A Stochastic Control Approach to Defined Contribution Plan Decumulation: "The Nastiest, Hardest Problem in Finance"
AA Stochastic Control Approach to Defined Contribution PlanDecumulation: “The Nastiest, Hardest Problem in Finance”
Peter A. Forsyth a August 12, 2020
Abstract
We pose the decumulation strategy for a Defined Contribution (DC) pension plan as aproblem in optimal stochastic control. The controls are the withdrawal amounts and the assetallocation strategy. We impose maximum and minimum constraints on the withdrawal amounts,and impose no-shorting no-leverage constraints on the asset allocation strategy. Our objectivefunction measures reward as the expected total withdrawals over the decumulation horizon, andrisk is measured by Expected Shortfall (ES) at the end of the decumulation period. We solvethe stochastic control problem numerically, based on a parametric model of market stochasticprocesses. We find that, compared to a fixed constant withdrawal strategy, with minimumwithdrawal set to the constant withdrawal amount, the optimal strategy has a significantlyhigher expected average withdrawal, at the cost of a very small increase in ES risk. Tests onbootstrapped resampled historical market data indicate that this strategy is robust to parametricmodel misspecification.
Keywords: optimal control, DC plan decumulation, variable withdrawal, Expected Shortfall,asset allocation, resampled backtests
JEL codes:
G11, G22
AMS codes:
The traditional Defined Benefit (DB) pension plan is in the process of disappearing for new entrantsinto the labour market. In some countries, notably Australia, DB plans have been replaced byDefined Contribution (DC) plans almost exclusively. Assuming the DC plan holder has accumulated a reasonable amount in her DC plan account,the retiree is faced with an enormous challenge. The retiree has to devise an investment policy anda withdrawal strategy during the decumulation phase. Nobel laureate William Sharpe has referredto DC plan decumulation as “the nastiest, hardest problem in finance” (Ritholz, 2017). a David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 34415. See, for example, “The extinction of defined-benefit plans is almost upon us,”
Globe and Mail,October 4, 2018. In Australia, DC plans have 86% of pension assets, compared with 14% in DB assets.(Towers-Watson, 2020) a r X i v : . [ q -f i n . C P ] A ug lthough it is often suggested that DC plan holders should purchase annuities upon retirement,this is rarely done (Peijnenburg et al., 2016). In fact, MacDonald et al. (2013) argue that in manyinstances, this entirely rational. Reasons for the lack of interest in annuities include meager returnsof annuities in the current low interest rate environment, poor annuity pricing, the lack of trueinflation protection, and no access to capital in the event of emergencies.For an extensive review of strategies for decumulation, we refer the reader to Bernhardt andDonnelly (2018) and MacDonald et al. (2013). A non-exhaustive list of the approaches discussedby these authors include use of traditional utility functions, practitioner rules of thumb, targetapproaches, minimizing probability of ruin and modern tontines. Previous decumulation strategiesare also summarized in Forsyth (2020b). Concerning the current state of DC plan decumulationstrategies, MacDonald et al. (2013) conclude “There is no solution that is appropriate for everyoneand neither is there a single solution for any individual.” We should mention that there is a standard rule of thumb for DC plan decumulation, termedthe four per cent rule . Based on historical backtests, Bengen (1994) suggests investing in a portfolioof 50% bonds and 50% stocks, and withdrawing 4% of the initial capital each year (adjusted forinflation). Over historical rolling year 30 year periods, this strategy would have never depleted theportfolio.Another recent strategy is based on the Annually Recalculated Virtual Annuity (ARVA) (Waringand Siegel, 2015; Westmacott and Daley, 2015; Forsyth et al., 2020). The ARVA strategy determinesthe yearly spending based on the theoretical value of a fixed term (virtual) annuity purchased withthe current portfolio wealth. This approach is efficient in the sense that the portfolio is exhaustedat the end of the investment horizon, but there is no guarantee of a yearly minimum withdrawalamount.A recent survey showed that a majority of pre-retirees fear exhausting their savings in retirementmore than death. In addition, it is considered axiomatic amongst practitioners that retirees desireto have minimum (real) cash flows each year to fund expenses (Tretiakova and Yamada, 2011).Typical (e.g. CRRA) utility function based objective functions do not directly focus on these twoissues.To address these two concerns, our objective in this article is to determine a decumulationstrategy which has the following characteristics.• Withdrawals can be variable, but with minimum and maximum constraints.• The risk of portfolio depletion is minimized.• The expected average withdrawal is maximized.• The asset allocation strategy can be dynamic and non-deterministic.We specify that the withdrawals are to take place over a fixed, lengthy (30 years) decumulationhorizon. We do not explicitly take into account longevity risk, which we recognize as a weakness ofthis strategy. However, this is mitigated (somewhat) by specifying a long decumulation period. Forexample, the probability that a 65-year old Canadian male attains the age of 95, is about . . We pose the decumulation problem as an exercise in optimal stochastic control. We have twocontrols: the asset allocation and the withdrawal amount. These controls are time and state de-pendent. Our objective function is composed of a measure of reward and risk. Our measure ofreward is the expected total withdrawals (EW) over the thirty year period. Our measure of risk is The induced time consistent strategy in this case is a target based shortfall. Theconcept of induced time consistent strategies is discussed in Strub et al. (2019). In fact, Forsyth(2020a) shows that enforcing a time consistent constraint on policies which use ES as a risk measurehas undesirable consequences. The relationship between pre-commitment and implementable targetbased schemes in the mean-variance context is discussed in Vigna (2014) and Menoncin and Vigna(2017).We assume that the retiree has an investment portfolio consisting of a stock index and a bondindex, and desires to maximize real (inflation adjusted) total withdrawals. We calibrate stochasticmodels of real stock and bond indexes to historical data over the 1926:1-2019:12 period. We assumeyearly withdrawals and rebalancing of the DC account. We term the market where the assets followthe parametric model fit to the historical data the synthetic market.We devise a numerical method for determining the optimal policies. We enforce realistic invest-ment constraints (no shorting, no leverage) and maximum and minimum constraints on the yearlywithdrawal amounts. Compared to a strategy with a fixed withdrawal amount per year, we find thata variable withdrawal strategy, with a minimum withdrawal set to the fixed withdrawal amount,has a significantly increased expected average withdrawal, with only a very small increase in ESrisk.We also test the robustness of the strategy computed in the synthetic market by carrying out testsusing bootstrap resampled historical data (the historical market ). The efficient EW-ES frontiers forboth synthetic and historical market tests are very close, indicating that the strategy computed inthe synthetic market is robust to model misspecification.
We assume that the investor has access to two funds: a broad market stock index fund and aconstant maturity bond index fund.The investment horizon is T . Let S t and B t respectively denote the real (inflation adjusted) amounts invested in the stock index and the bond index respectively. In general, these amountswill depend on the investor’s strategy over time, as well as changes in the real unit prices of theassets. In the absence of an investor determined control (i.e. cash withdrawals or rebalancing), allchanges in S t and B t result from changes in asset prices. We model the stock index as following ajump diffusion.In addition, we follow the usual practitioner approach and directly model the returns of theconstant maturity bond index as a stochastic process, see for example Lin et al. (2015); MacMinn An implementable strategy has the property that the investor has no incentive to deviate from the strategycomputed at time zero at later times (Forsyth, 2020a).
3t al. (2014). As in MacMinn et al. (2014), we assume that the constant maturity bond index followsa jump diffusion process as well.Let S t − = S ( t − (cid:15) ) , (cid:15) → + , i.e. t − is the instant of time before t , and let ξ s be a randomnumber representing a jump multiplier. When a jump occurs, S t = ξ s S t − . Allowing for jumpspermits modelling of non-normal asset returns. We assume that log( ξ s ) follows a double exponentialdistribution (Kou, 2002; Kou and Wang, 2004). If a jump occurs, p s u is the probability of an upwardjump, while − p s u is the chance of a downward jump. The density function for y = log( ξ s ) is f s ( y ) = p su η s e − η s y y ≥ + (1 − p su ) η s e η s y y< . (2.1)We also define κ s = E [ ξ s −
1] = p s u η s η s − − p s u ) η s η s + 1 − . (2.2)In the absence of control, S t evolves according to dS t S t − = (cid:0) µ s − λ sξ κ sξ (cid:1) dt + σ s dZ s + d π st (cid:88) i =1 ( ξ si − , (2.3)where µ s is the (uncompensated) drift rate, σ s is the volatility, dZ s is the increment of a Wienerprocess, π st is a Poisson process with positive intensity parameter λ sξ , and ξ si are i.i.d. positiverandom variables having distribution (2.1). Moreover, ξ si , π st , and Z s are assumed to all be mutuallyindependent.Similarly, let the amount in the bond index be B t − = B ( t − (cid:15) ) , (cid:15) → + . In the absence of control, B t evolves as dB t B t − = (cid:16) µ b − λ bξ κ bξ + µ bc { B t − < } (cid:17) dt + σ b dZ b + d π bt (cid:88) i =1 ( ξ bi − , (2.4)where the terms in equation (2.4) are defined analogously to equation (2.3). In particular, π bt is aPoisson process with positive intensity parameter λ bξ , and ξ bi has distribution f b ( y = log ξ b ) = p bu η b e − η b y y ≥ + (1 − p bu ) η b e η b y y< , (2.5)and κ bξ = E [ ξ b − . ξ bi , π bt , and Z b are assumed to all be mutually independent. The term µ bc { B t − < } in equation (2.4) represents the extra cost of borrowing (the spread).The diffusion processes are correlated, i.e. dZ s · dZ b = ρ sb dt . The stock and bond jump processesare assumed mutually independent. See Forsyth (2020b) for justification of the assumption of stock-bond jump independence. Remark 2.1 (Stock and Bond Processes) . An obvious generalization of processes (2.3) and (2.4)would be to include stochastic volatility effects. However, previous studies have shown that stochasticvolatility appears to have little consequences for long term investors (Ma and Forsyth, 2016). Asa robustness check, we will (i) determine the optimal controls using the parametric model based onequations (2.3) and (2.4) and (ii) use these controls on bootstrapped resampled historical data, whichmakes no assumptions about the underlying bond and stock stochastic processes.
4e define the investor’s total wealth at time t asTotal wealth ≡ W t = S t + B t . (2.6)We impose the constraints that (assuming solvency) shorting stock and using leverage (i.e. bor-rowing) are not permitted, which would be typical of a DC plan retirement savings account. Inthe event of insolvency (due to withdrawals), the portfolio is liquidated, trading ceases and debtaccumulates at the borrowing rate. Consider a set of discrete withdrawal/rebalancing times TT = { t = 0 < t < t < . . . < t M = T } (3.1)where we assume that t i − t i − = ∆ t = T /M is constant for simplicity. To avoid subscript clutter,in the following, we will occasionally use the notation S t ≡ S ( t ) , B t ≡ B ( t ) and W t ≡ W ( t ) . Letthe inception time of the investment be t = 0 . We let T be the set of withdrawal/rebalancingtimes, as defined in equation (3.1). At each rebalancing time t i , i = 0 , , . . . , M − , the investor(i) withdraws an amount of cash q i from the portfolio, and then (ii) rebalances the portfolio. At t M = T , the final cash flow q M occurs, and the portfolio is liquidated. In the following, given a timedependent function f ( t ) , then we will use the shorthand notation f ( t + i ) ≡ lim (cid:15) → + f ( t i + (cid:15) ) ; f ( t − i ) ≡ lim (cid:15) → + f ( t i − (cid:15) ) . (3.2)We assume that there are no taxes or other transaction costs, so that the condition W ( t + i ) = W ( t − i ) − q i ; t i ∈ T (3.3)holds. Typically, DC plan savings are held in a tax advantaged account, with no taxes triggeredby rebalancing. With infrequent (e.g. yearly) rebalancing, we also expect transaction costs to besmall, and hence can be ignored. It is possible to include transaction costs, but at the expense ofincreased computational cost (Staden et al., 2018).We denote by X ( t ) = ( S ( t ) , B ( t )) , t ∈ [0 ,T ] , the multi-dimensional controlled underlyingprocess, and by x = ( s, b ) the realized state of the system. Let the rebalancing control p i ( · ) be thefraction invested in the stock index at the rebalancing date t i , i.e. p i (cid:0) X ( t − i ) (cid:1) = p (cid:0) X ( t − i ) , t i (cid:1) = S ( t + i ) S ( t + i ) + B ( t + i ) . (3.4)Let the withdrawal control q i ( · ) be the amount withdrawn at time t i , i.e. q i (cid:0) X ( t − i ) (cid:1) = q (cid:0) X ( t − i ) , t i (cid:1) . Note that formally, the controls depend on the state of the investment portfolio,before the rebalancing occurs, i.e. p i ( · ) = p (cid:0) X ( t − i ) , t i ) (cid:1) = p (cid:0) X − i , t i (cid:1) , and q i ( · ) = q (cid:0) X ( t − i ) , t i ) (cid:1) = q (cid:0) X − i , t i (cid:1) , t i ∈ T , where T is the set of rebalancing times.However, it will be convenient to note that in our case, we find the optimal control p i ( · ) amongstall strategies with constant wealth (after withdrawal of cash). Hence, with some abuse of notation,we will now consider p i ( · ) to be function of wealth after withdrawal of cash p i ( · ) = p ( W ( t + i ) , t i ) W ( t + i ) = S ( t − i ) + B ( t − i ) − q i ( · ) S ( t + i ) = S + i = p i ( W + i ) W + i ; B ( t + i ) = B + i = (1 − p i ( W + i )) W + i . (3.5)5 emark 3.1 (Control depends on wealth only) . Note that we assume no transaction costs. Iftransaction costs are included, then the control p i ( · ) would in general be a function of the state ( S ( t − i ) , B ( t − i )) (Dang and Forsyth, 2014). Note that since p i ( · ) = p i ( W − i − q i ) , then it follows that q i ( · ) = q i ( W − i ) (3.6)which we will prove formally in a later section. Remark 3.2 (Instantaneous Rebalancing) . We assume that rebalancing occurs instantaneously.Informally, this has the consequence that no jumps occur in the unit prices of the stock and bondindexes over a rebalancing period ( t − i , t + i ) . A control at time t i , is then given by the pair ( q i ( · ) , p i ( · )) where the notation ( · ) denotes thatthe control is a function of the state.Let Z represent the set of admissible values of the controls ( q i ( · ) , p i ( · )) . As is typical for a DCplan savings account, we impose no-shorting, no-leverage constraints (assuming solvency). We alsoimpose maximum and minimum values for the withdrawals. We apply the constraint that in theevent of insolvency due to withdrawals ( W ( t + i ) < ), trading ceases and debt (negative wealth)accumulates at the appropriate bond rate of return (including a spread). We also specify that thestock assets are liquidated at t = t M .More precisely, let W + i be the wealth after withdrawal of cash, then define Z q = [ q min , q max ] ; t ∈ T , (3.7) Z p ( W + i ,t i ) = [0 , W + i > t i ∈ T ; t i (cid:54) = t M { } W + i ≤ t i ∈ T ; t i (cid:54) = t M { } t i = t M . (3.8)(3.9)The set of admissible values for ( q i ,p i ) , t i ∈ T , can then be written a ( q i ,p i ) ∈ Z ( W + i ,t i ) = Z q × Z p ( W + i ,t i ) . (3.10)For implementation purposes, we have written equation (3.10) in terms of the wealth after with-drawal of cash. However, we remind the reader that since W + i = W − i − q , the controls are formallya function of the state X ( t − i ) before the control is applied.The admissible control set A can then be written as A = (cid:26) ( q i , p i ) ≤ i ≤ M : ( p i , q i ) ∈ Z ( W + i ,t i ) (cid:27) (3.11)An admissible control P ∈ A , where A is the admissible control set, can be written as, P = { ( q i ( · ) , p i ( · )) : i = 0 , . . . , M } . (3.12)We also define P n ≡ P t n ⊂ P as the tail of the set of controls in [ t n , t n +1 , . . . , t M ] , i.e. P n = { ( q n ( · ) , p n ( · )) . . . , ( p M ( · ) , q M ( · )) } . (3.13)For notational completeness, we also define the tail of the admissible control set A n as A n = (cid:26) ( q i , p i ) n ≤ i ≤ M : ( q i , p i ) ∈ Z ( W + i ,t i ) (cid:27) (3.14)so that P n ∈ A n . 6 Risk and reward
Let g ( W T ) be the probability density function of wealth W T at t = T . Suppose (cid:90) W ∗ α −∞ g ( W T ) dW T = α, (4.1)i.e. Pr [ W T > W ∗ α ] = 1 − α . We can interpret W ∗ α as the Value at Risk (VAR) at level α . TheExpected Shortfall (ES) at level α is thenES α = (cid:82) W ∗ α −∞ W T g ( W T ) dW T α , (4.2)which is the mean of the worst α fraction of outcomes. Typically α ∈ { . , . } . Note that thedefinition of ES in equation (4.2) uses the probability density of the final wealth distribution, notthe density of loss . Hence, in our case, a larger value of ES (i.e. a larger value of average worst caseterminal wealth) is desired. The negative of ES is commonly referred to as Conditional Value atRisk (CVAR).Define X +0 = X ( t +0 ) , X − = X ( t − ) . Given an expectation under control P , E P [ · ] , as noted byRockafellar and Uryasev (2000), ES α can be alternatively written asES α ( X − , t − ) = sup W ∗ E X +0 ,t +0 P (cid:20) W ∗ + 1 α min( W T − W ∗ , (cid:21) . (4.3)The admissible set for W ∗ in equation (4.3) is over the set of possible values for W T .Note that the notation ES α ( X − , t − ) emphasizes that ES α is as seen at ( X − , t − ) . In other words,this is the pre-commitment ES α . A strategy based purely on optimizing the pre-commitment valueof ES α at time zero is time-inconsistent , hence has been termed by many as non-implementable , sincethe investor has an incentive to deviate from the the pre-commitment strategy at t > . However,in the following, we will consider the pre-commitment strategy merely as a device to determine anappropriate level of W ∗ in equation (4.3). If we fix W ∗ ∀ t > , then this strategy is the inducedtime consistent strategy (Strub et al., 2019), hence is implementable. We delay further discussionof this subtle point to later sections. We will use expected total withdrawals as a measure of reward in the following. More precisely, wedefine EW (expected withdrawals) asEW ( X − , t − ) = E X +0 ,t +0 P (cid:20) i = M (cid:88) i =0 q i (cid:21) . (4.4) Since expected withdrawals (EW) and expected shortfall (ES) are conflicting measures, we usea scalarization technique to find the Pareto points for this multi-objective optimization problem.Informally, for a given scalarization parameter κ > , we seek to find the control P that maximizesEW ( X − , t − ) + κ ES α ( X − , t − ) . (5.1) In practice, the negative of W ∗ α is often the reported VAR. ( P CEE t ( κ )) problem in terms ofthe value function J ( s,b,t − )( PCEE t ( κ )) : J (cid:0) s,b, t − (cid:1) = sup P ∈A sup W ∗ (cid:40) E X +0 ,t +0 P (cid:34) i = M (cid:88) i =0 q i + κ (cid:18) W ∗ + 1 α min( W T − W ∗ , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) (5.2)subject to ( S t , B t ) follow processes (2.3) and (2.4) ; t / ∈ T W + (cid:96) = S − (cid:96) + B − (cid:96) − q (cid:96) ; X + (cid:96) = ( S + (cid:96) , B + (cid:96) ) S + (cid:96) = p (cid:96) ( · ) W + (cid:96) ; B + (cid:96) = (1 − p (cid:96) ( · )) W + (cid:96) ( q (cid:96) ( · ) , p (cid:96) ( · )) ∈ Z ( W + (cid:96) ,t (cid:96) ) (cid:96) = 0 , . . . , M ; t (cid:96) ∈ T . (5.3)Interchange the sup sup in equation (5.2), so that value function J (cid:0) s,b, t − (cid:1) can be written as J (cid:0) s,b, t − (cid:1) = sup W ∗ sup P ∈A (cid:40) E X +0 ,t +0 P (cid:34) i = M (cid:88) i =0 q i + κ (cid:18) W ∗ + 1 α min( W T − W ∗ , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) . (5.4)Noting that the inner supremum in equation (5.4) is a continuous function of W ∗ , and noting thatthe optimal value of W ∗ in equation (5.4) is bounded , then define W ∗ ( s,b ) = arg max W ∗ (cid:26) sup P ∈A (cid:40) E X +0 ,t +0 P (cid:34) i = M (cid:88) i =0 q i + κ (cid:18) W ∗ + 1 α min( W T − W ∗ , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) . (5.5)We refer the reader to Forsyth (2020a) for an extensive discussion concerning pre-commitment andtime consistent ES strategies. We summarize the relevant results from that research here.Denote the investor’s initial wealth at t by W − . Then we have the following result. Proposition 5.1 (Pre-commitment strategy equivalence to a time consistent policy for an alterna-tive objective function) . The pre-commitment EW-ES strategy P ∗ determined by solving J (0 , W , t − ) (with W ∗ (0 , W − ) from equation (5.5)) is the time consistent strategy for the equivalent problem T CEQ (with fixed W ∗ (0 ,W − ) ), with value function ˜ J ( s,b,t ) defined by ( TCEQ t n ( κ/α )) : ˜ J (cid:0) s,b, t − n (cid:1) = sup P n ∈A (cid:40) E X + n ,t + n P n (cid:34) i = M (cid:88) i = n q i + κα min( W T − W ∗ (0 , W − ) , (cid:12)(cid:12)(cid:12)(cid:12) X ( t − n ) = ( s,b ) (cid:35)(cid:41) . (5.6) Proof.
This follows similar steps as in Forsyth (2020a), proof of Proposition 6.2, with the exceptionthat the reward in Forsyth (2020a) is expected terminal wealth, while here the reward is totalwithdrawals. This is the same as noting that a finite value at risk exists. This easily shown, assuming < α < , since ourinvestment strategy uses no leverage and no-shorting. emark 5.1 (An Implementable Strategy) . Given an initial level of wealth W − at t , then theoptimal control for the pre-commitment problem (5.2) is the same optimal control for the timeconsistent problem ( TCEQ t n ( κ/α )) (5.6), ∀ t > . Hence we can regard problem ( TCEQ t n ( κ/α )) as the EW-ES induced time consistent strategy . Thus, the induced strategy is implementable, in thesense that the investor has no incentive to deviate from the strategy computed at time zero, at latertimes (Forsyth, 2020a).
Remark 5.2 (EW-ES Induced Time Consistent Strategy) . In the following, we will consider theactual strategy followed by the investor for any t > as given by the induced time consistent strategy ( TCEQ t n ( κ/α )) in equation (5.6), with a fixed value of W ∗ (0 , W − ) , which is identical to the EW-ESstrategy at time zero. Hence, we will refer to this strategy in the following as the EW-ES strategy,with the understanding that this refers to strategy ( TCEQ t n ( κ/α )) for any t > . In order to solve problem ( P CEE t ( κ )) , our starting point is equation (5.4), where we have inter-changed the sup sup( · ) in equation (5.2). We expand the state space to ˆ X = ( s,b,W ∗ ) , and definethe auxiliary function V ( s, b, W ∗ , t ) ∈ Ω = [0 , ∞ ) × ( −∞ , + ∞ ) × ( −∞ , + ∞ ) × [0 , ∞ ) V ( s, b, W ∗ , t − n ) = sup P n ∈A n (cid:40) E ˆ X + n ,t + n P n (cid:34) i = M (cid:88) i = n q i + κ (cid:18) W ∗ + 1 α min(( W T − W ∗ ) , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ X ( t − n ) = ( s,b, W ∗ ) (cid:35)(cid:41) . (6.1)subject to ( S t , B t ) follow processes (2.3) and (2.4) ; t / ∈ T W + (cid:96) = S − (cid:96) + B − (cid:96) − q (cid:96) ; X + (cid:96) = ( S + (cid:96) , B + (cid:96) ) S + (cid:96) = p (cid:96) ( · ) W + (cid:96) ; B + (cid:96) = (1 − p (cid:96) ( · )) W + (cid:96) ( q (cid:96) ( · ) , p (cid:96) ( · )) ∈ Z ( W + (cid:96) ,t (cid:96) ) (cid:96) = n, . . . , M ; t (cid:96) ∈ T . (6.2)Equation (6.1) is a simple expectation. Hence we can solve this auxiliary problem using dy-namic programming. Recalling the definitions of Z p , Z q in equations (3.7-3.8), then the dynamicprogramming principle applied at t n ∈ T would then imply V ( s,b,W ∗ , t − n ) = sup q ∈Z q sup p ∈Z p ( w − − q,t ) (cid:26) q + (cid:20) V (( w − − q ) p, ( w − − q )(1 − p ) , W ∗ , t + n ) (cid:21)(cid:27) = sup q ∈ Z q (cid:26) q + (cid:20) sup p ∈Z p ( w − − q,t ) V (( w − − q ) p, ( w − − q )(1 − p ) , W ∗ , t + n ) (cid:21)(cid:27) w − = s + b . (6.3)Let V denote the upper semi-continuous envelope of V . The optimal control p n ( w,W ∗ ) at time t n is then determined from p n ( w, W ∗ ) = arg max p (cid:48) ∈ [0 , V ( wp (cid:48) , w (1 − p (cid:48) ) , W ∗ , t + n ) , w > t n (cid:54) = t M , w ≤ or t n = t M . (6.4)9he control for q is then determined from q n ( w,W ∗ ) = arg max q (cid:48) ∈Z q (cid:26) q (cid:48) + V (( w − q (cid:48) ) p n ( w − q (cid:48) ) , W ∗ ) , ( w − q (cid:48) )(1 − p n ( w − q (cid:48) )) , W ∗ ) ,t + n ) (cid:27) . (6.5)From the right hand sides of equation (6.4) and equation (6.5), we have the following result. Proposition 6.1 (Dependence of optimal controls) . For fixed W ∗ , the optimal control for q n ( · ) isa function only of the total portfolio wealth before withdrawals w − = s + b , i.e. q n = q n ( w − , W ∗ ) ,while the optimal control for p n ( · ) is a function only of the total portfolio wealth after withdrawals w + = w − − q n ( w − , W ∗ ) , i.e. p n = p n ( w + , W ∗ ) . The solution is advanced (backwards) across time t n by V ( s, b, W ∗ ,t − n ) = q n ( w − , W ∗ ) + V ( w + p n ( w + ,W ∗ ) , w + ( 1 − p n ( w + ,W ∗ ) ) , W ∗ , t + n ) w − = s + b ; w + = s + b − q n ( w − , W ∗ ) . (6.6)At t = T , we have V ( s, b, W ∗ ,T + ) = κ (cid:18) W ∗ + min(( s + b − W ∗ ) , α (cid:19) . (6.7)For t ∈ ( t n − ,t n ) , there are no cash flows, discounting (all quantities are inflation adjusted), orcontrols applied. Hence the tower property gives for < h < ( t n − t n − ) V ( s,b,W ∗ , t ) = E (cid:20) V ( S ( t + h ) , B ( t + h ) , W ∗ , t + h ) (cid:12)(cid:12) S ( t ) = s, B ( t ) = b (cid:21) ; t ∈ ( t n − , t n − h ) . (6.8)Applying Ito’s Lemma for jump processes (Tankov and Cont, 2009), noting equations (2.3) and(2.4), and letting h → gives, for t ∈ ( t n − , t n ) V t + ( σ s ) s V ss + ( µ s − λ sξ κ sξ ) sV s + λ sξ (cid:90) + ∞−∞ V ( e y s, b, t ) f s ( y ) dy + ( σ b ) b V bb + ( µ b − λ bξ κ bξ ) bV b + λ bξ (cid:90) + ∞−∞ V ( s, e y b, t ) f b ( y ) dy − ( λ sξ + λ bξ ) V + ρ sb σ s σ b sbV sb = 0 . (6.9) Proposition 6.2 (Equivalence of formulation (6.1-6.9) to problem ( P CEE t ( κ )) ) . Define J ( s,b,t − ) = sup W (cid:48) V ( s,b,W (cid:48) ,t − ) , (6.10) then formulation (6.1-6.9) is equivalent to problem ( P CEE t ( κ )) .Proof. Replace V ( s,b,W (cid:48) ,t − ) in equation (6.10) by the expressions in equations (6.1-6.9). Beginwith equation (6.7), and recursively work backwards in time, then we obtain equations (5.2-5.3), byinterchanging sup W (cid:48) sup P in the final step. 10 Continuous withdrawal/rebalancing limit
In order to develop some intuition about the nature of the optimal controls, we will examine thelimit as the rebalancing interval becomes vanishingly small.
Proposition 7.1 (Bang-bang withdrawal control in the continuous withdrawal limit) . Assume that• the stock and bond processes follow (2.3) and (2.4),• the portfolio is continuously rebalanced, and withdrawals occur at a continuous (finite) rate ˆ q ∈ [ˆ q min , ˆ q max ] ,• the HJB equation for the EW-ES problem in the continuous rebalancing limit has boundedderivatives w.r.t. total wealth,• in the event of ties for the control ˆ q , the smallest withdrawal is selected,then the optimal withdrawal control ˆ q ∗ ( · ) for the EW-ES problem ( P CEE t ( κ )) is bang-bang, ˆ q ∗ ∈{ ˆ q min , ˆ q max } .Proof. We consider (for ease of exposition) the case where the stock and bond funds follow geometricBrownian motion (i.e. no jumps). The analysis below can be easily (although tediously) extendedto the case of processes (2.3) and (2.4). Consequently, we assume that the stock S t and bond B t index processes are dS t S t = µ s dt + σ s dZ s ; dB t B t = µ b dt + σ b dZ b . (7.1)with dZ s · dZ b = ρ sb dt . Assume that rebalancing is carried out continuously, and let ˆ p ( W t , t ) = S t S t + B t , (7.2)with continuous withdrawal of cash at a rate of ˆ q ( W t , t ) . The SDE for the total wealth process W t = S t + B t is then dW t = ˆ pW t dS t S t + (1 − ˆ p ) W t dB t B t − ˆ q dt . (7.3)It is important to note that here ˆ q is a rate of cash withdrawal, whereas we have previously defined q as a finite amount of cash withdrawal. Define the following sets of admissible values of the controls ˆ Z q = [ˆ q min , ˆ q max ] ; t ∈ [0 ,T ] , (7.4) ˆ Z p ( W t ,t ) = [0 , W t > t ∈ [0 ,T ] ; t (cid:54) = T { } W t ≤ t ∈ [0 ,T ] ; t (cid:54) = T { } t = T . (7.5)We define the value function ˆ V ( w, W ∗ , t ) on the domain ˆΩ = ( −∞ , + ∞ ) × ( −∞ , + ∞ ) × (0 , ∞ ) forfixed W ∗ as ˆ V ( w, W ∗ , t ) =sup ˆ p ( · ) ∈ ˆ Z p sup ˆ q ( · ) ∈ ˆ Z q (cid:40) E ( W t ,W ∗ ,t )(ˆ p, ˆ q ) (cid:34)(cid:90) Tt ˆ q dt + κ (cid:18) W ∗ + 1 α min(( W T − W ∗ ) , (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( W t ,W ∗ ) = ( w, W ∗ ) (cid:35)(cid:41) . (7.6)11he continuous rebalancing, continuous withdrawal EW-ES problem is then posed as determining J ( w, t ) which is given by ˆ J ( w,t ) = sup W ∗ ˆ V ( w, W ∗ , t ) . (7.7)Following the usual arguments we obtain the Hamilton-Jacobi-Bellman PDE for ˆ V ˆ V t + sup ˆ p ∈ ˆ Z p sup ˆ q ∈ ˆ Z q (cid:26) w (cid:20) ˆ pµ s + (1 − ˆ p ) µ b (cid:21) ˆ V w − ˆ q ˆ V w + ˆ q + w (cid:20) (ˆ pσ s ) − ˆ p )ˆ pσ s σ b ρ sb + ((1 − ˆ p ) σ b ) (cid:21) ˆ V ww (cid:27) = 0 , (7.8)with terminal condition ˆ V ( w, T, W ∗ ) = κ (cid:18) W ∗ + 1 α min(( W T − W ∗ ) , (cid:19) . (7.9)In general, we seek the viscosity solution of equation (7.8), which does not require that the solution ˆ V be differentiable. However, we make the assumption that ˆ V w exists and is bounded.Rewriting equation (7.8) we have ˆ V t + sup ˆ p ∈ ˆ Z p (cid:26) w (cid:20) ˆ pµ s + (1 − ˆ p ) µ b (cid:21) ˆ V w + w (cid:20) (ˆ pσ s ) − ˆ p )ˆ pσ s σ b ρ sb + ((1 − ˆ p ) σ b ) (cid:21) ˆ V ww (cid:27) + sup ˆ q ∈ ˆ Z q (cid:26) ˆ q (1 − ˆ V w ) (cid:27) = 0 , (7.10)and therefore the optimal value of ˆ q is determined by maximizing sup ˆ q ∈ ˆ Z q ˆ q (1 − ˆ V w ) . (7.11)Breaking ties by choosing ˆ q = ˆ q min if (1 − ˆ V w ) = 0 , we then have that the optimal strategy ˆ q ∗ is ˆ q ∗ = (cid:40) ˆ q min ; (1 − ˆ V w ) ≤ q max ; (1 − ˆ V w ) > . (7.12)Equation 7.12 holds for any W ∗ and hence is also true for the optimal value of W ∗ in equation (7.7).We obtain the same result (after some algebraic complexity) if we assume that the stock and bondprocesses are given in equation (2.3) and equation (2.4). Remark 7.1 (Bang-bang control for discrete rebalancing/withdrawals) . Proposition 7.1 suggeststhat, for sufficiently small rebalancing intervals, we can expect the optimal q control (finite withdrawalamount) to be bang-bang. However, it is not clear that this will continue to be true for the case ofannual rebalancing (which we specify in our numerical examples). In fact, we do observe that the q control is very close to bang-bang in our numerical experiments, even for annual rebalancing. Weterm this control to be quasi-bang-bang . .1 Numerical algorithm: ( P CEE t ( κ )) We solve the auxiliary problem (6.1-6.2), with a fixed values of W ∗ , κ and α . We do not allowshorting of stock, so the amount in the stocks S ( t ) ≥ . We discretize the state space in s > using n ˆ x equally spaced nodes in the ˆ x = log s direction, on a finite localized domain s ∈ [ e ˆ x min , e ˆ x max ] . Wediscretize the state space in b > using n y equally spaced nodes in the y = log b direction, on a finitelocalized domain b ∈ [ b min , b max ] = [ e y min , e y max ] . We also define a b (cid:48) > grid, using n b equally spacednodes in the y (cid:48) = log b (cid:48) direction, on the localized domain with b (cid:48) ∈ [ b (cid:48) min , b (cid:48) max ] = [ e y min , e y max ] . Thegrid [ s min , s max ] × [ b min , b max ] represents cases where b ≥ . The grid [ s min , s max ] × [ b (cid:48) min , b (cid:48) max ] represents cases where b = − b (cid:48) < .We use the Fourier methods discussed in Forsyth and Labahn (2019) to solve PIDE (6.9) betweenrebalancing times. Further details concerning the Fourier method can be found in Forsyth (2020b).We choose the localized domain [ˆ x min , ˆ x max ] = [log(10 ) − , log(10 ) + 8] , with [ y min , y max ] =[ˆ x min , ˆ x max ] (units thousands of dollars). Wrap-around effects are minimized using the domainextension method in Forsyth and Labahn (2019). In our numerical experiments, we carried out testsreplacing [ˆ x min , ˆ x max ] by [ˆ x min − , ˆ x max +2] and similarly replacing [ y min , y max ] by [ y min − , y max +2] .In all cases, this resulted in changes to the summary statistics in at most the fifth digit, verifyingthat the localization error is small.We discretize the p controls using an equally spaced grid with n y values. We then solve the op-timization problem (6.4) using exhaustive search over the discretized p values, linearly interpolatingthe right hand side discrete values of V in equation (6.4) as required. We store the optimal controlfor p at n y discrete wealth nodes. We also discretize the controls for q in the range [ q min , q max ] inincrements of one thousand dollars, and determine the optimal control for q by exhaustive search.We then determine the optimal control for q using equation (6.5), at a set of n y discrete w nodes.We use the previously stored controls for p in order to evaluate the right hand side of equation (6.5),linearly interpolating the controls if necessary.We use a fixed discretization of the q controls since it is realistic to assume that retirees willchange withdrawal amounts in fairly coarse increments. As we shall see, as suggested by Proposition7.1, the q control turns out to be quasi-bang-bang, hence the discretization of the q control hardlymakes any difference to the solution.Finally, stored controls for q and p are then used to advance the solution in equation (6.6),linearly interpolating the controls and value function if required.Assume that n ˆ x = O ( n y ) . Then, the cost of using an FFT method to solve equation (6.9)between rebalancing times is O ( n y log n y ) . The cost of determining the optimal control for p in usingequation (6.6) at n y discrete w values, using exhaustive search, is O ( n y ) . The cost of determiningthe optimal q using equation (6.5) at n y discrete w values is O ( n y ) , since the number of discrete q controls is O (1) . In addition, the step (6.6) has complexity O ( n y ) . The total number of rebalancingtimes is fixed, hence the total complexity of the solution of problem (6.1) for a fixed value of W ∗ is O ( n y log n y ) . W ∗ Given an approximate solution of the auxiliary problem (6.1-6.2) at t = 0 , which we denote by V ( s, b, W ∗ , , we then determine the final solution for problem P CEE t ( κ )) in equations (5.2-5.3)13sing equation (6.10). More specifically, we solve J (0 , W , − ) = sup W (cid:48) V (0 , W , W (cid:48) , − ) W = initial wealth . (7.13)We solve the auxiliary problem on sequence of grids n ˆ x × n y . On the coarsest grid, we discretize W ∗ and solve problem (7.13) by exhaustive search. We use this optimal value of W ∗ as a startingpoint to a one dimensional optimization algorithm on a sequence of finer grids.This approach does not guarantee that we have the globally optimal solution to problem (7.13),since the problem is not guaranteed to be convex. However, we have made a few tests by carryingout a grid search on the finest grid, which suggest that we do indeed have the globally optimalsolution. If W t (cid:29) W ∗ , and t → T , then P r [ W T < W ∗ ] (cid:39) (recall that W ∗ is fixed for problem ( TCEQ t n ( κ/α )) (5.6) ). In addition, for large values of W t , the withdrawal will be capped at q max . In this fortu-itous situation for the retiree, the control only weakly effects the objective function. To avoid thisill-posedness for the controls, we changed the objective function (5.2) to J (cid:0) s,b, t − (cid:1) = sup P ∈A sup W ∗ (cid:40) E X +0 ,t +0 P (cid:34) i = M (cid:88) i =0 q i + κ (cid:18) W ∗ + 1 α min( W T − W ∗ , (cid:19) stabilization (cid:122) (cid:125)(cid:124) (cid:123) + (cid:15)W T (cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) . (7.14)We used the value (cid:15) = +10 − in the following test cases. Using a positive value for (cid:15) has the effectof forcing the strategy to invest in stocks when W t is very large, and t → T , when the controlproblem is ill-posed. In other words, when the probability that W T is less than W ∗ is very small,then the ES risk is practically zero, hence the investor might as well invest in risky assets. Thereis little to lose, and much to gain (at least for the retiree’s estate). Note that using this smallvalue of (cid:15) = 10 − gave the same results as (cid:15) = 0 for the summary statistics, to four digits. This issimply because the states with very large wealth have low probability. However, this stabilizationprocedure produced more smooth heat maps for large wealth values, without altering the summarystatistics appreciably. We use data from the Center for Research in Security Prices (CRSP) on a monthly basis over the1926:1-2019:12 period. Our base case tests use the CRSP 10 year US treasury index for the bondasset and the CRSP value-weighted total return index for the stock asset. This latter index includesall distributions for all domestic stocks trading on major U.S. exchanges. All of these various More specifically, results presented here were calculated based on data from Historical Indexes, ©2020 Center forResearch in Security Prices (CRSP), The University of Chicago Booth School of Business. Wharton Research DataServices was used in preparing this article. This service and the data available thereon constitute valuable intellectualproperty and trade secrets of WRDS and/or its third-party suppliers. The 10-year Treasury index was constructed from monthly returns from CRSP back to 1941. The data for1926-1941 were interpolated from annual returns in Homer and Sylla (2005). ρ sb is computed by removing anyreturns which occur at times corresponding to jumps in either series, and then using the samplecovariance. Further discussion of the validity of assuming that the stock and bond jumps areindependent is given in Forsyth (2020b).CRSP µ s σ s λ s p s up η s η s ρ sb µ b σ b λ b p b up η b η b ρ sb Table 8.1:
Estimated annualized parameters for double exponential jump diffusion model. Value-weighted CRSP index, 10-year US treasury index deflated by the CPI. Sample period 1926:1 to 2019:12.
Table 9.1 shows our base case investment scenario. We will use thousands as our units of wealth inthe following. For example, a withdrawal of per year corresponds to $40 , per year, with aninitial wealth of ( $1 , , ). Thus, a withdrawal of 40 per year would correspond to the useof the four per cent rule (Bengen, 1994).To make this example more concrete, this scenario would apply to a retiree who is yearsold, with a pre-retirement salary of $100,000 per year, with a total value of DC plan holdings atretirement of $1,000,000. In Canada, a retiree would be eligible for government benefits (indexed)of about $20,000 per year. If the investor targets withdrawing $40,000 per year from the DC plan,then this would result in total real income of about $60,000 per year, which is about 60% of pre-retirement salary. For risk management purposes, we will assume that the retiree owns mortgagefree real estate worth about $400,000, which will retain its value in real terms over 30 years. If ourmeasure of risk is Expected Shortfall at the 5% level, then we suppose that any ES which is greaterthan about -$200,000 (the negative of one half the value of the real estate) can be managed usinga reverse mortgage.Note that in Table 9.1 we have set the borrowing spread µ bc = 0 . The (real) drift rate of the10-year treasury index is about 200 bps larger than the 30-day T-bill index. Hence, borrowing atthe 10-year treasury rate is roughly comparable to borrowing at the short term rate plus a spread ofabout 200 bps, which we suppose to be a reasonable estimate for a well secured reverse mortgage. We fit the parameters for the parametric stock and bond processes (2.3 - 2.4) as described in Section8. We then compute and store the optimal controls based on the parametric market model. Finally,15nvestment horizon T (years) 30Equity market index CRSP Cap-weighted index (real)Bond index 10-year Treasury (US) (real)Initial portfolio value W t = 0 , , . . . , Withdrawal range [ q min , q max ] Equity fraction range [0 , Borrowing spread µ bc Table 9.1:
Input data for examples. Monetary units: thousands of dollars.
Data series Optimal expectedblock size ˆ b (months)Real 10-year Treasury index 4.2Real CRSP value-weighted index 3.1 Table 9.2:
Optimal expected blocksize ˆ b = 1 /v when the blocksize follows a geometric distribution P r ( b = k ) = (1 − v ) k − v . The algorithm in Patton et al. (2009) is used to determine ˆ b . Historicaldata range 1926:1-2019:12. we compute various statistical quantities by using the stored control, and then carrying out MonteCarlo simulations, based on processes (2.3 - 2.4). We compute and store the optimal controls based on the parametric model (2.3-2.4) as for thesynthetic market case. However, we compute statistical quantities using the stored controls, butusing bootstrapped historical return data directly. We remind the reader that all returns are inflationadjusted. We use the stationary block bootstrap method (Politis and Romano, 1994; Politis andWhite, 2004; Patton et al., 2009; Dichtl et al., 2016). A crucial parameter is the expected blocksize.Sampling the data in blocks accounts for serial correlation in the data series. We use the algorithmin Patton et al. (2009) to determine the optimal blocksize for the bond and stock returns separately,see Table 9.2. We use a paired sampling approach to simultaneously draw returns from both timeseries. In this case, a reasonable estimate for the blocksize for the paired resampling algorithmwould be about . years. We will give results for a range of blocksizes as a check on the robustnessof the bootstrap results. Detailed pseudo-code for block bootstrap resampling is given in Forsythand Vetzal (2019).
10 Synthetic and historical markets: constant withdrawals q = 40 ,constant proportion strategy We consider the scenario in Table 9.1. As a benchmark, we consider withdrawing at a constant rateof per year (units: thousands of dollars). This would correspond to the 4% rule suggested in16Bengen, 1994). We also assume that the portfolio is rebalanced to a constant weight in stocks eachyear. Table 10.1 shows the results for various equity weights in the synthetic market, while Table10.2 shows results for the bootstrapped historical market.Note that the results are roughly comparable for both synthetic and historical markets. However,none of the cases with constant withdrawals and constant equity weights meets our criteria of anES > − $200 , . Equity Weight Expected Shortfall (5%) M edian [ W T ] Table 10.1:
Synthetic market results assuming the scenario given in Table 9.1, with q max = q min =40 , and p (cid:96) = constant in equation (5.3). Stock index: real capitalization weighted CRSP stocks; bondindex: real 10-year US treasuries. Parameters from Table 8.1. Units: thousands of dollars. Statisticsbased on . × Monte Carlo simulation runs.
Equity Weight Expected Shortfall (5%)
M edian [ W T ] Table 10.2:
Historical market results (bootstrap resampling) assuming the scenario given in Ta-ble 9.1, except that q max = q min = 40 , and p (cid:96) = constant in equation (5.3). Stock index: realcapitalization weighted CRSP stocks; bond index: real 10-year US treasuries. Historical data in range1926:1-2019:12. Parameters from Table 8.1. Units: thousands of dollars. Statistics based on bootstrap resampling simulations. Expected blocksize . years.
11 Synthetic market
We carry out an initial test of convergence of our numerical method for the EW-ES problem (5.2).Table 11.1 shows the results for solution of the PDE on a sequence of grids. For each refinementlevel, we store the optimal control, and use this control in Monte Carlo simulations. The PDEsolution appears to converge at roughly a first order rate. However, the Monte Carlo simulations(based on the PDE controls) appear to be slightly more accurate. This effect has also been notedin Ma and Forsyth (2016). In the following, we will report results based on (i) determining thecontrol from the PDE solution (using the finest grid in Table 11.1) and (ii) using this control inMonte Carlo simulations. 17lgorithm in Section 6 Monte CarloGrid ES (5%) E [ (cid:80) i q i ] / ( M + 1) ES (5%) E [ (cid:80) i q i ] / ( M + 1)512 × -16.788 49.7470 -5.035 50.35 × -9.3609 49.8513 -4.511 49.86 × -7.6954 49.8998 -4.732 49.89 Table 11.1:
Convergence test, real stock index: deflated real capitalization weighted CRSP, real bondindex: deflated ten year Treasuries. Scenario in Table 9.1. Parameters in Table 8.1. The MonteCarlo method used . × simulations. κ = 1 . , α = . . Grid refers to the grid used in theAlgorithm in Section 6: n x × n b , where n x is the number of nodes in the log s direction, and n b isthe number of nodes in the log b direction. Units: thousands of dollars (real). ( M + 1) is the totalnumber of withdrawals. M is the number of rebalancing dates. q min = 35 . . q max = 60 . W ∗ = 204 . (equation(5.2)) on the finest grid, Algorithm in Section 6. As a benchmark strategy, we solve problem (5.2), scenario in Table 9.1, but force a constant with-drawal, i.e. we set q min = q max , but retain the optimal asset allocation control. The results areshown in Table 11.2. We can see from Table 11.2 that constant withdrawals of and peryear meet our objective that ES > − (recall that units are thousands of dollars). The strat-egy of withdrawing per year, coupled with optimal asset allocation, is a reasonable strategy,which meets both our income and risk targets. However, note that M edian [ W T ] = 717 , indicatingthat 50% of the time, we leave over $700,000 on the table at the end our investment horizon. Inother words, the constant withdrawal rate of $40,000 per year, while being reasonably safe over 30years, paradoxically also has a high probability of underspending. This leads us to then allow theadditional flexibility of variable spending. q max = q min ES (5%)
M edian [ W T ] (cid:80) i M edian ( p i ) /M
35 31.03 952.2 .27140 -196.1 716.6 .35745 -425.4 441.4 .424
Table 11.2:
Synthetic market results for optimal strategies, assuming the scenario given in Table 9.1.Stock index: real capitalization weighted CRSP stocks; bond index: real 10 year US treasuries. Pa-rameters from Table 8.1. Units: thousands of dollars. Statistics based on . × Monte Carlosimulation runs. Control is computed using the Algorithm in Section 6, stored, and then used in theMonte Carlo simulations. ( M + 1) is the number of withdrawals. M is the number of rebalancingdates. (cid:15) = 10 − . We solve problem (5.2), scenario in Table 9.1, and now allow the withdrawal to be determined fromour optimal strategy. We compute the efficient EW-ES frontiers for two cases: [ q min ,q max ] = [35 , and [ q min ,q max ] = [40 , , as shown in Figure 11.1. We also show the single points correspondingto constant withdrawals (from Table 11.2 for q = 35 , ) on the Figures as well. Detailed tables18
500 -400 -300 -200 -100 0 100
Expected Shortfall E [ ave r a g e w i t hd r a w a l ] Constantq=35qmin=35qmax=60 (a) q min = 35 , q max = 60 . -500 -400 -300 -200 -100 0 100 Expected Shortfall E [ ave r a g e w i t hd r a w a l ] Constantq=40 qmin=40qmax=65 (b) q min = 40 , q max = 65 . Figure 11.1:
EW-ES frontiers. Scenario in Table 9.1. Optimal control computed from problem(5.2), Parameters based on real CRSP index, real 10-year US treasuries (see Table 8.1). Controlcomputed and stored from the PDE solution (synthetic market). Frontier computed using . × MC simulations. Units: thousands of dollars. (cid:15) = 10 − . showing statistics for each point on the efficient frontier are given in Tables A.1 and A.2.For sufficiently large κ we expect that the the efficient frontier should converge to the constantwithdrawal with q = q min . However, numerically, we were not able to obtain accurate solutionsfor very large values of κ , hence the efficient frontiers are shown as ending above the constantwithdrawal points in Figure 11.1. The dotted lines represent the extrapolated values of the efficientfrontiers. Note that these dotted lines are almost vertical, indicating that very small decreases in EScause very large changes in EW. This is, of course, why it is hard to track the curve (numerically)along these points.Both of these efficient frontiers are qualitatively similar, so we focus on Figure 11.1(b). Comparethe variable withdrawal strategy to the fixed withdrawal strategy. The fixed withdrawal strategy q = 40 , from Table 11.2, has ES = − . . If we pick the point on the EW-ES curve withexpected average withdrawals of . , this corresponds to an ES = − . (from Table A.2). Inother words, by accepting a very small amount of extra risk (a smaller ES), we have a strategywhich never withdraws less than per year, but on average withdraws . per year. At firstsight, this seems to be a very counterintuitive result. However, from Table 11.2, we can see that M edian [ W T ] = 717 for constant q = 40 , while from Table A.2, the point ( EW,ES ) = (53 . , − . has M edian [ W T ] = 78 . . This means that the optimal variable withdrawal strategy is simply muchmore efficient in withdrawing cash over the 30 year horizon, in the event the investments do well. The percentiles of fraction in equities, wealth and withdrawals, for the point on the efficient frontier ( EW,ES ) = (51 . , − . are shown in Figure 11.2, for the case [ q min , q max ] = [35 , . The heatmaps of the controls for fraction in equities and optimal withdrawals are given in Figure 11.3. Thenormalized withdrawal is ( q − q min ) / ( q max − q min ) .Note the interesting feature of the median withdrawal in Figure 11.2(c). The median withdrawalstays at q = 35 for the first five years of retirement, then increases rapidly to q = 60 by year seven.This is a result of the fact that the optimal withdrawal is very close to a bang-bang type control,as seen in the heat map shown in Figure 11.3(b). This is not unexpected, due the fact that in thecontinuous withdrawal/rebalancing limit, the withdrawal control (for a rate of withdrawals) is infact bang-bang, as noted in Proposition 7.1. 19 ime (years) F r ac t i on i n s t o cks Median5thpercentile95thpercentile (a)
Percentiles fraction instocks
Time (years) W ea t l h ( Thou sa nd s ) (b) Percentiles wealth
Time (years) W i t hd r a w a l s (t hou sa nd s ) Median95thpercentile5th percentile (c)
Percentiles withdrawals
Figure 11.2:
Scenario in Table 9.1. Optimal control computed from problem (5.2). Parametersbased on the real CRSP index, and real 10-year treasuries (see Table 8.1). Control computed andstored from the PDE solution. Synthetic market, . × MC simulations. q min = 35 , q max = 60 , κ = 0 . . W ∗ = 177 . . (cid:15) = 10 − . Units: thousands of dollars. The corresponding percentiles and heat maps for the case where [ q min , q max ] = [40 , are givenin Figures 11.4 and 11.5, for the point on the EW-ES curve ( EW,ES ) = (54 . , − . . Thesefigures are qualitatively similar to the [ q min , q max ] = [35 , case. (a) Fraction in stocks (b)
Withdrawals
Figure 11.3:
Heat map of controls: fraction in stocks and withdrawals, computed from problem (5.2).cap-weighted real CRSP, real 10 year treasuries. Scenario given in Table 9.1. Control computed andstored from the PDE solution. q min = 35 , q max = 60 , κ = 0 . . W ∗ = 177 . . (cid:15) = 10 − . Normalizedwithdrawal ( q − q min ) / ( q max − q min ) . Units: thousands of dollars. ime (years) F r ac t i on i n s t o cks Median5thpercentile95thpercentile (a)
Percentiles fraction instocks
Time (years) W ea t l h ( Thou sa nd s ) (b) Percentiles wealth
Time (years) W i t hd r a w a l s (t hou sa nd s ) Median95thpercentile5th percentile (c)
Percentiles withdrawals
Figure 11.4:
Scenario in Table 9.1. Optimal control computed from problem (5.2). Parametersbased on the real CRSP index, and real 10-year treasuries (see Table 8.1). Control computed andstored using the PDE. Scenario given in Table 9.1. Synthetic market, . × MC simulations. q min = 40 , q max = 65 , κ = 1 . . W ∗ = − . . (cid:15) = 10 − . Units: thousands of dollars. (a) Fraction in stocks (b)
Withdrawals
Figure 11.5:
Heat map of controls: fraction in stocks and withdrawals, computed from problem (5.2).Parameters based on the real CRSP index, and real 10-year treasuries (see Table 8.1). Scenario givenin Table 9.1. q min = 40 , q max = 65 , κ = 1 . . W ∗ = − . . (cid:15) = 10 − . Normalized withdrawal ( q − q min ) / ( q max − q min ) . Units: thousands of dollars. We compute and store the optimal controls from Problem (5.2), and then use these controls in thebootstrapped historical market, as described in Section 9.2. Table B.1 shows the effect of usingdifferent blocksizes in the bootstrap simulations, compared to the synthetic market results. Theexpected withdrawals are all very close, for all blocksizes. There is more variability in the ES results,but this spread is acceptable for practical purposes. This indicates that the choice of blocksize willnot influence the qualitative results appreciably. In the following, we will report results using ablocksize of . years, which is justified from Table 9.2.The detailed bootstrapped efficient frontiers (using the controls computed in the synthetic mar-ket) are given in Tables C.1 and C.2. In Figure 12.1, we compare the EW-ES frontiers computed forthe case (i) controls computed in the synthetic market, frontier computed in the synthetic marketand (ii) controls computed in the synthetic market, control tested in the historical market. We cansee that the synthetic market frontiers are very close to the historical market frontiers. This indi-cates that the controls computed in the synthetic market are robust to uncertainty in the syntheticstochastic process model calibrated to historical data. -500 -400 -300 -200 -100 0 100 Expected Shortfall E [ ave r a g e w i t hd r a w a l ] HistoricalSynthetic (a) q min = 35 , q max = 60 . -500 -400 -300 -200 -100 0 100 Expected Shortfall E [ ave r a g e w i t hd r a w a l ] HistoricalSynthetic (b) q min = 40 , q max = 65 . Figure 12.1:
EW-ES frontiers, comparison of synthetic frontiers, and frontier generated from (i)controls computed in the synthetic market (ii) control tested in the historical (bootstrapped) market.Scenario in Table 9.1. Parameters based on real CRSP index, real 10-year US treasuries (see Table8.1). Control computed and stored, historical frontier computed using bootstrap resampled simu-lations, blocksize . years. Historical data in range 1926:1-2019:12. Units: thousands of dollars.
13 Discussion
Adding a variable withdrawal strategy, coupled with optimal asset allocation, can dramaticallyimprove the expected average withdrawal, compared with a constant withdrawal strategy. If theminimum withdrawal of the variable strategy is set equal to the constant withdrawal strategy, thenthis result still holds, requiring only a small increase in risk, as measured by expected shortfall (ES).At first sight, this result is almost too good to be true. However, this is easily explainable, dueto two effects.• The median final wealth of the variable withdrawal strategy is much lower than the constantwithdrawal policy. Hence, the variable withdrawal strategy is much more efficient in dis-bursing cash to the retiree over the investment horizon, while keeping the overall risk almost22nchanged. • Due to the quasi-bang-bang control for the variable withdrawal strategy, the median optimalpolicy is to withdraw at the minimum rate for the first few years, followed by withdrawingat the maximum rate for 20-25 years. This avoids large withdrawals in the early years,ameliorating sequence of return risk, with the benefits to be gained in later years.The downside of this strategy is that, although the average withdrawal is signficently improved,the first few years after retirement typically have the smallest (minimum) withdrawals. This maynot be desirable, if retirees are most active at this time, and may wish to have larger incomes.There are several ways to move spending earlier, but these all come at some cost in terms ofEW-ES efficiency. Recall that all quantities in this paper are real, hence we are always preservingreal spending power. However, we could add a real discounting multiplier to our measure of reward.This would change equation (4.4) toEW ( X − , t − ) = E X +0 ,t +0 P (cid:20) i = M (cid:88) i =0 e − βt i q i (cid:21) , (13.1)where β > is a discounting parameter. We experimented with this approach, and it did tend tomove more spending earlier, but at the expense of more risk. In fact, the results using a discountingfactor were similar to simply decreasing the scalarization parameter κ in equation (5.1). Thewithdrawal control in this case was also quasi-bang-bang. Adding a discount factor does not changethe bang-bang nature of the withdrawal control, at least in the continuous withdrawal limit. Thiscan be be verified by adding a discounting factor to equation (7.6).In order to produce a withdrawal control which is more gradual (not bang-bang), we need toadd a nonlinearity to the measure of reward. Let U ( · ) be a utility function, then our measure ofreward could be EW ( X − , t − ) = E X +0 ,t +0 P (cid:20) i = M (cid:88) i =0 U ( q i ) (cid:21) . (13.2)We experimented with various utility functions (e.g. log , power law), and this did have the effectof producing smoother controls as a function of wealth. However, this came at the cost of poorEW-ES efficiency. Recall that our initial objective in this work was to provide the retiree with fixedminimum cash flows, with small risk, while maximizing total withdrawals. Using a nonlinear utilityfunction would conflict with this criteria. We leave exploration of the use of a nonlinear utility inthe reward function as a topic for future work.
14 Conclusions
Our objective in this work was to provide a retiree with minimum fixed cash flows over a long timehorizon, with high probability of expected average withdrawals being signficently larger than theminimum withdrawal. In addition, we control the risk of this strategy as measured by expectedshortfall.The optimal controls consisted of a variable withdrawal rate (with minimum and maximumconstraints) and the asset allocation strategy. Allowing a variable withdrawal strategy (compared “If we have a good year, we take a trip to China,...if we have a bad year, we stay home and play canasta.” retired professor Peter Ponzo, discussing his DC plan withdrawal strategy
23o a fixed withdrawal) dramatically improved the expected average withdrawals, at the expense ofa very small increase in expected shortfall risk. However, the early withdrawals were (with highprobability) at the minimum level, and larger withdrawals were achieved later on in life.Note that the controls were computed in the synthetic market, i.e. a market based on a paramet-ric stochastic process model calibrated to data over the 1926:1-2019:12 period. However, bootstrapresampling tests showed that this strategy is robust to model and parameter uncertainty.An intriguing application of this research is the following. In many countries (Canada in partic-ular), there is a reward, in terms of increased cash flows, if the retiree delays receiving governmentbenefits, until later ages (e.g. 70 in Canada). The common advice is to delay receiving governmentbenefits, and offset this with larger drawdowns from the DC account in the early years of retirement.The argument here is that government benefits are indexed and certain, compared with uncertaininvestment cash flows.However, our results indicate that fairly small reductions in withdrawals from the DC account inearly years result in much larger withdrawals in later years, with a high probability. Hence a betterstrategy may be to take some government benefits earlier, allowing smaller withdrawals from theDC account in early years (reducing sequence of return risk). The smaller government benefits inlater years will (again, with high probability) be offset by these much larger withdrawals from theDC account. Of course, although this strategy has a high probability of success, it is not risk-free.
15 Acknowledgements
P. A. Forsyth’s work was supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC) grant RGPIN-2017-03760.
16 Conflicts of interest
The author has no conflicts of interest to report.24 ppendixA Detailed efficient frontiers: synthetic market
Tables A.1 and A.2 give the detailed results used to construct Figure 11.1. κ ES (5%) E [ (cid:80) i q i ] / ( M + 1) M edian [ W T ] (cid:80) i M edian ( p i ) /M Table A.1:
Synthetic market results for optimal strategies, assuming the scenario given in Table 9.1.Stock index: real capitalization weighted CRSP stocks; bond index: ten year treasuries. Parametersfrom Table 8.1. Units: thousands of dollars. Statistics based on . × Monte Carlo simulationruns. Control is computed using the Algorithm in Section 6, stored, and then used in the Monte Carlosimulations. q min = 35 . , q max = 60 . ( M + 1) is the number of withdrawals. M is the number ofrebalancing dates. (cid:15) = 10 − . κ ES (5%) E [ (cid:80) i q i ] / ( M + 1) M edian [ W T ] (cid:80) i M edian ( p i ) /M -196.2 50.63 202.1 .285 -196.1 48.91 316.0 .303 Table A.2:
Synthetic market results for optimal strategies, assuming the scenario given in Table 9.1.Stock index: real capitalization weighted CRSP stocks; bond index: ten year treasuries. Parametersfrom Table 8.1. Units: thousands of dollars. Statistics based on . × Monte Carlo simulationruns. Control is computed using the Algorithm in Section 6, stored, and then used in the Monte Carlosimulations. q min = 40 . , q max = 65 . ( M + 1) is the number of withdrawals. M is the number ofrebalancing dates. (cid:15) = 10 − . Effect of blocksize: stationary block bootstrap resampling
Table B.1 shows the effect of blocksize on the bootstrap resampling algorithm. κ ES (5%) E [ (cid:80) i q i ] / ( M + 1) M edian [ W T ] (cid:80) i M edian ( p i ) /M Synthetic Market0.5 -50.86 51.33 368.2 .363Historical Market: ˆ b = 0 . years.-17.28 51.19 340.6 .360Historical Market: ˆ b = 0 . years.-40.63 51.19 343.1 .361Historical Market: ˆ b = 1 years.-34.13 51.23 342.9 .361Synthetic Market1.0 4.730 49.89 406.3 .331Historical Market: ˆ b = 0 . years.20.47 49.72 381.0 .330Historical Market: ˆ b = 0 . years.-5.10 49.72 383.6 .331Historical Market: ˆ b = 1 years.-0.84 49.74 384.4 .331 Table B.1:
Historical market results for optimal strategy, q min = 35 , q max = 60 . The scenariois given in Table 9.1. Stock index: real capitalization weighted CRSP stocks; bond index: 10 yeartreasuries. Historical data in range 1926:1-2019:12. Units: thousands of dollars. Statistics based on bootstrap simulations. Control is computed using the algorithm in Section 6, stored, and thenused in the bootstrap resampling tests. ( M + 1) is the number of withdrawals. M is the number ofrebalancing dates. (cid:15) = 10 − . C Bootstrapped frontiers
Tables C.1 and C.2 show the detailed results for the EW-ES frontiers. The controls were computedin the synthetic market, and tested in the historical market.26
ES (5%) E [ (cid:80) i q i ] / ( M + 1) M edian [ W T ] (cid:80) i M edian ( p i ) /M q max = q min = 35 N/A 45.63 35.0 920.0 .269
Table C.1:
Control computed in the synthetic market, assuming the scenario given in Table 9.1.Stock index: real capitalization weighted CRSP stocks; bond index: ten year treasuries. Parametersfrom Table 8.1. Units: thousands of dollars. Statistics based on bootstrap resampling of thehistorical data. Historical data in range 1926:1-2019:12. Expected blocksize ˆ b = . years. q min = 35 . , q max = 60 . ( M + 1) is the number of withdrawals. M is the number of rebalancing dates. κ ES (5%) E [ (cid:80) i q i ] / ( M + 1) M edian [ W T ] (cid:80) i M edian ( p i ) /M -183.5 48.69 296.9 .300 q max = q min = 40 N/A -183.5 40.0 676.7 .340
Table C.2:
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