Abstract
In this chapter we lay the basis for actuarial valuation. We introduce the notion of financial risk and insurance technical risk. Such a split is crucial because it explains which risks can be hedged at financial markets and which risks cannot be hedged and need to be absorbed by the insurance company. This will result in the split of the filtration into a financial filtration and an insurance technical filtration which describe the corresponding flow of information. Moreover, the previously introduced state price deflator receives a deeper meaning in terms of a financial deflator and a probability distortion. The former describes price formation at financial markets, the latter describes margins for non-hedgeable risks.
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In Part I of this book we have introduced the basic discrete time stochastic model for the valuation of random cash flows (see Sect. 2.2) and for the pricing of basis financial instruments \(\mathfrak{A}^{(i)}\) of the financial market (see Sect. 5.2). The valuation was based on a fixed given state price deflator and has provided a φ-consistent pricing framework. This valuation has not distinguished between basis financial instruments \(\mathfrak{A}^{(i)}\) from the financial market and insurance cash flows X. In the following chapters we decouple insurance liabilities such that we can describe the hedgeable part of the insurance liabilities (using appropriate basis financial instruments). For the residual non-hedgeable part of the insurance liabilities we then calculate expected values and adequate risk margins. This requires to analyze in more depth the \(\mathbb{F}\)-adapted state price deflator φ and the filtration \(\mathbb{F}\) which contains (at the moment) both economic information from the financial market and insurance technical information about insurance liabilities. We consider models that allow to disentangle these financial variables and insurance technical variables.
1 Financial Market and Financial Filtration
We fix a filtered probability space and a state price deflator and we always work under this fixed given state price deflator φ. Our aim is to analyze the filtration \(\mathbb{F}\) and this state price deflator φ. We disentangle these terms into a financial part and an insurance technical part. The financial part will model economic and financial information that is available to the public. The insurance technical part will model the insurance liability related variables and the insurance technical flow of information.
Financial Market Model
Let describe the financial market of all basis financial instruments \(\mathfrak{A}^{(i)}\), . We assume that these basis financial instruments \(\mathfrak{A}^{(i)}\) have integrable and φ-consistent price processes denoted by , i.e. they satisfy the assumptions of Definition 5.4 for the given state price deflator .
Remarks
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The financial market describes the available basis financial instruments. These instruments are used for the description of the hedgeable part of the insurance liability cash flows.
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Throughout we assume that the financial market is sufficiently rich and contains at least all ZCBs \(\mathfrak{Z}^{(m)}\) with maturities (and henceforth also the bank account denoted by \(\mathfrak{B}\)).
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We assume that the price processes of all basis financial instruments are φ-consistent. As described in Sect. 5.2 this φ-consistency can be achieved in different ways, but it should reflect the price formation at the financial market . In Example 7.8 below we consider the statutory model which also provides a consistent pricing framework, resulting prices however are far from real financial market prices.
Remark on Liquidity
In financial practice and in the insurance industry financial assets that are used for the replication of insurance liabilities should have reliable market prices. Therefore, they should be traded in active financial markets. This means that admissible financial instruments for replication should meet the following requirements (see for instance QIS5 [64], p. 52): (i) a large number of assets can be transacted without significantly affecting the price of the financial instruments (deep); (ii) assets can be easily bought and sold (liquid); (iii) current trade and price information are normally readily available to the public (transparent).
In particular, the liquidity aspect has led to discussions among the actuarial community during the financial crisis 2008–2011. For example, the prices of two different corporate bonds that have the same marginal distributions for credit risk can substantially differ because one of the bonds is traded in a deep and liquid market and the other one is illiquid. One then says that the latter price contains an illiquidity spread. At the current stage we do not treat the particular triggers of the spreads (credit, liquidity, roll-over or funding risks). For the time being we simply assume that these risk factors are appropriately reflected in the state price deflator φ and prices match an equilibrium between supply and demand of particular assets.
We emphasize that there might be defaultable bonds that have the same risk profiles, i.e. the same marginal distributions for defaults under the real world probability measure, but their market prices differ. However, such bonds cannot have the same behavior in each future state of the world (they only have the same marginal distributions) because otherwise it would contradict the law-of-one-price principle which immediately would allow for arbitrage, see Wüthrich [163] and Danielsson et al. [48, 49] for more on this topic.
Financial Filtration
For describing the financial information we assume that there is second filtration on the filtered probability space with for all . The filtration \(\mathbb{A}\) describes the economic and financial market information. Therefore, we assume that the price processes are \(\mathbb{A}\)-adapted for all basis financial instruments \(\mathfrak{A}^{(i)}\), . This implies that
If we have then the basis financial instruments \(\mathfrak{A}^{(i)}\) already describe the entire economic and financial market flow of information. However, we do not necessarily assume this identity because there might be additional economic information which is not directly reflected at the financial market .
The filtration \(\mathbb{A}\) is called a financial filtration. Our aim is to study this financial filtration \(\mathbb{A}\) and a corresponding insurance technical filtration (denoted by \(\mathbb{T}\), see next section) for different actuarial models.
The introduction of the financial filtration \(\mathbb{A}\) and the \(\mathbb{A}\)-adaptedness of the φ-consistent price processes will also have implications on the state price deflator φ. In the next sections we are going to describe two different models.
2 Basic Actuarial Model
In the basic actuarial model we assume that we can decouple the financial filtration \(\mathbb{A}\) and the insurance technical filtration \(\mathbb{T}\) in an independent way. This independent decoupling has the advantage that we can study financial variables and insurance technical variables separately. A more general model is presented in the next section. The latter model has the disadvantage that it is analytically less tractable than the one presented in this section. Therefore, we mainly work in the framework of the current section.
Assumption 6.1
(Independent split of filtrations)
We assume that we have three filtrations , and on the given probability space with and
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(i)
is generated by and for all ,
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(ii)
\(\mathbb{A}\) and \(\mathbb{T}\) are independent w.r.t. the probability measure \(\mathbb{P}\).
Remarks 6.2
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The filtration \(\mathbb{A}\) models the financial events and the filtration \(\mathbb{T}\) models the insurance technical events. We have for all . Thus, financial information and insurance technical information are observable at time t w.r.t. .
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Assumption 6.1 imposes that financial variables and insurance technical variables develop independently. Illustratively, this gives the following picture:
(6.1)The development in (6.1) is such that the split into financial events \(\mathbb{A}\) and insurance technical events \(\mathbb{T}\) is independent. This independence assumption is the same as in Assumption 2.15 of Wüthrich et al. [168].
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It is important to realize that we assume that the financial variables and the insurance technical variables are independent. This does not imply that insurance liabilities are independent from economic and financial variables. Insurance liabilities are, in general, a function of financial variables and insurance technical variables, see for instance assumption (6.5) below.
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The combination of financial filtration \(\mathbb{A}\) and insurance technical filtration \(\mathbb{T}\) generate the whole flow of information \(\mathbb{F}\).
For the calculation of price processes of cash flows we introduce a product structure for both the given state price deflator φ and the cash flow X. In a first step, we split the given state price deflator into a financial deflator part φ A and into an insurance technical probability distortion part φ T.
We envisage the following three properties as desirable (the fast reader can directly go to Assumption 6.3 below where the basic actuarial model is defined):
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(1)
The state price deflator φ should have a product structure, i.e. for all
$$ \varphi_t= \varphi_t^A ~\varphi_t^T. $$(6.2) -
(2)
The financial deflator should be \(\mathbb{A}\)-adapted.
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(3)
The probability distortion should be a \(\mathbb{T}\)-adapted and normalized \((\mathbb{P},\mathbb{T})\)-martingale.
Product structure (6.2) facilitates calculation; φ A should explain price formation at the financial market, see Sect. 5.2; φ T should be a density process that provides a risk margin for non-hedgeable insurance technical risks. Desirable properties (1)–(3) imply under Assumption 6.1
i.e., for valuation based on the financial filtration \(\mathbb{A}\) only the financial deflator φ A of the state price deflator φ is relevant. Under the additional assumption that the φ-consistent price process of basis financial instrument \(\mathfrak{A}^{(i)}\) is \(\mathbb{A}\)-adapted we obtain
Hence, Assumption 6.1 and desirable properties (1)–(3) imply for φ-consistent and \(\mathbb{A}\)-adapted price processes
This tells us that the financial market is characterized by the filtered probability space with φ A-consistent price processes. In alignment with Sect. 5.2 we require that φ A and provide a sensible financial market model.
Summarizing these desirable properties motivates the following model assumption which is interpreted subsequently.
Assumption 6.3
(Basic actuarial model)
The three filtrations \(\mathbb{F}\), \(\mathbb{A}\) and \(\mathbb{T}\) fulfill Assumption 6.1. The given state price deflator has product structure (6.2) with \(\mathbb{A}\)-adapted financial deflator φ A and \(\mathbb{T}\)-adapted probability distortion φ T being a normalized \((\mathbb{P}, \mathbb{T})\)-martingale. The price processes of all basis financial instruments \(\mathfrak{A}^{(i)}\), , are \(\mathbb{A}\)-adapted, integrable and φ-consistent (according to Definition 5.4).
Remarks 6.4
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Both the \(\mathbb{A}\)-adapted financial deflator φ A and the \(\mathbb{T}\)-adapted probability distortion φ T are integrable and strictly positive. This is due to the assumption that is a state price deflator.
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The financial market is entirely characterized by the filtered probability space and φ A-consistent price processes, see (6.3). We could also have started with this financial market model and then extended this model to the basic actuarial model on with φ-consistent price processes for \(\mathbb{F}\)-adapted cash flows, which would have led to the product space described in Assumption 6.3.
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The financial deflator φ A and the financial filtration \(\mathbb{A}\) model the price processes of the basis financial instruments and the economic and financial flow of information, respectively. Typically the models presented in Part I of this book are used for this financial modeling part and financial market data is used for model calibration.
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The probability distortion can be viewed as a density process on the filtered probability space which allows for a measure transformation so that we can work with constant probability distortion equal to 1. That is, define the equivalent probability measure \(\mathbb{P}^{T} \sim\mathbb{P}\) via the Radon–Nikodym derivative
(6.4)Then \(\varphi^{T}_{n}\) changes (or distorts) \(\mathbb{P}\) to an equivalent probability measure \(\mathbb{P}^{T}\). This distortion will be used to calculate loadings for the non-hedgeable insurance technical risk part.
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The simplest choice of the probability distortion that fulfills Assumption 6.3 is φ T≡1. This will be the appropriate choice for the so-called pure risk premium and the best-estimate reserves, see (7.1) below.
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The interpretation of the probability distortion φ T and the insurance technical filtration \(\mathbb{T}\) is more sophisticated because there exist different levels of information and different modeling objectives. The insurance technical filtration \(\mathbb{T}\) can reach from very rough to very detailed insurance technical information. At the low end this is public available information which involves global factors like natural hazards, weather conditions, legal changes (as long as these factors do not influence \(\mathbb{A}\)) and insurance payments. In our context \(\mathbb{T}\) will model more detailed information on the level that it provides a regulatory solvency model which contains all insurance technical information available to the supervisor. That is, our aim is to build a regulatory valuation model that provides marked-to-model prices for insurance liability cash flows of the entire insurance market. The reason for this point of view is that the regulator aims to transfer insurance liabilities between different companies in case a specific company is in a distress situation. The probability distortion φ T then provides a regulatory loading for non-hedgeable risks that does not consider idiosyncratic company-specific knowledge, but discloses all information that is available to calculate marked-to-model prices for insurance portfolio transfers. We also refer to Remarks 8.4.
We summarize finding (6.3) in the following corollary.
Corollary 6.5
(Financial price processes)
Under Assumption 6.3 we obtain for any financial price process , , the following \((\mathbb{P},\mathbb{A})\)-martingale property
In particular this says that the price processes of basis financial instruments \(\mathfrak{A}^{(i)}\) are not only consistent w.r.t. φ and \(\mathbb{F}\) but also w.r.t. φ A and \(\mathbb{A}\).
Insurance Liability Cash Flows
Our aim is to analyze hedgeable and non-hedgeable parts of insurance liability cash flows X. Therefore, we assume a product structure for these insurance liability cash flows. Assume that the cash flow is for all given by
where Λ (k) is -measurable and \(U_{k}^{(k)}\) is -measurable. That is, the insurance liability cash flow X k consists of a hedgeable -measurable part and a non-hedgeable -measurable part. We describe these two components.
We assume that describes the \(\mathbb{A}\)-adapted price process of the financial portfolio \(\mathfrak{U}^{(k)}\) available at the financial market (further specified in (6.6)). The superscript k in \(U_{t}^{(k)}\) always denotes the fact that financial portfolio \(\mathfrak{U}^{(k)}\) supports cash flow X k and the subscript t in \(U_{t}^{(k)}\) denotes the fact that this is the price of \(\mathfrak{U}^{(k)}\) at time t. In our understanding the financial portfolios \(\mathfrak{U}^{(k)}\) are linear combinations of the underlying basis financial instruments \(\mathfrak{A}^{(i)}\), , i.e. for an appropriate choice the financial portfolio \(\mathfrak{U}^{(k)}\) is given by
and its price at time t is given by (using linearity)
Under Assumption 6.3 we know that the price processes are φ-consistent and therefore also the price process is φ-consistent (due to linearity).
The random variable Λ (k) describes the number of units of such financial portfolios \(\mathfrak{U}^{(k)}\) we need to purchase in order to fulfill the insurance liability cash flow X k at time k. Observe that to generate cash flow X k the financial portfolio \(\mathfrak{U}^{(k)}\) needs to be sold at time k which is indicated by the upper index k in \(\mathfrak{U}^{(k)}\) and , respectively.
For simplicity we consider the following insurance liability cash flow
where Λ (k) is -measurable and is the \(\mathbb{A}\)-adapted, φ-consistent price process of financial portfolio \(\mathfrak{U}^{(k)}\) of the form (6.6) (as outlined above).
Theorem 6.6
Under Assumption 6.3 the φ-consistent price process of an insurance liability cash flow of the form (6.7) is given by
at time t≤k.
Proof of Theorem 6.6
φ-consistency implies that is a \((\mathbb{P},\mathbb{F})\)-martingale. Thus, we obtain from Q k [X k ]=X k
where in the second last step we have used the independence of financial and insurance technical variables and in the last step we have used Corollary 6.5 as well as the linearity of price processes of financial portfolios (6.6). This completes the proof. □
Remarks 6.7
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Theorem 6.6 explains that for insurance liability cash flow of the form \(X_{k}=\varLambda^{(k)}~U_{k}^{(k)}\) we can consider two independent processes under Assumption 6.3:
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the \(\mathbb{A}\)-adapted price process of financial portfolio \(\mathfrak{U}^{(k)}\) which lives on the financial probability space and is φ A-consistent;
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the probability distorted insurance technical process
(6.8)which lives on the insurance technical probability space for -measurable Λ (k). Note is a \((\mathbb {P}, \mathbb{T} )\)-martingale. In view of (6.4) and Lemma 11.3 we can rewrite (6.8) for t<k as follows
which highlights the distortion character of φ T. Note that this distortion is non-linear which was already accentuated in the correlation statement (2.5).
This implies that the price processes have multiplicative structure
$$ Q_{t} [\mathbf{X}_k ] = \varLambda_t^{(k)}~U_{t}^{(k)}. $$(6.9)That is, we can separately study the price processes of financial portfolios \(\mathfrak{U}^{(k)}\) at the financial market and the probability distorted insurance technical processes .
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For general liability cash flows X we use the linearity of .
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The term reflects the hedgeable part of the insurance liability X k and Λ (k) the non-hedgeable part. The hedgeable part is modeled with an appropriate financial deflator φ A for the financial market . In addition, the insurance technical probability distortion φ T is used to calculate a risk margin for the non-hedgeable part, see also next bullet points.
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Once φ A is calibrated to the financial market, we still have the freedom of the choice of the probability distortion φ T (that fulfills Assumption 6.3). The simplest choice is \(\varphi_{t}^{T}\equiv1\) for all . In that case we obtain for X k the price at time t≤k (we denote this special case by \(Q^{0}_{t} [\cdot ]\))
(6.10)i.e., we purchase the expected number of financial portfolios \(\mathfrak{U}^{(k)}\) at time t≤k for replication. This means that the insurance technical risk Λ (k) is priced by its conditionally expected value. In insurance practice \(Q^{0}_{t}[\mathbf{X}_{k}]\) is called “best-estimate” discounted liability, where “best-estimate” refers to the conditional expectation . This is treated in detail in Chap. 7.
In conclusion, for \(\varphi_{t}^{T}\equiv1\) we hold at time t≤k the financial portfolio
(6.11)in order to replicate the insurance liability \(X_{k}= \varLambda^{(k)}~U_{k}^{(k)}\). Are we satisfied with this solution?
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The answer to the question in the last bullet point is NO! If Λ (k) would be -measurable then (6.11) would give a prefect replicating portfolio for X k at time t. However, in general, Λ (k) is not -measurable and we are facing insurance technical risks coming from possible adverse developments in Λ (k) at time k. These insurance technical risks are not hedgeable at the financial market (\(\mathbb{A}\) and \(\mathbb{T}\) are independent), and henceforth ask for a margin for insurance technical risks. This margin is the price for the risk beyond the expected value (6.10) and should reflect the marked-to-model reward for bearing these insurance liability run-off risks. Using a smart choice of the probability distortion \(\varphi_{k}^{T}\) generates such a margin for insurance technical risks:
Assume \(\varphi_{k}^{T}\) and Λ (k) are strictly positively correlated, given , for t<k. Then
(6.12)using the \((\mathbb{P},\mathbb{T})\)-martingale property of φ T in the last step. Henceforth, for strictly positive prices \(U_{t}^{(k)}>0\) we have at time t<k
That is, the risk-adjusted price Q t [X k ] at time t<k for risk bearing of the run-off of the insurance liability X k is strictly above its best-estimate price \(Q^{0}_{t}[\mathbf{X}_{k}]\). The difference
$$ Q_{t} [\mathbf{X}_k ] -Q^0_{t} [ \mathbf{X}_k ] >0 $$constitutes the margin for (at the financial market) non-hedgeable risks. This risk-adjusted value reflects the risk aversion of the risk bearer expressed by φ T (in our marked-to-model world). This is further discussed in Chap. 8 below.
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In Chap. 8 we give explicit (constructive) examples for probability distortions φ T which result in well-known premium calculation principles such as the Esscher premium, the cost-of-capital loading or the first order life tables.
The risk-adjusted price for X k at time t<k is given by \(Q_{t}[\mathbf{X}_{k}]= \varLambda_{t}^{(k)}U_{t}^{(k)}\) and the best-estimate price by . Under strictly positive correlation (6.12) we have a positive margin \(Q_{t}[\mathbf{X}_{k}] -Q^{0}_{t}[\mathbf{X}_{k}] >0\) for positive prices \(U_{t}^{(k)}>0\).
3 Improved Actuarial Model
The independent decoupling of the basic actuarial model (Assumption 6.1) does not always apply to practical problems (or such a separation is not obvious), we give examples in Remarks 6.9 below. Therefore, we define a more involved model in this section. For the improved actuarial model we assume that the financial filtration and the insurance technical filtration absorb all previous information. Often this is more realistic. However, this model has the limitation that it is only hardly analytically tractable.
Assumption 6.8
(Absorbing split of filtrations)
We assume that we have three filtrations , and on the given probability space with and
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(i)
is generated by and for all ,
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(ii)
for all ,
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(iii)
for all ,
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(iv)
conditionally given , and are independent w.r.t. \(\mathbb{P}\) for .
Remarks 6.9
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Assumption 6.8 imposes that and . This means that both, previous financial information and previous insurance technical information , are completely absorbed in both and . Illustratively this gives the following picture:
(6.13)The development in (6.13) is such that the split into financial events and insurance technical events is independent, conditionally given .
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Typical situations where such models apply are big insurance technical events that also influence financial prices in a non-trivial way. For instance an earthquake or a terrorist attack like 9/11 may influence financial prices also on a larger time scale and situation (6.13) applies.
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In life insurance modeling the situation can also be tricky. For example, lapse rates of life insurance policies depend on the economic environment. This fits into the modeling framework of Assumption 6.8. However, economically driven lapse rates may also fit into the modeling framework of Assumption 6.1, namely by the creation of a financial asset \(\mathfrak{A}^{(i)}\) that exactly reflects the economically driven lapses. This then leads to the separation of these lapses from pure insurance technical risks such as survival and death. The latter example also shows that financial assets for insurance liability replication can go beyond traded assets, resulting in financial (ALM) risks on the balance sheet. Thus, before working under Assumption 6.8 one should reflect whether one cannot introduce financial instruments so that one falls under Assumption 6.1, because modeling is simpler under the latter.
We could again consider (1)–(3) from Sect. 6.2 as desirable properties for the split of the state price deflator φ. However, then the product structure (6.2) implies some cumbersome properties. Therefore, we envisage the following two properties as desirable in this improved actuarial model.
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(4)
The \(\mathbb{A}\)-adapted financial deflator should be of the form
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(5)
The basis financial instruments \(\mathfrak{A}^{(i)}\) should satisfy
for all and .
Recall definition (2.9) of the span-deflator \(\breve{\varphi}_{t}\), . This implies for
For the improved actuarial model we choose a product structure for the span-deflator, i.e. we assume
where is \(\mathbb{A}\)-adapted and is \(\mathbb{T}\)-adapted.
This has the following consequences:
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(i)
Under Assumption 6.8 and (6.14) we see that φ is \(\mathbb{F}\)-adapted and has product structure
$$ \varphi_{t}= \varphi_{t-1}~\breve{\varphi}_{t}^A~ \breve{\varphi}_{t}^T = \cdots=\prod _{s=0}^t \breve{\varphi}_{s}^A~ \breve{\varphi}_{s}^T. $$ -
(ii)
Assumption 6.8, property (6.14) and the adaptedness properties imply
In view of desirable property (4) this suggests the following definition
$$ \varphi_t^A= \varphi_{t-1}~\breve{\varphi}_{t}^A, $$(6.15)which requires normalization .
Assumption 6.10
(Improved actuarial model)
The filtrations \(\mathbb{F}\), \(\mathbb{A}\) and \(\mathbb{T}\) fulfill Assumption 6.8. The span-deflator has product structure (6.14) with \(\mathbb{A}\)-adapted financial span-deflator \(\breve{\boldsymbol {\varphi}}^{A}\) and \(\mathbb{T}\)-adapted insurance technical span-deflator \(\breve{\boldsymbol{\varphi}}^{T}\) satisfying . The price processes of all basis financial instruments \(\mathfrak{A}^{(i)}\), , are \(\mathbb{A}\)-adapted, integrable and φ-consistent.
Similar to Corollary 6.5 we see in the next lemma that the deflated financial processes are \((\mathbb{P}, \mathbb{A})\)-martingales.
Lemma 6.11
(Financial price processes)
Under Assumption 6.10 we obtain for any financial price process , , the following \((\mathbb{P}, \mathbb{A})\)-martingale property
with financial deflator defined in (6.15).
Proof of Lemma 6.11
First we note that Assumption 6.10 implies
The φ-consistency and the \(\mathbb{A}\)-adaptedness imply
Now, we consider the conditional expectation w.r.t. on both sides of the last equality. This provides for and -measurable price \(A_{t}^{(i)}\) (in the third step we use (6.16))
This proves the claim. □
Under the basic actuarial model Assumption 6.3 we have assumed a product structure (6.5) for the cash flows X k . Under the improved actuarial model Assumption 6.10 things become more involved. Assume that the cash flow is given by
where Λ (k) is -measurable and \(U_{k}^{(k)}\) is -measurable. again describes the \(\mathbb{A}\)-adapted φ-consistent price process of the financial portfolio \(\mathfrak{U}^{(k)}\) available at the financial market , and Λ (k) is the number of units of such financial portfolios \(\mathfrak{U}^{(k)}\) we need to purchase in order to fulfill the insurance liability cash flow X k . However, this time we make these assumptions under the improved actuarial model Assumption 6.8. We obtain the following theorem.
Theorem 6.12
Under Assumption 6.10 the φ-consistent price at time k−1 of the insurance liability cash flow X k described above is given by
Proof of Theorem 6.12
We obtain at time k−1 (we use consistency w.r.t. φ in the first step, definition (6.15) in the second step, the conditional independence in the third step, and we use (6.16) and linearity for the financial price process in the fourth equality)
This proves the claim. □
The concept introduced in Assumption 6.10 is clear and the price
can easily be calculated. But the further derivations of the price process Q t [X k ] for t<k−1 are not straightforward. The reason therefore is that we would need to further describe the term , i.e. how this term is influenced by financial and insurance technical information.
Therefore, in the sequel our considerations will be done under the product structure of Assumption 6.3 (basic actuarial model). We especially also refer to the last bullet point in Remarks 6.9.
References
Danielsson J, De Jong F, Laeven R, Laux C, Perotti E, Wüthrich MV (2011) A prudential regulatory issue at the heart of solvency II. VoxEU, 31 March 2011. Available under http://www.voxeu.org
Danielsson J, Laeven R, Perotti E, Wüthrich MV, Ayadi R, Pelsser A (2012) Countercyclical regulation in solvency II: merits and flaws. VoxEU, 23 June 2012. Available under http://www.voxeu.org
European Commission (2010) QIS 5 technical specifications, annex to call for advice from CEIOPS on QIS5
Wüthrich MV (2011) An academic view on the illiquidity premium and market-consistent valuation in insurance. Eur Actuar J 1(1):93–105
Wüthrich MV, Bühlmann H, Furrer H (2010) Market-consistent actuarial valuation, 2nd edn. Springer, Berlin
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Wüthrich, M.V., Merz, M. (2013). Actuarial and Financial Modeling. In: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31392-9_6
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