Analytical Validation Formulas for Best Estimate Calculation in Traditional Life Insurance
aa r X i v : . [ q -f i n . R M ] J u l ANALYTICAL VALIDATION FORMULAS FOR BEST ESTIMATECALCULATION IN TRADITIONAL LIFE INSURANCE
SIMON HOCHGERNER, FLORIAN GACH
Abstract.
Within the context of traditional life insurance, a model-independent relationshipabout how the market value of assets is attributed to the best estimate, the value of in-forcebusiness and tax is established. This relationship holds true for any portfolio under run-offassumptions and can be used for the validation of models set up for Solvency II best estimatecalculation. Furthermore, we derive a lower bound for the value of future discretionary benefits.This lower bound formula is applied to publicly available insurance data to show how it canbe used for practical validation purposes. Introduction
Per 1 January 2016 the European Union has implemented a new insurance regulation schemecalled Solvency II. The own funds in this regime, essentially, equal the excess of the marketvalue of assets over the market value of liabilities. The legal basis for calculating market valuesin accordance with Solvency II is Art. 88 in [13]. One of the reasons that this is a difficultmatter is that insurance liabilities are not traded in an active market. See [20, 19, 22, 8, 17]for more information and a discussion.In the context of traditional life insurance products, with a well-defined profit sharing mech-anism, the market value of liabilities is dominated by the so-called best estimate. Indeed, forthe aggregated Austrian market we have, per year end 2016, the following situation (in billionEUR):Technical provisions - life (excluding health and index-linked and unit-linked) 59Best estimate 58Risk margin 1That is, the risk margin is only about 1 .
8% of the best estimate. For Germany we have asimilar situation:Technical provisions - life (excluding health and index-linked and unit-linked) 909Best estimate 894Risk margin 15Again, the risk margin amounts to 1 .
6% of the best estimate. All these numbers are takenfrom the EIOPA insurance statistics web-site [9].
Date : June 27, 2019.
Key words and phrases.
Solvency II, Best Estimate, Asset Liability Management, Market ConsistentValuation.
Thus, own funds of the company strongly depend on the value of the best estimate ( BE ).Unfortunately, best estimate calculation for traditional life insurance books is a very difficultproblem and its result generally subject to considerable modelling uncertainty.To get an idea of the impact of errors, or uncertainty, of BE calculation, consider thefollowing, not untypical, sand-box life insurance example in Table 1. In fact, these numbersBest Estimate 3’864Eligible Own Funds 572Solvency Capital Requirement ( SCR ) 292
Table 1. are not arbitrary, but represent (in billion EUR) the European aggregate values taken from[14] for the base-line scenario. The corresponding solvency ratio is thus 572 / ∼ = 196%.There are two (not unrelated) points to be made here: The eligible own funds are one orderof magnitude smaller than the technical provisions, which we have identified with the bestestimate (by disregarding the much smaller risk margin) for the sake of exposition. Secondly,an increase of the best estimate by only 1% would result in a decrease of solvency ratio by0 . · ′ / ∼ = 13 percentage points. An increase by 10% would yield a solvency ratiobelow 100%, i.e. insolvency. Here we assumed the SCR to remain unchanged by an increasein BE . In reality the SCR would increase with increasing BE , whence the resulting solvencyratio would be even smaller.In fact, [14] has considered mainly, but not exclusively, life insurance companies. If onewere to include only traditional life insurance books in the sample, the impact of best estimatechanges on solvency ratios should be expected to be even stronger.We mention these numbers in order to stress the importance of reliable and stable bestestimate calculation.In order to calculate a best estimate for traditional life insurance in accordance with SolvencyII regulation, one has to set up an asset liability model which generates cash-flows according tolocal generally accepted accounting principles. In particular, such a model has to keep track ofbook and market values of assets under management and generate a return on assets that is inline with realistic management actions. Furthermore, a stochastic economic scenario generatoris needed to model the asset portfolio and the relevant discount rates.It is therefore not surprising that there are no closed formulas to calculate best estimateliabilities associated to traditional life insurance contracts. Let us remark in this context, thatone of the outstanding features of the traditional life insurance business is that there exists amechanism of profit sharing between the firm’s surplus and the policy holder (see [12]).The goal of this paper is to present two results, Propositions 2.2 and 3.4, which providesimple and effective tools for best estimate validation. These validation tools are the nextbest thing to having a closed formula for best estimate calculation in the sense that they are analytical and model-independent . That is, they have to hold for any best estimate calculationin the context of life insurance with profit sharing – as long as certain assumptions are met.These assumptions are explicitly spelled out and subsequently discussed in Sections 2.C and3.D. Let us remark that these assumptions are tailor-made to the Austrian and German life NALYTICAL VALIDATION FORMULAS 3 insurance markets. We have not attempted a comprehensive study concerning the validity ofthe assumptions for other European markets.As an example, we apply the estimation formula of Proposition 3.4 to publicly availabledata of the Allianz Lebensversicherungs AG.Conclusions concerning our results are discussed in Section 4.
Acknowledgements.
We thank the anonymous reviewers for their detailed and helpfulinput.
Disclaimer.
The views expressed in this article reflect the authors’ personal opinions anddo not necessarily coincide with those of the Austrian Financial Market Authority (FMA).2.
A basic equation of traditional life insurance valuation
Traditional life insurance company.
We consider a company selling traditional lifeinsurance business. Traditional means in this context that there is a procedure of profitparticipation which is defined by local generally accepted accounting principles. The set ofinsurance contracts in the company’s liability book shall be denoted by X . Hence each x ∈ X corresponds to an individual policy holder. The types of contracts weconsider in this context are those of [12, Chapters 3 and 4]: life insurance and life annuities,subject to profit participation. Let us assume: Assumption 2.1.
The liability book consists of two items:(1)
T R , the technical reserves;(2) SF , the surplus fund.The technical reserves are made up of the individual statutory reserves, by which we meanthe contract specific reserves T R x with respect to local generally accepted accounting stan-dards. Thus to each insurance contract there corresponds a well-defined technical reserve T R x ,and T R is the sum of all the individual reserves. We emphasize that the technical reserves
T R x are assumed to be calculated as explained by [12], that is, with respect to a constant technicalinterest rate and with respect to prudentially chosen safety loadings. The surplus fund, onthe other hand, is an additional balance sheet item that belongs to the collection of insuredviewed as a whole, whence it belongs to the liability book. However, it cannot be assignedto any individual contracts. Its purpose is to smoothen the declaration of profit participationover time.The book values BV ( T R ) = X x ∈X BV ( T R x )and BV ( SF ) of T R and SF of the technical reserves and of the surplus fund, respectively,are well-defined balance sheet items in the realm of the local generally accepted accountingprinciples (see [4, § SIMON HOCHGERNER, FLORIAN GACH
The basic equation.
We work on a discrete time grid 0 , . . . , T . In practice time will bemeasured in years and T will be the time until the run-off of (a sufficiently large percentageof) the portfolio. Usually T is 60 years or even more. All our stochastic processes are assumedto be adapted to some underlying filtered probability space satisfying the usual assumptions.Assume we have fixed an interest rate model with corresponding bank account numeraire B t , t = 0 , . . . , T, B = 1 . Let Q be the risk neutral martingale measure associated to our interest rate model or, moregenerally, our model of the economy. For 0 ≤ t ≤ s let P ( t, s ) be the time t -value of a pay-outof one unit of currency at time s . Then, for any standard interest rate model, it is true that(2.1) B − t P ( t, s ) is a Q -martingalewith respect to the filtration underlying the interest rate model. For instance, this holds forshort rate models or Libor Market Models. We refer to [11] for more information on interestrate models. In fact, Condition (2.1) is the no-arbitrage condition as in, e.g., [21, Equ. (3.2)]. Interms of simple one-year forward rates F t , the bank account is given as the roll-over investment B t = Π ts =1 (1 + F s − ). The best estimate BE associated to X is then defined as(2.2) BE = E Q " T X t =1 B − t X x ∈X cf x,t where cf x,t is the cash-flow at time t generated by contract x . The cash flow has to take intoaccount all relevant premia, benefits and costs – see [6, Art. 28].Let(2.3) BV t = BV t ( T R ) + BV t ( SF )denote the book value of the liability portfolio at time t . The liability portfolio is covered byassets whose book value is assumed to equal BV t . That is, we make the following simplification: Assumption 2.2.
The company does not have equity in its statutory balance sheet, i.e. withrespect to local generally accepted accounting rules.In other words, Assumption 2.2 requires the book value of assets to be equal to the bookvalue of liablities, which is, of course, a very unrealistic assumption. Its meaning is discussedin Subsection 2.C below where we will see that no generality is lost. We denote the totalmarket value of the company’s assets at time t by M V t . The market value of assets
M V t will change from one period to the next due to marketmovements. The same is true, albeit in a more complicated manner, for the book valueof assets, BV t : it changes due to coupon payments, dividend yield, etc., as well as due torealization of unrealized gains (whose value depends again on market movements). The precisemanner in which this happens depends on the company’s management rules and valuationchoices such as the lower of cost or market principle. Regardless of the specific management The lower of cost or market principle, when applied in its strict form, requires a company to depreciatethe book value of an asset whenever its market value falls below its current book value. When applied in the
NALYTICAL VALIDATION FORMULAS 5 rules and choices, we denote the company’s book value return on BV t by roa t +1 and emphasize that this is the book value return before corporate tax.For the next statement we note that we call market value induced changes in book valuethose changes which follow from the application of valuation principles, such as the lower ofcost or market principle, or the realization of unrealized gains. Criterion 2.1 (Monetary conservation principle) . All changes in the book value of assets BV t are either due to a cash-flow or due to a market value induced change in book value. This property is fundamental. It can be viewed as a no-leakage and self-financing property:up to cash-flows, changes in the book value of the asset portfolio can only be due to interestrate or other (such as: stock) market effects. The above criterion is important in practiceas it provides a simple yet challenging validation test for the inspection of real models: seeProposition 2.2.Let us further elaborate on the no-leakage statement. To this end, we denote by tax t thecorporate tax and by sh t the shareholder gains at t . Now, since cf t = P x ∈X cf x,t alreadyincludes all policy holder and cost cash-flows, the no-leakage criterion amounts to BV t = BV t − − cf t − sh t − tax t + roa t (2.4)Let us define the unexpected return by ur t := roa t − F t − BV t − where F t − = B t B t − − t − t implied by the interest rate model.It follows that B − T BV T = BV + T X t =1 (cid:16) B − t BV t − B − t − BV t − (cid:17) = BV + T X t =1 (cid:16) B − t ( BV t − − cf t − sh t − tax t + roa t ) − B − t − BV t − (cid:17) = BV + T X t =1 (cid:16) ( B − t ( − cf t − sh t − tax t ) + B − t ( roa t − F t − BV t − ) (cid:17) = BV + T X t =1 (cid:16) ( B − t ( − cf t − sh t − tax t ) + B − t ur t (cid:17) . mild form, the book value is depreciated only if the market value is expected to remain below its current bookvalue for an extended period of time. SIMON HOCHGERNER, FLORIAN GACH
Taking the expected value with respect to the risk-neutral measure Q , this becomes(2.5) BV + E Q " T X t =1 B − t ur t = BE + V IF + T AX + E Q (cid:2) B − T BV T (cid:3) , where V IF = E Q " T X t =1 B − t sh t is the so-called value of in-force business and T AX = E Q " T X t =1 B − t tax t is the value of corporate tax payments. The value of in-force business is the model dependentpart of the market consistent embedded value M CEV which is generally expressed as
M CEV = V IF + F C , where F C is the market value of free capital at time t = 0. The M CEV is a measure forthe shareholder to determine how well the money is invested. Up to required capital andassociated frictional costs, this definition coincides with that of [5].Let us call
U G t := M V t − BV t the unrealized gains. Proposition 2.2 (Basic equation of market consistent valuation) . Let T be the projectionhorizon. Then (2.6) BV + U G = BE + V IF + T AX + E Q h B − T M V T i . Moreover, if BV t ( SF ) is bounded by BV t ( T R ) then E Q h B − T M V T i /M V ≈ . In this statement ≈ means equality for all practical valuation purposes. We remark thatthere is no advantage in defining the relation ≈ in a more mathematical manner. Practicallythe remainder term E Q h B − T M V T i /M V should be of the same order as the result of the leakagetest, which would be the difference of the two sides of Equation (2.6).Laimer [16] has verified in her diploma thesis that Equation (2.6) does indeed hold with E Q [ B − T M V T ] /A = 0 up to numerical errors. To do so, she employed a best estimate calcula-tion tool proprietary to the FMA Austria and used several concrete traditional life insuranceportfolios. Proof.
Let A t denote the company’s set of assets under management at time t . It follows that ur t = P a ∈A t − ur a,t where ur a,t is the contribution stemming from asset a and where the sum As a rule of thumb the relative error should not exceed 1 ‰ , otherwise the impact of potential error onown funds would be too large – compare with the sand box example in Table 1. NALYTICAL VALIDATION FORMULAS 7 is over all assets in the portfolio at time t −
1. Now, for any asset a ∈ A , we have T X t =1 B − t ur a,t = T X t =1 B − t (cid:16) cf a,t + a ∗ t − (1 + F t − ) a ∗ t − (cid:17) = T X t =1 (cid:16) B − t cf a,t + B − t a ∗ t − B − t − a ∗ t − (cid:17) = T X t =1 B − t cf a,t + B − T a ∗ T − a ∗ = T X t =1 B − t cf a,t + B − T a T − a ∗ − B − T ( a T − a ∗ T )where a ∗ t (resp. a t ) is the time t book (resp. market) value of a and cf a,t is the asset’s cash-flow (coupon, dividend payment, etc.) at t . If a has a maturity T a within the projectionhorizon such that T a ≤ T , then book and market value coincide at T a such that a T a = a ∗ T a ,and cf a,t = a t = a ∗ t = 0 for T a < t ≤ T . On the other hand, if T a > T , then a T − a ∗ T = U G a,T are the unrealized gains of a at time T . Because of (2.1) we have a = E Q " T X t =1 B − t cf a,t + B − T a T , which implies that(2.7) E Q h T X t =1 B − t ur a,t i = a − a ∗ − E Q (cid:2) B − T ( a T − a ∗ T ) (cid:3) = U G a, − E Q (cid:2) B − T U G a,T (cid:3) . Assets that are bought in the course of reinvestment at t >
U G a,t = 0, because bookvalue and market value coincide at time of purchase. It follows that E Q h T X t =1 B − t ur t i = E Q h T X t =1 X a ∈A t − B − t ur a,t i = U G − E Q (cid:2) B − T U G T (cid:3) , where A t denotes the asset portfolio at time t , and the result is independent of the particularreinvestment strategy. The statement now follows from (2.5). (cid:3) Discussion of assumptions.
Assumption 2.1 is very generic. All we need at this pointare well-defined book and market values for the liability side of the balance sheet. If the cash-flows in Equation (2.2) depend on additional provisions, these should be accordingly added to BV and U G in (2.6).Assumption 2.2 is only at first sight a strong constraint. Actual companies will hold strictlypositive equity. The relevant position should be added to the left hand side of (2.6). For theright hand side, however, the statement E Q h B − T M V T i /M V ≈ M V T will still contain own funds. For actual validation purposes regarding (2.6)it is thus advisable to keep track of equity separately. SIMON HOCHGERNER, FLORIAN GACH An analytic lower bound for future discretionary benefits
The goal of this section is to derive an analytic, i.e. model-independent, formula for thevalue of future discretionary benefits. This is achieved under certain assumptions which arediscussed in Section 3.D below.3.A.
The lower bound formula.
The best estimate can be written as BE = GB + F DB where GB stands for guaranteed benefits and is the value of all future cash flows that arealready guaranteed at time of calculation t = 0. Thus GB is independent from all futuredevelopments and therefore purely deterministic. In particular, its value is independent ofall management actions and economic scenarios. On the other hand, F DB stands for futurediscretionary benefits and denotes the value of those cash flows that arise via the (future)profit sharing mechanism. Solvency II requires that
F DB be determined and reported as astand-alone part of the best estimate: see [6, Art. 25].
Remark 3.1.
We emphasize that this does not mean that the guaranteed benefits are fixedfrom the policy holder’s perspective. In reality, benefits could be influenced by time of sur-render, time of death, interest rate movements, tax incentives or other unknown variables.However, the point is that these variables contribute to the guaranteed benefits (as defined bythe reporting template [7, Template S.12.01.01]) with their expected values.
It is only in thissense, that the guaranteed benefits are model-independent and fixed cash-flows.The profit sharing mechanism dictates that the collective of policy holders receives a yearlyaccounting flow ph ∗ t .We make a few generic assumptions: Assumption 3.1.
We assume that the profit sharing mechanism is clearly defined by legisla-ture and stable management rules.
Assumption 3.2.
The gross policy holder profit participation rate gph is constant. This isthe rate with which the policy holder participates in the company’s declared gross surplus gs ∗ (if positive). It does not say anything about the company’s return.We emphasize that ph ∗ t is an accounting flow and not a cash flow. As such it is not paid outto the policy holder at time t , but rather increases the book value of liabilities. Observe that P H ∗ := E Q h X B − t ph ∗ t i (3.8) = gph E Q h X B − t gs ∗ + ( t ) i ≥ gph E Q h X B − t gs ∗ ( t ) i = gph ( V IF + P H ∗ + T AX )where sh t , tax t are the respective shareholder, tax cash flows, and x + = max( x, gs ∗ t = sh t + ph ∗ t + tax t together with the definitions of V IF and
T AX We reiterate that we only consider with-profit contracts. All assumptions are discussed in Section 3.D.
NALYTICAL VALIDATION FORMULAS 9 from Section 2.B. Note that sh t can be negative, which corresponds to the case of shareholdercapital injection. Assumption 3.3.
The policy holder participation ph ∗ t is negatively correlated with discountrate movements; i.e., policy holder participation will generally increase when interest ratesincrease: Corr [ B − t , ph ∗ t ] < Assumption 3.4.
We assume, for the purpose of this section, that the liability book consistsof only one contract and that the time to maturity of this contract is M . Assumption 3.5.
The technical reserves
T R t evolve deterministically. Insurance technicalgains are deterministic; i.e., we do not consider stochastic mortality modelling or stochastic(and/or dynamical) surrender behavior.Notice that the future discretionary benefits received by the policy holder at time of maturity M are exactly the sum P t ≤ M ph ∗ t of accumulated policy holder profits. This is actually a trickypoint and holds only because we assume that policy holder survival probabilities (mortality,surrender, etc.) have already been taken into account. At the same time we do not list thispoint as an assumption, because it only means that we regard cash flows of surviving policyholders.Notice that F DB = E Q h B − M X t ≤ M ph ∗ t i = E Q h B − M i · E Q h X t ≤ M ph ∗ t i + Cov h B − M , X t ≤ M ph ∗ t i ≥ P (0 , M ) · E Q h X t ≤ M ph ∗ t i − · SD h B − M i · SD h X t ≤ M ph ∗ t i (3.9)where P (0 , M ) = E Q [ B − M ] is the discount factor at time 0 and we have made use of the factthat the correlation Corr [ B − M , P t ≤ M ph ∗ t ] is bounded from below by − Remark 3.2 (Standard deviation) . For a random variable X , we shall denote the standarddeviation by SD [ X ] = p E [( X − E [ X ]) ]. In formula (3.9) the standard deviation is under-stood with respect to the risk neutral measure Q . Often the standard deviation is denoted by σ when viewed as a parameter to be inferred from a financial or econometric model. We havechosen the notation SD [ · ] to reflect the purely statistical approach of Assumption 3.12. Ofcourse, the statistical approach could also be replaced by a parametric model. However, sinceformula (3.11) depends on the product of two standard deviations, the gain in accuracy of thelower bound (3.12) by using a more refined model to estimate SD [ · ] is limited.On the other hand, Corr [ B − t , ph ∗ t ] < P H ∗ = E Q h X t ≤ T B − t ph ∗ t i ≤ X t ≤ T E Q [ B − t ] · E Q [ ph ∗ t ] ≤ max ≤ t ≤ T P (0 , t ) · E Q h X t ≤ T ph ∗ t i . (3.10)Note that, if forward rates are positive, this maximum is simply max ≤ t ≤ M P (0 , t ) = (1+ F ) − .Currently interest rates are negative at the short end. Nevertheless, for the EIOPA curve peryear-end 2017 as displayed in Table 8, the term max ≤ t ≤ M P (0 , t ) ∼ = 1 .
005 is very close to 1, and we will simply set it equal to 1 for better readability and because the error is negligible.The point is, in any case, that this term is deterministic and can be calculated from the initialforward curve.
Proposition 3.3.
Let A = M V = BV + U G , A T = E Q [ B − T M V T ] and (3.11) η := P (0 , M ) · (cid:16) − SD [ B − M ] P (0 , M ) · SD [ P t F DB ≥ η η ( A − A T − GB ) Proof. The inequality follows from equations (3.8), (3.9), (3.10) and Proposition 2.2. (cid:3) A lower bound formula for various maturities. For the purpose of this section,we shall keep all of the above assumptions except Assumption 3.4. We replace the latter asfollows: Assumption 3.6. Assets are not attributed to individual contracts.Let X denote the (finite) set of all contracts in the liability book. For each x ∈ X define η x according to formula (3.11) with respect to the contract’s maturity M x . Further, let D x := η x η x . Recall from Assumption 2.1 that the statutory technical reserves are the sum of the individualreserves, BV ( T R ) = P x ∈X BV ( T R x ). Consider A x := A · BV ( T R x ) /BV ( T R ), which is a proportional attribution of market values toindividual contracts according to the principle (3.6); GB x denotes the value of guaranteed benefits (at time 0) of contract x , and GB = P x ∈X GB x ; F DB = P x ∈X F DB x . Assumption 3.7. Suppose that Proposition 3.3 can be applied to each contract such that F DB x ≥ D x ( A x − A xM x − GB x ) . Notice that we do not assume A xM x = 0. This is to allow for cross-financing between contracts,after time M x . Define the weights w x := A x − GB x A − GB and the weighted depreciation factor (3.13) D := X x ∈X w x D x . The attribution can only depend on the book value, since T R x is a statutory reserve and therefore doesnot have a market value. NALYTICAL VALIDATION FORMULAS 11 It follows that F DB = X x ∈X F DB x ≥ X x ∈X D x ( A x − A xM x − GB x )= D X x ∈X ( A x − A xM x − GB x ) − X x ∈X ( D − D x )( A x − A xM x − GB x )= D ( A − GB ) − X x ∈X D x A xM x where we use that the weights w x are chosen such that P x ∈X ( D − D x )( A x − GB x ) = 0. Thequantities A xM x correspond to the fraction of A x which remains in the model after time M x .Unfortunately, these quantities are model dependent and are, therefore, a priori unknown. Theterm P x ∈X D x A xM x accounts for the cross-financing. If all of A x were to be accounted for bya cash-flow up to, and including, time M x , then A xM x = 0. Proposition 3.4. Assume that there exists F > such that P x ∈X D x A xM x ≤ F . Then (3.14) F DB ≥ D ( A − GB ) − F. Existence of such an F means that cross-financing is bounded by an a priori estimatedquantity F . See Section 3.C for a concrete application of this formula.3.C. Concrete numbers. Let us apply formula (3.14) to publicly available data from theAllianz Lebensversicherungs AG. Allianz Lebensversicherungs AG is a German life insurancecompany which has profit sharing contracts in its liability book. The data in Table 2 is takenfrom the publicly available reports [2] and [1], which concern the accounting year 2017. Hencethe applied interest rate information from Table 7 is also with respect to year-end 2017. Symbol Value Source Name in source BV U G SF GB F DB Table 2. Allianz Lebensversicherungs AG: public data. Values are in billion Euros. Assumption 3.8. Suppose the weighted depreciation factor (3.13) is equal to the one withmaturity M = 15, so that D = η / (1 + η ) with(3.15) η := P (0 , · (cid:16) − SD [ B − ] P (0 , · SD [ P t< ph ∗ t ] E Q [ P t< ph ∗ t ] (cid:17) · gph − gph . Variations of this assumption are shown in Tables 4 and 5.For concrete validation purposes, companies would have the data to explicitly calculate D according to formula (3.13), since w x and D x are quantities which are known a priori . Assumption 3.9. Assumption 3.2 is made concrete by setting gph = 0 . 80. In Table 3 weshall show results for gph = 0 . gph = 0 . 80 and gph = 0 . Assumption 3.10. Surplus funds are bounded by the technical reserve, i.e., SF t ≤ θT R t where θ > Assumption 3.11. The variance of ph ∗ t is not very high, that is, we assume SD [ P ph ∗ t ] ∼ =5% · E [ P ph ∗ t ]. Assumption 3.12. Interest rate variance should also be reasonably bounded. When estimatedon monthly historical data from year-end 2014 to year 2017 as shown in Table 7, we find, forthe coefficient of variation, that SD [ B − ] /P (0 , ∼ = 4%.With the discount factor P (0 , ∼ = 84% from Table 8, we insert these numbers in (3.11) toobtain η ∼ = 84% · (cid:16) − · (cid:17) · . − . ∼ = 3 . LB := η η (cid:16) A − GB (cid:17) ∼ = 77% · (cid:16) . . − . (cid:17) ∼ = 62 . F DB = 48 . 6, we have to subtract two quantitiesfrom the lower bound LB :(1) According to [13, Art. 91] the surplus fund, SF = 10 . 4, is not part of the Solvency II valueof liabilities if this article is authorized by national law . This is the case for Germany (see[3]), whence the surplus fund is to be subtracted from the future discretionary benefitswhich are calculated by the company (and this deductible is not part of the reported F DB ).(2) The cross-financing term F from Proposition 3.4.Therefore, the resulting lower bound is(3.17) LB = LB − SF − F ∼ = 52 . − F. To estimate F , we have to attribute A to individual contracts and say something about thecross-financing effects. This information is not publicly available. We therefore separate A into buckets A x ( t )0 , where A x ( t )0 belongs to those contracts x ( t ) which mature at time t . Thuscontracts are bundled according to their time of maturity. We have to make an assumptionconcerning the run-off of the portfolio: Assumption 3.13. The portfolio run-off is roughly geometric, so that the value of reserves isreduced by a factor of every 10 years until the end of the projection. This is formalized asthe requirement A x ( t )0 := (cid:16) − t − − − t (cid:17) A for t = 1 , . . . , T − A x ( T )0 := 2 − T − A , and we shall assume that T = 60.Notice that A = P x ∈X A x = P t =1 A x ( t )0 and the corresponding run-off is given by(3.18) A − s X t =1 A x ( t )0 = A · − s/ NALYTICAL VALIDATION FORMULAS 13 for s < T .Furthermore, we have to make an assumption concerning the policy holder cross-financingterm A tM t from Proposition 3.4 (where M t = t ). This term has to be a fraction of A x ( t )0 , but thisfraction need not be the same for all contracts. For example, it is conceivable that contractswith a low technical interest rate will yield more cross-financing in comparison to contractswith a high technical interest rate. Moreover, the cross-financing effect need not be constantin time. Assumption 3.14. The cross-financing factor is a decreasing function of time, but does notdepend on other contract properties (such as technical interest rate): A x ( t ) t = C ( t ) A x ( t )0 where C ( t ) = C T − tT ,T = 60; and we will consider C = 1%, C = 3% and C = 5%.With the above assumptions it is possible to calculate the term F from (3.14). Indeed, weset(3.19) F = X x ∈X D x A xt = C X t =1 η x ( t ) η x ( t ) T − tT A x ( t )0 where(3.20) η x ( t ) = P (0 , t ) (cid:16) − SD [ B − t ] P (0 , t ) · (cid:17) gph − gph . Now, P (0 , t ) is taken from Table 8 and SD [ B − t ] is estimated from Table 7. With gph = 80%and C = 3% this yields F ∼ = 4 . 1, whence we obtain the lower bound(3.21) LB = LB − SF − F ∼ = A C48 . bn. This number should be compared with the value of F DB = A C48 . bn in Table 2. The assump-tions leading to (3.21) and, in particular, the choices M = 15, gph = 80% and C = 3% arediscussed in Section 3.D. See Section 5 for further choices of M , gph and C .3.D. Discussion of assumptions. In the following we shall provide justification for the aboveassumptions. Nevertheless, we stress that these remain unproved (in the precise mathemati-cal sense) assumptions based on heuristic arguments and expert judgment. For some of theassumptions we can provide a sensitivity analysis in Section 5.Assumption 3.1 is one that is necessary for any asset liability model that could be employedfor best estimate calculation. It is also necessary for our derivation of the lower bound formula.The net policy holder participation fraction shall be denoted by nph. Under Austrian law ([23, § at least nph = 85% of theirnet profits with policy holders. As the surplus fund belongs to the liability side, this sharingmechanism does not imply that 85% of net profits are directly declared to specific policy holderaccounts. Rather the profits are shared with the surplus fund and then may be used in thefuture, according to discretionary management rules, to increase policy holder profits. We also remark that the situation is very similar in Germany. The Solvency II requirements forrealistic modelling of future management actions are given in [6, Art. 23].Assumption 3.2 means that the profit sharing rate and the corporate tax rate are constant.In Austria this rate is 25%. If this rate is not constant, one would have to use a mean rate toderive the corresponding gph from nph .Assumption 3.3 can be seen as a consequence of Solvency II’s going concern hypothesis [13,Art. 101]. Indeed, if interest rates go up, one would assume that companies increase theirpolicy holder profit declarations in order to remain a competitive participant in the market.Assumption 3.4 is only used to derive the preliminary result stated in Proposition 3.3, andis then removed in Section 3.B.Assumption 3.5 remains unjustified as we do not know of any publicly available data tosupport it.Assumption 3.6 means that assets are shared equally among policy holders. For instance,this is the case for Germany ([18, § § x with maturity M x . This statement means that weassume the cross-financing between x and other contracts, which occurs before M x , can be ne-glected on average . Alternatively, one could separately consider the cross-financing before time M x and then assume that this cross-financing fully contributes to the F DB in Equation (3.14),whence the lower bound remains unchanged.Assumption 3.8: To calculate the weighted depreciation factor (3.13), one would need port-folio specific data. According to [14, Fig. A. II.1] the distribution of liability duration forGermany is between 14 and 22 years. Since formula (3.11) includes a discounting term P (0 , t ),we have chosen M = 15 to reflect the fact that contributions from later times give rise to alesser weight. The sensitivity with respect to this assumption is shown in Tables 4 and 5 for M = 10 and M = 20, respectively.Assumption 3.9: German legislature [18, § 4] distinguishes between two different values forthe net policy holder participation nph . For most sources of profit [18, § 4] dictates that nph = 90%. To arrive at the corresponding gph one has to account for tax payments suchthat “ nph times net profit” equals “ gph times gross profit”. This assumption thus amounts tousing gph = 80% as an approximating average factor. The sensitivity on this assumption isshown in Tables 3, 4 and 5.Assumption 3.10 means that gains as well as hidden reserves cannot be kept from the policyholder indefinitely. This assumption is necessary in order to apply Proposition (2.2) with E Q [ B − T M V T ] = 0. For modelling purposes, one could apply the following future managementaction: SF t ≤ θM V t ( T R ) where the concrete value of θ < SD [ P ph ∗ t ]would not have significant impact on the lower bound (3.21). Hence this assumption reflectsthe assumption that management would try to follow interest rate movements in profit sharingdeclaration but, at the same time, would try to avoid strong jumps in the declaration. The corresponding Austrian value for gph can be calculated to be 80 . NALYTICAL VALIDATION FORMULAS 15 Assumption 3.12 is a consequence of the historical data in Tables 7 and 8. Nevertheless, wehave listed this as an assumption since there are many different estimators and time series whichone could use to find the coefficient of variation SD [ B − t ] /P (0 , t ). As with Assumption 3.11,we remark that the sensitivity on this assumption is relatively low, since we are dealing witha product of two standard deviations.Assumption 3.13 is a formalization of the idea that the portfolio run-off is, roughly, homo-geneous in time: according to [2, p. 20] the number of policies in the with-profit business wentfrom approximately 10.5 million to approximately 10 million in the year 2017 (disregardingnew business). This is a reduction of about 5%. Now, time-homogeneity in this context shouldmean that the number of policies will be reduced by a factor of (1 − t after t years. Thusthe run-off is geometric and we take this as a justification for the assumed portfolio reduc-tion (3.18). However, in (3.18) we are concerned with the run-off of technical provisions insteadof policy numbers. Therefore, this argument is only a partial justification for (3.18). We stressthat companies would have the necessary information in this context, whence Assumption 3.13would not be needed in real applications (or could be tested against real data).Assumption 3.14 is related to Assumption 3.10: if we did not assume that the surplus fundwas bounded by the decreasing value of technical reserves, then the remainder term A x ( t ) t couldincrease the surplus fund at time t and never be converted to a cash-flow. On the other hand,if we allow for an upper bound rule such as Assumption 3.10, then A x ( t ) t has to decrease withtime. A good first approximation for C could be SF /A = 4 . C = 3%to indicate that not all of the cross-financing A x ( t ) t has to be kept within the surplus fund, buta part may also be directly declared to other policy holders (compare with Assumption 3.7)at times s ≤ t . Sensitivity on this assumption is shown in Tables 3, 4 and 5.4. Discussion of results The basic equation. Best estimate calculation for with-profit life insurance productsis a numerical task. This is due to the fact that there are no closed formula solutions to obtaina best estimate, whence Monte Carlo techniques are employed. This involves, in particular,setting up a full asset liability model such that all relevant cash-flows are generated. Thegeneral calculation process is discussed in [17]. Given the degree of sophistication of theseasset liability models, one should not expect a closed formula for best estimate calculation.The virtue of Equation (2.6) is thus that it is a closed formula and has to hold for all assetliability models (that are market consistent and satisfy Assumptions 2.1 and 2.2).Of course, this does not mean that we can compute the best estimate solely based onEquation (2.6). The terms V IF and T AX are just as hard to compute as BE . The practicalvalue of this equation rather is tied to the fact that it gives a very straightforward validationtest for any market consistent best estimate calculation model.This validation procedure can be viewed as a leakage test . It is a necessary condition fortwo model properties:All cash flows are accounted for – nothing is “lost” by the numerical model.The (numerically generated) economic scenarios are free of arbitrage. It seems to us that this test is well-known to at least parts of the applied insurance mathematicscommunity as a kind of “folklore wisdom”. We remark here that we have found this formulaindependently. More importantly, we do not know of any literature which explicitly states thisformula and, much less, of any references where the assumptions have been spelled out in arigorous manner.4.B. The lower bound formula. The best estimate can be expressed as a sum of guar-anteed cash flows GB and future discretionary benefits F DB . Moreover, it is a Solvency IIrequirement to report F DB separately as part of the best estimate ([7, Template S.12.01.01]).The guaranteed benefits depend on second order assumptions concerning, e.g., mortalityand surrender but are otherwise completely model-independent . In particular, they do notdepend on future management actions or economic scenarios. Indeed, these are fixed cashflows whence they can be valuated deterministically using only the initial risk-free (EIOPA-)interest rate curve. On the other hand, the calculation of future discretionary benefits F DB involves all theintricacies of best estimate valuation: asset liability model, management rules, economic sce-narios, etc. – see [15, 17, 22]. Hence one should not expect a closed forumla solution for the F DB . However, the next best thing to a closed formula is a lower bound, and this is whatProposition 3.4 achieves.Clearly this lower bound cannot be used for reporting purposes, since the true F DB couldbe considerably larger. We see three important applications for this estimate:It can be used by the company as an immediate test for their F DB calculated bymeans of a numerical asset liabilty model.It can be used by the company as a target towards which to optimize the F DB byan appropriate choice of admissible management rules (reinvestment strategy, profitsharing, surplus fund management, etc.)It can be used by the regulator as a simple on- or off-site plausibility check for reported F DB s.All of this works only if the assumptions and generic management rules discussed in Sec-tion 3.D are either directly met or suitably amended.To show how Proposition 3.4 could be used in practice we have, in Section 3.C, applied thelower bound formula to publicly available data of Allianz Lebensversicherung AG [1, 2]. InEquation (3.21) this yields a lower bound of A C48 . bn , while the reported number, included inTable 2, is F DB = A C48 . bn . The assumptions leading to the result (3.21) are discussed inSection 3.D. Finally, we have included results for the lower bound with respect to variationsof some of the key assumptions in Tables 3, 4 and 5.5. Appendix Table 7 is used to calculate the standard deviation SD [ B − t ] of the observed discount factor P (0 , t ) from Table 8. Compare with Remark 3.2. See Remark 3.1. NALYTICAL VALIDATION FORMULAS 17 M = 15 gph = 75% gph = 80% gph = 85% C = 1% 46.9 50.9 55.7 C = 3% 44.3 48.2 52.8 C = 5% 41.8 45.4 49.8 Table 3. Lower bound for the F DB . Values are in billion Euros. These resultsshould be compared to the reported value of F DB = 48 . M = 10 gph = 75% gph = 80% gph = 85% C = 1% 47.7 52.5 56.5 C = 3% 45.1 49.8 53.6 C = 5% 42.6 47.0 50.6 Table 4. Lower bound for the F DB . Values are in billion Euros. These resultsshould be compared to the reported value of F DB = 48 . M = 20 gph = 75% gph = 80% gph = 85% C = 1% 44.5 49.3 54.0 C = 3% 41.9 46.6 51.1 C = 5% 39.4 43.8 48.1 Table 5. Lower bound for the F DB . Values are in billion Euros. These resultsshould be compared to the reported value of F DB = 48 . C = 1% C = 3% C = 5% gph = 75% 1 . . . gph = 80% 1 . . . gph = 85% 1 . . . Table 6. Values for the cross-financing term F . Values are in billion Euros. Table 7. Euro base curve, 15-year spot rates. Source: [10]. t P ,t t P ,t t P ,t t P ,t t P ,t t P ,t Table 8. Euro discount rates at 31.12.2017. Source: [10]. References [1] Allianz Lebensversicherungs AG, Bericht ¨uber Solvabilit¨at und Finanzlage 2017: [2] Allianz Lebensversicherungs AG, Gesch¨aftsbericht 2017: [3] BaFin Auslegungsentscheidungen, ¨Uberschussfonds nach Art. 91 der Solvency II Richtlinie , [4] Bundesgesetz ¨uber den Betrieb und die Beaufsichtigung der Vertragsversicherung (Versicherungsaufsichts-gesetz 2016 – VAG 2016)[5] CFO Forum, Market Consistent Embedded Value Principles – April 2016. Downloaded from [6] Commission Delegated Regulation (EU) 2015/35 of 10 October 2014 supplementing Directive2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the businessof Insurance and Reinsurance (Solvency II).[7] Commission Implementing Regulation (EU) 2015/2450 of 2 December 2015 laying down implementingtechnical standards with regard to the templates for the submission of information to the supervisoryauthorities according to Directive 2009/138/EC of the European Parliament and of the Council[8] L. Delong, Practical and theoretical aspects of market-consistent valuation and hedging of insurance liabil-ities , Bank i Kredyt, Narodowy Bank Polski, vol. 42(1) (2011), pages 49-78.[9] EIOPA Statistics, https://eiopa.europa.eu/Pages/Financial-stability-and-crisis-prevention/Insurance-Statistics.aspx .[10] EIOPA, Risk-free interest rate term structure: https://eiopa.europa.eu/regulation-supervision/insurance/solvency-ii-technical-information/risk-free-interest-rate-term-structures .[11] D. Filipovic, Term-Structure Models , Springer Finance Textbooks, 2009.[12] H. Gerber, Life insurance mathematics , Springer 1997.[13] Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on thetaking-up and pursuit of the business of Insurance and Reinsurance (Solvency II).[14] EIOPA-BoS-16/302, 2016 EIOPA Insurance Stress Test Report.[15] Kemp, M. Market Consistency: Model Calibration in Imperfect Markets , Wiley 2009.[16] K. Laimer, Zinsszenarien und Best Estimate in der Lebensversicherung , Diploma Thesis, TU Vienna,2015.[17] Laurent, J.-P., Norberg, R., and Planchet, F., Modelling in Life Insurance A Management Perspective ,EAA Series (2016), Springer International Publishing.[18] Mindestzuf¨uhrungsverordnung (MindZV): [19] C. O’Brien, Valuation of Life Insurance Liabilities on a Market-Consistent Basis: Experience from theUnited Kingdom , Actuarial Practice Forum (2009).[20] T. Sheldon, A. Smith, Market Consistent Valuation of Life Assurance Business , British Actuarial Journal (3) (2004), 543-605. NALYTICAL VALIDATION FORMULAS 19 [21] J. Teichmann, M. W¨uthrich, Consistent long-term yield prediction , arXiv:1203.2017 .[22] J. Vedani, N. El Karoui, S. Loisel, J.-L. Prigent, Market inconsistencies of market-consistent Europeanlife insurance economic valuations: pitfalls and practical solutions , Eur. Actuar. J. (2017).[23] Verordnung der Finanzmarktaufsichtsbeh¨orde (FMA) ¨uber die Gewinnbeteiligung in der Lebensver-sicherung. Austrian Financial Market Authority (FMA), Otto-Wagner Platz 5, A-1090 Vienna E-mail address : [email protected] E-mail address ::