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## Abstract

The article’s primary contribution is applying a dynamic decision theory approach to investment management and adding spending flexibility to improve incomes for retirees for whom saving more and working longer are no longer options. We investigate whether a self-driving portfolio, engineered to protect against market risk, can deliver more income than other investment approaches while minimizing the risk of ruin. We consider whether a dynamic strategy, based on a stochastically dominant decision theory algorithm and tested with fixed and variable spending policies, can minimize the chance of running out of money before dying. Outcomes are compared with a number of static approaches.

We find that the dynamic strategy provides an average of 51% more income than a target date fund through retirement when combined with a hybrid spending rule over a relevant probability of ruin range of 1% through 25%. Furthermore, when rebalanced to constant risk rather than to fixed asset mixes, sequence of return risk can be reduced. Variable spending policies add effectiveness and should be considered regardless of investment strategy. Importantly, dynamic decision theory establishes a framework for goal-seeking solutions that may have application in other phases of the investor’s life cycle.

This article is organized into five parts. First, we examine the road to retirement and the accelerating need for post-work income solutions for those entering the workplace as well as for Baby Boomers retiring with anxiety. Second, we discuss the importance of destination as a fundamental objective of both self-driving automobiles and retirement portfolio management. We propose addressing portfolio construction by using two principles of self-driving automobiles: protection and navigation. Third, we explain our methodology and intention first to test post-work investment strategies alone to evaluate their effectiveness discreetly, and second to apply variable spending schemes to test for added value. We establish our strategic assumptions and approaches, set income goals, and outline the spending schemes to be tested. Fourth, we catalogue our capital market, longevity, and testing assumptions and describe our observations. Fifth, we summarize the observations and offer conclusions about variable spending and dynamic decision theory.

## ROAD THROUGH RETIREMENT

This article is about decumulation strategy. Managing resources in retirement has become acutely important because the changing nature of work has made accumulating pension assets increasingly difficult. We start by reviewing these challenges because more workers may reach retirement with insufficient savings to fund the lifestyle they want or need, putting more pressure on already overburdened social welfare systems globally.

Even the most carefully designed withdrawal scheme will not ensure adequate income in retirement if savings are too small. Indeed, 48% of the retirement-age population globally has no pension. The estimated gap between government, employer, and individual retirement savings and a 70% income replacement for the six largest pension systems in the world (Australia, Canada, Japan, Netherlands, United Kingdom, United States) plus China and India stood at $70 trillion in 2015 and is expected to grow to $400 trillion by 2050. The gap for the United States alone is estimated to be $27.8 trillion (2015), growing to $136.8 trillion by 2050 (Wheeler [2017]).

If that were not bad enough, employment patterns in the United States make it difficult for younger workers to establish meaningful pension plans because jobs with good pension benefits are less accessible. Unions represent only 10.7% of wage and salary workers (2016), compared with 20.1% a generation ago (1983) (Bureau of Labor Statistics [2017]). Public sector jobs are available, but with U.S. federal, state, and municipal pension plans’ underfunded liabilities at $7 trillion (Moody’s Investor Services [2016]), sustainability is an issue. Large companies offer better pension coverage than smaller ones, but 64% of job creation between 1993 and 2011 was from small firms (Small Business Association [2012]) and virtually all net new jobs are from new companies (Wiens and Jackson [2015]). One of the best ways to accumulate pension benefits has been to stay with the same employer for a long time, but Generation Y (those born after 1981) are expected to have 15–20 jobs over their working lives (Meister [2012]), making pension savings difficult.

Shifting social patterns offer little help. Married people are more likely to include retirement in their savings plans and to have an individual retirement account (IRA) or participate in a defined contribution (DC) pension plan (Knoll, Tamborini, and Whitman [2012])—perhaps retirement is less abstract for those planning a family. However, with only 50% of adults married in 2014 compared with 69% in 1970 and with cohabitation up 29% from 2007 to 2016 (Stepler [2017]), this social motivation for pension savings is less strong than it used to be (Exhibits 1 and 2).

Despite growing concern about pension adequacy, the immediate demand for post-work replacement income solutions is from Baby Boomers, who are retiring at the rate of 10,000 per day in the United States. Annuities remain unpopular with retirees (Babbel and Merrill [2006]; Pashchenko [2012]). There are unfortunately few systematic strategies other than annuities to help manage the drawdown of hard-earned savings, particularly if the individual has not saved enough to support an adequate retirement (Blanchett [2015]). Some 61% of those aged 44–75 surveyed by Allianz [2010] were more afraid of running out of money than of dying. A conservative diversified fund is a likely option, but its static asset allocation resigns the investor to whatever returns the market provides without regard for investing time horizon or personal circumstances. At retirement or termination, 58% of those fortunate enough to be members of defined contribution pension plans take lump sum payouts (AON Hewitt [2014]). Making things worse, they move from lower-cost institutional funds and invest in higher-cost retail funds. Welcome to retirement.

## LIFE IS BASICALLY A JOURNEY; RETIREMENT INVESTING IS CONCERNED WITH ITS DESTINATION

Saving more, working longer, and (up to a point) taking on more risk are three accepted ways to improve retirement incomes. However, the discouragement of approaching retirement without sufficient savings to fund the desired lifestyle can turn to devastation when realizing that health and time have diminished access to one or more of these escape routes.

Getting to the desired financial goal is all that should matter to an investor; however, the pension industry’s incentives are not aligned with this purpose. Rather than focusing on a retirement benefit target, the industry concentrates on growing assets, growing membership, and maximizing returns (Van Wyk [2012]). Except for the very wealthy who can afford to self-fund their retirement, pensions are an asset–liability-matching problem, not a return-maximizing one, and should address the contingent liability of the specific future need: retirement income (Kim [2017]).

The self-driving automobile, a developing technology that seeks to overcome the auto industry’s weakest link, the driver, holds clues for the investment and pension industries and their weakest link, the investor. Both industries share a common purpose: to get their passenger/investor to their desired destination as safely and efficiently as possible. Self-driving automobiles approach this challenge by protecting passengers from surrounding hazards and navigating to a predetermined destination. Ironically, protection and navigation are two things the investment industry does not do particularly well.

The investment industry in general and the personal retirement savings industry in particular, as represented by DC pension plans, seem preoccupied with maximizing returns, risk-adjusted or otherwise, but Google Maps users know that the best way to a destination is not always the shortest route. UPS routes its van drivers so they can avoid making turns against oncoming traffic (in North America, no left-hand turns) to reduce both accidents and delays caused by waiting for a gap in traffic, which has led to fuel cost savings, smaller required fleet sizes, and shorter overall delivery times (Kendall [2017]). Traditionally, investors in a diversified fund, target date fund (TDF), or managed account are encouraged to rebalance to the most aggressive suitable asset mix allowable, such as 60% equities and 40% bonds, or a laddered mix (i.e., a portfolio of bonds that matures in stages as the holder ages), regardless of market conditions. If portfolios were automobiles, this would be akin to driving with the pedal to the metal (i.e., as fast as the vehicle can go) at all times without regard for terrain, weather, or traffic conditions. This hardly represents protection nor, we will argue, efficiency.

Investment planning relies upon historical risk and return data to project future capital market results. This is like navigating an automobile by peering through its rear-view mirror. We investigate a way to control exposure to risks along the road to and through retirement. Instead of maintaining a fixed asset allocation regardless of market conditions, we seek a way to identify risks and take evasive action. We stay on course by making decisions to increase or decrease portfolio risk based on a goal-focused navigation system.

Global positioning system (GPS) navigation systems triangulate the position of the vehicle and that of the target destination. During a pension’s accumulation phase, the destination is target capital. During decumulation, triangulation should focus on target income, payout time table, and mortality.

## METHOD

First, to measure the effectiveness of each approach in isolation in the test environment, we propose the use of a novel investment strategy that maintains a constant portfolio risk as measured by its standard deviation rather than rebalancing to a constant asset mix or a static laddered mix like a target date fund. This dynamic constant risk (DCR) approach reduces portfolio risk in response to high or rising market risk and conversely increases portfolio risk when market risk is low or falling. Dynamic risk adjustments are then made based on progress toward a predetermined capital accumulation goal based on remaining life expectancy. We study the performance of the DCR and other benchmark strategies’ ability to deliver target income and evaluate the risk of ruin for each. We assume anyone saving for retirement wants a reliable stream of income in retirement like that provided by a defined benefit (DB) pension (Tretiakova and Yamada [2011]). Accordingly, we use fixed withdrawals adjusted for inflation because they most closely resemble a DB payout.

Second, we test the DCR and other benchmark strategies using five variable spending schemes to observe how the probability of ruin varies among them. Although investors using a variable spending overlay are willing to forgo some annual income as a trade-off for making funds last longer, we assume that there are practical limits to this sacrifice. For consistency, we therefore set a maximum decline of 20% of the initial withdrawal rate as the spending floor across all the tested strategies. For example, if we target 4% for all strategies, we will withdraw at least 3.2% of the first-year capital in real terms each year. All spending rule amounts are adjusted for inflation except for the three rules, those of Zolt [2013] and Guyton (Guyton and William [2006]) and the hybrid spending rule, which specifically use inflation as an adjustment mechanism. Withdrawal rates are recalculated annually at the end of the year. We correct for any bias or distortion from these differences by survivor-weighting withdrawals and adding any residual capital weighted by the probability of mortality to create a uniform basis for comparison.

### Basic Assumptions

We employ a dynamic strategy used in earlier work that we found effective for targeting outcomes (Tretiakova and Yamada [2013]). The DCR approach makes changes to the asset mix of a portfolio based on the two objectives of protection and navigation.

**Retirement age.** Although the official age for retirement is being extended in a number of countries, including Australia, France, Germany, Denmark, and the United States, and working longer is an increasingly popular and necessary option for many, we assume a retirement age of 65 for the purposes of this article.

**Starting capital.** Because our primary focus is on sustaining a withdrawal rate (not pension adequacy), we assume starting capital of $100,000 throughout.

**Protection: keeping portfolio risk constant.** Using volatility as a proxy for threats to portfolio safety, we use the methodology described for the S&P Dow Jones Indices Daily Risk Control Series to rebalance portfolios to a consistent risk level^{1} (Banerjee, Srivastava, and Cheng [2016]).

This approach reduces portfolio risk when market volatility increases and vice versa. A strategy like this is crucial in retirement because employment income may become sporadic or disappear altogether as full-time employment gives way to part-time work and finally to full retirement. Without inflows, investors are particularly vulnerable to sequence-of-return risk that exacerbates the probability of ruin when investors make withdrawals during weak markets early in retirement, making recovery difficult. One approach to reducing sequencing risk is to decrease the volatility of the investment. However, this lower volatility is usually achieved by reducing equity exposure, which increases longevity risk by reducing the potential for growth. Sequencing risk therefore needs to be balanced with longevity risk.

Using a constant risk adjustment strategy, as described earlier, helps to control volatility. This strategy allows retirees to avoid extreme downturns. We argue that adjusting for constant volatility would mitigate periods of extreme turbulence and reduce sequencing risk. By controlling volatility with a constant risk approach, the investor need not sacrifice returns by switching to a lower equity allocation (see the Appendix for more about sequencing risk). Because of volatility’s persistence (Bollerslev, Chou, and Kroner [1992]), constant risk provides a stable input for dynamic decision theory, which addresses longevity risk directly. Constant risk management furthermore allows for the harvesting of volatility (Bouchey et al. [2012]).

### Navigation: GPS

In considering target income, payout time table, and mortality, we use a practical adaptation of research by Moore and Young [2006] to adjust asset allocation for expected surpluses and shortfalls resulting from market changes that incorporates both market and longevity risks. To make the discrete rebalancing methodology implementable, we constrain the algorithm to a maximum of 60% equities for behavioral reasons (rather than the unlimited leverage that Moore and Young applied to extremely underfunded portfolios). We assume no one over the age of 65 years would be comfortable holding more equities. We calculate the allocation π to the risky asset using the following formula:

1where μ is the return on the risky asset, σ is standard deviation, and *r* is the risk-free rate. The ruin probability Ψ is a function of current capital; consumption, as described later; and life expectancy (Moore and Young [2006]). Ψ_{w} and Ψ_{ww} are first and second partial derivatives with respect to *w*. Note that because Ψ is a function of consumption, it depends in part on a withdrawal rule. The ruin probability Ψ and the allocation π to the risky asset decrease as the wealth *w* increases. In particular, probability of ruin is zero for safe wealth level ω^{u}. Thus, once wealth reaches the safe level ω^{u}, the individual invests everything in a risk-free asset. Both π and Ψ decrease as life expectancy or consumption decrease.

The basic intuition behind this process is that the allocation to equities is higher when the investor is behind the goal (the probability of ruin is higher). Conversely, the allocation to equities decreases when the investor is on target. As shown by Browne [1997], a similar type of goal-seeking strategy is found to be stochastically dominant when using unconstrained leverage, guaranteeing that no other strategy can achieve the goal with lower probability of ruin.

### Benchmark Strategies

As a way of evaluating the performance and soundness of our strategy, we test the performance of diversified funds with various asset mixes using equities (S&P 500 Index), bonds (Lehman Aggregate 10-year and 10-year U.S. Treasury bonds), and TDFs as calculated by Morningstar (both to and through retirement). “To” TDFs refer to those with glide paths that end at retirement. “Through” TDFs refer to those with glide paths that continue after retirement date, usually with gradually reducing equity exposure. Benchmark strategies tested include the following:

• All cash

• Combinations of equities and bonds from 100% bonds to 100% equities in 10% increments (100% bonds/90% bonds, 10% equities/80% bonds, 20% equities, etc.)

• Morningstar through-retirement TDFs

• Morningstar to-retirement TDFs, which have a constant equity allocation of 31%

The shape of the through-retirement TDF is shown in Exhibit 3.

### Mortality

As noted, marriage is less popular today than it has been in the past. Nevertheless, we assume 50% of Americans still need their retirement funds to last as long as one spouse survives. Accordingly, the joint mortality experience of couples is estimated using longevity factors calculated from the U.S. Social Security Administration 2014 Period Life Table for age 65, assuming spouses of the same age.

### Hazard Rate

For the hazard rate λ(*T*), the risk of dying soon, we use the estimation choice that Moore and Young [2006] suggested offers the best results with low probability of ruin:

where *E*(*T*) is the expected lifetime estimated from mortality tables. The rapidly increasing risk of dying within a year as each year passes is reflected by this function.

### Capital Market Assumptions

To be relevant, we need to make plausible capital market assumptions. We do not expect the future to experience the same 36 years of declining real interest rates that has been the history of the most recent past, so we focus sampling on the periods from January 1936 to September 1981 and December 2012 to December 2016 that feature low to rising rates. Our sampling methodology selects monthly returns from the pool of historical returns without replacement. We draw contemporaneous equity, bond, inflation, and so on, to preserve the correlation structure in the historical data (Exhibit 4).

### Benchmark Fixed Withdrawal

We set the fixed withdrawal amount as a percentage of the first-year capital adjusted only for inflation. Annual payouts are tested as a percentage of rule-defined capital from 1% to 10% in 0.1% increments for each strategy.

### Variable Spending Rules

Variable spending rules try to minimize the number of spending cuts while protecting capital. An excellent survey of these strategies is given by Pfau [2015]. The idea of linking spending (payout) to the performance of the portfolio—allowing an increase in the rate when markets are buoyant, and vice versa—is a sound one. These are the strategies we have chosen to test:

1. Bengen: Best known for his introduction of a 4% annual withdrawal rate, Bengen [1994] also described a version that effectively creates a band around first-year withdrawal—a 90% floor and 125% ceiling. For example, if the first year withdrawal is $10,000, the floor would be $9,000 and the ceiling $12,500. If the portfolio doubles in year two, the withdrawal would be limited to $12,500, the ceiling amount. We use this latter constrained floor-and-ceiling version of Bengen’s rule.

2. Yale: The withdrawal rate is 80% of the previous year’s withdrawal amount plus 20% of target withdrawal rate based on the portfolio value adjusted for inflation with floor and ceiling constraints. This formula is based on the spending rule for the Yale Endowment.

3. Zolt: Withdrawal rates are not adjusted for inflation when remaining wealth falls behind the target value calculated under a fixed-return assumption and 45-year time horizon. The goal is defined as a target rate of return that ensures that spending needs can be met. It is then compared to the historical portfolio return to determine whether the inflation adjustment in that period can be made. We use Zolt’s assumptions (horizon: 45 years; return: 8.60%; inflation: 3%; real return: 5.6%) to calculate a critical path for how much wealth should remain in each year of retirement. For any year in which remaining wealth is higher than the critical number from this calculation, spending adjusts for inflation. Conversely, when the wealth falls below the critical number, no inflation adjustment is made.

4. Guyton: This strategy suspends inflation adjustment and reduces or increases withdrawals based on prosperity, capital preservation, and modified inflation adjustment rules. The starting value of the initial withdrawal rate varies and depends on the retiree’s desired level of spending. Guyton stipulates three rules. First, the prosperity rule increases spending by 10% for any year in which the current withdrawal rate falls below 80% of its initial level. For example, a $100,000 portfolio and a 3% withdrawal rule suggest $3,000 will be withdrawn. Assume the portfolio rises 50% to $150,000. The $3,000 withdrawal amount represents 2.0% of the portfolio’s value, or 66% of the initial level. Being below 80% of the initial level, the prosperity rule would increase spending by 10% (to $3,300). Second, the capital preservation rule cuts spending by 10% if the current withdrawal rate rises to more than 120% of its initial level. For example, if the same $100,000 portfolio falls 20% to $80,000, the $3,000 initial withdrawal amount is 3.75%, or 125% of 3%, which is above 120%. Spending is cut 10% to $2,700. Third, the inflation adjustment rule includes a spending adjustment for inflation unless the portfolio had a negative return in the previous year and this year’s withdrawal rate is higher than the initial withdrawal rate. For example, if the portfolio return in the previous year is -5.0% and the current withdrawal rate is 3.5% compared with the starting withdrawal rate of 3%, the inflation adjustment is skipped. Overall spending can increase faster than inflation when the markets are doing well and can fall when the portfolio is losing value.

5. Hybrid: This is a modified version of Zolt’s approach, which we find attractive because income never declines in nominal terms and the penalty—forgoing inflation adjustment—although potentially hazardous during periods of high inflation, has proven viable in actual implementation (Ontario Teachers’ Pension Plan [2012]). We modified the definition of being behind the goal as defined in Zolt’s rule by forgoing the inflation adjustment if the implied probability of ruin (calculated annually) exceeds an acceptable threshold, which we set at 10%.

## OBSERVATIONS

### Strategies

Exhibit 5 shows the withdrawal rates achievable by various strategies over a range of probabilities of ruin. The DCR strategy dominates other strategies, except for 100% equities over a 6% withdrawal rate, a strategy unlikely to be palatable in retirement.

We calculate the withdrawal rates possible for each strategy and the accompanying probability of ruin. Note that bond withdrawal rates are actually below those of cash because in a low and rising interest rate environment real interest rates are negative and bond prices are falling.

The DCR strategy offered the highest withdrawal rates of all strategies, including the 60% equity portfolio that represents the upper bound of equity exposure assumed for the DCR strategy.

Our findings are consistent with Pfau’s recommendation of a 3% withdrawal rate in the current low interest rate environment because this rate results in an acceptably low probability of ruin for most investment strategies (Finke, Pfau, and Blanchett [2013]; Pfau [2015]).

### Making DCR Practical

Although the strategy is allowed to invest up to 60% in equities, Exhibit 6 indicates that, in realized paths, significantly less risk is assumed. Importantly, the strategy has the flexibility to adjust the risk exposure if higher income is required. However, even more aggressive income targets result in more conservative risk profiles than the maximum 60% equity.

We explore how frequently the strategy resorts to maximum equity exposure. Exhibit 6 shows the range of percentiles of equity exposure by age. Of interest is how infrequently the 60% equity exposure was needed, particularly for lower target income strategies.

Another measure of an implementable strategy is whether the implied asset mix changes are practical. Exhibit 7 shows that the change in portfolio weight for the DCR is in the range of ±6 percentage points per annum, narrowing in later years. This low range of asset allocation changes shows that wild swings in asset allocation are not indicated, and the only real constraint to make the approach practicable is the maximum 60% equity weight.

### Add Spending Rules

The actuarial value of withdrawals depends on when they are made. To make the spending rules comparable in terms of efficiency, we need to normalize for mortality risk and take into account residual capital. Overall, we propose the following approach to normalization:

• To compare spending policies fairly, we give greater weight to withdrawal amounts available earlier in retirement compared with those available later. Mortality weighting suggests that $1 available at age 70 is weighted 98.1% (the probability that a 65 year old will survive to age 70) and $1 at age 100 is weighted 0.9% (the probability that a 65 year old will survive to age 100). We sum the amounts to calculate the total mortality-weighted withdrawals for each policy.

• We calculate average capital weighted by mortality to estimate legacy. Because capital becomes legacy after death, we weight capital in each year by the complement of survival probability and take the average across all years. For example, at age 80, $100,000 capital has a 15% probability of becoming legacy. The legacy value is thus $15,086 for that year. At age 100, $100,000 capital has a 99% probability of becoming legacy, so the legacy value is $99,180. We take the average of legacy amounts for each year.

• Having used survivor-weighted withdrawals to more fairly value cash flows and mortality-weighted capital to more fairly value the residual or legacy amount remaining, we sum the mortality-weighted withdrawals and survival-weighted residual capital legacy to calculate the total longevity-weighted income stream of each policy plotted with its respective probability of ruin compared with the fixed 4% fixed withdrawal (Exhibits 7 and 8). The approach has a motivation similar to that of a discounted cash flow approach in which we incorporate the cash flow (retirement income) as well as the terminal value (legacy) to arrive at a single value. In this case, the flows are mortality weighted to account for the unknown time horizon.

The fixed spending rule generates longevity-weighted income (withdrawal and residual capital) comparable to that of the other five rules (>$76,000) but with a higher probability of ruin (5.8%). The hybrid rule, which is a modified version of Zolt, demonstrated the highest mortality-weighted income stream at $84,249 with a probability of ruin of 1.9%. Spending that adjusts to portfolio performance makes a valuable contribution to reducing the probability of ruin.

Exhibit 9 shows that the DCR strategy successfully provides an average of 51% more income than a TDF (with a glide path that continues to decline through retirement) when combined with the hybrid spending rule over a probability of ruin ranging from 1% through 25%. This compares with an average of 47% more when using fixed withdrawal rates alone (mortality weighted, as described). From the perspective of retirement income, if the TDF could pay out a 4% withdrawal rate on the sample $100,000 portfolio (or $4,000), the DCR strategy could pay out 47% more ($1,880) for a total of $5,880 before applying a variable spending rule. If the hybrid spending rule is used with the DCR strategy, a withdrawal rate 51% higher ($2,040, or $6,040 total) is possible (Exhibit 9).

The DCR strategy combined with the hybrid spending rule produced an average of 40% more income than a conservative diversified fund (40% equity and 60% bonds) over the same range of probabilities of ruin. This compares with 35% more income from DCR alone.

## SUMMARY

The main purpose of this study has been to devise an investment strategy and a spending rule to maximize the sustainable withdrawal rate for the retiree while minimizing the chance that he or she runs out of money before dying.

We found that the practice of establishing a policy asset mix and routinely rebalancing to a fixed allocation, like a diversified fund, or to a static laddered mix, like a target date fund, can expose retirees to periods of unnecessary risk at a period in their lives when time is a more rapidly diminishing asset. Reducing exposure to these periods of added risk, which we define as market volatility, added incremental returns and allowed for higher withdrawal rates with the same risk of running out of money within the test environment.

In the Strategies subsection, we observed that targeting a specific income replacement goal using GPS-like navigation (dynamic decision theory) coupled with road hazard–like protection (constant risk) reduced the probability of ruin to 5.8% compared with 15.6% for a conservative diversified (40% equity, 60% bond) portfolio and 24.8% for a target date fund (TDF) for a 4% annual withdrawal rate. Viewed another way, DCR adds 35% over the sustainable withdrawal rate of the diversified (40% equity, 60% bond) portfolio and adds 47% to the sustainable withdrawal rate of target date funds for the same likelihood of ruin from 1% through 25%.

In the Add Spending Rules subsection, using variable spending schemes further reduced the likelihood of ruin for DCR from 5.8% to a range of 1.5%–2.0%. These variable spending strategies also reduced the likelihood of ruin for the diversified 50:50 portfolio to 4.3%–6.3% (from 15.6%) and for target date fund portfolios to 8%–12.5% (from 24.8%). The hybrid rule provided the highest mortality-weighted income stream.

## CONCLUSION

Managing money in retirement presents many challenges, primarily because there are so many variables and unknowns. How long will I live? How healthy will I be? How will my needs change? When and by how much will capital markets move relative to my need for income and capital? These personal variables compound the challenge for asset managers, who must find a scalable mass-customized solution to address what Martellini [2016] described as cross-sectional and time series dimensions: addressing the needs of different investors starting at the same time and others starting at different times, respectively. The decision-making process is complex. However, by using “safe” and “risky” portfolios (Martellini [2016]) and taking the further step of maintaining each at constant risk to mitigate the impact of sequencing risk, both dimensions can be addressed. Investors must control what they can because so many things are beyond their power. The self-driving automobile must also deal with uncertainty and an array of choices, but decisions always focus on getting safely to the destination. We show that investors can learn from this focus. The practical application of decision theory to an asset allocation that aims to keep the portfolio travelling toward income goals rather than trying to maximize returns differs from common asset management practice. Rebalancing to constant risk to protect portfolios rather than to a predetermined asset mix is also different from what most of the industry does. These two approaches, which address protection and destination, are things investors can control in a list of factors that they cannot.

## APPENDIX

### SEQUENCE OF RETURN RISK

We carry out a simple simulation based on the 20-year period from December 31, 1996, to December 31, 2016. We take 20 annual total returns for S&P 500. The average geometric return for the period was 7.7%. We also use the constant-risk-adjusted S&P 500 that returned 8.4% with lower downside deviation at 0% (10.5% versus 12.6% for S&P 500). To assess the impact of sequencing risk, we change the order of the 20 annual returns. There are 2.4 × 10^{18} possible orderings. We generate 100,000 random sequences from the 2.4 × 10^{18} possible ones. We assume the investor starts with $100,000 and withdraws 5%, or $5,000, at the end of every year. For each of the simulations, we calculate the remaining capital in the last year. Pfau [2013] showed that in the presence of sequencing risk, the range of sustainable withdrawal rates can be from 1.9% to 10.9%.

A tighter bound on the outcomes would suggest reduced sequencing risk. To assess the impact of sequencing risk, we calculate the money-weighted return of the worst one percentile of outcomes as well as the coefficient of variation of the last year’s capital. Sequencing risk results in dispersion of possible outcomes in the last year that can be estimated by the coefficient of variation. To account for the higher historical return achieved by the constant risk strategy, we standardize the dispersion of outcomes caused by sequencing risk by dividing the standard deviation over the mean return. This allows for the comparison of outcome dispersion between the S&P 500 and the constant-risk S&P 500 strategies. The strategy with lower sequencing risk would have a lower coefficient of variation.

The worst 1-percentile money-weighted return for the S&P 500 investor was 2.1%; for the constant-risk S&P 500, it was 3.4%. Exhibit A1 shows that the capital over time in the worst 1% of outcomes is higher throughout the time horizon for the Constant Risk strategy compared to the S&P 500. The coefficient of variation of the last-year capital was 26% for the constant-risk S&P and 30% for S&P 500. The tighter range of outcomes shows that a simple adjustment of constant volatility can mitigate the impact of sequencing risk.

Kitces [2014] described a decision-rule-based asset allocation that limits the sale of equities when the stock market is down, drawing from cash and bonds in these circumstances. This increases portfolio risk that addresses longevity risk, a potentially good thing, but it does so in an uncontrolled manner that is dependent on the size the withdrawals.

## ENDNOTE

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^{1}The S&P risk control methodology estimates realized standard deviation as the maximum of two exponentially weighted moving averages measuring short-term and long-term variance.

- © 2017 Institutional Investor, Inc.