Optimal control of the decumulation of a retirement portfolio with variable spending and dynamic asset allocation
OOptimal control of the decumulation of a retirement portfoliowith variable spending and dynamic asset allocation
Peter A. Forsyth a Kenneth R. Vetzal b Graham Westmacott c January 6, 2021
Abstract
We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for re-tirement savings decumulation (Waring and Siegel, 2015) to include a cap and a floor onwithdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the riskof depleting the investment portfolio. We determine the dynamic asset allocation strategywhich maximizes a weighted combination of expected total withdrawals (EW) and expectedshortfall (ES), defined as the average of the worst five per cent of the outcomes of realterminal wealth. We compare the performance of our dynamic strategy to simpler alter-natives which maintain constant asset allocation weights over time accompanied by eitherour same modified ARVA spending rule or withdrawals that are constant over time in realterms. Tests are carried out using both a parametric model of historical asset returns aswell as bootstrap resampling of historical data. Consistent with previous literature thathas used different measures of reward and risk than EW and ES, we find that allowingsome variability in withdrawals leads to large improvements in efficiency. However, unlikethe prior literature, we also demonstrate that further significant enhancements are possiblethrough incorporating a dynamic asset allocation strategy rather than simply keeping assetallocation weights constant throughout retirement.
Keywords:
Finance, risk management, optimal asset allocation, decumulation, definedcontribution plan
JEL codes:
G11, G22
AMS codes:
Declarations of interest:
None
Funding:
Access to Wharton Research Data Services and historical data from the Centerfor Research in Security Prices was provided through an institutional subscription paid forby the University of Waterloo. Peter Forsyth was also supported by the Natural Sciencesand Engineering Research Council of Canada (NSERC) under grant RGPIN-2017-03760. a David R. Cheriton School of Computer Science, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 34415. b School of Accounting and Finance, University of Waterloo, Waterloo ON, Canada N2L 3G1, [email protected] , +1 519 888 4567 ext. 46518. c PWL Capital, 20 Erb Street W., Suite 506, Waterloo, ON, Canada N2L 1T2, [email protected] ,+1 519 880 0888. a r X i v : . [ q -f i n . C P ] J a n Introduction
Defined Benefit (DB) pension plans are disappearing, being replaced by Defined Contribution(DC) plans. According to a recent study by the Organization for Economic Co-operation andDevelopment (OECD), less than 50% of pension assets in 2018 were held in DB plans in over 80%of reporting jurisdictions. Moreover, in more than 75% of reporting countries the proportion ofpension assets in DB plans was lower in 2018 relative to its level a decade earlier (OECD, 2019).Note that the proportion of assets in DB plans is a lagging indicator of the shift to DC plansbecause employees who were historically covered by traditional DB plans have had more time toamass retirement savings. For example, in Israel the proportion of pension assets in DB plansdropped from 84% in 2008 to 56% in 2018. However, DB plans in that country were closed tonew members in 1995 (OECD, 2019). Almost 25 years later, over half of pension assets in Israelare still in DB plans.The shift to DC plans is an inevitable consequence of corporations and governments beingunwilling (or unable) to manage the risks associated with DB plans. In contrast, in DC plans themanagement of the financial assets is left up to individual investors. Given the long-term natureof retirement savings, this is a challenging task for most people. Assuming that investors do man-age to accumulate healthy balances in their DC accounts, the situation gets even more complexupon retirement. Individuals must continue to manage their financial assets, and also determinea decumulation strategy to withdraw assets and fund spending with uncertain longevity. Whileit is often suggested that retirees should purchase annuities, this rarely happens in practice. Forexample, Milevsky and Young (2007) report findings from a survey of U.S. retirees indicatingthat only 8% of respondents who were DC plan members and less than 2% of all respondentschose to annuitize. More recently, it has been reported that only around 4% of retirees with DCplans at a prominent Canadian insurer opted to annuitize (Carrick, 2020).The reluctance of retirees to annuitize is sometimes called a puzzle, since standard life cycleeconomic models based on utility maximization suggest that annuitization is optimal (Peijnen-burg et al., 2016). However, the overwhelming aversion to annuitization by retirees suggests thatthese economic models are missing something important. In practice, there are many reasonswhy retirees do not annuitize. MacDonald et al. (2013) list dozens of real-world factors includinglack of true inflation protection, loss of control over capital, expensive pricing, the availabilityof other sources of guaranteed income such as government benefits, and paltry payments undersome financial market conditions such as the current low interest rate environment.Assuming that purchasing an annuity is undesirable, retirees must devise suitable decumu-lation strategies. A major component of these plans is how much money to withdraw over time.Retirees who withdraw fairly large sums run the risk of outliving their resources, i.e. the risk of“ruin”. On the other hand those who take out relatively small amounts may have less enjoyableretirements and leave their heirs with (unintended) large bequests.Absent any annuitization, decumulation strategies can generally be classified as having fixedor variable withdrawals. Within these categories, several variations have been proposed. Mac-Donald et al. (2013) provide a nice summary of the various possibilities. In a fixed scheme, the MacDonald et al. (2013) also discuss hybrid strategies, which combine some level of annuitization with a(fixed or variable) decumulation scheme. We concentrate on strategies involving cash flows in the absence of any
4% rule due to Bengen (1994). Thisfixed scheme states that retirees with an annually rebalanced portfolio split evenly betweenbonds and stocks can withdraw 4% of their initial wealth each year in real terms. Backtestingthis rule on U.S. data showed that retirees would never have run out of funds, over any rollinghistorical 30-year period considered (Bengen, 1994).Backtesting using rolling historical periods is common in the practitioner literature. However,in general this approach seriously underestimates risk. Any two adjacent 30-year periods willhave 29 years in common, any two 30-year periods beginning two years apart will have 28 yearsin common, etc. Consequently, the overall results will tend to be highly correlated, and thiscould be very misleading. The findings reported by Bengen (1994) address the question ofwhat the historical experience would have been over a long period for someone who retired ina particular year and then followed the 4% rule. In other words, using rolling historical periodsonly considers what did happen, giving zero weight to any other plausible scenario that might have happened, and which could occur in the future. Two alternatives which can give a bettersense of the risk involved are (i) to fit a parametric model to the historical data and then runa large number of Monte Carlo simulations, and (ii) to use block bootstrap resampling of thedata (Politis and Romano, 1994), which involves randomly drawing (with replacement) shorterperiods of data and chaining them together over the decumulation horizon. We use both of theseapproaches below and find that the risk of using the 4% rule is quite significant. As mentioned above, practitioners have proposed several variable schemes that allow spend-ing to fluctuate in response to portfolio returns. These strategies typically permit higher initialwithdrawal rates compared to fixed schemes such as the 4% rule. These enhanced withdrawal actual level of annuitization, so we ignore hybrid strategies in this work. There are other reasons to think that Bengen (1994) understated the risk of the 4% rule. One is that datapast 1992 was extrapolated using historical averages for financial market returns each year. For example, the30-year performance of the rule given a retirement date of 1976 was assessed using 16 years of actual marketdata, followed by 14 years in which the returns for stocks and bonds and the inflation rate were constant eachyear at their long-term average values. This clearly understates the strategy’s risk for cases with several years inretirement after 1992. A more fundamental issue from today’s perspective is the reliability of the 4% rule duringa lengthy period of very low interest rates. Finke et al. (2013) considered bond market conditions early in 2013and estimated that the failure rate for the 4% rule assuming 10 years of below average bond returns and a 50%stock allocation was 32%, strongly suggesting that 4% is too high a withdrawal rate. Given that interest rateshave continued to trend downwards more recently, there are solid grounds for pessimism about the viability ofthe 4% rule today. T . Dang et al. (2017) assume that most 65-year olds can expect tolive for 20 years with high probability, and thus set a wealth target of one-half of the initialwealth at T = 20 years (after retirement). The idea is that retirees can decide how to hedgelongevity risk at age , expecting to have spent one-half of their original wealth up to then.Irlam (2014) uses dynamic programming methods to determine asset allocation, given anobjective of maximizing the number of years of solvency divided by the number of years lived.This is the only study we are aware of in the practitioner literature for which the asset allocationdepends on a specified financial objective. Irlam (2014) concludes that asset allocation rules thatdepend only on time such as “age in bonds” or various target-date fund glide paths require ahigher amount of investment in order to obtain the same withdrawal rates in retirement, ascompared to his approach where the asset allocation is time and state-dependent. However,Irlam (2014) only considers a fixed annual withdrawal amount in retirement.In this work we further explore the effect of a variable spending rule in combination with anasset allocation strategy tailored to optimizing a financial objective. In particular, we use anARVA spending rule augmented by constraints on minimum and maximum annual withdrawals.The minimum withdrawal constraint means that there is risk of depleting the portfolio entirelyprior to the end of the investment horizon. We use the Expected Shortfall (ES) of the terminalportfolio value as a measure of risk. The ES at level x % is the mean of the worst x % of outcomes,and is thus a measure of tail risk. As a measure of reward, we use total Expected Withdrawals(EW). Based on a parametric model calibrated to historical market data, we determine theportfolio allocation strategy that optimizes the multi-objective Expected Withdrawals-ExpectedShortfall (EW-ES) objective function. A similar measure of risk and reward for DC plan decumulation is used in Forsyth (2020c).However, Forsyth (2020c) uses the withdrawal amount as a control, rather than an ARVAspending rule. In this case, Forsyth (2020c) shows that the withdrawal control is essentiallya bang-bang type control, with minimum withdrawals during the earlier years of retirement.Use of the ARVA spending rule (with constraints) provides more control over the timing ofwithdrawals.We verify the robustness of this strategy through tests using bootstrap resampling of histori-cal return data. Our tests show that the ARVA spending rule coupled with an optimal allocationstrategy is always more efficient than a constant withdrawal, constant weight strategy. In fact,our optimal dynamic ARVA strategy outperforms this alternative even when the minimum with-drawal under ARVA is equal to the constant withdrawal with constant weights. This verifiesthat allowing some variability in withdrawals sharply reduces the risk of depleted savings, con-sistent with Pfau (2015) and Tretiakova and Yamada (2017). In addition, we demonstrate thatsolving an optimal stochastic control problem to specify the asset allocation can provide furthersignificant benefits beyond those obtained by permitting withdrawal variability alone. Forsyth et al. (2020) use the same measure of reward, but minimize the downside variability of withdrawalsfor an ARVA type spending rule, i.e. the risk measure is downward withdrawal variability. There are some othernoteworthy differences between this work and that of Forsyth et al. (2020). First, we impose upper and lowerbounds on annual withdrawals. Second, the assumed underlying financial model is more complex here, as itincorporates stochastic bond market returns. ARVA Spending Rule
Consider the following spending rule. Each year, a virtual (hypothetical) fixed term annuityis constructed, based on the current portfolio value, the number of remaining years of requiredcash flows, and a real (inflation adjusted) interest rate. The investor then withdraws an amountbased on the hypothetical payment of this virtual annuity. Clearly, the annual payments willbe variable, since the virtual annuity is recalculated each year, and is a function of the currentportfolio value. The portfolio is liquidated at the end of the investment horizon. A surplus willbe returned to the investor (or the investor’s estate). Any shortfall must be settled at this timeas well.We are now faced with the choice of determining a timespan for the virtual fixed term annuity.Rather than specifying a maximum possible lifespan (which would be overly conservative), weassume that retirees are in the top 20% of the population in terms of conditional expectedlongevity (Westmacott, 2017). Consider a retiree who is x years old at t = 0 . Assuming thatthe x + t year old retiree is alive at time t , let T ∗ x ( t ) be the time at which 80% of the cohort of x + t year olds are expected to have passed away, conditional on all members of the cohort beingalive at time t . At time t , the fixed term of the virtual annuity is then T ∗ x ( t ) − t . This mortalityassumption has the effect of providing increased spending during the early years of retirement.By varying the fraction of the cohort assumed to have passed away, we can increase/decreasespending in early retirement years at the cost of decreased/increased spending in later years.Note that our ARVA withdrawal amount is not generally the same as would be obtained from acurrently purchased life annuity.Given the real interest rate r , the present value of an annuity which pays continuously at arate of one unit per year for T ∗ x ( t ) − t years is denoted by the annuity factor a ( t ) = 1 − exp[ − r ( T ∗ x ( t ) − t )] r . (2.1)It follows that W ( t ) /a ( t ) is the continuous real annuity payment for ( T ∗ x ( t ) − t ) years, which canbe purchased with wealth W ( t ) at time t . We make the assumption that withdrawals occur atdiscrete times in T ≡ { t = 0 < t < · · · < t M = T } , (2.2)where t denotes the time that the x year old retiree begins to withdraw money from the DCplan. We assume the times in T are equally spaced with t i − t i − = ∆ t = T /M , i = 1 , . . . , M .We let ∆ t = one year. We determine the cash withdrawal at time t i by converting the continuouspayment above into a lump sum received in advance of the interval [ t i , t i +1 ] . This lump sumwithdrawal at t i is W ( t i ) A ( t i ) , where A ( t i ) = (cid:90) t i +1 t i e − r ( t (cid:48) − t i ) a ( t (cid:48) ) dt (cid:48) . (2.3)In this work, we will compute equation (2.3) based on the CPM 2014 mortality tables (male)from the Canadian Institute of Actuaries to compute T ∗ x ( t ) with x = 65 . Further discussion of We assume that the investment portfolio consists of two index funds. These funds include astock market index fund and a constant maturity bond index fund. Let the investment horizonbe T , and S t and B t respectively denote the real (inflation adjusted) amounts invested in thestock index and the bond index. These amounts can change due to (i) changes in the real unitprices and (ii) the investor’s asset allocation strategy. In the absence of the application of aninvestor’s control, all changes in S t and B t result from changes in asset prices.We model the stock index (in the absence of an applied control) as following a jump diffusionprocess. 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 randomjump multiplier. When a jump occurs, S t = ξ s S t − . Use of jump processes allows for modellingof fat-tailed (non-normal) asset returns. We assume that log( ξ s ) follows a double exponentialdistribution (Kou and Wang, 2004). The probability of an upward jump is p s u , with − p s u beingthe probability 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< . (3.1)Define κ sξ = E [ ξ s −
1] = p s u η s η s − − p s u ) η s η s + 1 − . (3.2)Without an applied control, 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 − , (3.3)where µ s is the (uncompensated) drift rate, σ s is the diffusive volatility, Z s is a Brownian motion, π st is a Poisson process with intensity parameter λ sξ , and ξ si are i.i.d. positive random variableshaving distribution (3.1). Moreover, ξ si , π st , and Z s are assumed to all be mutually independent.As in MacMinn et al. (2014) and Lin et al. (2015), we use a common practitioner approachand model the returns of the constant maturity bond index (absent an applied control) as astochastic process. This approach has the advantage that estimating model parameters frommarket data is quite straightforward, without the need to devise a parametric process for realinterest rates. As in MacMinn et al. (2014), we assume that the constant maturity bond indexfollows a jump diffusion process. In particular, B t − = B ( t − (cid:15) ) , (cid:15) → + . In the absence ofcontrol, 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 − , (3.4)where the terms in equation (3.4) are defined analogously to equation (3.3). In particular, π bt is Appendix A documents evidence of leptokurtic behavior for both of the indexes that we use in our tests. 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< , (3.5)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 (3.4) represents an additional cost of borrowing ( B t < ), i.e. a spreadbetween borrowing and lending rates. We assume that the diffusive components of S t and B t are correlated, i.e. dZ s · dZ b = ρ sb dt . However, the jump process terms for these two indexesare assumed to be mutually independent. It is possible to include more complex stock and bond processes, such as stochastic volatilityfor example. However, Ma and Forsyth (2016) have shown that including stochastic volatilityeffects does not have a significant effect on the results for long term investors. In order to verifythe robustness of the strategies, we will determine the optimal controls using the parametricmodel based on equations (3.3) and (3.4). We then test these controls on bootstrapped resampledhistorical data. This is quite a strict test, since the bootstrapped resampling algorithm makesno assumptions about the underlying bond and stock stochastic processes.We define the investor’s total wealth at time t as W t ≡ S t + B t . We impose the constraintsthat (assuming solvency) shorting stock and using leverage (i.e. borrowing) are not allowed.Insolvency can arise from withdrawals. If this happens, the portfolio is liquidated and debtaccumulates at the borrowing rate. The borrowing rate is taken to be the return on the constantmaturity bond index plus a spread µ bc . For ease of explanation, we will occasionally use the notation S t ≡ S ( t ) , B t ≡ B ( t ) and W t ≡ W ( t ) . Earlier in equation (2.2) we specified a set of times T for which withdrawals are permitted.We now expand the scope of T so that portfolio rebalances are also allowed at those times, i.e. T is the set of withdrawal/rebalancing times. More specifically, let the inception time of theinvestment be t = 0 . At each withdrawal/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 portfolio is liquidated and the final cash flow q M occurs.Given a time dependent function f ( t ) , we use the shorthand notation f ( t + i ) ≡ lim (cid:15) → + f ( t i + (cid:15) ) and f ( t − i ) ≡ lim (cid:15) → + f ( t i − (cid:15) ) . We assume that no taxes are triggered by rebalancing. Thiswould normally be the case in a tax-advantaged DC savings account. Since we assume yearlyapplication of the controls (rebalancing), we expect transaction costs to be small and hence theycan be safely ignored. With no taxes or transaction costs, it follows that W ( t + i ) = W ( t − i ) − q i .The multi-dimensional controlled underlying process is denoted by X ( t ) = ( S ( t ) , B ( t )) ,with t ∈ [0 ,T ] . The realized state of the system is x = ( s,b ) . Let the rebalancing control p i ( · ) See Forsyth (2020b) for a discussion of the evidence for stock and bond price jump independence. It is possible to include transaction costs, but this will increase computational cost (Van Staden et al., 2018).
8e the fraction invested in the stock index at 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 ) . (4.1)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) , t i ∈ T . We search for the optimal strategies amongst allcontrols with constant wealth after cash withdrawal, 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 . (4.2)We assume that rebalancing occurs instantaneously, with the implication that the probabilityof a jump occurring in either index is zero during the rebalancing period ( t − i , t + i ) .Let Z represent the set of admissible values of the control p i ( · ) . An admissible control P ∈ A ,where A is the admissible control set, can be written as P = { p i ( · ) ∈ Z : i = 0 , . . . , M − } .We impose no-shorting and no-leverage constraints by specifying Z = [0 , . (4.3)We also apply the constraint that if W ( t + i ) < , the stock index holding is liquidated, p ( W ( t + i ) , t i ) = 0 if W ( t + i ) < , (4.4)and no further stock purchases are permitted, with the result that debt accumulates at the bondreturn plus a spread. In addition, we 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 = { p n ( · ) , . . . , p M − ( · ) } . Initially, we describe our measure of risk. Suppose g ( W T ) is the probability density function ofterminal wealth W T at t = T , and let (cid:90) W ∗ α −∞ g ( W T ) dW T = α, (5.1)so that Prob [ W T > W ∗ α ] = 1 − α . We can interpret W ∗ α as the Value at Risk (VAR) at level α .The Expected Shortfall (ES) at level α is thenES α = (cid:82) W ∗ α −∞ W T g ( W T ) dW T α , (5.2)which is the mean of the worst α fraction of outcomes. Usually, α ∈ { . , . } . We emphasizethat the definition of ES in equation (5.2) uses the probability density of the final wealth distri-9ution, not the density of loss . This has the implication that a larger value of ES is desirable(the worst case average portfolio value at T ). Define X +0 = X ( t +0 ) , X − = X ( t − ) . Given an expectation under control P , E P [ · ] , Rockafellarand Uryasev (2000) show that 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) . (5.3)The notation ES α ( X − , t − ) indicates that ES α is as seen at ( X − , t − ) . This definition is then the pre-commitment ES. A strategy based on optimizing the pre-commitment ES at time zero is timeinconsistent , since the investor may have an incentive to deviate from the strategy at t > . Thus,some authors have described pre-commitment strategies as being non-implementable . However,this is really a matter of interpretation: we consider the pre-commitment strategy as a usefultechnique to compute an appropriate value of W ∗ in equation (5.3). In fact, the strategy whichfixes W ∗ ∀ t > , is the induced time consistent strategy (Strub et al., 2019), and is consequentlyimplementable. We delay further discussion of this point to Section 6.Our measure of reward is expected total withdrawals (EW), defined asEW ( X − ,t − ) = E X +0 ,t +0 P (cid:20) i = M (cid:88) i =0 q i (cid:21) . (5.4)Note that we do not discount withdrawals, with either a market-based measure of the appropriaterisk-adjusted discount rate or with a subjective discount rate. This reflects a desire to avoidbasing our strategy on parameters that are difficult to estimate. Since the portfolio weights willdepend on realized investment returns and withdrawals over time, it is problematic to estimatethe appropriate risk-adjusted discount rate. Moreover, it is likely to be difficult to determine asubjective discount rate, which could easily vary across investors and/or over time. However,we observe that the economic effect of discounting the withdrawals would be to make earlierwithdrawals more desirable. We have already incorporated a similar effect through the mortalityboost to the spending rule discussed in Section 2 above. Our overall approach involves a statistical tradeoff between reward and risk, similar to mean-variance portfolio analysis but with different measures of reward and risk. The main alternativewould be a standard life cycle approach, where we would maximize a specified utility function.This would raise concerns related to estimating parameters such as risk aversion or elasticitiesof intertemporal substitution, similar to the subjective discount rate discussed in the precedingparagraph. However, this would pose more of a problem since the appropriate form of theutility function itself is open to question. The most popular specification in the literature ispower utility, which implies constant relative risk aversion. However, a recent empirical studyby Meeuwis (2020) of the portfolio holdings and income of millions of US retirement investors The negative of ES is often called Conditional Value at Risk (CVAR), which has been used as a risk measurein several prior asset allocation studies (e.g. Gao et al., 2016; Cui et al., 2019; Forsyth, 2020a). κ > , we seek the control P that maximizesEW ( X − , t − ) + κ ES α ( X − , t − ) . (6.1)More precisely, we define the pre-commitment EW-ES problem in terms of the value function J (cid:0) s,b,t − (cid:1) = sup P ∈A sup W ∗ (cid:40) E X +0 ,t +0 P (cid:34) M (cid:88) i =0 q i + κ (cid:18) W ∗ + min( W T − W ∗ , α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) (6.2)and the constraints ( S t , B t ) follow processes (3.3) and (3.4) ; t / ∈ T W + (cid:96) = S − (cid:96) + B − (cid:96) = q (cid:96) ; X + (cid:96) = (cid:0) S + (cid:96) ,B + (cid:96) (cid:1) S + (cid:96) = p (cid:96) ( · ) W + (cid:96) ; B + (cid:96) = (1 − p (cid:96) ( · )) W + (cid:96) p (cid:96) ( · ) ∈ Z = [0 , if W + (cid:96) > p (cid:96) ( · ) = 0 if W + (cid:96) ≤ (cid:96) = 0 , . . . , M − t (cid:96) ∈ T . (6.3)By reversing the order of the sup sup in equation (6.2), the value function 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 ∗ + min( W T − W ∗ , α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) . (6.4)Denote the investor’s initial wealth at t by W − = S − + B − . Observe that the inner supremumin equation (6.4) is a continuous function of W ∗ . Then, assuming that the domain of W ∗ iscompact, we define W ∗ (0 ,W − ) = arg max W ∗ (cid:40) 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 ∗ + min( W T − W ∗ , α (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = (0 ,W − ) (cid:35)(cid:41)(cid:41) . (6.5)11egarding W ∗ (0 ,W − ) as fixed ∀ t > , the following proposition follows immediately: Proposition 6.1 (Pre-commitment strategy equivalence to a time consistent policy for analternative objective function) . The pre-commitment EW-ES strategy P ∗ determined by solving J (0 ,W ,t − ) with W ∗ (0 ,W − ) from equation (6.5) is the time consistent strategy for an equivalentproblem with fixed W ∗ (0 ,W − ) and value function ˜ J ( s,b,t ) defined by ˜ J ( s,b,t − n ) = sup P n ∈A (cid:40) E X + n ,t + n P n (cid:34) i = M (cid:88) i = n q i + κ min( W T − W ∗ (0 ,W − ) , α (cid:12)(cid:12)(cid:12)(cid:12) X ( t − n ) = ( s,b ) (cid:35)(cid:41) . (6.6) Remark 6.1 (EW-ES induced time consistent strategy: an implementable control) . In thefollowing, we consider the actual strategy followed by the investor for any t > as given by theinduced time consistent strategy that solves problem (6.6) with the fixed value of W ∗ (0 ,W − ) from equation (6.5). This strategy is identical to the EW-ES strategy at time zero. Hence, werefer to this strategy as the EW-ES strategy. It is understood that this refers to the strategy thatsolves the time consistent equivalent problem (6.6) for any t > . Consequently, this strategy isimplementable (Forsyth, 2020a) (the investor has no incentive to deviate from this control for t > ). To solve the pre-commitment EW-ES problem (6.2), we start by interchanging the sup sup toarrive at equation (6.4). We expand the state space to ˆ X = ( s,b,W ∗ ) , and define the auxiliaryvalue function V ( s,b,W ∗ ,t − n ) = sup P n ∈A (cid:40) E ˆ X + n ,t + n P n (cid:34) M (cid:88) i = n q i + κ (cid:18) W ∗ + min( W T − W ∗ , α (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ˆ X ( t − n ) = ( s,b,W ∗ ) (cid:35)(cid:41) (7.1)and slightly revised constraints ( S t , B t ) follow processes (3.3) and (3.4) ; t / ∈ T W (cid:96) = S − (cid:96) + B − (cid:96) = q (cid:96) ; ˆ X + (cid:96) = (cid:0) S + (cid:96) ,B + (cid:96) ,W ∗ (cid:1) S + (cid:96) = p (cid:96) ( · ) W + (cid:96) ; B + (cid:96) = (1 − p (cid:96) ( · )) W + (cid:96) p (cid:96) ( · ) ∈ Z = [0 , if W + (cid:96) > p (cid:96) ( · ) = 0 if W + (cid:96) ≤ (cid:96) = 0 , . . . , M − t (cid:96) ∈ T . (7.2)We can solve auxiliary problem (7.1) using dynamic programming. The optimal control p n ( w,W ∗ ) at time t n is determined from p n ( w, W ∗ ) = arg max p (cid:48) ∈Z V ( wp (cid:48) ,w (1 − p (cid:48) ) ,W ∗ ,t + n ) if w > if w ≤ . (7.3) See Strub et al. (2019) for a discussion of induced time consistent strategies. t n via V ( s,b,W ∗ ,t − n ) = V ( w + p n ( w + ,W ∗ ) , w + (cid:0) − p n ( w + ,W ∗ ) (cid:1) , W ∗ , t + n ) + q n ( w − ,W ∗ ) , (7.4)where w − = s + b , and w + = w − − q n . q n ( w − ,W ∗ ) is based on our ARVA spending rule (seeSection 9 for a precise specification). Note that the spending rule will be a function of wealthbefore withdrawal. At t = T , we have V ( s,b,W ∗ ,T + ) = κ (cid:18) W ∗ + min( s + b − W ∗ , α (cid:19) . (7.5)For times t ∈ ( t n − ,t n ) , there are no cash flows or controls applied. Recall that all quantities arereal, and that there is no discounting. The iterated expectation property combined with Itô’sLemma for jump processes in equations (3.3-3.4) then gives 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 ; t ∈ ( t n − , t n ) (7.6)Define J ( s,b,t − ) = sup W (cid:48) V ( s,b,W (cid:48) ,t − ) . (7.7)It is then straightforward to see that formulation (7.1-7.6) is equivalent to problem (6.2). We briefly describe our numerical solution approach. We refer the reader to Forsyth andLabahn (2019) and Forsyth (2020b) for further details. We start by solving the auxiliary problem(7.1-7.2) with fixed values of W ∗ , κ and α . Since shorting of the stock index is not allowed, S ( t ) ≥ . We localize the domain s > on a finite localized domain s ∈ [ e ˆ x min ,e ˆ x max ] . The computationaldomain for s is discretized using n ˆ x equally spaced nodes in the ˆ x = log s direction. Similarly,we define the localized domain for b > to be b ∈ [ b min , b max ] = [ e y min ,e y max ] . The computationaldomain for b > is discretized using n y equally spaced nodes in the y = log b direction. Sincethe investor can become insolvent due to withdrawals, we also define a mirror image grid for b < (Forsyth, 2020b).We use the Fourier methods described in Forsyth and Labahn (2019) to solve PIDE (7.6)between rebalancing times. Wrap-around errors are minimized using the domain extensiontechnique in Forsyth and Labahn (2019). 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 for e ˆ x are thousands of dollars). Numericaltests showed that the errors involved in this domain localization were at most in the fifth digit.At rebalancing times, we discretize the equity fraction p ∈ [0 , using n y equally spacednodes and evaluate the right hand side of equation (7.3) using linear interpolation. We thensolve the optimization problem (7.3) using exhaustive search over the discretized p values. See Forsyth (2020a) for discussion of a similar problem. t = 0 , which we denoteby V ( s, b, W ∗ , , we then compute the solution of problem (6.2) using equation (7.7). Morespecifically, we solve J (0 ,W , − ) = sup W (cid:48) V (0 , W , W (cid:48) , − ) (7.8)given initial wealth W . We solve this outer optimization problem using a one-dimensionaloptimization algorithm. If W t (cid:29) W ∗ and t → T , then Prob [ W T < W ∗ ] (cid:39) . In addition, for large values of W t thewithdrawal is capped at q max . As a result the objective function is almost independent of thecontrol, and thus determination of the control becomes ill-posed. To avoid this, we change theobjective function (6.2) by adding a stabilizing term (cid:15)W T , giving J ( s,b,t − ) = 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 ∗ + min( W T − W ∗ , α (cid:19) + (cid:15)W T (cid:12)(cid:12)(cid:12)(cid:12) X ( t − ) = ( s,b ) (cid:35)(cid:41) . (7.9)A negative value for (cid:15) forces the strategy to invest in the bond index when W t is very largeand t → T , where the original control problem is ill-posed. This choice is consistent with de-risking retirement assets as soon as possible (Merton, 2014). Setting (cid:15) = − − gave the sameresults as setting (cid:15) = 0 to four digits for the summary statistics of the problem solution. This isdue to the fact that outcomes with very large terminal wealth are highly unlikely. As mentioned above, our model assumes that the retiree’s portfolio is allocated to either astock index or a constant maturity bond index. In order to have a long history encompassingexpansions, recessions, stock market booms and crashes, and different levels of interest rates,we use US financial market data. In particular, the stock index is taken to be the Center forResearch in Security Prices (CRSP) Value-Weighted Index , while the bond index is the CRSP30-Day Treasury bill (T-bill) Index. Both indexes are measured on a monthly basis from January1926 through December 2018, giving a total of 1,116 observations. To work in real terms, wedeflate both indexes by the Consumer Price Index (CPI), which was also provided by CRSP. We use the threshold technique (Mancini, 2009; Cont and Mancini, 2011; Dang and Forsyth,2016) to estimate the parameters for the stochastic process models (3.3-3.4) (see Appendix A).All estimated parameters reflect real (inflation adjusted) returns. Table 8.1 shows the annualizedparameter estimates. For reference, the table also gives the estimated parameters for the two time Since the problem is not guaranteed to be convex, we cannot be sure that we converge to the global maximum.Additional testing based on a search over the finest grid suggests that we do indeed have the globally optimalsolution. This is a total return index of the broad US stock market, reflecting both distributions such as dividends andcapital gains/losses due to price changes. The CRSP data used in this study was obtained through Wharton Research Data Services (WRDS). Thisservice and the data available thereon constitute valuable intellectual property and trade secrets of WRDS and/orits third party suppliers. µ s σ s λ s p s up η s η s ρ sb Threshold ( β = 3 ) .08607 .14600 .32258 .23333 4.3578 5.5089 .08311GBM .08044 .18460 N/A N/A N/A N/A .05870Real 30 Day T-bill IndexMethod µ b σ b λ b p b up η b η b ρ sb Threshold ( β = 3 ) .00454 .01301 .51610 0.39580 65.875 57.737 .08311GBM .00448 .01814 N/A N/A N/A N/A .05870 Table 8.1:
Estimated annualized parameters for the double exponential jump diffusion model (3.3-3.4). Sample period 1926:1 to 2018:12. GBM refers to a geometric Brownian motion model (i.e.no jumps). The threshold method is described in Appendix A. series assuming geometric Brownian motion (GBM). For the threshold case, after removingany returns which occur at times corresponding to jumps in either series, the correlation ρ sb isthen estimated using the remaining sample covariance.The annualized real value-weighted stock index parameters in Table 8.1 for the double ex-ponential jump diffusion model correspond to an (uncompensated) drift rate of 8.6% and adiffusive volatility of 14.6%. Jumps in the stock index are estimated to occur about once everythree years. Conditional on a jump occurring, a downwards jump is about 3 times more likelythan an upwards jump. The mean jump size is about 23% in the upward direction and 18%in the downward direction. Since the standard deviation is equal to the mean for an exponen-tially distributed random variable, the magnitudes of both upward and downward jumps canvary considerably. The corresponding GBM parameter estimates imply a drift of about 8% perannum, with a volatility of 18.5%. This volatility is higher than the diffusive volatility for thejump model since in the GBM case this term effectively combines the effects of volatility due toboth diffusion and jumps.Turning to the T-bill index, the annualized jump model parameters correspond to a real(uncompensated) drift of approximately 0.45% and a diffusive volatility of about 1.3%. Jumpsare estimated to occur about every 2 years, slightly more often than for the stock index. Down-ward jumps are again more likely than upward jumps, though somewhat less so compared to thestock index. The mean jump size is around 1.5% in the upward direction, and about 1.7% in thedownward direction. The GBM parameter estimates indicate a drift that is also about 0.45%,and a volatility of approximately 1.8%. Finally, the correlation between the diffusive terms forthe two indexes is quite low, around .083 for the jump model and .059 for the GBM case. In order to focus exclusively on decumulation, we consider an investor just entering retirementat age 65 with savings of $1 million. Our investor is assumed to have the life expectancycharacteristics of a Canadian male. According to the CPM 2014 mortality table, this investor The GBM parameter estimates are calculated using maximum likelihood estimation. T (years) 30Investor ( t = 0 ) 65-year old Canadian maleMortality table CPM 2014Equity market index CRSP value-weighted index (real)Bond index 30-day T-bill index (real)Initial portfolio value W t = 0 , , . . . , q max q min W t < µ bc = . Interest rate for ARVA computation (2.3) µ b = 0 . Rebalancing interval (years) 1Market parameters See Table 8.1
Table 9.1:
Base case input data. Monetary units: thousands of dollars. The CPM 2014 mortalitytable is from the Canadian Institute of Actuaries. has a 13% probability of attaining the age of 95 and a 2% probability of reaching the centurymark. We set the investment horizon T to be 30 years.We alter the standard ARVA spending rule so as to include an annual floor of q min = $30 , and an annual cap of q max = $80 , . Recall that all quantities are expressed in real (i.e.inflation-adjusted) terms. Our modified ARVA spending rule is then q i = max (cid:2) q min , min (cid:0) A ( t i ) W − i , q max (cid:1)(cid:3) (9.1)where A ( t i ) is given in equation (2.3). To provide more context, a Canadian male who hasworked for 40 years in a high-earning occupation can expect to receive slightly over $20,000 peryear in government benefits. Hence, we are assuming that the minimum total amount neededper year is about $30 ,
000 + $20 ,
000 = $50 , per year. Of course, the investor would like towithdraw more than the minimum amount of $30,000. However, as noted we also place a capof $80,000 per year on withdrawals. The cap prevents the retiree from reducing savings veryquickly, establishing a buffer against potential poor investment returns. We are thus effectivelyassuming that our retiree has no need for income above $80 ,
000 + $20 ,
000 = $100 , per year. Our retired investor withdraws cash and rebalances his portfolio at the start of each year,beginning immediately. The interest rate used in the ARVA calculation (2.3) is set equal to theestimated value of µ b , which is given in Table 8.1 as 0.454%. Table 9.1 summarizes the basecase investment scenario. Note that monetary units here and in the following tables and plotsare expressed in thousands of (real) dollars.Since the investor uses a risky portfolio to fund minimum cash flows annually, there is clearlyno guarantee that he will not run out of savings if he has survived to age 95. As outlined above,we seek an investment strategy that minimizes risk as measured by expected shortfall (ES), asdefined by equation (5.2). We use α = 5% , so we are trying to minimize the adverse consequencesmeasured by the average outcome in the worst 5% of the distribution. As indicated in Table 9.1, It is also worth noting that Canadian government benefits are reduced when total income exceeds about$80,000 per year, providing further incentive to not withdraw more than the specified cap. W t < we assume that debt accumulates at the rate given by the current return on 30-dayT-bills plus a spread of µ bc = 2% .We focus solely on measured outcomes for the investment account, but it is easy to imaginethat our retiree also owns real estate such as a home. In this case, the ES risk could behedged using a reverse mortgage with the home as collateral. However, we assume that theinvestor wants to avoid using a reverse mortgage if at all possible, so we seek an investmentstrategy that minimizes the magnitude of ES risk on its own. Our scenario shares some featureswith the behavioural life cycle approach originally described in Shefrin and Thaler (1988). Inthis framework, investors divide their wealth into separate “mental accounts” containing fundsintended for different purposes such as current spending or future needs. The standard life cycleapproach assumes that wealth is completely fungible across any such accounts, so that the sameincrease in wealth from any source (e.g. positive returns for a financial market portfolio, anincrease in the value of one’s house, lottery winnings, etc.) has the same effect on consumption.In contrast, in the behavioral approach wealth is not completely fungible, so the effects ofincreased wealth depend on the source of the increase. In our case, even if the investor’s wealthrises because the value of his real estate has increased, there will be no impact on the amountwithdrawn from the retirement portfolio. The real estate account will only be accessed as a lastresort. It is assumed to be there in the background if needed, but it is ignored in our analysis.
10 Numerical Results: Synthetic Market
We evaluate the performance of three alternative strategies based on the scenario described byTable 9.1: (i) constant withdrawals and investment portfolio rebalanced to maintain constantasset allocation weights (in particular, we set q min = q max = 40 instead of the values givenin Table 9.1 so that this strategy corresponds to the 4% rule of Bengen (1994)); (ii) ARVAwithdrawals as indicated in Table 9.1 and investment portfolio rebalanced to maintain constantasset allocation weights; and (iii) ARVA withdrawals as indicated in Table 9.1 and investmentportfolio rebalanced to optimal asset allocation weights, in accordance with solving the pre-commitment EW-ES problem (6.2) by the methods described in Section 7. In each case, theperformance evaluation is based on Monte Carlo simulated paths of market returns based onthe parametric model (3.3-3.4), with statistics of interest calculated across all paths. We referto this as a synthetic market , since the data used is generated by simulation of the parametricmodel rather than taken directly from actual historical market returns. We begin with the first strategy described above: constant withdrawals based on the 4%rule ( q max = q min = 40 ) and constant weights, i.e. p (cid:96) = constant in equation (6.3). The resultsfor the equity index weight p (cid:96) = 0 . , . , . , . . . , . are shown in Table 10.1. This table alsodisplays the results for p (cid:96) = 0 . , since this is approximately the equity weight which results inthe maximum ES. We conjecture that this low equity weight is due to our use of ES to measurerisk, compared to the more typical standard deviation. As p (cid:96) increases past 0.15, the magnitudeof ES increases strongly. Taking on more equity market risk results obviously leads to higherES. Of course reward also rises, as shown by the median value of terminal wealth W T . We provide results based on historical market returns below in Section 11 and Appendix B. In general, our measure of reward is total expected withdrawals. However, in this case the withdrawals are p (cid:96) ES ( α = 5% ) Median[ W T ]0.00 − . − . − . − . − .
28 22 . − .
32 108 . − .
05 310 . − .
62 550 . − .
55 828 . − .
24 1143 . − .
67 1490 . − .
08 1862 . − .
57 2249 . − .
37 2637 . Table 10.1:
Synthetic market results for constant withdrawals with constant weights, i.e. assum-ing the scenario from Table 9.1 except that q max = q min = 40 and p (cid:96) = constant in equation (6.3).Units: thousands of dollars. Statistics are based on . × Monte Carlo simulated paths.
To see the benefit of the ARVA withdrawal strategy, we repeat the Monte Carlo simulationsfrom above, except that here the ARVA spending strategy (2.3) is used with the constraints q min = 30 and q max = 80 . The results are shown in Table 10.2, which has an additional columncompared to Table 10.1. This extra column shows the expected average withdrawals over thedecumulation period, EW / ( M + 1) = (cid:80) i q i /M . In Table 10.2 the largest ES is − . for p (cid:96) = 0 . . This equity weight gives an expected annual withdrawal of . . Recall that thelargest ES from Table 10.1 was − , with constant annual withdrawals of 40. There is adramatic improvement in ES, despite higher average withdrawals. As another observation, inTable 10.2 the strategy with p (cid:96) = 0 . has better ES than the best result in Table 10.1, whilethe average expected withdrawal is . , again compared to the constant withdrawal of q = 40 .Overall, our comparison between strategies with constant asset weights and constant vs. variablespending (the ARVA rule augmented with a floor and a cap) is consistent with the results instudies such as Pfau (2015), albeit with different measures of risk and reward: a variable spendingrule allows for both higher average withdrawals and lower risk as measured by ES.We next consider our third strategy of ARVA withdrawals with optimal asset allocation. Inparticular, we consider the scenario described in Table 9.1 and solve for the optimal control p ( W,t ) for the pre-commitment EW-ES problem given by equation (6.2) using the methodsdiscussed in Section 7. We store the optimal control and then carry out Monte Carlo simulationsto calculate statistical properties as above but with applying p ( W,t ) along each path rather thanrebalancing to constant weights. We reiterate that for all times t > , this corresponds to theinduced time consistent strategy that solves equation (6.6).Before presenting the main results, we first verify the convergence of the algorithm given inSection 7 that is used to solve the optimal control problem given by equation (6.2). Table 10.3 fixed, so wealth is drawn down slowly given a sufficiently high p (cid:96) and decent equity market returns, resulting inrelatively high values for W T . This column was excluded from Table 10.1 because in that case the annual withdrawals were constant at 40. p (cid:96) ES ( α = 5% ) EW / ( M + 1) Median[ W T ]0.0 − .
89 34 . − . − .
60 37 .
85 31 . − .
43 42 .
07 64 . − .
01 46 .
95 90 . − .
92 51 .
46 111 . − .
19 54 .
95 138 . − .
92 57 .
42 179 . − .
69 59 .
13 275 . − .
78 60 .
30 486 . − .
96 61 .
07 739 . − .
67 61 .
56 1013 . Table 10.2:
Synthetic market results for ARVA withdrawals with constant weights, i.e. assumingthe scenario from Table 9.1 except that p (cid:96) = constant in equation (6.3). There are M = 30 rebalancing dates and M + 1 withdrawals. Units: thousands of dollars. Statistics are based on . × Monte Carlo simulated paths.
Algorithm in Section 7 Monte CarloValueGrid ES ( α = 5% ) EW / ( M + 1) Function ES ( α = 5% ) EW / ( M + 1)512 × − .
633 54 . . − .
326 54 . × − .
305 54 . . − .
381 54 . × − .
196 54 . . − .
469 54 . Table 10.3:
Convergence test for the algorithm from Section 7 used to determine the optimalasset allocation strategy to solve the pre-commitment EW-ES problem (6.2) with κ = 2 . for thescenario from Table 9.1. The Monte Carlo method used . × simulated paths. The grid isreported as n x × n b , where n x is the number of nodes in the log s direction and n b is the number ofnodes in the log b direction. There are M = 30 rebalancing dates and M + 1 withdrawals. Units:thousands of dollars. The value of W ∗ in equation (6.2) is 4.13 on the finest grid. shows a test with various levels of grid refinement for a fixed value of κ = 2 . in equation (6.2). Ateach grid refinement, we compute and store the optimal controls which are then used in MonteCarlo simulations. The algorithm in Section 7 and the Monte Carlo simulations are in goodagreement. As expected, the value function appears to be converging at almost a quadraticrate. The other quantities ES and expected average withdrawals which are derived from thealgorithm in Section 7 converge a bit more erratically. Results reported below for all cases withoptimal asset allocation are calculated using the finest grid from Table 10.3.Table 10.4 shows the results for the ARVA spending rule with optimal asset allocation fromsolving the pre-commitment EW-ES problem (6.2) for various values of κ . In addition to ES,expected average withdrawals EW / ( M + 1) , and median W T , Table 10.4 shows the averagethroughout the investment horizon of the median value of the fraction of the portfolio investedin equities in the furthest right column. This gives a rough indication of the equity market risktaken on over the period. As indicated by equation (6.1), increasing κ places more emphasis onrisk relative to reward. As a result, the optimal equity allocation tends to decrease with κ . This19 ES ( α = 5% ) EW / ( M + 1) Median[ W T ] (cid:80) i Median ( p i ) /M − .
93 63 .
01 266 . . − .
26 61 .
67 258 . . − .
63 60 .
15 250 . . − .
10 57 .
91 237 . . − .
02 56 .
04 208 . . − .
47 54 .
81 180 . . − .
91 52 .
35 129 . . − .
90 49 .
59 93 . . − .
78 46 .
82 66 . . − .
98 42 .
35 44 . . − .
74 40 .
30 39 . . Table 10.4:
Synthetic market results for ARVA withdrawals with optimal asset allocation basedon the scenario from Table 9.1 for various values of κ . The optimal control that solves the pre-commitment EW-ES problem (6.2) is computed using the algorithm given in Section 7, stored,and then applied in the Monte Carlo simulations. There are M = 30 rebalancing dates and M + 1 withdrawals. Units: thousands of dollars. Statistics are based on . × Monte Carlo simulatedpaths. The stabilization parameter in equation (7.9) is (cid:15) = − − . is also reflected in reduced median W T and expected average withdrawals. The benefit fromhigher κ is a lower magnitude of ES. Consider the case here with κ = 5 which results in ES of − . , expected average withdrawals of 52.35, and median W T of 129.97. This strategy has anaverage median equity allocation of 0.34. Contrast this with the result reported in Table 10.2for p (cid:96) = 0 . , which had about the same ES ( − . ), but expected average withdrawals ofjust 42.07 and median terminal wealth of 64.31. In this case, using an optimal asset allocationstrategy compared to a constant weight strategy results in about the same ES but significantlyhigher average withdrawals and about twice as much median W T . This attests to the benefitsof optimizing the asset allocation strategy, in addition to allowing for variable withdrawals.To further investigate the benefits of using an optimal asset allocation strategy, we plot theefficient frontiers of expected average withdrawals EW / ( M + 1) vs. ES in Figure 10.1(a). Weshow these frontiers for (i) the ARVA spending rule with optimal asset allocation as computedby solving the pre-commitment EW-ES problem (6.2), with results provided in Table 10.4; (ii)the ARVA spending rule with a constant weight asset allocation strategy, with results shownin Table 10.2; and (iii) a constant withdrawal of q = 40 with a constant weight strategy, withjust the best result (i.e. highest ES) from Table 10.1. Note that we have removed all non-Pareto points from these frontiers for plotting purposes. Figure 10.1(a) shows that even withconstant asset allocation weights the ARVA spending rule is much more efficient than a constantwithdrawal strategy which also has constant asset allocation weights. In fact, ARVA aloneprovides about 50% higher expected average withdrawals for the same ES achieved by a constantwithdrawal strategy by allowing for a higher stock allocation and limited income variability. Thecase with optimal asset allocation with the ARVA spending rule plots above the correspondingcase with constant asset allocation, with a larger gap between them for higher values of ES. This last case leads to just a single point in our plot since withdrawals are fixed at 40 regardless of the assetallocation and all other constant equity weights lead to lower ES.
500 -400 -300 -200 -100 0
Expected Shortfall E [ a v e r age w i t hd r a w a l ] ConstantWeight OptimalConst q=40Const p (a)
ARVA withdrawals with optimal and constantweight asset allocation, and the single best pointfor a constant withdrawal strategy with q = 40 andconstant weight asset allocation. For this point, p (cid:96) = 0 . . -500 -400 -300 -200 -100 0 Expected Shortfall E [ a v e r age w i t hd r a w a l ] qmin=30qmin=35qmin=40Const q=40 (b) ARVA withdrawals with optimal asset alloca-tion with q max = 80 and various values for q min ,and the single best point for a constant withdrawalstrategy with q = 40 and constant weight asset al-location. For this point, p (cid:96) = 0 . . Figure 10.1:
Efficient frontiers in the synthetic market for the scenario from Table 9.1. Allnon-Pareto points have been removed. Units: thousands of dollars.
To see the impact of the minimum required withdrawals, Figure 10.1(b) displays efficientfrontiers for the ARVA spending rule with optimal asset allocation for various values of q min ,keeping q max = 80 . As a point of comparison, we also show the point corresponding to theconstant weight strategy with p (cid:96) = 0 . , which gives the highest ES for constant withdrawals of q = 40 . As q min rises the efficient frontiers move down and to the left, as expected. However, evenfor q min = 40 , the efficient frontier plots well above the best point for constant withdrawals of q =40 with constant asset weights. This indicates that much larger expected average withdrawalscan be attained at no cost in terms of higher ES through the use of the ARVA spending ruleand optimal asset allocation. Surprisingly, Figure 10.1(b) shows that the combination of ARVAand optimal control increases EW by 25%, even when income is constrained to be no less thanfor the constant withdrawal case.Additional insight into the properties of the ARVA spending rule in conjunction with anoptimal asset allocation strategy can be gleaned from Figure 10.2 showing the 5th, 50th, and95th percentiles of the fraction of the retiree’s portfolio invested in the stock index, withdrawals,and wealth throughout the 30-year decumulation period. The optimal controls are computedby solving the pre-commitment EW-ES problem (6.2) with κ = 2 . and then used in MonteCarlo simulations to generate these plots. The general trend is for the equity index weight todecline over time, but there are cases where it rises significantly instead. Median withdrawalsincrease for the first 25 years, before falling off a bit. The 5th percentile of withdrawals quicklydrops to q min = 30 and remains there. On the other hand, the 95th percentile of withdrawalsrises sharply for about the first 5 years, and then stays at q max = 80 . Median wealth trendsdownward consistently over time, as does the 5th percentile of wealth. The 95th percentile ofwealth rises over the first several years, before also falling off fairly sharply.Recall that Proposition 6.1 states that the solution of the pre-commitment EW-ES prob-lem (6.2) has the same controls at time zero as the induced time consistent problem (6.6).21 ime (years) F r ac t i on i n s t o cks Median5thpercentile95thpercentile (a)
Percentiles of the fractioninvested in the stock index.
Time (years) W i t hd r a w a l s (t hou sa nd s ) Median95thpercentile5th percentile (b)
Percentiles of with-drawals.
Time (years) W ea t l h ( Thou sa nd s ) (c) Percentiles of wealth.
Figure 10.2:
Percentiles in the synthetic market of the fraction invested in the stock index,withdrawals, and wealth for the scenario from Table 9.1 with ARVA withdrawals and optimal assetallocation. Based on . × Monte Carlo simulated paths. Units: thousands of dollars.
Given any point in ( W t n , t ) space ( t n are the rebalancing times), maximizing ˜ J ( s,b,t − n ) = sup P n ∈A (cid:40) E X + n ,t + n P n (cid:34) i = M (cid:88) i =1 q i + κ min( W T − W ∗ , α + (cid:15)W T (cid:12)(cid:12)(cid:12)(cid:12) X ( t − n ) = ( s,b ) (cid:35)(cid:41) (10.1)leads to the optimal strategy depicted in the heat map contained in Figure 10.3. For thisexample, if we set κ = 2 . in problem (6.2), then W ∗ = 4 . . Recall that W ∗ is set to be thevalue such that Prob [ W T < W ∗ ] = α as determined at time zero . The structure of the heat map can be understood as follows. As t → T , there are multiply-connected regions of all bond and all stock portfolios. For small values of wealth, the optimalstrategy is to be fully invested in stocks, thus attempting to maximize ES. As wealth increases, Prob [ W T < W ∗ ] is small, and the investor switches to a portfolio that is heavily weighted towardsthe bond index to protect against the ES risk. If wealth increases further, the investor moves toinvesting more in stocks, in order to maximize withdrawals. Finally, for large values of wealth,there is little chance that W T < W ∗ . Since the withdrawals are capped at 80 per year, thereis no incentive to take on any more risk. In this case, the stabilization term (cid:15)W T in equation(10.1) comes into effect. Since (cid:15) = − − < , this forces the strategy back into bonds.It is useful to examine Figure 10.3 with reference to the median wealth shown in Figure10.2(c). The initial wealth of is in the green region, with equity weight (cid:39) . . As t → T ,the optimal control attempts to guide real wealth into the sweet spot between the lower bluezone and the upper red zone. The lower blue zone then acts as a barrier to lower wealth (i.e.running out of cash), since the portfolio becomes very stable with a large fraction of bonds.Above the lower blue zone, the allocation can vary considerably in an effort to maximize thetotal withdrawals, especially with a short time remaining.Figure 10.3 also shows the effect of different starting values of wealth W , keeping a minimumwithdrawal of q min = 30 . For example, with W = 400 the investor has no choice but to startwith an investment of 100% in stocks and hope for the best. This is essentially a “Hail Mary”strategy, with little chance of success. On the other hand, if W = 2000 , the investor will start In all of our examples, we maximize ES at the α = . level. igure 10.3: Heat map of controls computed from solving the pre-commitment EW-ES prob-lem (6.2) for κ = 2 . with ARVA withdrawals based on the scenario from Table 9.1. The stabi-lization parameter in equation (7.9) is (cid:15) = − − . off being completely invested in bonds with very high probability of success.
11 Numerical Results: Historical Market
We continue to compute and store the optimal controls based on the parametric model (3.3-3.4) as in the synthetic market case. As a robustness test, we now calculate statistics usingthese stored controls, but with bootstrapped historical real return data rather than Monte Carlosimulations following the parametric model. We employ the stationary block bootstrap method(Politis and Romano, 1994; Politis and White, 2004) to generate many bootstrap simulatedpaths. A single path entails sampling randomly sized blocks from the historical data withreplacement and pasting them together to cover the entire decumulation period of T = 30 years. The blocksize is generated randomly according to a geometric distribution with expectedblocksize ˆ b , which helps to mitigate the effects of a fixed block size.We implement an algorithm from Patton et al. (2009) to determine the optimal expectedblocksize ˆ b for the bond and stock indexes separately. This indicates that the optimal expectedblocksizes are 0.25 and 4.2 years for the stock and bond indexes respectively. However, to allowfor possible contemporaneous dependence between the two indexes we use paired sampling tosimultaneously draw returns from both series. Given the large difference in optimal expectedblocksize for the two indexes, it is not obvious what should be done for paired sampling. Onepossibility is to use an average of the two estimates, suggesting about 2 years. We do this, butwe also give results for a range of expected blocksizes as a robustness check. In these bootstrap simulations, we continue to use the average historical real (uncompen- Sampling in blocks helps to incorporate any serial correlation that is present in the data. Detailed pseudo-code for block bootstrap resampling can be found in Forsyth and Vetzal (2019). b ES ( α = 5% ) EW / ( M + 1) Median [ W T ] (cid:80) i Median ( p i ) /M Synthetic Market (from Table 10.4)N/A -59.47 .
81 180 . . Historical Market0.25 years -43.93 .
66 169 . . .
88 174 . . .
07 178 . . .
15 180 . . .
14 182 . . Table 11.1:
Historical market results for ARVA withdrawals with optimal asset allocation basedon the scenario from Table 9.1 for various expected blocksizes ˆ b . The optimal control that solves thepre-commitment EW-ES problem (6.2) is computed using the algorithm given in Section 7, stored,and then applied to bootstrap resamples of the monthly data from 1926:1 to 2018:12. Statisticsare based on bootstrapped paths. There are M = 30 rebalancing dates and M + 1 withdrawals.The scalarization parameter in equation (6.2) is κ = 2 . and the stabilization parameter in equa-tion (7.9) is (cid:15) = − − . Units: thousands of dollars. sated) drift for the T-bill index µ b as the interest rate in the ARVA computation (2.3). Thisavoids the problem of fluctuating withdrawal amounts which are driven just by the bootstrapresampling methods. It is also a conservative approach since µ b (cid:39) .We first explore the effect of the expected blocksize ˆ b . Table 11.1 shows the results computedby solving the pre-commitment EW-EW problem (6.2) in the synthetic market with κ = 2 . andthen using this control with block bootstrap resampling having various expected blocksizes ˆ b . Forease of comparison, the table also provides the results for κ = 2 . in the synthetic market thatwere previously shown in Table 10.4. The historical market results in Table 11.1 are generallysimilar to the corresponding synthetic market result, at least for values of ˆ b between 0.5 and2 years. The reported ES values for the historical market are consistently a bit better thanin the synthetic market, while expected average withdrawals and median terminal wealth arequite comparable. However, the average of the median value of the equity weight is a bit higher,clustering at or above 0.4 for the historical market compared to 0.375 for the synthetic market.Results reported below use ˆ b = 2 years, as this is (approximately) the average of the optimalexpected blocksizes for the two indexes.Figure 11.1 shows the percentiles of the optimal controls, withdrawals and wealth throughoutthe decumulation period in the historical market with ˆ b = 2 years. Figure 11.1 is very similarto the corresponding Figure 10.2 for the synthetic market. The median fraction invested in thestock index increases a little more sharply in Figure 11.1, and the 5th percentile of this fractionreaches zero a little later, but these are almost the only discernible differences. Overall, the closecorrespondence between the various panels of these two figures suggests that the parametricmodel used when solving for the optimal control is fairly robust as the historical market makesno assumptions about the processes followed by the stock and bond indexes. We now compare in the historical market the same three strategies that were considered However, this is not always true. In this case, ES (see Table 11.1 with ˆ b = 2 years) is about − . As we willsee below, if we try to increase ES to higher values than this, then the controls do not appear to be robust. ime (years) F r ac t i on i n s t o cks Median5thpercentile95thpercentile (a)
Percentiles of the fractioninvested in the stock index.
Time (years) W i t hd r a w a l s (t hou sa nd s ) Median95thpercentile5th percentile (b)
Percentiles of with-drawals.
Time (years) W ea t l h ( Thou sa nd s ) (c) Percentiles of wealth.
Figure 11.1:
Percentiles over time in the historical market of the fraction invested in the stockindex, withdrawals, and wealth for the scenario from Table 9.1 with ARVA withdrawals and optimalasset allocation. The scalarization parameter in equation (6.2) is κ = 2 . and the stabilizationparameter in equation (7.9) is (cid:15) = − − . Based on bootstrap resamples of the monthly datafrom 1926:1 to 2018:12. Units: thousands of dollars. previously in the synthetic market of Section 10, i.e. constant withdrawals of q = 40 withconstant asset allocation weights, ARVA withdrawals with constant asset allocation weights,and ARVA withdrawals with optimal asset allocation. Appendix B provides tables of results forthese strategies in the historical market with ˆ b = 2 years; here we present plots based on thoseresults.The efficient frontiers of expected average withdrawals vs. ES in the historical market areplotted in Figure 11.2(a), which is analogous to Figure 10.1(a) for the synthetic market. Asin Figure 10.1(a), Figure 11.2(a) shows that the ARVA withdrawal with constant weight assetallocation is a major improvement over the constant withdrawal with constant asset allocationweights. As expected, the optimal ARVA withdrawal strategy with optimal asset allocationcontinues to plot above the ARVA withdrawal strategy with constant weight asset allocation,indicating that optimal asset allocation can provide further significant enhancements. Althoughthe general picture is the same here in the historical market as it was in the synthetic market,it is worth pointing out a couple of specific differences. First, consider the constant withdrawalstrategy with constant asset allocation. In the synthetic market, the highest ES of about − for an equity weight of 0.15 (see Table 10.1). This is the best available point, since withdrawalsare constant. In the historical market, the corresponding ES is about − for an equity weightof 0.40 (see Table B.1). However, Figure 10.1(a) indicates that in the synthetic market anES of − can be attained with expected average withdrawals of about 58 for the constantweight case and about 60 for the optimal asset allocation case. The corresponding values forthe historical market in Figure 11.2(a) with an ES of − are a little higher, about 61 for theconstant weight case and around 63 for optimal asset allocation. These values do not constitutethe largest gap between these two frontiers, but they do indicate that ARVA withdrawals (witheither constant weight or optimal asset allocation) perform a bit better in the historical marketrelative to the synthetic market, at least for this level of ES. On the other hand, the performanceof the constant withdrawal strategy is notably worse in the historical market.A more direct comparison between the synthetic and historical markets is given in Fig-25
500 -400 -300 -200 -100 0
Expected Shortfall E [ a v e r age w i t hd r a w a l ] Optimal ConstantWeightConst q=40Const p (a)
ARVA withdrawals with optimal and constantweight asset allocation, and the single best pointfor a constant withdrawal strategy with q = 40 andconstant weight asset allocation. For this point, p (cid:96) = 0 . . -500 -400 -300 -200 -100 0 Expected Shortfall E [ a v e r age w i t hd r a w a l ] Synthetic ControlsSynthetic MarketSynthetic ControlsHistoric Market (b)
ARVA withdrawals with optimal asset alloca-tion, for both the historical and synthetic markets.
Figure 11.2:
Efficient frontiers in the historical market for the scenario from Table 9.1. Allnon-Pareto points have been removed. Units: thousands of dollars. ure 11.2(b) which plots the efficient frontiers of expected average withdrawals vs. ES for ARVAwithdrawals with optimal asset allocation in both markets, with the optimal controls having ofcourse been determined in the synthetic market. The frontier for the historical market plotsabove the frontier for the synthetic market if ES < − . However, the situation is reversed forES > − . This suggests that it is unreliable to try to achieve very low ES risk in the actualmarket. This is not unreasonable, since in order to obtain ES values close to zero the optimalstrategy will depend greatly on the stochastic market structure. Consequently, it appears thatthe synthetic market controls are not robust to parameter uncertainty for ES > − , althoughthe controls do appear to be robust otherwise.
12 Conclusions
For both parametric model simulations and bootstrap resampling of the historical data, theARVA withdrawal strategy with constant asset weights and minimum/maximum withdrawalconstraints outperforms a constant withdrawal strategy with constant asset weights based onexpected average withdrawals and expected shortfall criteria. This is consistent with resultsfrom the practitioner literature (e.g. Pfau, 2015) which show that withdrawal variability cansignificantly improve performance in cases with constant weight asset allocation. However,we also show that the ARVA withdrawal strategy can be further improved by dynamicallychoosing the equity weight. This strategy is determined by maximizing an expected total with-drawals/expected shortfall objective function using dynamic programming, assuming a para-metric model of historical asset returns. As long as the desired expected shortfall is not unre-alistically large, this strategy is robust to parameter misspecification, as verified by tests usingbootstrapped resampled historical data.Remarkably, the optimal dynamic ARVA strategy continues to outperform the constantwithdrawal/constant weight strategy, even if the minimum
ARVA withdrawal is set equal to26 robability Density: Stocks -6 -4 -2 0 2 4 6
Return scaled to zero mean, unit standard deviation
HistoricalReturnsStandardNormalJumpDiffusion (a)
Log returns and densities, stock index.
Probability Density: T-Bills -6 -4 -2 0 2 4 6
Return scaled to zero mean, unit standard deviation
HistoricalReturnsJumpDiffusionStandardNormal (b)
Log returns and densities, T-bill index.
Figure A.1:
Actual and fitted log returns for the CRSP value-weighted equity index and 30-day T-bill indexes. Monthly data from 1926:1-2018:12, scaled to zero mean and unit standarddeviation. A standard normal density and the fitted double exponential jump diffusion density(threshold, β = 3 ) are also shown. the constant withdrawal in the latter strategy. These results indicate that if an investor in thedecumulation stage of a DC plan is prepared to allow some variability in withdrawals, significantimprovements can be obtained in both expected total withdrawals and expected shortfall. AppendixA Calibration of Model Parameters
This appendix discusses the estimation of the parameters of the jump diffusion processes for thestock and bond indexes given by equations (3.1), (3.3), (3.4), and (3.5). Recall that the equityindex is the CRSP value-weighted stock index while the bond index is the CRSP 30-day T-billindex, and that both of these indexes are adjusted for inflation by using the CPI.Jumps in the data are identified using the thresholding technique described in Mancini (2009)and Cont and Mancini (2011). Let ∆ ˆ X i be the detrended log return in period i , with periodtime interval ∆ t . Suppose we have an estimate for the diffusive volatility component ˆ σ . Thenwe detect a jump in period i if (cid:12)(cid:12)(cid:12) ∆ ˆ X i (cid:12)(cid:12)(cid:12) > β ˆ σ √ ∆ t . We choose β = 3 in this paper (note that ∆ t is fixed). For justification for this parameter selection, see (Shimizu, 2013; Dang and Forsyth,2016; Forsyth and Vetzal, 2017). For details describing the recursive algorithm used to determine ˆ σ , see Forsyth and Vetzal (2017).Figure A.1(a) shows a histogram of the monthly log returns from the value-weighted CRSPstock index, scaled to zero mean and unit standard deviation. We superimpose a standardnormal density onto this histogram, as well as the fitted density for the double exponential jumpdiffusion model. Figure A.1(b) shows the equivalent plot for the 30-day T-bill index.During the sample period of 1926:1-2018:12 (monthly), the filtering algorithm identified 30stock index jumps and 48 T-bill index jumps. Of these cases, just 5 were identified as occurringin the same month for both stocks and bonds, all in the 1930s. This supports our modellingassumption of no dependence between the jump intensities or jump distributions of the twoindexes, though we do allow for correlated Brownian motion terms in the parametric model.27 Historical Market: Detailed Results
This appendix presents detailed results for the historical market bootstrap resampling testswith expected blocksize ˆ b = 2 years. Table B.1 shows the results for a constant withdrawal( q = 40 ) strategy with constant equity weight asset allocation, analogous to Table 10.1 in thesynthetic market. Table B.2 gives results for ARVA withdrawals with constant equity weightasset allocation, analogous to Table 10.2 in the synthetic market. Finally, Table B.3 presentsresults in the historical market for ARVA withdrawals and optimal asset allocation (the optimalcontrol is computed by solving the pre-commitment EW-ES problem (6.2) in the syntheticmarket). This table is analogous to Table 10.4 for the synthetic market.Equity Weight p (cid:96) ES ( α = 5% ) Median [ W T ] − . − . − . − . − .
73 113 . − .
56 317 . − .
67 562 . − .
58 850 . − .
71 1177 . − .
42 1548 . − .
29 1956 . − .
39 2381 . − .
09 2823 . Table B.1:
Historical market results for constant withdrawals with constant weights, i.e. as-suming the scenario given in Table 9.1 except that q max = q min = 40 , and p (cid:96) = constant inequation (6.3). Units: thousands of dollars. Statistics based on bootstrap resamples of themonthly data from 1926:1 to 2018:12 with expected blocksize ˆ b = 2 years. Equity Weight p (cid:96) ES ( α = 5% ) EW / ( M + 1) Median[ W T ]0.0 − .
41 35 . − . − .
74 38 .
53 31 . − .
37 42 .
27 64 . − .
44 46 .
79 90 . − .
86 51 .
37 111 . − .
20 55 .
20 137 . − .
58 58 .
02 170 . − .
23 59 .
93 269 . − .
74 61 .
34 493 . − .
09 62 .
23 766 . − .
60 62 .
80 1069 . Table B.2:
Historical market results for ARVA withdrawals with constant weights, i.e. assumingthe scenario given in Table 9.1 except that p (cid:96) = constant in equation (6.3). There are M = 30 rebalancing dates and M + 1 withdrawals. Units: thousands of dollars. Statistics based on bootstrap resamples of the monthly data from 1926:1 to 2018:12 with expected blocksize ˆ b = 2 years. ES ( α = 5% ) EW / ( M + 1) Median[ W T ] (cid:80) i Median ( p i ) /M − .
50 64 .
05 258 . . − .
76 63 .
09 253 . . − .
43 61 .
74 247 . . − .
02 59 .
81 239 . . − .
23 58 .
86 230 . . − .
17 56 .
48 204 . . − .
80 55 .
15 180 . . − .
96 52 .
26 135 . . − .
34 49 .
77 101 . . − .
87 43 .
22 53 . . Table B.3:
Historical market results for ARVA withdrawals with optimal asset allocation basedon the scenario given in Table 9.1 for various values of κ . The optimal control that solves thepre-commitment EW-ES problem (6.2) is computed in the synthetic market using the algorithmgiven in Section 7, stored, and then applied to bootstrap resamples of the historical data. There are M = 30 rebalancing dates and M +1 withdrawals. Units: thousands of dollars. Statistics based on bootstrap resamples of the monthly data from 1926:1 to 2018:12 with expected blocksize ˆ b = 2 years. The stabilization parameter in equation (7.9) is (cid:15) = − − . References
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