Rule-based Strategies for Dynamic Life Cycle Investment
T.R.B. den Haan, K.W. Chau, M. van der Schans, C.W. Oosterlee
aa r X i v : . [ q -f i n . P M ] N ov Rule-based Strategies for Dynamic Life CycleInvestment
T.R.B. den Haan ∗ , K.W. Chau , M. van der Schans , andC.W. Oosterlee Delft University of Technology, DIAM - Delft Institute of AppliedMathematics, Delft, the Netherlands CWI - The Center for Mathematics and Computer Science,Amsterdam Ortec Finance, Rotterdam, the Netherlands University of Groningen, Department of Economics, Econometricsand Finance, Groningen, the Netherlands2020-11-06
Nowadays, many people invest their retirement savings in a defined contributionpension scheme. In such a scheme, the contributions are agreed upon and are,e.g., a percentage of one’s salary. The pension, however, is uncertain as itdepends on the returns on investment. At retirement, the accumulated wealthis converted to a pension income that intends to replace a proportion of theinvestor’s income, typically about 70%, which is referred to as the replacementratio. In this paper, we propose a dynamic strategy that optimally steers theinvestor towards a replacement ratio target.Our dynamic strategy will reduce risk after several years of good returnson investment. It presumes that upward potential concurs with downside risk.Our pension investor is only interested in reaching her replacement ratio target,i.e., not making the target is considered downside risk and she feels indifferentabout any two values above the target. We will show that, in this sense, thedesigned dynamic strategy outperforms static life cycle strategies. By decreasingrisk after several good years, our dynamic strategy prevents unnecessary risktaking.A well-known static life cycle strategy is known as Bogle’s rule (Bogle), whichprescribes to invest 100% minus one’s age in risky assets. Decreasing risk in thecourse of the life cycle in such a way is called a glide path. When the glide pathis known in advance up to retirement, the strategy is static and does not adjustas events unfold. Therefore, static strategies may take unnecessary risk when ∗ Electronic address:
[email protected] ; Corresponding author
To demonstrate the rule-based strategy’s practical value, we will consider a spe-cific pension investor (we will choose typical retirement data from the Nether-lands). At t = 0, the 26 year old investor will start saving up to retirementat time t = T , coinciding here with a retirement age of 67 years. She intendsto replace 70% of her income by her pension (including government allowancesfor old age). Although, in practice, an investor might be interested in insur-ing longevity risk or be interested in employing advanced withdrawal strategies,Blanchett, Kowara, and Chen illustrates that simple withdrawal strategies canperform well, e.g., based on an annuity with a maturity roughly equal to an in-vestor’s life expectancy. Therefore, as we focus on accumulating wealth beforeretirement, we simply assume the investor buys an annuity that indexes withthe expected inflation, i.e., a bond which, apart from indexation for expectedinflation, equals annual payoffs, for a period of N , say 20, years after retirement.Whichever withdrawal strategy an investor might follow, the assumption hereis that this annuity gives a good estimate of, at least, the investor’s income inher first year after retirement, and, thereby, to what extend she can replace hersalary for 70% with a pension.The investor can invest her wealth W t in a risky, equity-like, asset, whichis called the return portfolio, or in a safe, bond-like, asset with annual payoffsduring retirement, the matching portfolio. In our setting, the strategy will usethe matching portfolio to protect the current gains, and it grows with inflation.Therefore, the matching portfolio also carries risk. Put differently, we assumethe investor doesn’t hedge inflation risk with inflation protected securities asthe market for inflation protected is illiquid and strategies that hedge againstinflation are not straightforward to follow in practice (Martellini, Milhau, andTarelli). Finally, we assume there is no risk-free rate to invest money in.The investor annually manages her portfolio, i.e., decisions, contributionsand pension payments are made in discrete time, which runs up to retirement,from t = 0 to t = T . The pension payments start at t = T and run up to t = T + N −
1. At time t ≤ T before retirement, she invests a fraction α t ofher wealth W t in the return portfolio. The investor is not allowed to short-sellassets or borrow money, so that 0 ≤ α t ≤ . (1)In the dynamic programming literature, α t is referred to as the control (asdecisions intend to give the investor control over the outcome). A strategymaps information Z t available at time t , e.g., past returns and current wealth W t , to the desired allocation: α t : R K ∋ Z t α t ( Z t ) ∈ [0 , . (2)Here Z t is adapted to a filtration F t , governing the underlying stochastic pro-cesses. Before time t , the information Z t is not yet available, and α t is thus a3tochastic quantity. In a static strategy, such as Bogle’s rule, α t only dependson time and is known, i.e., not stochastic, not even when the information Z t isnot yet available. In practice, risk is reduced towards retirement, meaning that α t typically decreases over time.Just before rebalancing, the investor makes a contribution c t to the portfolio.These contributions resemble an age-dependent percentage p t , see Table 3, of theinvestor’s salary s t which she earned in the period t − t . We assume thatthe investor’s salary s t follows a deterministic career path, i.e., it increases withage. The investor’s salary also increases stochastically with the wage inflation w t , see Appendix A.The investor’s objective is to achieve her 70% replacement ratio target atretirement without encurring too much downside risk. The replacement ratioat retirement, R T , is given by R T = W T M T · T + 1 T P t =0 s t T Q τ = t +1 (1 + π τ ) . (3)Here, the second term divides the investor’s average wage in nominal amountsindexed with inflation π t to retirement at t = T , and M t is the market valuefactor that discounts N future pension payments indexed by expected inflationto time t ≤ T : M t = T + N − X τ = T (1 + r τ − tt ) τ − t E t " τ Y τ ′ = t +1 (1 + π τ ′ ) , (4)where E t is the expectation, conditional on F t (i.e., conditional on the infor-mation available at time t ), and r τ − tt represents the market rates that discountpayments from τ − t years into the future back to the present time. Using themarket value factor M T at retirement, the first term in (3) converts the accu-mulated wealth W T to N annual income payments indexed for expected futureinflation.To measure whether a strategy achieves the investor’s objective, we usea utility function, U , which, whenever decisions are to be taken, intends tomaximize the following expression in expectation:max α t ,...,α T − E [ U ( Z T ) | F t ] , (5)where F t represents current market information, α t is as in (2) and Z T is avector with outcomes including the terminal replacement ratio. Although otherchoices are possible, we choose U ( . ) to be the shortfall below the investor’starget replacement ratio of 70%: U ( Z T ) = min( RR T − , , (6)where there is no shortfall in replacement ratio if it ends above 70%. Notethat this measure is not conditional on the shortfall. So, additionally, we willalso evaluate a strategy’s performance using the 10% conditional value at riskCVaR . ( RR T ) of the replacement ratio, i.e., the expectation of the 10% worstcase outcomes as defined byCVaR α ( RR T ) = E (cid:2) RR T (cid:12)(cid:12) RR T ≤ F − RR T ( α ) (cid:3) , (7)4here F − RR T ( α ) is the inverse cumulative distribution function of terminal re-placement ratio RR T and represents the α -th quantile below which are the worstcase outcomes. For general applicability, we require that the designed strategies are not definedin terms of the governing stochastic model parameters. That is, the strategiescan be applied when different governing stochastic models would be used. Wemerely assume that the governing stochastic model can be simulated by meansof a Monte Carlo simulation. To make this explicit, we choose to use a stan-dard model developed to make risk analyses comparable between Dutch pensionfunds, see (Koijen, Nijman, and Werker). The model and its calibration are welldocumented (Draper). Calibration on recent market data and a Monte Carlosimulation of the model are publicly available at the website of the Dutch Cen-tral Bank (DNB). In this paper, we use the set of 2017 (quarter 1), which iscalibrated on data up to ultimo 2016 and start simulating from there.In discrete time, the model is a VAR(1) model with normally distributedincrements, see Muns for a short summary of the model specification. In thecalibration, some structure is imposed to achieve realistic market dynamics.Based on the model, sample paths are generated for the following variables: • Equity returns x t , which are used for the return portfolio; • Inflation π t ; • Wage inflation w t , which equals inflation π t plus 0 . • A yield curve with interest rates r mt containing rates for each maturity m .The matching portfolio is tailored to the investor’s retirement age. Its returns m t equal the rate of change in the market value factor: m t = M t M t − − , (8)where M t is defined in (4). Note that the matching portfolio protects the in-vestor against expected future inflation. To determine the expected future in-flation, we use the least squares Monte Carlo technique, as presented in SectionB.3.Table 1 gives the annual return statistics of the variables. Due to the fluc-tuating market price of future pension payments, the standard deviation of thematching returns is very similar to the one of the equity returns. Although thematching portfolio follows these fluctuations, it is considered less risky, in termsof the investor’s goals. By investing in the matching portfolio, the pensionerwill receive the corresponding amount from the annuity, no matter the futuremarket prices. 5 t m t r t π t w t Mean 6.1% 3.4% 2.5% 1.6% 2.1%Standard deviation 18.3% 18.5% 2.4% 1.5% 1.5%CorrelationsEquity return ( x t ) 1.00Matching return ( m t ) -0.06 1.0010 year interest ( r t ) 0.17 -0.17 1.00Inflation ( π t ) 0.11 -0.04 0.82 1.00 1.00Wage inflation ( w t ) 0.11 -0.04 0.82 1.00 1.00Table 1: Annual statistics of the underlying stochastic model calculated on asample that combines all sample paths. In this section, we define three rule-based strategies: a cumulative target strat-egy that decreases risk once it reaches a cumulative target for the contributionspaid so far, an individual target strategy that tracks the investments of thecontributions separately and decreases risk once it reaches the target for thatcontribution, and a combination strategy that combines the two with dynamicprogramming. The strategies all intend to steer towards a target replacementratio of 70%, and decrease risk when return on investment develops well. Thestrategies differ in their views on when return on investment has been developingwell enough to decrease risk.
The cumulative target strategy that we consider here has similarities with thestrategies studied in Zhang et al. and Forsyth and Vetzal: risk is reduced oncewealth exceeds a pre-defined wealth target. Contrary to Zhang et al. and Forsythand Vetzal, however, our investor saves for retirement and we relate the wealthtarget to the price of a bond with payoff equal to the desired pension.Given a density forecast for the matching and return portfolios, see Section2.2, the strategy depends on two parameters: a required real rate of return r (before retirement) and a discount rate δ (after retirement) to discount pensionpayments after retirement to the retirement date. At time t before retirement,i.e., t ∈ . . . T , the investor contributes c t to her pension savings, see Table3. The contributions c τ up to time t , i.e., τ = 0 , . . . , t , are supposed to growwith inflation π , plus the real rate of return r , to a target wealth c τ E t F τ atretirement, where F t is given by F τ = T Y τ ′ = τ +1 (1 + r + π τ ′ ) , (9)and the conditional expectation, E t , enforces that the realized inflation is usedbefore time t and the expected inflation is used beyond time t . The wealthtargets at retirement for all contributions c τ up to time t are combined andconverted into a target pension using a discount factor, ˜ M T , which is based on6he discount rate δ : ˜ M T = T + N − X τ = T δ ) τ − T . (10)Using the market value factor M t , as defined in (4), this gives us the followingcurrent target wealth ˜ W t : ˜ W t = M t ˜ M T t X τ =0 c τ E t F τ , (11)where the summation represents the combined wealth targets at retirement forall contributions c τ up to time t .The cumulative target strategy starts by investing new contributions c t inthe risky asset. If the current wealth W t , including the current contribution c t , exceeds the target wealth ˜ W t , risk is reduced and W t is transferred to thematching portfolio. For the matching portfolio, the investor follows a buy andhold strategy. New contributions invested in the risky asset, will also be trans-ferred to the matching portfolio if the current wealth W t , which consists of thecurrent contribution c t , the value of the matching portfolio and the value of thereturn portfolio, exceeds the target wealth ˜ W t . In other words, at t = 0, thecontrol α is given by α = ( W ≥ ˜ W , (12a)1 otherwise, (12b)and, for t = 1 . . . T , the control α t is given by α t = W t ≥ ˜ W t , (13a) α t − (1 + x t ) α t − (1 + x t ) + (1 − α t − )(1 + m t ) otherwise. (13b) Contrary to the cumulative target strategy, the individual target strategy, whichis the second strategy we will analyze here, defines a wealth target per contribu-tion and invests each contribution separately, i.e., the wealth W t is seen a sumof the individual wealth components resulting from investing the contributionsseparately: W t = t X τ =0 W t,τ , (14)where W t,τ is the wealth component from investing the contribution c τ . As in(11), a wealth target ˜ W t,τ , at time t for a contribution invested at time τ ≤ t ,is given by ˜ W t,τ = M t ˜ M T c τ E t F τ . (15)Apart from this, the strategy works similarly: the individual contributions areinvested in the risky asset until the invested amount exceeds the wealth target forthat contribution, in which case they are transferred to the matching portfolio7ntil retirement. Thus, the control α t,τ for investing contribution c t,τ is givenby α t,τ = ( τ ′ = τ . . . t we have W τ ′ ,t ≥ ˜ W τ ′ ,τ , (16a)1 otherwise. (16b)At the aggregated level, the control α t is now given by α t = 1 W t t X τ =0 W t,τ α t,τ . (17)Conceptually, the difference between the cumulative target strategy and theindividual target strategy is what triggers the risk reduction. Contrary to theindividual target strategy, in the cumulative target strategy new investmentshave to make up for insufficient past returns before a transfer to the matchingportfolio can take place. On the other hand, in the cumulative target strategygood past returns may cause new contributions to be transferred immediately tothe matching portfolio. With the individual target strategy, each contributionhas to generate sufficient return on investment before such a transfer takes place. Both the cumulative and the individual target strategy either reduce risk byswitching completely to the matching portfolio or don’t reduce risk at all. In-stead of completely switching or not switching at all, the combination strategy,which is the third strategy considered, combines the individual target strat-egy with dynamic programming to dynamically steer the wealth W t,τ resultingfrom the contribution c τ above its wealth target ˜ W t,τ . For this, we define thefollowing wealth to target ratio, Z t,τ := W t,τ ˜ W t,τ , (18)and solve V ( z, t, τ ) = sup A t,τ E (cid:2) ˇ U ( Z T,τ ) | Z t,τ = z (cid:3) , (19)where ˇ U is a utility function, V ( z, t, τ ) is the value function in the dynamicprogramming problem and the control A t,τ consists of the future investmentdecisions: A t,τ = { α t,τ , . . . , α T,τ } . (20)Using the dynamic programming principle, it follows that the optimal control, A ∗ t,τ , satisfies A ∗ t,τ = { α ∗ t,τ , A ∗ t +1 ,τ } , (21)which allows us to solve for the optimal control problem for A ∗ t,τ , backwards intime.In this context, we choose a utility function that steers the ratio Z T,τ inbetween the bounds z ∗ min and z ∗ max . This is in line with the investor’s goalof minimizing downside risk, and with our assumption that upward potential8 1 2 3 4 5 6 − −
100 State U t ili t y Utility function z ∗ min z ∗ max Figure 1: Plot of (22) with z ∗ min = 1 and z ∗ max = 3.comes with downside risk. The utility function should be positive concave andtakes here the following functional form:ˇ U ( z ) = − ( z − β ) − ( z − z ∗ min ) z , (22)where β = q z ∗ max ) − ( z ∗ min ) , see Figure 1 (note that this is a different utility function than U ( · ) from (6).Utility function ˇ U ( · ) is clearly concave and continuous on the domain R > . Weset z ∗ min = 1 and z ∗ max = 3, as this choice fits well with the investor’s replacementratio target and, as we will show in Section 4.1, is sufficient to demonstrate thestrategy’s added value.Now, we will show that the ratio Z t,τ , between the current wealth W t,τ and its target ˜ W t,τ , evolves in time by making returns on investment in thenominator and updating the inflation expectation in the denominator. Sincethis time evolution is independent of τ , we can show that the optimal control α ∗ t,τ is independent of τ , i.e., once the optimal control is found, it can be appliedto all contributions. Lemma 1.
The optimal control α ∗ t,τ of dynamic programming problem (19) isindependent of the contribution c τ and the time τ at which the contribution ismade.Proof. The portfolio wealth W t,τ , accumulated by investing contribution c τ ,increases with the return on investment and, therefore, satisfies W t,τ = [(1 + x t ) α t − ,τ + (1 + m t )(1 − α t − ,τ )] W t − ,τ . (23)From (15), (9) and (8), it follows that the wealth target, ˜ W t,τ , satisfies˜ W t,τ = E t F t − E t − F t − (1 + m t ) ˜ W t − ,τ . (24)9ubstitution of (23) and (24) in (19) yields that the optimal controls α ∗ t,τ solvesup A t,τ E ˇ U z T Y τ ′ = t +1 (1 + x τ ′ +1 ) α τ ′ ,τ + (1 + m τ ′ +1 )(1 − α τ ′ ,τ ) E τ ′ +1 F τ ′ E τ ′ F τ ′ (1 + m τ ′ +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t,τ = z . (25)This shows that both the value function V ( z, t, τ ) and the optimal control α ∗ t,τ are independent of τ .Lemma 1 implies that, theoretically, the dynamic programming problem hasto be solved only once, i.e., the investment decisions for the first contribution c can be used for all other contributions.For the practical implementation for the dynamic programming algorithm,readers may refer to Appendix B. The variable r , used in the construction of the wealth target, can be interpretedin multiple ways. First of all, it serves as a discount rate, which is used tocompute the present value of contributions that are made in the future. It canalso be viewed as an annual return requirement: each contribution is requiredto have an average annual return of r . A third interpretation of r is that of afuture expected annual return. The computation of the expected replacementratio requires a future annual return assumption.Let t ∈ T and let F t be the corresponding filtration. The expected replace-ment ratio R t is defined as R t := E [ P |F t ] E (cid:20) T P t =0 s t T Q τ = t +1 (1 + π τ ) |F t (cid:21) , where E [ P |F t ] = E [ W T |F t ] E [ M T |F t ] , with E [ W T |F t ] = [1 + r + I ( T ; t )] T − t W t + T − X k = t +1 [1 + r + I ( T ; k ]) T − k E [ c k |F t ] . Computation of the expected replacement ratio requires four different esti-mators. The discount rate r is used as an estimator for the future expectedannual return. The estimator for the future inflation, I ( T ; t ), has to be esti-mated through regression between the future and the past cumulative inflation,as shown in Equation (27). Future salaries are based on the information fromTable 3. Lastly, the estimator for the market value factor at the end of the in-vestment horizon, E [ M T |F t ], is based on regression between M t and M T , withΦ = { , x } . See Appendix B.3 for details of the regression method used.10he market value factor is considered to be independent of the discount rate r , inflation and wage inflation (the division operator can therefore be taken outof the expected value operator).The computation of the target replacement ratio at time t is similar to thecomputation of the expected replacement ratio. The only difference is thatthe portfolio wealth, W t , is replaced by the target terminal wealth, W ∗ ( t ).The target wealth definition causes the target replacement ratio, R ∗ ( t ), to beindependent of the market value factor: R ∗ ( t ) = E [ P ∗ |F t ] E (cid:20) T P t =0 s t T Q τ = t +1 (1 + π τ ) |F t (cid:21) , by using the independence of the market factor M T and the wealth process andthe definition of the current target wealth ˜ W t , E [ P ∗ |F t ] = E [ W ∗ ( T ) |F t ] E [ M T |F t ] = ˜ W t . As mentioned, the investor’s target is to reach a replacement ratio of 70%.To translate this target into the wealth target in terms of portfolio wealth, wehave: ˜ W t = R ∗ ( t ) E " T X t =0 s t T Y τ = t +1 (1 + π τ ) |F t . To steer towards a fixed replacement ratio target, ˜ W ( t ) would have to bealtered for each scenario. It is, however, easier to differ the quantity R ∗ ( t )slightly between scenarios, from a computational point of view. Instead, ˜ W t isdefined as in Equation (11) and r is set to the required annual return.Numerically, we find that the target replacement ratio within a scenariois almost constant throughout time, as can be seen in the bottom-left plot ofFigure 2. Small alterations are caused by the estimators for the inflation andthe wage inflation. Alterations of up to 0 .
01 within a scenario are observed for adiscount rate of 2 . . . . In this section, we apply the rule-based strategies described in Section 3 to thepension investor introduced in Section 2.1, using the governing stochastic modeldescribed in Section 2.2.
To illustrate the dynamics of the rule-based strategies, Figure 2 shows one of the2000 sample paths for the investor’s portfolio dynamics. In particular, the topleft figure shows the investor’s wealth W t when following the cumulative targetstrategy (orange) and when the investor’s wealth exceeds the target ˜ W t (yellow).Note that when this occurs, the investments are transferred to the matching11ortfolio (orange line, bottom right figure). The individual target strategy (ingreen) works similarly, but, as discussed, uses a target per contribution, so that,typically, only part of the wealth is transferred to the matching portfolio (greenline, bottom right figure). The bottom left figure illustrates that, in this samplepath, the rule-based strategies outperform the optimal static strategy in terms ofexpected replacement ratio - although only in the first 10 years of the investmentthe rule-based strategies take substantially more risk, i.e., have a substantiallyhigher allocation to the return portfolio. Therefore, in this particular samplepath, one could argue that the better performance comes from the rule-basedstrategies and not from increased exposure to risk.The combination strategy is best illustrated by means of the resulting in-vestment decisions, i.e., the optimal control α t,τ as defined by equation (25).Figure 3 illustrates the optimal allocation to the matching portfolio, 1 − α t, , forthe first contribution c as a function of time t and the wealth to target ratio, asdefined by (18). In this example, allocations are restricted to multiples of 20%.Note that, contrary to the rule-based strategies, in the combination strategyrisk can be increased and investments can be transferred from the matching tothe return portfolio. All together, this makes the combination strategy morerefined than the rule-based strategies, which follow a “risk on” or “risk off”approach, in terms of their allocation.Figure 4 compares the distribution of the terminal replacement ratio for thefollowing best performing strategies in terms of the expected shortfall below theinvestor’s 70% replacement ratio target: two rule-based strategies, a combina-tion strategy and a static strategy. The figure illustrates that, as intended, thedynamic strategies reduce downward risk at the expense of upward potential,i.e., the dynamic strategies are centered more around the target replacementratio of 70%.A comparison of all strategies is best made by comparing the strategy suc-cesses, i.e., whether a strategy achieves the intended 70% replacement ratiotarget, versus its downside risk, and parametrize the strategies by the param-eters that control the strategy’s risk appetite, see Figure 5. From this figure,we conclude that all the dynamic strategies clearly outperform the traditionalstatic strategies. Together with the intuitive rationale to reduce risk after sev-eral good years, we believe this sufficiently demonstrates the added value ofthese dynamic strategies. We do, however, find these simulations insufficient torank the dynamic strategies based on their effectiveness. It is well-known thatthe relative performance of dynamic strategies can be sensitive to the charac-teristics of the underlying stochastic model. As such, the characteristics are notcompletely objective, and we believe that use of the strategies in practice is anappropriate way to test the strategies further (which lies beyond the scope ofthis research). One of the intended advantages of a static life cycle strategy is the reducedrisk close to the retirement, meaning that one can provide the investor withan accurate estimate of her retirement income in the years before retirement.Table 2 provides a comparison of the dynamic strategies and traditional lifecycle strategies. In particular, the table lists the standard deviations of thedifference between the expected replacement ratio 5 years before retirement and12 10 20 30 4010 t W ( t ) − . − . . . . . t P o r t f o li o r e t u r n . . t E t RR T t − α t Figure 2: Sample paths of wealth, return of the matching and return portfolio,expected replacement ratio, and allocation to the matching portfolio applied tothe pension investor introduced in Section 2.1 following rule-based strategieswith the discount rates r = 2% and π = 2 . W t for respectively thecumulative target strategy (orange), its wealth target ˜ W t (yellow), individualtarget strategy (green), the optimal static strategy (blue) and cumulative con-tribution (dark blue). Top right: return of the matching portfolio (blue) andreturn portfolio (orange). Bottom left: 70% replacement ratio target (yellow)together with the expected replacement ratio of the cumulative target strategy(orange), individual target strategy (green) and optimal static strategy (blue).Bottom right: allocation 1 − α t to the matching portfolio for the cumulativetarget strategy (orange), individual target strategy (green) and optimal staticstrategy (blue). 13 5 10 15 20 25 30 35 4001234 Time (years) W e a l t h t o t a r g e t r a t i o . . . . . . . . . − α t, to the matching portfolio, as a function ofwealth to target ratio W t, / ˜ W t, (y-axis), as defined by equation (18), and time t (x-axis) for the first contribution of an investor following the combinationstrategy, discussed in Section 3.3, and using the stochastic model of section 2.2.Allocations are restricted to multiples of 20%.the replacement ratio at retirement. We conclude that when following the rule-based strategies the investor can be provided with a similarly accurate estimateof the replacement ratio before retirement.14 0 . . . . . . . . F r e q u e n c y Figure 4: Distribution of the replacement ratio for, respectively, the cumulativetarget strategy with r = 3 .
06% (orange), individual target strategy r = 2 . r = 1% (yellow) and a static strategy with46 .
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12 0 . . . . . . . Shortage % C V a R Figure 5: Expected shortfall below the investor’s 70% replacement ratio target,see equation (6), versus 10% CVaR of the terminal replacement ratio, as definedby (7), for: the cumulative target strategy (orange), the individual target strat-egy (green), the combination strategy (yellow), all parametrized by the real rateof return r (yellow), and, also, several annually rebalanced allocations (blue),and several default life cycles reducing risk with the investor’s age.15 tatic mix Static life cycle Dynamic strategies
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07 0 . r = 3 .
06% for the combination strategy, r = 2 .
99% for theindividual strategy, and r = 1% for the combination strategy.Although the rule-based strategies outperform other strategies in our exam-ples, we wish to point that there are also disadvantages in the all-or-nothingapproach, e.g, the portfolio remains 100% invested in the more risky returnportfolio when targets are not reached. Such truly worst case scenarios appearto have a minor influence, but are, e.g., illustrated in the far left lower tailin Figure 5. The individual target strategy presented in Section 3.2 suffers lessfrom the all-or-nothing disadvantages, as it defines a target per contribution. Asa result, inferior past performance does not influence the required performanceof current and future contributions.Compared to the rule-based strategies, the combination strategy does notexploit the fact that the matching portfolio can grow an investment securelyto its intended target (indexed by expected inflation) until retirement. As therule-based strategies explicitly made use of this, the combination strategy couldbe further improved.One advantage of the combination strategy in practical use is that the cor-responding asset allocation is much more smooth than for the rule-based strate-gies. The necessity of large turnovers is difficult to explain and investors mightbe uncomfortable to follow such a drastic strategy to the end. In this paper, we discussed several dynamic strategies, suitable for pension in-vestors that aim to replace a proportion of their salary with a retirement in-come. The strategies reduce risk after several good years and steer the investorto her target. By having the allocation depend on return on investment, theapproaches exploit a freedom which is typically not used by traditional staticapproaches. We have shown that the dynamic approaches may outperform sometraditional static approaches and prevent unnecessary risk taking.16wo simple and intuitive rule-based strategies were introduced that secureinvestments in a cash flow matching portfolio once they yielded sufficient return.Although both rule-based strategies can straightforwardly be implemented inpractice, we recommend to also investigate alternatives where the investor, e.g.,switches between an aggressive traditional life cycle and a matching portfolio torule out very aggressive portfolios close to retirement.The rule-based strategies were further refined into a combination strategybased on dynamic programming. In the current setup, the combination strategymay not be superior and we even found that the rule-based strategies outperformthe combination strategy in a numerical example. We certainly believe thatdynamic strategies based on dynamic programming can be further improved, asalso this research clearly demonstrates their added value to pension investors.A most suitable dynamic strategy is hard to determine objectively as itsperformance depends on the governing stochastic model. Also, such a dynamicstrategy should fit well with practical requirements, such as whether an investorwill follow through on the strategy or will feel the need to combine such astrategy with her own judgement, and whether such strategies comply withregulations. This research, however, demonstrates the added value of dynamicstrategies to pension investors. In summary, such strategies exploit freedomthat is not used by traditional approaches, can steer a pension investor to hertarget and prevent unnecessary risk taking.
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Career path and contribution rate
Age Salary ContributionRate Euro Rate Euro
25 3% 29403 7.8% 127026 3% 30285 7.8% 133927 3% 31193 7.8% 140928 3% 32129 7.8% 148229 3% 33093 7.8% 155830 3% 34086 9.0% 188731 3% 35108 9.0% 197932 3% 36162 9.0% 207333 3% 37247 9.0% 217134 3% 38364 9.0% 227235 2% 39131 10.5% 273136 2% 39914 10.5% 281337 2% 40712 10.5% 289738 2% 41526 10.5% 298239 2% 42357 10.5% 307040 2% 43204 12.2% 367041 2% 44068 12.2% 377542 2% 44950 12.2% 388343 2% 45849 12.2% 399344 2% 46765 12.2% 410445 1% 47233 14.2% 4844
Age Salary ContributionRate Euro Rate Euro
46 1% 47705 14.2% 491147 1% 48183 14.2% 497848 1% 48664 14.2% 504749 1% 49151 14.2% 511650 1% 49642 16.5% 602651 1% 50139 16.5% 610852 1% 50640 16.5% 619053 1% 51147 16.5% 627454 1% 51658 16.5% 635855 0% 51658 19.4% 747656 0% 51658 19.4% 747657 0% 51658 19.4% 747658 0% 51658 19.4% 747659 0% 51658 19.4% 747660 0% 51658 23.0% 886361 0% 51658 23.0% 886362 0% 51658 23.0% 886363 0% 51658 23.0% 886364 0% 51658 23.0% 886365 0% 51658 26.0% 1001966 0% 51658 26.0% 10019Table 3: Salary in nominal amount, annual salary increase as rate and contribu-tion in both nominal amount and as percentage by age at t = 0, correspondingto ultimo 2016. All is based on statistics reported for the Netherlands in 2016.Salary for a 25 year old is reported by Eurostat as the average Dutch salaryunder 30 years old in 2014 and is indexed with 2 years of wage inflation asreported by the Dutch Central Bureau of Statistics. The salary increases corre-spond to a career path used in actuarial calculations of the Dutch government(available under document number blg-230018). Contribution percentages areas prescribed under Dutch law in 2016 and are applied to the difference be-tween the salary and a franchise of 13123 Euro. For t >
0, nominal amountsare indexed with wage inflation and rates are kept fixed.
B Dynamic Programming Implementation
In this section, we will give a brief description of various computational insightsfrom our implementation. The algorithm for the combination strategy includesthe dynamic programming design, the selection of the state variable and the uselocal regression. The local regression technique in this section is also used bythe target strategies. 20 .1 Dynamic Programming Algorithm
Asset allocations for a dynamic programming strategy follow from Algorithm1. The algorithm runs backward in time, similar to the algorithms in Cong andOosterlee and Binsbergen and Brandt.The solution space is discretized such that a suitable algorithm can be usedto solve the dynamic programming problem: it is assumed that S : T × R > → { a , a , . . . , a k } , (26)with the values of the control, a j ∈ [0 ,
1] for j ∈ { , . . . , k } . The investor canchoose between at most k asset allocations at each time t .Algorithm 1 solves the optimal control problem backward in time by calcu-lating the expected utility, E (cid:2) ˇ U ( Z T ) |F t (cid:3) , with the state variable Z t (see SectionB.2) by the local regression technique (see Section B.3). The use of local re-gression is similar to the use of bundling in Cong and Oosterlee: neighborhoodpoints are used in the local regressions for each step of the algorithm.Solving the sub-problem at time t i changes the future states Z t i +1 , . . . , Z t n − .These states have previously been used in the local regressions to find the op-timal solution for the sub-problems at times t i +1 , . . . , t n − . Thus, the optimalsolutions for sub-problems at t i +1 , . . . , t n − may be different after solving thesub-problem at time t i . This is why Algorithm 1 follows a “snake-like pattern”through time: after the sub-problem at time t i is solved, future sub-problemsare first updated in a forward fashion in time. Sub-problems are subsequentlyupdated backwards in time at the next step until the sub-problem at time t i − is solved for the first time.Once all sub-problems have been solved, the solution can be further improvedby repeating the procedure. Algorithm 1 restarts at the beginning of the snake-like pattern through time. Each iteration of Algorithm 1 follows the snake-likepattern from T to t once. B.2 State Variable
Individual wealth targets are not constant over the time horizon. They are de-fined using the expected inflation and the market value factor at time t (see alsoSection 3.4). The individual wealth targets are also not constant between thedifferent scenarios, as they are dependent on the contribution. The dynamicprogramming approach requires, however, a fixed wealth target in order to eval-uate the expected utility. This issue is resolved by using the ratio betweenthe portfolio wealth and the wealth target Z t , defined in Equation (18), as thestate variable of the dynamic programming algorithm. The target state is nowconstant in time: Z ∗ t = 1.An advantage of choosing this state variable is that, theoretically, the dy-namic programming solution only has to be computed once. The investmentdecisions for the first contribution of the investor can be used for all other con-tributions. Dynamic programming results for the first contribution span thetime horizon [ t , T ]. At each rebalancing time, the optimal investment decision,depending on state Z t , has already been computed. Because a ratio is used,these investment decisions can also be used for the contributions in later years.This is, however, not the case in practice. Algorithm 1 does not fully con-verge due to the limited sample size and due to sets of observations changing21 lgorithm 1: Dynamic programming investment strategy
Data:
Scenario Set, asset allocations a , . . . , a k Result:
Optimal admissible asset control A ∗ Initialization: create initial solution by choosing α i = a for all t i ∈ T = { t , . . . , t n = T } for all scenarios. for m = 1 , . . . , number of iterations do for j = 1 , . . . , k do Set α n − = a j for all scenarios Determine the expected utility function f j,n − , f j,n − ( z ) = E (cid:2) ˇ U ( Z T ) | Z n − = z, α n − = a j (cid:3) ,by using local regression for Z n − and ˇ U ( Z T ) (Least SquaresMonte Carlo method) Determine optimal asset allocation decision at time t n − for allscenarios: m = arg max j ∈{ ,...,m } f j,n − ( Z n − ) α ∗ n − = a m for t i = t n − , t n − , . . . , t do Update optimal allocation decisions (backward update): for τ = n, . . . , i + 1 do Update expected utility function f j,τ , for j = 1 , . . . , k Update optimal asset allocation decision α ∗ τ , for all scenarios Determine optimal asset allocation decision at time t i for thefirst time for j = 1 , . . . , k do Set α i = a j for all scenarios Determine the expected utility function f j,i , f j,i ( z ) = E (cid:2) ˇ U ( Z T ) | Z i = z, α i = a j , A i +1 = A ∗ i +1 (cid:3) ,by using local regression for Z i and ˇ U ( Z T ) (Least SquaresMonte Carlo method) Determine optimal asset allocation decision at time t , for allscenarios: m = arg max j ∈{ ,...,k } f j,i ( Z i ) α ∗ i = a m Update optimal allocation decisions (forward update): for τ = i + 1 , . . . , n − do Update expected utility function f j,τ , for j = 1 , . . . , k Update optimal asset allocation decision α ∗ τ , for all scenarios22ultiple times per iteration. Running Algorithm 1 separately for each contribu-tion will give better decision rules. Separate runs of the algorithm will providea better approximation of the optimal solution on average. B.3 Least squares Monte Carlo method
At each optimization step in Algorithm 1, a least squares Monte Carlo methodis used to avoid nested simulations (and the, related, exploding computationtimes). The least squares Monte Carlo method was introduced by Longstaffand Schwartz as a simple method for pricing American options by simulation.The conditional expectation of the pay-off under the assumption of not exercis-ing the option is estimated by using cross-sectional information already availablein the simulation. Realized pay-offs from continuation (or, in the pension invest-ment setting, from final utility ˇ U ( Z T )) are regressed on functions of the statevariables. The fitted value provides an estimate of the conditional expectation.The regression employed is based on a so-called regress-now strategy, andspecifically on a local regression version. Regress-now estimates the expectation E (cid:2) Y t i +1 | X t i (cid:3) , X t i ∈ F t i by using a set of basis functions, Φ, with index set J : Y t i +1 ≈ X j ∈J c j ϕ j ( X t i ) , with c j coefficients found by using least squares regression and ϕ j ∈ Φ. Substi-tution gives E (cid:2) Y t i +1 | X t i (cid:3) ≈ E X j ∈J c j ϕ j ( X t i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t i = X j ∈J c j ϕ J ( X t i ) . Local regression, introduced by Cleveland, estimates a linear or quadraticpolynomial fit at x by using a weighted least squares regression. Weights for anobservation ( x i , y i ) are dependent on the distance between x i and x (Clevelandand Grosse). The smoothness of the fit is dependent on the percentage ofobservations that are taken into account when evaluating at x .Let n be the number of observations and let 0 < d ≤ k = d · n , rounded up toan integer value, ∆ i ( x ) be the Euclidean distance of x to x i , and ∆ ( i ) ( x ) be thevalues of these distances, ordered from smallest to largest.The weight ζ t for an observation ( x i , y i ) is then equal to ζ i ( x ) = T (cid:18) ∆ i ( x )∆ ( k ) ( x ) (cid:19) , with T ( u ) = (cid:26)(cid:0) − u (cid:1) , for 0 ≤ u < , , for u ≥ , also known as the tri-cube weight function.23ot only is the regress-now strategy used to approximate the expected utility,this strategy is also used to estimate the expected annual inflation E t π τ , withΦ = { , x } . The future cumulative inflation, Y t i +1 , is regressed on the pastcumulative inflation, X t i , to estimate the future annual inflation at time t i : x j = j Y k = t (1 + π k ) ,y j = T Y k = j +1 (1 + π k ) , with x j ∈ X t i and y j ∈ Y t i +1 for each j in the scenario set.The resulting regression function is of the form f = ξ x + ξ . The expectedeffective annual inflation for the time period k to T is now equal to: I ( T ; k ) := T − ( k +1) p ξ x k + ξ − ..