Insurance valuation: a computable multi-period cost-of-capital approach
We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-of-capital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capital margin and relate it to dynamic risk measurement. Moreover, we present explicit formulas and properties of the cost-of-capital margin under further assumptions on the model for the liability cash flow and on the conditional risk measures and utility functions. Finally, we highlight computational aspects of the cost-of-capital margin, and related quantities, in terms of an example from life insurance.
Keywords: valuation of insurance liabilities, multi-period valuation, market-consistent valuation, cost of capital, risk margin, dynamic risk measurement
The current solvency regulatory framework Solvency II emphasizes market-consistent valuation of liabilities; it is explicitly stated that liabilities should be “valued at the amount for which they could be … transferred or settled … between knowledgeable and willing parties in an arm’s length transaction”. Solvency assessment of an insurance company is based on future net values of assets and liabilities, and market-consistent valuation enables solvency assessments that takes dependence between future values of assets and liabilities into account. Moreover, current regulatory frameworks emphasize risk measurement over a one-year period. In particular, at any given time, the whole liability cash flow is taken into account in terms of the cash flow during the next one-year period and the market-consistent value at the end of the one-year period of the remaining liability cash flow. However, liability cash flows are typically not replicable by financial instruments. Therefore, the contribution to the liability value from the residual cash flow resulting from imperfect replication must be determined.
Given an aggregate liability cash flow of an insurance company, portfolios may be formed that generate cash flows with expected values matching that of the liability cash flow. The traditional actuarial practice of reserving provides an example of such a portfolio consisting of default-free bonds. In case of dependence between the liability cash flow and market values of financial instruments, more sophisticated replicating portfolios may be more suitable. However, the mismatch between the cash flow of such a portfolio and that of the original liability cash flow is typically substantial. The residual liability cash flow must be handled throughout the life of the liability cash flow by making sure that sufficient additional capital is available at all times. Capital providers, such as share holders, require compensation for providing buffer capital, which should be taken into account in the liability valuation. In particular, capital costs should be accounted for.
In Solvency II, the so-called technical provisions correspond to the aggregate liability value and is defined as the sum of a best estimate, corresponding to a discounted actuarial fair value, and a so-called risk margin aimed at capturing capital costs. Unfortunately, the risk margin in the current regulatory framework lacks a proper definition and theoretical foundation, and different approximation formulas for this ill-defined object have been suggested. Criticism of the risk margin and suggestions for better notions of cost-of-capital margins or market-value margins are found in e.g. , , ,  and .
This paper addresses valuation of an aggregate liability cash flow of an insurance company, although the problem and our suggested solutions apply to liability valuation in other contexts as well. We present an approach that, in many aspects, is similar to current practice and has wide-ranging applicability. The approach we propose is inspired by  and , where the cost-of-capital margin for valuing aggregate liability cash flows is analyzed. The framework for liability cash flow valuation presented in  combines financial replication arguments with cost-of-capital considerations. Our proposed framework is on the one hand a generalization of that framework and with more attention paid to mathematical details. On the other hand, for the repeated one-period replication of the residual cash flow, we severely restrict the allowed replication instruments compared to . Such a simplification of the problem allows us to derive much stronger results which in turn yields a framework that can be easily adopted, and it avoids many of the computational difficulties highlighted in  without sacrificing conceptual consistency. From a practical perspective, it leads to an approach to valuation of liability cash flows that does not rely heavily on subjective choices of joint dynamics for market prices of possible replication instruments. The use of financial valuation principles in insurance is inevitable given the principle of market-consistent valuation of liabilities, in particular for liability cash flows with long durations and products with guarantees. For more on financial and actuarial valuation of insurance liabilities aimed at solvency assessment, see .
Conditional monetary risk measures and utility functions are important basic building blocks in the approach to liability cash flow valuation considered here. Extensions from static, or one-period, risk measurement to dynamic risk measurement has been studied extensively for more than a decade following the seminal paper  on one-period risk measurement, see e.g. , , , ,  and . Much of the analysis of dynamic risk measurement has focused on dynamic measurement of risk corresponding to a single cash flow, such as the cash flow of a derivative payoff, at a fixed future time. In  a market-value margin for valuation of insurance liabilities is presented based on multi-period mean-variance hedging. However, this valuation framework is not directly applicable to the problem we consider since the liability cash flow considered in  occur at a terminal time whereas we consider cash flows at all times up to a terminal time. For the problem we consider, there is no natural way to roll cash flows forward and thereby reducing the dynamic risk measurement problem to a considerably simpler problem. The frameworks for dynamic risk measurement developed in  and  are well-suited to handle liability valuation problems of the type we consider. However, we do not want to restrict the liability cash flows to bounded stochastic processes. Moreover, the liability valuation problem we consider corresponds to repeated one-period replication rather than truly multi-period replication. Another important aspect is that we assume that the capital provider has limited liability and that causes the cost-of-capital margin to lack the convexity properties that are essential for the so-called risk-adjusted values analyzed in  and similarly for the dynamic risk measures analyzed in . Risk measurement for multi-period income streams are studied in  and , where the dynamic risk measurement problem is formulated as a stochastic optimization problem. There, computational aspects of multi-period risk measurement are clarified and illustrated. Although our approach to liability valuation differ substantially from that in  and , computability is an essential feature.
This paper is organized as follows.
Section 2 gives a nontechnical derivation of the cost-of-capital margin, as we believe that it should be defined, by economic arguments.
Section 3 presents a mathematical framework that allows the cost-of-capital margin to be defined rigorously, and establishes its fundamental properties. We also describe how the cost-of-capital margin is related to conditional monetary utility functions in the sense of , showing that the cost-of-capital margin is conceptually consistent with dynamic monetary utility functions and risk-adjusted values as defined in  and . There is however a major difference. The limited liability property of capital providers is an essential ingredient in our definition of the cost-of-capital margin and may cause violation of concavity/convexity properties. We define the cost-of-capital margin in terms of repeated one-period replication similar to  and allow capital requirements to be given in terms of conditional versions of nonconvex risk measures such as Value-at-Risk which is the current industry practice for insurance markets subject to the Solvency II regulation. Therefore, convexity properties are not assumed and not essential to us. Time-consistency is however an essential property. For the cost-of-capital margin, this property essentially follows immediately from the definition. Towards the end of Section 3 we consider a family of conditional risk measures that include commonly used risk measures such as Value-at-Risk and spectral risk measures, and we show that this family of risk measures are particularly useful for ensuring stronger properties and explicit formulas for the cost-of-capital margin when the liability cash flow is restricted to certain families of stochastic processes. It is well known, see e.g.  and , that conditional or dynamic versions of Value-at-Risk and spectral risk measures are not time-consistent when applied to time periods of varying lengths. In our setting, only repeated conditional single-period risk measurement appears. Therefore, time-inconsistency of risk measurement over time periods of varying lengths does not cause problems for the time-consistency of the cost-of-capital margin.
Section 4 considers specific models for the liability cash flow and the filtration representing the flow of information over time about the remaining cash flow until complete runoff of the liability. Specifically, we consider models of autoregressive type and Gaussian models. We show that when combined with the general framework presented in Section 3, these models allow for explicit formulas and stronger results concerning the effects of properties of the chosen filtration. We believe that the explicit expressions presented here constitute candidates for standard formulas for cost-of-capital margin computation that may be adopted in improved future solvency regulation.
Finally, Section 5 presents a life-insurance example that illustrates features of the cost-of-capital margin and clarifies computational aspects.
2. The cost-of-capital margin
In this section we derive the cost-of-capital margin without mathematical details, they are found in Section 3.
We consider time periods (years) , corresponding time points , and a filtered probability space , where with .
A liability cash flow corresponds to an -adapted stochastic process interpreted as a cash flow from an aggregate insurance liability in runoff. Our aim is to give a precise meaning to the market-consistent value of the liability by taking capital costs into account, and provide results that allow this value to be computed.
When the value of an insurance liability cash flow includes capital costs from capital requirements based on future values of both assets and liabilities, the liability value depends on the future values of all assets, including assets held for investment purpose only. In particular, two companies with identical liability cash flows would assign different market-consistent values to the two identical cash flows. This has undesired implications. Instead, as is done in e.g.  and prescribed by EIOPA, see [8, Article 38], we take the point of view that an aggregate liability cash flow should be valued by considering a hypothetical transfer of the liability to a separate entity, a so-called reference undertaking, whose assets have the sole purpose of matching the value of the liability as well as possible.
We will give a meaning to the liability value by a particular two-stage valuation procedure: the first stage corresponds to choosing a replicating portfolio of traded financial instruments, the second stage corresponds to managing the residual cash flow from imperfect replication in the first stage.
At time , a portfolio is purchased with the aim of generating a cash flow replicating the cash flow . This static replicating portfolio has a market price and generates the cash flow . We use the wording “static” in order the emphasize that, for the purpose of valuation, it is a portfolio strategy that is fixed throughout the life of the liability cash flow. However, the replicating portfolio may be “dynamic” in the sense that its cash flow depends on events not known at time . This is completely in line with pricing a financial derivative in terms of the initial market price of a self-financing hedging strategy. If is independent of financial asset prices, then the canonical example is a portfolio of zero-coupon bonds generating the cash flow .
The value of the original liability is defined as the sum of the market price of the replicating portfolio and the value of the residual cash flow from repeated one-period replication using only cash funds provided by a capital provider with limited liability requiring compensation for capital costs. That is, the cash flow will be re-valued at all times . We call the value of the residual cash flow the cost-of-capital margin. Note that we below refer also to as the cost-of-capital margin for all . We emphasize that repeated one-period replication is done with cash only. Allowing for repeated one-period replication using a mix of assets such as bonds with short time to maturity inevitably makes the value of the liability cash flow depend on subjective views on the development over time of spot rates over different time horizons.
Next we will present economic arguments that lead to a recursion defining in terms of and , capital requirements and the acceptability condition of the capital provider. At time , the insurance company is required to hold the capital , where is a conditional monetary risk measure, see Definition 1, quantifying the risk from liability cash flow during year and the value at time of the remaining residual cash flow . The capital provider is asked to provide the capital
The amount is the difference between the required capital and the value of the residual liability cash flow at time . If the capital provider accepts providing at time , then at time the capital is available. If this amount exceeds the value of the liability at time , then the capital provider collects the excess capital as a compensation for providing the buffer capital at time . Moreover, the capital provider has limited liability: if
then the capital provider has no obligation to provide further capital to offset the deficit.
The capital provider’s acceptability condition at time is expressed in terms of a conditional monetary utility function , see Definition 2, and a value quantifying the size of the intended compensation to the capital provider for making capital available:
where . may be chosen as the conditional expectation although alternatives that take the risk aversion of the capital provider into account may be more appropriate.
If the inequality above is strict, then the capital provider obtains a better-than-required investment opportunity at the expense of policy holders who are obliged to pay higher-than-needed premiums. Therefore, we define the value of the cash flow as the smallest value for which the capital provider finds the investment opportunity acceptable. That is, we replace the inequality above by an equality:
Recall that the market-consistent value we assign to the original cash flow is the sum of the market price of a replicating portfolio set up at time and the cost-of-capital margin . Notice that if perfect initial replication is possible, then as there is no residual cash flow, and consequently since no capital funds for repeated one-period replication are needed.
3. The valuation framework
We consider time periods , corresponding time points , and a filtered probability space , where with . Let denote the vector space of all real-valued -measurable random variables, and let be the subset of of random variables taking values in . is simply the set of constants, i.e. . For , let and let denotes the subset of of random variables taking values in . Finally, consists of the essentially bounded -measurable random variables: such that
We say that two random variables are equal if they coincide -almost surely (a.s.). All equalities and inequalities between random variables are interpreted in the -a.s. sense.
In order to determine the cost-of-capital margin we consider conditional monetary risk measures and conditional monetary utility functions . We express values and cash flows via a numéraire which we take to be a money market account that pays no interest, where money can be deposited and later withdrawn. We have no need for and do not assume risk-free borrowing. Choosing the numéraire to be a money market account paying stochastic interest rates would not pose mathematical difficulties but would force us to pay more attention to the interpretation of the cash flows.
By a dynamic monetary risk measure quantifying one-period capital requirements we mean the following:
For , a dynamic monetary risk measure is a sequence of mappings satisfying
By a dynamic monetary utility function quantifying one-period acceptability for capital providers we mean the following:
For , a dynamic monetary utility function is a sequence of mappings satisfying
The following proposition provides the basis for defining the cost-of-capital margin in (3) rigorously.
Proof of Proposition 1.
Since is -measurable and takes values in , it follows directly from the definitions of and that is a mapping from to . The properties (11) and (13) for follow immediately from the corresponding properties of and . It remains to verify property (12) for . Take in . Then
which further implies
and verifies the property (12). Finally, we verify the representation (14) of in terms of and . If for , then and the right-hand side in (14) is well-defined. Repeated application of (11) now verifies the representation (14). ∎
Notice that we may express (15) as .
Let be -adapted cash flows with for every .
(i) Let , let be a -dimensional vector with components in , and let for each . Then, for every ,
(ii) The cost-of-capital margins are time consistent in the sense that for every pair of times with , the two conditions and together imply .
Proof of Proposition 2.
The dynamic version of Value-at-Risk presented in the example below is an example of a dynamic monetary risk measure with . In Section 3.3 further examples of dynamic monetary risk measures and utility functions are presented and their properties are investigated for use in Section 4 together with specific models for the liability cash flows.
In the static or one-period setting, Value-at-Risk at time at level of a value is defined as
where denotes the distribution of , and . The natural dynamic version of Value-at-Risk at time at level of a value is
where “” denotes the greatest lower bound of a family of random variables (with respect to -almost sure inequality). Alternatively, we may define in terms of a conditional distribution of with respect to : for each , is a probability measure on the Borel subsets of , and for each Borel set , is a version of . Define by
and notice that -almost surely. Notice that satisfies the properties in Definition 1 for .
3.1. Model-invariant bounds
Consider a dynamic risk measure and a dynamic monetary utility function , and given by (10). Consider also an -adapted cash flow with for every . With ,
Further assumptions clearly enable sharper bounds. Suppose that and take, for every , to be the conditional expectation given . It follows from Jensen’s inequality for conditional expectations that the conditional expectation is well defined as a mapping for . In particular,
Applying this inequality repeatedly together with the tower property of conditional expectation yields
Notice that if for all , then
Notice that if, further, the static replicating portfolio is chosen at time such that , then the residual cash flow has zero mean. In particular, then the second sum in the above upper bound vanishes, i.e.
Notice that the upper bound for in (16) can be rewritten as
Repeated application of this inequality and use of the tower property of conditional expectation yields
Hence, if further for all and , then
Notice the difference between the two upper bounds (17) and (18): the former is formulated in terms of expected future capital requirement whereas the latter is formulated in terms of expected future buffer capital provided by capital providers.
We end the discussion of model-invariant bounds for the cost-of-capital margin with a comment on the Solvency II risk margin. In [8, Article 37] it is stated that the risk margin should be computed as
where , denotes the basic risk-free interest rate for the maturity of years, and denotes the Solvency Capital Requirement after years. In our setting, for since there is no liability cash flow beyond that time. One may criticize several aspects of the Solvency II risk margin. First, for , is a random variable as seen from time . Secondly, does not take capital costs into account, and the discounting in the computation of and in the computation of the risk margin are not conceptually consistent.
The upper bound in (18) is somewhat similar to the formula for the Solvency II risk margin. For , take to be a dynamic risk measure in the sense of Definition 1, and take the to be conditional expectations . If we define the SCR-like quantity
For an in-depth comparison between a conceptually consistent notion of cost-of-capital margin and the Solvency II risk margin, see Section 5 in .
3.2. Dynamic monetary utility functions
The notions of conditional and dynamic monetary utility functions in  and risk-adjusted values in  are closely connected to the cost-of-capital margin considered here. In  and , cumulative cash flows and value processes are considered whereas we consider incremental cash flows and liability processes corresponding to liability values for future cash flows. We will now proceed to establish the connection between (and ) and the work of  and .
For , let denote the space of all -adapted stochastic processes with for every . For , define the projection by
and . Let be as in Proposition 1. For all , let be given by
and similarly, .
We now verify that in (19) is a conditional monetary utility function in the sense of Definition 3.1 in , excluding the concavity axiom, and that is time-consistent in the sense of Definition 4.2 in .
The mappings in (19) are conditional monetary utility functions in the sense
and is time-consistent in the sense
for every and all .
Proof of Proposition 3.
Since for ,
Due to the fact that for . For such that ,
Noting that and using, repeatedly, the monotonicity property of for we arrive at . Finally we note that
where the second equality follows from the translation invariance of for . Time consistency in the sense of (20) follows almost directly from the definition.
where the last equality is due to the translation invariance of for combined with the fact that is -measurable for and noting that for . From the definition (19) of we get
3.3. Risk measures and utility functions based on conditional quantiles
For , write for its conditional distribution given : is a probability measure on the Borel subsets of , and is a version of . We may write the conditional distribution and quantile functions of given , respectively, as
For a probability measures and on , define, for ,
Suppose there exist and such that, for ,