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A measure to analyse the interaction of contracts in a heterogeneous life insurance portfolio

Abstract

Because of the long-term nature of life insurance policies including interest rate guarantees and the current low interest rate environment, the fair valuation of insurance contracts is of particular interest. Fair valuation is often discussed on a single contract basis or from the viewpoint of a homogenous portfolio, i.e. a portfolio with identical policies. However, insurance portfolios are heterogeneous, i.e. consist of many different contracts. These contracts interact, e.g. because they share reserves, profits and the risk of default of the insurance company. In this paper, we introduce a methodology how interactions within heterogeneous insurance portfolios can be measured and provide some sample analyses showing how different contracts may subsidize each other. This methodology also allows for a check, whether a contract is fairly calculated in a heterogeneous portfolio.

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Notes

  1. 1.

    We call a contract fair, if \(V_{{t_{0} }} = E_{Q} \left[ {\left. {\frac{{B_{{t_{0} }} }}{{B_{{t_{0} + T}} }} \cdot L_{{t_{0} + T}} } \right|{\mathcal{F}}_{{t_{0} }} } \right] = L_{{t_{0} }} \left( { = P_{{t_{0} }} } \right)\) using the risk-neutral pricing measure \(Q\) where \(\left( {B_{t} } \right)_{t \ge 0}\) denotes the risk-free asset applied as the discount factor in our setting and further introduced in Sect. 3.

  2. 2.

    Definition BE: cf. Article 77 of the [6]; definition PVFP: cf. Principle 6 of the CFO [5].

  3. 3.

    Any new business after time \(t^{*}\) is not considered here.

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Appendices

Appendix A: Proof of Proposition 1

Proposition 1

Assume the model setup of Sect3.1. The sum of the collective boni of in-force contracts \(i \in N_{2}\) is equal to the difference of the best estimate of liabilities and the premium payments before time \(t^{*}\) invested in the reference portfolio \(\left( {F_{t} } \right)_{{t \ge t_{0}^{1} }} .\)

$$BE_{{t^{*} }} - \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} = \mathop \sum \limits_{{i \in N_{2} }} CB_{{t^{*} }}^{i}$$

Proof

To follow the proof easier the main transformations are marked in bold.

First, we consider the definition of \(L_{t}^{i} .\) If \(t \in \left\{ {t_{0}^{i} , \ldots ,t_{0}^{i} + T^{i} - 1} \right\},\) from Eq. (3) follows:

$$\begin{aligned} L_{t}^{i} & = \sum\limits_{{l = t_{0}^{i} }}^{t} {P_{l}^{i} } \cdot \prod\limits_{j = l}^{t - 1} {\left( {1 + f_{j + 1}^{i} } \right)} \\ & = P_{t}^{i} + \sum\limits_{{l = t_{0}^{i} }}^{t - 1} {P_{l}^{i} } \cdot \prod\limits_{j = l}^{t - 1} {\left( {1 + f_{j + 1}^{i} } \right)} \\ & = P_{t}^{i} + \left( {1 + f_{t}^{i} } \right)\left( {\sum\limits_{{l = t_{0}^{i} }}^{t - 1} {P_{l}^{i} } \cdot \prod\limits_{j = l}^{t - 2} {\left( {1 + f_{j + 1}^{i} } \right)} } \right) \\ & = P_{t}^{i} + \left( {1 + f_{t}^{i} } \right) \cdot L_{t - 1}^{i} \\ \end{aligned}$$

For t = \(t_{0}^{i} + T^{i}\) we analogously get (there is no premium payment at \(t_{0}^{i} + T^{i} )\):

$$\begin{aligned} L_{{t_{0}^{i} + T^{i} }}^{i} & = \sum\limits_{{l = t_{0}^{i} }}^{{t_{0}^{i} + T^{i} - 1}} {P_{l}^{i} } \cdot \prod\limits_{j = l}^{{t_{0}^{i} + T^{i} - 1}} {\left( {1 + f_{j + 1}^{i} } \right)} = \left( {1 + f_{{t_{0}^{i} + T^{i} }}^{i} } \right)\left( {\sum\limits_{{l = t_{0}^{i} }}^{{t_{0}^{i} + T^{i} - 1}} {P_{l}^{i} } \cdot \prod\limits_{j = l}^{{t_{0}^{i} + T^{i} - 2}} {\left( {1 + f_{j + 1}^{i} } \right)} } \right) \\ & = \left( {1 + f_{{t_{0}^{i} + T^{i} }}^{i} } \right) \cdot L_{{t_{0}^{i} + T^{i} - 1}}^{i} \\ \end{aligned}$$

With the definition of the best estimate of liabilities we get

$$\begin{aligned} BE_{{t^{*} }} & = \sum\limits_{{i \in N_{2} }} B E_{{t^{*} }}^{i} \hfill \mathop { \, = }\limits^{\text{Equation} \left( 4 \right)} \\ & = \sum\limits_{{i \in N_{2} }} {E_{Q} } \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot L_{{t_{0}^{i} + T^{i} }}^{i} - \sum\limits_{{j = t^{*} + 1}}^{{t_{0}^{i} + T^{i} - 1}} {\frac{{B_{{t^{*} }} }}{{B_{j} }}} \cdot P_{j}^{i} } \right] \hfill \\ \end{aligned}$$

Next we use the alternative calculation of the policyholder account (cf. above):

$$\begin{aligned} BE_{{t^{*} }} & = \sum\limits_{{i \in N_{2} }} {E_{Q} } \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot L_{{t_{0}^{i} + T^{i} - 1}}^{i} \cdot \left( {1 + f_{{T^{i} + t_{0}^{i} }}^{i} } \right) - \sum\limits_{{j = t^{*} + 1}}^{{t_{0}^{i} + T^{i} - 1}} {\frac{{B_{{t^{*} }} }}{{B_{j} }}} \cdot P_{j}^{i} } \right] \\ & = \sum\limits_{{i \in N_{2} }} {{{E}}_{{Q}} } \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} }} }}{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} }} + \frac{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} }} }}{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} }}} \right) - \sum\limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} {\frac{{B_{{t^{*} }} }}{{B_{j} }}} \cdot {{P}}_{{j}}^{{i}} } \right] \\ & = \sum\limits_{{i \in N_{2} }} {{{E}}_{{Q}} } \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right)} \right] + {{E}}_{{Q}} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right] \\ & - {{E}}_{{Q}} \left[ {\sum\limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} {\frac{{B_{{t^{*} }} }}{{B_{j} }}} \cdot {{P}}_{{j}}^{{i}} } \right] \\ \end{aligned}$$

Next, we apply the tower property

$$\begin{aligned} {{BE}}_{{{{t}}^{ *} }} & = \sum\limits_{{i \in N_{2} }} {{{E}}_{{Q}} } \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right)} \right] \\ & + {\text{E}}_{{Q}} \left[ {{{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \frac{{B_{{t^{*} }} \cdot {{E}}_{Q} \left[ {B_{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} }}^{ - 1} \cdot {{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} }} |{\mathcal{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} } \right]}}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right] - {{E}}_{{Q}} \left[ {\sum\limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} {\frac{{B_{{t^{*} }} }}{{B_{j} }}} \cdot {{P}}_{{j}}^{{i}} } \right] \\ \end{aligned}$$

and use the martingale property of the reference portfolio:

$$\begin{aligned} {{BE}}_{{{{t}}^{ *} }} & = \mathop \sum \limits_{{i \in N_{2} }} {{E}}_{{Q}} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right)} \right] \\ & + {{E}}_{{Q}} \left[ {{{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \frac{{B_{{t^{*} }} \cdot {B}_{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 1}}^{ - 1} \cdot {{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right] - {{E}}_{{Q}} \left[ {\mathop \sum \limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} \frac{{B_{{t^{*} }} }}{{B_{j} }} \cdot {{P}}_{{j}}^{{i}} } \right] \\ & = \mathop \sum \limits_{{i \in N_{2} }} {{E}}_{{Q}} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right)} \right] + {{E}}_{{Q}} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} - 1}} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}}^{{i}} - \mathop \sum \limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} \frac{{B_{{t^{*} }} }}{{B_{j} }} \cdot {{P}}_{{j}}^{{i}} } \right] \\ \end{aligned}$$

For \(T^{i} + t_{0}^{i} - 1\) we do the same transformations as above. We use again the alternative calculation of the policyholder account:

$$ \begin{aligned} {{BE}}_{{{{t}}^{ *} }} & = \sum\limits_{{i \in N_{2} }} {E_{Q} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right)} \right.} \\ & + \frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} - 1}} }}\left( {{{L}}_{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 2}}^{{{i}}} \cdot \left( {1 + {{f}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}}^{{{i}}} } \right) + {{P}}_{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 1}} } \right)\left. { - \mathop \sum \limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} \frac{{B_{{t^{*} }} }}{{B_{j} }} \cdot {{P}}_{{j}}^{{i}} } \right] \\ & = \sum\limits_{{i \in N_{2} }} {E_{Q} \left[ {\frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} }} }} \cdot {{L}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}}^{{i}} \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }}^{{i}} - \frac{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} }} }}{{{{F}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}} }}} \right) + \frac{{B_{{t^{*} }} }}{{B_{{t_{0}^{i} + T^{i} - 1}} }} \cdot {{L}}_{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 2}}^{{i}} } \right.} \\ & \quad \left. { \cdot \left( {1 + {{f}}_{{{{T}}^{{i}} + {{t}}_{0}^{{i}} - 1}}^{{i}} - \frac{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} }}{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 2}} }} + \frac{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 1}} }}{{{{F}}_{{{{T}}^{{{i}}} + {{t}}_{0}^{{{i}}} - 2}} }}} \right) - \mathop \sum \limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 2}} \frac{{B_{{t^{*} }} }}{{B_{j} }} \cdot {{P}}_{{j}}^{{i}} } \right] \\ \end{aligned} $$

Tower property and the martingale property of the reference portfolio yield

$${{BE}}_{{{{t}}^{ *} }} = \sum\limits_{{i \in N_{2} }} {{{E}}_{{Q}} \left[ {\mathop \sum \limits_{{{{j}} = {{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 2}}^{{{{t}}_{0}^{{{i}}} + {{T}}^{{{i}}} - 1}} \frac{{B_{{t^{*} }} }}{{B_{j + 1} }} \cdot {{L}}_{{j}}^{{i}} \cdot \left( {1 + {{f}}_{{{{j}} + 1}}^{{i}} - \frac{{{{F}}_{{{{j}} + 1}} }}{{F_{j} }}} \right) + \frac{{{B}_{{{t}^{{*}} }} }}{{{B}_{{{t}_{0}^{{i}} + {T}^{{i}} - 2}} }} \cdot {L}_{{{t}_{0}^{{i}} + {T}^{{i}} - 2}}^{{i}} - \mathop \sum \limits_{{{{j}} = {{t}}^{ *} + 1}}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 2}} \frac{{B_{{t^{*} }} }}{{B_{j} }} \cdot {{P}}_{{j}}^{{i}} } \right]}$$

The same transformations as above are repeated until \(t^{*}\)

$${{BE}}_{{{{t}}^{ *} }} = \mathop \sum \limits_{{i \in N_{2} }} {{L}}_{{{{t}}^{ *} }}^{{i}} + {{E}}_{{Q}} \left[ {\mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{{{j}} = {{t}}^{ *} }}^{{{{t}}_{0}^{{i}} + {{T}}^{{i}} - 1}} \frac{{B_{{t^{*} }} }}{{B_{j + 1} }} \cdot {{L}}_{{j}}^{{i}} \cdot \left( {1 + {{f}}_{{{{j}} + 1}}^{{i}} - \frac{{{{F}}_{{{{j}} + 1}} }}{{F_{j} }}} \right)} \right]$$

Finally we use the definition of the collective bonus (cf. Eq. (6))

$$ {{BE}}_{{{{t}}^{ *} }} = \mathop \sum \limits_{{i \in N_{2} }} {{CB}}_{{{{t}}^{ *} }}^{{i}} + \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{{{j}} = {{t}}_{0}^{{i}} }}^{{{{t}}^{ *} }} {{P}}_{{j}}^{{i}} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} $$

Thus, the collective bonus can also be calculated using the best estimate of liabilities and the premium payments before time \(t^{*}\).\(\square\)

The result of Proposition 1 is on an aggregate level (sum of all contracts of \(N_{2}\)). A direct consequence of Proposition 1 is the following result for each contract \(i \in N_{2}\):

$$BE_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} = CB_{{t^{*} }}^{i}.$$
(8)

Appendix B: Proof of Proposition 2

Proposition 2

Assume the model setup of Sect3.1. The market value of assets is equal to the sum of present value of future profits and best estimate of liabilities if and only if the sum of the collective boni is zero, i.e.

$$A_{{t^{*} }} = BE_{{t^{*} }} + PVFP_{{t^{*} }} \Leftrightarrow CB_{{t^{*} }} + CB_{{t^{*} }}^{sh} = 0.$$

Proof

We show by induction that \(A_{{t^{*} }}\) for \(t\in{t_0^1, \ldots,T}\) is given as

$$A_{{t^{*} }} = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }}.$$

First, \(t^{*} = t_{0}^{1}\):

$$A_{{t_{0}^{1} }} = \mathop \sum \limits_{{i \in N_{2} = \left\{ {1, \ldots ,N} \right\}}} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t_{0}^{1} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t_{0}^{1} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} = \emptyset }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t_{0}^{1} }} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t_{0}^{1} }} \frac{{F_{l} }}{{F_{l - 1} }} = P_{{t_{0}^{1} }}$$

Next, \(t^{*} \in \left\{ {t_{0}^{1} + 1, \ldots ,T} \right\}\). We apply the definition of the market value of assets (see Eq. (5)) and distinguish different cases for \(t^{*}\) in the following.

Case 1

\(t^{*} - 1 \ne t_{0}^{i} + T^{i} , i \in N_{1}\), i.e. there is no contract which matures at \(t^{*} - 1\):

$$\begin{aligned} A_{{t^{*} }} & = \left( {A_{{t^{*} - 1}} - \mathop \sum \limits_{{i \in N_{1} }} L_{{t_{0}^{i} + T^{i} }}^{i} \cdot 1_{{\left\{ {t^{*} - 1 = t_{0}^{i} + T^{i} } \right\}}} } \right)\frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \\ & = A_{{t^{*} - 1}} \cdot \frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \\ & = \left( {\mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} - 1}}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} - 1}} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} - 1}} \frac{{F_{l} }}{{F_{l - 1} }}} \right) \cdot \frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \\ & = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} - 1}} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \\ & = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }}. \\ \end{aligned}$$

Case 2

\(t^{*} - 1 = t_{0}^{i} + T^{i}\), for at least one contract \(i \in N_{1} ,\) i.e. there is at least one account i which is paid out to the policyholder at \(t^{*} - 1\). We define

$$N_{1}^{*} = \left\{ {i \in N_{1} |t^{*} - 1 = t_{0}^{i} + T^{i} } \right\}.$$

Consequently, the insurer’s portfolio consists of the contracts \(i \in N_{1}^{*} \cup N_{2}\) at time \(t^{*} - 1.\)

$$\begin{aligned} A_{{t^{*} }} & = \left( {A_{{t^{*} - 1}} - \mathop \sum \limits_{{i \in N_{1} }} L_{{t_{0}^{i} + T^{i} }}^{i} \cdot 1_{{\left\{ {t^{*} - 1 = t_{0}^{i} + T^{i} } \right\}}} } \right)\frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \hfill \\ & = \left( {\mathop \sum \limits_{{i \in N_{1}^{*} \cup N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} \backslash N_{1}^{*} }} CB_{{t^{*} - 1}}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} - 1}} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} - 1}} \frac{{F_{l} }}{{F_{l - 1} }} - \mathop \sum \limits_{{i \in N_{1}^{*} }} L_{{t_{0}^{i} + T^{i} }}^{i} } \right)\frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \hfill \\ &= \left( {\mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} \backslash N_{1}^{*} }} CB_{{t^{*} - 1}}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} - 1}} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} - 1}} \frac{{F_{l} }}{{F_{l - 1} }}} \right. \hfill \\ & \quad - \left. {\sum\limits_{{i \in N_{1}^{*} }} \left( {L_{{t_{0}^{i} + T^{i} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }}} \right) } \right)\frac{{F_{t^{*}} }}{{F_{t^{*} - 1} }} + \mathop \sum \limits_{{i \in N_{2} }} P_{{t^{*} }}^{i} - X_{{t^{*} }} \hfill \\ & = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} \backslash N_{1}^{*} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} \hfill \\ & \quad - \mathop \sum \limits_{{i \in N_{1}^{*} }} \left( {L_{{t_{0}^{i} + T^{i} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 1}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }}} \right) \cdot \frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} \hfill \\ \end{aligned}$$

Because of \(P_{{t^{*} - 1}}^{i} = 0,\, i \in N_{1}^{*}\) (no premium payment at maturity), it follows that

$$\begin{aligned} & = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} \backslash N_{1}^{*} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} \hfill \\ & \quad - \mathop \sum \limits_{{i \in N_{1}^{*} }} \left( {L_{{t_{0}^{i} + T^{i} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} - 2}} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} - 1}} \frac{{F_{k} }}{{F_{k - 1} }}} \right) \cdot \frac{{F_{{t^{*} }} }}{{F_{{t^{*} - 1}} }} \hfill \\ & = \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} }}^{i} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} \hfill \\ \end{aligned}$$

Now we can prove:

$$\begin{aligned} BE_{{t^{*} }} & + PVFP_{{t^{*} }} - A_{{t^{*} }} \mathop = \limits^{\text{Proposition 1}} \hfill \\ &= \mathop \sum \limits_{{i \in N_{2} }} CB_{{t^{*} }}^{i} + \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} + PVFP_{{t^{*} }} - A_{{t^{*} }} \hfill \\ & \mathop = \limits^{ } \mathop \sum \limits_{{i \in N_{2} }} CB_{{t^{*} }}^{i} + \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} + CB_{{t^{*} }}^{sh} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} - A_{{t^{*} }} \hfill \\ & = CB_{{t^{*} }} - \mathop \sum \limits_{{i \in N_{1} }} CB_{{t^{*} }}^{i} + \mathop \sum \limits_{{i \in N_{2} }} \mathop \sum \limits_{{j = t_{0}^{i} }}^{{t^{*} }} P_{j}^{i} \cdot \mathop \prod \limits_{k = j + 1}^{{t^{*} }} \frac{{F_{k} }}{{F_{k - 1} }} + CB_{{t^{*} }}^{sh} - \mathop \sum \limits_{{j = t_{0}^{1} + 1}}^{{t^{*} }} X_{j} \cdot \mathop \prod \limits_{l = j + 1}^{{t^{*} }} \frac{{F_{l} }}{{F_{l - 1} }} - A_{{t^{*} }} \hfill \\ & = CB_{{t^{*} }} + CB_{{t^{*} }}^{sh} \hfill. \\ \end{aligned}$$

\(\square\)

Appendix C: Average net return of the German insurance companies

Year Average net return of the
German insurance companies
1990 6.78%
1991 7.44%
1992 7.39%
1993 7.59%
1994 7.15%
1995 7.37%
1996 7.37%
1997 7.46%
1998 7.57%
1999 7.58%
2000 7.51%
2001 6.12%
2002 4.68%
2003 5.05%
2004 4.90%
2005 5.18%
2006 4.82%
2007 4.65%
2008 3.54%
2009 4.18%
2010 4.27%
2011 4.13%
2012 4.59%
2013 4.68%
2014 4.63%
  1. Reference: Gesamtverband der deutschen Versicherungswirtschaft (GDV)

Glossary

This is a comparison of the notation in the present paper and the paper [12] as well as Døskeland and Nordahl [7], we use in Sect. 4.

This paper Døskeland and Nordahl [7]
F A
\(T^{i}\) \(T\)
This paper Hieber et al. [12]
N K
\(g^{i}\) \(g^{\left( i \right)}\)
\(T^{i}\) \(T^{\left( i \right)}\)
\(P^{i}\) \(P^{\left( i \right)}\)
\(\alpha^{i}\) \(\alpha^{\left( i \right)}\)
\(\beta^{i}\) \(\beta^{\left( i \right)}\)
L P
F A

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Eckert, J., Graf, S. & Kling, A. A measure to analyse the interaction of contracts in a heterogeneous life insurance portfolio. Eur. Actuar. J. 11, 87–112 (2021). https://doi.org/10.1007/s13385-020-00225-2

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Keywords

  • Life insurance
  • Heterogeneous portfolio
  • Participating contracts
  • Risk-neutral valuation
  • Interaction of contracts