Collectivised Post-Retirement Investment
CCollectivised Post-Retirement Investment
John Armstrong, Cristin Buescu
Abstract
We quantify the benefit of collectivised investment funds, in whichthe assets of members who die are shared among the survivors. Forour model, with realistic parameter choices, an annuity or individualfund requires approximately 20% more initial capital to provide asgood an outcome as a collectivised investment fund. We demonstratethe importance of the new concept of pension adequacy in defininginvestor preferences and determining optimal fund management. Weshow how to manage heterogeneous funds of investors with diverseneeds. Our framework can be applied to existing pension products,such as Collective Defined Contribution schemes.
We study investment funds where all investors agree that any funds left whena member dies should be shared among the survivors. We call this new fi-nancial product a collectivised investment fund. An investor may choose toinvest in such a collectivised fund at retirement instead of purchasing an an-nuity or managing an individual fund. We will argue that such funds shouldbe particularly attractive to investors and will quantify the benefit of such afund. While various types of risk-sharing funds have been proposed before,our proposal differs by providing a clear framework to describe the optimalmanagement of the fund, and by describing how risk should be shared.While collectivised investment funds can be viewed either as an interest-ing financial product in their own right, they also provide a model for thepost retirement phase of a Collective Defined Contribution (CDC) pensionscheme. Such schemes are still very new and the precise manner in whichCDC funds are managed varies from scheme to scheme. Broadly speaking aCDC fund is one which is managed for the benefit of a group of individualsand endeavours to obtain a good pension for all its members, but which doesnot promise a precisely defined pension. Examples of such schemes include1 a r X i v : . [ q -f i n . P M ] A p r he New Brunswick Hospitals’ plan in Canada [34], a number of Dutch pen-sion schemes [11, 35] and the planned new pension scheme for Royal Mail inthe UK [39].CDC funds have emerged for two reasons. Firstly, low yields and tighterregulation have made traditional employer pension schemes unattractive.Secondly, separate legislative changes, such as the Pension Schemes Act 2015in the UK, allow much greater flexibility in how pensions can be invested.Focusing on the UK as an example, pension funds have historically beeneither defined-benefit (DB) funds or defined-contribution (DC) funds. In a DBfund an employer promises to pay their employees and their partners a pre-specified income from retirement until death. Often these benefits would beindex linked, i.e. they would provide a constant real-terms income. In a DCfund an employee has a personal fund of pension investments. Historicallythe assets in a DC fund were used to purchase an annuity at retirement,i.e. a financial product that would pay a pre-specified income to the retireeand their partner until death (typically a constant real-terms income). Asa result of the 2015 pension reform, it is now possible to receive the taxbenefits afforded to pension investment without being restricted to such anarrow range of investments. The legislation introduces a new framework of“defined ambition” schemes into which CDC schemes fit.In this paper, we will study the benefits of collectivised pensions to em-ployees. We note, however, that CDC funds are also very attractive to em-ployers as they remove the liabilities inherent in a DB scheme from theirbalance sheet, but we will not attempt to quantify this. We note also thatthere will be legal and taxation considerations that one should take into ac-count in order to develop collective funds as a new financial product, butthese are jurisdiction specific and beyond the scope of this paper.There are a number of reasons why one would expect a collectivised in-vestment fund to yield better pension outcomes than an annuity, and for aCDC fund should outperform a DB fund.1. For large funds, ignoring systematic longevity risk, one can assume thatan employer’s DB liabilities depend only on interest rates. Assuming atypical risk-neutral pricing model for interest rate products, this meansthat a fully-funded DB scheme will not use equity investments, and socannot benefit from the equity risk-premium.2. A constant real-terms income does not benefit from the possibility of2 ntertemporal substitution . This is the observation that if one is willingto delay consumption in favour of investing for longer, one may beable to obtain a higher rate of consumption in the future leading to apreferable outcome.3. Due to changes in the level of the state pension, the optimal determin-istic real-terms income will change over time.We will also argue that there is an additional, less obvious, reason why con-stant consumption is suboptimal.4. A constant real-terms income ignores the risk of dying young and notenjoying any consumption. It also ignores the risk of living on aninadequate pension for many years.A collectivised investment fund should also outperform an individualfund, and a CDC fund should also outperform a DC fund. This is becausecollectivisation should reduce idiosyncratic longevity risk.The primary research questions that this paper seeks to answer are: (i)How significant are these various effects are in practice? (ii) How should afund be managed to best exploit these effects?In order to highlight the effects of collectivisation, we will answer theseresearch questions in the context of the simplest market and longevity modelthat is capable of modelling all these effects. Our market model will bea Black–Scholes–Merton model, and we will assume that future mortalitydistributions are known so that there is no systematic longevity risk. Forthe same reasons, we consider only the post-retirement investment and con-sumption and we do not model any form of bequests. Pension schemes areoften intended to provide for a couple in retirement. We model this as twoindividuals purchasing independent shares in a collectivised fund.A first contribution of this paper is to compute detailed numerical exam-ples of pension outcomes based on reasonable market assumptions in orderto quantify the benefits of collectivisation. We find that either an annuityor an individual fund requires 20% more initial capital to be as effective asa collectivised fund.A second contribution of this paper is an understanding of how to modelinvestor preferences over pension outcomes. The model we choose for in-vestor preferences will play a critical role in our theory. We will argue that3he classical model for preferences, intertemporally additive von Neumann–Morgernstern utility, (used, for example, in [28]) are essentially risk-neutral.Conversely annuities appear to be designed for the infinitely risk-averse. Wefind that using preferences which allow for a moderate level of risk-aversionleads to much more plausible results, which we believe will be more appeal-ing to investors. We first study preferences from a theoretical point of viewbuilding on works such as [29], [27], [25] and [18] but incorporating mortality.We then compare the possible preference models numerically. Together ourtheoretical and numerical results leads to a clear choice of the most appropri-ate model for our problem. We will see that it is essential to use a preferencemodel which allows for flexible specification of what we call the adequacylevel . Our results also highlight the importance of risk-aversion, leading tothe somewhat unexpected phenomenon outlined in point (4) above.A third contribution of this paper is an algorithm for managing heteroge-neous funds, i.e. funds containing individuals with diverse preferences, wealthand mortality distributions. This is not a standard optimal control problem,as it is not clear how to define an objetive for the fund as a whole. Never-theless we showed in [3] that it is possible to bound the utility that can beobtained by a collective fund. In the example tested numerically in this pa-per, our proposed algorithm achieves 98% of the maximum potential benefitof collectivisation for a heterogeneous fund of only 100 members.Together with the technical results of [3], these last two contributions pro-vide a rigorous framework for understanding collective pension investment.The potential advantages of such pension products has been observed before,leading to the development of with-profits annuities. However, without a rig-orous mathematical underpinning, it has been unclear how to manage suchproducts in the investor’s best interests and as a result it has been unclearhow best to define contracts to guarantee that this is done [38]. We believeour framework is capable of remedying these issues.While we have studied only post-retirement investment, it follows fromour results that a CDC fund (managed optimally) will outperform a DB fundfrom the point of view of the pensioner. To see this note that if one had aDB fund giving a guaranteed income, a collective could sell this guaranteedincome stream and then use the proceeds to pursue the optimal investmentstrategy of this paper. We emphasize that this argument only applies toan optimally managed CDC fund. Existing CDC funds are designed andmanaged using a variety of heuristics rather than by solving an optimalinvestment problem. Thus this paper has significant implications for the4anagement of such funds.Having described our findings, let us now describe the structure of thepaper.In Section 1 we introduce the topic of preferences over pension outcomes,before proposing two concrete models for preferences in Sections 1.2 and 1.3.In Section 2 we select concrete market, preference and longevity models cali-brated to the UK pension market. In Section 2.1 we use this to quantify thebenefits of collectivisation for infinite funds of identical investors using ourpreferred preference model. In Section 2.3 we examine how our results changeif we use a different preference model, and use this to identify the most ap-propriate choice to use in practice. In Section 3 we drop the assumption thatthe fund contains infinitely many identical investors and give an algorithmfor managing heterogeneous funds. We test the efficacy of this algorithm.We end in Section 4 with a summary of the financial consequences. We must choose a model of investor preferences. The choice of preferenceswill determine the meaning of optimal fund management, and so this choiceplays a central role in this paper. We should, therefore, justify our preferencemodel in some detail.If one ignores mortality for the moment, in the current literature, ho-mogeneous Epstein–Zin preferences seem to be the most popular model forpreferences over consumption streams. These preferences were introducedby [18] and have been successfully applied to provide potential resolutions tovarious asset pricing puzzles [6, 5, 7, 8]. These preferences are homogeneous in the sense that if one prefers income γ to γ then one will prefer λγ to λγ for all positive reals, λ . Homogeneity is a symmetry which ultimately resultsin a dimension reduction of the Hamilton Jacobi Bellman (HJB) equationyielding a more tractable model. Given the success of this preference model,it is very natural to incorporate mortality into homogeneous Epstein–Zinpreferences, and it is easy to see how to do this in a manner that preservesthe homogeneity property. This approach has already been taken in a num-ber of papers such as [19, 20, 10, 16] which deal with mortality directly, aswell as mathematical works such as [4] which discusses random horizons ingeneral. 5owever, we will argue that for pension investment problems, homogene-ity of preferences is undesirable. We will define a notion of pension adequacy for an individual’s preferences. Any finite, non-zero choice of adequacy levelwill automatically break the homogeneity of the preferences. We will laterconfirm numerically that considering finite, non-zero adequacy levels leads toquite different numerical results to those obtained with homogeneous prefer-ences.In Section 1.1 we will establish the necessary technical vocabulary for atheoretical comparison by defining the notion of pension outcomes and ofpreferences over such outcomes. We define a number of desiderata for apreference model and define pension adequacy.In Section 1.2 we identify a preference model from first principles whichwe call exponential Kihlstrom–Mirman preferences with mortality , as it in-corporates mortality into the preference models of [25, 24]. We will see thatthis model meets all our desiderata.In Section 1.3 we present the alternative approach of homogeneous Epstein–Zin preferences with mortality. We explain how this model could be modifiedto incorporate a flexible model for pension adequacy, but at the cost of ho-mogeneity. We model a “pension outcome” as a pair ( γ, τ ) consisting of a stochasticprocess γ t , representing the rate at which payments are received at time t ,and a random variable τ representing the time of death. The underlyingfiltered probability space will be denoted by (Ω , F , F t , P ). The units of γ t should be taken to be in real terms to ensure that our models for inflationand preferences are separate.We consider both discrete and continuous cashflow processes γ t . We write T for the set of time indices which may be either [0 , T ) or the evenly spacedtime grid { , δt, δt, δt, . . . , T − δt } where T is an upper bound on anindividual’s possible age which may be infinite. We write d T ( t ) for themeasure determined by the index set: this will be the Lebesgue measureon [0 , ∞ ) in the continuous case, or the sum of Dirac masses of mass δt ateach point in T for the discrete case. It will occasionally be convenient toallow the cashflow γ t to be non-zero when t > τ , but this cash will not beconsumed. In the discrete case we assume that cashflow at the moment ofdeath γ τ is still consumed. So the total consumption over the lifetime of an6ndividual is (cid:90) τ γ t d T ( t ) . We wish to describe an individual’s preferences over pension outcomes.This will be represented by an ordering (cid:22) on the set of pairs ( γ, τ ). Theoutcome ( γ, τ ) is considered preferable to the outcome (˜ γ, ˜ τ ) if (˜ γ, ˜ τ ) (cid:22) ( γ, τ ).We define (cid:23) in the obvious way and write x ∼ y if x (cid:22) y and y (cid:22) x . Weassume that an individual is indifferent to cashflows after death. This canbe expressed mathematically as( ∀ t ≤ τ γ t = ˜ γ t ) = ⇒ ( γ t , τ ) ∼ (˜ γ t , τ ) . We will now define various properties that a preference relation may pos-sess and which may be considered desirable.
Definition 1.1.
The preferences are monotonic if ( γ, τ ) (cid:22) ( γ (cid:48) , τ ) if γ t ≤ γ (cid:48) t for all t ∈ T .This simply reflects preference for consuming more.We would like the preferences to depend only on the probabilistic prop-erties of γ and τ and not on any extraneous data. We will formalize thisrequirement as the concept of invariance . To define this, we first recall thata mod 0 isomorphism is a measure preserving bijection from a full subsetof a probability space Ω onto a full subset of another probability space Ω (cid:48) with measurable inverse. An automorphism of a filtered probability space isa mod 0 isomorphism of probability spaces that acts as a mod 0 isomorphismon each element of the filtration. Definition 1.2.
The preferences (cid:22) are invariant if for any automorphism, φ , of the filtered probability space (Ω , F t , P ) we have that φ preserves (cid:22) . Our framework is very similar to the descriptive model presented in Sec-tion 4 of [27], but differs in that we consider preferences over random variablesrather than preferences over distributions of random variables. Requiring in-variance acts as a substitute for defining the preferences over distributions.We will not repeat the axioms of [27], but we note that the specific preferencemodels we will ultimately use in this paper will also satisfy their axioms.
Definition 1.3.
The preferences (cid:22) are law-invariant if they depend only onthe joint distribution of ( γ t , τ ) .
7f preferences depend upon the time at which information becomes avail-able, they may be invariant but not law-invariant. [27] developed their theoryto allow the study of preference relations which depend upon the timing ofthe resolution of uncertainty and [18] provides some discussion of when thismay be desirable. However, for normative pension investment with no exoge-nous investment opportunities, law-invariance seems a desirable feature as itis hard to justify why one might be willing to pay to receive (or not receive)information which one cannot act upon.
Definition 1.4.
The preferences (cid:22) are convex if for any γ and τ the set { ˜ γ | ( γ, τ ) (cid:22) (˜ γ, τ ) } is convex.Convex preferences are mathematically desirable as one may then ap-ply the tools of convex analysis. As we shall see, convex preferences arisenaturally as a consequence of the concepts of satiation and risk-aversion.Given cashflows γ t defined on an interval t ∈ [ a, b ) and cashflows ˜ γ t definedon an interval t ∈ [ b, c ) we define the concatenated cashflow on [ a, c ) by( γ ⊕ ˜ γ ) t = [ a,b ) ( t ) γ t + [ b,c ) ( t ) ˜ γ t . Definition 1.5.
The preferences (cid:22) are
Markovian if for any cashflows γ α,t , γ β,t defined on the finite interval [0 , a ) with a of measure zero (i.e. a is not agrid point in the discrete case) and any cashflows γ ,t , γ ,t defined on [ a, ∞ )( γ α ⊕ γ , τ ) (cid:22) ( γ α ⊕ γ , τ ) ⇐⇒ ( γ β ⊕ γ , τ ) (cid:22) ( γ β ⊕ γ , τ )This definition captures the case when future preferences do not dependupon the past. There is no logical reason to insist that preferences shouldbehave in this way: for example if one has purchased a house, the anticipatedcost of housing repairs might well affect one’s future preferences.Markovian preferences are desirable mathematically because they resultin more tractable problems: if one has non-Markovian preferences then onemust keep track of additional state variables when solving optimal controlproblems and this increases the dimension of the HJB equation. Markovianpreferences are desirable from the point of view of parsimony as one need notchoose an initial state. 8 efinition 1.6. The preferences (cid:22) are stationary if for all a there exists anisomorphism of filtered probability spaces φ : (Ω , F , ( F t ) t ≥ a , P ) → (Ω , F a , P ) × (Ω , F , ( F t ) t ≥ , P )such that for any cashflows γ α,t defined on a finite interval [0 , a ) with a ofmeasure 0, any cashflows γ ,t , γ ,t defined on [0 , ∞ )( γ α ⊕ ( γ ◦ φ ) , τ ◦ φ + a ) (cid:22) ( γ α ⊕ ( γ ◦ φ ) , τ ◦ φ + a ) ⇐⇒ ( γ , τ ) (cid:22) ( γ , τ )This definition captures the case when preferences over future cashflowsremain constant in time. The isomorphism φ is required in order to definepreferences at future times in terms of preferences at time 0. Stationarityimplies Markovianity. Stationary preferences are particularly parsimoniousas one does not have to justify how the preferences vary in time. Stationarypreferences are very attractive in infinite-horizon problems as they lead to atime-symmetry of the HJB equation, which then allows the dimension to bereduced.Our notion of stationary preferences corresponds to “stationarity of pref-erence” in [26] (we say “corresponds to” because our set-up is slightly dif-ferent). It is related to the concept called “intertemporal consistency ofpreference” ([27, 23]) and “recursive preferences” ([18]). However, as wehave not specified preferences at future times, we have no need for an axiomof intertemporal consistency. The preferences at future times are describedimplicitly by preferences at time 0 and the requirement of temporal consis-tency (as explained by [27]). Stationarity then requires that these implicitpreferences at future times are isomorphic to the preferences at time 0. Definition 1.7. An adequacy level for preferences (cid:22) is a random process suchthat one is indifferent between dying at a particular time and living longerwhile receiving an income at the adequacy level. Formally, an F t -adapted,process a t is an adequacy level for the preferences (cid:22) if1. τ < ˜ τ ;2. ∀ t ∈ [0 , τ ] : γ t = ˜ γ t ;3. and ∀ t ∈ ( τ, ˜ τ ] : ˜ γ t = a t .together imply ( γ, τ ) ∼ (˜ γ, ˜ τ ). If death is better than any finite cashflowswe will say that the adequacy level is ∞ . If death is worse than any finitecashflows we will say that the adequacy level is −∞ .9 efinition 1.8. Inter-temporally additive von Neumann–Morgernstern pref-erences with mortality are determined by a choice of concave, increasing util-ity function u : R → R and a discount rate b . The preferences for ( u, b ) onpension outcomes with non-negative cashflows are( γ, τ ) (cid:22) (˜ γ, ˜ τ ) ⇐⇒ E (cid:18)(cid:90) τ e − bt u ( γ t )d T ( t ) (cid:19) ≤ E (cid:18)(cid:90) ˜ τ e − bt u (˜ γ t )d T ( t ) (cid:19) . (1.1)This definition is based on [29]. These preferences are montonic, convex,invariant, law-invariant, Markovian and stationary with an adequacy level of u − (0). In control problems where one cannot control mortality, the adequacylevel of these preferences is unimportant. To see why, observe that E (cid:18)(cid:90) τ ( u ( γ t ) + c ) d T ( t ) (cid:19) = E (cid:18)(cid:90) τ u ( γ t ) d T ( t ) (cid:19) + E (cid:18)(cid:90) τ c d T ( t ) (cid:19) . The term on the right is independent of the cashflows γ and so the pref-erences are unchanged when one adds a constant c to the utility function u . However, the adequacy level would become important in problems where τ could be controlled, for example, in a problem where one may choose toincrease health-care expenditure to increase life-expectancy.Although inter-temporally additive von Neumann–Morgenstern prefer-ences have many attractive properties, we will argue in the next section thatthey fail to adequately model risk-aversion. In models which include risk-aversion, we will find that the adequacy level plays a role even if one cannotcontrol mortality. We will now see how a number of simple considerations lead to a particularform of preference model which we call exponential Kihlstrom–Mirman pref-erences. We will suppose that there is an upper bound T on the durationof an individual’s life, so only τ ≤ T are considered admissible and we shallwork with continuous time consumption.Let us consider an individual’s preferences over deterministic outcomes( γ, τ ). We will assume the individual is order indifferent , which we defineto mean that their preferences depend only on the distribution function of γ F γ ( x ) = (cid:90) T γ t ≤ x d T ( t ) . Let us also suppose that their is a fixed, deterministic adequacy level a . Letus write a for a constant income stream at the adequacy level. We may outany deterministic consumption stream ( γ, τ ) to the right at the adequacylevel to obtain an equally preferable outcome ( γ ⊕ a , T ). So an individual’spreferences over deterministic consumption streams are then determined bytheir preferences over the distribution functions of γ ⊕ a .Preferences over distribution functions were studied by [29] who showedthat under modest axioms, preferences over distribution functions are deter-mined by an expected utility. Although they had probability distributions inmind, rather than temporal distributions, the two problems are mathemati-cally identical.Rather than duplicate the axioms of von Neumann and Morgernstern, wepropose a single axiom which captures their results together with the notionsof order indifference and a deterministic adequacy level. Axiom P. references between deterministic outcomes ( γ, τ ) and (˜ γ, ˜ τ ) aredescribed by a utility function u : R ∪ { a } → R with ( γ, T ) (cid:22) (˜ γ, T ) ⇐⇒ s ( γ t , τ ) ≤ s (˜ γ t , ˜ τ ) where s t := (cid:90) τ u ( γ t ) d T ( t ) . We call s t the satisfaction associated with (˜ γ, ˜ τ ) . Note that u is only deter-mined up to scale. Axiom A. n individual’s preference over pension outcomes ( γ, τ ) are givenby a von Neumann–Morgernstern preference relation over satisfaction. This discussion leads to the following definition.
Definition 1.9.
Kihlstrom–Mirman preferences with mortality are deter-mined by a choice of concave, increasing utility function u : R → R , asecond increasing function w : R → R and a discount rate b . The preferencesfor ( u, w, b ) on pension outcomes are( γ, τ ) (cid:22) (˜ γ, ˜ τ ) ⇐⇒ E (cid:18) w (cid:18)(cid:90) τ e − bt u ( γ t ) d T ( t ) (cid:19)(cid:19) ≤ E (cid:18) w (cid:18)(cid:90) ˜ τ e − bt u (˜ γ t ) d T ( t ) (cid:19)(cid:19) . on Neumann-Morgernstern preferences with mortality arise in the specialcase w ( x ) = x . We will call the case w ( x ) = − e − x and b = 0 exponentialKihlstrom–Mirman preferences .If one replaces τ with a deterministic time T one obtains the preferenceswithout mortality of [25]. We see that if an individual’s preferences satisfyAxioms P and A then their preferences must be Kihlstrom–Mirman prefer-ences with mortality and the discount rate b must equal 0.We believe that the most important assumption we have made is orderindifference. This is an important assumption for our normative pensionsmodel. We make this assumption because we believe that an individual’spension in old age should be given equal weight to their pension at retirement.This assumption is controversial and so merits further discussion. Thereare a number of reasons why one might include discounting in a preferencemodel. Firstly, one might use discounting as a proxy for directly modellingmortality. This idea is justified in [1] where it is shown for a specific modelthat the force of mortality and the discount factor play mathematically equiv-alent roles. However, our model includes mortality endogenously. Secondly,in a descriptive model, one might use discounting to model an irrationalbias towards early consumption. However, our model is normative. Third,one might use discounting to represent exogenous investment opportunities.However, we seek to model the entire market endogenously.As well as assuming that equal weight is given to all ages, the assumptionof order indifference requires that the utility function and the adequacy levelremain constant over time. We believe this is reasonable if one asks what forma preferences should take in a model that consciously chooses to ignore anyfeatures of pension outcomes other than mortality and cashflows. However,it may be beneficial to relax these requirements to allow for more flexiblemodelling. For example, we will allow the adequacy level to change overtime in our numerical work in order to model a non-constant deterministicstate pension.We will say that Kihlstrom–Mirman preferences are monotone if both u and w are monotone increasing, so that preferences are increasing as a func-tion of satisfaction, and satisfaction is increasing as a function of consump-tion. We will say that they model satiation if u is concave, in which case therewill be diminishing returns at higher levels of consumption as u (cid:48) ( c ) ≤ u (cid:48) ( c )if c < c . The term “satiation” is non-standard with most authors prefer-ring to talk in terms of intertemporal substitutability (e.g. [25, 18, 17, 40]).12e prefer the term satiation partly because it is more intuitive and easier tosay. It also refers to the preferences themselves: by contrast intertemporalsubstitution refers to the resulting behaviour when interest rates are changedand so incorporates the market model into the terminology for preferences.We will say the preferences are satisfaction-risk-averse if w is concave.This is the assumption that we would prefer to receive the satisfaction E ( s )with certainty than to receive a random satisfaction s . Since Axiom P pre-supposes that satisfaction, being an integral, has additive properties, it isreasonable to take expectations of satisfaction. This is important becausethe concept of risk-aversion is not topologically invariant and depends uponthe additive structure of R .There is an alternative additive structure one could consider, namely thestructure defined by the additivity of cash values and this gives rise to analternative concept of risk-aversion. Given preferences satisfying our axioms,we may define the constant cash equivalent of a deterministic cashflow ( γ, τ )by c ( γ, τ ) = u − (cid:18) T (cid:90) τ u ( γ t ) d T ( t ) (cid:19) . We may then write our preferences over non-deterministic cashflows as( γ, τ ) (cid:22) ( γ (cid:48) , τ (cid:48) ) ⇐⇒ E ( w ( T u ( c ( γ, τ ))) ≤ E ( w ( T u ( c ( γ (cid:48) , τ (cid:48) ))) . This leads to the definition that these preferences are monetary-risk-averse if the function x → w ( T u ( x )) is concave.We see that Kihlstrom–Mirman preferences successfully separate an in-dividual’s satiation preferences and an individual’s risk preferences.If one agrees that satisfaction-risk-aversion is the correct operational-ization of the intuitive concept of risk aversion, one is lead to the con-clusion that inter-temporally additive von Neumann–Morgernstern prefer-ences do not model risk aversion at all. This is a rather stronger state-ment than the more familiar observation that inter-temporally additive vonNeumann–Morgernstern preferences fail to disentangle risk aversion and sa-tiation [17, 40].If one insists on Markovianity, then, as was observed in [18], one mayidentify the function w . Lemma 1.10.
Kihlstrom–Mirman preferences with mortality in continuoustime are Markovian if and only if w takes the either the form w ( x ) = c exp( c x ) + c or w ( x ) = c x + c or some constants c , c , c ∈ R for x ∈ U defined by on the set U defined by U = ( M inf u, M sup u ) , M = (cid:90) T e − bt d T ( t ) They are stationary only if one additionally has b = 0 .Proof. The function w in Kihlstrom–Mirman preferences is determined bythe preferences up to positive affine transformation. Hence the preferenceswill be Markovian if and only if for any admissible γ α,t and γ β,t defined on[0 , a ) and γ t defined on [ a, ∞ ) we can find A > B such that E (cid:18) w (cid:18)(cid:90) a e − bt u ( γ α,t )d T ( t ) + (cid:90) ∞ a e − bt u ( γ t ) d T ( t ) (cid:19)(cid:19) = A E (cid:18) w (cid:18)(cid:90) a e − bt γ β,t d T ( t ) + (cid:90) ∞ a e − bt u ( γ t ) d T ( t ) (cid:19)(cid:19) + B. (1.2)The “if” statement for Markovian preferences is now clear.To prove the “only if” part of the same statement, we may assume withoutloss of generality that 0 ∈ U since a shift in the definition of u can beaccommodated by the choice of constants. Let us choose deterministic γ α,t , γ β,t and γ t and introduce variables x , (cid:15) and yx + (cid:15) := (cid:90) a e − bt u ( γ α,t ) d T ( t ) ,x := (cid:90) a e − bt u ( γ β,t ) d T ( t ) ,y := (cid:90) Ta e − bt u ( γ t ) d T ( t ) . We may then rewrite equation (1.2) as w ( x + (cid:15) + y ) = Aw ( x + y ) + B. The constants A and B may depend upon x and (cid:15) , but they are independentof y . Taking x as a fixed point in U , our assumption on u allows us to choose (cid:15) to be arbitrarily small and to choose y arbitrarily in some interval I around0. Taking y = ( n − (cid:15) we find w ( x + n(cid:15) ) = Aw ( x + ( n − (cid:15) ) + B. n such that n(cid:15) ∈ I , we have w ( x + n(cid:15) ) = (cid:40) A n w ( x ) + − A n − A w ( x ) B A (cid:54) = 1 w ( x ) + nB A = 1 . This gives the result on the grid of points in I starting at x separated by adistance (cid:15) . The result for points of the form x + q ∈ I for rational Q followsby refining the grid and the general case of x ∈ U is now clear.We now specialise to the case where w ( x ) = − exp( − x ). E (cid:18) w (cid:18)(cid:90) a e − bt γ α,t d T ( t ) + (cid:90) ∞ a e − bt γ t d T ( t ) (cid:19)(cid:19) = A E (cid:18) w (cid:18)(cid:90) ∞ e − b ( t − a ) γ t − a d T ( t ) (cid:19)(cid:19) . The preferences will be stationary if and only if this is equal to some affinetransformation applied to E (cid:18) w (cid:18)(cid:90) ∞ e − bt γ t − a d T ( t ) (cid:19)(cid:19) . The result for stationary preferences follows.Let us summarize the properties of exponential Kihlstrom–Mirman pref-erences with mortality.
Lemma 1.11.
Exponential Kihlstrom–Mirman preferences with mortalityare the only continuous time preferences with mortality which1. satisfy Axioms P to A2. are strictly risk-averse,3. and are Markovian.If u is monotone increasing and concave, then exponential preferences arealso monotonic, invariant, law-invariant, stationary and convex. Kihlstrom–Mirman preferences are not the most popular choice to modelpreferences in economic literature. Many authors prefer to consider
Epstein–Zin preferences , which we will describe in this section.15he theoretical observation that makes Epstein–Zin preferences morepopular than Kihlstrom–Mirman preferences is that Lemma 1.10 shows Kihlstrom–Mirman preferences are not stationary if one incorporates discounting. Forexample, in [18] it is remarked:Finally, note that if indifference to timing [of information ar-rival] and the intertemporal consistency of preferences are bothassumed, then ([14]) an expected utility ordering is implied.Although they do not emphasize discounting here, elsewhere in their pa-per, Epstein and Zin do make their implicit assumption on b clear. In ourterminology, [14] show that discounting, law-invariance and stationarity areincompatible.Our focus in this paper is on normative pension investment, but Epsteinand Zin’s focus is wider. For example, they remark that when choosing apreference model, ultimately one should “let the data speak” suggesting theirpriorities are descriptive.Since discounting is crucial in many economic models, this motivatedEpstein and Zin to propose dropping law-invariance and so consider modelswhere the time at which information is received is important. [18] describesa general theory of such stationary preferences, extending the work of [27] tothe infinite time setting. In this regard, their work extends the homogeneouspreferences proposed by [13] and [15]. As we shall see, the resulting theoryseparates satiation and risk in a very similar manner to Kihlstrom–Mirmanpreferences.Although these considerations are important, a second reason for thepopularity of Epstein–Zin preferences is that they incorporate homogeneouspreferences. In this context, Lemma 1.10 tells us that Kihlstrom–Mirmanpreferences cannot simultaneously have the symmetries of homogeneity (cor-responding to using a power function for w ) and stationarity. This provides amotivation for considering homogeneous Epstein–Zin preferences even if onebelieves that discounting is not required for the problems we are considering.The general form of Epstein–Zin preferences for a sequence of positivescalar cashflows γ t is given by Z t ( γ ) = [ γ ρt + βµ t ( Z t + δt ( γ )) ρ ] ρ . (1.3)where µ is a certainty equivalent operator and ρ ∈ ( −∞ , \ { } and 0 <β <
1. Sometimes a normalization constant (1 − β ) is included in front ofthe γ ρt , but this is not essential. 16ince the sequence of cashflows γ t is infinite, the equation (1.3) onlydefines the utility as the solution of a fixed point problem. The discountfactor β plays an important role in the proof that the fixed point exists.Given an adequacy level a and a pension outcome ( γ, τ ) we define γ at to the stream of cashflows equal to γ t up to death and a after death. Wemay then define the Epstein–Zin utility with mortality to be given by thestandard Epstein–Zin utility of γ a .We will be primarily interested in the case where µ t ( Z t +1 ( γ )) = E t ( Z t +1 ( γ ) α ) α where α ∈ ( −∞ , \ { } . We refer to this case as homogeneous Epstein–Zinpreferences as they have the property that for λ > Z t ( λγ ) = λZ t ( γ )This symmetry yields a dimension reduction of the HJB, as shown in somegenerality in [40, 4]. This allows some interesting pension problems to besolved analytically, as demonstrated in [12]. In [2], a companion paper tothis article, we use symmetry to compute the optimal investment strategy forhomogeneous Epstein–Zin preferences in the Black–Scholes model for bothindividual and collective investment funds when consumption occurs in dis-crete time.If we assume that all individuals will eventually die, we may give a simplerdefinition for homogeneous Epstein–Zin preferences with mortality which hasthe additional advantage of allowing the case β = 1. Definition 1.12.
Homogeneous Epstein–Zin utility with mortality is definedin discrete time and depends on parameters α ∈ ( −∞ , \ { } , ρ ∈ ( −∞ , \{ } , and 0 < β = e − bt ≤
1. It is the R ≥ ∪ {∞} -valued stochastic processdefined recursively by Z t ( γ, τ ) = t > τ ; α > ∞ t > τ ; α < (cid:2) γ ρt + β E t ( Z t + δt ( γ, τ ) α ) ρα (cid:3) ρ otherwise . (1.4)To interpret this formula we use the convention ∞ α = 0 for α <
0. Assum-ing γ is deterministic, Z is deterministic, so we may define homogeneousEpstein–Zin preferences with mortality by( γ, τ ) (cid:22) (˜ γ, ˜ τ ) ⇐⇒ Z ( γ, τ ) (cid:22) Z (˜ γ, ˜ τ ) . t > τ is determined by therequirement that positive homogeneity still holds.Although the defining formula (1.3) is elegant, Epstein–Zin preferencesare a little easier to understand if one defines the signed power function bysp γ ( x ) = (cid:40) x γ when γ > − x γ when γ < Epstein–Zin satisfaction , z t by z t = sp ρ ( Z t , ρ ) . We may then write the defining equations of homogeneous Epstein–Zin pref-erences as follows z t = γ ρt + β sp − αρ (cid:16) E t (cid:16) sp αρ ( z t + δt ) (cid:17)(cid:17) . (1.5)Written in this form it becomes clear that the Epstein–Zin satisfaction is anadditive quantity (as indicated by the plus sign). For deterministic cashflows,these preferences simplify to z t = γ ρt + βz t + δt = ∞ (cid:88) i =0 β i γ ρiδt . Hence ρ is a parameter measuring satiation. We also see from (1.5) that thecombination of parameters αρ can be interpreted as satisfaction-risk-aversionparameter. The preferences are satisfaction-risk-averse if αρ ≤
1, i.e. if α < ρ .In the case that α = ρ the preferences are satisfaction-risk-neutral and degen-erate to inter-temporally additive von Neumann–Morgernstern preferences.With this interpretation in place, we may return to the Epstein–Zin utilityitself Z t . We now see that this is the instantaneous cash equivalent value of z t . Hence Z t can be interpreted as a cash value and we see that the parameter α is a monetary-risk-aversion parameter.We note that the choice of utility value for τ > t is forced upon us by therequirement that our preferences are positive homogeneous and independentof any cashflows that occur after death. This is a limitation of homogeneousEpstein–Zin preferences with mortality. For α < α > α = ρ the pension adequacy level willnot affect investment decisions.We summarize the properties of these preferences. Lemma 1.13.
Homogeneous Epstein–Zin preferences with mortality are mono-tone, convex, invariant, Markovian and stationary, but are only law-invariantif α = ρ . The key advantages of these preferences are analytic tractability and thepotential to include discounting.We have only described Epstein–Zin preferences in discrete time, but onemay also formulate a continuous time theory [17, 37, 40, 4]. However, thistheory is considerably more complex than the continuous time theory forexponential Kihlstrom–Mirman preferences.It is instructive to note that discrete time exponential Kihlstrom–Mirmanpreferences satisfy a similar equation to (1.5). If we define˜ z t := − log (cid:18) E (cid:18) exp (cid:18)(cid:90) τ u ( γ t )d T ( t ) (cid:19)(cid:19)(cid:19) . then one easily checks that˜ z t = u ( γ t δt ) + v − ( E t ( v (˜ z t + δt ))) (1.6)where v ( x ) = − exp( − x ). Thus these preferences also fit into the recursivepreferences framework of [27], but are inhomogeneous.To define Epstein–Zin type preferences with mortality that allow a flexiblespecification of the adequacy level there are two approaches. Either one coulddefine the utility via equation (1.6), taking u ( x ) = x ρ + α for some α and v =SP αρ . Alternatively one could leave the refining recursion equation unchangedand modify the value of a , the utility when dead. This latter approachwould require choosing a value of β less than 1. Since it is difficult to decidehow to do this for our normative investment questions, the former approachseems preferable. Whichever approach one takes, breaking homogeneity isinevitable. 19his completes our theoretical discussion of preference models. To makea final decision on which model is the most appropriate we must wait untilwe can examine our numerical results. Then we will know which models yieldreasonable investment/consumption strategies. We wish to compare numerically the performance of annuities, individualfunds and collective funds. To do this we must now choose precise market,mortality and preference models. We choose market and mortality modelswhich are as close as possible to the model used in [32], which in turn isbased on modelling assumptions of [31]. We use a version of exponentialKihlstrom–Mirman preferences modified to incorporate a deterministic statepension.We will work in continuous time for investments, but consumption willbe assumed to take place in discrete time, with δt taken to be 1 year.We specialise to the case of the Black–Scholes–Merton model. That is,we suppose that there is a risk free asset S t growing at a risk free rate r anda risky asset S t which follows geometric Brownian motion with drift µ andvolatility σ : d S t = S t ( r d t ) , S d S t = S t ( µ d t + σ d W t ) , S . (2.1)We emphasize that all values are quoted in real terms. In particular r isthe difference between the nominal interest rate and the rate of inflation.Similarly µ measures real returns.There are many well-known limitations to the Black–Scholes–Merton model.We believe that the most important limitation of the model for pension mod-elling is the assumption of a fixed deterministic interest rate. For example,the low interest rates that have prevailed over the last decade have had adramatic impact upon pension outcomes. Nevertheless for the purposes ofthis paper (estimating the potential benefits of collectivised pensions) we be-lieve that this limitation is acceptable. We aim to extend our approach tostochastic interest rate models in future research.Currently in the UK, the state pension grows in real terms due to theso-called “triple lock”. The UK Office of Budget Responsibility uses a de-20erministic model of the state pension growing at a rate of average earningsgrowth plus 0 . . J ( γ, τ ) = E ( − exp ( − s γ,τ )) (2.2)where the satisfaction, s γ,τ is given by s γ,τ = (cid:88) t ∈T ,t ≤ τ u ( γ t , t ) δt and where u is given by u ( γ, t ) = (cid:40) a ( γ t + SP t ) ρ − a (AL t + SP t ) ρ γ t ≥ ∀ t, −∞ otherwise . (2.3)The parameter SP t is a deterministic state pension at time t , and AL t is theadequacy level for the private pension. Thus if the individual consumes at arate γ t = AL t at all times, their overall satisfaction will be 0. The parameter ρ < a is a satisfaction-risk-aversion parameter.If ρ < a < a >
0. Our gain function ensures that consumptionmust always be non-negative.We note that incorporating the state pension into the model will in-evitably break any homogeneity properties of the problem and, if the statepension is time varying, this will also break any translation invariance proper-ties. This is why we have selected to use a time varying-version of exponentialKihlstrom–Mirman preferences in this model.The numerical value of the parameter a in our gain function will dependupon the units of currency. To remedy this we first define X AL to be thecurrency value required at time 0 in order to fund a deterministic pension of AL t := max { AL t , } X AL = (cid:90) τ e − rt AL t d T ( t ) . We now define a parameter λ by λ = lim (cid:15) → s (1+ (cid:15) ) AL t ,t − s AL t ,t (cid:15) . λ therefore measures the rate of increase in satisfaction as oneproportionately increases a deterministic pension set at the adequacy level.It provides a dimensionless parameter proportional to a .We must choose all the parameters in our model in order to perform thecomparison.Table 1 contains a summary of all our parameter assumptions.The market parameters are mostly calibrated using the assumptions of[31]. For equity returns we used the assumptions of the report [32] whichwere designed to be compatible with those of the OBR. To estimate equityvolatility, σ , we used data for the FTSE All Share Total Return Index fromDecember 1985 to June 2019 obtained from [9].The mortality distribution p t was obtained using the model CMI 2018described in [30]. We used this model to find the mortality distribution forwomen of UK retirement aged 65 in 2019 (65 being the UK retirement ageas of 2019). The CMI model requires one to choose a parameter determiningthe long-term rate of mortality improvement. We chose a long-term rateof 1 . − . The resulting distribution is shown in Figure 1. $ J H p t Figure 1: Probability density p t for the random variable τ . Data for UKwomen aged 65 in 2019 using the model CMI 2018 F [1.5%] (see [30]).The parameters determining the utility function are subjective and will22 a r a m e t e r V a l u e D e s c r i p t i o n J u s t i fi c a t i o n S P t £ e x p ( r T L t ) A nnu a l s t a t e p e n s i o n . £ . p e r w ee k a s o f J u l y . G r o w t h a s i n O B R . A L t £ − S P t . A d e q u a c y l e v e l.[ ] a nd [ ] r C P I . C P I g r o w t h . O B R . r . − r C P I G il tr e t u r n s . O B R . r T L . − r C P I S t a t e p e n s i o n g r o w t h . O B R . µ . − r C P I E q u i t y g r o w t h . B a s e d o n . % i nd e x g r o w t h a s i n O B R t og e t h e r w i t h a d i v i d e nd y i e l d o f . % a s a ss u m e d i n [ ]. σ . [ ] X X A L = £ I n i t i a l f und v a l u e . ρ − S a t i a t i o np a r a m e t e r .[ ]. λ S a t i s f a c t i o n - r i s k - a v e r s i o n p a r a m e t e r . I ll u s tr a t i v ec h o i ce . p t C M I F [ . % ] M o rt a li t y d i s tr i bu t i o n C M I T a b l e : P a r a m e t e r s u mm a r y ρ . We know that in the case of von-Neumann Morgernsternpreferences with utility function u ( x ), the value of ρ is closely relatedto the elasticity of inter-temporal substitution. Since von-NeumannMorgernstern preferences are a limiting case of exponential Kihlstrom–Mirman preferences, this suggests we calibrate ρ from empirically ob-served inter-temporal substitution. The mean elasticity observed in themeta-analysis [21] is 0 .
5. We compute the elasticity of inter-temporalsubstitution in the case of homogeneous Epstein–Zin preferences forour market model in [2]. Together these results suggests we choose ρ such that − ρ = 0 .
5, hence we take ρ = −
1. We emphasize that thevalue of ρ will likely vary between individuals. The standard method-ology used to estimate the elasticity of inter-temporal substitution as-sumes a simple market model and is compatible with our choice of theBlack–Scholes model. For more sophisticated models with time vary-ing market price of risk [6] suggest the the elasticity of inter-temporalsubstitution may be closed to 1 . ρ = . Bychoosing the smaller value ρ = − X . We choose the initial budget X to equal X AL . Thusour illustrative individual can just afford a deterministic pension at theadequacy level.3. Choice of AL t . We are referring to the parameter AL t as the adequacylevel because it is the obvious generalization of the notion of adequacylevel given in Section 1 to the form of gain function we are using in thissection. However, the term “adequacy” has already been used in thepension literature and we will insert quotation marks around the word“adequacy” when the term should be understood in this broad sense.Various definitions for “adequacy” have been proposed. For example,one may choose an “adequacy” level based on absolute poverty world-wide, relative poverty within one’s country or relative to one’s own life-time earnings. See [36] for a fuller discussion. There is no a priori reason24hy our formal notion of adequacy should correspond to any particularnotion of “adequacy”. Indeed most notions of “adequacy” depend onlyon the age, nationality and income of the individual whereas our notionof adequacy depends on preferences and so is likely to vary between in-dividuals of identical age, income and nationality. Nevertheless, we willchoose one specific model of “adequacy” to determine AL: specificallywe will use the target replacement rates given in [33] to determine the“adequacy” level as a proportion of final earnings.The usual notion of “adequacy” refers to the required total pension.Since we have modelled the state pension by making a horizontal shift ofour utility function, our notion of adequacy is correspondingly reducedby the state pension.With this understood, we assume that our individual is earning £ , £ ,
800 per annum for the total income from private and statepension.4. Choice of λ . We take λ = 1 as an illustrative example. To decide on areasonable value for λ , one can look at the resulting range in the levelof consumption when one simulates the investment strategy. We willplot a fan-diagram of the consumption in the next section (Figure 2)and it can be seen from this diagram that λ = 1 gives a reasonableresult. In practice one might try to calibrate λ for an individual usinga risk questionnaire, but we will not attempt to consider how such aquestionnaire could be designed. Using the model with parameters as described in Section 2 we are able tocompute the optimal consumption for an individual fund ( n = 1), a collectivefund ( n = ∞ ) and to compare this with an annuity. The problem may bewritten formally as an optimal control problem, and this is done in [1] and,25oreover, that paper describes a numerical method to solve the problem.The resulting pattern of consumption is shown in Figure 2. $ J H & R Q V X P S W L R Q , Q G L Y L G X D O &