Collectivised Pension Investment with Homogeneous Epstein-Zin Preferences
CCollectivised Pension Investment withHomogeneous Epstein–Zin Preferences
John Armstrong, Cristin BuescuNovember 25, 2019
Abstract
In a collectivised pension fund, investors agree that any money re-maining in the fund when they die can be shared among the survivors.We compute analytically the optimal investment-consumption strat-egy for a fund of n identical investors with homogeneous Epstein–Zinpreferences, investing in the Black–Scholes market in continuous time butconsuming in discrete time. Our result holds for arbitrary mortality dis-tributions.We also compute the optimal strategy for an infinite fund of investors,and prove the convergence of the optimal strategy as n → ∞ . The proofof convergence shows that effective strategies for inhomogeneous fundscan be obtained using the optimal strategies found in this paper for ho-mogeneous funds, using the results of [2].We find that a constant consumption strategy is suboptimal even forinfinite collectives investing in markets where assets provide no return solong as investors are “satisfaction risk-averse.” This suggests that annu-ities and defined benefit investments will always be suboptimal invest-ments.We present numerical results examining the importance of the fundsize, n , and the market parameters. Introduction
A group of individuals may group together and invest income for their retirementin a collective fund. When an individual dies, any funds associated with thatindividual are then divided among the survivors. The paper [2] shows how tomodel the management of these funds mathematically and argues that theyshould yield significantly better results for investors than traditional pensioninvestment models.This paper complements [2] by computing the optimal investment strategyfor collective investment under the following assumptions:(i) There are n identical investors in the collective.1 a r X i v : . [ q -f i n . P M ] N ov ii) The fund may invest in continuous time in a Black–Scholes–Merton marketwith one risky asset.(iii) The mortality of the individuals is independent: that is there is no system-atic longevity risk. Mortality occurs with a known probability distribution.(iv) The preferences of each individual are given by homogeneous Epstein–Zinpreferences with mortality (as defined in [2]).(v) Consumption occurs in discrete time (for example once per year).We formulate the problem mathematically and are able to give analytical for-mulae for the optimal consumption and investment at each time.Assumption (i), that the investors are identical, is not a significant restric-tion. The paper [2] shows that once one knows how to manage a fund of identicalindividuals, it is easy to devise very effective management strategies for inho-mogeneous funds , i.e. funds of diverse individuals.Assumption (ii), that the market is a Black–Scholes–Merton market is, ofcourse, restrictive. This is the simplest type of market we could consider. We aretrading full realism for analytic tractability. The assumption that the markethas only one risky asset is not restrictive. It follows from the mutual fundtheorem arguments of [1] that essentially the same strategy can be used in aBlack–Scholes–Merton market with n stocks.Assumption (iv), that the preferences are given by homogeneous Epstein–Zin utility, is a central assumption to this paper, and is key to the analytictractability of the problem. We consider different possible models for prefer-ences over consumption with mortality in [2] and find that two models standout as having particularly attractive properties. These models are called homo-geneous Epstein–Zin preferences and exponential Kihlstrom–Mirman preferences in [2] and we will use the same terminology. As the results of this paper demon-strate, homogeneous Epstein–Zin preferences have the tremendous advantageof being analytically tractable. This stems from the homogeneity property ofthese preferences. By contrast, we can only expect to solve analogous problemswith exponential Kihlstrom–Mirman preferences using numerical methods (weshow how to do this in [3]). It is possible to define inhomogeneous Epstein–Zinpreferences with mortality (see [2]). We believe that the techniques of [3] couldbe applied to such problems, but that one cannot, in general, expect analyticalresults without homogeneity.Assumption (v), that consumption occurs in discrete time is not a significantrestriction. Indeed one might argue it adds to the realism of the model.The apparent mismatch between discrete time consumption and continuoustime investment is the key technical trick required to obtain our analytic re-sults. As is explained in [1], the Black–Scholes–Merton market is isomorphicto a linear market in continuous time and this explains the analytic tractabil-ity of many problems involving this market. However, in discrete time theBlack–Scholes–Merton market is fundamentally non-linear, and this explainsthe analytic intractability of problems such as Merton’s investment problem in2iscrete time. We deduce that we must allow continuous time investment toobtain analytic results.On the other hand, if we are to allow arbitrary mortality distributions andcontinuous time consumption, then there is no hope of obtaining analytic re-sults as one wouldn’t even be able to write down the mortality distribution ingeneral. Our assumption of discrete time consumption is essentially equivalentto assuming that mortality occurs in discrete time, and so restricts the set ofmortality distributions we are considering to ensure tractability.The advantage of analytic tractability is the insight it gives us into optimalpension investment.For example, we can analyse how consumption varies over time. We findthat except for very special cases, constant consumption is never optimal. Thisis interesting because many people see a defined-benefit pension which providesconstant real-terms income as the “gold standard” for a pension fund. Ourresult shows that chasing constant income is, in fact, suboptimal.It perhaps isn’t so surprising that if market returns are non-zero there areadvantages to taking some risk by investing in equities.Nor should it be surprising that delaying consumption to benefit from marketreturns can also be advantageous. We are able to make this precise by computingthe elasticity of intertemporal substitution in our model.It is perhaps, surprising that even if one assumes that both equities andbonds provide no return, it is still not optimal to receive a constant income, if oneis satisfaction risk-averse (see below for a definition). Under these circumstancesit can be optimal to spend earlier (if the primary risk one perceives is the risk ofdying before one can consume one’s pension) or to spend later (if the primaryrisk one perceives is the risk of living for a long time on an inadequate pension).Let us now describe the structure of the paper.Section 1 reviews the definition of homogeneous Epstein–Zin preferences withmortality.Section 2 states the optimal investment problems we will solve. The problemdepends upon the number of individuals n . We will also state an investmentproblem for a fund which is intended to represent the limiting case n = ∞ .Section 3 solves the optimal investment problems analytically in the caseswhere n = 1 and n = ∞ .Section 4 computes how consumption and wealth vary over time, givinganalytic descriptions of their probability distributions.Section 5 generalizes the results to arbitrary fund sizes n .Section 6 uses our results to provide a rigorous justification for the claimthat our models for finite n converge to the case n = ∞ . This is of obvioustheoretical interest in its own right, but we remark that the proof is essential todemonstrating that the strategies for inhomogeneous funds of described in [2]will be effective. 3 Homogeneous Epstein–Zin utility with mor-tality
Let us recall the definition of homogeneous Epstein–Zin utility with mortalitygiven in [2]. In order to give a crisp definition, we first define a convention forhow we will we handle algebra using infinite and infinitesimal values.
Definition 1.1.
The extended positive reals R ++ is the set R ++ = R + ∪ { (cid:15) α | α ∈ R \ { }} where (cid:15) is a symbol representing an infinitesimal value. We extend addition,multiplication and raising to a real power to R ++ in the obvious way: x + (cid:15) α = (cid:40) x when α > (cid:15) α otherwise (cid:15) α + (cid:15) β = (cid:15) min { α,β } x(cid:15) α = (cid:15) α (cid:15) α (cid:15) β = (cid:15) α + β ( (cid:15) α ) β = (cid:15) αβ We now assume that we are given a time grid T = { t , t + δt, t + 2 δt, t +3 δt . . . , T − δt } where t is some initial time, δt is a fixed time step and T isthe time by which we assume mortality is certain.We will model the individual’s consumption as a non-negative stochastic pro-cess ( γ t ) t ∈T in a filtered probability space (Ω , F , F t , P ). The time of death τ isa stopping time taking values in T . Our convention is that any consumption upto and including time τ may effect the individual’s utility, but any consumptionoccurring after time τ will be ignored. Definition 1.2.
Homogeneous Epstein–Zin utility with mortality is defined for anon-negative consumption process ( γ t ) t ∈T and a stopping time τ taking valuesin T . It depends on parameters α ∈ ( −∞ , \ { } , ρ ∈ ( −∞ , \ { } , and0 < β = e − bt ≤
1. It is the R ++ -valued random process defined recursively by Z t ( γ, τ ) = (cid:40) (cid:15) α t > τ ; (cid:2) γ ρt + β E t ( Z t + δt ( γ, τ ) α ) ρα (cid:3) ρ otherwise . (1.1) A general optimal investment problem for a homogeneous collective fund wasdescribed mathematically in [2]. In this section we summarize the formulationof [2], specializing to the case of interest for this paper.We model a collective fund of n investors. The fund may invest in either ariskless bond which grows at a risk-free rate of r or in a stock whose price at4ime t , is denoted S t . The stock price at time t is given. At subsequent times S t obeys the SDE d S t = S t ( µ d t + σ d W t ) S t (2.1)for a constant drift µ and volatility σ , and a 1-dimensional Brownian motion W t .We will model consumption taking place in discrete time on a grid T as de-scribed in the previous section. Since we are modelling consumption in discretetime, we may safely model mortality in discrete time. We let τ be a random vari-able modelling the time of death of a representative individual. We assume that τ has a probability distribution given by p t d T ( t ) where d T ( t ) is the measuregiven by adding the Dirac measures associated to the grid points of T .Let s t denote the survival probability between times t and t + δt . That is s t = (cid:80) ∞ i =1 p t + iδt (cid:80) ∞ i =0 p t + iδt (2.2)Write n t for the number of individuals individuals with a time of death greaterthan or equal to t . The process ( n t ) t ∈T is a Markov process, with initial value n t . Note, however, that n t + δt will be F t measurable. We choose this conventionfor n t as it works well with our existing convention that individuals who die atage t still consume at time t .For the case of a finite number of individuals, the transition probability of n t moving from a value of n at time t to the value i at time t + δt is given by S t ( n, i ) := (cid:18) ni (cid:19) ( s t ) i (1 − s t ) n − i . (2.3)We also wish to write down a formal optimal investment problem for thecase of a fund with n = ∞ investors. In this case we will define n t = ∞ for alltimes up to T .We will write a t for proportion of the fund invested in stock at time t . Wewill write X t for the value of the fund per survivor at time t before consumption.We will define X t = 0 if n t = 0. Similarly, we will write X t for the value of thefund per survivor after consumption. We note that X t = lim h (cid:37) t X h at timepoints t ∈ T . At intermediate times, t ∈ [ iδt, ( i + 1) δt ), X t obeys the SDEd X t = X t ( a t µ + (1 − a t ) r ) d t + X t a t σ d W t , with initial condition given by the budget equation X t = (cid:40) n t n t + δt ( X t − γ t ) n finite s − t ( X t − γ t ) n = ∞ (2.4)unless n t + δt = 0 in which case we define X t to be zero on [ t, t + δt ). Note thatthis formula is based on our convention that an individual who dies at a time t still consumes at that time and the corresponding convention for n t whichensures n t + δt is F t measurable. 5et τ i denote the time of death of individual i .For finite n we let (Ω , F , F t , P ) be the filtered probability space generatedby W t and the time of death variables τ i . We define τ = τ to be the time ofdeath variable for one specific individual whose time of death is greater than orequal to t .For n = ∞ we let τ be any random variable with the distribution p t d T ( t ).We let (Ω , F , F t , P ) be the filtered probability space generated by W t and τ .Let ˜ A ( x, t ) denote the space of admissible controls ( γ t , a t ): that is F t -predictable processes such that 0 ≤ γ t ≤ X t and with X t = x . We define thevalue function of our problem starting at time t to be v n ( x, t ) = sup ( γ,a ) ∈ ˜ A ( x,t ) Z t ( γ, τ ) . (2.5)where Z t is an Epstein–Zin utility function. n = 1 and n = ∞ To highlight the key ideas we will consider only the case when n = 1 and n = ∞ in this section, leaving the case of general n until Section 5. We define C todistinguish these cases as follows C = (cid:40) n = 11 n = ∞ . (3.1)We write z t := v n (1 , t ) so that the positive-homogeneity of Z t implies that v n ( x, t ) = x z t . (3.2)We note that we have not yet shown whether z is finite, but equation (3.2) canstill be interpreted for infinite values of z .Let c t denote the consumption rate at time t , so γ t = c t X t for an individualwho is still alive at time t . The budget equation (2.4) can then be written as X t = s − Ct (1 − c t ) X t . (3.3)We now let A t + δt ( x, c t ) be the set of random variables X t + δt representingthe value of our investments at time t + δt that can be obtained by a continuoustime trading strategy with an initial budget given by (3.3) with X t = x . Thus A t + δt ( x, c t ) is the set of admissible investment returns given the budget and theconsumption.The Markovianity of Epstein–Zin preferences and the definition of Z t tocompute allow us to apply the dynamic programming principle to compute v n ( x, t ) = sup c t ≥ ,X t + δt ∈A t + δt ( x,c t ) [( x c t ) ρ + β { s t E t ( v n ( X t + δt , t + δt ) α ) } ρα ] ρ = sup c t ≥ ,X t + δt ∈A t + δt ( x,c t ) [( x c t ) ρ + βz ρt + δt { s t E t ( X αt + δt ) } ρα ] ρ . (3.4)6he second line follows from the first by the positive homogeneity of Epstein–Zinutility (3.2).If α >
0, we may compute the value ofsup X t ∈A t + δt ( c t ) E t ( X αt + δt )by solving the Merton problem for optimal investment over time period [ t, t + δt ],with initial budget X t , no consumption, and terminal utility function u ( x ) = x α .We find sup X t ∈A t + δt ( c t ) E t ( X αt + δt ) = (exp( ξ δt ) X t ) α (3.5)where ξ = sup a ∈ R [ a ( µ − r ) + r − a (1 − α ) σ ]= a ∗ ( µ − r ) + r −
12 ( a ∗ ) (1 − α ) σ , with a ∗ := µ − r (1 − α ) σ (3.6)For details see Merton’s paper [6] or [9] equations (3.47) and (3.48). Moreoverthe proportion invested in stock is given by a ∗ which is a constant determinedentirely by α and the market. In the case where α < X t ∈A t + δt ( c t ) E t ( X αt + δt ) . However, apart from the change of a sup to an inf everything is algebraicallyidentical, so the same formulae emerge.Putting the value (3.5) into our expression (3.4) for the value function weobtain v n ( x, t ) = sup c t ≥ [( x c t ) ρ + z ρt + δt β ( s t exp( αξ δt ) X αt ) ρα ] ρ = sup c t ≥ [( x c t ) ρ + β ( z t + δt exp( ξ δt ) s ( α − C ) t (1 − c t ) X t ) ρ ] ρ (3.7)where the last line follows from the budget equation (3.3). We define φ t := β ρ exp( ξδt ) s ( α − C ) t θ t := φ t z t + δt (3.8)so that (3.7) may be written as z t = sup c t ≥ [( c t ) ρ + θ ρt (1 − c t ) ρ ] ρ (3.9)Differentiating the expression in square brackets on the right-hand side, wesee that the supremum is achieved in equation (3.7) when c t = c ∗ t , where c ∗ t satisfies ρ ( c ∗ t ) ρ − − ρθ ρt ((1 − c ∗ t ) ρ − = 0 , c ∗ t , we should take c ∗ t = 0. Simplifying wemust have (cid:18) c ∗ t − c ∗ t (cid:19) ρ − = θ ρt . (3.10)So c ∗ t = (1 + θ ρ − ρ t ) − . (3.11)This expression for c ∗ t is non-negative, so it always gives the argument for thesupremum in (3.7). We obtain z t = [( c ∗ t ) ρ + θ ρt (1 − c ∗ t ) ρ ] ρ = c ∗ t (cid:20) θ ρt (cid:18) − c ∗ t c ∗ t (cid:19) ρ (cid:21) ρ = c ∗ t (cid:20) θ ρt (cid:16) θ ρρ − t (cid:17) − ρ (cid:21) ρ , by (3.10) , = c ∗ t (cid:104) θ − ρρ − t ) ρ (cid:105) ρ = c ∗ t (cid:104) θ ρ − ρ t (cid:105) ρ = (1 + θ ρρ − t ) − (1 + θ ρ − ρ t ) ρ , by (3.11) , = (1 + θ ρ − ρ t ) − ρρ . We conclude that z ρ − ρ t = 1 + θ ρ − ρ t = 1 + φ ρ − ρ t z ρ − ρ t + δt (3.12)where φ is given by equation (3.8). We observe also that equation (3.11) for theoptimal consumption rate per survivor simplifies to c ∗ t = z ρρ − t . (3.13)We summarize our findings below. Theorem 3.1 (Optimal investment with Epstein–Zin preferences) . Supposethat an individual has probability p t δt of dying in the interval [ t, t + δt ] . Indi-viduals consume at fixed time points i δt . By the time N δt , death is certain.Between time points, we may trade in the Black–Scholes–Merton market (2.1) .Let C = 0 if we are interested in optimizing the consumption of an individualand C = 1 if we are interested in the collectivised problem. The utility of eachindividual is given by Epstein–Zin utility of the form (1.1) . Then(i) The optimal proportion of stock investments is determined entirely by themonetary-risk-aversion α and the market. In particular it is independentof time and wealth. It is given by the value a ∗ given in equation (3.6) . ii) The optimal Epstein–Zin utility is given by xz t where z t obeys the differ-ence equation (3.12) . The value of φ t is given in equation (3.8) . Theoptimal Epstein–Zin utility may be computed recursively since z Nδt = 0 α .(iii) The consumption for each survivor at time t is given by γ t = x t c ∗ t where c ∗ t is given by (3.13) . It is interesting to see how wealth and the consumption per individual vary overtime.
Theorem 4.1.
Under the same conditions as 3.1, the optimal fund value persurvivor at time t , X t , follows a log normal distribution. Write µ Xt and σ Xt forthe mean and standard deviation of log X t so that log X t ∼ N ( µ Xt , ( σ Xt ) ) . (4.1) The standard deviation is given by σ Xt = σa ∗ √ t. (4.2) where a ∗ is given by (3.6) . The mean satisfies the difference equation µ Xt + δt = µ Xt + log( s − Ct ) + log (cid:16) − z ρρ − t (cid:17) + ˜ ξδt, µ X = log( x ) (4.3) where z t is given by (3.12) and where we define ˜ ξ by the same formula used todefine ξ but with α set to zero, i.e. ˜ ξ := a ∗ ( µ − r ) + r −
12 ( a ∗ ) σ . (4.4) The optimal consumption per survivor γ t at time t is also log normally dis-tributed with log γ t ∼ N ( ρρ − log( z t ) + µ Xt , ( σ Xt ) ) . (4.5) The mean of the log consumption per survivor satisfies the equation E (log γ t + δt | γ t ) = log( γ t ) + log( s − Ct ) + ρ − ρ log( φ t ) + ˜ ξδt (4.6) where φ t is given by equation (3.8) .Proof. We suppose as induction hypothesis that the distribution of X t is asdescribed at time t .The budget equation (3.3) then tells us that the wealth per survivor afterconsumption, X t , satisfies X t = s − Ct (1 − z ρρ − t ) X t . X t ∼ N ( µ Xt + log( s − Ct (1 − z ρρ − t )) , ( σ Xt ) ) . Our investment strategy from t to ( t + δt ) is a continuous time trading strategywhere we hold a fixed proportion of our wealth in stocks at all time. So, in theinterval ( t, t + δt ], X t satisfies the SDEd X t = (1 − a ∗ ) rX t d t + a ∗ X t ( µ d t + σ d W t ) , X t = X t . By Itˆo’s lemma we findd(log X ) t = (1 − a ∗ ) r d t + a ∗ (( µ − a ∗ σ ) d t + σ d W t ) , log X t = log X t = ˜ ξ d t + a ∗ σ d W t . (4.7)We deduce that (log X t + δt − log X t ) conditioned on the value of X t will follow anormal distribution with mean ˜ ξ δt and standard deviation a ∗ σ √ δt . Moreoverthe random variable (log X t + δt − log X t ) is independent of X t .The sum of independent normally distributed random increments yields anew normally distributed random variable, and one can compute the mean andvariance by adding the mean and variance of the increments. Hencelog X t + δt ∼ N ( µ Xt + δt , ( σ Xt + δt ) )where µ Xt + δt = µ Xt + log( s − Ct (1 − z ρρ − t )) + ˜ ξ δt (4.8)and ( σ Xt + δt ) = ( σ Xt ) + ( a ∗ ) σ δt. (4.9)Solving the recursion (4.9) yields equation (4.2). The result for X t now followsby induction.Equation (4.5) follows from equation (3.13). Using (4.5), (4.3) we calculate E (log γ t + δt | γ t ) − log( γ t )= ρρ − z t + δt ) − log( z t )) + log( s − Ct ) + log(1 − z ρρ − t ) + ˜ ξδt = ρ − ρ (log( z t ) − log( z t + δt )) + log( s − Ct ) + log (cid:32) z ρ − ρ t − z ρ − ρ t (cid:33) + ˜ ξδt = log( s − Ct ) + log (cid:16) φ ρ − ρ t (cid:17) + ˜ ξδt by equation (3.12).This completes the proof.To interpret Theorem 4.1 we specialize to the case of a market where µ = r = 0 and to preferences with β = 1. This represents the problem of consuminga fixed lump sum over time when there is no inflation but also no investmentopportunities. While not financially reasonable, this problem highlights how10ongevity risk affects consumption, when considered in isolation from marketrisk.In this case γ t is a deterministic function. We find from equations (4.6) and(3.8) that γ t + δt = (cid:18) s α − Cρ t (cid:19) ρ − ρ γ t . (4.10)We note that s t is a non-zero probability, so 0 < s t ≤
1. We may use equation(4.10) to compute whether γ increases or decreasing over time. We summarizethe results in Table 4.Perhaps surprisingly, we find that sometimes consumption increases overtime rather than decreases. In the collectivised case, this can be explained bythe fact that the pension will always be inadequate when α <
0. We say thata pension is inadequate if living an extra year on that pension decreases utility.If α <
0, living longer is considered negative by homogeneous Epstein–Zinpreferences, so we may wish to compensate individuals who have the misfortuneto live longer. We cannot identify these individuals in advance, so the only wayto provide this compensation is to increase consumption over time. Thus theincreasing consumption arises from the fact that when α <
0, the primary risk isthe inadequacy of the pension, when α >
0, the primary risk is dying young. Webelieve that this mixing of the notion of pension adequacy with the risk aversionparameter is a significant shortcoming of homogeneous Epstein–Zin preferences.It is the price one must pay for analytic tractability.In the individual case, the concern that one will die young is much moreserious. This is why for the individual problem, the fear of an inadequatepension only dominates when both α < < ρ < α = ρ corresponds to the case of standard von-NeumannMorgernstern preferences, in which case constant consumption is optimal in thecollectivised case as was proved in [2].More significantly, our result also shows the converse. Constant consumptionfrom one period to the next is only optimal if and only if either (i) the survivalprobability is one, or (ii) α = ρ so one is satisfaction risk-neutral. Hence, evenignoring market effects constant consumption will be suboptimal for any realisticparameter choices.We have not shown the case α > ρ in Table 4 as in this case one hasmonetary-risk-aversion but not satisfaction-risk-aversion. We found the result-ing behaviour to be difficult to interpret as rational, cautious (as understoodintuitively) strategies. We see this simply as a evidence that satisfaction-risk-aversion is the correct operationalization of the intuitive notion of risk-aversion.It is also interesting to calculate how consumption changes according to theavailable investment opportunities. If interest rates increase one may chooseto defer consumption to a later date to benefit from the increased rate. Toquantify this behaviour one wishes to calculate the elasticity of intertemporalsubstitution which is defined as follows. Definition 4.2.
The elasticity of intertemporal substitution at time t is defined11ollectivised Individual α < ρ < < α < < ρ < < α < ρ < α = ρ < < < α = ρ < µ = r = 0, β = 1 and α ≤ ρ .to be EIS t := 1 δt dd r ( E (log( γ t +1 )) − log( γ t )) . When γ t is deterministic, this definition corresponds with the standard definition[5]. Theorem 4.1 allows us to calculate this elasticity. Corollary 4.3.
For the optimal investment strategy of Theorem (3.1) we have
EIS t = 11 − ρ (cid:18) − ( µ − r )(1 + α ( ρ − α − σ (cid:19) . If µ = r this simplifies to EIS t = 11 − ρ . In the case of von Neumann-Morgernstern utility we have
EIS t = 11 − ρ (cid:18) − µ − rσ (cid:19) . Proof.
From (4.6) and (3.8) we immediately findEIS t = dd r (cid:18) ρ − ρ ξ − ˜ ξ (cid:19) . The result is now a simple calculation from (3.6) and (4.4).
Let v n be the value function (2.5) for the optimal investment problem for n in-dividuals with homogeneous Epstein–Zin preferences. By positive homogeneitywe may define z n,t := v n (1 , t ), so that v n ( x, t ) = xz n,t .Let I i,t + δt denote the event that both12i) there are i survivors at time t + δt , i.e. n t + δt = i .(ii) the individual whose utility we wish to calculate is one of those survivors,so τ ι > t .Recall that X t denotes the value of the fund per survivor at time t beforeconsumption and mortality. E t ( Z αt + δt | I i,t + δt ) = E t ( v i ( X t + δt , t + δt ) α | I i,t + δt ) . Hence E t ( Z αt + δt ) = n t (cid:88) i =1 in t S t ( n t , i ) E t ( v i ( X t + δt , t + δt ) α | I i,t + δt ) . We now let A t + δt ( x, c t ) be the set of random variables X t + δt representingthe value of the fund per survivor at time t + δt before consumption thatcan be obtained by a continuous time trading strategy given initial capital X t = n t + δt n t (1 − c t ) X t per survivor when X t = x . Writing c t for the rate ofconsumption and using the dynamic programming principle we find v n ( x, t ) = sup ct ≥ X t + δt ∈A t + δt ( x,c t ) ( x c t ) ρ + β (cid:40) n (cid:88) i =1 in S t ( n, i ) E t ( v i ( X t + δt , t + δt ) α | I i,t + δt ) (cid:41) ρα ρ = sup ct ≥ X t + δt ∈A t + δt ( x,c t ) ( x c t ) ρ + β (cid:40)(cid:32) n (cid:88) i =1 in S t ( n, i ) z αi,t + δt E t ( X αt + δt | I i,t + δt ) (cid:33)(cid:41) ρα ρ . (5.1)We used positive homogeneity to obtain the last line. The argument of Section3 tells us how to optimize over X t . Hence we find z n,t = sup c t ≥ ( c t ) ρ + β (cid:32) n (cid:88) i =1 (cid:18) in (cid:19) − α S t ( n, i ) z αi,t + δt (cid:33) ρα (exp( ξ δt )(1 − c t )) ρ ρ where ξ is as defined in equation (3.6). The optimal investment policy is alsodescribed in equation (3.6), and as before it depends only on the market andthe monetary-risk-aversion parameter α .We may rewrite our expression for z n,t as follows: z n,t = sup c t ≥ (cid:104) ( c t ) ρ + ˜ θ ρn,t (1 − c t ) ρ (cid:105) ρ (5.2)where ˜ θ n,t = β ρ exp( ξ δt ) (cid:32) n (cid:88) i =1 (cid:18) in (cid:19) − α S t ( n, i ) z αi,t + δt (cid:33) α . (5.3)13quation (5.2) is structurally identical to equation (3.9). Hence from equation(3.12) we may deduce similarly that z ρ − ρ n,t = 1 + ˜ θ ρ − ρ n,t . (5.4)We state our results as a theorem. Theorem 5.1.
Let z n,t denote the optimal Epstein–Zin utility, (2.5) , for acollective of n individuals investing an amount at time t . The collective isallowed to invest in the Black–Scholes–Merton market (2.1) in continuous time.Individuals have independent mortality, with survival probability given by (2.2) .Then equations (5.4) and (5.3) together with the initial condition z n,T = 0 α allow us to compute the optimal Epstein–Zin utility by recursion. The value of ξ is given in equation (3.6) and the value of S ( n, i ) is given in equation (2.3)It is reassuring to check that ˜ θ ,t in equation (5.3) coincides with the valueof θ t for the individual problem given by equation (3.8). v n as n → ∞ In this section we will analyse the behaviour as n → ∞ to prove the following. Theorem 6.1.
Let z ∞ ,t denote the maximum Epstein–Zin utility at time t forthe infinitely collectivized case then z n,t = z ∞ ,t + O ( n − ) . Our proof strategy will be to approximate an expectation involving the bi-nomial distribution with a Gaussian integral which we can then estimate usingLaplace’s method. To get a precise convergence result, we need some estimateson the rate of convergence in the Central Limit Theorem. The estimates givenin [4] suit our purposes well. For the reader’s convenience we will summarizethe result we will need.We begin with some definitions. A random variable X is said to satisfyCram´er’s condition if its characteristic function f X satisfiessup {| f X ( t ) | | t > η } < η . Let Φ be the standard normal distribution. Given a set A ⊆ R ,and a function g , ω g ( A ) is defined to equal ω g ( A ) := sup {| g ( x ) − g ( y ) | | x, y ∈ A } . The set B (cid:15) ( x ) is the ball of radius (cid:15) around x .Let Q n be the appropriately normalized n -th partial sum of a sequence ofindependent identically distributed random variables X ( i ) for which Cram´er’scondition holds and which have finite moments have all orders. Appropriatelynormalized means normalized such that the central limit theorem implies Q n c such that for anybounded measurable function g | (cid:90) R g d( Q n − Φ) | ≤ c ω g ( R ) n − + (cid:90) ω g ( B cn − ( x ))dΦ( x ) . (6.2)The full result given in [4] is more general and more precise than we need. Letus explain how the statements are related. We have simplified our statement tothe one-dimensional case, we have assumed the X ( i ) are identically distributedand we have assumed all moments of X (1) exist. The statement in [4] is thereforemore complex, and in particular involves additional terms called ρ s,n defined inthe one-dimensional case by ρ s,n := 1 n ( n (cid:88) i =1 E | σ n X ( i ) | s )where σ n := n (cid:32) n (cid:88) i =1 Var( X ( i ) ) (cid:33) − . Our assumptions on X guarantee that ρ s,n is independent of n and so we havebeen able to absorb these terms into the constant c . In addition, our statementuses Theorem 1 of [4], together with remarks at the end of the second paragraphon page 242 about Cram´er’s condition.We are now ready to prove the desired convergence result. Proof of Theorem 6.1.
We proceed by a backward induction on t . The result istrivial for the case t = T . We now assume the induction hypothesis z n,t + δt = z ∞ ,t + δt + O ( n − ) . We wish to compute ˜ θ n,t , but only the sum in the expression (5.3) is difficult tocompute. We will call this sum λ n,t , so λ n,t := n (cid:88) i =1 (cid:18) in (cid:19) − α S t ( n, i ) z αi,t + δt . (6.3)Heuristically, one can approximate with a Gaussian integral using the Cen-tral Limit Theorem and then apply Laplace’s method to compute the limit as n → ∞ . This motivates the idea of decomposing the sum above into a “lefttail” for small values of i , a central term for values of i near the mean of theBinomial distribution np , and a “right tail” for larger values of i . We will in factbound the tails separately (Steps 1 and 2, below), and then we will be able torigorously apply a Central Limit Theorem argument to the central term (Step3). 15e compute S t ( n, i − S t ( n, i ) = is t (1 − i + n )(1 − s t ) . (6.4)We note that i ≤ ( n + 1)(1 − s t )( λ − s t + 1 = ⇒ is t (1 − i + n )(1 − s t ) ≤ λ = ⇒ S t ( i − , n ) S t ( i, n ) ≤ λ . So we define an integer N λ,t,n by N λ,t,n := (cid:22) ( n + 1)(1 − s t )( λ − s t + 1 (cid:23) and then equation (6.4) will ensure that we have exponential decay of S t ( i, n )as i decreases i ≤ N λ,t,n = ⇒ S ( n, i ) ≤ λ i − N λ,t,n S ( n, N λ,t,n ) . (6.5) Step 1.
We can now estimate the left tail of (6.3). There exists a constants C ,t , C ,t such that N ,t,n (cid:88) i =1 (cid:18) in (cid:19) − α S t ( n, i ) ≤ C ,t (cid:90) N ,t,n − N ,t,n + i S ( n, N ,t,n ) (cid:32)(cid:18) n (cid:19) − α + 1 (cid:33) d i ≤ C ,t S ( n, N ,t,n ) (cid:32)(cid:18) n (cid:19) − α + 1 (cid:33) . (6.6)To estimate this term, we observe that N ,t,n − N ,t,n = (cid:22) ( n + 1)(1 − s t ) s t + 1 (cid:23) − (cid:22) ( n + 1)(1 − s t )2 s t + 1 (cid:23) ≥ (cid:22) ( n + 1)(1 − s t ) s t + 1 − ( n + 1)(1 − s t )2 s t + 1 (cid:23) − (cid:22) ( n + 1) s t (1 − s t )( s t + 1)(2 s t + 1) (cid:23) − N ,t,n (cid:88) i =1 (cid:18) in (cid:19) − α S t ( n, i ) ≤ C ,t S ( N ,t,n , n )2 − (cid:106) ( n +1) st (1 − st )( st +1)(2 st +1) (cid:107) +2 (cid:32)(cid:18) n (cid:19) − α + 1 (cid:33) ≤ C ,t − (cid:106) ( n +1) st (1 − st )( st +1)(2 st +1) (cid:107) +2 (cid:32)(cid:18) n (cid:19) − α + 1 (cid:33) . which decays exponentially as n → ∞ . Our induction hypothesis ensures thatthe z αi,t + δt are bounded, so we may safely conclude that λ n,t := n (cid:88) i = N ,t,n (cid:18) in (cid:19) − α S t ( n, i ) z αi,t + δt + O ( n − ) (6.7)16 tep 2. We apply the same strategy to the right tail. This time we compute S t ( n, i + 1) S t ( n, i ) = (1 − s t )( n − i )( i + 1) s t we define M λ,t,n = (cid:24) λn (1 − s t ) − s t λ (1 − s t ) + s t (cid:25) to ensure that i ≥ M λ,t,n = ⇒ S t ( n, i + 1) S t ( n, i ) ≤ λ . Repeating the same argument as for the left tail tells us that λ n,t := M ,t,n (cid:88) i = N ,t,n (cid:18) in (cid:19) − α S t ( n, i ) z αi,t + δt + O ( n − ) (6.8)We note that (cid:0) in (cid:1) − α is monotonic in i and that N ,t,n n and N ,t,n n tend to finite,non-zero limits as n → ∞ . We deduce that there exists a constant C ,t suchthat N ,t,n ≤ i ≤ M ,t,n = ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) in (cid:19) − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ,t . (6.9)This implies that M ,t,n (cid:88) i = N ,t,n (cid:18) in (cid:19) − α S t ( n, i ) ≤ C ,t M ,t,n (cid:88) i = N ,t,n S t ( n, i ) ≤ C ,t Using this together with our induction hypothesis, we may obtain from (6.8)that λ n,t := M ,t,n (cid:88) i = N ,t,n (cid:18) in (cid:19) − α S t ( n, i ) z α ∞ ,t + δt + O ( n − ) . (6.10) Step 3.
In order to apply the bound (6.2), we define a Bernoulli randomvariable X i,t which takes the value 1 if the i -th individual survives at time t and 0 otherwise. Thus S t ( n, i ) is the probability that (cid:80) nj =1 X j,n = i . We definescaled random variables ˜ X j,t of mean 0 and standard deviation 1 by˜ X j,t = X j − s t (cid:112) s t (1 − s t ) , and so the appropriately scaled partial sum Q n is given by Q n = 1 √ n n (cid:88) i =1 ˜ X n = 1 √ n n (cid:88) j =1 X j − s t (cid:112) s t (1 − s t ) = ( (cid:80) ni =1 X i ) − ns t (cid:112) ns t (1 − s t ) .
17e now wish to rewrite (6.10) as an integral. λ n,t = (cid:90) [ N ,t,n ,M ,t,n ] ( i ) (cid:18) in (cid:19) − α z α ∞ ,t + δt d( n (cid:88) j =1 X j,t )( i ) + O ( n − ) . We make the substitution i = ns t + x (cid:112) ns t (1 − s t ) to find λ n,t = (cid:90) (cid:104) [ N ,t,n ,M ,t,n ] ( ns t + x (cid:112) ns t (1 − s t )) × (cid:32) ns t + x (cid:112) ns t (1 − s t ) n (cid:33) − α z α ∞ ,t + δt (cid:105) d Q n ( x ) + O ( n − ) . (6.11)We may rewrite this as λ n,t = (cid:90) [ (cid:96) n,t ,u n,t ] ( x ) (cid:32) ns t + x (cid:112) ns t (1 − s t ) n (cid:33) − α z α ∞ ,t + δt d Q n ( x ) + O ( n − )(6.12)where (cid:96) n,t := ( N ,t,n − ns ) / (cid:112) ns (1 − s ) , u n,t := ( M ,t,n − ns ) / (cid:112) ns (1 − s ) . From our expressions for N ,t,n and M ,t,n one readily sees that (cid:96) n,t tends to −∞ at a rate proportional to O ( −√ n ) as n → ∞ . Likewise u n,t tends to + ∞ at a rate O ( √ n ) as n → ∞ . We will assume that n is large enough to ensurethat (cid:96) n,t < < u n,t .Let us define g by g = [ (cid:96) n,t ,u n,t ] ( x ) (cid:32) ns t + x (cid:112) ns t (1 − s t ) n (cid:33) − α . (6.13)By (6.9), g is bounded by a constant independent of n . Hence ω g ( R ) is boundedindependent of n . We can bound the derivative of g inside the interval ( (cid:96) n,t , u n,t ),independent of n . Hence for any x ∈ ( (cid:96) n,t , u n,t ) and for sufficiently large n , ω g ( B cn − ( x )) < C ,t n − for some constant C ,t independent of n . It followsthat (cid:90) [ (cid:96) n,t , u n,t ] ( x ) ω g ( B cn − ( x )) dΦ( x ) = O ( n − ) . (6.14)Since (cid:96) n,t tends to −∞ at a rate proportional to O ( −√ n ), since g is bounded,and since the normal distribution has super-exponential decay in the tails, wehave (cid:90) ( −∞ ,(cid:96) n,t ] ω g ( B cn − ( x )) dΦ( x ) = O ( n − ) (6.15)and similarly (cid:90) [ u n,t , ∞ ) ω g ( B cn − ( x )) dΦ( x ) = O ( n − ) . (6.16)18stimates (6.14), (6.15), (6.16) together with the bound on ω g ( R ) allow usto apply the Central Limit Theorem estimate (6.2) to (6.12). We note thatCram´er’s condition holds. The result is λ n,t = (cid:90) [ (cid:96) n,t ,u n,t ] ( x ) (cid:32) ns t + x (cid:112) ns t (1 − s t ) n (cid:33) − α z α ∞ ,t + δt dΦ( x ) + O ( n − )We now apply Laplace’s method to estimate this Gaussian integral (seeProposition 2.1, page 323 in [10] ) and obtain λ n,t = s − αt + O ( n − )From the definition of ˜ θ in equation (5.3) and our definition of λ n,t in (6.3) weobtain ˜ θ n,t = β ρ exp( ξ δt ) s − αα t z ∞ ,t + δt + O ( n − ) . We may now compare this with the definition of θ t given in (3.8) for the infinitelycollectivised case C = 1. We see that in this case˜ θ n,t = θ t + O ( n − ) . It now follows from the recursion relations for z n,t and z ∞ ,t (given in (3.12) and(5.4) respectively) together with our induction hypothesis that z n,t = z ∞ ,t + O ( n − ) . This completes the induction step and the proof.
We illustrate our results with some numerical examples. We will restrict ourattention to the case of von Neumann–Morgernstern preferences in this section.We refer the reader to the numerical results of [2] where we give numerical resultsfor more general homogeneous Epstein–Zin preferences. In that paper we alsocompare the results with those obtained using exponential Kihlstrom–Mirmanpreferences.For ease of comparison with [2] (and other pension models based on [8]) wechoose the parameter values given in Table 2. Due to the positive homogeneityof our model, the choice of value for X is unimportant.The mortality distribution we use is for women retiring at age 65 in 2019. Weobtained this distribution using the model “CMI 2018 F [1.5%]” as describedin [7].We define the annuity equivalent value of each investment-consumption ap-proach. We define this to be the price of an annuity which would give the samegain. We define the annuity outperformance byannuity outperformance := annuity equivalentbudget − . This gives a measure of the performance of the strategy relative to an annuityof the same cost. 19arameter Value r CPI . r . − r CPI µ . − r CPI σ . ρ − n In Figure 1 we show how the annuity outperformance depends upon the numberof individuals n in the collective. This plot shows that as few as 40 individualsare required to obtain most of the benefits of collectivisation. One does notneed a large fund to benefit from a collective investment: simply sharing riskwith one’s partner brings a substantial benefit. n $ Q Q X L W \ R X W S H U I R U P D Q F H , Q G L Y L G X D O &