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An Age-Based, Three-Dimensional Distribution Model Incorporating Sequence and Risks by Larry R. Frank Sr., CFP®; John B. Mitchell, D.B.A.; and David M. Blanchett, CFP®, CLU, AIFA®, QPA, CFA

Larry R. Frank Sr., CFP®, is a registered investment adviser with an M.B.A. and B.S. cum laude in physics, Executive Summary living in Rocklin, California. (LarryFrankSr@ BetterFinancialEducation.com) • The authors develop an age-based, tion periods. The model developed three-dimensional distribution in this paper demonstrates: John B. Mitchell, D.B.A., is professor of at Central model that illustrates a retiree’s » How longevity probability can be Michigan University. ([email protected]) yearly transition through retirement, developed into dynamic, yearly including superannuated years (very adjustable distribution periods David M. Blanchett, CFP®, CLU, AIFA®, QPA, CFA, is a ), to demonstrate the impact » These dynamic distribution research consultant for Morningstar Investment Man- of longevity risk periods can then be combined agement in Chicago, Illinois. Blanchett has published • The model builds on “Probability- with stochastic (Monte Carlo) more than 30 papers in various industry journals and has of-Failure-Based Decision Rules real returns an M.B.A. from the University of Chicago, Booth School to Manage Sequence Risk in • The distribution period (DP) can be of Business. ([email protected]) Retirement” by Frank, Mitchell, and dynamically managed, as the retiree Blanchett (2011) to simultaneously: ages, by varying the percentiles of » Establish an age-based distribu- longevity ithdrawal rates are sensi- tion model • The withdrawal rate is very sensitive tive to two fundamental » Incorporate current-age life to DP, and the DP is very sensitive Wcomponents: the distribu- expectancy to ; thus more atten- tion period (DP) and the market returns’ » Address survivorship into tion should be paid to longevity risk effect on portfolio values (through the superannuated ages than has been in past research basic formula in which withdrawal rate » Address market sequence risk • The probability of failure (POF) equals the annual dollars distributed » Develop a method to rationally remains a useful metric to evaluate divided by the portfolio value). How incorporate retiree goals of either exposure of a retiree’s portfolio to do retirement planners separate these consumption or inheritance, and market sequence risk, and a POF- two components when helping retirees switch between the two as desired based decision rule applies to both make decisions about the sustainability • Past research has primarily focused the asset allocation effect and the of their distributions? Frank, Mitchell, on fixed, non-age-specific distribu- age effect on withdrawal rates and Blanchett (hereinafter FMB) (2011) described how to measure the market returns component. This paper builds on life expectancies define the length of the risk decision rule in which portfolio that work by addressing the distribution current distribution period (DP) based withdrawal amounts are adjusted based on period component. on the current age of the retiree. The market returns. Rather than use generic fixed distribu- current DP changes dynamically each tion periods not necessarily associated year as the retiree ages and/or by adjusting Brief Overview with a specific retirement age, the authors the longevity percentile (how long the This paper develops an age-based, three- have developed an age-based model that retiree is expected to live). Additionally, dimensional distribution model for retire- uses period life table statistics so that the authors introduce a market sequence ment. The model plots the results of many

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simulations relative to each other—each periods. For the application of withdrawal The model developed in this paper will simulation represents a transient state rate research in any given country, the help the reader determine what the effect a retiree may dynamically pass through expected longevity of that country should on withdrawal rate, more importantly as time and portfolio values change. The be used. POF, is of surviving into the “next” concept of transitory states suggests a Making decisions based on a single distribution period (DP)—the “next” dynamic process of change the retiree’s simulation into perpetuity is unrealistic DP is the then current life expectancy for decumulation goes through over time because conditions continually change the then current retiree’s age. The model and market changes. A range of transient with time. Proactive or strategic decision dynamically adjusts as the retiree ages states is okay for a retiree, but beyond making requires a larger context of how a through rolling distribution periods that that range, spending retrenchment (with decision to solve one issue (for example, decrease slightly in length each year. bad market returns, commonly referred longevity risk) may affect another issue The authors’ hypothesis is that sequence to as sequence risk) is recommended to (for example, sequence risk), and may risk occurs more often and throughout increase the probability the portfolio does need to include other factors relevant to retirement relative to the occurrence not fail during that DP. The transient overall strategic decisions such as asset of longevity risk; hence the POF-based state concept was initially developed and allocation, time frame, retiree goals, etc.— techniques that manage exposure to explained by FMB (2011). all of which come from within a model and sequence risk also simultaneously manage The lack of a specific distribution period methodology to evaluate choices. In other aging into and through superannuation. based on the retiree’s current age is prob- words, how do a multitude of possible The authors’ model no longer shows lematic because, as the authors will show, coexisting transient states relate to each generic distribution periods. Instead, the withdrawal rates are sensitive to distribu- other? Second, models can be developed current age of the retiree is used as the tion periods. What is an age-appropriate to compare strategies; for example, starting point for each year during simula- DP? The model here removes much of this different retrenchment strategy models to tion. The DP is derived by subtracting the DP uncertainty. The authors also evaluate the baseline model or allocation change current age from the percentile longevity the transition between “early” retiree ages strategy models to the baseline model. age (from periodic life tables). Joint (with corresponding expected longevities) Last, the methodology should be consis- couples are assumed to be the same age to “older” retiree ages (superannuated); tent throughout all models and include for concept development purposes. For specifically, how outliving early/younger the three dimensions of distributions example, a 65-year-old couple in which DP estimates affects later/older distribu- (allocation, withdrawal rate, time) with a either, or both, have a 20 percent chance tion feasibility. focus on probability of failure1 (POF)—a to live beyond age 95: DP = 95 – 65 = The sustainable withdrawal rate variable that does not depend on time and 30 years. This DP calculation is updated available from a retiree’s portfolio depends is therefore a proper variable with which to annually within the simulation. on a number of factors. The withdrawal make key decisions (FMB 2011). The overall effect of using joint data rate itself is a variable that depends The model and methodology in this initially is for longer DPs in the model, directly on time because it changes as paper extend the work of Mitchell (2010 thus making the model more conservative the retiree ages—it goes up as the retiree and 2011) and combine Mitchell’s work than would be the case for either single ages or the expected distribution period with the methodology first developed female or single male scenarios. Further shortens. Thus, longevity probability is a by FMB (2011) to address the follow- model development and research is very important variable when determin- ing issues simultaneously: (1) applying needed for situations of singles, although ing proper age-specific distribution expected longevity as the distribution the effect would be slightly shorter DPs periods. Life expectancies in this paper are period to an age-specific model, (2) compared with joint data, thus resulting in based on the Social Security Administra- evaluating sequence risk as the retiree slightly higher withdrawal rates. Various tion’s 2006 periodic life table. Annual ages through this model, (3) developing age differences within couples would also changes to the life table have not been a method to address early on the effect affect DPs and therefore withdrawal rates. material, and therefore should not make a of a retiree continuing to survive beyond The authors combine male and large difference in findings and methodol- longevity estimates, and (4) incorporating female life expectancy statistics and ogy. However, just as market data may decisions for both poor or good market assume independence to determine joint update results over time, periodic life table effects (sequence risk). The objective is probabilities. For example, if the prob- adjustments should also be applied so to develop a more complete model and ability of a 65-year-old male living to age that both market and longevity effects are methodology that address all phases and 85 is 40 percent, and the probability of a slowly captured in this model over longer issues of distributions. 65-year-old female living to age 85 is 50 www.FPAnet.org/Journal March 2012 | Journal of Financial Planning 53 Contributions F r a n k | M i t c h e l l | B l a n c h e t t

Time Sequencing and Simulation Periods Table 1: Period Life Table, 2006 The authors use a 3D method of graph- Male Female ing to illustrate the dynamic concept of Exact Death Number of Life Death Number of Life transient states, in which each data point Age Probability* Lives** Expectancy Probability* Lives** Expectancy represents a possible withdrawal state 50 0.00566 92,041 28.78 0.003275 95,460 32.49 for a retiree at that specific point in time, 55 0.007936 89,037 24.66 0.00464 93,659 28.07 and how that transient state may relate to 60 0.011599 85,026 20.7 0.007219 91,109 23.78 65 0.017161 79,354 17 0.011009 87,217 19.72 other possible transient states the retiree 70 0.026212 71,586 13.55 0.017646 81,571 15.9 may experience given changing market 75 0.041267 60,942 10.46 0.028247 73,134 12.43 return sequences, retiree withdrawals, 80 0.066266 47,073 7.78 0.046337 61,289 9.33 and/or asset allocation. 85 0.107951 30,778 5.56 0.079984 45,395 6.68 For Phase 1, a 10,000-run Monte Carlo 90 0.177636 15,051 3.84 0.138938 26,608 4.62 95 0.277945 4,445 2.67 0.226885 10,324 3.2 generator was built in Microsoft Excel. 100 0.371724 657 2.01 0.317702 2,223 2.35 The distribution is assumed to be taken 105 0.474424 46 1.51 0.425157 241 1.7 from the portfolio at the beginning of 110 0.605499 1 1.11 0.568956 9 1.19 each year. All returns are in real terms so 115 0.772787 0 0.78 0.761392 0 0.79 that a constant real withdrawal amount *Probability of dying within one year. is assumed to be taken from the portfolio **Number of survivors out of 100,000 born. Source: Social Security Administration period life table (2006), http://www.ssa.gov/OACT/ during each year in retirement. Returns STATS/table4c6.html. are generated assuming a normal distribu- tion based on the average historical annual percent, the probability that either or both Index for All Urban Consumers, from the real returns and standard deviation, would live to that age would be 1 minus Bureau of Labor Statistics. not the actual historical series and not the probability that both are dead, which bootstrapped. The success of a portfolio or would be 70 percent (1 – [(1 – .4) x (1 – Cash: 30-day T-bill withdrawal is calculated by determining .5)]); or alternatively, a 30 percent chance Bonds: Ibbotson Associates Long-Term how many portfolios had positive values at to outlive 85 (Goodman 2002). In this Corporate Bond Index the end of the year. A positive value would manner, specific ages are combined with Domestic Large-Cap Equity: S&P indicate the portfolio was successful for specific life expectancies to move away 500 Index that year. Withdrawal rates are tested in from generic distribution periods. Various Domestic Small-Cap Equity: Ibbotson .05 percent increments from 0 percent thresholds of life expectancy are used in Associates U.S. Small Stock Index to 25 percent, in .10 percent increments this paper, for example, 75th percentile (75 International Large-Cap Equity: Global from 25 percent to 50 percent, and in .25 percent chance of outliving the distribu- Financial Data Global ex USA Index percent increments from 50 percent to tion period), 50th percentile, and 25th from January 1926 until December 1969 100 percent. The simulation is based on percentile, in order to evaluate directly and then the MSCI EAFE from January a period dependent on the retiree’s prob- how longevity percentile affects the model. 1970 until December 2010 ability of outliving the distribution period The use of life expectancy as the current as described for Table 1. DP as the retirees dynamically age will Portfolios are constructed of two compo- be discussed below. A data cross section nents: cash/fixed and equity. The cash/ Phase 1 (Table 1) for the DPs is extracted from the fixed component is 25 percent cash and Rather than illustrate distribution periods latest (2006) Social Security period life 75 percent bond. The equity component is on a figure’s time axis, this paper intro- table. The model uses the 50th percentile 50 percent domestic large-cap equity, 25 duces the retiree’s age on the time axis as a proxy for expected longevity. percent domestic small-cap equity, and 25 with each simulation distribution period percent international large-cap equity. For from the expected longevity for each Asset Classes example, a 60/40 portfolio (60 percent retiree age from Table 1, resulting in an Returns are based on five asset classes from equity and 40 percent cash/fixed) would age-based model. January 1926 until December 2010. All have 30 percent domestic large-cap equity, The model (Figure 1) in this phase returns are converted into “real returns” 15 percent domestic small-cap equity, 15 most closely represents how advisers (adjusted for inflation); inflation is defined percent international large-cap equity, 10 envision retirement withdrawals, now in as the increase in the Consumer Price percent cash, and 30 percent bond. 3D. However, after the distribution period

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begins, few advisers annually adjust the Figure 1: 3D Probability-of-Failure Landscapes distribution period dynamically as the retiree ages. The DPs in Figure 1 represent 50th Longevity Percentile, POF 50% the estimated expected longevity of the retiree (50th longevity percentile). The 35% 30% withdrawal rate at the beginning of each 25% 20% DP represents the starting withdrawal rate 15% 10% for that age-specificDP. The fundamental 5% 0% difference for this paper is the adjustment –5% 95 100 90 Withdrawal Rate 0% 85

of the distribution period each year the 80 10% 20% 75 30% 70 40% 50% 65 60% 70%

retiree ages—rolling DPs (decreasing 80% 60 90% lengths) as the retiree ages each year with Equity A 100% Retiree Age llocation a subsequent rolling recalculation of the withdrawal rate; an annual determination of the current transient state. All solutions 50th Longevity Percentile, POF 5% on the 5 percent POF landscape represent transient states (solutions) with 5 percent 35% 30% or fewer of the simulations failing to 25% 20% reach the end of the DP; similarly for the 15% 10% 50 percent POF. Other POF landscapes 5% exist between 5 percent and 50 percent, 0% Withdrawal Rate 0% 95 10% 100 90

illustrated in Figure 2, and represent the 20% 85 30% 40% 80 50% 75 60% 70 range of POFs that coexist (FMB (2011)). 70% 80% 65 90% Equit 60

Figure 2 illustrates age (distribution time 100% y Alloc Retiree Age remaining) cross sections through Figure ation 1. For example, the same 10 percent POF landscape from Figure 1 has a 4.1 percent Withdrawal Rate withdrawal rate at age 60, a 5.3 percent 0%–5% 5%–10% 10%–15% 15%–20% 20%–25% 25%–30% 30%–35% withdrawal rate at age 70, and an 8.1 percent withdrawal rate at age 80 (all 60 percent represents people who outlive a period • Of interest: the differences between equity allocation and 50th percentile shorter than expected longevity (50th outliving expected longevity percen- longevity expectancy), and transient states percentile), and the 25th longevity tiles increase as the retiree ages. coexist for the 30 percent POF landscape, percentile represents people who outlive The last observation will be the focus of respectively: 5.1 percent at 60, 6.5 percent at a period longer than expected longevity. strategy application in Phase 2, in which 70, and 9.5 percent at age 80. This withdrawal rate shift, as a result of the longevity percentile variable will also Most advisers think in terms of constant changing the longevity percentile, will be be dynamically altered during simulations. asset allocation and illustrate their figures further explained in Phase 2. along the asset allocation plane. The Some Phase 1 observations include: Phase 2 authors show cross sections to demon- • Withdrawal rates change naturally When the distribution period used is strate how the withdrawal rate is actually as the retiree ages. Thus, a set “safe” expected longevity, the question then a time-dependent variable and also show withdrawal rate (0 percent POF), for arises: what happens to retiree withdrawal that the time-independent variable is the example, 4 percent, is not optimal for values when the retiree lives beyond probability of failure. all retirees because not all retirees are expected longevity? Phase 2 seeks to solve POF landscapes would shift the the same age. this fundamental problem—it is uncertain withdrawal rates up (not illustrated) as • Dispersion of POF in Figure 2 which retirees in the population will the longevity percentile increased.2 This appears to be a function of volatility experience superannuation. Clearly, those occurs because distribution periods are in which POF dispersion is “narrow” who are already ill are unlikely to outlive shorter for higher longevity percentiles for lower equity allocations and expected longevity. What about those who and longer for lower longevity percentiles. “wider” for higher equity allocations. exhibit no sign of illness? Phase 2 of this For example, the 75th longevity percentile This is an area for further research. project evaluates how the methodology www.FPAnet.org/Journal March 2012 | Journal of Financial Planning 55 Contributions F r a n k | M i t c h e l l | B l a n c h e t t

Figure 2: Three Age Cross Sections Through Figure 1 Landscapes General Initial Reflections Figures 3 and 4 reflect serially connected, annually calculated cash-flow percentiles as Age 60, 50th Longevity Percentile cash-flow snapshots of a retiree aging and 11% use the longevity percentile expectation to 10% e t 9% establish the length of the DP. This process a 8% determines an optimal withdrawal rate for

al R 7%

w 6%

a that specific annual transient state, based 5% 4% on setting the expected POF at 10 percent. ithd r 3% W 2% The resulting withdrawal rate is the same 1% across all cash-flow percentiles at each age; 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% for example, 4.65 percent (Figure 3) or Equity Allocation 3.60 percent (Figure 4), both at age 62— thus withdrawal rates are directly sensitive Age 70, 50th Longevity Percentile to the longevity-percentile-calculated DP. 11% What changes is the portfolio value based 10% e t 9% on “good” or “bad” simulated markets. The a 8% 5th (best one in twenty) and 25th (best one

al R 7%

w 6%

a in four) percentiles represent the “good” 5% 4% market sequence cash flows and the 75th ithd r 3% W 2% (worst one of four) and 95th (worst one 1% of twenty) percentiles represent the “bad” 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% market sequence cash flows, all from Equity Allocation different annual starting portfolio values (transient states). Note that a retiree may Age 80, 50th Longevity Percentile experience “good” markets and be above 11% the 50th percentile during one period e

t 10% a 9% (age 60 simulation) and experience “bad” 8% al R 7% markets and be below the 50th percentile w 6% a 5% during another period (age 62, or any sub- 4%

ithd r 3% sequent age simulation) because sequence

W 2% 1% risk is ever-present during retirement. 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% The tendency is to look at graphs or Equity Allocation calculations as “prophetic.” Observe that at age 61, for example, it is not certain at age Probability of Failure 50% 40% 30% 20% 10% 70 how the markets may have performed in the interim. Only at age 70 can the tran- applies to a retiree who continues to “age outlive expected longevity. The meth- sient state for a 70-year-old be assessed. through” the period life table. odology uses DPs based on the length of There are corresponding portfolio Cash-flow values are expressed in real lifetimes as adjusted by a dynamic change values that support their respective terms, thus representing the same level to the percentage of the population annual percentile cash flows. Note that of income replacement when factoring expected to outlive expected longevity. there are now two sets of percentiles: one in inflation. The effect is to shorten the DPs, thus for longevity (to determine distribution Klinger (2007, 2010) researches increasing withdrawal rates relative periods) and another for different possible higher withdrawal rates early in to static DPs, and provide a rational market sequences. retirement and reduces them later in method to assign DPs to specific retiree FMB (2011) demonstrates how using retirement, and he suggests a method to ages. The result: an age-based model POF in a 3D model assists in evaluating do this. This paper develops a method that also allows for differences between exposure to sequence risk. It also helps to reduce withdrawal rates through retiree goals of early consumption determine when to make a decision evaluating the percentage who may versus later consumption (inheritance). about changing the withdrawal amount

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Figure 3: Consumption-Oriented Retiree Cash Flows, Ages 60, 70, 80, and 90

Retiree Age 60 Cash Flows (Percentiles) Retiree Age 70 Cash Flows (Percentiles) $250,000 $250,000 $225,000 $225,000 $200,000 $200,000 s

$175,000 s $175,000

l ow $150,000 $150,000 l ow $125,000 $125,000 ash F

$100,000 ash F $100,000 $75,000 $75,000 $50,000 $50,000 nnual C nnual C A $25,000 $25,000 $0 A $0 61 66 71 76 81 86 91 96 101 106 71 76 81 86 91 96 101 106 Retiree Age Retiree Age

Retiree Age 80 Cash Flows (Percentiles) Retiree Age 90 Cash Flows (Percentiles) $250,000 $250,000 $225,000 $225,000 $200,000 $200,000 s $175,000 s $175,000 $150,000 $150,000 l ow l ow $125,000 $125,000 ash F $100,000 ash F $100,000 $75,000 $75,000 $50,000 $50,000 nnual C $25,000 nnual C $25,000 A $0 A $0 81 86 91 96 101 106 91 96 101 106 Retiree Age Retiree Age

Market Returns Percentiles Note: Includes serially connected, annually adjusted withdrawals. 95th 75th 50th 25th 5th All points calculated at 10% POF, 60% equity. by allowing POF to float as a result of should be viewed as a dynamic rather 30th at 85, 20th at 90, and 10th at 95 and the withdrawal rate changes because of than set-and-forget exercise. The DP for beyond. This scenario reflects reality in that market returns, before either a retrench- expecting 75 percent of the retirees to the 75th percentile is not used throughout ment (bad market) or additional spending outlive the time horizon results in a higher the simulation, but instead for a brief (good market) decision is made. A single withdrawal rate because the DP is shorter period. Cash flows are calculated annually asset allocation (60 percent) was used for relative to the 50th longevity percentile, and the withdrawal rate is adjusted annu- Figures 3 and 4; however, similar results which is shorter relative to the 25th lon- ally based on the annually recalculated DP, emerge with any other allocation with gevity percentile. The shorter DP results in with all simulations constrained to have differences only in cash-flow amounts and a higher possible relative withdrawal rate, 10 percent or fewer fail to reach the end of general trends in percentiles because of all within the same POF landscape. that current year’s DP. allocations expected from heavier bond or As the retiree ages in Figure 3, the This is a dynamic, serially connected, equity exposures. longevity percentile (longevity “lever”) is annual rolling adjustment that slowly reduced periodically (every five years in extends the DP so the retiree has a smaller Consumption-Oriented Retiree this model), thus dynamically extending probability of outliving the DP from his A “consumption-oriented” retiree can be the DP. For example, a 60-year-old retiree or her current retirement age when the demonstrated through the interaction and may start with the 75th longevity percen- transient state is re-evaluated. This model utility of longevity percentile to determine tile, then adjust to 65th percentile at age is thus a longevity-based methodology the starting DP variable. Retirement 65, 55th at age 70, 50th at 75, 40th at 80, that investigates how to achieve Klinger’s www.FPAnet.org/Journal March 2012 | Journal of Financial Planning 57 Contributions F r a n k | M i t c h e l l | B l a n c h e t t

Figure 4: Inheritance-Oriented Retiree Cash Flows, Ages 60, 70, 80, and 90

Retiree Age 60 Cash Flows (Percentiles) Retiree Age 70 Cash Flows (Percentiles)

$250,000 $250,000 $225,000 $225,000 s

$200,000 s $200,000

l ow $175,000 $175,000 $150,000 l ow $150,000 ash F

$125,000 ash F $125,000 $100,000 $100,000 $75,000 $75,000 nnual C nnual C A $50,000 $50,000 $25,000 A $25,000 $0 $0 61 66 71 76 81 86 91 96 101 106 71 76 81 86 91 96 101 106

Retiree Age Retiree Age

Retiree Age 80 Cash Flows (Percentiles) Retiree Age 90 Cash Flows (Percentiles) $250,000 $250,000 $225,000 $225,000 s $200,000 s $200,000 $175,000 $175,000 l ow $150,000 l ow $150,000

ash F $125,000 ash F $125,000 $100,000 $100,000 $75,000 $75,000 nnual C $50,000 nnual C $50,000 A $25,000 A $25,000 $0 $0 81 86 91 96 101 106 91 96 101 106

Retiree Age Retiree Age

Market Returns Percentiles Note: Includes serially connected, annually adjusted withdrawals. 95th 75th 50th 25th 5th All points calculated at 10% POF, 60% equity.

objective in which retirees begin with retiree who experiences better than longevity percentile) to determine the a higher withdrawal rate and adjust the 50th market returns percentiles. Market DP. The authors label such a retiree withdrawal rate as they age. Also note returns do not separate retirees by the year “inheritance oriented,” which is illus- that the longevity percentile reduction is they retire; rather, the POF function is trated in Figure 4. an example for concept demonstration best used as a method to evaluate exposure purposes and not represented as an to market returns sequence risk based Inheritance-Oriented Retiree optimal solution. on FMB (2011)—when POF increases, Figure 4 illustrates a low, constant 10th Consumption-oriented retirees may the withdrawal dollar amount should longevity percentile across all ages for an find they are “unlucky” and experience be adjusted to reduce the probability “inheritance-oriented” retiree. Note that market returns under the 50th market of exhausting the portfolio within the the cash flows are shifted toward later returns percentile (Figure 3). These retir- current DP (retrenchment), and vice versa retirement ages as compared with those ees would need to adjust their withdrawal if POF decreases. of the consumption-oriented retiree. dollar amounts down. Alternatively, Alternatively, retirees may choose to be The use of longevity percentiles allows a retiree may choose lower longevity more conservative from the beginning of the practitioner to measure and manage percentile values, the impact of which will retirement so that more of their portfolio either “pulling” retirement consumption be to extend the DP, thus reducing the value extends into their later years. This into early years or “pushing” retirement withdrawal rate with the same target POF. is accomplished by using low longevity consumption into later years—or the The opposite would apply to a “lucky” percentile values (for example, the 10th ability to switch between either goal at

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any time. This pull or push is visualized though there is a reduction in the changes. A retiree moves between the by comparing the retiree age at which the number of expected remaining years. POF landscapes with every market “hump” occurs between Figures 3 and 4. move and every change in time. The Phase 2 combined uncertain market Sequence Risk simulation that tells us where the returns (probability of the portfolio) with How does sequence risk, with subsequent retiree is within the landscapes is uncertain longevity (probability of the adjustment to the withdrawal amount, associated with conditions in that person) into one age-based distribution affect the distributions for retirees who brief moment (transient state). Redo model for goals of consumption or inheri- continue to live beyond expected longevity? the simulation again at a later time tance that is also three dimensional when Because withdrawal rates naturally with a different portfolio value, and asset allocation and POF are incorporated, increase for retirees as they age as a result you arrive at a different place in the as demonstrated in Phase 1. Some Phase 2 of a decreasing DP, it is difficult to separate landscapes. Many of the transient observations include: out an increasing withdrawal rate because states are okay. As they approach the • Using higher longevity percentile of the aging effect from a bad market 30 percent POF boundary (because values (shorter DPs early on), and sequence effect. The withdrawal rate also of poor markets or large lump-sum moving toward lower values as the is affected to some degree by asset alloca- withdrawal, FMB 2011), then a retiree ages (longer DPs later), tends tion. Because market values change more decision should be made to pull them to maximize the retiree income rapidly relative to aging, the sequence risk away from that boundary. As they over the retiree’s lifetime. This is an effect is observed more often. move away from that boundary they example of using longevity percentile The bow wave in Figures 3 and 4, which are better off (good markets). Pulling values for a consumption-oriented emanates and widens from each simula- them away from a high POF boundary retiree described in Figure 3. The tion starting age, illustrates sequence risk may be accomplished by reducing the longevity percentile values are for and represents the possible distribution withdrawal dollar amount or reducing concept demonstration of polar of unknown outcomes that exist at any the longevity percentile value. retiree strategies, not necessarily simulation’s starting age. For example, the • The transient state perspective leads suggested values. illustration for the 71-year-old is simply to serially connected, constantly • Frank and Blanchett (2010) found one possible sample of what a 61-year-old adjusting annual distribution periods that exposure to sequence risk never retiree may experience when he or she rather than a disconnected fixed goes away. The results from the reaches age 71. It remains unknown distribution period based on no methodology in this paper support just where market sequences may take determinable end point. this observation—percentile ranges a retiree, even for the subsequent year. • The DP may be dynamically man- at all ages had either “good” or “bad” These cash-flow percentiles represent aged as the retiree ages by varying cash-flow values regardless of the transient states that may be monitored the percentage who outlive expected retiree’s age (percentile “bow waves” through POF, and decisions can then be longevity (longevity percentile). in Figures 3 and 4. Bow waves are made, according to FMB (2011), to reduce Shorter DPs result in higher with- discussed below in the Sequence Risk the dollar amount withdrawn as POF drawal rates, and by evaluation of section). approaches or exceeds 30 percent. the longevity percentile, DP lengths • The later retirement age cash flows may change dynamically. Longevity decline primarily because of the Overall Conclusions percentile is a useful “lever” to effect of ever-shorter DPs as a result The development of an age-based model manage distribution periods. of fewer years of life expectancy. Fur- is more meaningful than generic distribu- • Mitchell (2010) demonstrates that ther research should look specifically tion period analysis because an age-based the uncertainty of remaining life at developing methods to evaluate model incorporates life expectancy uncer- span increases as the retiree ages later-age (superannuation) cash-flow tainty in addition to market uncertainty. even though there is a reduction in strategies according to retiree goal This paper demonstrates a methodology the number of expected remaining (consumption versus inheritance). to unify into a single model the various years. Both consumption- and • Possible insight into further superan- factors that affect decumulation. inheritance-oriented retirees experi- nuation strategies comes from • The concept of transient states ence decreasing cash flow in the Mitchell (2010), who demonstrates suggests that the retiree’s decumula- superannuated years as a result of the that the uncertainty of remaining life tion goes through a dynamic process higher withdrawal rates associated span increases as the retiree ages even of change over time and market with short DPs corresponding to very www.FPAnet.org/Journal March 2012 | Journal of Financial Planning 59 Contributions F r a n k | M i t c h e l l | B l a n c h e t t

short expected longevity. This is a lifetimes. Modifying longevity statistics the FMB (2011) probability-of-failure future area of research. for health reasons is problematic unless evaluation, as a more comprehensive The withdrawal rate is a poor metric for reliable statistics are available for a client’s model for retiree distributions. retirement success because it is sensitive specific health condition. This paper to DP as well as market returns on portfo- provides a conceptual methodology to lio values. The DP is not a fixed period—it better assess where a retiree may be found may be changed dynamically as the retiree in a distribution model that incorporates Endnotes ages by changing the longevity percentile longevity statistics directly to determine 1. Probability of failure (POF) comes from prior for the retiree’s current age. The longevity distribution periods. conventional use of the term. Percentage of percentile is also not a fixed value; it too failure would be more accurate because the may be changed. The POF is best used as Practical Application results represent the percentage of simulations a measure of exposure to market sequence Practitioners can apply two components that do not reach the end of the simulation risk. This paper, combined with FMB of this paper while software developers period in question. (2011), demonstrates that withdrawal rate consider and catch up with the methodol- 2. A working paper, which includes appendices is a dynamic function of other variables, ogy studied here. Until software directly with data that demonstrate how these figures including portfolio allocation, each of applies longevity percentiles to determine are developed, and additional data, is available at which should be evaluated three dimen- DP lengths, practitioners should access lon- SSRN: www.ssrn.com/abstract=1849983. sionally in relation to each other to access gevity tables for expected longevity (50th the current transient state of a retiree percentile) ages, and adjust that age to then References during decumulation years. determine the current DP. From a transient Frank, L. R., and D. M. Blanchett. 2010. “The Researchers have sought a single “safe” state perspective, the DP would apply to Dynamic Implications of Sequence Risk on withdrawal rate that would last the retiree the retiree’s current age for the current DP a Distribution Portfolio.” Journal of Financial from retirement to death. The dynamic and would be readjusted in subsequent Planning 23: 52–61. model in this paper, combined with the years when simulations are rerun. After Frank, L. R., J. B. Mitchell, and D. M. Blanchett. authors’ prior paper, however, develops the DP has been calculated, the current 2011. “Probability-of-Failure-Based Decision a methodology to monitor, evaluate, and software available to practitioners may be Rules to Manage Sequence Risk in Retirement.” react as necessary to transient states as well used to determine the current POF. Journal of Financial Planning 24: 46–55. as base the model on age-specific expected Note the tendency to interpret figures Goodman, Marina. 2002. “Applications of Actuarial longevity rather than generic, , or and calculations as being predictive. The Math to Financial Planning.” Journal of Financial unanchored distribution periods. figures here illustrate that the future is Planning (September). Two sustainable ranges of withdrawal unknown. People do not know whether Klinger, W. 2007. “Using Decision Rules to Create rates exist: the first is based on the longev- they will be in the 75th percentile or 25th Retirement Withdrawal Profiles.”Journal of ity percentiles’ effect on DP bounded by percentile, or any other possible percen- Financial Planning 20: 60–67. consumption (upper range) or inheritance tile. The current transient state and trend Klinger, W. 2010. “Creating Safe, Aggressive (lower range) goals; the second is is determined through evaluation of the Retirement Income Profiles.” Journal of Financial determined within the first’s DP based current POF. If the POF is rising, for Planning 23: 44–53. on market fluctuation effects on portfolio example from 10 percent to 15 percent, Mitchell, J. B. 2010. “A Modified Life Expectancy values bounded by high POF/high with- this is an indication of an increased prob- Approach to Withdrawal Rate Management.” Pre- drawal rate (upper range because of “bad” ability that the current withdrawals may sented at Academy of Financial Services, Denver, sequence) or low POF/low withdrawal rate not be sustainable. From FMB (2011), Colorado. www.ssrn.com/abstract=1703948. (lower range because of “good” sequence). the retiree can pre-calculate and consider Mitchell, J. B. 2011. “Withdrawal Rate Strategies The methodology presented here a change of the withdrawal dollar amount for Retirement Portfolios: Preventive Reduc- provides a model to directly incorporate to reduce the POF, then adjust back to a tions and Risk Management.” Financial Services longevity risk and provides a rational more successful track. This adjustment Review 20: 45–59. alternative to guarantees, implemented is still not predictive because we do not here through index funds, with the option know what track will result from future Join FPA’s webinar with co-author to switch to a guarantee at any later time. markets. Thus, the concept demonstrated David Blanchett as he discusses A dynamic model provides more assur- here is a transient model that uses annuities on March 8. Find out ance that retirees may be able to maximize both probabilities of the portfolio and more at www.FPAnet.org/VLC. their cash distributions over their probabilities of longevity, combined with

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