Resilience and Intergenerational Fairness in Collective Defined Contribution Pension Funds

Forschung am ivwKöln Band 7/2018 Resilience and Intergenerational Fairness in Collective Defined Contribution Pension Funds Oskar Goecke Forschung am ivwKöln, Band 7/2018 Oskar Goecke Forschungsstelle FaRis Resilience and Intergenerational Fairness in Collective Defined Contribution Pension Funds Abstract A pension system is resilient if it able to absorb external (temporal) shocks and if it is able to adapt to (longterm) shifts of the socio -economic environment. Defined benefit (DB) and defined contribution pension plans behave contrastingly with respect to capital market shocks and shifts: while DB -plan benefits are not affected by external shocks they totally lack adaptability with respect to fundamental changes; DC -plans automatically adjust to a changing environment but any external shock has a direct impact on the (expected) pensions. By adding a collective component to DC-plans one can make these collective DC (CDC) -plans shock absorbing - at least to a certain degree. In our CDC pension model we build a collective reserve of assets that serves as a buffer to capital market shocks, e.g. stock ma rket crashes. The idea is to transfer money from the collective reserve to the individual pension accounts whenever capital markets slump and to feed the collective reserve whenever capital market are booming. This mechanism is particular valuable for age cohorts that are close to retirement. It is clear that withdrawing assets from or adding assets to the collective reserve is essentially a transfer of assets between the age cohorts. In our near reality model we investigate the effect of stock market shock s and interest rate (and mortality) shifts on a CDCpension system. We are particularly interested in the question, to what extend a CDC -pension system is actually able to absorb shocks and whether the intergenerational transfer of assets via the collecti ve reserve can be regarded as fair. - I - Resilience and Intergenerational Fairness in Collective Defined Contribution Pension Funds Oskar Goecke - II - Resilience and Intergenerational Fairness in Collective Defined Contribution Pension Funds Content 1 Introduction .............................................................................. 1 2 Basic Model ............................................................................. 3 2.1 Population Model ............................................................................ 3 2.1.1 CDC Pension Fund ....................................................................................... 3 2.1.2 Steady State Population and Population Dynamics ...................................... 5 2.2 Liabilities ......................................................................................... 7 2.3 Assets ............................................................................................. 10 3 Asset Liability Management .................................................. 12 3.1 Basic Relations .............................................................................. 12 3.2 ALM – Strategies .......................................................................... 17 3.3 Individual Saving and Dissaving ................................................... 24 3.4 Steady State Analysis .................................................................... 25 4 Resilience Test ....................................................................... 28 4.1 Steady State Original Position ....................................................... 29 4.2 Capital Market Shock .................................................................... 30 4.2.1 Capital Market Shock Effect on IDC-Plans ............................................... 30 4.2.2 Capital Market Shock Effect on CDC-Plans .............................................. 31 4.3 Capital Market Shift ...................................................................... 35 4.3.1 Capital Market Shift Effect on IDC-Plans .................................................. 35 4.3.2 Capital Market Shift Effect on CDC-Plans ................................................ 36 4.4 Mortality Shift ............................................................................... 40 4.4.1 Mortality Shift Effect on IDC-Plans ........................................................... 42 4.4.2 Mortality Shift Effect on CDC-Plans ......................................................... 43 5 Concluding Remarks .............................................................. 51 References .................................................................................... 52 Figures and Tables ....................................................................... 54 Contact ......................................................................................... 56 - 1 - 1 Introduction All over the world defined benefit pension plans (DB-plans) are in retreat, meaning that young employees entering working life must accept defined contribution pension plans (DC-plans).1 There are several reasons for this development, including: increased risk awareness among employers, intensified regulation and a low interest rate environment. Employees and labour unions regard the shift from DB to DC as a massive reduction of labour rights since the investment risk is put on the weak shoulders of employees. This fact cannot be denied. However, one can also argue that the transition from DB to DC is just proof that DB plans are unsustainable in the sense that they lack flexibility to adjust to a changed economic environment. As a consequence, inevitable adjustments had to be made by closing old DB systems and in doing so putting the financial burden of the obsolete DB plans on the shoulders of the younger generation.2 This generation is hit twice since at the same time the social security pension systems are under reconstruction with the obvious outcome for the young.3 Compared to DB-plans, pure (individual) DC-plans are “over-reactive” in the sense that pension benefits are directly linked to the time value of the pension pot. Equity market shocks, shifts of the yield curve or changing life expectancy instantaneously hit the expected pension or the pension in payment. The idea behind collective DC- (CDC-) plans is to introduce a collective component to a DC-plan to buffer external shocks or shifts in order to stabilise (expected) pension payments. The collective reserve in a CDC system can be regarded as an unallocated fund of assets. This fund must be fed by contributions or asset returns. Payments into and withdrawals from the collective reserve constitute an intergenerational transfer of assets. In the following we present a multi generation CDC-pension model including rules for when and how the intergenerational transfer is to be carried out. The main purpose of this paper is to apply the concept of resilience to a pension system. Resilience is the ability of a system to absorb (single) external shocks and to adapt to (permanent) shifts of the socio-economic environment. Our approach allows us to explicitly measure the intergenerational transfer. 1 Cf. [OECD 2011], p. 15. 2 We have the same effect if the benefits of a DB plan remain untouched but the contributions are adjusted. 3 Cf. [House of Commons 2016] p. 15-16. - 2 - The utility increasing effect of intergenerational risk transfer has been proven by many authors using different methods. [Gordon/ Varian 1988] use a stylised overlapping generation model to prove that the government should play an active role by borrowing or saving in the capital market to improve risk allocation between generations. [Gollier 2007] addresses the intergenerational risk transfer in a pension fund with a stable number of new young workers replacing the retirees who get a lump sum payment as pension benefit. Using expected utility theory, Gollier can prove that if all generations save into a common pension fund the expected utility for every generation can be increased. [Westerhout 2011] discusses the question of how the intergenerational risk transfer in a pension system can be designed in such a way that every generation really takes advantage of the system. [Cui e.a. 2011] argue in the same spirit as [Gollier 2007], however their pension model is more realistic in the sense that their model works with current pension payments (instead of lump sum benefits) and they introduce an absorbing funding surplus, which finances the intergenerational transfer. Furthermore [Cui e.a. 2011] use option price techniques to value the intergenerational transfer. Our contribution is to discuss the resilience of a CDC pension scheme with respect to intergenerational fairness. We say that a pension scheme is resilient, if it is able to absorb external (single) shocks (e.g. a crash of market value of equities) and it is able to adjust to (permanent) shifts (e.g. shift of interest rates or mortality). It is desirable that a single stock market crash does not affect pensions in payment to full extend. However, as in defined contribution system with no external sponsor any protection of the group of pensioners is implicitly financed by an intergenerational transfer from the young to the old. Young participants will regard this kind of intergenerational transfer as fair because they expect that sooner or later the effects of the down shock will be compensated by an up shock. However, if e.g. the risk-free interest rate shifts to a new lower level, say combined with a lower inflation rate, then the understanding of intergenerational fairness could be that all age cohorts have to bear the consequences. Under these circumstances a waving of pension adjustments or a cut of pensions in payment could be compelling from the perspective of intergenerational fairness. The setup of this paper is as follows. Following this introduction, section 2 introduces our basic pension model and section 3 the asset liability management (ALM) rules. The resilience test in section 4 constitutes the main part of this paper. To test the resilience of the pension system we have to define a steady state position (section 4.1). Then we apply capital market shock (section 4.2) and capital market shift (section 4.3) scenarios to the system. Finally in section 4.4 we discuss the effects of a mortality shift. - 3 - 2 Basic Model 2.1 Population Model 2.1.1 CDC Pension Fund We consider a pension fund for active and retired employees. The active employees pay periodic contributions to build up a pension capital. At a certain retirement age z the individual pension capitals are converted into a life annuity. The pension fund is exclusively financed by the regular contributions; there is no external entity that could step in if the pension fund runs out of assets. Examples of such scenarios would be if assets do not perform as expected or if the retirees live longer than expected resulting in pension benefits having to be adjusted. In extreme cases pension payments may have to be cut. On the other hand, overperforming assets or declining life expectancy eventually result in higher pension benefits. In the case of a defined contribution (DC) pension fund, the contributions determine the pension benefits. If observed asset returns or mortality rates deviate from the expected values the pension benefits have to be adjusted while contributions remain unchanged. In contrast, in a definded benefit (DB) scheme, the contributions would be adjusted but not the promised benefits. The standard design of a DC schemes is an individual DC scheme, where each participant pays contributions into a personal pension pot, at retirement the accrued capital of the pension pot determines the paid benefits. To our understanding the characteristic feature of a collective DC (CDC) pension fund is that there is a collective reserve, i.e. part of the total assets can be used to balance unexpected losses on the asset side or actuarial losses on the liability side. The following FIGURE 1 shows the stylised balance sheet of the pension fund. We have to explain when and how the collective reserve is deployed and refilled. FIGURE 1: Stylised Balance Sheet - 4 - We assume that employees enter the system at a fixed entry age x0 and that they remain in the population until death. If an employee dies before age z the balance of the personal account is paid out. From the retirement age of z onwards an annuity is paid until the person dies. Here we list some basic notations with respect to the population model: t: time index t = 0, 1, …, T x0: fixed entry age, if not stated otherwise we set x0 = 20 z: fixed retirement age, if not stated otherwise we set z = 65 ω : maximal age, if not stated otherwise we set 115 ω = L(t, x): number of persons of the (t, x)-cohort, i.e. the number of persons who are x years old at time t. We assume that each age cohort is homogeneous, i.e. all members share the same mortality risk and have the same pension entitlements. (, ) ( 1, 1)/ (, ) ptx Lt x Ltx=++  : survival probability for the (t, x)-cohort. This is a random variable conditioned to the avaible information at time t, observable at time t+1. ˆ(, )ptx : estimated survival probability for the (t, x)-cohort for the time interval [t, t+1] based on the information up to time t (, ) a p tx : actuarial survival probability for the (t, x)-cohort. These values are used to calculated the actuarial reserve for pensions due. The actuarial survival probalities could be best or prudent estimates. We do not model an ongoing updating of pa (t, x) to match the experienced mortality rate up a certain date. However, in the course of our discussion we will also examine the effect of a mortality shift. By definition of ω we have ˆ (, ) (, ) (, ) 0 a pt pt p t ωω ω = = =  for all t. We do not model the idiosyncratic mortality risk, i.e. the risk that a single person dies in a certain time period. Instead, we allow for non integer L(t, x) and assume that ( 1, 1) (, ) (, )Lt x Ltx ptx+ +=  , where the random variable (, )ptx  represents the systematic mortality risk. - 5 - We think of ˆ(, )ptx as any reasonable best estimate for (, )ptx  . In practice, the phrase best estimate does not necessarily imply that ( ) ˆ(, ) (, )ptx ptx=E .4 We distinguish between (, ) a p tx and ˆ(, )ptx to allow for safety margins with respect to mortality rates. We regard the initial population ( ) 0 (0, ) : L xx x ω ≤≤ and the new entrants ( ) 0 ( , ): 0Ltx t≥ as deterministic. 0 1 ( ): ( , ) z A xx L t Ltx − = = ∑ : total number of active employees at time t ( ): ( , ) R xz L t Ltx ω = =∑ : total number of retirees at time t (): () () AR Lt Lt Lt= + : total population at time t. For convenience we define L(t, x):= L(t, x0) for all x < x0 and L(-1, x):= L(0, x) for all x, assuming that before time t =0 we had a stable population. If not stated otherwise we calibrate our model population such that L(0, 20) = 1000. 2.1.2 Steady State Population and Population Dynamics The best estimate probabilities ˆ(, ) ptx are taken from the mortality tables Richttafeln 2005G, 5 which are the generally accepted standard tables for calculating book reserves for DBplans in Germany. The entry age of the Richttafeln 2005G is x0 = 20 and the terminal age is ω = 115, i.e. ˆ( ,115) 0pt = for all t. The Richttafeln 2005G are derived from social security data for male and female employees and comprise tables for all birth cohorts between 1891 and 2005. If indicated we will present separate results for a male and a female population. However, most calculations are performed on the basis of a hybrid male/ female population . To this end we define hybrid survival probabilities by ( ) () ( ) 1 2 ˆˆ ˆ (, ) (, ) (, ) male female ptx p tx p tx= + . One should be aware of the fact that the resulting hybrid population is not the population of a 50 - 50 mixed male/female population. 4 For example, in the stochastic CDB-model (as described in the [Cairns e.a. 2006]) the “natural” best estimate is not necessarily an unbiased estimator. 5 “Reference tables” [Heubeck et al. 2006] - 12 - ( ) 2 1 12 ( ) (1 )exp( ) exp( ) t t MM PP β µβ µ σ + =−++E . If we define exp( ) 1 A i µ = − and 2 1 2 exp( ) 1 S MM i µσ = +− , we get 1 (1 ) (1 ) (1 ) 1 ( ) t A S A SA t Pi i i ii P ββ β +  =− + + + =++ −   E . However, one may convince oneself that ( ) 1 ( 1) : ln / tt t PP µ + +=  cannot be decomposed as in (Eq. 2). 3 Asset Liability Management 3.1 Basic Relations We define () (): ln ()/ ()t Pt Vt ρ = - the log-reserve ratio or simply the reserve ratio. He have ρ (t) > 0 iff P(t) > V(t). In the following ρ (t) will be the fundamental control variable for the asset liability management (ALM). For practioners, the cover ratio P(t)/V(t) rather than ρ (t) is taken as the indicator of the “wellbeing” of a pension fund. Clearly, it makes no difference whether we control ρ (t) or P(t)/V(t). But, as we will see, ρ (t) simplifies notations. Note that for P(t)/V(t) ≈ 1 (say 0.8< P(t)/V(t) < 1.2) we have 1 + ρ (t) ≈ P(t)/V(t). At time t (i.e. based on the information up to time t) the pension manager has to dedide on σ t , the risk exposure for the coming time period [t, t +1]. If we apply a prospective declaration, then also η (t+1) and ε (t+1) are determined at time t. It is clear that if we want to guarantee a minimum cover ratio (or reserve ratio) then we must apply a retrospective declaration. For the following propositions we define for t ≥ 0: ( ) ( ) (, ) (, ) (, ) 1 (, ) (, ) (, ) 1 ( , ): () () () ( ) RR z Ltxbtx atx Ltxbtx atx wt x Vt Vt Bt Vt −− = = +− +   (for x ≥ z) 1(, ) ( 1) : ln ( , ) (, ) xz a ptx t wt x p tx ω π − =  +=−   ∑  1 ˆ(, ) ˆ( 1) : ln ( , ) (, ) xz a ptx t wt x p tx ω π − =  +=−   ∑ - 13 - () ( ): () CF t tVt λ = ( ) 1 ()exp () () () () ( ) : ln ln ln () () () 1 () tt Pt CFt Pt tVt CFt Vt t λρ δλ −−     − = −=     −−    () () () () ( ): () () ( ) RR z V t V t Bt Vt tVt CFt Vt γ +− + = = −+ . Remarks 1. Since ( ) 1 ( ) (, ) (, ) (, ) 1 R xz V t Ltxbtx atx ω − = += − ∑ , w(t,x) is the relative weight of the (t, x)-cohort in VR(t+). Note that 1 (, ) 1 xz wt x ω − = = ∑ . 2. ˆ( 1)t π + is the weighted safety margin if the actuarial assumptions with respect to the survival probabilities are set so that ˆ (, ) (, ) a p tx ptx> . If we use best estimate survival probabilities for actuarial valuation we have ˆ( 1) 0t π += . 3. ( 1) t π +  and ˆ( 1)t π + only depend on the survival probabilities for the cohort of retirees. ( 1)t π +  measures to what extent the experienced and the actuarially presupposed mortality rates diverge. If the actuarial assumptions include safety margins then ( 1)t π +  is expected to be positive. We regard ˆ( 1) t π + as the best estimate for ( 1)t π +  based on information up to time t. As practitioners we do use the phrase “best estimate” rather generously. In particular we do not stipulate that ( ) ˆ ( 1) ( 1)tt ππ += +E . One should note that ( ) ˆ(, ) (, )ptx ptx=E for all x and t does not imply that ( ) ˆ ( 1) ( 1)tt ππ += +E . 4. If ˆ (, ) (, ) a p tx ptx= then ˆ( 1) 0t π += and 1 (, ) ( 1) ln ( , ) ˆ(, ) xz ptx t wt x ptx ω π − =  +=−  ∑   . ( 1) t π +  can be interpreted as the weighted mortality effect. 5. λ (t) can be interpreted as the liquidity ratio, the ratio of outgoing money to the total liabilities. Note that λ (t) < 1 since CF(t) < V(t). 6. Since V(t) > CF(t) (by definition) δ (t) is well defined provided P(t) > CF(t). 7. Note that ρ (t) = 0 implies δ (t) = 0. For ρ (t) > 0 δ (t) is positive and increasing in CF(t) and for ρ (t) < 0 δ (t) is negative and decreasing in CF(t). δ (t) - 14 - measures the effect of the cashflow CF(t) on the reserve ratio ρ (t). CF(t) has no effect on the absolute value of the reserve P(t) – V(t), but CF(t) ≠ 0 effects the reserve ratio. If CF(t) < 0, which is typical for a young population, the reserve ratio will decrease. This effect is similar to the stock dilution effect when additional common shares are issued. A share buy-back program has an opposite effect. So we call δ (t) the stock effect. As we will see below, the stock effect will be positive if the pension system is in a steady state. It is also positive if the pension system is unwinding. 8. γ (t) can be interpreted as the weighted age burden. γ (t) = 0 means that there are no pension liabilities, and γ (t) = 1 implies that there are no liabilities for active workers. Proposition 1 If η (t +1) is the profit participation for the individual pension accounts and if the pensions are adjusted by ε (t +1), then we have the following recursions for the liabilities: ( ) ( 1) exp ( 1) ( ) AA Vt t Vt η += + + (Eq. 4) ( ) ( 1) exp ( 1) ( 1) ( ) RR a Vt t t Vt ε µπ += ++ − + +  (Eq. 5) If ( 1) ( 1) ( 1) a tt t ε η µπ += +− + +  , then ()( ) ( ) ( 1) exp ( 1) ( ) ( ) exp ( 1) ( )Vt t Vt CFt t Vt ηη += + − = + + (Eq. 6) ( 1) ( ) ( 1) ( 1) ( )t tt t t ρ ρµ η δ +− = +− ++  . (Eq. 7) Proof To prove (Eq. 4) we use definition (Eq. 1) and the fact that v(t, x0) = 0: 01 ( 1) ( , 1) ( 1, ) z A xx V t Ltx vt x = + += − + ∑ ( ) ( ) 01 exp ( 1) ( , 1) ( , 1) z xx t Ltx vtx c η = + = + − −+ ∑ ( ) 0 1 exp ( 1) () (, ) (, ) z xx t Ct Ltxvtx η − =  =++   ∑ ( ) ( ) exp ( 1) () () () () A z t V t Dt V t Ct η = + −−+ . - 15 - To verify (Eq. 5) we take (Eq. 2) and use the definition of w(t, x) and ( 1)t π +  and the recursion for ä (t, x): ( ) ( ) ( ) ( ) ( ) 1 1 1 ( 1) ( 1, 1) ( 1, 1) ( 1, 1) (, ) 1 exp ( 1) (, ) (, ) (, ) (, ) (, ) exp ( 1) () () () (, ) (, ) exp ( 1) ( 1) () () () . R xz a xz a R az xz a R az Vt Ltxbtxatx atx t ptxLtxbtx p tx ptx t V t V t Bt wt x p tx t t V t V t Bt ω ω ω µε µε µε π − = − = − = += ++ ++ ++ − = ++ = ++ + − = + +− + + − ∑ ∑ ∑      (Eq. 6) and (Eq. 7) follow directly from (Eq. 4) and (Eq. 5) and the definition of δ (t). ♦ Remark If we determine η (t +1) and ε (t +1) retrospectively, i.e. on the basis of information up to time t+1, then according to (Eq. 7) η (t +1) and ε (t +1) can be defined such that any predeterminded reserve level ρ (t +1) can be reached. For example, if we define ( 1) ( 1) ( )t tt η µδ += ++  and ( 1) ( 1) ( 1) a tt t ε η µπ += +− + +  , then ( 1) ( )tt ρρ += . If this was our ALM-strategy, we wouldn’t need a collective reserve! However, in this setting capital market risks and the mortality risk would directly affect the individual accounts or pensions. The main benefit of a collective system, namely the intergenerational risk sharing, would then not be enabled. Fixing η (t+1) and ε (t+1) at time t (and not at time t +1) reflects the idea of defined ambition.6 This is attractive for savers and retirees because they know in advance, how their contributions are accrued and how the pensions are adjusted. In Proposition 1 we have set ( 1) ( 1) ( 1) a tt t ε η µπ += +− + +  , which can only be determined retrospectively. Thus for a prospective declaration we have to replace ( 1)t π +  by ˆ( 1)t π + . Proposition 2 If in the situation of Proposition 1 we define ˆ ( 1) ( 1) ( 1) a tt t ε η µπ += +− + + then ( )( ) 1 ( 1) exp ( 1) ( ) ( ) t Vt t Y Vt CFt η + += ++ − (Eq. 8) 6 c.f. [Day et al. 2014] - 16 - 11 ( 1) ( ) ( ) ( ) ( 1) tt t t t t XY t t ρ ρ σ µσ δ η ++ +− = − + + − + , (Eq. 9) where () () ( ) 1 ˆ : ln 1 ( ) exp ( 1) ( 1) 1 t Y t tt γ ππ + = + +− + −  . Proof The definition of ε (t+1) together with Proposition 1 shows that () ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( 1) ( 1) ( 1) ˆ exp ( 1) ( ) exp ( 1) ( 1) ( 1) ( ) () () ˆ exp ( 1) ( ) exp ( 1) ( 1) () () ˆ exp ( 1) ( ) 1 ( ) exp ( 1) ( 1) 1 exp ( 1) ( ) . AR AR AR t Vt Vt Vt t Vt t t t Vt Vt Vt t Vt t t Vt Vt t Vt t t t t Y Vt η ηππ η ππ η γ ππ η + += ++ + = + ++ ++ +− + +  ++ = + + + +− +  ++  = + + + +− + −   = ++ +    (Eq. 9) is a direct consequence of (Eq. 8) and the definition von δ (t). ♦ Remark 1. (Eq. 9) will be the basis for the ALM-strategies which are presented in the next section. The change of the reserve ratio, ρ (t+1)- ρ (t), can be broken down into  the stochastic capital market effect 1 ( 1) ( ) t tt tX µ µσ σ + += +   the stochastic longevity effect Yt+1  the structural stock effect δ (t)  the profit participation η (t). 2. Admittedly, the definition of Yt+1 is a little bit clumsy, but it serves perfectly to isolate the longevity risk. Yt+1 depends on the weighted age burden γ (t) and the difference between the estimated and the observed longevity effect ˆ( 1) ( 1)tt ππ +− +  . If γ (t) = 0 then Yt+1 = 0, and if γ (t) = 1 then 1ˆ( 1) ( 1) t Yt t ππ += +− +  . Using the 2nd order Taylor approximation for the function ( ) ( ) ln 1 ( ) exp( ) 1t γ ∆ + ∆− , we get the following approximation: ( ) ( ) 1 12 ()1 1 () t Yt t γγ +≈∆ + ∆ − with ˆ : ( 1) ( 1)tt ππ ∆= + − +  . (Eq. 10) 3. If ˆ( 1) 0t π += , and especially if ˆ (, ) (, ) a p tx ptx= for all x, then 1 1 (, ) ln 1 ( ) 1 ( , ) (, ) t xz a ptx Y t wt x p tx ω γ − + =   =+−      ∑ . - 17 - 4. The pension adjustment ˆ ( 1) ( 1) ( 1) a tt t ε η µπ += +− + + can be regarded as fair, since there is no systematic transfer of capital between the young and the old. If the actuarial surviving probabilities pa(t, x) are calculated with safety margins, then the initial pensions b(t, z) are lower compared to a best estimate pension. Then ˆ( 1)t π + ensures that the safety margins are (on average) refunded to the cohort of retirees. However, within the cohort of retirees high safety margins with respect to pa(t, x) do have a redistributional effect, since higher pension adjustments are unilaterally favourable for long living retirees. 3.2 ALM – Strategies We now come to the question of how to control the CDC-pension fund described above. Capital market opportunities and risks, mortality rates and the number of new entrants are exogenous variables, of which only the capital market risk can be controlled to a certain extent. Our CDC-pension fund is self financing in the sense that there is no outside institution that can step in if capital markets perform extremely badly or people live much longer than expected. On the other hand, the pension member can be sure that every contribution paid into the system is exclusively used for death or pension benefits. Since the pension fund itself does not guarantee any benefits, there must be some good arguments for employees to entrust their contribution to such a system. Actually, the only good reason to enter such a collective system is that the employees have a good chance to get a better risk-return profile than in an individual saving and dissaving arrangement. Before presenting ALM-rules for the CDC pension fund, let us state some principles that the ALM has to comply with: Principle 1: The benefits a person receives are calculated on the basis of their personal pension capital at retirement age. Especially all pension members within an age cohort are treated equally. The idea behind this principle is that the sole purpose of the collective element in the CDC plan is to enable an intertemporal risk transfer. Thus, in the absence of risk a CDC plan should be nothing but a simple DC plan with a one-to-one correspondence between contributions and benefits on the individual level. Our CDC model complies with Principle 1 since the pensions are calculated on the basis of accumulated contribution and furthermore, η (t) and ε (t) apply equally to active workers and retirees respectively. - 18 - Principle 2: It must be ensured that P(t) ≥ V(t), i.e. ρ (t) ≥ 0. We think of a capital funded system, which in general means that pension benefits are prefunded by regular contributions. In contrast, in a pay-as-you-go provision system the currently paid benefits are covered by currently incoming contributions. Instead of P(t) ≥ V(t) for all t, we could require that at any time all pension liabilities can be settled even if there are no further contributions. However, in a system with no guarantees the expression “all pension liabilities” is rather vague or has to be made precise. In our model the understanding of VR(t) is that this is the actuarial reserve under the assumption that the currently paid pensions are kept constant in future. Note that VR(t) is not the market consistent value of the pension liabilities since we do not price the pension fund’s implicit option to increase or reduce future payments if circumstances require. In our model CDC pension system we can ensure P(t) ≥ V(t) only if we allow for a retrospective declaration. In case of a prospective declaration P(t) ≥ V(t) can only be ensured with a certain degree of probability. Thus, in the case of a prospective declaration we have to take a weakened version of Principle 2: Principle 2’: It must be ensured that P(t) ≥ V(t), but for a transitional period P(t) < V(t) is accepted provided measures are taken to restore full funding. Principle 3: No age cohort is systematically preferred or put at a disadvantage compared to others (intergenerational equity). The requirement of generational equity is fundamental for any pension system - capital funded or pay-as-yougo. This issue is widely discussed in literature.7 Admittedly, Principles 2, 2’ and 3 are put in rather vague terms. They convey the idea of a “fair” pension system, but fairness is not an actuarial concept. At this point, it is worth to mention the fundamental concept, which John Rawls (1921-2002) worked out in his seminal book “A Theory of Justice”. He addresses the problem of justice between generations from an abstract perspective so that his rules are not directly applicable to a funded pension scheme.8 However, his idea of a social contract agreed upon behind the “veil of ignorance” (“in the original position”) can be applied to the question of a fair pension scheme. Behind the veil of ignorance people do not know in advance whether their generation will be lucky or unlucky with respect to the individual life span and to the future development of capital markets. 7 Cf. [European Union 2016] IORP II Directive, Article 7. 8 Rawls explicitly addressed the issue of intergenerational fairness – cf. [Rawls 1971], Chapter 44, pp. 251-258. - 19 - Principle 2 requires that the ALM has to control the cover ratio P(t) / V(t) or - which is equivalent - the reserve ratio ρ (t) . If ρ (t) threatens to fall below zero or some threshold ρ min measures have to be taken , e.g. pension cuts and/ or the reduction of the risk exposure on the asset side. Intergenerational equity (Principle 3) requires that ρ (t) is also capped above, since an unreasonable large reserve ratio indicates that there is a systematic transfer from the old to the young. As pointed out, generational equity requires that ˆ ( 1) ( 1) ( 1) a tt t ε η µπ += +− + + , otherwise there would be a systematic income transfer between old and young. In our setting a feasible retrospective ALM rule is a rule which at time t (on the information up to time t) determines η (t) and σ t, such that Principles 1, 2 and 3 are satisfied. A feasible prospective ALM-rule is a rule which at time t (on the information up to time t) determines η (t+1) and σ t, such that Principles 1, 2’ and 3 are satisfied. We can think of a wide range of ALM-strategies that comply with the above principles. The ALM rules we use here are taken from [Goecke 2013]. The model presented there is time continuous and restricted to the accumulation phase. But the basic features can be transferred the discrete case. In particular we adopt the idea of a strategic reserve ratio ˆ ρ and a strategic risk exposure ˆ σ . The pair ˆˆ (, ) ρσ represents a state of equilibrium in the sense that if we observe a reserve ratio ˆ ()t ρρ = then we choose ˆ t σσ = as the risk exposure. η (t) is chosen such that the reserve ratio remains unchanged provided capital market returns and mortality rates are just as expected. Another feature taken from [Goecke 2013] is that whenever ˆ ()t ρρ ≠ we adjust t σ and ( 1) t η + in dependence of the reserve gap ˆ ()t ρρ − . We now state our basic ALM-strategy in the prospective version. For real numbers max ˆˆ (, ,, , )a ρσ θσ we define (ALM 1) ( ) max ˆ ˆ( ) and 0 tt at σσ ρ ρ σσ =+ − ≤≤ (ALM 2) ( ) ˆ ( 1) ( ) () () t t tt η µσ δ θ ρ ρ += + + − (ALM 3) ˆ ( 1) ( 1) ( 1) . a tt t ε η µπ += +− + + Remarks: 1. The reserve gap ˆ ()t ρρ − rather than the reserve ratio ρ (t) is the decisive control variable of the CDC-pension system. However, due to the stock effect δ (t) the absolute level of ρ (t) does have influence on the process. - 20 - 2. (ALM 1) is motivated by the following considerations. Suppose at time t we determine the risk exposure σ t under the side constraint, that with probability 1α the reserve ratio does not fall below ρ min, i.e. ( ) min ( 1)t ρ ρα +≤ ≤ P . For ( ) ˆ ( 1) ( ) () () t t tt η µσ δ θ ρ ρ += + + − - c.f. (ALM 2) - this is equivalent to () () 1 1 min ˆ () () tt t XY t t σ ρ ρ θρ ρ α ++ −≤ − + −≤ P . Let VaR α > 0 denote the α -value at risk of Xt+1, i.e. ( ) 1t X VaR αα +≤− =P , then for Yt+1 = 0 (i.e. neglecting the mortality risk) we get () min ˆˆ (1 ) ( ) t t VaR α ρρ θρ ρ σ − +− − ≤ . Thus, if we seek maximal risk exposure under the constraint ( ) min ( 1)t ρ ρα +≤ ≤P , then we have to define ( ) ˆ ˆ() tat σσ ρ ρ =+− with min ˆ1 ˆ,a VaR VaR αα ρρ θ σ −− = = . If we use the Black Scholes framework for the capital market (cf. Remark 1 of section 2.3), then Xt is normally distributed with variance 1 and expectation 0. On the basis of the Solvency 2 security level of 1α = 99.5% we get VaR α = 2.5758. Suppose that the regulator allows a temporary underfunding of 90% and a “normal” funding ratio of 115%, then ρ min = ln(0.9) = -10.54%, ˆ ρ = ln(1.15) = 13.98%, and min ˆ1 ˆ0.0952 and a VaR VaR αα ρρ θ σ −− = = = . Assuming that a broadly diversified portfolio of stocks has a volatility of about 19%, ˆ0.095 σ = corresponds to an equity ratio of about 50%. The question of how to calibrate θ , we will answer in view of Prop. 3, below. 3. Parameter a in (ALM 1) determines the adjustment speed with respect to the risk exposure. For a = 0 we have a constant mix strategy throughout the time horizon. If a > 0 then the risk appetite for the asset allocation changes in line with the positive or negative reserve gap ˆ () t ρρ − . The case a < 0 corresponds to a massive anti cyclic investment strategy, because we then increase the risk exposure after bad experience with the pension assets. However, this strategy massively increases the risk of encountering negative reserve ratios. - 21 - 4. The side constraint 0 ≤ σ t ≤ σ max allows us to keep the risk exposure within reasonable limits. In our setting σ t = 0 implies a risk free investment. We should be aware that even a portfolio of AAAgovernment bonds is not risk free, since bond prices are driven by market interest rates. So in practice we must choose a σ t not below some σ min > 0. We could skip the upper bound σ max if we allowed for leverage instruments. However usually these instruments are prohibited for pension funds. 5. According to (ALM 1) and (ALM 2) the risk exposure and the profit participation are linear function of the reserve gap. Since P(t)/V(t) ≈ 1+ ρ (t) for 0.8 ≤ P(t)/V(t) ≤ 1.2, we can say that risk exposure and profit participation are approximately linearly dependent of the reserve gap. Since a low cover ratio P(t)/V(t) << 1 is generally regarded as more critical than a high cover ratio, the transition from P(t)/V(t) to ln(P(t)/V(t)) is at least plausible. 6. In (ALM 2) η (t+1) has three components:  µ (σt) ensures a fair participation in the portfolio returns. All pension members directly share the expected asset returns.  δ (t) ensures that the capital returns from the collective reserve are evenly redistributed to the pension members.  The term ( ) ˆ ()t θρ ρ − represents an intergeneration risk transfer. If the observed reserve ratio falls behind the target ratio, then all members have to put extra money aside to fill the gap. If there is a positive reserve gap then all members get an equal share. It is obvious that the generation of young employees would prefer a strong reserve because this allows a higher risk exposure and, in the long run, a higher return on investment. The pensioners would be rather reluctant to strengthen the collective reserve. In [Goecke 2013] this term (for θ > 0) ensures the mean reverting property of the stochastic process ρ (t). Economically, θ < 0 makes no sense; it is also clear that with θ = 0 we had no control over the reserve. The case θ > 1 implies an overreaction – cf. Prop. 3 below. 7. As pointed out, the prospective declaration in (ALM 2) cannot ensure that P(t) ≥ V(t). In order to safeguard a minimum reserve ratio ρ min, we can define a retrospective variant of (ALM 2) by ( ) ( ) () 1 1 min ˆ ( 1) ( ) () () , () retro t tt t t t Min t X Y t η µσδ θρρσ ρρ ++ += + + − − + − . - 28 - The comparison of CDCand IDC-pension arrangements must take into account that members of a CDCplan receive additional returns from the collective reserve, namely the stock effect δ which is positive provided ρ >0. To measure the effect we calculate TVIDC(x) and TVCDC(x), the time value of future (death and pension) benefits minus future contributions for members the x-cohort in the IDC and CDC-case. Then TVCDC(x) - TVIDC(x) measures the effect of the extra return of ln(e e e ) ρ µ ρµ δ + =− +− from the collective reserve. Figure 5 illustrates this for µ = 0.025, µ a = 0.01, ρ =0.15 and δ ≈ 0.0041. For example, an employee, aged x0 = 20, entering the CDC plan will receive more benefits with a time value of about 4.39 contribution rates. This is exactly the time value of the additional return of δ . FIGURE 5: Value added per head (TVCDC(x) - TVIDC(x)) in a CDC-pension scheme in steady state with a constant capital market return of µ =0.025, a reserve ratio of ρ = 0.15 and a contribution rate of c = 1. 4 Resilience Test A pension system is resilient, if it is able to absorb external (single) shocks and adapt to a (lasting) shift of the economic environment. Our resilience test works as follows: We start from a steady state situation and then apply a shock or a shift scenario and analyze the effects on the pension benefits. In a DC pension system all disturbances from outside must be compensated by adjusting the pension benefits. In the IDC-version we do not allow for risk transfer between generations, so the IDC-version will serve as a reference model to evaluate different ALM-strategies of the CDCmodel. - 29 - 4.1 Steady State Original Position We assume that our pension system starts from a steady state position11 with following parameters:  annual contributions c = 1 payable from age x0= 20 until age z-1 = 64  constant capital market returns ( ) 0.025t µµ = =   a stationary population with time independent survival probabilities ( 1) ˆ (, ) (, ) (, ) ( ) () a Lx ptx ptx p tx px Lx + = = = =  , where p(x) are the male/ female hybrid survival probabilities as described in section 2.1.2. We then have ˆ () () 0 tt ππ = =  for all t.  constant number of new entrants L(x0) = L(20) = 1000  fixed actuarial interest rate of µ a = 0.01 and annuity factors for x ≥ z 0 () ( ) : exp( ) () x a k Lx k ax k Lx ωµ − = + = − ∑  ; ä(z) = 17.9249  pensions in payment are adjusted at the rate of ε = µ - µ a = 0.015. For the IDC-model the accrued pension capital at the age of x: x0 ≤ x ≤ z = 65 is then ( ) 0 exp ( ) 1 ( ): 1 exp( ) xx vx µ µ −− =−− , and the initial steady state pension is ( ) 84.2531 ( ) 4.7003 17.9249() vz bz az = = =  . To make IDCand CDC-plans comparable we assume that in the CDC-case we start with a steady state resevere ratio ˆ0 ρ = . Then in the steady state situation pensions and death benefits are identical for IDCand CDC-plans. We define 0 () () for :() () () for x Lx vx x x z PLxrxax z x ω ≤<  =≤≤  and ( ) 0 ( 1) () ()for :0 for x Lx Lx vx x x z Dzx ω −− <≤   =<≤   to be the steady state pension capital for the x-cohort and the death benefit for those who die between age x-1 and x. We denote by P the total steady state pension capital and V the total steady state pension liabilities. Under the assumption that ˆ0 ρ = we 11 cf. section 3.4 - 30 - calculate 00 11 z xx xx xx PV P D ω =+=+ = = + ∑∑ = 2 530 615 + 5 851 = 2 536 466. Note that by our convention P and V comprise the death benefits for the decedents of the foregoing year. 4.2 Capital Market Shock A capital market shock is associated with an equity market crash or boom. Starting from a steady state situation with a constant investment return of µ = 0.025 we assume that at time T0 (i.e. at the end of [T0-1, T0]) we observe a return of µ + µ ∆ with µ ∆ = +0.2 (“up-scenario”) or µ ∆ = -0.2 (“down-scenario”). In the following our wording always refers to the down-scenario, however the derived formulars apply to either cases. 4.2.1 Capital Market Shock Effect on IDC-Plans Instantly upon observation of the capital market shock the individual pension accounts and the annuities are adjusted. Consider the (T0, x)-cohort, i.e. the generation of persons aged x at time T0. For x0 ≤ x ≤ z the personal pension capital at T0 will be v’(x):= exp( µ ∆) v(x) instead of v(x). After T0 the annual return is again µ , therefore the resulting annuity (z-x years later) will be cut by factor ( )( ) () 1 exp ( ) 1 exp( ) () vx zx vz µµ ∆ − −− - cf. FIGURE 6. For x > z the due pension will be b’(x) = exp( µ ∆) b(x) instead of b(x). From time T0+1 onwards pensions will again be adjusted by ε = µ - µ a . The capital market shock has the strongest effect on persons aged z or older. Their benefits would be cut by about 18% compared to the pre-shock level. - 31 - FIGURE 6: Down-shock scenario ( µ ∆= -0.2) for IDC-plans: Effect on the expected pension level, depending on the age x at time T0. 4.2.2 Capital Market Shock Effect on CDC-Plans In the steady state scenario we have a constant expected return ˆ () µσ µ = and no external disturbances, i.e. Xt = Yt = 0 and ˆ0 ρ = . Then ρ (t) = δ (t) = 0 and η (t+1) = µ for all t < T0. In the steady state original position for all ages x the cohort pension capital Px and the individual pension capital Px /L(x) coincide with the time value of future benefits minus contributions. Now consider a single interest rate shock at time T0 (i.e. 0 ˆT X σµ ∆ = ). Applying rule (ALM 2) we have ( 1) () ()t tt η µ δ θρ +=+ + for all t and η (T0) = µ . Therefore at time T0 neither the individual accounts v(x) nor the due pensions r(x) are affected. However, the total pension capital at time T0 falls to exp( µ ∆) P and ρ (T0) = µ ∆ and 0 ( ) ln( )T ee e µ µµ µ δ ∆∆ −− = +− . By Proposition 3 we know that 0 ( ) (1 )k Tk ρ θµ ∆ +=− , so for 0< θ < 2 ρ (t) converges to ˆ0 ρ = . In the special case θ =1 we get 0 ( 1) 0T ρ += and η (T0+1) = µ + µ ∆ + δ (T0) = ( ) ln 1 ( 1)ee µ µ µ ∆ ++ − . Note that ε (t) = η (t) - µ a . Due to the non-trivial stock effect δ (t) there is no simple formula for η (t). Therefore we just illustrate η (t) for the down-scenario ( µ ∆ = -0.2) for different levels of θ - cf. FIGURE 7. - 32 - FIGURE 7: Effect of a capital market down-shock ( µ ∆= - 0.2) at time T0 on the reserve ratio ρ (t) (top chart) and the profit participation η (t) (bottom chart) for alternative levels of θ . For θ = 0 the reserve ratio will remain at the level of ρ = -0.2 forever. This means that all future generations have to pay the bill: Due to the negative stock effect we have η = µ - δ = µ - ln( )ee e µ µµ µ ∆∆ −− +− = 2.04% instead of µ = 2.5%. For 0 < θ < 2 the reserve ratio returns to the steady state level. If we wanted to avoid a negative profit participation, we would have to choose θ ≤ 0.1 with the consequence that it takes about 7 years to halve the after-shock reserve gap of 20%. Our goal is to measure the intergenerational effects of a CDC-plan compared to an IDC-plan in a shock scenario. To this end for each (T0, x)-cohort we calculate TVCDC (T0, x), the time value of future benefits minus future contributions. Note that for IDC-plans the time value equals the cohort’s pension capital i.e. 0 ( ) ( ) for ( , ) exp( ) exp( ) ()()() for IDC x Lxvx x z TV T x P Lxrxax x z µµ ∆∆ <  = = ≥  . - 33 - We take 00 (): ( ,) ( ,) CDC IDC TV x TV T x TV T x∆= − as a measure of the intergenerational asset transfer for the (T0, x)-cohort, and ∆TV(x)/L(x) as the individual effect. Fig. 8 illustrates the intergenerational redistribution in the down-shock scenario. Let us consider the CDC-plan with θ = 0.2. Then the steady state pension capital is maximal for the (T0, z)-cohort – we get Pz = 75798. In the IDC-case the pension capital falls to exp( µ ∆) Pz = 62058. In the CDC-case the pension capital of the (T0, z)-cohort remains unchanged after the shock, but the time value of future pensions reduces to 64206. This means that the CDC-plan causes an intergenerational redistribution of ∆TV(z) = 64206 - 62059 = 2147 in favour of the (T0, z)-cohort. FIGURE 8: ∆TV(x) for age cohorts 0 ≤ x ≤ 115 for different levels of θ for a downshock scenario ( µ ∆= - 0.2). There is an additional (small) redistribution effect in favour of the death benefits payable at time T0 after the shock. While in the CDC-case the total death benefit is not affected at T0 , in the IDC-case the death benefit is reduced by factor exp( µ ∆). It is clear that the total effect over all generations (including future generations of new entrants) must be zero. If we look at the effects per capita we see that the positive or negative redistribution effects amounts to a multiple of the regular contribution (which is 1 in our calculations) – cf. Fig. 9. For example, in the case θ = 0.2 and µ ∆ = -0.2 each single member of the (T0, z)-cohort receives a transfer of ∆TV(z)/ L(65) = 2.39. In the extreme case θ = 0 the reserve will remain at the after shock level of -0.2 for ever so that all future generations will be charged. This extreme case again shows that a CDC-system could be misused by the generation 50+, who may have strong influence on ALM-decisions and who are prone to postpone unpleasant decisions. - 34 - FIGURE 9: Redistribution effect per head ( ()/ ()TV x L x∆ ) for different levels of θ for a down-shock scenario ( µ ∆= - 0.2). The following table shows the intergenerational redistribution for a single capital market down and up shock in relation to the pre-shock total pension capital. Note that for θ = 0 the reserve ratio will remain at the level at time directly after the shock. i.e ρ (t) = µ ∆= - 0.2 for all t ≥ T0. Capital Market Down Shock ( µ ∆ = -0.2) Capital Market Up Shock ( µ ∆ = +0.2) θ Redistribution to the older generation Beneficiary age cohorts Redistribution to the younger generation Beneficiary age cohorts 0 9.11% ≥46 11.86% ≤58 0.1 3.52% ≥59 4.67% ≤61 0.2 2.17% ≥62 2.86% ≤63 0.3 1.56% ≥64 2.06% ≤63 0.4 1.22% ≥64 1.62% ≤64 0.5 1.00% ≥65 1.33% ≤64 0.6 0.86% ≥65 1.13% ≤64 0.7 0.75% ≥65 0.99% ≤64 0.8 0.66% ≥65 0.87% ≤64 0.9 0.60% ≥65 0.79% ≤64 1.0 0.54% ≥65 0.71% ≤64 TABLE 1: Overall redistribution effect in % of total pre-shock pension capital in favour of the older generation (down-scenario) or younger generation (up-scenario) for different levels of θ . - 35 - 4.3 Capital Market Shift We now want to analyse the effect of an interest rate shift on a pure bond portfolio. We analyse a sudden but permanent interest rate shift from µ to µ ':= µ + µ shift from some time T0 onwards. We associate this stylised situation with a non-expected decision of the central bank to adjust interest rates.12 This interest rate shift has then two effects: Firstly, new fixed income investments bear an interest rate of µ ' instead of µ , and secondly, there is a price effect on existing bond investments. If the interest rate shift occurs at the beginning of the time period [T0, T0 +1] then the market value of a bond portfolio with an average duration of D will chance by factor f ≈ exp(-D µ shift) instantly after the shift. From time T0 onwards all assets including new investments will have a return of µ '. If D > 0 then the interest rate up/down shift results in a single down/up shock followed by a permanent up/down shift. We want to check how IDCand CDC-plans adapt to this permanent change of the capital market. In our wording we concentrate on a down shift scenario ( µ shift < 0). It is quiet obvious that, cum grano salis, in an up-shift scenario the same happens in the other direction. To keep the variants of our calculations in limits we do not adjust the actuarial interest rate µ a , so that the annuitisation factors ä(x) remain unchanged. If not stated otherwise our numerical examples are calculated on the basis of µ = 2.5%, µ shift = -1.0%, µ '= µ + µ shift = 1.5%, µ a = 1%, ε ' = µ ' - µ a = 0.5%. Furthermore, we consider the price effect due to the interest rate shift by assuming that the time value of assets change by factor fD := exp(-D⋅ µ shift) for D = 0, D = 5 and D = 10. 4.3.1 Capital Market Shift Effect on IDC-Plans From time T0 onwards the individual pension capital bears interest at the lower rate µ ' = 1.5%, pensions in payment are adjusted by ε ' = 0.5%. We illustrate the effect for the group of active members – cf. FIGURE 10 below. For example a person aged x = 20 or younger at time T0 will be affected most, because they experience the lower interest rates for the whole accumulation phase. Their pension capital at age z = 65 will be 64.75 instead of 84.25, that is about 77% of the preshift level. This is independent of the duration of the underlying assets. For older 12 Actually central banks can only determine the short term interest rates, long term interest rates can only be influenced indirectly. - 36 - members the positive duration effect for D > 0 can overcompensate the reduced future returns. However, after retirement the pensions are only adjusted by ε ' = 0.5% instead of ε = 1.5%. Pension in payment will experience a single increase by factor fD followed by reduced pension adjustments. For older pensioners the duration effect at time T0 might overcompensate the reduced adjustment rate. FIGURE 10: Down-shift scenario ( µ shift = -1%) for IDC-plans: Effect on the expected pension level at age z, depending on the age x at time T0 and the duration. Let TVIDC (x, T0) denote the time value of future benefits (pension and death benefit) minus contributions at time T0 immediately after the shift. If we want to calculate TVIDC (x, T0) market consistently it must be calculated on the basis of the shifted discount rate µ + µ shift. Then clearly TVIDC (T0, x) = fD Px , where Px denotes the pre-shift pension-capital for the x-cohort. 4.3.2 Capital Market Shift Effect on CDC-Plans We assume that the pension management instantly recognises the interest rate shift as permanent. According to (ALM 2) for t ≥ T0 ( ) ˆ ( 1) () ()t tt η µ δ θρ ρ ′ += + + − . Due to the interest rate shift, at time T0 the assets have to be revalued. As above, we assume that P':= fD P is the value of assets immediately after revaluation. Accordingly, after revaluation we have 0 ( ) ln shift P TD V ρµ ′  = = −   and - 37 - ( ) 0 ( ) ln ln ln 1 exp( )(1 ) D shift P CF P T fD V CF V δ µµ ′′ −    = − =− −+    −    . Here we used the fact that in the steady state situation CF = (1-exp(- µ )) P. Following (ALM 2) we get ()() 0 00 ˆ ( 1) ( ) ( ) ln 1 exp( )(1 ) (1 ) D shift T TT f D η µ δ θρ ρ µ µ θ µ ′′ + = + + − = + − − +− and 00 0 0 00 0 ( )exp( ) ( 1) () ln () ( )exp( ( 1)) ( 1) () (). P CF TT T V CF T TT T µ ρρ ρ η µ η δ θρ  ′′ − +− = −  −+  ′ = − ++ =− We could have derived this directly from (Eq. 11) of Prop. 3. More generally we get 0 ( ) (1 ) k shift Tk D ρ θµ + =−− . For D = 0 the reserve ratio is not affected at all. Due to the non-trivial stock effect for D > 0 there is no simple formula for η (t) for t > T0+1. So we just present numerical results – cf. FIGURE 11. For D > 0 we observe an increase of the reserve ratio at time T0. After T0 the reserve is drawn down depending on the speed parameter θ . - 44 - 1 ( 1) (1 ) ( ) t tY t ρ θρ + +=− +− , ( 1) () ()t tt η µ δ θρ +=+ + and ( 1) ( 1) a tt εηµ += +− . (Eq. 19) In particular 0 0 () T TY ρ = − and 0 ()T ηµ = . Different from the IDC-case, here all age cohorts are treated equally and the cohort of active workers becomes involved. The effect on the profit participation η (t) and the reserve ratio ρ (t) is more complex than in the case of a capital market shift, since the age profile of the population changes and it takes ω – x0 = 95 years until a new steady state population is reached - see FIGURE 15. FIGURE 15: Effect of a mortality down-shift (∆ =+0.5) on the reserve ratio ρ (t) (top chart) and profit participation η (t) (bottom chart) in a CDC-pension system for Strategy 1 for alternative levels of θ . - 45 - Since from time T0 onwards, the age burden γ (t) and the weights w(t, x) deviate from the steady state values, the Yt -process is not trivial. Clearly, for θ = 0 ρ (t) does not converge, neither does Yt. One may check that 0 0 0 ( ) (1 ) k j T kj j Tk Y ρθ +− = +=− − ∑ . So we can deduce that for 0 < θ < 2, ρ (t) converges provided Yt converges. The following table shows the new steady state values for ρ and η . E.g. for θ = 0.2 and ∆ = +0.5 the CDC-system will converge to a new steady state with a permanent negative reserve of ρ = -3.17%. The steady state profit participation ( η = 1.79%) falls behind the capital market return ( µ = 2.50%) because the mortality shift has to be financed year by year and furthermore due to the negative reserve we have a negative stock effect (in this case δ = -0.0788%). Δ = + 0.5 Δ = -0.5 θ η ρ TV (x0=20) η ρ TV (x0=20) 10% 1.72% -6.25% -3.8082 3.47% 7.66% 5.7557 20% 1.79% -3.17% -3.2344 3.35% 3.76% 4.3964 30% 1.81% -2.12% -3.0320 3.31% 2.49% 3.9811 40% 1.82% -1.59% -2.9286 3.29% 1.86% 3.7799 50% 1.83% -1.28% -2.8658 3.28% 1.49% 3.6611 60% 1.83% -1.06% -2.8237 3.27% 1.24% 3.5828 70% 1.84% -0.93% -2.7934 3.27% 1.08% 3.5272 80% 1.84% -0.80% -2.7706 3.26% 0.93% 3.4857 90% 1.84% -0.71% -2.7529 3.26% 0.82% 3.4535 100% 1.84% -0.64% -2.7386 3.26% 0.74% 3.4279 TABLE 4: Strategy 1: Profit participation ( η ), reserve ratio ( ρ ) and time value of future benefits minus contributions (TW) for new entrants (x0=20) in the adjusted steady state after a mortality shift of ∆ = +/- 0.5. We now turn to the question of to what extent a mortality shift induces a transfer of wealth between the age cohorts. To this end we first calculate the time value TVCDC (T0, x) of future benefits (including death benefits) minus future contributions for each (T0, x)-cohort immediately after the shift occurred. Then the difference TVCDC (T0, x) - TVIDC (T0, x) is a suitable figure to measure the intergenerational wealth transfer. We also calculate 00 0 ( ,) ( ,) ( ,) CDC IDC TV T x TV T x LT x − , the individual contribution (positive or negative) to the intergenerational transfer. Consider for example the (T0, z)-cohort. At time T0 we observe more survivors than expected, namely ( 1) () () ( 1) pz L z Lz pz ∆ ∆ − =− instead of L(z). The total pension capital - 46 - for this cohort is L∆(x) v(x) = 76059.39 for both, the IDC and the CDC-case. In the IDC-case all future pensions are paid from this capital stock. In the CDC-case the pensions are adjusted according to (Eq.19). For θ = 0.2 the time value of all pensions paid to the (T0, x)-cohort amounts to 83171.67. The difference 6706.07 is the intergenerational wealth transfer in favour of the (T0, x)-cohort, which comes to an individual transfer of 7.88. In other words, each single member receives a subsidy of about eight contribution rates. Let’s now look at the (T0, x0)-cohort. At time T0 the pension capital is zero. In the IDC-case all members of this cohort know that every Euro they pay into the system bears an interest rate of µ = 0.025 and will be paid back – at least on average. In the CDC-regime ( θ = 0.2) we get TVCDC (T0, x0) = -3 528.47 and an individual transfer of -3.53. This means that a new entrant has to realise that more than 3 of the future contribution rates are transferred to the old generation. The following FIGURE 16 illustrates the intergenerational transfer on cohort-level including cohorts of unborn. It is clear that the total sum taken over all existing and future generations must add up to zero. FIGURE 16: Intergenerational redistribution after a mortality down-shift (∆ = +0.5) in a CDC-pension system for Strategy 1 for age cohorts x ≥ -80 and for θ = 0/ 0.1/ 0.2/ 0.4/ 1. FIGURE 17 shows the transfer on individual level for x ≥ 0. Since the old age cohorts have fewer members the individual effect is more significant. - 47 - FIGURE 17: Intergenerational redistribution per head after a mortality down-shift (∆ = +0.5) in a CDC-pension system for Strategy 1 for ages x ≥ 0 and for θ =0.1/ 0.2/ 0.6/ 0.8. Strategy 2 (Instant Recognition) Instant recognition means that at time T0 the liabilities in the balance sheet are adjusted to comply with the new survival probabilities. But the benefits payable at T0 (pensions and death benefits) remain unchanged i.e. η (T0) = µ and ε (T0) = µ - µ a . Furthermore, we assume that the new pensions for the (T0, z)-cohort are calculated on the basis of ä(z). However from T0+1 onwards we apply äΔ(z). Let P(t) resp. V(t) denote the total of assets resp. liabilities at time t ≥ T0 . By our convention P(t) and V(t) include the death benefit payable in t for active workers who die in [t-1, t]. Thus we have P(T0) = P, the steady state value of assets. Let us denote by L´(x) the number survivors of the (T0-1, x-1)-cohort after the mortality shift. Proposition 5 ( ) 0 ( 1) ( ) ( ) () () () 1 () ()() ( 1) ( ) xz px ax VT V L z Lz vz Lxaxrx px ax ω ∆∆ =  − ′ =+− + −  −  ∑   (Eq. 20) Proof ( ) 0 1 0 ( ) ( 1)() () ( 1) () () ()() z xx xz VT Lx vx L z Lz vz L xa xrx ω − ∆ = = ′′ = − + −− + ∑∑  ( ) ( ) 0 1 ( 1)() () ( 1) () ()()() ( 1) () () () () () ()() ( 1) z xx xz xz Lx vx Lz Lz vz Lxaxrx px L z Lz vz a x ax Lxrx px ω ω − = = ∆∆ = = − + −− +  − ′ +− + −  −  ∑∑ ∑    - 48 - Since ( ) 0 1 ( 1)() () ( 1) () ()()() z xx xz V Lx vx Lz Lz vz Lxaxrx ω − = = = − + −− + ∑∑  we get (Eq. 20). ♦ Note that for ∆ > 0 ( ) () () () 0 L z Lz vz ′−< and ( 1) ( ) 10 ( 1) ( ) px ax px ax ∆∆  −−>  −    . If the mortality shift is recognised instantly, the effect on the reserve ratio and the profit participation strongly resembles the situation after a capital market down shock. We illustrate the effects in FIGURE 18 below. As in Figure 7 we see that for θ > 0 the reserve ratio will gradually return to the steady state level ˆ0 ρ = . FIGURE 18: Effect of a mortality down-shift (∆ =+0.5) on the reserve ratio ρ (t) (top chart) and profit participation η (t) (bottom chart) in a CDC-pension system for Strategy 2 (instant recognition) for alternative levels of θ . As for Strategy 1 we measure the intergenerational wealth transfer by comparing the time value of future benefits minus contributions for the (T0, x)-cohorts. The instant recognition of the mortality shift (∆ = +0.5) has a mild effect on the (T0, x)-cohorts - 49 - for x ≥ z since their pensions are only indirectly affected via reduced ε (t). However those who enter retirement at T0+1 or later have to endure a double impact: firstly the profit participation and future pension increases will go down to refill the reserve and secondly, their initial pensions are calculated on the basis of the shifted mortality. This is illustrated in FIGURE 19 and 20 below. We notice a sharp cut at age x = 65 which is a result of the fact that due to the instant recognition of the mortality shift, from time T0 onwards all new pensions are calculated on the basis of the shifted mortality rates. We see that the redistributional effect of a mortality shift differs clearly from that of a capital market shockcompare Figure 19/ 20 and Figure 8/ 9. FIGURE 19: Intergenerational redistribution after a mortality down-shift (∆ = +0.5) in a CDC-pension system for Strategy 2 for age cohorts x ≥ 0 and for θ =0.1/ 0.2/ 0.6/ 0.8. FIGURE 20: Intergenerational redistribution per head after a mortality down-shift (∆ = +0.5) in a CDC-pension system for Strategy 2 for ages x ≥ 0 and for θ =0.1/ 0.2/ 0.6/ 0.8. - 50 - Comparison of Strategy 1/ 2 (delayed/ instant recognition) The avoidance of cutting pensions in payment seems to be a touchstone of a pension plan. Accordingly, the managers of a pension plan will be very reluctant to actually cut pensions. If we look at the effect of Strategy 1 or 2 on the pensions in payment (cf. FIGURE 21) then it is clear that the “procrastination policy” (Strategy 1) is very attractive. We know from the analysis above that Strategy 1 shifts the burden of longer life expectance to future generations, who inherit an eternal loan from the old. FIGURE 21: Pension level for pensions in payment after a mortality down-shift (∆ = +0.5) for Strategy 1 and 2 for different levels of θ . 100% marks the pre-shift pension level. Both strategies imply a massive wealth transfer between the generations. TABLE 5 below shows the wealth transfer in favour of the older generations as a proportion of the steady state total pension capital (= 2543840). Strategy 1 turns out to produce a stronger transfer than Strategy 2. If we compare the age cohorts that profit from the transfer we see that Strategy 1 is attractive for active employees aged 54 and over. One may guess that for many pension plans these age cohorts are dominant in the representative bodies, so one might expect that in real life there will be a strong tendency to postpone the updating of the mortality tables. - 51 - Mortality Shift (∆ = +0.5) Strategy 1 (delay recognition) Strategy 2 (instant recognition) θ Redistribution in % of total pension assets burdened age cohorts Redistribution in % of total pension assets burdened age cohorts 0 13.78% x≤40 8.88% x≤64 0.1 10.12% x≤50 6.97% x≤64 0.2 9.22% x≤52 6.31% x≤64 0.3 8.81% x≤53 6.00% x≤64 0.4 8.58% x≤53 5.82% x≤64 0.5 8.44% x≤54 5.71% x≤64 0.6 8.34% x≤54 5.63% x≤64 0.7 8.27% x≤54 5.57% x≤64 0.8 8.21% x≤54 5.53% x≤64 0.9 8.17% x≤54 5.49% x≤64 1.0 8.13% x≤54 5.46% x≤64 TABLE 5: Overall redistribution effect from young to old of a mortality shift (∆ = +0.5) for Strategy 1 and Strategy 2 in % of total pre-shift pension capital. Neither Strategy 1 nor 2 should be the choice in practice! There are good arguments to apply a mixed strategy by adjusting mortality rates step by step. 5 Concluding Remarks The primary purpose of collective DC-plans is smooth away the ups and downs of capital market returns, which are particularly volatile for stock markets. If there is no external institution to step in if equities slump, the smoothing can only be done by some kind of intergenerational risk transfer. Intergenerational risk transfer is going on since decades but in general unilaterally at the cost of the younger generation. The shift from DBto DC-plans is only one example. So the challenge is to find rules that allow for a fair risk transfer between age cohorts. Our proposal for such rules is guided by the concept of resilience. We apply these rules to several capital market shock and shift scenarios and to mortality shift scenarios. We measure the intergenerational effects; so we have instrument to measure intergenerational equity. 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