Deep neural network for optimal retirement consumption in defined contribution pension system
DDeep neural network for optimal retirement consumptionin defined contribution pension system
Wen Chen and Nicolas Langrené Commonwealth Scientific and Industrial Research Organisation, Data61, RiskLab Australia Corresponding author. Email: [email protected]
July 28, 2020
Abstract
In this paper, we develop a deep neural network approach to solve a lifetime expected mortality-weighted utility-based model for optimal consumption in the decumulation phase of a defined contributionpension system. We formulate this problem as a multi-period finite-horizon stochastic control problem andtrain a deep neural network policy representing consumption decisions. The optimal consumption policyis determined by personal information about the retiree such as age, wealth, risk aversion and bequestmotive, as well as a series of economic and financial variables including inflation rates and asset returnsjointly simulated from a proposed seven-factor economic scenario generator calibrated from market data.We use the Australian pension system as an example, with consideration of the government-funded means-tested Age Pension and other practical aspects such as fund management fees. The key findings from ournumerical tests are as follows. First, our deep neural network optimal consumption policy, which adaptsto changes in market conditions, outperforms deterministic drawdown rules proposed in the literature.Moreover, the out-of-sample outperformance ratios increase as the number of training iterations increases,eventually reaching outperformance on all testing scenarios after less than 10 minutes of training. Second,a sensitivity analysis is performed to reveal how risk aversion and bequest motives change the consumptionover a retiree’s lifetime under this utility framework. Our results show that stronger risk aversion generatesa flatter consumption pattern; however, there is not much difference in consumption with or withoutbequest until age 103. Third, we provide the optimal consumption rate with different starting wealthbalances. We observe that optimal consumption rates are not proportional to initial wealth due to theAge Pension payment. Forth, with the same initial wealth balance and utility parameter settings, theoptimal consumption level is different between males and females due to gender differences in mortality.Specifically, the optimal consumption level is slightly lower for females until age 84.
Keywords : decumulation, retirement income, deep learning, stochastic control, economic scenario gen-erator, defined-contribution pension, optimal consumption
JEL Classification : C45, D81, E21, C53
MSC Classification : 91G60, 62P05, 62M45, 93E20, 91B70, 65C05, 91B16
The global trend of transition from defined benefit (DB) to defined contribution (DC) system means that re-sponsibility switched from employers to employees, making life-cycle management an important and relevanttopic to every individuals. In Australia, the DC system established in 1992, requires employers to contributea minimum percentage (Superannuation Guarantee rate ∗ ) of employee’s earnings to a superannuation fund.Over 28 years, the Australia DC system has reached a mature stage, and achieved retirement savings of ∗ The SG rate was 9.5% on 1 July 2014, and will remain 9.5% rate until 30 June 2021, and is set to have five annual increases,where the SG rate will increase to 12% by July 2025. a r X i v : . [ q -f i n . GN ] J u l round $3 trillion (AUD) in asset, and has the highest proportion in DC assets (86%) relative to DB assets(14%) (Towers Watson, 2020). More new retirees now own significant superannuation savings accumulatedduring their working lives. However, there is little guidance on how to achieve the best strategies after re-tirement. Also due to reasons such as fear of ruin and bequest motive, most retirees tend to only withdrawthe statutory minimum rate from their retirement account (Sneddon et al. 2016, De Ravin et al. 2019), thusnot optimising benefit from their retirement savings. Designing income solution becomes a major challengefor the Australian superannuation fund industry. The retirement income solution in the decumulation phaseis underdeveloped. In the academic as well as the industrial practitioner literature, the utility frameworkhas been used to analyse complex life-cycle management problems which involve consumption, asset alloca-tion and possibly other decisions. The utility function reduces the dimension of the problem by providing aquantitative measure of uncertain retirement outcome satisfaction resulting from different deterministic ordynamic financial strategies, such as drawdown strategies and investment strategies. Since the seminal con-tribution of Yaari (1964, 1965) on utility based life-cycle models in asset allocation and optimal consumption,alternative methods and extensions have been investigated.For example, Thorp et al. (2007) investigated optimal investment and annuitisation strategies for the decu-mulation phase using a Hyperbolic Absolute Risk Aversion (HARA) utility function. Huang et al. (2012)proposed a partial differential equation (PDE) framework for optimal consumption with stochastic force ofmortality and deterministic investment returns to maximise discounted expected Constant Relative RiskAversion (CRRA) utility over a random time horizon. Mao et al. (2014) also looked at the optimal consump-tion problem during decumulation, in conjunction with related decision problems such as optimal retirementage and optimal leisure time, using CRRA utility. Ding et al. (2014) used a CRRA utility, focusing onoptimal portfolio allocation and the effect of bequest. Andréasson et al. (2017) investigated the optimal con-sumption, investment and housing decisions with means-tested public pension in retirement, using a HARAutility function discounted by inflation. Butt et al. (2018) considered the optimal consumption and asset al-location problems with Age Pension taken into account, using CRRA utility. De Ravin et al. (2019) designeda mortality weighted CRRA utility-based metric called the Members Default Utility Function (MDUF) toassist industry to design retirement outcome solutions for the decumulation phase. Finally, a recent attemptto avoid the expected utility approach is Forsyth et al. (2019), in which a CVaR approach is used to man-age depletion risk via asset allocation throughout both the accumulation and decumulation phases in a DCsystem.In practice, the most popular approach for solving such optimal decumulation stochastic control problems isnumerical dynamic programming with state discretisation and interpolation (Rust 1996, Andréasson et al.2017, Butt et al. 2018, De Ravin et al. 2019, Jin et al. 2020). Other popular approaches include analytical orsemi-analytical solutions (Yaari 1965, Ding et al. 2014) and numerical schemes for Hamilton-Jacobi-Bellman(HJB) PDE (Cairns et al. 2006, Huang et al. 2012, Forsyth et al. 2019).The analytical solution approach is the most attractive, but is unfortunately usually infeasible or requirestoo significant simplifications of the original problem to remain useful in practice. The numerical dynamicprogramming and PDE approaches are able to solve more realistic problem with a greater set of features, butboth suffer from the curse of dimensionality, which puts a limit on the number of stochastic factors whichcan be accounted for.In order to break the limitations of these classical numerical approaches, we propose in this paper a deep neuralnetwork approach (DNNs, Goodfellow et al. 2016). The principle is to model the unknown consumption policyby a DNN and to optimise it directly by modern gradient descent techniques. This bypasses the dynamicprogramming principle altogether, overcomes the curse of dimensionality (Poggio et al. 2017, Han et al. 2018),and makes it possible to consider as many realistic features and stochastic factors as deemed necessary toobtain informative consumption advice in the decumulation phase in a DC system. Being able to accountfor a great range of customised features and personal objectives in the decision-making process with machinelearning technology shall improve member engagement in the DC system (Fry, 2019).Approximating policies by DNNs have been well explored in the reinforcement learning field, see Sutton andBarto (2018, Chapter 13), with great practical success as illustrated for example by the famous AlphaGo andAlphaGo Zero programs Sutton and Barto (2018, Chapter 16). For the most part, reinforcement learning isconcerned with infinite time horizon problems, coupled with discrete-valued control spaces. By contrast, thispaper addresses a finite time horizon stochastic control problem (due to mortality) with continuous policies2namely consumption). In this context, optimal policies should explicitly depend on time, either time sinceinception t or time to maturity T − t , where T is the maturity.Classically, the optimal policies of stochastic control problems with finite time horizon are estimated in abackward manner, taking advantage of the dynamic programming principle (Beckmann, 1968). Recently,DNNs have been used in this context in Huré et al. (2018), Bachouch et al. (2018) and Fécamp et al. (2019)for approximating value functions, policy functions or both. As an alternative to the classical backwarddynamic programming approach, Han and E (2016) suggested to use one DNN policy function for eachdecision time t i , yielding a collection of sub-networks, indexed by time, and trained simultaneously. Thisglobal policy learning approach has been used in Fécamp et al. (2019) for financial option hedging problems,in Guo et al. (2019) for robust portfolio allocation, and has been adapted to optimal stopping problems inBecker et al. (2019) with application to exotic option pricing.One modification of this global approach is to consider one single DNN and include time as part of theinputs, in addition to all the other state variables. The very same DNN can then be used at all decisiontimes. Fécamp et al. (2019) also implemented and tested this modification, and Li and Forsyth (2019) used itfor an asset allocation problem. This single DNN approach can be thought of as the adaptation of recurrentneural networks (RNN) to finite-horizon problems, for which policies do change over time and therefore timeneeds to be part of the inputs of this single DNN. As such a modification is much more parsimonious, easierto train and better suited to problems for which policies vary in a smooth manner over time, we make use ofthis approach in this paper to model and learn optimal consumption policies.The main contribution of this paper is the proposed use of a DNN-policy approach to realistic utility maximi-sation problems driven by multi-factor economic stochastic model. Our approach is very general: as it doesnot rely on dynamic programming, the objective function is not limited to expected utility and is allowed tobe more sophisticated an personalised; as it is simulation-based, all the realistic features of the problem, suchas fees and Age Pension policy can be easily accounted for; finally, the ability of DNNs to handle large-scaleproblems means that the economic scenario generator (ESG) used to represent the state of the economy cancontain as many stochastic variables as deemed necessary. The second contribution of this paper is a proposed7-factor ESG which we use to perform numerical experiments and obtain findings on realistic decumulationtest cases in the Australian DC pension system.The paper is organised as follows. Section 2 formulates the problem and introduces useful notations. Section2.2 introduces the ESG used for our numerical tests. Section 3 details the DNN numerical approach used tosolve our utility maximisation problem. Section 4 presents our numerical results on several test cases basedon the Australian superannuation system. Finally, Section 5 concludes the paper and suggests several areasof future work. In this section, we introduce our utility-based objective function (subsection 2.1), a stochastic economicmodel in the form of an Economic Scenario Generator (ESG, subsection 2.2) which provides us with MonteCarlo simulations of economic variables, means-tested Age Pension (subsection 2.3) and wealth dynamics(subsection 2.4).
The objective of this problem is to maximise the mortality-weighted expected utility through a dynamic con-sumption policy in order to obtain the optimal consumption level in the future under any market conditions.We formulate the objective function similar to the MDUF proposed in Bell (2017); De Ravin et al. (2019),with the wealth of the retiree following the dynamics defined in subsection 2.4.When entering retirement at time t = 0, the total utility of one single person is defined as follows. V ( w ) = max { c t } ≤ t ≤ T E " T X t =0 (cid:8) t p x u ( c t ) + t − | q x v ( w t ) (cid:9) (2.1)3ubject to c t ∈ [0 , w t + a t ] (2.2)where x is the retirement age (in years) which is set to be 67, and the maximum age is set to be 108, thereforethe time horizon is T = 41. c t is the annual consumption, w t is the wealth and a t is the Age Pension paymentsubjected to means test at decision time t , where t = 0 , , ..., T . All these three variables are expressed inreal terms, adjusted for inflation, when computing the utilities. The consumption c t should be positive andless than the total wealth plus the Age Pension payment at any time t . The consumption utility function u ( c t ) is a CRRA type with consumption risk aversion parameter ρ ( ρ = 0 being risk neutral) defined as u ( c t ) = c − ρt − ρ , (2.3)and the final utility of the residual wealth v ( w t ) if the person dies between t − t is defined as v ( w t ) = w − ρt − ρ (cid:18) φ − φ (cid:19) ρ , (2.4)with strength of bequest motive φ ∈ [0 , φ , the stronger the bequest motive. Conversely, φ = 0means there is no desire to leave wealth unspent after death. t p x is the probability of surviving at age x + t conditional on being alive at age x . It can approximatethe state of health. t − | q x is the probability of death during ( x + t − , x + t ] conditional on being aliveat age x . We assume that the male and female retirees are subject to the mortality rates in AustralianLife Tables 2015-17 published by Australian Government Actuary (AGA, Australian Government Actuary2017). Since the problem involves ageing and the time span covers several decades, it is important toallow for future improvements in mortality rates. We assume the mortality rates decrease with the 25-yearimprovement factor as published by the AGA, to project the mortality rates that could be expected to occurover an individual’s lifetime ∗ . We also assume that mortality rates are independent of retirement wealth,consumption and portfolio returns. To complete the definition of the utility maximisation problem (2.1), we need a model to describe economicvariables such as inflation and the asset returns which affect the wealth of the retiree as well as its age pensionentitlements. In Bell (2017); De Ravin et al. (2019), the authors used a fixed risk-free rate and assume thereturns of the risky asset follow a normal distribution. Such constant or fixed investment return distributionassumptions are common and convenient for computational reasons, but can be deemed too basic to properlycapture uncertainties inherent in the financial market for retirement management. In the present work,one convenient consequence of the proposed deep learning approach for solving (2.1) (Section 3) is that thespecific choice of model for the economic variables of the problem does not restrict the numerical feasibilityof the utility maximisation problem (2.1).In this paper, we propose to use a multivariate stochastic investment model in the form of an EconomicScenario Generator (ESG, Moudiki and Planchet 2016), which are commonly applied in the actuarial fieldto simulate future economic and financial variables. A well-known ESG is Wilkie’s four-factor investmentmodel (Wilkie, 1984) which models inflation rates, equity returns and bond returns as stochastic time seriesthrough a cascading structure to describe the investment returns. A series of updates and extensions of thismodel have been proposed on both practical and theoretical aspects (Wilkie, 1995; Şahin et al., 2008; Wilkieand Şahin, 2017, 2019). The Ahlgrim model (Ahlgrim et al., 2005) developed for the Casualty ActuarialSociety extends the model to include real estate prices. Zhang et al. (2018) revisit Wilkie’s model andexamine the model performance for the United States. Based on the model in Wilkie (1995), Butt and Deng(2012) propose an ESG model to investigate investment strategies for the Australian DC pension system.Chen et al. (2020a) extended the SUPA model of Sneddon et al. (2016) to a 14-factor SUPA model for theAustralian system, and use it to quantify uncertainty and model downside risks with retirement savings inthe accumulation and decumulation phases. ∗ The details on how to use life tables and the mathematical form of incorporating future improvements can be found at http://aga.gov.au/publications/life_table_2010-12/07-Part3.asp .2.1 Seven-factor economic scenario generator We design a seven-factor ESG covering inflation and six asset classes, which is sufficient for completing thedescription of the optimal decumulation problem (2.1).Our ESG models the dynamics of inflation q ( t ), risky asset returns such as domestic and international totalequity returns e ( t ) and n ( t ), real estate returns h ( t ), and defensive asset returns such as interest rates s ( t ) anddomestic and international bond returns b ( t ) and o ( t ). Similar to Wilkie’s model (Wilkie, 1984), we assumethe inflation rate q ( t ) follows a discretised mean-reverting Ornstein-Uhlenbeck process (a.k.a. AR(1) process).From there, the specific dynamics of each economic factor and their dependence are summarised in Table A.2in Appendix A. The model is calibrated to historical data from year 1992 when the Superannuation Guaranteestarted, to year 2020. The data are obtained from the Reserve Bank of Australia (RBA), Australian Bureauof Statistics (ABS) and Bloomberg. Table A.2 also reports the calibrated values on the considered dataset.The simulated inflation allows us to project the future Age Pension rates and the mean-test thresholds values,so we can compute the future Age Pension eligible payment under the current pension policy. We also useinflation-adjusted consumption and wealth to compute their utilities. The simulated asset returns allow usto compute the returns of the predefined portfolios described in the next subsection 2.2.2. We build investment portfolios across the aforementioned six different asset classes. According to the Gov-ernment website MoneySmart, there are four common types of life-cycle investment strategies:
Cash , Con-servative , Balanced and
Growth with 0%, 30%, 70% and 85% invested in growth assets respectively. In thispaper, we adopt the portfolios settings in Chen et al. (2020b) where the growth (risky) portfolio includes50% Australian equity (domestic) e ( t ), 30% international equity (excluding Australia) n ( t ) and 20% property h ( t ), and the defensive portfolio includes 30% risk-free term deposit s ( t ), 50% domestic bond b ( t ) and 20%international bond o ( t ) † . Then the returns of the growth and defensive portfolios are given as follows: R growth ( t ) = 50% e ( t ) + 30% n ( t ) + 20% h ( t ) ,R defensive ( t ) = 30% s ( t ) + 50% b ( t ) + 20% o ( t ) . In our numerical examples, we select the
Balanced investment strategy, with ω = 70% invested in growthassets and 30% invested in defensive assets. R ( t ) = ω · R growth ( t ) + (1 − ω ) R defensive ( t ) . (2.5)We keep these weights fixed throughout retirement.In Australia, all superannuation funds charge management fees, and the impact of fees should not be ignoredwhen modelling retirement consumption. Australian Government Productivity Commission (2018) reportedthat “ balances are eroded by fees and insurance ” and that fees were “ the biggest drain on net returns ”. Thetotal management fee consists of at least administration fee, super fund member fee with an indirect costratio depending on the chosen fund level, and investment fee depending on the chosen investment strategyand balance. In some superannuation funds, the investment fee is also associated with its performance, andthe total cost could include other fees, such as advice fees, exit fees and brokerage fees.Total fees vary significantly by superannuation funds level, balance and investment option. In our example,we use the rates provided by the MoneySmart superannuation calculator of the Australian Securities andInvestments Commission (ASIC) ‡ . We choose a medium-level fund with annual administration fee of $50,indirect cost ratio of 0 .
6% and
Balanced investment option with investment fee of 0 . † These weights reflect a lower exposure to both growth assets (50% v 66%) and international assets (25% v 36%) on theaverage Australian Prudential Regulation Authority (APRA) regulated superannuation fund asset allocations as at December2019, reflecting the conservative nature of typical retiree allocations. More information is available at (Retrieved June 4, 2020). ‡ More details about the superannuation fund fees and cost can be found at .3 Age Pension simulation The Age Pension is a government payment scheme which provides income to help Australian retirees to covertheir cost of living. It is paid to people who meet the retirement age requirement, subject to an income testand an asset test. The payment rates also depend on the family (single or couple) and homeownership status.The pension age in 2020 is 66 and will be gradually increased to 67 by 2023. The payment rates and thethresholds are adjusted by changes in Consumer Price Index (CPI) or the Pensioner and Beneficiary LivingCost Index twice a year § .In the following, we consider a 67-year-old single homeowner who converted all liquid assets into the account-based pension and has no other income stream. We assume the payment rates and test thresholds are onlyadjusted for inflation rates annually. Based on these assumptions, the payment is determined by the wealth,deemed income from the liquid assets, and the compound inflation Q t = e Σ ts =0 q s . (2.6)The Age Pension A t this person can receive at time t is the minimum of the payment under the asset test A At and income test A It , that is A t ( W t , Q t ) = min (cid:0) A At , A It (cid:1) where A At = max (cid:0) A max t − τ A max (cid:2)(cid:0) W t − W At (cid:1) , (cid:3)(cid:1) A It = max (cid:0) A max t − τ I max (cid:2) r min (cid:0) W t , W It (cid:1) + r max (cid:0) W t − W It , (cid:1) − I t , (cid:3) , (cid:1) .A max t is the maximum Age Pension, W At is the asset test threshold for full pension, I t is the income test cutoffpoint . W It is the lower deeming asset threshold at time t . These four variables are adjusted for inflationwith x t = x Q t for x ∈ (cid:2) A max , W A , W I , I (cid:3) and t ∈ [0 , T ] . τ A and τ I are the taper rates for asset and incometest; r and r are the lower and higher deeming rates; r < r . These four rates are determined by the AgePension policy, therefore they are assumed to be constants in this paper. The parameters are listed in Table2.1. Parameter Value Parameter ValueFull Age Pension A max t $24 ,
619 Income test limit I t $4536Asset test limit W At $263 ,
250 Lower deeming rate limit W It $51 , τ A .
3% Lower deeming rate r . τ I
50% Higher deeming rate r . Table 2.1:
Means-tested Age Pension rates published as of June 2020.
In our model, the wealth is an endogenous stochastic variables which is determined by the market changesand the consumption decisions. The wealth at time t + 1 is given as follows: W t +1 = ( W t + A t ( W t , Q t ) − C t − Fee t ) e R t (2.7)where W t , A t , C t and Fee t are the future (non-deflated) value of wealth, Age Pension, consumption andfund management fees. The discounted variables w t = W t /Q t , a t ( W t , Q t ) = A t ( W t , Q t ) /Q t and c t = C t /Q t are the wealth, Age Pension and consumption in real terms. Note that the total consumption consists ofthe drawdown from the wealth plus the Age Pension payment. When evaluating the utility of the futureconsumption and residual wealth, we used consumption and wealth in real terms. After the problem description, in this section we focus on the numerical method and algorithm. We firstintroduce the objective function and then the DNN control policy which is a parametric function determiningthe optimal consumption at any time in any possible scenario. § More details about Age Pension can be found at .1 Empirical objective function Problem (2.1) is a discrete-time finite horizon stochastic control problem. Its optimal policy (consumption) isaffected in feedback form by the stochastic state variables of the ESG described in Subsection 2.2 and by thewealth (Subsection (2.4)). More specifically, the state variables of Problem (2.1) are the wealth W (equation(2.7)), the investment return R (equation (2.5)) and the compound inflation discount factor Q (equation(2.6)). We use a parametric consumption policy approach, meaning that we model the set of possible policiesby a class of functions ( t, x ) ∈ R d +1 (cid:1) c ( t, x ; β ) indexed by a parameter β ∈ R p , where p is the number ofparameters. For a fixed parameter β , the dynamics of the controlled state process X c = ( W t , R t , Q t ), valuedin R , is X c = x = ( W , R , Q ) X ct +1 = T ( X ct , c t ( t, X ct ; β ) , (cid:15) t +1 ) , t = 0 , ..., T − (cid:15) t ) is a sequence of i.i.d. random variables, c = c t ( t, X ct ; β ) is the consumption policy and T is aknown transition function given by equations (2.7)-(2.5)-(2.6) and the ESG dynamics in Table A.2.In practice, we use Monte Carlo simulations to estimate the expectation involved in the objective function(2.1). Let M be the number of Monte Carlo simulations. For each m = 1 , , . . . , M , we compute a simulationpath X c,m = x = ( W , R , Q ) X c,mt +1 = T (cid:0) X c,mt , c t ( t, X c,mt ; β ) , (cid:15) mt +1 (cid:1) , t = 0 , ..., T − (cid:15) mt , m = 1 , , . . . , M , are i.i.d. realisations of the random variable (cid:15) t . The counterpart of problem (2.1)with sample averaging and parametric control is given byˆ V M = max β ∈ R p M M X m =1 " T X t =0 (cid:8) t p x u ( c t ( t, X c,mt ; β )) + t − | q x v ( w c,mt ) (cid:9) (3.1) In order to solve the empirical problem (3.1) in practice, we still need to choose a specific class of parametricfunctions ( t, x ) ∈ R d +1 (cid:1) c ( t, x ; β ) to model the consumption policy. We choose to model the control policyby a deep neural network (DNN, Goodfellow et al. 2016). In other words, we consider functions c defined asa composition of linear combinations and nonlinear activation functions: c ( t, W, R, Q ; β ) = ( W + A t ( W, Q )) × S out b (3) + K X k =1 w (3) k ϕ (2) k ! (3.2) ϕ (2) k = S b (2) k + K X k =1 w (2) k ,k ϕ (1) k ! , k = 1 , , . . . , K (3.3) ϕ (1) k = S b (1) k + K X k =1 w (1) k ,k ϕ (0) k ! , k = 1 , , . . . , K (3.4) ϕ (0) k = S in (cid:16) b (0) k + w (0)0 ,k t + w (0)1 ,k W + w (0)2 ,k R + w (0)3 ,k Q (cid:17) , k = 1 , , . . . , K (3.5)for any input values ( t, W, R, Q ). Equations (3.2)-(3.3)-(3.4)-(3.5) give an example of fully connected deepneural network with four layers, including two hidden layers. The input layer (3.5) takes ( t, x ) as inputsare returns K “neurons” ϕ (0) k , referring to nonlinear transforms of linear combinations of the inputs. Thenonlinear transformation is performed by the activation function S in . Then, the first hidden layer (3.4)is obtained by a linear combination of the input neurons composed with the activation function S . Ina similar manner, the second hidden layer (3.3) is obtained by a linear combination of the neurons (3.4)composed with the activation function S . Finally, the output layer (3.2) is obtained by a linear combination7f the neurons (3.3) composed with the output activation function S out . We choose a sigmoid function S out ( x ) = 1 / (1 + e − x ) for the output activation function, scaled by the factor ( W + A t ( W, Q )) to enforce theconsumption constraint (2.2). For the other activation functions, we choose Rectified Linear Unit (ReLU)activation functions S in ( x ) = S ( x ) = S ( x ) = max(0 , x ).The set of parameters β contains all the weights w ( ‘ ) and biases b ( ‘ ) , ‘ = 0 , . . . β = (cid:26)(cid:16) b (0) k , w (0) k ,k (cid:17) k =1 ,...,K k =0 ,..., , (cid:16) b (1) k , w (1) k ,k (cid:17) k =1 ,...,K k =1 ,...,K , (cid:16) b (2) k , w (2) k ,k (cid:17) k =1 ,...,K k =1 ,...,K , (cid:16) b (3) , w (3) k (cid:17) k =1 ,...,K (cid:27) In order to train the parameters (i.e. finding the optimal parameters β ∗ maximising equation (3.1)), weperform a gradient descent using the Adam optimiser (Adaptive Moment Estimation, Kingma and Ba 2014),with parameter initialisation of He et al. (2015), using the Python machine learning library PyTorch (Paszkeet al., 2019), which takes care of gradient computations by automatic differentiation (Paszke et al., 2017).In practice, we consider a global consumption DNN with four layers as described by equations (3.2)-(3.3)-(3.4)-(3.5). There are four input dimensions (time t , wealth W , return R and inflation discount factor Q )and one output (consumption). We choose K = K = K = 20 neurons for every layer, set the optimiserlearning rate to 5 × − and the maximum number of iterations to 100 , M = 100 ,
000 simulations using the ESG described in subsection2.2. We perform our numerical tests on an Intel® CPU i7-7700 @ 3.60GHz ∗ and NVIDIA® GPU GeForce®GTX 1070 † , taking advantage of PyTorch’s built-in support for CUDA® ‡ .The next section describes our numerical results. In this section, we demonstrate numerical results from our trained DNN consumption policy, which estimatesthe optimal consumption taking into account age, gender, wealth, inflation, portfolio returns and the paymentsfrom means-tested Age Pension. In the numerical test, the individual is assumed to be aged 67 in year 2020,a single male homeowner with total wealth of A$500 ,
000 in account-based pension and no other testableassets or financial asset. He chooses the
Balanced investment strategy, with 70% growth assets in domestic,international equities and property, and 30% in defensive assets in deposit, domestic and equity bonds asdescribed in subsection 2.2.2. This person is entitled to the Australian mean-tested Age Pension. Themaximum age is set to 108 years, so the total time horizon is 41 years.We use our ESG to simulate M = 100 ,
000 scenarios for each state variable to train our DNN optimal con-sumption policy. We simulate an independent testing set of M scenarios to compute the realised wealthtrajectories, Age Pension eligibility and realised lifetime utility (equation (3.1)) from the trained DNN con-sumption policy. We compare the consumption policy from our DNN approach with other alternative draw-down strategies using the lifetime utility with risk aversion ρ = 5 and bequest motive φ = 0 .
5. We alsoconduct sensitivity analysis for retirees who are less risk averse ( ρ = 2), and those who have no bequestmotive ( φ = 0). We also show how this policy performs with different initial starting wealth. Finally, weillustrate how the difference in mortality rates between males and females affect their respective optimalconsumption policies. At time t , the trained DNN consumption policy estimates the optimal consumption level based on the availableinformation from simulated inflation, portfolio returns, wealth and the Age Pension eligibility which dependson the wealth and compound inflation before time t . In Figure 1, we randomly pick one realisation todemonstrate how our trained DNN policy works in the face of changes in inflation and portfolio returns. ∗ † ‡ https://pytorch.org/docs/stable/notes/cuda.html ,
917 with partial Age Pensionpayment, the resulting wealth balance at age 68 is $440 ,
506 after being adjusted for inflation according to Eq.2.7. At age 73 and 84, the huge drops in the market returns R t have impact on the wealth and consumptionat age 74 and 85. The consumption decreases over time as mortality rate increases. At age 107, the adjustedconsumption is only $22 , ,
929 at age 108. Note that after age 67,the wealth at time t > t and t + 1. Figure 1:
One realisation of consumption and wealth under one simulated inflation and portfolio return with DNN optimalconsumption policy.
With the lifetime utility measure with risk aversion ρ = 5 and bequest motive θ = 0 .
5, we compare ourdynamic DNN consumption strategy with six alternative deterministic drawdown strategies discussed inChen et al. (2020b) which are: (1) the mandated age-related minimum drawdown rules ∗ ; (2) of the initialbalance in real term (Bengen, 1994); (3) the Rule of Thumb ( RoT ), which is the minimum of the first digitof the age of the individual as the drawdown rate, plus 2% if the wealth is between A$250,000 and A$500,000(Bell, 2017; De Ravin et al., 2019); (4) the Association of Super Funds Australia’s (ASFA, 2020)
Modest $28 ,
220 and (5)
Comfortable $44 ,
183 lifestyle as of Mar 2020; and (6) a
Luxury consumption of $50 ,
000 peryear.We demonstrate such a comparison using one simulated path in Figure 2. The realised lifetime utilitiesfor this path under DNN,
RoT , minimum , modest , , comfortable and luxury drawdown strategies are: U DNN = − . × , U RoT = − . × , U min = − . × , U mod = − . × , U = − . × , U cmft = − . × , U lux = − . × respectively. By definition, the realised lifetime utilities arenegative, and the higher (i.e. the closer to zero) the better. For this particular scenario, the DNN strategyoutperforms all the other strategies, yielding the highest lifetime utility. The RoT strategies, which is a simplerule developed under similar lifetime utility framework, takes the second place, followed by the mandated minimum and modest target drawdown strategies. Using drawdown rule with an annual consumption of$20 ,
000 in real term, the remaining wealth keeps growing. It is not optimal as such a consumption level is ∗ The minimum withdrawal rate is available at
9o low that the accumulated wealth makes this person no longer eligible to receive any Age Pension paymentafter age 79 due to the asset test, and this person has to live solely on the 4% withdrawal from the account-based pension. The comfortable and luxury target consumption strategies are not sustainable under thisutility measure, as the wealth runs out at age 106 and 96 respectively.
Figure 2:
The simulated consumption and wealth paths under seven different drawdown strategies for one realisation of inflationand portfolio returns.
To investigate the utility differences across the 100 ,
000 test simulations, we plot the kernel density estimateof the utility differences U DNN − U i with i ∈ { RoT , min , mod , cmft , , lux } between the DNN policy andthe six deterministic drawdown strategies in Figure 3. The horizontal axis of each subplot is log-scaled. TheDNN policy performs better than all these strategies. The RoT strategy yields the least utility differenceand performs relatively better than the rest of the fives strategies.10 igure 4:
Number of DNN outperformance paths over the number of training iterations (in log-scale).
Figure 3:
The kernel density estimate of the utility differences between the DNN drawdown and six deterministic drawdownstrategies.
In addition, the performance of the DNN policy can also be determined with respect to the number of trainingiterations. In order to minimise the loss function (3.1), the DNN policy is trained on M = 100 ,
000 MonteCarlo scenarios and 100 ,
000 iterations. To show the relationship between the performance of the DNN policyand the number of training iterations, we plot on Figure 4 the total number of testing scenarios for which theDNN consumption policy outperforms alternative strategies versus the number of iterations. Starting withrandom weights, our trained DNN policy begins to outperform the modest , luxury and drawdown rulesin more than 80% of the scenarios after a few iterations, and in more than 90% of the scenarios after 200iterations (completed in 2 minutes) except for the RoT strategy. Eventually, after less than 1000 iterations(completed in 10 minutes), the trained DNN policy outperforms the other six strategies on all 100 ,
000 testingpaths. 11 .3 Utility parameters sensitivity
In this utility maximisation problem (2.1), the optimal policy also depends on the choice of utility parame-ters. We compare the differences in median consumption and median wealth between two consumption riskaversions ρ and two bequest motives φ for a male retiree. We use ρ = 5 and φ = 0 . ρ = 2 reported in Figure 5, and a lower bequestmotive φ = 0 reported in Figure 6. From the upper subplot of Figure 5, we can see that under lower riskaversion ρ = 2, the median of the first year optimal annual consumption (dashed orange curve) can startfrom a higher level of $59 , ,
910 for higher risk aversion ρ = 5 (solid orange curve). Thelower subplot of Figure 5, shows that the consumption gap decreases first and changes of sign after age 87.To sum up, retirees with higher consumption risk aversion tend to save more for later consumption to avoidpotential shortfalls, and their consumption curve is flatter.The upper subplot of Figure 6 shows the median consumption and median wealth with ( φ = 0 .
5) and without( φ = 0) bequest motive. Without any bequest motive, the medians of initial consumption c is $52 , ,
910 with bequest motive. The lower subplot of Figure 6shows that there is no significant difference in consumption until age 102 when the person can start to spendmore if no desire to leave any legacy. Such results may be counter-intuitive, as one might think that a lotmore can be spent in early retirement. However, our results indicate that even without any bequest motive,overspending at early retirement is not optimal. The reason is that capital returns play a very important rolein the wealth aggregation in retirement as there is a tradeoff between consumption level and sustainabilityof the wealth. Therefore, the optimal consumption is less sensitive to the bequest motive parameter than tothe risk aversion parameter.
Figure 5:
Comparison of the median consumption and medianwealth with risk aversion parameter ρ = 5 and ρ = 2, withbequest motive φ = 0 .
5, and the consumption difference over aretiree’s lifetime.
Figure 6:
Comparison of the median consumption and medianwealth with φ = 0 . φ = 0 bequest motive, withrisk aversion ρ = 5, and the consumption difference over aretiree’s lifetime. We now investigate the effect of the initial wealth balance. We demonstrate the medians of the simulatedwealth and consumption with starting balances of $300 , , , ,
000 in Figure 7 and theconsumption rates (consumption divided by wealth) in Figure 8. Because of the eligibility to receive fullor partial Age Pension from the government for the retirees who own asset below the asset test limit, theretirement consumption is not proportional to the initial wealth. Retirees with lower wealth can draw downat a higher rate as there exist a pension safety net . 12 igure 7:
Comparison of the wealth and consumption over aretiree’s lifetime with different initial wealth w Figure 8:
Comparison of the consumption rate over aretiree’s lifetime with different initial wealth w Finally, we investigate the effect of gender on optimal consumption decisions during retirement. The LifeTable 2015-2017 from ABS shows that the mortality of males is higher than that of females. In 2020, thelife expectancy with 25 year improvement factor for age 65 is 85.5 for male and 87.9 for female (AustralianGovernment Actuary, 2017). To obtain realistic measure of longevity, we consider the possible future im-provements in mortality that may occur in the future. We implement the adjustment for mortality with the25 year improvement factor † to project the mortality that could be expected to occur over a retiree’s lifetime.In Figure 9, we plot the median of the simulated optimal consumption controlled by the DNN policy and theresulting wealth for a male and a female aged 67 with initial balance w = 500 , In this paper, we propose a deep neural network (DNN) optimal consumption policy analysis for the de-cumulation phase in a defined contribution (DC) pension system under a lifetime utility framework. Thetrained DNN can estimate the optimal consumption level for a retiree under any financial conditions thatcould happen during retirement. This DNN approach is independent of the simulation tool used to representthe possible future economic conditions. In practice, we propose and use a specific seven-factor EconomicScenario Generator (ESG), sufficient for decumulation analysis, and calibrate it to Australian market data.We demonstrate that a DNN-based decumulation policy can successfully be trained from a realistic mul-tivariate economic scenario generator in a matter of minutes. Our numerical tests demonstrate that theDNN-based policy outperforms six common deterministic decumulation rules by yielding higher lifetime util-ity for all testing scenarios after less than 1000 training iterations (completed in 10 minutes). We also reportthe densities of lifetime utility improvement provided by the DNN consumption policy compared to thesedeterministic policies. A sensitivity analysis with respect to the risk aversion parameter ρ reveals that ahigher consumption risk aversion generates a smoother and flatter consumption pattern, with less variationover time. This contrasts with the bequest motive parameter φ , which is shown to have a negligible effect on † Details about the mortality improvement factors can be found on the Australian Government Actuary website igure 9: Comparison of the median consumption and median wealth between a male and a female with ( φ = 0 .
5, left) andwithout ( φ = 0, right) bequest motive under consumption risk aversion ρ = 5. optimal consumption until very late in retirement. Due to the safety net provided by the means-tested AgePension payment, we observe that the optimal consumption is not proportional to the wealth of the retiree.Finally, gender has a small effect on optimal consumption due to the difference in mortality between malesand females. Our results suggest that females should consume slightly less in early retirement (before age84) to mitigate their greater longevity risk.The DNN-policy approach developed in this paper for Australia can be applied to similar DC systems inother countries (e.g. UK DC system, USA 401(k)) with the necessary country adjustments and taking intoaccount the government funded pension payment policy. Moreover, it can be extended to incorporate otherfinancial decisions such as asset allocation or partial annuitisation, which we leave for future research. References
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A Calibration of ESG parameters
This Appendix details the historical data used to calibrate the ESG and provides the calibrated parametervalues in order to make our results replicable. The ESG parameters are calibrated by ordinary least squaresmethod using historical data from 1992, the year when the compulsory contribution system started, to 2020as shown in Table A.1. The short-term interest rates s ( t ) are the 90-day bank accepted bills from the RBAwebsite. Most data are indices, except for the short-term interest rates s ( t ), therefore we first compute thereturns using the log-ratio of two consecutive indices in time. The inflation rate q ( t ) = log (CPI (t) / CPI (t-1))is the log-ratio of the all groups Consumer Prices Index CPI( t ) from the RBA. The domestic e ( t ) andinternational equity total returns n ( t ) are computed from the S&P/ASX200 Accumulation Index E ( t ) andMSCI World ex-Australia Net Index (in Australian Dollar) N ( t ) respectively. While the domestic b ( t ) andinternational bond returns o ( t ) are computed from the AusBond Index B ( t ) and Citigroup World GovernmentBond Index O ( t ). These four indices are downloaded from Bloomberg. The house price growth h ( t ) comesfrom the House Price Index HPI( t ) which is the Established House Price Index (weighted average of 8 capitalcities) from the ABS website ∗ .We assume that the residuals are conditionally independent. Figure 10 shows the lower triangular part oftheir correlation matrix which exhibit low correlations as expected.We plot the histogram and the kernel density estimate of the empirical errors. The empirical distributionsof the residuals look not too far from being centered, unimodal, symmetric, and can be approximated bynormal variables. ∗ The ABS has compiled a House Price Index (HPI) since 1986. A significant review of the HPI occurred in 2004 and a newseries of HPI was introduced in 2005. We scaled the HPI before 2005 to keep the house price returns consistent with the databefore 2005. More details about the HPI is available at t ) s ( t ) % E ( t ) N ( t ) B ( t ) O ( t ) HPI( t )1992 59.7 6.42 5808.3 902.2 1692.5 299.6 28.731993 60.8 5.25 6291.6 1602.4 1928.3 373.1 29.501994 61.9 5.47 7252.1 1591.3 1906.5 357.9 30.491995 64.7 7.57 7756.2 1817.0 2133.0 438.2 30.901996 66.7 7.59 8866.4 1937.9 2334.6 396.1 31.231997 66.9 5.28 11313.3 2491.4 2725.8 432.3 32.111998 67.4 5.32 11542.2 3541.8 3022.4 545.5 34.891999 68.1 4.93 13251.6 3830.8 3121.7 531.5 36.922000 70.2 6.23 15628.0 4742.7 3314.3 604.5 40.502001 74.5 4.97 17044.8 4457.9 3559.7 592.3 43.822002 76.6 5.07 16245.3 3410.4 3781.3 714.1 52.12003 78.6 4.67 15966.7 2778.2 4151.3 696.6 61.22004 80.6 5.49 19416.7 3316.6 4247.9 712.2 70.92005 82.6 5.66 24533.9 3318.6 4578.8 641.5 71.02006 85.9 5.96 30405.1 3978.3 4735.1 712.2 73.82007 87.7 6.42 39119.1 4287.3 4923.0 641.5 80.92008 91.6 7.81 33875.3 3375.9 5141.6 663.5 91.82009 92.9 3.25 27053.6 2827.8 5698.0 819.1 86.82010 95.8 4.89 30610.0 2975.3 6145.6 807.7 103.12011 99.2 4.99 34200.7 3054.4 6486.5 704.4 103.22012 100.4 3.49 31904.5 3039.0 7290.8 755.4 99.72013 102.8 2.80 39163.3 4045.1 7492.6 807.9 103.12014 105.9 2.70 45991.2 4870.5 7950.1 837.1 114.42015 107.5 2.15 48602.3 4686.0 8397.5 935.2 123.02016 108.6 1.99 48872.4 4557.9 8986.8 1074.1 132.12017 110.7 1.72 55758.6 5386.9 9009.23 999.5 147.32018 113.5 2.07 63015.4 5987.6 9287.2 1057.4 150.62019 114.8 1.29 70291.8 6636.4 10176.2 1174.3 138.12020 116.6 0.102 64116.9 6583.0 10601.6 1262.5 150.3 Table A.1:
Australian historical data from June 1992 to June 2020.
Figure 11:
Kernel density estimate of empirical errors. q ( t ) µ q . q ( t ) = (1 − φ q ) µ q + φ q q ( t −
1) + (cid:15) q ( t ) φ q . σ q . s ( t ) µ S . S ( t ) = φ S S ( t −
1) + (1 − φ S ) ( µ S − µ q ) + (cid:15) s ( t ) φ S . s ( t ) = S ( t ) + q ( t ) σ S . e ( t ) µ e . e ( t ) = φ e e ( t −
1) + (1 − φ e ) µ e + (cid:15) e φ e . σ e . n ( t ) ψ n, − . n ( t ) = ψ n, + ψ n, n ( t −
1) + ψ n, e ( t ) + (cid:15) n ( t ) ψ n, . ψ n, . σ n . b ( t ) ψ b, . ψ b, − . b ( t ) = ψ b, + ψ b, b ( t −
1) + ψ b, n ( t ) + (cid:15) b ( t ) ψ b, − . σ b . o ( t ) ψ o, − . ψ o, . o ( t ) = ψ o, + ψ o, e ( t ) + ψ o, n ( t ) + (cid:15) o ( t ) ψ o, − . σ o . h ( t ) ψ h, . h ( t ) = ψ h, + ψ h, q ( t ) + ψ h, b ( t ) + (cid:15) h ( t ) ψ h, − . ψ h, . σ h . Table A.2: