Asymptotically Optimal Management of Heterogeneous Collectivised Investment Funds
aa r X i v : . [ q -f i n . P M ] A p r Asymptotically Optimal Management ofHeterogeneous Collectivised Investment Funds
John Armstrong, Cristin BuescuApril 6, 2020
Abstract
A collectivised fund is a proposed form of pension investment, inwhich all investors agree that any funds associated with deceased mem-bers should be split among survivors. For this to be a viable financialproduct, it is necessary to know how to manage the fund even when it isheterogeneous: that is when different investors have different preferences,wealth and mortality. There is no obvious way to define a single objectivefor a heterogeneous fund, so this is not an optimal control problem. Inlieu of an objective function, we take an axiomatic approach. Subject toour axioms on the management of the fund, we find an upper bound onthe utility that can be achieved for each investor, assuming a completemarkets and the absence of systematic longevity risk. We give a strat-egy for the management of such heterogeneous funds which achieves thisbound asymptotically as the number of investors tends to infinity.
Introduction
We study the problem of maximizing the benefit one can obtain from one’spension if one is willing to pursue a collective strategy. This is a strategy inwhich a group of individuals agree that all assets left by an individual who diesare shared among the survivors.By collectivising investments one counters idiosyncratic longevity risk. Un-like an annuity which guarantees a fixed real-terms income until retirement, acollectivised fund may pursue a risky strategy to take advantage of the equityrisk-premium and so yield a higher utility for investors. Additionally, a fundmay also exploit intertemporal substitution of consumption, by delaying con-sumption until investments have grown. These considerations imply that a col-lectivised fund should out-perform both an annuity and a individually managedpension funder. For a more detailed discussion of the benefits of collectivisedpension investment, see [4].We will model a collective fund of n individuals who have already retired andinvested their capital in the fund. We will then consider the optimal pattern ofinvestment and consumption. 1or simplicity we assume that the only stochastic risk factors are given bythe market and idiosyncratic mortality risk. Our modelling for the market,mortality and individual preferences is described in detail in 1. We will alsogive a number of remarks on how systematic longevity risk could be included inthe analysis.In Section 2 we give a mathematical description of the optimal investmentproblem for a homogeneous fund: that is to say, a collective fund where all theindividuals are isomorphic. We also show that an overall objective function forthe fund can be derived using the ideas of robust optimization or the notionof “The Veil of Ignorance” introduced in [17]. Both approaches yield the sameobjective function. The management of the fund may then be written as anoptimal control problem. We state the optimal control problems for both thecases of continuous and discrete time consumption. We do not solve theseproblem in this paper. The discrete time problem is solved analytically in thecase of homogeneous Epstein–Zin preferences in the Black–Scholes model in [3]building on the techniques of [16, 10]. A numerical algorithm for exponentialKihlstrom–Mirman preferences is given in [2].In Section 3 we prove some basic properties of collectivised pension invest-ment. Specifically we prove that, under very mild assumptions, collectivisationis always beneficial. We also give a sufficient condition for constant consump-tion to be the optimal strategy. These conditions are extremely stringent andindicate that an annuity will typically be a suboptimal investment.The mathematical novelty in the paper appears in Section 4 where we con-sider how to manage heterogeneous funds. These are funds where preferences,capital and mortality distributions vary between investors. This problem is chal-lenging because it is not obvious how to combine each individual’s preferencesto obtain a single objective for the fund as a whole. Rather than attempt this,we take an axiomatic approach describing the properties that any managementstrategy for the fund should possess. From these axioms we are able to deducean upper bound on the utility of each investor in a heterogeneous fund, on theassumption that the market is complete. Given the times-scale of pension in-vestments this assumption is a very reasonable approximation to reality. Weare then able to describe a strategy that asymptotically achieves this bound asthe number of investors tends to infinity. We will assume that a fund may invest in a market determined by a filteredprobability space (Ω M , F , F Mt ∈ R + , P M ). In particular we will only consider con-tinuous time markets.We assume there are k available assets and that the price of asset j at time t in real terms is given by S jt . Arbitrary quantities of assets can be bought or soldat these prices. We assume that the asset 1 is risk free, so that d S t = r t S t d t r t is the short-rate. We assume as our no-arbitrage condition that therethere is an equivalent measure Q M such that ( S t ) − S it is a Q -Martingale foreach i . We will call such a market an infinitely-liquid market. We will later assume that consumption only occurs at times in a set T which maybe either [0 , T ) or the evenly spaced time grid { , δt, δt, δt, . . . , T − δt } where T is an upper bound on an individual’s possible age which may be infinite. Asa result we may assume that mortality events are also restricted to T . We writed T ( t ) for the measure determined by the index set: this will be the Lebesguemeasure on [0 , ∞ ) in the continuous case, or the sum of Dirac masses of mass δt at each point in T for the discrete case.We assume that the random variables τ i representing the time of death ofindividual i are independent and absolutely continuous with respect to d T ( t ),with distribution given by p it d T ( t ). We will write (Ω L , F L , F Lt , P L ) for thefiltered probability space generated by all the τ i , the filtration is obtained byrequiring that each τ i is a stopping time. We will write F τ ( t ) for the distributionfunction of τ .The assumption of independence means that we are only considering idiosyn-cratic longevity risk, but we will give some remarks on how systematic longevityrisk will affect our findings later in the paper.We will write n t for the number of survivors at time t , that is the number ofindividuals whose time of death is greater than or equal to t . This conventionensures that n = n and works well with our convention that cashflows receivedat the time of death are still consumed. Note, however, that n t + δt will be F Lt measurable. We assume that there are n individuals. We model a “pension outcome” forindividual i as a pair ( γ i , τ i ) consisting of a stochastic process γ it , representingthe rate of pension payments to individual i at time t , and the random variable τ i representing the time of death. The underlying filtered probability space willbe denoted by (Ω , F , F t , P ) and is assumed to satisfy the usual conditions. Theunits of γ t should be taken to be in real terms which ensures that our modelsfor inflation and preferences are separate.We will later assume that consumption only occurs at times in a set T . Itwill occasionally be convenient to allow the cashflow γ t to be non-zero when t > τ , but this cash will not be consumed. In the discrete case we assume thatcashflow at the moment of death γ τ is still consumed. So the total consumptionover the lifetime of an individual is Z τ γ t d T ( t ) .
3e will assume that the preferences of our individuals are given by a R ∪{±∞} valued gain function J i ( γ, τ i ) which acts on pension outcomes. Theindividual prefers pension outcomes which yield higher values of the gain func-tion. We will assume that the gain function takes the value −∞ whenever P ( γ t t ≤ τ ( t ) < >
0. Following [4] we make the following definitions.
Definition 1.1.
The preferences are said to be invariant if they are invariantunder filtration preserving automorphisms of the probability space.
Definition 1.2.
We will say that J i is concave if it is concave as a function of γ for all τ i . Definition 1.3.
The preferences are monotonic if J i ( γ, τ ) ≤ J i ( γ ′ , τ ) if γ t ≤ γ ′ t for all t ∈ T . Definition 1.4.
A gain function does not saturate if whenever J ( γ, τ ) is finitewe have J ( γ + ǫ, τ ) > J ( γ, τ ) for all positive ǫ . Example 1.5.
Von Neumann–Morgernstern preferences with mortality are de-termined by a choice of concave, increasing utility function u : R ≥ → R and adiscount rate b . The gain function is given by J ( γ, τ ) = E (cid:18)Z τ e − bt u ( γ t ) d T ( t ) (cid:19) . Example 1.6.
Exponential Kihlstrom–Mirman preferences with mortality aredetermined by a choice of concave, increasing utility function u : R ≥ → R . Thegain function is given by J ( γ, τ ) = E (cid:18) − exp (cid:18) − Z τ u ( γ t ) d T ( t ) (cid:19)(cid:19) . Both of these gain functions give rise to concave, invariant, monotonic pref-erences. As is explained in detail in [4], these gain functions are also stationary (meaning that the preferences do not depend upon the time period being con-sidered) and law-invariant (meaning that the preferences depend only on theprobability law of the consumption and not the time at which information isreceived).A larger class of stationary preferences over consumption may be obtained ifone drops the requirement that preferences are law-invariant. Such preferenceswere studied in discrete time by Kreps and Porteus, [15], and the particularcase of Epstein–Zin preferences, introduced in [13], has proved popular in appli-cations as they allow for separate risk-aversion and intertemporal substitutionparameters while maintaining mathematical tractability. Such preferences havebeen used to resolve a number of asset pricing puzzles, see for example [7, 6, 8, 9].In [18], a continuous time analogue of Epstein–Zin preferences is defined us-ing BSDEs on probability spaces equipped with a filtration generated by Brown-ian motion, building on the continuous time analogues of Kreps–Porteus prefer-ences introduced in [11]. A detailed literature review of the development of suchpreferences is given in [18]. Motivated by [18] we may define continuous-timeEpstein–Zin preferences with mortality as follows.4 xample 1.7.
Suppose we are working with continuous time mortality. Supposethat
Ω = Ω M × Ω L and that the filtered probability space Ω M is generated byd-dimensional Brownian motion. Let N it = τ i ≤ t be the jump process associatedwith the death of individual i . Let Λ it be the predictable compensator of N it givenby Λ it = Z t ∧ τ i λ s d s. (1.1) where λ t is the force of mortality. Let M it be the compensated martingale processdefined by M it = N it − Λ it .Let S denote the set of R valued progressively measurable processes { Y t } (0 ≤ t ≤ T ) such that k Y k S := E [sup t ≥ | Y t ∧ T | ] < ∞ . Let L ( W ) denote the set of R d valued progressively measurable processes Z t (0 ≤ t ≤ T ) such that k Z k L ( W ) := E [ Z ∞ k Z t k d t ] < ∞ . Let L ( λ ) denote the set of R n valued progressively measurable processes ζ t (0 ≤ t ≤ T ) such that k ζ k L ( λ ) := E [ n X i =1 Z t ∧ τ i | ζ is | λ s d s ] < ∞ . Let b > be a discount rate. Let = ρ < and = α < be parametersdetermining the individual’s satiation and risk preferences. The Epstein–Zinaggregator f : [0 , ∞ ) × ( −∞ , → R is defined by f ( γ, v ) := b αvρ (cid:18)(cid:18) γ ( αv ) α (cid:19) ρ − (cid:19) . We will assume further that α < and < ρ < . These parameter restrictionsare justified in [18] on the grounds of empirical relevance. We require oneadditional parameter a > which we call the adequacy level .A stopping time τ is admissible if τ < T almost surely. Given an admissiblestopping time τ , the set of admissible consumption streams is defined by C := { γ ∈ R + | E [ Z τ e − bs γ s d s ] < ∞} where R + is the set of all non-negative progressively measurable processes. Givenan admissible consumption and mortality ( γ t , τ ) we define ( V t , Z t , ζ t ) to be the [18] uses a slightly different parameterization. Their parameters δ , γ and ψ are relatedto ours by δ = b , γ = 1 − α and ψ = − ρ . olution of the BSDE d V t = f ( γ t , V t ) t ≤ τ d t − Z t d W t − n X i =1 ζ it d M it , ≤ t ≤ T ; V T = U (1.2) where ( V, Z, ζ ) ∈ S × L ( W ) × L (˜ λ ) and where we set U = a α α . The existenceand uniqueness of solutions of the BSDE (1.2) may be established using thetechnique of the proof of Proposition 2.2 of [18] combined with the propertiesof BSDEs with default jumps established in [12]. Since we assume that τ isbounded above by T , the techniques of [5] are not required.We may then define the Epstein–Zin utility with mortality for adequacy level a associated with ( γ, τ ) by J ( γ, τ ) := V . In the classical Epstein–Zin utility without mortality, the term U in (1.2) is a F τ -measurable random variable given by a U = c ατ α where c τ denotes a bequestat time τ . Such a formulation is attractive mathematically as it ensures thepreferences are positively homogeneous. This symmetry then allows one toreduce the dimension of control problems involving such preferences.However, we note that when α <
0, this modelling choice would associatean infinitely negative utility to dying without a bequest. No amount of lifetimeconsumption would be sufficient to overcome this. This does not seem a plausiblemodel for typical pension preferences.In our model, we call a an adequacy level because the outcome ( γ, τ ) hasthe same utility as (˜ γ, T ) where ˜ γ is equal to γ up to time τ and equal to a thereafter (this is because f ( a, U ) = 0). Thus an investor is indifferent to dyingor living at the adequacy level. The importance of the adequacy level in pensionmodelling is studied in [4].The techniques of [18] allow one to prove that Epstein–Zin preferences withmortality are concave and monotone. Lemma 1.8.
Epstein–Zin preferences with mortality are invariant.Proof.
Let φ : Ω → Ω be a filtration automorphism. Let (
V, Z, ζ ) ∈ S × L ( W ) × L (˜ λ ) be the solution to 1.2. Then we haved( V ◦ φ ) t = f (( γ ◦ φ ) t , ( V ◦ φ ) t ) t ≤ τ d t − ( Z ◦ φ ) t d( W ◦ φ ) t − n X i =1 ( ζ ◦ φ ) it d( M i ◦ φ ) t for all 0 ≤ t ≤ T and V T ◦ φ = U . By the martingale representation theoremfor processes with default jumps (see e.g. [14]) we may find ˜ Z t ∈ L ( W ) and˜ ζ ∈ L (˜ λ ) such thatd( V ◦ φ ) t = f (( γ ◦ φ ) t , ( V ◦ φ ) t ) t ≤ τ d t − ˜ Z t d W t − n X i =1 ˜ ζ it d M it , ≤ t ≤ T. Hence the Epstein–Zin utility with mortality for γ ◦ φ is equal to ( V ◦ φ ) = V (the last equality follows since F = {∅ , Ω } and so V ( ω ) is independent of ω ). 6 Managing homogeneous funds
In this section we consider how to manage homogeneous funds. We call a fundhomogeneous if all individuals in the fund have identical preferences, wealth andmortality. We will consider the management of heterogeneous funds in 4.Since we assume that each individual has an identical mortality distributionwe may define p t := p it .Let us first consider the case of finite n .We wish to decide how to manage a collective pension fund where individualcontributes an amount X at time 0. Individual i will receive an income γ it attime t , with γ it = 0 if the individual is dead at that time. Any cash that is yet tobe consumed is invested in the market. There is no bequest when an individualdies, all remaining cash is shared with the fund.We need to choose an objective for the fund itself. We informally outlinetwo proposals which we will then explain in more detail.(i) Suppose that the individual gain function is law-invariant and so dependsonly on the distribution of outcomes. We may understand “distribution”to mean distribution in both probability and distribution across the pop-ulation. In this way the individual gain function gives rise to an objectivefor the entire fund. We will call this the “distribution approach”.(ii) We follow the robust optimization approach to managing the fund: wemaximize the infimum of the individual gain functions.Let us explain the mathematical detail required for the distribution ap-proach. We suppose that the individual gain function is law-invariant. Wedefine a discrete uniformly distributed random variable ι which takes values in { , . . . , n } . We write (Ω ι , σ ι , P ι ) for the probability space generated by ι . Wedefine a filtration F ιt ∈ R + ∪{∞} by F ιt = ( { Ω ι , ∅} t < ∞ σ ι t = ∞ . Thus Ω ι represents a random choice of individual made at time ∞ . If we have alaw-invariant individual gain function J ι defined relative to a probability spaceΩ, we can then define a gain function relative to Ω × Ω ι by requiring J D ( γ ) = J ι ( γ ι , τ ι ) . (2.1)Note that since τ ι is not a stopping time, this gain function can only be givena meaning for law-invariant individual gain functions J ι .The gain function for the robust approach is given by J R ( γ ) := inf i ∈ I J i ( γ i , τ i ) . (2.2)7hese two approaches are not equivalent in general. Consider the Biblicalproblem faced by Solomon of distributing a child among two women who claimto be its mother. In the distribution approach, giving the child to a randomlyselected woman would be optimal. In the robust approach, giving neither womanthe child would be an equally optimal alternative. For concave individual gainfunctions, Solomon’s recommended approach of splitting the child in two wouldbe ideal.Having decided on a gain function J for our fund, we can write down theassociated optimization problem.We may also augment our probability space with a filtered probability spaceΩ ⊥ so that any arbitrary decisions that need to be made (such as choosing arandom woman to give the child) can be made using random variables definedon this space. We then take Ω = Ω M × Ω L × Ω ι × Ω ⊥ equipped with the productfiltration F t and product measure P . As one might expect, under reasonableconditions, Ω ⊥ proves to be irrelevant, and the optimal strategies can be takento be Ω M × Ω L × Ω ι measurable. See Remark 3.5 below. Note that when workingwith Epstein–Zin preferences we will always assume Ω ⊥ is trivial to ensure thatthe preferences are defined.We then wish to choose progressively measurable consumption streams γ it ≥ t ∈ T and investment proportions α jt in asset j with t ∈ R + . We require P kj =1 α jt ≤
1. We have an inequality in this equation, because it is acceptable,if sub-optimal, to simply discard assets. We must choose these processes suchthat the total wealth of the fund is always non-negative. We write A for theresulting set of admissible controls ( γ , α ), and we will now give a fully precisemathematical description of this set.First let us write down the dynamics of the fund value. For continuous timeconsumption the fund value satisfies the SDEd F t = k X i =1 α is F s d S is − n X i =1 γ it d t (2.3)with F = P ni =1 B i , where B i is the initial budget of individual i (in this sectionwe are assuming that all individuals have identical initial budget B i = X , butthe equations for the more general case will be useful later). For discrete timeconsumption, let F t denote the fund value before consumption and F t denotethe fund value after consumption. We then have the following budget equationsfor the dynamics of F t and F t . F t = P ni =1 B i t = 0lim h → F t − h t ∈ T \ { } F t otherwise. F t = ( F t − P ni =1 γ it F t ′ + P ki =1 R tt ′ α is F s d S is t ′ ∈ T and t ′ ≤ t < t ′ + δt. (2.4)A strategy is ( γ , α ) is admissible if it ensures F t ≥ F t ≥ A = { ( γ , α ) ∈ PM | F t ≥ F t ≥ , ∀ t } (2.5)where PM is the set of progressively measurable R n × R k valued processes forthe probability space Ω.Our objective is to compute v n = sup ( γ , α ) ∈A J ( γ ) . (2.6)and to find ( γ , α ) achieving this supremum.We will henceforth assume that our individual gain functions are concave.Since the market is positively homogeneous, given a strategy ( γ , α ) we may forma new strategy (¯ γ , α ) which assigns the mean consumption to all survivors:¯ γ it = ( τ i < t n t P nj =1 γ jt otherwise . The concavity of our individual gain functions ensures that for our fund’s gainfunction we have J (¯ γ i , α ) ≥ J ( γ i , α ) . So we may assume without loss of generality that all survivors consume thesame amount γ t at time t . Under this assumption we have that J D = J R ,so the distinction between the distributional approach and the robust approachwill not in fact be important.Our motivation for introducing the two approaches is that we believe thedistributional approach corresponds more closely to the intuitive notion of op-timal investment for a collective fund, however the robust approach is requiredif we wish to use Epstein–Zin utility as the individual gain function. To justifyour claim that the distributional approach is more intuitive we first note theexample of Solomon above. A second justification is given by the concept of the“Veil of Ignorance” described in [17]. This concept suggests that when makingcollective decisions one should make those decisions as though the identity ofthe individuals was unknown. Our filtered probability space Ω ι represents thisveil of ignorance: the veil being lifted at time ∞ , but the control ( γ, α ) is chosenin a state of ignorance.Having decided that the consumption, γ t , should be the same for all indi-viduals, we may now consider how to model infinite collectives ( n = ∞ ). Whenperforming accounting calculations in this case, we will perform all calculationson a per-individual basis. For example, rather than keep track of the total fundvalue which would be infinite, we keep track of the fund value per individualwhich will be finite. We will assume that a deterministic proportion of the orig-inal individuals dies over each time interval given by the integral of p t d T ( t )over that interval. This assumption allows us to include mortality within ouraccounting. 9et us express this precisely. Let Y t represent the fund value per individualat time t before consumption or mortality, and let Y t represent the fund valueper individual after consumption. Then in the continuous time case we haved Y t = k X i =1 α is Y s d S is − π t γ t d t (2.7)where π t is the proportion of individuals surviving to time t . In the discretetime case we have Y t = X t = 0lim h → Y t − h t ∈ T \ { } Y t otherwise. Y t = ( Y t − π t γ t t ∈ T Y t ′ + P ki =1 R tt ′ α is Y s d S is t ′ ∈ T and t ′ ≤ t < t ′ + δt. (2.8)For the case n = ∞ , equations (2.7) and (2.8) define the process Y t . In thecase n = ∞ , π t = 1 − F τ ( t ) is deterministic. For the case of finite n , equations(2.7) and (2.8) follow from (2.3) and (2.4). In the case of finite n , π t = n t n is arandom variable.In the case n = ∞ we take as the gain function for our fund J ( γ ) := J ( γ, τ ) . (2.9)This is reasonable since we have assumed that all living individuals receive thesame consumption stream γ t and have isomorphic preferences. Alternatively ifthe individual gain function is law-invariant we may define a random variable τ ι which is measurable only at ∞ and which has distribution p t d t . We may thenwrite the gain function as J ( γ ) := J ι ( γ, τ ι ) . (2.10)These two formulations will be equivalent, but (2.10) seems a more intuitiveformulation.We define the set A of admissible strategies in the case n = ∞ by sayingthat a strategy is admissible if Y t ≥ v ∞ viaequation (2.6) as before.We note that our approach to treating of the case n = ∞ is at present ratherheuristic, but it will be justified rigorously via limiting arguments in the nextsection. It is intuitively clear that collectivisation should be beneficial. We give a formalproof within our model. 10he probability spaces we have defined depend upon the number of individu-als, n , but if n < m there is an obvious way to map random variables defined onthe probability space for n individuals to random variables defined on the prob-ability space for m individuals that preserves the random variables defining themarket and the mortality of the first n individuals. We shall assume henceforththat the individual gain functions are preserved under this mapping. This willhappen automatically if we define our gain functions using a specification suchas Epstein–Zin preferences with mortality with parameter values ( α, ρ, b, a ). Theorem 3.1.
If the individual gain functions J i are concave then v n ≤ v m if n ≤ m < ∞ . Moreover v n ≤ v ∞ in complete markets. In place of assuming that the marketis complete, one may instead require that admissible strategies are integrable.Proof. We recall that, by assumption, the individual gain functions J i are in-variant and isomorphic. We note that to each permutation of the individualswe can associate an automorphism of the probability space which permutes theindividuals. Hence a strategy which is effective for one set of n individuals willgive rise to an isomorphic strategy for another set of n individuals.Suppose that m is finite. Suppose we set up (cid:0) mn (cid:1) funds corresponding to eachpossible choice of n individuals from the full set of m individuals and allocatethe initial budget equally to each of these funds. Given an admissible strategywhich achieves a value v for the gain function for the first n individuals, we canuse it in each of these (cid:0) mn (cid:1) funds. By the concavity of the gain function, we seethat the resulting strategy will have a value greater than or equal to v . Theresult follows.Now consider the case when m = ∞ . Let ( γ , α ) be an admissible strategy fora collective of n investors. Our budget constraint together with the requirementthat the discounted asset prices are Q -measure martingales ensures that E Q × P L n n X i =1 Z γ it S t d T ( t ) ! ≤ X . Note that γ it is non-negative. Hence by Fubini’s theorem we may define astochastic process γ t by γ t S t := E P L n n X i =1 Z γ it S t d T ( t ) | S t ! ≤ X . where S t is the vector of asset prices at time t . This will be progressivelymeasurable with respect to the filtered probability space of the market (Ω M )and moreover will satisfy E Q (cid:18)Z γ t S t d T ( t ) (cid:19) ≤ X .
11t follows from our assumption that the market is complete that we can then find α which ensures that ( γ, α ) is an admissible strategy for an infinite collective.The concavity of the gain function now ensures that J i ( γ i , τ i ) ≤ J ι ( γ, τ ι ) . Hence v ∞ ≥ v n .If the market is not complete, we instead use the integrability of α in orderto define α by averaging.To prove that v n → v ∞ as n → ∞ we require a little more than concavityand invariance, one also needs some degree of continuity in the gain functionwhen it is perturbed by a low probability event. As an example we prove thefollowing in Appendix A. Theorem 3.2.
If preferences are given by Epstein–Zin preferences with mor-tality with < ρ < and α < then v n → v ∞ as n → ∞ . See [2] for the case of exponential utility.
Remark 3.3.
The proof of Theorem 3.1 will continue to work if one introducessystematic longevity risk. The main challenge in extending 3.2 lies in definingthe utility function in a richer probability space. Nevertheless, the continuityrequired to establish 3.2 can be expected for any gain function that yields a well-posed optimization problem.
Defined benefit pensions and annuities typically aim to provide constant con-sumption stream in real terms. It is therefore natural to ask when constantconsumption is optimal. Our next result gives sufficient conditions. We assumefor technical reasons that the asset price processes S it are all given by diffu-sion processes. We will call a market that satisfies this assumption a diffusionmarket. Theorem 3.4.
Constant consumption is optimal for von Neumann–Morgernsternpreferences with b = 0 in complete diffusion markets with P = Q and r t = 0 when n = ∞ . If p t = 0 for all times except T − δt then constant consumptionis optimal for all n . For example, in a Black–Scholes–Merton market with nodrift and risk free rate of zero one has P = Q .Proof. We will give our proof using symmetry arguments rather than directcalculation. Our aim in using this approach is to explain why this result feels“obvious”.First consider the case n = ∞ .By the classification of standard probability spaces, the filtered probabilityspace (Ω , F t ∈T , P ) is isomorphic to the Cartesian product ( S ) T where S is thecircle of circumference 1. Note that we are using the assumption that we are12n a diffusion market to ensure that these probability spaces are standard andatomless. Thus rotations of the circles give rise to market isomorphisms (see [1]for a definition of market isomorphism). This gives an action of the Lie group( S ) T on our market and hence on the space of admissible investment strategies.Given any strategy we may apply these rotations to obtain new, equivalentstrategies. Choose an invariant metric on our Lie group, so that we may definethe average strategy. Note that we are using market completeness here, justas we did in the proof of Theorem 3.1, to ensure that the average strategyexists. By the concavity of our maximization problem and Jensen’s inequality,this averaged strategy will outperform the original strategy. But the averagedstrategy will be invariant under the group action and hence deterministic.Since we may assume our strategy is deterministic, it will be a pure bondinvestment so Y t + δt = ˜ Y t . From our dynamics (2.8) we have Y t + δt = Y t − (1 − F τ ( t )) γ t Solving this difference equation yields the (intuitively obvious) budget constraint0 ≤ X = Z T (1 − F τ ( t )) γ t d T ( t ) . (3.1)Where F τ is the distribution function of τ . The expected utility is Z T (1 − F τ ( t )) u ( γ t ) d T ( t ) . The result for n = ∞ is now easy to prove by brute force, but we wish to usesymmetry.Given a measurable space with non-negative, standard measure µ we canconsider the more general problemmaximize γ ∈ L ( µ ) Z u ( γ ) µ subject to γ ≥ Z γ µ = X (3.2)The optimization we wish to solve is just the special case with µ = (1 − F τ ( t )) d T ( t ). If µ is isomorphic to an invariant measure m on the circle S ,we can use the rotation argument above to prove that the optimizer is given byconstant γ ∗ , specifically γ ∗ = X R µ .For more general µ , suppose we are given a function γ which satisfies theconstraints. We let γ × be the function on the space with product measure µ × λ where λ is the Lebesgue measure on [0 , µ × λ is an atomless,positive probability measure so is isomorphic to an invariant measure m on thecircle. Hence γ × is outperformed by the constant function γ ∗ on µ × λ . Thiscan be projected to a constant function γ ∗ on µ . Since the objective functions13n µ × λ and µ are equal for functions which are constant on the λ factor, thismeans that γ itself is outperformed by γ ∗ . Hence the constant function γ ∗ isoptimal on µ itself.In the case when p t = 0 for all times except T − δt , the optimization problemsfor different values of n are all equivalent, so the result follows.This result provides a mathematical explanation for the intuitive appeal ofannuities and defined benefit pensions. However, there is the obvious caveatthat the assumptions are extremely strong. One expects that with any slightweakening of these assumptions, constant cashflows will no longer be optimal.We prove such a converse in the case of Epstein–Zin preferences in the paper[3]. One interpretation of this result that is worth noting is that it suggeststaking the discount rate b = 0 in our preferences will accord better with investorexpectations than choosing other values of b . This accords with the theoreticaldiscussion of discount rates in pension modelling in [4]. Remark 3.5.
We note that the same averaging argument can be used to showthat the probability space Ω ⊥ can be safely ignored in a complete market withconcave preferences. We now wish to consider how to manage consumption and investment when theindividuals are not identical.For simplicity, let us assume that there are a finite number, ℓ , of possibleinitial types of investor { ζ , . . . , ζ ℓ } . We write Z for the set of types of investor.Each type ζ describes the initial capital, mortality distribution and preferencesof the individual. Our argument will carry through with appropriate modifica-tions to the case of a compact space of types rather than a finite set of types.However, giving such an account would increase the technical burden to littlereal purpose.It is hypothetically appealing to identify the optimal investment strategyfor the heterogeneous fund. However, we would need to define optimality, andthis will require us to specify preferences over outcomes between different typesof investor. Describing such preferences seems challenging and the choice ofpreferences would be controversial.Instead we will take an axiomatic approach. Rather than find an optimalscheme for managing a fund we merely seek an acceptable management scheme.We will define the notion of acceptable axiomatically, and will show that forlarge funds all acceptable management schemes yield similar outcomes for theinvestors.We assume that the preferences of an individual of type ζ are given by aconcave, invariant gain function J ζ ( γ, τ ). Let us write v ( n, ζ ) for the valuefunction for individuals of type ζ investing in this market in a collective of size n as modelled in Section 2. 14e assume that lim n →∞ v ( n, ζ ) = v ( ∞ , ζ ) . Sufficient conditions to ensure this are described in the papers [2] and [3].The initial population is determined by a vector ζ ∈ Z n , where component i of ζ denotes the type of the i -th individual. A management scheme M is afunction acting on vectors ζ of arbitrary length n and which yields a strategy( γ , α ) where γ t is a vector of consumptions of length n with i -th componentbeing the consumption of individual i , and α is an investment strategy. Werequire that the combined consumption and investment are self-financing. Thus M : ∞ G n =1 Z n → ∞ G n =1 L ( R n × R k ) . and M maps the i -th component of the first union into the i -th component ofthe second union.Our first axiom for M is one of fairness, which as we have seen is a propertythat arises automatically in any concave maximization problem. Axiom I1.
All surviving individuals of type ζ consume the same amount. Thatis(i) if M ( ζ ) = ( γ , α ) then γ it = γ jt when ζ ( i ) = ζ ( j ) , t ≤ τ i and t ≤ τ j .(ii) If ζ ′ is obtained by permuting the elements of ζ , then M ( ζ ′ ) is obtainedby the corresponding permutation of the consumption streams of M ( ζ )Let the proportions of different individuals prevailing in the population begiven by a vector of weights ω ( ζ ) = ( ω ζ , . . . , ω ζ ℓ ) with 0 < ω ζ < ω ζ rationaland P ζ ∈Z ω ζ = 1. Let lcm( ω ) denote the lowest common multiple of thedenominators of the fractions ω i , so we know that the population is some integermultiple of lcm( ω ).By Axiom I1 we may define a M ( ω, n, ζ )to be the value of the gain function achieved by an individual of type ζ for apopulation of size n . This is defined if n is any integer multiple of lcm( ω ).If a M ( ω, n, ζ ) < v ( n ω ζ , ζ ) then the investment strategy for the heteroge-neous fund will not be able to attract investors of type ζ as they would bebetter off following the strategy of Section 2. Theorem 3.1 then suggests thatthis would be to the detriment of all other investors in the heterogeneous fund,as increasing collectivisation should always be beneficial. These observationsmotivate the following axioms. Axiom I2.
A management scheme is monotone if: a M ( ω, m, ζ ) ≤ a M ( ω, n, ζ ) if m ≤ n . xiom I3. A management scheme achieves the performance standard if a M ( ω, n, ζ ) ≥ v ( n ω ζ , ζ ) . for all n and ζ . Definition 4.1.
A management scheme is acceptable if it satisfies Axioms I1,I2 and I3.By the monotonicity property, we may unambiguously define a M ( ω, ∞ , ζ ) = lim n →∞ a M ( ω, n, ζ ) . By the performance standard and our assumption on the convergence of v ( n, ζ )as n → ∞ we see that for any acceptable management scheme a M ( ω, ∞ , ζ ) ≥ v ( ∞ , ζ ) . (4.1)The scheme of simply grouping all individuals of a given type together into ahomogeneous fund and managing that according to the model of Section 2 willyield an acceptable management scheme which achieves the lower bound (4.1).We call this the basic management scheme .We wish to show that the basic management scheme is asymptotically op-timal in complete markets. The key observation is that collective investmentprovides no substantive benefit for the investment problem in a complete marketwithout mortality.Let us consider how to write a collective investment problem for n individ-uals without mortality investing in our market. We choose consumption γ it forindividual i , and overall investment proportions α t . The total fund value will begiven by the budget equations (2.4), and hence the set of admissible controls forthis problem will be given by (2.5). The difference will be that the preferences ofthe individual will depend only upon cashflows received and not upon mortality,so we will suppose that for individual i we have a gain function ˆ J i ( γ it ) whichdepends only upon the cashflows. We define the value functionˆ v i := sup A ˆ J i ( γ it )where A n is the set of acceptable admissible controls for n individuals. Althoughwe have defined the set of admissible controls for n individuals, we do not writedown an optimization problem for n individuals as we do not know what theobjective should be across a heterogeneous collective.Our next result shows that in a complete market without mortality thereis no real benefit in considering collectivised problems as, however we select anadmissible control (by solving an optimization problem or otherwise), it willnever bring a substantive advantage to any individual unless it also gives asubstantive disadvantage to some other individual. In other words, one cannotpay Paul without robbing Peter. 16 emma 4.2. Suppose the gain functions ˆ J i ( γ t ) are concave, monotone and donot saturate and satisfy ˆ J i ( γ t ) > −∞ for positive cashflows γ t . Suppose that ˆ J i ( γ t ) = −∞ whenever γ t is negative on a set of finite measure.Let i ∗ be an individual, then for any ǫ > there exists ǫ > such thatfor any admissible investment consumption strategy for all the investors with γ i satisfying ˆ J i ∗ ( γ i ∗ t ) ≥ ˆ v i ∗ + ǫ (4.2) there is an investor i such that ˆ J i ( γ it ) ≤ ˆ v i − ǫ . Proof.
Let us write v i ( b ) for the value function for individual i as a function oftheir budget b . By Lemma (4.3) below, v i is continuous as a function of b forany b >
0. We write B i for the budget of individual i .In a complete market, we call a measurable non-negative cashflow γ t a deriva-tives contract and we call the discounted Q -measure expectation of γ t the priceof this contract. This price may be infinite, which means that the cashflows can-not be super-replicated by any admissible trading strategy. If the price is finite,the contract can be replicated by an admissible trading strategy with initialbudget given by the price. Note that the requirement that γ t is non-negativeensures that the price of derivative contracts is additive: if negative infinitieswere allowed as prices this would not be the case.If we write D b for the set of derivatives contracts of price less than or equalto b , we have: ˆ v i ( b ) = sup γ ∈D b ˆ J ( γ ) . Here we have used our assumption that negative cashflows yield a value of −∞ for the gain function.By the continuity of v i ∗ , there is a price, δ , such that if ˆ J i ∗ ( γ i ∗ t ) ≥ v i ∗ + ǫ ,then the price of the derivative contract with payoff γ i ∗ t is at least B i ∗ + δ .By the monotonicity and non-saturation of the gain functions, together withthe concavity given by Lemma (4.3) belowˆ v i (cid:18) B i − δℓ − (cid:19) < ˆ v i ( B i ) . Let ǫ = inf i = i ∗ (cid:26) ˆ v i ( B i ) − ˆ v i (cid:18) B i − δ ℓ − (cid:19)(cid:27) . Any derivative contract with cashflows γ i satisfyingˆ J i ( γ it ) ≤ ˆ v i − ǫ . The definitions of these terms were given for gain functions over cashflows with mortality,but the corresponding definitions for ˆ J should be obvious B i − δ ℓ − . Hence the total cost of an investment strategyyielding all the cashflows γ i is at least B i + δ + X i = i ∗ (cid:18) B i − δ ℓ − (cid:19) = X i B i + δ Lemma 4.3.
Consider investment without mortality in a homogeneous market.Suppose an individual’s gain function over consumption, ˆ J ( γ ) , is concave andmonotone. Let A b be the set of admissible consumption, investment strategieswith initial budget b . Define ˆ v ( b ) = sup ( γ,α ) ∈A b ˆ J i ( γ ) . The function v is concave and hence v is continuous on any open set on whichit is finite.Proof. Given two budgets b , b > b then let ( γ i , α i ) be admissible strategiesfor each budget. So for any λ ∈ [0 , λγ + (1 − λ ) γ , λα + (1 − λ ) α ) is anadmissible strategy with budget λb + (1 − λ ) b . By the concavity of ˆ J ,ˆ J ( λγ + (1 − λ ) γ ) ≥ λ ˆ J ( γ ) + (1 − λ ) ˆ J ( γ ) . Hence ˆ v ( λb + (1 − λ ) b ) ≥ λ ˆ v ( b ) + (1 − λ )ˆ v ( b ) . Remark 4.4.
Lemma 4.2 is only true in complete markets. If two individualshave different risk or consumption preferences in an incomplete market then itcan often be beneficial to design a derivatives contract to the mutual advantage ofboth parties. This is the essential purpose of derivatives contracts and explainswhy there is a market for such contracts.
We are now ready to prove the main result of this section.
Theorem 4.5.
For any acceptable management scheme in a complete market a M ( ω, ∞ , ζ ) = v ( ∞ , ζ ) so long as all individual gain functions are concave, monotone, invariant anddo no saturate, and so long as lim n →∞ v ( n, ζ ) = v ( ∞ , ζ ) > −∞ . (4.3) for any positive budget. roof. We will prove the result in the case of discrete time consumption. Thecase of continuous time consumption is similar.By Axiom I3, lim n →∞ a M ( ω, n, ζ ) ≥ lim n →∞ v ( nω ζ , ζ ). By assumptionlim n →∞ v ( n, ζ ) = v ( ∞ , ζ ). Hence a ( ω, ∞ , ζ ) = lim n →∞ a M ( ω, n, ζ ) ≥ v ( ∞ , ζ ).Let us now define the meaning of an admissible strategy for an infinite het-erogeneous collective where the types of each individual are given by the pro-portions ω . We will assume that surviving individuals of a given type, ζ , allconsume γ ζt at time t . Hence the total consumption per person (i.e. of any type,including both survivors and the deceased) is X ζ ∈Z γ ζt π ζt ω ζ where π ζt denotes the proportion of individuals of type ζ who survive to time t .Note that π ζt = 1 − F τ ζ ( t ) where τ ζ is a random variable distributed accordingto the time-of-death distribution for individuals of type ζ .If we let Y t denote the fund value per person before consumption and Y t denote the fund value per person after consumption we have the following bud-get equations for the dynamics of Y t and Y t . Let us write B ζ for the initialbudget of individuals of type ζ . Then the fund value per person for the infiniteheterogenous collective should be defined to follow the dynamics Y t = P ζ ∈Z ω ζ B ζ t = 0lim h → Y t − h t ∈ T \ { } Y t otherwise. Y t = ( Y t − P ζ ∈Z γ ζt π ζt ω ζ t ∈ T Y t ′ + P ki =1 R tt ′ α is Y s d S is t ′ ∈ T and t ′ ≤ t < t ′ + δt. (4.4)We may alternatively view the equations above as describing the dynamicsof a fund of “virtual individuals” where each virtual individual represents theinterests of an infinite fund of individuals all of type ζ . To see this, we defineˆ γ ζt = γ ζt π ζt ω ζ . We say that virtual individual ζ consumes an amount ˆ γ ζt at eachtime t ∈ T . We say that the initial budget of virtual individual ζ is ω ζ B ζ . Wemay now view equation (4.4) as giving the dynamics of the total fund valuebefore consumption Y t of a collective of these heterogeneous virtual individuals.This observation will allow us to apply Lemma 4.2 to the collective investmentproblem with mortality.Let us define a gain function (without mortality) ˆ J ζ for the virtual individual ζ by ˆ J ζ (ˆ γ ζt ) := J ζ ( γ ζt , τ ζ )Let us write ˆ v ζ ( b ) for the value function of a virtual individual with this gainfunction with an initial budget b . We see that ˆ v ζ ( ω ζ B ζ ) = v ( ∞ , ζ ).Suppose ( γ , α ) is a strategy for the infinite heterogeneous collective and let ζ ∗ be a chosen type of individual. It follows by Lemma 4.2 that for any given ǫ > J ζ ∗ (ˆ γ ζ ∗ t ) ≥ v ( ∞ , ζ ∗ ) + ǫ (4.5)19here exists ǫ > ζ such thatˆ J ζ (ˆ γ ζt ) ≤ v ( ∞ , ζ ) − ǫ . (4.6)Hence given any ǫ > J ζ ∗ ( γ ζ ∗ t , τ ζ ∗ ) ≥ v ( ∞ , ζ ∗ ) + ǫ (4.7)there exists ǫ > ζ such that J ζ ( γ ζt , τ ζ ) ≤ v ( ∞ , ζ ) − ǫ . (4.8)Now let us suppose for a contradiction that for some ǫ > ζ ∗ and N we have a ( ω, N, ζ ∗ ) ≥ v ( ∞ , ζ ∗ )+2 ǫ . Note that we then have a ( ω, n, ζ ∗ ) ≥ v ( ∞ , ζ ∗ )+2 ǫ for all n ≥ N . We may then choose ǫ > ζ and we know lim n →∞ a M ( ω, n, ζ ) ≥ v ( ∞ , ζ ). So for sufficiently large n we may assume that a M ( ω, n, ζ ) ≥ v ( ∞ , ζ ) − ǫ . Hence for such n we may find an investment-consumption strategy ( γ , α ) for acollective of n individuals which yields the same consumption for all survivingindividuals of a given type and which satisfies J ζ ∗ ( γ ζ ∗ t , τ ζ ∗ ) ≥ v ( ∞ , ζ ∗ ) + ǫ and which also satisfies J ζ ( γ ζt , ζ ) ≥ v ( ∞ , ζ ) − ǫ for all types ζ .Since the discounted asset prices S it are Q -martingales we have E Q × P L X ζ ∈Z Z e − rt ω ζ γ ζt d T ( t ) ≤ X ζ ∈Z ω ζ B ζ . Since the consumption γ t is non-negative we have by Fubini’s theorem that γ ζt := E P L (cid:18)Z e − rt ω ζ γ ζt d T ( t ) | S t (cid:19) is a progressively measurable process for the filtered probability space (Ω M , F t , P M )and satisfies X ζ ∈Z E Q (cid:18)Z e − rt ω ζ γ ζt d T ( t ) (cid:19) ≤ X ζ ∈Z ω ζ B ζ .
20y the complete market assumption we can then find an investment strategy α that funds γ . Hence we have found an admissible strategy ( γ , α ) for the infinitehomogeneous collective which by the concavity of the preferences will satisfy J ζ ∗ ( γ ζ ∗ t , τ ζ ∗ ) ≥ v ( ∞ , ζ ∗ ) + ǫ and which also satisfies J ζ ( γ ζt , ζ ) ≥ v ( ∞ , ζ ) − ǫ for all types ζ . This contradicts equations (4.7) and (4.8).We deduce that a ( ω, N, ζ ) ≤ v ( ∞ , ζ ) for all ζ , which gives the result.The financial significance of Theorem 4.5 is that all acceptable strategies areasymptotically equivalent in a complete market, in particular this applies to theBlack–Scholes–Merton market.The result is analogous to the classical result that in the Black–Scholes–Merton market any derivative has a unique price independent of the preferencesof the investor. This too is an asymptotic result, in the sense that it assumescontinuous time and zero transaction costs.In practice no two individuals are alike, and so our assumption of a finitenumber of distinct types can be criticised. However, simple modifications ofour strategy for heterogeneous funds will yield good results if there are a largenumber of similar individuals. We give a concrete algorithm in [4] and showthat it achieves approximately 98% of the maximum benefit of collectivisationfor a heterogeneous fund of only 100 investors. Remark 4.6.
It is natural to ask how our results extend if one incorporatessystematic longevity risk. The simplest approach would be to assume that thereis a complete market of contracts on the risk factors determining systematiclongevity risk, in which case a similar result can be expected to hold. Longevityderivatives do exist, but at present the market is not very liquid. Our resultssuggest that each type of individual will have different views on the attractivenessof longevity risk, which would imply that the development of collective fundsshould result in a more liquid market in longevity derivatives.
A Proof of Theorem 3.2
Proof of Theorem 3.2.
Recall that when working with Epstein–Zin preferenceswe assume Ω ⊥ is trivial.To prove the existence of Epstein–Zin utility with mortality following themethods of [18] one first makes the change of variables ( ˜ V t , ˜ Z t , ˜ ζ t ) = αe − bαtρ ( V T , Z t , ζ t )to transform 1.2 to the BSDEd ˜ V t = F ( t, γ t , ˜ V t ) t ≤ τ d t − ˜ Z t d W t − n X i =1 ˜ ζ it d M it , ≤ t ≤ T ; ˜ V T = e − bTαρ αU (A.1)21here F ( t, γ t , v ) := bα exp( − bt ) ρ γ ρt v − ρα . To prove the existence and uniqueness of this BSDE, one next solves the familyof BSDEs indexed by m ∈ R ≥ ,d ˜ V mt = F m ( t, γ t , ˜ V mt ) t ≤ τ d t − ˜ Z t d W t − n X i =1 ˜ ζ it d M it , ≤ t ≤ T ; ˜ V T = e − bTαρ αU (A.2)where F m ( t, γ t , v ) := b αρ exp( − bt )( γ ρt ∧ m )( v ∧ m ) − ρα . The driver F m has the property that y → F m ( t, γ t , y ) is Lipschitz (since ourparameter assumptions ensure 1 − ρα >
1) and so one can use the theory of[12] to obtain the existence of a unique solution to this BSDE. The comparisontheorem ensures that ˜ V m is non-negative and decreasing in m . The proof ofProposition 2.2 in [18] then shows that the ˜ V = lim m →∞ ˜ V m can be extendedto give a solution to (A.1).Let us define EZ m ( γ, τ ) := α ˜ V m . We have that EZ m ( γ, τ ) is finite, non-positive and increasing in m with limit given by the Epstein–Zin utility withmortality which we denote EZ( γ, τ ).Let M < v ∞ and ǫ > γ ∞ t for the problem of an infinite collective with initial budget B suchthat EZ( γ ∞ t , τ ι ) ≥ M .Since EZ is concave and finite, for any admissible γ , the map λ → EZ( λγ ∞ t , τ ι )is continuous on R ≥ . Hence we may choose λ ∈ (0 ,
1) such that EZ( λγ ∞ t , τ ι ) ≥ M − ǫ . By the convergence of EZ m in m , we may then find m such thatEZ m ( λγ ∞ t , τ ι ) ≥ M − ǫ. (A.3)Suppose that we have an initial budget of B for the problem of a collectiveof n investors. Recall that n t denotes the number of survivors at time t . Let G n,t be the event defined by G n,t = { ω | n s ( ω ) ≤ λ E ( n s ) ∀ ≤ s ≤ t } . We write ¬ G n,t for its compliment. The consumption stream λ G n,t γ ∞ t will beadmissible, as the consumption per survivor will always be less than γ t .Define F m, ∞ ( t, v ) := F m ( t, λγ ∞ t , v ) and F m,n ( t, v ) := F m ( t, λ G n,t γ ∞ t , v )for finite n . Write ( ˜ V m,n , ˜ Z m,n , ˜ ζ m,n ) for the solution to (A.2) for the driver F m,n for both finite and infinite values of n , and define EZ m,n := α ˜ V m,n Differentiating the expression for F m,n ( t, v ) with respect to v , we obtain thefollowing Lipschitz estimate for the drivers | F m,n ( t, y ) − F m,n ( t, y ) | ≤ (cid:16) − ρα (cid:17) b | α | ρ m − ρα | y − y | =: C m | y − y | C m .Let us define e = | ˜ V m, ∞ − ˜ V m,n | . By Proposition 2.4 of [12], we obtain thefollowing bound on e : e ≤ η m E [ Z T e β m s | F m, ∞ ( s, ˜ V m, ∞ ) − F m,n ( s, ˜ V m,n ) | d s ] , where η m := C m and β m := 3 C m + 2 C m . Inserting the definitions of F m,n weobtain e ≤ η m E [ Z T bαρ e β m s (( λ ¬ G n,s γ ∞ s ) ρ ∧ m )( ˜ V m, ∞ ∧ m ) − ρα d s. Splitting the integral into two regions, we find that for δ ∈ (0 , T ), e ≤ η m bαm − ρα ρ e β m T E [ Z T − δ ¬ G n,s d s + Z TT − δ d s ] ≤ η m bαm − ρα ρ e β m T E [ Z T − δ ¬ G n,T − δ d s + δ ] as ¬ G n,s ⊆ ¬ G n,t if s ≤ t ≤ ˜ C m ( T P ( ¬ G n,T − δ ) + δ ) (A.4)for an appropriately defined C m . Choose δ such that ˜ C m δ ≤ α ǫ . Now usethe fact that P ( ¬ G n,T − δ ) → n → ∞ (as follows readily from Lemma A.1,below) to choose n such that ˜ C m ( T P ( ¬ G n,T − δ ) ≤ α ǫ . By (A.4) we will thenhave that e ≤ | α | ǫ . We then have v n ≥ EZ( λ G n,t γ ∞ t ) ≥ | EZ m,n | ≥ | EZ m, ∞ | − e | α | ≥ | EZ m, ∞ | − ǫ ≥ M − ǫ. where we have used in sequence: the definition of v n ; the convergence of theincreasing sequence in m given by EZ m,n ; the definitions of e and EZ m,n ; ourbound for e ; equation (A.3). So by Theorem 3.1, lim n →∞ v n = v ∞ .We now prove the Lemma used in the proof. Lemma A.1.
Let T ∗ be minimum time by which an individual is almost sureto have died. For a continuous mortality distribution, given a time point ≤ t < T ∗ , for any ǫ ∈ (0 , there exists a finite set of points t i ∈ [0 , t ) , indexedby i ∈ I such that P ∀ t ∈ [0 , t ] : n t ≤ (cid:18) − ǫ (cid:19) E ( n t ) ! ≥ P (cid:18) ∀ i ∈ I : n t i ≤ (cid:18) − ǫ (cid:19) E ( n t i ) (cid:19) . Proof.
We define t i inductively. If t i − = 0, we are done and take the index set I = { , , . . . , i − } . Otherwise define t i = inf (cid:26) t | t = 0 or E ( n t ) ≤ − ǫ E ( n t i − ) (cid:27) . t i = 0 we see E ( n t i − ) ≥ − ǫ E ( n t i ), so for sufficiently large i we must have t i = 0. Hence the index set, I , is finite.Given t ∈ [0 , t ] we can find i ∈ I with t i ≤ t ≤ t i − . Suppose n t > (cid:18) − ǫ (cid:19) E ( n t )then we have n t > (cid:18) − ǫ (cid:19) E ( n t i ) ≥ (cid:18) − ǫ (cid:19) E ( n t i − ) ≥ (cid:18) − ǫ (cid:19) E ( n t ) . References [1] John Armstrong. Classifying markets up to isomorphism. arXiv preprintarXiv:1810.03546 , 2018.[2] John Armstrong and Cristin Buescu. Collectivised pension invest-ment with exponential Kihlstrom–Mirman preferences. arXiv preprintarXiv:1911.02296 , 2019.[3] John Armstrong and Cristin Buescu. Collectivised pension investment withhomogeneous Epstein–Zin preferences. arXiv preprint arXiv:1911.10047 ,2019.[4] John Armstrong and Cristin Buescu. Collectivised post-retirement invest-ment. arXiv preprint arXiv:1909.12730 , 2019.[5] Joshua Aurand and Yu-Jui Huang. Epstein-Zin utility maximization onrandom horizons. arXiv preprint arXiv:1903.08782 , 2019.[6] Ravi Bansal. Long-run risks and financial markets. Technical report, Na-tional Bureau of Economic Research Cambridge, Mass., USA, 2007.[7] Ravi Bansal and Amir Yaron. Risks for the long run: A potential resolutionof asset pricing puzzles.
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