Estimation of future discretionary benefits in traditional life insurance
aa r X i v : . [ s t a t . M E ] J a n ESTIMATION OF FUTURE DISCRETIONARY BENEFITS IN TRADITIONAL LIFEINSURANCE
FLORIAN GACH & SIMON HOCHGERNER
Abstract.
In the context of traditional life insurance, the future discretionary benefits (
F DB ), which area central item for Solvency II reporting, are generally calculated by computationally expensive Monte Carloalgorithms. We derive analytic formulas for lower and upper bounds for the
F DB . This yields an estimationinterval for the
F DB , and the average of lower and upper bound is a simple estimator. These formulae aredesigned for real world applications, and we compare the results to publicly available reporting data. Introduction
Market consistent valuation.
Regulatory regimes increasingly require insurance companies to assignmarket consistent values to all items in their balance sheets. While market values for assets can often beobtained from data providers (“mark-to-market”) or inferred from well-established models (“mark-to-model”),no active (“deep and liquid”) market exists for trading insurance liabilities. Hence liabilities have to be valuatedby appropriately designed models.Specifically, when future cash flows depend on economic scenarios and discretionary management rules(concerning, e.g., reinvestment, realization of unrealized gains, level of profit declarations) are involved, and,on top of that, there is a minimum guarantee rate, developing such a model is not an easy task. All of theforegoing difficulties are characteristics of traditional life insurance, and the corresponding market consistentvaluation is a relatively novel problem in the realm of insurance mathematics. Existing literature concerningthe market consistent valuation of insurance technical provisions includes [1, 4, 6, 7, 8, 5], as-well as referencestherein. In particular, [4, 6] follow the general idea of disentangling insurance cash flows into a hedgeableand a non-hedgeable part. The value of the hedgeable, or financial, cash flow then follows from methodsof financial mathematics (“no-arbitrage pricing”), and the value of the non-hedgeable part is determined byactuarial techniques. We mention that [4, 6] study the problem of fair valuation from a general point of view,without regard to a specific regulatory regime.In this paper, we take this approach one step further. To do so, we define “market consistent valuation” tobe the calculation of the best estimate in the sense of Solvency II [12]. This is the regulatory framework thathas been implemented per 1. January 2016 and is relevant for companies operating in the European EconomicArea. In principle, the market consistent value corresponds to the sum of a best estimate and a risk margin.However, for life insurance liabilities, the risk margin is usually very small compared to the best estimate. Werefer to the introduction of [7] for concrete numbers showing that the risk margin generally does not amountto more than 2 % of the best estimate, whence we focus on the latter.1.B.
Description of results.
In determining the best estimate of a traditional life insurance liability portfoliowe can use two additional sources of information:(1) The best estimate, BE , is a sum of guaranteed benefits, GB , and future discretionary benefits, F DB .That is, BE = GB + F DB . The guaranteed benefits are certain at time of valuation, whence GB isa purely deterministic quantity that can be calculated by actuarial techniques. Moreover, Solvency II([14]) requires that both, BE and F DB , are reported.(2) Cash flows follow from local generally accepted accounting principles (local GAAP).
Austrian Financial Market Authority (FMA), Otto-Wagner Platz 5, A-1090 Vienna
E-mail address : [email protected], [email protected] . Key words and phrases.
Solvency II, Future Discretionary Benefits, Market Consistent Valuation.
In this context, certainty of guaranteed benefit cash flows is understood with respect to financial variableswhile actuarial assumptions (assumed to be independent of economic scenarios) are taken into account viatheir expected values. Thus there is an easy separation into guaranteed benefits, whose value follows fromactuarial methods, and future discretionary benefits. The
F DB cannot be calculated by a purely financialapproach. This follows from Item (2) above and is a consequence of the fact that the profit sharing mechanismis (for many European markets such as Germany, France or Austria) dictated by book values and not bymarket values. (See [7, 8, 5].) However, the crucial observation is now that the accounting principles can beused to further unravel the future discretionary benefit cash flows into items of cascading magnitude.Therefore, we rephrase the disentanglement approach as consisting of an actuarial part (for GB ) and afinancial part that depends on book values and accounting flows (for F DB ). The second part is in general nothedgeable either and we cannot give an exact solution for
F DB that involves anything less than a full MonteCarlo model of the balance sheet projection. However, by carefully analyzing the accounting flows that lead tothe future discretionary benefits, we can identify those constituents of
F DB which are of large, medium andsmall magnitude (compared to BE ), respectively. The smallest items are then neglected, the medium itemsare approximated, and it turns out that the largest items can be expressed in terms of known balance sheetquantities (which do not involve the F DB ).As a result, we obtain analytical formulas for lower and upper bounds, LB and U B , such that(Theorem 4.2) LB ≤ F DB ≤ U B.
The bounds are given by LB := SF + gph · ( LP − GB + U G ) − (1 − gph ) · SF · T X t =1 P (0 , t ) F t − (4.36) − (1 − gph ) gph (1 + θ ) · LP · T X t =2 F t − t − X s =0 P ( t − s − , t ) · O + t − s − · − ( t − s − /h U B := SF + gph · ( LP − GB + U G ) + gph · (1 + θ ) · LP · T X t =1 − ( t − /h O − t (4.37)where SF is the book value of the surplus fund, LP refers to the local GAAP life assurance provisions (excluding SF ), and U G are the unrealized gains, all at valuation time t = 0. Further, gph is the gross policyholder participation rate (Section 8.C), θ = 5 % is the fixed fraction in Assumption 1.2, F t is the prevailingforward rate curve and O ± t are the values (4.35) of certain caplets/floorlets. Finally, T is the projection horizonand h is the liability half life which is used as a proxy for the average time to maturity of contracts.The formulas for LB and U B arise as a combination of estimating accounting flows as a function of finan-cial variables, company specific information and generic management rules. The financial variables are theprevailing interest rate curve and implied volatility data. This caters to the intuition that, at the end of theday, accounting flows do depend on market movements to a certain extent . Secondly, the company specificinformation includes, for example, the average guaranteed interest rate, the average time to maturity of theasset and liability portfolios, and the value of unrealized gains associated to the relevant portfolio of assets.Thirdly, the management rules are chosen so as to reflect generic management decisions such as minimizingshareholder capital injections or providing a “reasonable” level of profit participation (in order to remain acompetitive vendor in the life insurance market).Theorem 4.2 is a statement about the value of the full liability portfolio. In particular, we do not assign fairvalues to individual contracts. In this context we note that it is not clear how to assign fair values to individualcontracts in a market consistent full balance sheet approach. Indeed, the future discretionary benefits dependnot only on individual contract features but also on properties of both the asset and liability portfolios: the
F DB depends on the amount of available unrealized gains,
U G , and on the (book value of the) surplus fund, SF , both viewed at valuation time, t = 0. The unrealized gains are the difference of market and book valuesof the full asset portfolio that covers local GAAP technical provisions, i.e. U G = M V − BV . Neither U G nor SF are assigned to individual contracts, and using any kind of theoretical proportional assignment willgenerally lead to a completely unrealistic duration gap between assets and liabilities. Consequently Solvency II STIMATION OF FUTURE DISCRETIONARY BENEFITS 3 reporting also does not ask for best estimates of individual contracts (although for some lines of business –other than traditional life insurance – these may be provided).The terminology “benefits” is slightly misleading in GB and in F DB . Both quantities contain also premiaand costs. Nevertheless, we stick to the established Solvency II nomenclature.For the purpose of this paper, the use of the guaranteed benefits has the added advantage that we do notneed to set up an actuarial model and specify contract details. All these details are already contained in GB , and for our concrete applications in Section 5 to publicly available data of the German insurer AllianzLebensversicherungs-AG the guaranteed benefits are a given input.We apply Theorem 4.2 with gph = 75 . T = 60 and h = 12 to publicly available data of the Germancompany Allianz Lebensversicherungs-AG for the years 2017, 2018 and 2019. Table 1 is a snapshot of theresults in Section 5. YE GB F DB \ F DB LB U B ∆2017 154.1 48.6 47.99 43.45 52.53 -0.30 %2018 158.8 46.2 46.71 41.74 51.67 0.25 %2019 195.2 47.4 49.93 43.95 55.90 1.04 %
Table 1.
Values are in billion euros. The items GB and F DB are the reported values. Theestimated values are LB , U B and \ F DB := ( LB + U B ) /
2. The difference ∆ = \ F DB − F DB is in percent of BE = GB + F DB .Once an actuarial model has been established to calculate GB , the bounds (4.36) and (4.37) yield anestimator, d BE , and an estimation interval for the best estimate:(1.1) d BE = ( GB + \ F DB ) ± δ, \ F DB = (
U B + LB ) / , δ = ( U B − LB ) / GB is deterministic since guaranteed cash flows do not dependon financial variables. In Table 1 we have δ < . · BE . The immateriality threshold for model uncertainties(such as leakage) is in practice often taken to be ± · BE . Thus a deterministic routine which yields amaximal error of ± . F DB intocontributions of varying magnitude. This process is repeated until the contributions which cannot be easilycomputed are of sufficiently small magnitude (compared to BE ) such that crude approximations are admissiblewithout completely distorting the final result.These approximations (and the determination of what is negligible) cannot be carried out in an abstractmathematical manner but have to be seen within the context of traditional life insurance business. Indeed, it isvery easy to generate pathological examples where F DB = 0 or
F DB is arbitrarily large; for the first considerthe case where gph = 0 and SF = 0, and for the second a shareholder that is willing to inject arbitrarily largeamounts of capital to increase policy holder participation. Clearly, both these scenarios are unrealistic, and inorder to derive meaningful bounds on the F DB one has to exclude such possibilities.One of the main observations of this paper is that a set of generic management rules can be found such that,firstly, meaningful bounds follow and, secondly, these bounds are extreme within the set of meaningful bounds.By the second point we mean that the upper bound can be reduced by additional management actions, butnot increased; likewise the lower bound can be increased, but not reduced.Thus we have chosen the assumptions so that the bounds continue to hold for other (realistic) managementrules. The main assumptions are as follows:
Assumption 1.1.
Management rules are designed to prevent excessive shareholder injections.
FLORIAN GACH & SIMON HOCHGERNER
This precludes the above mentioned possibility of the shareholder generating policyholder profits by capitalinjections. But we also interpret it to mean that there is no profit declaration target that is attained byrealizing unrealized gains. In this sense our lower bound, LB , is minimal. If there were such a target, LB would have to increase. Conversely, we have not used this non-realization rule in the derivation of the upperbound, U B . Hence
U B may be further decreased, but not increased, by assuming that negative gross surpluscan be covered (to a certain extent) by releasing unrealized gains.
Assumption 1.2. (Cf. [7, A. 3.10]) Insurance companies are subject to market competition and thus inclined todeclare profits to policy holder accounts. In technical terms: The surplus fund is bounded, due to managementrules which hold for all scenarios, by a fraction of the technical reserves. That is, SF t ≤ θ · LP t . In Section 5we will assume θ = 5 %.In this assumption we distinguish between profit sharing and profit declaration. This distinction is importantbecause profit sharing is, for traditional life insurance, mandatory. This is the process where a company sharesa legally specified minimum of its net profits with the collective of policy holders. However, in the legallyrelevant meaning of the term, sharing does not necessarily imply that the policy holder will receive moneyfrom the company’s surplus. Indeed, the company may decide at its own discretion how much of the shared profit is declared (or credited) to the policy holders’ accounts and how much is not declared , i.e. “parked”in the surplus fund. Hypothetically, it would be possible for a company to park all of the shared profits inthe surplus fund, and hence never pay any profit sharing benefits. Practically, this is not the case because ofmarket competition (and, at least in some jurisdictions, tax incentives). A low fraction of declared profits isnot good for the reputation and subsequently for the new business margin. Note that, tacitly, this argumentrelies on Solvency II’s going concern hypothesis.The purpose of the surplus fund, SF , is to obtain a smooth and steady stream of declarations. Hence, if thecompany’s profit has increased, the declaration factor is likely to decrease while, if the profit has decreased,the factor may increase. See equation (2.11) for details. The result of this mechanism is the following: Assumption 1.3.
Management rules are designed to reduce variation of policy holder profit declarations.Technically, we use this assumption to justify some of the approximations in Section 4 where the variation ofthe life assurance provisions, LP , is viewed as a secondary effect compared to the variation in the forward rate.These approximations are crude but reasonable at this point since the approximated quantities are already (atleast) one order of magnitude smaller than BE .1.C. Structure of the paper.
Section 2 introduces the basic notions and terminology regarding the account-ing model and the corresponding accounting flows (which are not cash flows).The main statement of Section 3 is Theorem 3.15 which is an exact decomposition of the
F DB in termsof quantities which are known (and large), quantities which are unknown and need to be approximated,and quantities which are sufficiently small to be neglected. To derive Theorem 3.15, we use the no-leakageprinciple (3.16) formulated systematically in [7, Prop. 2.2].Section 4 contains the main result, i.e. Theorem 4.2 with the formulae for LB and U B .In Section 5 we apply Theorem 4.2 to publicly available data of Allianz Lebensversicherungs-AG for theyears 2017, 2018 and 2019.Section 6 draws conclusions.Appendix 7 is a presentation of the gross surplus from the actuarial point of view. Finally, Appendix 8 is acollection of all the publicly available data, together with their sources, that are used in Section 5.
Acknowledgements.
We thank our colleagues Wolfgang Herold and Sanela Omerovic for valuable commentsand discussions.
Disclaimer.
The opinions expressed in this article are those of the authors and do not necessarily reflect theofficial position of the Austrian Financial Market Authority.
STIMATION OF FUTURE DISCRETIONARY BENEFITS 5 The accounting model
Consider a European insurance company which sells traditional life insurance. The company is subject tothe Solvency II regulatory regime and therefore has to report, on a quarterly basis, the best estimate, BE ,associated to its liabilities. In the reporting, the best estimate is split into two components, the guaranteedbenefits GB and the future discretionary benefits F DB , i.e.(2.2) BE = GB + F DB.
This splitting is required by Solvency II ([13, Art. 25] and [14]).2.A.
Assumptions.
Let us fix a yearly time grid 0 , , . . . , T and remark that t = 0 is the valuation time while T corresponds to the run-off time of the liability portfolio. In practice T may be as large as 100 years. Assumption 2.1.
The end of projection, T , coincides with the time to run off of the liability portfolio.Let F mt denote the simple forward rate at time t valid on the accrual period [ t + m, t + m + 1]. If m = 0 we write F t = F t . The initial forward curve is the set ( F m ) T − m =0 . We assume further that we have chosen an arbitragefree interest rate model with respect to a risk neutral measure Q (and a filtered probability space satisfyingall the usual assumptions), and consequently denote the corresponding (stochastic) one-year forward rate at t by F t . The associated discount factor, for t ≥
1, is B t = Q t − s =0 (1 + F s ) − , and we set B = 1. Expected valuesare throughout understood to be with respect to Q , E [ · ] = E Q [ · ]. The deterministic discount factor is denotedby P (0 , t ) = E [ B − t ].According to Solvency II [12, 13], the best estimate is the expectation of all future cash flows which arerelated to existing business. These cash flows are benefits, bf t , premium income, pr t , and costs, co t . The bestestimate is thus(2.3) BE = E " T X t =1 B − t ( bf t + co t − pr t ) The benefits can be further decomposed into guaranteed benefits, gb t , and discretionary benefits, ph t . Thatis,(2.4) bf t = gb t + ph t , and gb t is independent of asset movements. Thus gb t is a deterministic quantity and can be calculated byactuarial methods. It follows that(2.5) F DB = E " T X t =1 B − t ph t . Assumption 2.2. (Cf. [7, A. 2.1]) The liabilities (technical provisions under local GAAP) at time t arecomposed of two items:(1) life assurance provision, LP t ;(2) surplus fund, SF t .These are book value items, and if L t is the book value of the liabilities at t , then L t = LP t + SF t . The lifeassurance provision is the sum of the individual life assurance provisions, that is,(2.6) LP t = X x ∈X LP xt where X is the set of policies at valuation date 0. Hence LP xt = 0 if t is bigger than the maturity of x . FLORIAN GACH & SIMON HOCHGERNER
The surplus fund at time t , SF t , comprises those profits that have not (yet) been declared to policyholdersat time t . As opposed to LP t , SF t belongs to the collective of policyholders and cannot be attributed toindividual contracts.The book value of assets at time t is denoted by BV t and their market value by M V t . Unrealised gains orlosses are defined as residual U G t = M V t − BV t . Assumption 2.3. (Cf. [7, A. 2.2]) The book value of assets is equal to the book value of liabilities,(2.7) BV t = L t = LP t + SF t . For a discussion of why this assumption is, in fact, not restrictive see [7, Section 2.3].Let gs ∗ t denote the company’s gross surplus at time t . Assumption 2.4. (Cf. [7, A. 3.9]) The gross of tax policyholder profit participation rate gph is constant. Thisis the rate with which the policyholder participates in the company’s gross surplus gs ∗ , if the latter is positive.The profit participation in this assumption refers to sharing but does not say anything about profit dec-larations (see also the discussion following Assumption 1.2). The profit sharing mechanism in traditional lifeinsurance means that a part of gs ∗ t is declared to each one of the policy holders’ accounts. This declaration,denoted by db ∗ t only takes place when the surplus is positive and is therefore of the form(2.8) db ∗ t = θ t · gph · ( gs ∗ t ) + where θ t is the fraction that is to be declared to the collective at t and ( gs ∗ t ) + = max( gs ∗ t , θ t depends on complex model rules such as future management actions.Quantities with a star ∗ denote accounting flows, as opposed to cash flows. Let(2.9) ph ∗ t := gph · ( gs ∗ t ) + and note that this is an accounting flow (not a cash flow), whence it is not (immediately) paid out to policyholders but rather increases the book value of liabilities. For x ∈ R let x = x + − x − be the decomposition intopositive and negative parts. For further reference we observe that P H ∗ := E h X B − t ph ∗ t i = gph · E h X B − t ( gs ∗ t ) + i (2.10) = gph · E h X B − t gs ∗ t i + gph · E h X B − t ( gs ∗ t ) − i = gph · ( V IF + P H ∗ + T AX ) + gph · COG.
Here we have used the splitting gs ∗ t = sh t + ph ∗ t + tax t , where sh t and tax t are the shareholder and tax cashflows, together with the definitions V IF = E " T X t =1 B − t sh t , COG = E " T X t =1 B − t ( sh t ) − , T AX = E " T X t =1 B − t tax t . Note that sh t can be negative, which corresponds to the case of shareholder capital injections.2.B. Projection of the surplus fund SF . Consider the surplus fund SF t − at t −
1. Going one time stepfurther, it is increased by allocating (1 − θ t ) · gph · ( gs ∗ t ) + to the fund and decreased by declaring η t · SF t − to the policy holders’ accounts. The fraction η t ∈ [0 ,
1] depends, just like θ t , on modeling assumptions. Wethus obtain an iterative evolution of the form(2.11) SF t = SF t − + (1 − θ t ) · gph · ( gs ∗ t ) + − η t · SF t − where SF is known. STIMATION OF FUTURE DISCRETIONARY BENEFITS 7
Projection of declared bonuses DB . Let DB t be the sum of all profit declarations that have occurredat times 0 < s < t and belong to contracts x ∈ X which are active at t . We set DB = 0 . Passing from t − t , DB t − isincreased by declarations η t · SF t − from the surplus fund,increased by direct policy holder declarations θ t · gph · ( gs ∗ t ) + ,decreased by cash flows, ph t , to policy holders whose contracts terminate at t , anddecreased by accounting flows, χ t − DB t − , with 0 ≤ χ t − ≤ χ t − DB t − is freed up, in the sense that it is not attributed to specific contracts anymore andcontributes to the annual gross surplus.Therefore, we have the iterative relation(2.12) DB t = DB t − + η t · SF t − + θ t · gph · ( gs ∗ t ) + − ph t − χ t − DB t − . Together with (2.11) this yields(2.13) DB t = DB t − + SF t − − SF t + ph ∗ t − ph t − χ t − DB t − where we use definition (2.9). Notice that the model dependent fractions θ t and η t disappear.3. A general formula for future discretionary benefits
The run off assumption 2.1 implies that all declarations are paid out at T , or earlier, i.e. DB T = 0 = DB .Using (2.13), it follows that B − T SF T = SF + T X t =1 (cid:16) B − t ( DB t + SF t ) − B − t − ( DB t − + SF t − ) (cid:17) = SF + T X t =1 (cid:16) ( B − t − B − t − )( DB t − + SF t − ) + B − t ph ∗ t − B − t ph t − B − t χ t − DB t − (cid:17) = SF + T X t =1 B − t ph ∗ t − T X t =1 B − t ph t − T X t =1 B − t F t − ( DB t − + SF t − ) − T X t =1 B − t χ t − DB t − . Definitions (2.5) and (2.10) imply that the expected values satisfy
F DB = SF + P H ∗ − E h T X t =1 B − t F t − ( DB t − + SF t − ) i (3.14) − E h B − T SF T i − E h T X t =1 B − t χ t − DB t − i . Theorem 3.1.
The value of future discretionary benefits,
F DB , satisfies (3.15)
F DB = SF + gph · ( LP − GB + U G ) + gph · COG − I − II − III where I := E (cid:2) B − T SF T (cid:3) + gph · E (cid:2) B − T U G T (cid:3) II := (1 − gph ) E " T X t =2 B − t χ t − DB t − III := (1 − gph ) E " T X t =1 B − t F t − ( DB t − + SF t − ) . FLORIAN GACH & SIMON HOCHGERNER
Proof.
Equality (2.10) implies
P H ∗ = gph − gph ( V IF + T AX + COG ) . Hence we can use the no-leakage principle(3.16) BE + V IF + T AX = BV + U G − E [ B − T M V T ]in [7, Prop. 2.2] to conclude(3.17) P H ∗ = gph − gph (cid:16) BV + U G − GB − F DB − E [ B − T M V T ] + COG (cid:17) which, together with (3.14), BV − SF = LP and using that BV T = SF T at run-off time T , yields (3.15). (cid:3) Remark 3.2.
Terms I , II and III in the theorem turn out to be quantities that have to be subtracted from SF + gph · ( LP + U G − GB ) because they are not assigned to policyholders. Term III results from thedelay between the point in time when profits to policyholders originate as accounting flows and the point whenthese profits are actually paid out as cash flows. In fact:Term I is related to the policyholder share of assets that remain in the company after run-off of theliability portfolio;Term II is the tax and shareholder share in the gross surplus due to the fraction of declared futureprofits, DB t , that is freed up because of mortality or policy holder behaviour.Term III captures the tax and shareholder shares in interests on allocated profits as well as on thesurplus fund.
Remark 3.3.
The estimation in this theorem regards the
F DB as calculated with the stochastic cash flowmodel. In those EEA member states that have authorised Article 91(2) of Directive 2009/138/EC [12] thesurplus fund (more exactly, the part that is not used to compensate losses) is not considered as a liability, andtherefore not part of the technical reserves. Let SF SII0 denote the part of the surplus fund that is not usedused to absorb losses (in the risk neutral average over all scenarios). This is deducted from the
F DB to yieldthe reported future discretionary benefits,
F DB rep = F DB − SF SII0 .4.
Analytical lower and upper bounds for future discretionary benefits
The model dependent quantities in Theorem 3.1 are I , II , III and
COG . Calculating these explicitly isjust as difficult as calculating the
F DB . The purpose of this section is therefore to derive model independentand analytical bounds for these quantities. In doing so some quantities will be neglected and others will beapproximated. The overall guiding principle in this approach is that errors below 1 % · BE are immaterial.4.A. Estimating I . Assumption 1.2 prevents the surplus fund from becoming arbitrarily large. In theorythis would be possible, as there is no legal obligation to share the profits accumulated in the surplus fund.In reality the insurance market forces the companies to actually credit these profits to policyholders. ThusAssumption 1.2 is a realistic management rule.The run-off assumption 2.1 now implies that Term I is negligible compared to the F DB . Thus we obtainthe very simple estimate(4.18) I ∼ = 0 . Estimating II . The expression χ t DB t in Term II corresponds to the fraction of DB t that is freed upeach year due to policy holder mortality or behaviour (surrender or conversion to paid-up policy).This fraction is converted into the gross surplus and makes up a part of the company’s technical margin. SeeAppendix 7 for details. In Section 8.A the technical gains of the Allianz Lebensversicherungs-AG are estimatedto correspond to 0 .
77 % of the technical reserves. Since the inherited reserves are a part of this, a reasonableestimate may be(4.19) χ t DB t . .
25 % · LP t . STIMATION OF FUTURE DISCRETIONARY BENEFITS 9
In fact, this estimate seems rather generous since by definition DB t starts at DB = 0, whence the inheritedfraction from DB t will initially only make a very small contribution to the technical gains. Assumption 4.1. (Cf. [7, A. 3.13]) The expected technical reserves, E [ LP t ], decrease geometrically as E [ LP t ] = LP · − t/h with fixed h >
0. (See also [7, Assumption 3.13])With h = 12, T = 60, gph = 75 . II ≤ .
25 % · .
245 % · T X t =2 − t/h P (0 , t ) · LP = 0 .
72 % · BE.
Neglecting contributions smaller than 1 % · BE , this leads to the estimate(4.21) II ∼ = 0 . Since there is also a dependence on the interest rate curve we will show the result of estimate (4.20) for eachof the years under consideration in Section 5.A.4.C.
Estimating
III . The essential idea is to use equation (2.13) to obtain a recursive inequality relationwhich allows to estimate term
III .We first analyze the contributions to the gross surplus gs ∗ t . Let K t − denote a deterministic spread whichreflects technical gains and fixed coupon payments at time t and depends on unrealized gains of the assetportfolio at time t − κ t to be a decreasing function which reflects the reduction of initial unrealized gains due toassets reaching their maturities. Thus κ t depends on the duration of the portfolio. We set(4.22) κ t − = 2 − ( t − /d − − t/d such that U G t ∼ κ t U G . Here d is assumed to be smaller than the half-life h of the technical reserves to reflectthe duration gap between assets and liabilities.Notice that Assumption 2.3 implies that U G consists of unrealized gains corresponding to the full bookvalue BV = LP + SF .Legislative requirements for traditional life insurance ([9] for Germany and [11] for Austria) imply thatthe policy holder collective participates, at each time step t , in the return on assets which cover BV t − = LP t − + SF t − (Assumption 2.3). At the same time the company has to guarantee that gains are not below ρ t · LP t − , where ρ t is the average technical interest rate at time t . Using equation (7.49) we therefore estimate ph ∗ t ≤ gph · (cid:16) F t − + K t − − ρ t θ (cid:17) + · (1 + θ ) · LP t − + gphρ t ( DB t − + DB ≤ t − ) , (4.23)where K t − = 2 ( t − /h κ t − U G LP + γ and where γ and DB ≤ t − are defined in Appendix 7.Inequality (4.23) and equation (2.13) yield DB t − + SF t − = DB t − + SF t − + ph ∗ t − − ph t − − χ t − DB t − ≤ DB t − + SF t − + gph · (cid:16) F t − + K t − − ρ t − θ (cid:17) + · (1 + θ ) · LP t − + gphρ t − ( DB t − + DB ≤ t − ) − ph t − − χ t − DB t − . Because DB = 0 and SF is known, we thus obtain a recursive inequality relation for DB t + SF t . Note thatwe do not need separate estimates for DB t or SF t . This is a very convenient feature since any splitting of DB t + SF t into DB t and SF t would depend on management rules, compare with equation (2.11) for SF t . Onthe other hand, as noted just below equation (2.13), the sum is independent of such choices. Indeed, DB t − + SF t − ≤ DB t − + SF t − + gph · (cid:16) F t − + K t − − ρ t − θ (cid:17) + · (1 + θ ) · LP t − + gphρ t − ( DB t − + DB ≤ t − ) − ph t − − χ t − DB t − ≤ DB t − + SF t − + gph · (cid:16) F t − + K t − − ρ t − θ (cid:17) + · (1 + θ ) · LP t − + gph · (cid:16) F t − + K t − − ρ t − θ (cid:17) + · (1 + θ ) · LP t − + X s =0 (cid:16) gph · ρ t − s − ( DB t − s − + DB ≤ t − s − ) − ph t − s − − χ t − s − DB t − s − (cid:17) ... ≤ SF + gph · (1 + θ ) · t − X s =0 (cid:16) F t − − s + K t − − s − ρ t − − s θ (cid:17) + · LP t − − s + t − X s =0 (cid:16) gphρ t − s − ( DB t − s − + DB ≤ t − s − ) − ph t − s − − χ t − s − DB t − s − (cid:17) whence F t − B − t ( DB t − + SF t − ) ≤ (cid:16) B − t − − B − t (cid:17) · SF + gph (1 + θ ) t − X s =0 B t − s − (cid:16) B − t − − B − t (cid:17) B − t − − s (cid:16) F t − − s + K t − − s − ρ t − − s θ (cid:17) + · LP t − − s + (cid:16) B − t − − B − t (cid:17) t − X s =0 (cid:16) gph · ρ t − s − ( DB t − s − + DB ≤ t − s − ) − ph t − s − − χ t − s − DB t − s − (cid:17) (4.24)The quantity (cid:16) F t − − ρ t − K t − θ (cid:17) + is the payoff of a caplet option with strike ( ρ t − K t − ) / (1 + θ ), maturity t − t . Its value(price) at valuation time t = 0 is(4.25) O + t := E h B − t (cid:16) F t − − ρ t − K t − θ (cid:17) + i . Let P ( s, t ) = E [ B s B − t ] for s ≤ t . Assumption 1.3 says that the coefficient of variation of LP t can beneglected in comparison to that of F t − . We therefore approximate LP . in (4.24) by its expectation. Thisyields E h B t − s − ( B − t − − B − t ) · B − t − − s (cid:16) F t − − s + K t − − s − ρ t − − s θ (cid:17) + · LP t − − s i (4.26) ∼ = (cid:16) P ( t − s − , t − − P ( t − s − , t ) (cid:17) · O + t − s − · E h LP t − − s i where E [ LP . ] is calculated according to Assumption 4.1. STIMATION OF FUTURE DISCRETIONARY BENEFITS 11
Remark 4.1.
In (4.26) we have also simplified E h B t − s − ( B − t − − B − t ) · B − t − − s (cid:16) F t − − s + K t − − s − ρ t − − s θ (cid:17) + i ∼ = (cid:16) P ( t − s − , t − − P ( t − s − , t ) (cid:17) · O + t − s − Alternatively one could also evaluate the expectation of the left hand side of (4.26) as E h(cid:16) B − t − − B − t (cid:17) · (cid:16) F t − − s + K t − − s − ρ t − − s θ (cid:17) + i · E h LP t − − s i , and use a Monte Carlo routine for the first factor and Assumption 4.1 for the second. The path dependentpayoff ( F t − − s + K t − − s − ρ t − − s θ ) + viewed at settlement times t − t is that of a deferred caplet. Ananalytic, but approximate, pricing formula for E [( B − t − − B − t ) · ( F t − − s + K t − − s − ρ t − − s θ ) + ] is derived in [3,Sec. 13.4.1] by using a drift freezing technique. From an actuarial perspective the interpretation as a deferredcaplet is more accurate. However, for our purposes the simplicity of evaluating O + via the Black formulaoutweighs the disadvantage of the slightly crude approximation (4.26). In any case, the approximation errorthat arises at this point is not that of a discounted caplet versus a deferred caplet, but that of a difference ofa differently discounted caplet versus a difference of deferred caplets.The parameter h is the number of years for the portfolio to reduce to half of its size. The difference h − d > K t , can now be explicitlyspecified as(4.27) K t − = γ + (cid:16) − ( t − /d − − t/d (cid:17) ( t − /h · U G LP . Owing to Assumption 4.1 and estimate (4.24) we therefore have
III ≤ (1 − gph ) · SF · T X t =1 P (0 , t ) F t − (4.28) + (1 − gph ) gph (1 + θ ) · LP · T X t =2 t − X s =0 (cid:16) P ( t − s − , t − − P ( t − s − , t ) (cid:17) · O + t − s − · − ( t − s − /h − ε where ε is defined as ε := ε + − ε − (4.29) ε + := (1 − gph ) E h T X t =2 (cid:16) B − t − − B − t (cid:17) t − X s =0 (cid:16) ph t − s − + χ t − s − DB t − s − (cid:17)i ε − := gph (1 − gph ) E h T X t =2 (cid:16) B − t − − B − t (cid:17) t − X s =0 (cid:16) ρ t − s − ( DB t − s − + DB ≤ t − s − ) (cid:17)i Estimating ε . The negative part of ε , i.e. ε − depends on the initial declared profits, DB ≤ , and thetechnical interest rate, ρ , while the positive part, ε + , does not. Hence, if DB ≤ or ρ were to become arbitrarilylarge, ε would become arbitrarily negative, and from a mathematical point of view there is no argument topreclude this possibility. However, a realistic model of an insurance company does not need to allow forpathologically high technical rates or initial profits. In this section we use this observation to argue that 0 . ε . Assumption 4.2.
On average, the total declared profits are a constant fraction of the technical reserves. Thatis, E [ DB t + DB ≤ t ] = σE [ LP t ] for a constant 0 ≤ σ ≤ Typical values for T , gph , σ , ρ and P (0 , t − − P (0 , t ) are 60, 75 % (Table 17), 20 %, 2 . ε − should be of theorder ε − /LP ∼
25 % ·
75 % · . ·
20 % · T X t =2 ( P (0 , t − − P (0 , t )) t − X s =0 − ( t − s − / ∼ .
20 %(4.30)This may be slightly larger than the immateriality threshold of 1 % · BE . Further, since the estimate (4.30)uses many assumptions which are difficult to justify rigorously, we should check that ε is at least positive fora range of reasonably conservative values.To this end, let ph t = ψ t − DB t − and assume that DB t = c ( t ) · σ · LP t where c ( t ) = (cid:26) t/ h t < h t ≥ h (cid:27) In conjunction with Assumption 4.2 this captures the fact that DB = 0 and DB ≤ t is expected to convergeto something negligible as t increases.We obtain the following estimator for a lower bound of ε :(4.31) b ε = (1 − gph ) T X t =2 (cid:16) P (0 , t − − P (0 , t ) (cid:17) t − X s =0 (cid:16) ( ψσ + χ ) c ( t − s − − gph · ρσ (cid:17) · − ( t − s − /h · LP where the time dependence in ψ t , χ t and ρ t has been omitted. The assumption ψ = 3 % is very conservativesince the pay-out fraction is expected to increase over time for a run-off portfolio, and larger ψ contributespositively to b ε . Therefore, b ε need not be a sharp lower bound for ε . But this is not necessary since positivevalues are allowed.In the concrete examples in Section 5 we check that (4.31) is indeed small or positive ( b ε/BE ≥ − σ = 20 %, ψ = 3 %, χ = 0 .
25 % (as in estimate (4.19)) and the given value for ρ .This gives a posteriori justification, both in the mathematical and economic sense, to omitting ε in (4.28).The resulting estimate for III is III ≤ (1 − gph ) · SF · T X t =1 P (0 , t ) F t − (4.32) + (1 − gph ) gph (1 + θ ) · LP · T X t =2 t − X s =0 (cid:16) P ( t − s − , t − − P ( t − s − , t ) (cid:17) · O + t − s − · − ( t − s − /h This estimate should always be used in conjunction with the verification that (4.31) is positive (or sufficientlysmall). In concrete applications, insurance companies could use their known (range of) values for h , χ , ψ t and σ .4.E. Estimating
COG . To estimate
COG , notice that
COG = T X t =1 E h B − t ( gs ∗ t ) − i ∼ = T X t =1 E h B − t (cid:16) F t − + S t + K t − − ρ t θ (cid:17) − · LP t − i where S t ≥ t , that can be generated at the company’s discretion by realizing unrealizedgains. Indeed, here we use Assumption 1.1.Therefore, invoking Assumptions 1.3 and 4.1 again, we find COG = T X t =1 E h B − t ( gs ∗ t ) − i ≤ T X t =1 E h B − t (cid:16) F t − + K t − − ρ t θ (cid:17) − i · − ( t − /h · (1 + θ ) · LP . (4.33) STIMATION OF FUTURE DISCRETIONARY BENEFITS 13
The expression(4.34) O − t := E h B − t (cid:16) F t − − ρ t − K t − θ (cid:17) − i is the value of a floorlet with strike ( ρ t − K t − ) / (1 + θ ), maturity t − t .4.F. Estimating
F DB . To evaluate O + t and O − t , we use the Black formulas associated to the normal model.If interest rates are positive one may also use the log-normal model ([2, 3]). In Section 5 we are interested inthe years 2017, 2018 and 2019 where rates are negative at the short end. The normal Black formulas are(4.35) O ± t = P (0 , t ) · (cid:16) ± ( F t − − ρ t − K t − θ )Φ( ± d ) + IV t √ tφ ( ± d ) (cid:17) where Φ and φ are the normal cumulative distribution and density functions, respectively. Further, d = (1 + θ ) F t − − ρ t + K t − (1 + θ )IV t √ t and IV t is the caplet implied volatility.Let LB := SF + gph · ( LP − GB + U G ) − (1 − gph ) · SF · T X t =1 P (0 , t ) F t − (4.36) − (1 − gph ) gph (1 + θ ) · LP · T X t =2 F t − t − X s =0 P ( t − s − , t ) · O + t − s − · − ( t − s − /h and(4.37) U B := SF + gph · ( LP − GB + U G ) + gph · (1 + θ ) · LP · T X t =1 − ( t − /h O − t . Assume it has been verified that (4.31) is positive or negligible. Together with Theorem 3.1, the respectiveestimates (4.18), (4.21), (4.32) and (4.33) for I , II , III and
COG yield:
Theorem 4.2.
The value of future discretionary benefits,
F DB , is bounded from below and above as LB ≤ F DB ≤ U B.
Remark 4.3.
In line with Remark 3.3, if Article 91 of [12] has been ratified by the local jurisdiction, then(the part not used for loss absorption of) SF has to be subtracted from the future discretionary benefits toobtain the reported value, F DB rep = F DB − SF SII0 ; this part is denoted by SF SII0 and satisfies . Thus werestate Theorem 4.2 as(4.38) LB − SF ≤ F DB rep ≤ U B − SF . In the restatement we use SF instead of SF SII0 since the only the former is a model independent quantity.For the lower bound this is clearly admissible since SF SII0 ≤ SF . For the upper bound one could also retain U B instead of
U B − SF . However, our assumptions leading to U B are already quite generous so that thededuction seems reasonable. In Table 12 the values of the surplus fund, SF , and the Solvency II values of thesurplus fund, SF SII) , as used in the example of Section 5 are given. The differences are quite small.
Remark 4.4.
When gph = 0 then
U B = SF . This makes sense since SF already belongs to the collective ofpolicy holders. Thus no additional profits are shared and, in particular, the policy holders also do not receivethe unrealized gains proportional to SF , whence only the book value remains to be paid out.Likewise, when gph = 0 we have LB = (1 − P Tt =1 P (0 , t ) F t − ) SF , that is, the lower bound for F DB isgiven by the amount of depreciation that arises because SF is not paid out at t = 0 but at later times, at therate of decrease, determined by h , of the liability portfolio, and the corresponding loss of order 1 − gph = 1 ininterest rate margin incurred by the policy holders. Conversely, when gph = 1 we have LB = SF + LP − GB + U G . In this case all future profits go to thepolicy holders with absolute certainty, and the value of these profits must be equal to the difference of the fullinitial market value, LP + SF + U G , and the already guaranteed benefits, GB . That is, LB = F DB ≤ U B = LB + (1 + θ ) · LP · P Tt =1 − ( t − /h O − t .5. Application to reported values
We apply the results of the previous section to publicly available data from the Allianz Lebensversicherungs-AG and publicly accessible market data (average technical gains, discount rates, etc.). The relevant data iscollected in Appendix 8. Since Germany has ratified Article 91 of [12], we apply Theorem 4.2 in the form ofestimates (4.38). Unfortunately interest rate implied volatilities are not publicly available. Therefore, we fix an artificialvolatility structure that approximately reflects the correct behavior (Section 5.A) and provide sensitivities withrespect to this choice (Section 5.B). The results below are generated with respect to the following volatilitystructure: IV t = (cid:26)
10 + 2( t −
1) 1 ≤ t ≤ t > (cid:27) Base case.
The results for the base case are shown in Tables 2 and 3, in absolute numbers and normalizedas a percentage of BE = GB + F DB , respectively. The chosen parameters are as follows: T = 60 , θ = 5 % , d = 8 , h = 12 , gph = 75 . ρ γ LP SF U G GB F DB \ F DB LB U B
Table 2.
Values are in billion euros, except for ρ and γ which are percentage rates.YE LP SF U G GB F DB \ F DB LB U B δ ∆ II b ε Table 3.
Values are in percent of BE = 100 %. The terms II and b ε are calculated with(4.20) and (4.31), respectively. Further, ∆ = \ F DB − F DB and δ = ( U B − LB ) /
2, also inpercent of BE .5.B. Sensitivity: volatility ± . The values obtained by reducing the implied volatilities by 50 % arepresented in Table 4. In line with Assumption 1.3, this may equally be viewed as the base case. Indeed,management actions are usually designed to reduce variation in the gross surplus and in profit declarations,and thus one can make the argument that the
F DB will not be subject to the full market volatility. In thissense Table 4 may actually be a more realistic base case scenario than Table 2. The formulas have been implemented in R . The code can be provided upon request. STIMATION OF FUTURE DISCRETIONARY BENEFITS 15 YE F DB \ F DB LB U B δ ∆2017 23.98 23.32 21.58 25.05 1.74 -0.662018 22.54 22.36 20.53 24.19 1.83 -0.172019 19.54 19.96 18.37 21.56 1.60 0.42
Table 4.
Values are in percent of BE = 100 %. Sensitivity: 50 % · IV t .Table 5 shows the effect of a relative increase in implied volatilities by 50 %. Note that the estimationuncertainty δ increases. YE F DB \ F DB LB U B δ ∆2017 23.98 24.29 21.18 27.40 3.11 0.312018 22.54 23.46 20.08 26.83 3.38 0.922019 19.54 21.43 17.78 25.07 3.64 1.89
Table 5.
Values are in percent of BE = 100 %. Sensitivity: 150 % · IV t .5.C. Sensitivity: technical interest rate ±
10 % . Tables 6 and 7 show the effects of a relative decrease andincrease of the technical interest rate by 10 %.YE
F DB \ F DB LB U B δ ∆2017 23.98 23.43 21.14 25.72 2.29 -0.552018 22.54 22.30 20.04 24.56 2.27 -0.242019 19.54 20.42 18.01 22.84 2.42 0.88
Table 6.
Values are in percent of BE = 100 %. Sensitivity: 90 % · ρ .YE F DB \ F DB LB U B δ ∆2017 23.98 24.15 21.77 26.53 2.38 0.172018 22.54 23.34 20.65 26.04 2.70 0.812019 19.54 20.83 18.28 23.39 2.56 1.29
Table 7.
Values are in percent of BE = 100 %. Sensitivity: 110 % · ρ .5.D. Sensitivity: h ± . Tables 8 and 9 show the effects of choosing h = 10 and h = 14, respectively.YE F DB \ F DB LB U B δ ∆2017 23.98 23.59 21.64 25.55 1.96 -0.382018 22.54 22.70 20.60 24.79 2.10 0.162019 19.54 20.43 18.36 22.50 2.07 0.89
Table 8.
Values are in percent of BE = 100 %. Sensitivity: h = 10. YE F DB \ F DB LB U B δ ∆2017 23.98 23.76 21.24 26.28 2.27 -0.222018 22.54 22.87 20.14 25.59 2.73 0.332019 19.54 20.76 17.91 23.61 2.85 1.22
Table 9.
Values are in percent of BE = 100 %. Sensitivity: h = 14.6. Conclusions
Concerning the estimation error in LB . An exact calculation of ε requires a Monte Carlo evaluationof (4.29), and this is just as difficult as numerically calculating the F DB . This means that, with respect tofinding a lower bound, all of the difficulties of the
F DB calculation are transferred to the computation of ε .This transfer is a three step process:(1) Express F DB in terms of I ∼ = 0, II ∼ = 0 and III , as in (3.15).(2) Find an estimate for
III , as in (4.28).(3) Find an estimate for ε .Then it turns out that general actuarial arguments imply that ε is positive or very small (i.e. ε ≥ − · BE ).Hence it can be omitted from (4.28).6.B. Concerning LB and U B . The upper and lower bounds in Theorem 4.2 depend only on four kinds ofquantities:(1) Balance sheet items SF , LP and U G . These are known at valuation time and clearly model inde-pendent.(2) The value of guaranteed cash flows, GB : This value has to be calculated, but is a deterministic quantity(with respect to financial variables) and therefore independent of all model choices or management rules.Further, being deterministic, it is scenario-independent and can therefore be calculated with respectto the initial yield curve.(3) Company specific information such as the average technical interest rate ( ρ ), the expected amount oftechnical gains ( γ ), time to maturities of assets ( d ) and liabilities ( h ), the bound on the surplus fund( θ ), and the gross policy holder participation ( gph ).(4) Interest rate related quantities: F t , O − t and O + t . The first follows directly from the yield curveat valuation time and the latter two can be either inferred from known option prices or calculatedanalytically via Black’s formula ([2]).Since the upper and lower bounds depend on the optionalities O − t and O + t , respectively, it follows that theestimation interval [ U B, LB ] widens as the interest rate volatility increases. Thus the usefulness of the bounds LB ≤ F DB ≤ U B will deteriorate with increasing volatility level. This is consistent with the intuition thatthe quality of any closed formula approximation of a stochastic quantity should depend on the magnitude ofthe variance.6.C.
Comparison with the lower bound formula in [7] . We have already presented an analytic lowerbound formula for the
F DB in [7, Prop. 3.4]. This previous lower bound, g LB , differs from LB in Theorem 4.2in the following aspects:(1) The starting point for the derivation of g LB is a single contract picture. This is then subsequentlygeneralized to a realistic liability portfolio consisting of multiple contracts. But this generalization usesadditional assumptions and suffers also from the defect that the attribution of non-contract specificbalance sheet items is not well-defined (without yet additional assumptions). The non-contract specificbalance sheet items are the surplus fund, SF t and the unrealized gains, U G t . These shortcomings areremoved by the present approach. STIMATION OF FUTURE DISCRETIONARY BENEFITS 17 (2) The formula for g LB involves a cross financing term, F . This term is estimated by ad-hoc expertjudgement. In the present context, the cross financing is included in the terms I , II and III , andthese are estimated by a careful analysis of the accounting flows (Section 4.C). A consequence ofthis analysis is that the estimate (4.32) for
III also depends on interest rate volatility, as should beexpected, whereas the formula for F was volatility-independent.Furthermore, we now also have estimate (4.37) for an upper bound of the F DB . This is completely new andyields the simple estimator \ F DB = ( LB + U B ) / F DB , which is useful whenever (
U B − LB ) /BE is not too large.6.D. Possible applications.
The bounds LB and U B in Theorem 4.2 can be readily applied to real worlddata, as we have shown in Section 5. Immediate practical applications therefore include the following:(1) Internal validation: companies may use LB and U B to validate their
F DB calculations, and thus theirvaluation models.(2) External validation by parent companies: holdings may wish to validate the valuation models in theirsubsidiaries.(3) External validation by supervisors or auditors.Clearly, the validation of the best estimate will be most effective when the control via Theorem 4.2 is pairedwith a statistical analysis of the second order assumptions leading to GB and a verification that the contractspecific features, which give rise to the guaranteed benefit cash flows, are correctly implemented.7. Appendix: actuarial approximation of gross surplus
The aim of this appendix is to derive the approximate formula (7.49) from an actuarial point of view.7.A.
The gross surplus.
To this end, we first write out the gross surplus in detail (where we assume premi-ums, costs and benefits to be paid at time t ):(7.40) gs ∗ t = − ∆ V t − ∆ DB t − ∆ DB t − + cf ∗ assets, t + pr t − co t − bf t + S t , where: ∆ V t = V t − V t − is the difference of mathematical reserve at t and t − cf ∗ assets, t = cf assets, t + BV t − BV t − is the accounting flow of assets before application of management rules; cf assets, t denotes the cash flow ofall assets from time t − t before application of management rules; S t ≥ DB t := − ph t − χ t − DB t − is the difference of DB t and DB t − in (2.13) if η t = 0 and θ t = 0, as is the case with regard to the calculationof the gross surplus; and ∆ DB ≤ t := DB ≤ t − DB ≤ t − = − ph ≤ t − χ ≤ t − DB ≤ t − . The last quantity is defined as DB ≤ t := LP t − V t − DB t . At valuation time t = 0, DB ≤ = LP − V is equal to the accumulated profits thathave been shared with the policy holders and that have been set aside as part of the technical reserves LP .The benefits, bf t , are the sum of all benefits paid out at t , including ph t and ph ≤ t . We can thus rewrite (7.40)as gs ∗ t = S t + cf ∗ assets ,t − ρ t V t − + (cost margin)+ (mortality margin)+ (surrender margin)Notice that, if first and second order assumptions coincide, then gs ∗ t = 0. The sum(cost margin) + (mortality margin) + (surrender margin)is generally referred to as the technical gains . As opposed to the interest rate gains, S t + cf ∗ assets ,t − ρ t V t − ,the technical gains are a relatively stable source of income. Approximations.
Since technical gains are supposed to be stable and independent of interest ratemovements, we approximate (technical gains) ∼ γ · LP t − (7.41)where γ is assumed to be constant. See also Section 8.A for a concrete estimation of γ .We further approximate the interest margin as cf ∗ assets, t = cf assets, t + BV t − BV t − = cf assets, t + M V t − M V t − + U G t − − U G t ∼ F t − M V t − + U G t − − U G t (7.42) = F t − BV t − + (1 + F t − ) U G t − − U G t ∼ F t − BV t − + κ t − U G (7.43) ∼ F t − BV t − + 2 ( t − /h κ t − U G LP · LP t − , (7.44)where we have used the definition of κ t − in (4.22) and, leaning on Assumption 4.1, that LP t − ∼ − ( t − /h · LP .The approximation (7.43) and definition (4.22) make sense since we want to capture the effect of the incomedue to unrealized gains and, on average, most of this income is expected to come from the initial position U G .In a risk neutral projection, assets do not generate unrealized gains on average. However, along individualscenarios unrealized gains or losses do arise; these are omitted in the above approximation. Moreover, sincemost of the portfolio is expected to be invested in fixed income instruments, the additional income due tounrealized gains is expected to be predominantly generated by assets progressing towards their maturities andthe corresponding convergence of market and book values.Therefore, from Assumption 1.2 we obtain(7.45) gs ∗ t ∼ (1 + θ ) · F t − · LP t − + S t + ρ t · ( DB t − + DB ≤ t − ) + κ t − · U G − ( ρ t − γ ) · LP t − Estimations.
Assumption 1.1, which implies that unrealized gains are not realized to generate additionalsurplus, thus yields(7.46) (cid:16) gs ∗ t (cid:17) + ≤ (cid:16) (1 + θ ) F t − LP t − + κ t − U G − ( ρ t − γ ) LP t − (cid:17) + + ρ t ( DB t − + DB ≤ t − )and (cid:16) gs ∗ t (cid:17) − ∼ (cid:16) (1 + θ ) · F t − · LP t − + S t + ρ t · ( DB t − + DB ≤ t − )(7.47) + κ t − · U G − ρ t · LP t − (cid:17) − ≤ (cid:16) (1 + θ ) F t − LP t − + κ t − U G − ( ρ t − γ ) LP t − (cid:17) − (7.48)since S t ≥ ρ t − γ ) · ( DB t − + DB ≤ t − ) ≥ ph ∗ t = gph · (cid:16) gs ∗ t (cid:17) + ≤ gph · (1 + θ ) · LP t − · (cid:16) F t − − ρ t − ( t − /h κ t − UG LP − γ θ (cid:17) + (7.49) + gph · ρ t · ( DB t − + DB ≤ t − ) . Appendix: public data
Estimating technical gains from market data.
For the German life insurance market, the factor γ in (7.41) can be determined from Tables 130 and 141 in [23]. The relevant items are stated below: STIMATION OF FUTURE DISCRETIONARY BENEFITS 19
Item Value as of 2017 Value as of 2018 Value as of 2019¨Uberschuss a b c d e f Table 10.
BaFin: public data for 2017-2019. Values are in billion euros. a gross surplus net of direct policy holder declarations b direct policy holder declarations c share of gross surplus allocated to the surplus fund d assets - interest margin e Tabelle 130, Versicherungstechnische R¨uckstellungen - selbst abgeschlossenes Gesch¨aft - brutto (technical provisions for directbusiness, gross of reinsurance) f Tabelle 130, Versicherungstechnische R¨uckstellungen - soweit das Anlagerisiko vom Versicherungsnehmer getragen wird - brutto(technical provisions, gross of reinsurance, of those contracts where the investment risk is carried by the policyholder)
In fact,(8.50) γ = ¨Uberschuss + Direktgutschrift − Kapitalanlageergebnis 1 b)Bilanzposten 6 brutto − Bilanzposten a) 6 bruttoWe obtain: Value as of 2017 Value as of 2018 Value as of 2019 γ Table 11.
Values of γ for 2017-2019.For the estimation of Term III we fix γ = 0 .
77 % as the average of the values in Table 11.8.B.
Allianz Lebensversicherungs-AG: publicly reported values.
The data in Table 12 is taken fromthe publicly available reports for the accounting years 2017-2019. For the relevant references see Table 13; andfor explanations of the symbols see Table 14.Symbol Value as of 2017 Value as of 2018 Value as of 2019 L U G SF SF GB F DB
Table 12.
Allianz Lebensversicherungs-AG: public data for 2017-2019. Values are in billion euros.The value of
U G is already scaled to L , which is in line with Assumption 2.3. The reason behind thisscaling is that according to [9, §
3] only the fraction of the capital gains, corresponding to the assets scaled tocover the average value of liabilities in the accounting year under consideration, contribute to the gross surplus.As for L , we adjust the local GAAP value of life insurance with profit participation for necessary regroupingof business, as explained in [18, p. 52], [21, p. 46], [22, p. 46] for the different accounting years 2017–2019. Symbol Source for 2017 Source for 2018 Source for 2019 L [18, p. 46, 52] [21, p. 42, 46] [22, p. 42, 46] U G [17, p. 46] [19, p. 42] [20, p. 46] SF [17, p. 55] [19, p. 51] [20, p. 55]Solvency II value of SF [18, p. 52] [21, p. 46] [22, p. 46] GB [18, p. 46] [21, p. 42] [22, p. 42] F DB [18, p. 46] [21, p. 42] [22, p. 42]
Table 13.
Allianz Lebensversicherungs-AG: references for the data in Table 12.Symbol Technical term L Versicherung mit ¨Uberschussbeteiligung
U G Stille Reserven der einzubeziehenden Kapitalanlagen SF R¨uckstellung f¨ur Beitragsr¨uckerstattung abz¨uglichfestgelegte, aber noch nicht zugeteilte TeileSolvency II value of SF ¨Uberschussfonds GB Bester Sch¨atzwert: Wert f¨ur garantierte Leistungen
F DB
Bester Sch¨atzwert: zuk¨unftige ¨Uberschussbeteiligung
Table 14.
Allianz Lebensversicherungs-AG: technical terms used in any of the sources pro-vided in Table 13.The average technical interest rate, ρ , of the Allianz Lebensversicherungs-AG can be derived from thedistribution of the technical reserves over the guaranteed interest rates, with the following results:Value as of 2017 Value as of 2018 Value as of 2019 ρ Table 15.
Values of ρ for 2017-2019.The respective values can be found in [17, p. 34], [19, p. 33], and [20, p. 37]. For the base case calculationsin Section 5.A we use the corresponding (time independent) values. In reality slightly smaller rates may bemore appropriate since the level of guarantee rates is expected to decline over time. A sensitivity which showsthe effect of a 10 % reduction in ρ is provided in Section 5.C.8.C. Estimating gph . The net policy holder shares, nph for the accounting years 2017-2019 can be obtainedvia(8.51) nph = Zuf¨uhrung zur RfB + DirektgutschriftBrutto¨uberschuss . from the values collected in the table below (see [17, p. 9], [19, p. 9], and [20, p. 8] for accounting years2017-2019). Item Value as of 2017 Value as of 2018 Value as of 2019Brutto¨uberschuss 2.6 3.1 3.6Zuf¨uhrung zur RfB 2.0 2.3 2.9Direktgutschrift 0.1 0.1 0.2 Table 16.
Values are in billion euros.
STIMATION OF FUTURE DISCRETIONARY BENEFITS 21
The gross policy holder share, gph is calculated from nph according to the relation(8.52) gph = (1 − τ ) nph − τ · nph . Applying the German tax rate of τ = 29 . Table 17.
Values of gph for 2017-2019.For the estimation of Term III we fix gph = 75 . Discount rates.
The following are the publicly available EIOPA discount rates for 2017, 2018 and 2019. t P ,t t P ,t t P ,t t P ,t t P ,t t P ,t Table 18.
Euro discount rates as of 31.12.2017. The rates are with volatility adjustment.Source: [16] t P ,t t P ,t t P ,t t P ,t t P ,t t P ,t Table 19.
Euro discount rates as of 31.12.2018. The rates are with volatility adjustment.Source: [16] t P ,t t P ,t t P ,t t P ,t t P ,t t P ,t Table 20.
Euro discount rates as of 31.12.2019. The rates are with volatility adjustment.Source: [16]
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