Coping with Sequence Risk: How Variable Withdrawal and Annuitization Improve Retirement Outcomes

Monte Carlo analysis

A popular research approach for testing the sustainability of retirement withdrawal strategies involves using Monte Carlo simulations. The random generation of annual investment returns will produce some “bad” sequences, some “good” sequences, and others that are more average. This is the approach I will use to test fixed versus variable withdrawal strategies. However, to provide supplementary information on sequence risk, I’ll also run Monte Carlo simulations where I tilt the return generation toward good and bad sequences, but keep the lifetime average returns about the same.

Other key aspects of my modeling approach are:

• The assumption of lower-than-historical returns for both stocks and bonds
• Stochastic mortality to generate a randomized year of death for each Monte Carlo run rather than assuming a fixed retirement period

The modeling approach and assumptions are described in detail in the Appendix.

The particular example I use is for a 65-year-old female with a remaining life expectancy of 25 years. I assume she has $1 million in retirement savings allocated 50/50 stock/bond and will be receiving $30,000 per year from Social Security. I assume her essential living expenses are $50,000 per year, so she needs to generate at least an additional $20,000 per year from her savings portfolio to avoid plan failure. Her primary goal is to generate retirement income; leaving a bequest is of secondary importance. The analysis is pre-tax, and all dollar figures are in 2017 real dollars.

Variable versus fixed withdrawals

The chart below compares performance measures for the 4% rule (fixed) and an endowment approach (variable) based on withdrawing 4.9% of the portfolio at the beginning of each year. I highlight in bold what I consider to be the three key measures of outcomes. I show the comparison for normal sequences using the standard Monte Carlo approach as well as where the return generation has been tilted toward bad and good sequences.

Average bequests are essentially the same for the two withdrawal strategies. What is dramatically different between the two strategies is the average shortfall, which measures the combined impact of the percentage of Monte Carlo simulations that fail to cover essential expenses and the dollar magnitude of such failures. The shortfalls are roughly 10-times worse under the 4% rule, and this is true for normal, good, and bad sequences. What’s happening here is that the 4% rule is generating retirement withdrawals that ignore what’s happening with the underlying portfolio. So if things are going badly, withdrawals continue at the same real level until the portfolio is depleted. With the endowment withdrawals, when things go so badly the dollar amount of withdrawals is reduced, there is less late-in-life shock.

Based on these measures, I reach the same conclusion as Cotton that variable beat fixed withdrawals. There are other takeaways from the chart.

The good and bad sequences were calibrated to produce the same average lifetime returns, but there is a clear difference in outcomes for bad sequences with lower early returns than for good sequences. This is simply another way of demonstrating that for retirement outcomes both the level of returns and the sequence matter.

The general superiority of variable over fixed withdrawals applies regardless of the type of sequence – normal, bad, or good. The only exception is in the bequests where, as Cotton also demonstrated, variable withdrawals produce a tighter distribution of bequests. As a result, the good sequences average bequest is less for the endowment approach.

Cotton noted that the variable approach resulted in more year-to-year variability in withdrawals. Although not shown in the chart, I measured variability – for the normal sequences it was 0.6% for the 4% rule and 4.6% for the endowment approach. One might wonder why the 4% rule with steadier withdrawals produced an inferior CE. The answer is that 28.2% of the Monte Carlo simulations ran out of money, causing a steady $70,000 of annual real consumption ($30,000 plus 4% of $1 million) to drop precipitously to $30,000. This resulted in a greater CE penalty than endowment withdrawals bouncing around from year-to-year in a narrower range.

Improving on the endowment method

Lifecycle economics is allied with the concept of utility maximization via consumption smoothing. For retirement withdrawals this means spreading consumption over the expected remaining lifetime, or in very approximate terms determining consumption by dividing savings by expected remaining life. Since expected remaining life shortens as one ages, this implies a path of increasing withdrawal rates, rather than the level percentage under the endowment method. One can attempt exact calculations using an appropriate actuarial table and assumed investment returns, or a simpler approach is to rely on the IRS requires minimum distribution (RMD) tables. Utilizing the table that applies for most individuals, the withdrawal rate is 3.65% at age 70, 5.35% at age 80, 8.77% at age 90 and tops out at over 50% at extremely advanced ages.

In the following chart, I compare outcomes under endowment and RMD approaches.

For this comparison I raised the endowment percentage to 5.2% to roughly match average consumption under normal sequences. The results we see are a slight increase in CEs and slight decreases in average bequests – favorable if the retiree’s main focus is consumption. But the biggest differences are in the average shortfalls and these strongly favor RMDs, particularly for normal and bad sequences. (I should note that this is a single test with one endowment percentage and one asset allocation, so further testing would be needed for a full comparison of RMDs versus the endowment approach.)

Substitution of SPIAs for bonds

In the chart below, I show the results of purchasing an inflation-adjusted SPIA at age 65 to produce $20,000 of annual income to fill the gap between Social Security and essential expenses. Based on pricing from the annuity information service CANNEX, such a SPIA would cost $468,384, and doing a fixed income substitution of the SPIA for bonds would use up most of the 50% bond allocation on the $1 million of savings. RMDs were applied to remaining savings.

With the SPIA purchase we significantly increase consumption and CEs while bringing down average bequests. Again, if the client’s primary focus is consumption, this is a positive change. We are able to push bequest money into consumption and accomplish this while eliminating the negative outcomes. We get these positive results regardless of the sequence of return scenario.

Final word

Besides demonstrating how variable withdrawals can mitigate sequence of returns risk, I’ve shown that the choice of variable approach and annuitization can make a significant improvement. There a lot of variable approaches that researchers have proposed, so it would be worth further testing, particularly under bad sequences, to see which variable approaches perform best.

Joe Tomlinson, an actuary and financial planner, is managing director of Tomlinson Financial Planning, LLC in Greenville, Maine. Most of his current work involves research and writing on financial planning and investment topics.

Appendix

This modeling is based on a 65-year-old retired female with a remaining life expectancy of 25 years and $1 million in savings that she can dedicate to generating retirement income. Her basic living expenses are $50,000 per year, increasing with inflation, and she will receive an inflation-adjusted $30,000 annually from Social Security. She will utilize withdrawals from savings to cover the gap between basic living expenses and guaranteed lifetime income and additional withdrawals can be used for discretionary spending. The analysis is pre-tax.

Investment assumptions: For the Monte Carlo analysis, stocks are assumed to earn an arithmetic average real return of 5% with a 20% standard deviation, and bonds (TIPS) are assumed to earn 0% with a 7% standard deviation. These returns are significantly lower than historical averages, reflecting current interest rates and a lower-than-historical equity risk premium. The modeling assumes a 50/50 stock/bond mix, and allocations are rebalanced annually to maintain this allocation.

Modeling sequence of returns risk: I compare “bad” sequences to “good” sequences. I create bad sequences by subtracting 5% from the Monte Carlo generated first-year stock return and grade this reduction to zero at half of life expectancy and then grade up from there to a 5% addition at life expectancy. The good sequences are the mirror image. The basic idea was to create bad and good sequences that produced similar average returns over expected lifetimes, but with markedly different tilts.

Withdrawal methods: I test following approaches to generate funds for retirement consumption:

• • The venerable 4% rule, where the initial annual withdrawal is set at 4% of the initial savings portfolio
  • An approach with annual withdrawals equal to 4.9% of the current portfolio (with the 4.9% chosen to produce roughly the same lifetime average annual withdrawals as the 4% rule)
  • An approach with annual withdrawals equal to 5.2% of the current portfolio (with the 5.2% chosen to produce roughly the same lifetime average annual withdrawals as IRS required minimum distributions--RMDs)
  • An approach where withdrawals equal RMDs (and withdrawals prior to age 70 at the age-70 percentage – 3.65%)

Modeling methodology: For each retirement income approach that I model, I generate 10,000 simulated retirements. Withdrawals each year are determined based on the particular withdrawal approach being modeled. Investment returns for each year are generated randomly based on average return and standard deviation characteristics. The date of death for each of the 10,000 simulations is randomly determined based on the Gompertz mortality function.

Although methods that determine withdrawals as a percentage of the current portfolio balance will never completely deplete savings, I use a slightly different approach. If calculated consumption is not sufficient to cover basic living expenses but there are remaining savings, I take from the savings to cover the expense gap until savings are depleted. This way I can determine the percentage of simulations where savings are depleted.

Performance measures

Average consumption: Consumption equals Social Security income ($30,000) plus annual withdrawals and SPIA payments, if any. For each of the 10,000 simulations, I compute the lifetime average of the annual consumption amounts and then take the average of these averages.

Average CE consumption: This certainty equivalent (CE) measure is based on an economic utility calculation that converts variable year-by-year consumption into a level amount that the recipient would view as equivalent. The CE amount depends on what economists refer to as the recipient’s level of risk aversion. For example, if annual consumption bounced around randomly between $60,000 and $80,000, an individual with low risk aversion would demand close to $70,000 if offered a trade to level consumption. A highly risk averse individual would be willing to accept an amount closer to $60,000.

For this analysis, I have assumed a medium aversion to variable consumption – an individual would be willing to accept annual consumption of a level $66,600 in trade for consumption that bounced around randomly between $60,000 and $80,000. This translates to a risk aversion coefficient of 5 based on a CRRA utility function of the form U = (1/(1-RA))*C^(1-RA) to convert consumption into utility. For each of the 10,000 Monte Carlo iterations, I convert each year’s consumption into utility, average the utilities based on the number of years in each iteration and convert to a CE using the inverse of the utility function.

Average bequest: Each of the 10,000 simulations produces a remaining savings at time of death, which can be zero or a positive amount. I show the average of the 10,000 bequest amounts.

Average shortfall: For each simulation that “fails,” I calculate the amount of additional funds that would have been needed to pay for basic living expenses until the end of life. The sum of these amounts is divided by 10,000 to determine an average shortfall for all the simulations (including those with zero shortfall). This is a more useful failure measure than failure percentage, because it incorporates both frequency and magnitude. However, I also show failure percentage because it is a more commonly used measure.

Failure percentage: This represents the percentage of the 10,000 simulations where savings are insufficient to fully pay for the assumed $50,000 of basic living expenses.

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