x}+n-k+1:k-1) \\ + i\_{\mathbf{x}+n-k}A^{i}(\mathbf{x}+n-k+1:k-1) \end{pmatrix}\_{\prime} \quad k=2,3,\ldots,n,\tag{81}$$
 
$$A^{i}(\mathbf{x}+n-k:k) = \upsilon \cdot \left(q\_{\mathbf{x}+n-k}^{i} + p\_{\mathbf{x}+n-k}^{i}A^{i}(\mathbf{x}+n-k+1:k-1)\right)\_{\prime} \quad k=2,3,\ldots,n-1,$$

with the starting values ( 1 : 1) ( 1 : 1) *a i A x n Ax n v* . One sees that the second relationship in (81) is satisfied by the formula

$$A^i(\mathbf{x} + \mathbf{n} - k : k) = \mathbf{1} - d \cdot a^i(\mathbf{x} + \mathbf{n} - k : k), \quad k = \mathbf{1}, \mathbf{2}, \dots, \mathbf{n} - \mathbf{1}, \tag{82}$$

which reminds one of the usual formula for an endowment insurance with disability as single cause of decrement, e.g. Gerber [10], formula (2.15), p.37. Inserting (77) and rearranging one obtains the corresponding explicit sum representation

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 23

$$A^i(\mathbf{x} + \mathbf{n} - k \colon k) = \sum\_{j=1}^{k-1} \mathbf{v}^j \cdot \underset{j-1}{\operatorname{p}} \mathbf{p}^i\_{\mathbf{x} + \mathbf{n} - k} \cdot \mathbf{q}^j\_{\mathbf{x} + \mathbf{n} - k + j - 1} + \mathbf{v}^k \cdot \underset{k-1}{\operatorname{p}} \mathbf{p}^i\_{\mathbf{x} + \mathbf{n} - k'} \quad k = \mathbf{2}, \mathbf{3}, \dots, \mathbf{n} - 1. \tag{83}$$

Using these results and proceeding through backward induction, one obtains the following explicit formula for the evaluation of the APV of future benefits for a life in the active state

22 Risk Management – Current Issues and Challenges

at age *x* is denoted by (:)

One has <sup>2</sup>

1

*<sup>n</sup> <sup>a</sup> k a*

*k a xn v p* 

0 (:) , for the

1

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

*k axn v p* 

0 (:) , for the

*k x*

*k x*

premium is derived. The backward recursive reserve formulas are given by

*a a a*

*k xk xk k n*

( : ), 2,3,..., 1.

Inserting (79) and (80) into (78) one obtains the backward recursions for APVs

*a aa*

*i xnk*

*xnk xnk*

rearranging one obtains the corresponding explicit sum representation

*a a a*

*V Ax kn k k n*

*a xnk xnk*

*i i ii*

with the starting values ( 1 : 1) ( 1 : 1)

relationship in (81) is satisfied by the formula

<sup>0</sup> *<sup>V</sup>* 0 because the life is in the state "a" at contract issue, <sup>2</sup>

*V vq p V k n V* 

*i i*

future premiums one has further the relationships

*i*

1 2

*k*

*k*

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

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

> 1 1 2 1 1 1 22 2 1

can only be in the state "i" after at least one year and then no actuarial reserve is available, and <sup>3</sup> 0, 0,1,..., *Vk n <sup>k</sup>* , because no actuarial reserve is required in case the insured life has died. Since actuarial reserves represent differences between APVs of future benefits and

*V A x kn k P xn a x kn k k n*

On the other hand the APVs of the active life annuities in (79) satisfy the recursions

( : )1 ( 1 : 1), 0,1,..., 1. *<sup>a</sup> a a*

( : ) 1 ( : ), 1,2,..., 1,

which reminds one of the usual formula for an endowment insurance with disability as single cause of decrement, e.g. Gerber [10], formula (2.15), p.37. Inserting (77) and

( 1 : 1) ( :) , 2,3,..., , ( 1 : 1) ( :) ( 1 : 1) , 2,3,..., 1,

*q p Ax n k k <sup>A</sup> x n kk v k n i Ax n k k Ax n kk v q p Ax n k k k n*

*k xk xk k xk k n*

*V vq p V i V P k n V*

, 2,3,..., 1, 1.

( : ) ( : ) ( : ), 0,1,..., 1,

*x k a x kn k vp a x k n k k n* (80)

*a i A x n Ax n v* . One sees that the second

*i i A x n kk da x n kk k n* (82)

*<sup>a</sup> A x n* (respectively (:)

*n year active life annuity* (76)

*n year disabled life annuity* (77)

*<sup>i</sup> A x n* ). Using the backward recursive

(78)

(79)

(81)

<sup>1</sup> *V* 0 because the life

, 0,1,..., 1, 1,

$$\begin{split} A^a(\mathbf{x};n) &= \sum\_{k=1}^{n-1} \boldsymbol{v}^k \cdot \boldsymbol{\uprho}\_{\cdot -1}^a \cdot \boldsymbol{q}\_{\times +k-1}^a + \boldsymbol{v}^n \cdot \boldsymbol{\uprho}\_{\cdot -2}^a \cdot (\mathbf{1} - \boldsymbol{q}\_{\times +n-2}^a) \\ &+ \sum\_{k=2}^{n-1} \boldsymbol{v}^k \cdot \left(\sum\_{j=0}^{k-2} \boldsymbol{p}\_x^a \cdot \boldsymbol{i}\_{\times +j} \cdot \boldsymbol{\uprho}\_{\cdot -2 - j} \boldsymbol{p}\_{x+j+1}^i\right) \cdot \boldsymbol{q}\_{\times +k-1}^i + \boldsymbol{v}^n \cdot \sum\_{j=0}^{n-3} \boldsymbol{p}\_x^a \cdot \boldsymbol{i}\_{x+j} \cdot \boldsymbol{\uprho}\_{\cdot -2 - j} \boldsymbol{p}\_{x+j+1}^i \end{split} \tag{84}$$

The net level premium is determined by the actuarial equivalence principle, which states that at contract issue <sup>1</sup> <sup>0</sup> *V* 0 . Using (79) one obtains the explicit formula (use (76), (84))

$$P^{a}(\mathbf{x}:\mathfrak{n}) = A^{a}(\mathbf{x}:\mathfrak{n}) / \mathfrak{a}^{a}(\mathbf{x}:\mathfrak{n}).\tag{85}$$

#### **7.3. Conditional mean and variance of the prospective insurance loss**

We determine the conditional mean and variance given survival of the time-*t* prospective insurance loss 1 0 , 0,1,..., 1. *n t <sup>k</sup> t t k k Z vC t n* Besides the state dependent actuarial reserves (79), we consider the net premium reserve (35), which coincides with the conditional mean

$$\begin{split} V\_t^Z &:= \mathbb{E}\left[\mathbf{Z}\_t \middle| \mathcal{E}\_t \right] = V\_t^1 \cdot P\left(\mathbf{X}\_t = a \middle| \mathcal{E}\_t \right) + V\_t^2 \cdot P\left(\mathbf{X}\_t = i \middle| \mathcal{E}\_t \right) \\ &= V\_t^1 \cdot \frac{P\left(\mathbf{X}\_t = a\right)}{P\left(\mathcal{E}\_t \right)} + V\_t^2 \cdot \frac{P\left(\mathbf{X}\_t = i \right)}{P(\mathcal{E}\_t)} = \frac{V\_t^1 \cdot i\_t p\_x^a + V\_t^2 \cdot i\_x \cdot i\_{t-1} p\_{x+1}^i}{} \\ &\qquad \cdot \left(\mathcal{P}\left(\mathbf{X}\_t = a\right) = \operatorname\*{arg\,min}\_{t \in \mathcal{P}\_t} \cdot t = 0, 1, 2, \dots, n-1, \\ &\qquad P\left(\mathbf{X}\_t = i\right) = i\_x \cdot i\_{t-1} p\_{x+1}^i \quad t = 1, 2, \dots, n-1, \quad P\left(\mathbf{X}\_0 = i \right) = 0, \\ &\qquad P(\mathcal{E}\_t) = i\_x p\_x^a + i\_x \cdot i\_{t-1} p\_{x+1}^i, \quad t = 1, 2, \dots, n-1, \quad P(\mathcal{E}\_0) = 0. \end{split} \tag{87}$$

Furthermore, a calculation based on Theorem 5.1 yields the following conditional variances

$$\begin{split} \left(\sigma\_{t}^{Z}\right)^{2} & \coloneqq \operatorname{Var}\left[Z\_{t}\middle|\mathcal{E}\_{t}\right] = \sum\_{k=0}^{n-1-k} v^{2^{k}} \cdot \left(\frac{\operatorname{E}\left[\big|\mathcal{C}\_{t+k}^{2}\right] \cdot P(E\_{k}) - \pi^{S}(t+k)^{2}}{P(E\_{k})}\right), \quad t = 0, 1, \ldots, n-1, \\ \operatorname{E}\left[\mathcal{C}\_{0}^{2}\right] & = P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})^{2} + v\big(\boldsymbol{v} + P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})\big) q\_{\boldsymbol{x}\boldsymbol{x}}^{a} \\ \operatorname{E}\left[\mathcal{C}\_{k}^{2}\right] & = \left[P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})^{2} + v\big(\boldsymbol{v} + P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})\big) q\_{\boldsymbol{x}+k}^{a}\right]\_{k} p\_{\boldsymbol{x}}^{a} + v^{2} i\_{\boldsymbol{x}} \cdot \boldsymbol{k}\_{k} p\_{\boldsymbol{x}+k}^{i} q\_{\boldsymbol{x}+k'}^{i} \quad k = 1, \ldots, n-2, \\ \operatorname{E}\left[\mathcal{C}\_{n-1}^{2}\right] & = \left[P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})^{2} + v\big(\boldsymbol{v} + P^{a}(\boldsymbol{x} \cdot \boldsymbol{n})\big)(1 - p\_{\boldsymbol{x}+n-1}^{a}\right]\_{n-1} p\_{\boldsymbol{x}}^{a} + v^{2} i\_{\boldsymbol{x}} \cdot \boldsymbol{n}\_{n-2} p\_{\boldsymbol{x}+1}^{i} q\_{\boldsymbol{x}+n-1}^{i} \end{split} \tag{88}$$

where the savings premiums are determined by the formulas

$$\begin{aligned} \pi^{\mathbb{S}}(\mathbf{0}) &= \upsilon V\_1^1, \\ \pi^{\mathbb{S}}(k) &= \upsilon \Bigl(V\_{k+1}^1 + A^i(\mathbf{x} + k + 1:n-k-1)\Bigr) - \Bigl(V\_k^1 + A^i(\mathbf{x} + k:n-k)\Bigr), \quad k = 1, \dots, n-1, \end{aligned} \tag{89}$$

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 25

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

**x \*qax qax \*ix ix qix**

**30** 0.183% 0.106% 0.064% 0.141% 4.348% 31 0.181% 0.097% 0.064% 0.148% 4.324% 32 0.178% 0.087% 0.064% 0.154% 4.300% 33 0.176% 0.078% 0.063% 0.161% 4.276% 34 0.173% 0.069% 0.063% 0.167% 4.252% **35** 0.171% 0.060% 0.063% 0.174% 4.228% 36 0.174% 0.059% 0.072% 0.186% 4.204% 37 0.177% 0.059% 0.080% 0.198% 4.180% 38 0.180% 0.059% 0.089% 0.210% 4.156% 39 0.183% 0.058% 0.097% 0.222% 4.132% **40** 0.186% 0.058% 0.106% 0.234% 4.108% 41 0.207% 0.079% 0.129% 0.257% 4.084% 42 0.227% 0.099% 0.153% 0.280% 4.061% 43 0.248% 0.120% 0.176% 0.304% 4.037% 44 0.268% 0.141% 0.200% 0.327% 4.014% **45** 0.289% 0.161% 0.223% 0.350% 3.990% 46 0.335% 0.218% 0.285% 0.401% 3.966% 47 0.382% 0.275% 0.347% 0.452% 3.942% 48 0.428% 0.331% 0.408% 0.504% 3.919% 49 0.475% 0.388% 0.470% 0.555% 3.895% **50** 0.521% 0.444% 0.532% 0.606% 3.871% 51 0.586% 0.532% 0.694% 0.743% 3.848% 52 0.650% 0.620% 0.856% 0.881% 3.824% 53 0.715% 0.708% 1.019% 1.018% 3.801% 54 0.779% 0.795% 1.181% 1.156% 3.777% **55** 0.844% 0.883% 1.343% 1.293% 3.754% 56 0.909% 0.892% 1.765% 1.766% 3.731% 57 0.974% 0.900% 2.187% 2.240% 3.707% 58 1.040% 0.908% 2.609% 2.713% 3.684% 59 1.105% 0.916% 3.031% 3.187% 3.660% **60** 1.170% 0.923% 3.453% 3.660% 3.637%

**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

corresponds to a 99.5% Solvency II calibration).

and the probabilities ( ) *<sup>k</sup> P E* are defined in (87). Neglecting the probabilistic terms of second order 1 1 *i i x k x xk i pq* , one obtains the following simpler approximations to (88):

$$\begin{split} &E\left[\boldsymbol{C}\_{k}^{2}\right] = \left\{P^{a}\left(\boldsymbol{x}:\boldsymbol{n}\right)^{2} + \upsilon\left(\upsilon + P^{a}\left(\boldsymbol{x}:\boldsymbol{n}\right)\right)q\_{\boldsymbol{x}+k}^{a}\right\}\cdot\_{k}p\_{\boldsymbol{x}'}^{a} \quad k = 0,\ldots,n-2, \\ &E\left[\boldsymbol{C}\_{n-1}^{2}\right] = \left\{P^{a}\left(\boldsymbol{x}:\boldsymbol{n}\right)^{2} + \upsilon\left(\upsilon + P^{a}\left(\boldsymbol{x}:\boldsymbol{n}\right)\right)(1 - p\_{\boldsymbol{x}+n-1}^{a})\right\}\cdot\_{n-1}p\_{\boldsymbol{x}'}^{a} .\end{split} \tag{90}$$

#### **7.4. 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),

$$(1 - q\_{x+t}^a - i\_{x+t} = \left(1 - \prescript{\*}{}{q}\_{x+t}^a\right) \cdot \left(1 - \prescript{\*}{}{i}\_{x+t}\right) \cdot \tag{91}$$

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 in Saxer [26], p.240. The entries \* *<sup>a</sup> x t <sup>q</sup>* , \* *x t <sup>i</sup>* and *<sup>i</sup> x t q* are taken from the "EVK Table 1950" and the entry *x t i* is taken from the "VZ Table 1950", where VZ stands for "**V**ersicherungskasse **Z**ürich". The missing entries between the 5-year ages are linearly interpolated.

While the standard solvency capital ratio does not depend on the initial cohort size, this is the case for the stochastic approaches. The age at contract issue is either 30 *x* or 40 *x* and the contract duration is 20 *n* . We compare the stochastic approach with the standard approach for the contract times *t* 0,1,2,3,4,5,10,15,18 . 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).

24 Risk Management – Current Issues and Challenges

1 1

(0) ,

*i i x k x xk*

**7.4. Numerical illustration** 

*vV*

*S*

order 1 1

\* *a*

*x t*

p.240. The entries \* *<sup>a</sup>*

\* *x t* 1

2 2

2 2

[26], Section 2.4, or Bowers et al. [19], Section 9.5, denoted by

*x t q* : one-year independent rate of active mortality at time t

*i* : one-year independent rate of disability at time t

*x t <sup>q</sup>* , \*

*x t <sup>i</sup>* and *<sup>i</sup>*

The missing entries between the 5-year ages are linearly interpolated.

where the savings premiums are determined by the formulas

1 1

*S i i k k*

*i pq* , one obtains the following simpler approximations to (88):

*a a aa k xk k x*

*EC P x n vv P x n p p*

*EC P x n vv P x n q p k n*

1 1 1

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

The independent rates of decrement are linked to the probabilities of active mortality and

\* \* 1 1 1. *a a*

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 in Saxer [26],

*i* is taken from the "VZ Table 1950", where VZ stands for "**V**ersicherungskasse **Z**ürich".

While the standard solvency capital ratio does not depend on the initial cohort size, this is the case for the stochastic approaches. The age at contract issue is either 30 *x* or 40 *x* and the contract duration is 20 *n* . We compare the stochastic approach with the standard approach for the contract times *t* 0,1,2,3,4,5,10,15,18 . 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

*xt xt xt xt qi q i* . (91)

*x t q* are taken from the "EVK Table 1950" and the entry

disability through the relationship, e.g. Saxer [26], formulas (2.5.1) and (2.5.2),

( ) ( 1 : 1) ( : ) , 1,..., 1,

( : ) ( : ) , 0,..., 2,

( : ) ( : ) (1 ) .

*a a aa n xn n x* (89)

(90)

*k vV A x k n k V A x k n k k n*

and the probabilities ( ) *<sup>k</sup> P E* are defined in (87). Neglecting the probabilistic terms of second


**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].

Biometric Solvency Risk for Portfolios of General Life Contracts (II) The Markov Chain Approach 27

lower bound 100 lives 500 lives 1'000 lives 10'000 lives 100'000 lives

**Normal approximation to VaR SR with varying cohort size**

**Figure 2.** Time evolution of VaR solvency capital ratios, ( , ) (40,20) *x n*

0 5 10 15

**contract time**

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

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

**8. Conclusions and outlook** 

0%

50%

100%

150%

**solvency capital ratio**

200%

250%

probabilities.


**Table 2.** Coefficients of variation of the shifted prospective insurance loss


**Table 3.** VaR solvency capital ratios for the endowment with waiver of premium

**Figure 2.** Time evolution of VaR solvency capital ratios, ( , ) (40,20) *x n*
