### A New Framework for Comparing Withdrawal Strategies

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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.