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A number of different strategies are available for living off assets in retirement. This article examines the performance of nine different decumulation methods that can be categorized into three broad classes. In the first class, constant-rate methods, the annual withdrawal is obtained by multiplying the amount of capital by a percentage, traditionally 4%. In the second class, mortality methods the mortality tables are used to determine anticipated lifetime, which decreases with age and is used to increase the percentage withdrawal each year. The third class, mortgage methods, increases the withdrawal rate over time based on maximum lifetime rather than expected lifetime.
Tilting – A directional nudge
A universal problem with decumulation is that we never know in advance what market returns and longevity will be. A good decumulation strategy adapts to market conditions by adjusting the withdrawal rate to reflect observed results and adapts to longevity improvements by conservatively preserving residual capital and income.
One option for adjusting withdrawals to market returns is to modify (tilt) the withdrawals based on the current value of the remaining capital relative to a metric that measures whether the client is on track. A positive tilt spends not only less, but also proportionally less when the portfolio value is below the metric and proportionally more when the portfolio is above the metric. This sacrifices income stability in favor of capital stability but maximizes income if returns are favorable. A small positive tilt helps to gently nudge the remaining portfolio back to the desired trajectory, albeit at a cost of more income variability. Negative tilts sacrifice capital stability in favor of a smoother income stream.
A simplified example helps explain this. If we assume a constant rate of return and income is drawn from one of the constant base rate strategies (see the equations for tilt given below), then, if tilt>0 the system eventually comes to a new equilibrium where the portfolio and income are in balance even if the constant base rate (4% in the example) does not match actual returns. Figure 1 solves the relevant equations for the equilibrium income for different values of return and tilt, assuming a base rate of 4%. For 0 tilt (the endowment method), the capital and income slowly drift upward or downward with no limit. The endowment method still works for retirement withdrawals because the drift is slow relative to most lifetimes.
As tilt increases above zero, a new equilibrium is eventually reached. At tilt>2 the equilibrium income approaches the actual rate of market return. At tilt of positive infinity spending equals the actual rate of return. In the real world, where returns are highly variable and equilibrium may take more than a lifetime to reach, a positive tilt gives a gentle nudge (0 < tilt < 1) or a hard shove (tilt > 1) back toward capital stability and sustainable income.
Positive tilt, however applied, is a form of self-correction to the portfolio and income. It reduces longevity risk and acts like a good financial advisor to put the portfolio back on track.
Figure 1. Equilibrium income from an initial million-dollar portfolio
(if the base withdrawal rate is 4% and market returns are as shown given different amounts of tilt).
My previous article introduced the concept of tilting. I intentionally chose the original implementation of tilting for simplicity. In this article, tilt is implemented differently, by taking a ratio of current capital divided by the capital that would keep the client on track and then raising the ratio by an exponent. A positive exponent preserves capital at the expense of income stability, and a negative exponent preserves income at the expense of capital. Two of the most common retirement income methods (4% rule and endowment method) occur as special cases of the constant base rate equation with tilting. For perspective, the variations provided by tilting are compared to three mortality based methods, a popular strategy not amenable to the concept of tilting. The mortgage method is illustrated with and without tilt.
Methods
Four of the methods simulated are based upon a constant rate of withdrawal (4%) with different values for tilt. The tilt is obtained by taking a ratio of current real capital divided by the capital the client should have at that point in retirement and taking the ratio to an exponent called the tilt (see appendix). A constant annual amount (the 4% rule) is obtained when tilt equals negative one. The endowment method, 4% of current balance, is obtained when tilt is equal to zero. To more fully illustrate the impact of tilt values of +1/3 and -1/3 are also included. Tilt could be set anywhere from minus one to positive infinity.
Three methods based upon statistically derived remaining lifetime estimates are labeled as mortality methods. The concept behind each of them is that the information contained in mortality statistics can be used to enhance or optimize withdrawal rates. The simplest method is to make the withdrawal rate equal to one divided by anticipated lifetime plus a margin of safety. Another option is to use the Internal Revenue Service Required Minimum Distribution tables. A third option is to base the withdrawal rate on an average of remaining life and maximum life using real interest rates.
The final class of methods is labeled as mortgage. The fixed term annuity equation is used to set the annual income and create a table of how much capital should remain each year if the client is on track (just like remaining principal in a mortgage). The table is used as the basis for tilting. The annual withdrawal rate is analogous to the fraction of a mortgage payment going toward the principal. Importantly the time for the annuity is based upon maximum lifetime, not anticipated lifetime. The withdrawal is revised each year based upon the remaining capital and assumed interest rate.
Simulations
Monte-Carlo simulations were used to compare the different withdrawal options. All calculations are in real (constant) dollars. Lifetime is based on the single female mortality tables with a retirement age of 65 and income taken at the beginning of the year. The mortality results for a single female are intermediate between single male and male/female couple, making the single female a good basis for comparison of methods.
Returns are based on Vanguard, December 2015 projections from an 80% stock, 20% bond portfolio. The Vanguard Figure IV-2 projections are roughly consistent with annual real return of 05.3% and standard deviation 11%. A high stock portfolio is assumed because, in general, bonds in retirement portfolios are inferior to annuities. The high standard deviation will result in a wide range of outcomes that illustrate even higher and lower returns. Simulations were also performed after reducing the annual return to 0.033 and gave the same conclusions.
Results
Comparison of different retirement income methods is difficult. Two aspects are important:
- Remaining capital at any point in time provides not only money for bequests, but also income security. If significant capital remains and income is inadequate, the client can always capitulate and purchase an annuity for more income or just “cheat” on the income plan. If all the capital is gone, so are the options. In the absence of an annuity, capital is safety.
- Income amount and income stability are critical. Does income vary from year to year? Is there a secular change in income over time? Can income ever go to zero? Is longevity risk an issue?
The initial comparison uses the CRRA utility function with risk aversion of four (Figure 2). This function is effective at sorting through the more than two million person-years simulated for every method and flagging periods of low income or capital (i.e., identifying risk). Two functions are applied: income over time and capital over time. They are labeled respectively as income and safety. The income stream is time discounted at -2%/year to reflect the observed pattern of lower spending with age. The upper points moving from left to right represent the optimal methods, the “efficient frontier.”
With these low return projections, the positive tilts are all above and left of comparable un-tilted or less tilted options (i.e., cyan is to the left of red, and green is to the right of red). The constant income does not plot on the graph since it sometimes leads to bankruptcy (utility function goes to minus infinity). It is not an acceptable method based on risk. The RMD method is not competitive and the 1/(life+7) method requires a large drop in capital safety for a small income gain. Three of the nine methods can be eliminated from further consideration based on risk.
Figure 2. Constant Relative Risk Aversion (CRRA) utility function with relative risk aversion of 4, average of all simulated years.
Figure 3 represents median (50%) results, the expected case. The capital is the median amount remaining at death, which occurs at any age between 65 and 105 based on mortality. Again the upper surface is the efficient frontier. Any point that is below and to the left of another is dominated by the upper-right method.
The best methods display a clear tradeoff between income and remaining capital. The positive tilt methods have all moved in the direction of increasing income (green to the left of red, cyan to the right of red; yellow is to the right of brown). The constant method (-1 tilt, AKA the 4% rule) works well with median investment results, although the amount of remaining capital is low relative to income.
Figure 3. Median of mean lifetime income versus median capital at death.
Figure 4 has the 5% worst-case results – the most important results if one is concerned with risk. The constant withdrawal method has supplied income the entire time but is almost broke, an unacceptable risk. The positive tilt methods have all migrated up and to the left to preserve capital. The negative tilt methods move to the right to preserve income. The mortality-plus-annuity method is significantly off the efficient frontier. The other two mortality methods give up $100-200K of capital for only about $2K/year in additional income, a poor tradeoff. All of the mortality methods perform poorly under unfavorable (mostly negative in this case) market returns.
Figure 4. Lower 5% final income versus lower 5% final capital.
Withdrawal methods should handle better than anticipated market results as well as poor ones. The 90% results reflect markets that performed better than anticipated. All the methods work well with a tradeoff between income and remaining assets.
The positive tilt methods have shifted down and to the right to automatically increase income. The constant method has accumulated significant assets since it doesn’t respond to market results. The efficient frontier is dominated by the three mortality methods. They all tend to delay peak spending until the 80’s and therefore benefit from good market returns, at least by these metrics.
Figure 5. 90% mean lifetime income versus 90% final capital.
Figure 6 is the 10% income over time during the years of retirement up to age 105 (105 = 65 at retirement + 40 years of retirement). These are not time histories of individual clients, but rather a slice of the income distribution of the survivors. They represent increasingly low overall probabilities at advanced age because few women live that long. The figure illustrates the problem with the three mortality methods that was picked up by the CRRA utility function, the median results and the 5% results.
All three mortality methods form a camel’s hump shape over time with low withdrawals at first, highest withdrawals in the 80s and the potential for running short of money in the 90s if the client lives that long. The mortality-plus-annuity method, given current real interest rates, provides low income until relatively late in life. These three methods are all inconsistent with data indicating slow decreases in spending with age.
The constant withdrawal method has gone broke for the lower 10% of survivors after 31 years of retirement. The endowment method and +1 tilt have a secular decline in income; however, these methods have relatively high levels of remaining capital and thus the option of purchasing additional income at any time (e.g., a late in life annuity). The un-tilted and tilted mortgage methods supply a slowly declining income for approximately the first 15 years, and then plateau; they also have significant remaining capital at death (Figures 3-5). The positively tilted methods are nearly flat at advanced ages because they have approximately equilibrated the withdrawal rate to the actual (lower 10%) returns, thereby negating longevity risk.
Figure 6. Lower 10% income during each year of retirement.
All the mortality based methods have a camel’s hump problem because the mortality curve doesn’t follow the same shape with time as a fixed-term annuity and thus leads to a problem of too much income in the middle with lower amounts at the beginning and end. The mortality methods are based on the subtle error of applying concepts that work only for groups to the isolated individual. The mortgage-plus-annuity method as implemented follows the average of the mortality and annuity curves, giving in-between performance. The mortgage method follows the fixed term annuity equation shape and conservatively increases income based on a maximum life expectancy. The simple tilts also all work well as long as the tilt does not become too negative (near -1).
The successful methods (constant rate and mortgage), combined with tilts >-1, offer a clear tradeoff between capital and income stability. They eliminate sequence-of-returns risk and minimize longevity risk. A future article will illustrate how different amounts of tilt and single premium immediate annuity (SPIA) can be used to place clients anywhere desired along the capital preservation versus income stability continuum.
Appendix
Constant base rate
The withdrawal rate at any point in time is given by: rate = base rate *current capital* (current capital/initial capital)^tilt
Where the base rate is the chosen withdrawal rate, usually 4%, and the tilt is an exponent. The constant rate methods in this article are all shown with a 4% base rate.
This formula has convenient simplifications. If tilt = -1 then it reduces to: rate = base rate * initial capital, a constant amount per year. If the base rate is 4% this is the classic 4% rule. Label: “constant,-1tilt”
If tilt=0 it reduces to: rate = base rate * current capital, usually called the endowment method. Label: “endow, 0 tilt”)
If tilt is positive then we have an inverted withdrawal rate, where when capital is above the initial capital not only more, but also a higher percentage is taken out. Labels: “+0.33 tilt”, “-0.33 tilt”
Mortality
A different class of strategies attempts to take credit for limited mortality to increase income. As the client ages, the remaining lifetime decreases. Instead of sticking to a constant base rate (e.g., 4%), why not just use rate = (1/expected lifetime)? Since half of each cohort will outlive the expected lifetime, typically a margin of safety is added: rate = 1/(expected lifetime + n), where n is the extra years of safety. In this article n=7 is used, giving: rate = 1/(expected lifetime +7). Label: “mortality,1/(life+7”
A variant of this method uses a virtual annuity based on life expectancy plus a safety factor, the real interest rate, and the current capital updated each year. The source paper is more a discussion of options than a discrete method. For comparison, a real interest rate of 0.05% was assumed, and ((expected lifetime + 120)/2) was used as the annuity time. The conclusions are not sensitive to the interest rate chosen within the range of current real rates. Label:“mortality+annuity”
A third mortality strategy is to use the IRS Required Minimum Distributions table. If a client has most of his/her wealth in a regular IRA/ 401K or 403B then this is the amount that must be withdrawn every year after age 70.5. Why not just make RMDs the spending plan? The RMD tables don’t extend for ages younger than 70.5. From 65 to 70 the initial table entry of 1/27.4 years was assumed in the simulations. Label: “mortality,RMD”
Mortgage
The mortality methods all update annually based on remaining lifetime. Consider instead taking the withdrawal rate from a fixed-term annuity started at the date of retirement based on maximum lifetime as opposed to expected lifetime. This follows the slope of a fixed-term annuity instead of the slope of mortality. The withdrawal rate is the fraction of payment going to principal in a fixed mortgage. It starts off low and gradually increases with time.
The annual withdrawal, shown in Excel format is: rate = (=-PMT(0.03525,120-age at retirement, current capital, 0, 1)) * (current capital/capital that should remain)^tilt
The capital that remains at any future date can be calculated at the start of retirement and used each year to decide if the client’s retirement is on track (i.e., as the basis for tilting). This is calculated from the present value equation for a fixed term annuity (=-PV(0.03525,120-current age, 40000,0,1)). The fixed-term annuity equations are given as Excel functions for brevity and to illustrate how to apply them, the simulations use Mathematica. Spreadsheet examples of how to apply all the tilted methods are here.
For fair comparison, the interest rate assumed could either be the same 4% as the constant rate methods or adjusted to provide the same initial income as the constant rate methods ($40K). The latter was used in the simulations (3.525%); both work well. The assumed interest rate could be changed over time, but it is constant in these simulations. This is referred to as the mortgage method because most clients are familiar with how mortgages work. A version with no tilt is considered (Label: “mortgage”) and one with +0.33 tilt. Label: “mortgage,+0.33 tilt”
John Walton is an engineering professor at the University of Texas at El Paso, where he has taught for 24 years. He has a Ph.D. in Chemical Engineering. His interest in financial planning came from considering options for his coming retirement.
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