Quantifying the Levers of Retirement Success: An Empirical Study

By in Asset Allocation, Institutional, Retail, Retirement

*Special thanks to Michael Guan at the SystematicEdge blog who wrote most of the code for the following analysis.

According to the Pew Research Center, in 2011 the oldest members of the baby boomer generation turned 65. Further, they assert that each and every day between 2011 and 2030 10,000 baby boomers will turn 65. Currently, just 13% of Americans are age 65 or older, but by the time all of the Boomers have turned 65, 18% of the population will have reached retirement age. Nothwithstanding that the average Boomer claims to feel 9 years younger than his or her chronological age, the tidal wave of aging Boomers will alter the economic and financial landscape in ways not yet contemplated by many people in wealth management.
Not a week goes by that we don’t see an article in a newspaper, trade journal or web journal related to retirement. Most articles relay hearsay rules of thumb, or address a single dimension of the retirement equation in isolation, without regard for how changes in that dimension affect the other parts of the equation.
Safe Withdrawal Rate
At root, what retired or retiring investors care about is their Safe Withdrawal Rate (SWR). This rate represents the pre-tax amount that can be drawn from a portfolio each year, after adjusting for inflation. The amount is typically fixed in real terms as of the investor’s retirement date, to allow retired persons to budget consistent income over time as if they were drawing from a fully indexed pension.
For example, imagine a couple retiring today with $1,000,000 in portfolio assets. They want to ‘pensionize’ their nest egg by drawing a consistent annual income from their portfolio, and adjusting their income each year to account for inflation. Typically, they want to know where to set their permanent income level; but what they are really asking is, “What is my Safe Withdrawal Rate?”
More often, a working couple is trying to discover how much money they need at retirement to fund their lifestyle. A substantial part of the process involves optimizing their taxable situation by deciding which accounts to draw from and when, but once that is complete the couple is left with an amount they wish to draw pre-tax from their retirement portfolio. To answer this question, they also need to know their safe withdrawal rate because their retirement savings target is equal to their pre-tax income needs divided by their SWR.
For example, a couple who needs $100,000 per year, pre-tax, to fund their lifestyle might discover that their SWR of 5%; they would then need $100,000 / 5%, or $2,000,000 in savings.
Jim Otar at Otar Retirement Solutions first proposed an analysis of actual market history as the most realistic model for retirement simulation, which he called an Aftcast. While many others have proposed complex, coherent and comprehensive actuarial frameworks for quantifying probable retirement outcomes, the aftcast framework is perhaps uniquely suited to the problem because it successfully aggregates the feedback dynamics between equity returns, interest rates and inflation.
Most theoretical approaches, even sophisticated ones, model the distribution of returns, interest rates and inflation independently because it is extremely complex to link these variables in a randomized simulation such as Monte Carlo. Monte-Carlo may be an inferior approach however, because the relationship between these variables is very important, and it can change dramatically over time and across market regimes.
Aftcasts link actual investment returns and inflation results through time, which offers the most realistic snapshot of how retirees would have fared throughout history. While this method does not offer the opportunity to create ‘alternative histories’ through randomization, it does preserve the important relationship between the three variables described above. Further, with over 100 years of data, representing over 1300 months, we can still construct a wide variety of actual histories through time.
An aftcast quantifies the experience of a theoretical investor who retires at the beginning of every month through history over a pre-determined investment horizon, with a specified withdrawal rate. For example, an investor is assumed to have retired on Jan 1, 1900, after which he lived for 20 years withdrawing 5% per year, adjusted for inflation using the realized CPI. We assume he was invested in some mix of U.S. stocks (S&P 500) and U.S. Treasuries, and model this retiree’s experience to see if he succeeded in outliving his money. We then perform the same simulation assuming retirement on Feb 1, 1900 and track the outcome. Next we simulate the experience for a person retiring on March 1, 1900, and repeat this process for all rolling 20 year (240 month) periods through August 2012. Altogether, we generate 1088 actual 20-year retirement ‘experiences’ using realized historical monthly total return and inflation numbers.
If we were to plot the total portfolio values through time for all 1088 tests starting at month 1 and ending at month 240, the chart would look like Chart 1. All of the lines that ended the period below the black ‘zero’ line would represent failed outcomes, because in those histories the portfolio ran out of money before the end of the 20 year retirement horizon. Lines ending above the line represent successes; in this case, which assumed a 70/30 stock/bond allocation, a 5% initial withdrawal rate, and a 0.5% annual fee over a 20 year retirement horizon, 93.5 percent of theoretical retirees were successful.
Chart 1. Aftcast for 5% withdrawal rate, 70/30 stock/bond asset allocation, 20 year horizon
Source: Shiller, Federal Reserve
In other words, throughout the 112 years of history that we have for stocks, bonds and inflation, an investor with a 20 year retirement horizon who withdrew 5% per year from a 70/30 portfolio, and adjusted withdrawals each year for CPI inflation, would have had a successful retirement 93.5% of the time. This 93.5% success rate represents the level of confidence that we might assume for this specific set of inputs: 20 years of retirement lifespan, 5% withdrawal rate, and 70/30 asset allocation. The next sections will explore how sensitive this confidence rate is to changes in these inputs.
The 4 Levers of Retirement
As discussed above, the calculation of Safe Withdrawal Rates is impacted by four parameters that the investor controls:
  • Age of retirement
  • Investment risk tolerance
  • Longevity risk tolerance
  • Failure risk tolerance

Some retirees are unable to control when they retire from their job because their employers mandate fixed retirement dates. However, many employees have a fair degree of control over when they want to retire, and those who don’t can often find other gainful activity for some time after they retire from their job. To the extent that employees can control when they start drawing from their retirement savings, this is an important lever to apply in optimizing the retirement equation. Obviously, delaying full retirement has the benefit of reducing the number of years one will need to draw on savings, but working a few more years also often means saving a few more years, and every dollar counts.People tend to worry most about the investment risk of their retirement portfolio, which is just the risk of poor returns to the portfolio over the retirement investment horizon. This risk is generally managed by adjusting the asset allocation in portfolios. Portfolios with a higher percentage allocation to stocks are expected to have a higher return, but this exposure also carries higher risk. This risk can impact retirees in two ways: first, a high exposure to equities carries a risk of large sustained losses that can have profound negative impacts on retirement sustainability, especially if the losses are incurred early on. Secondly, high volatility increases the risk of anxiety and poor decision-making; investors have a habit of panicking at precisely the wrong time, and volatility amplifies this propensity dramatically. However, while the risks of overexposure to equities are important to understand, the risks of underexposure are no less grim. Investors with too little exposure to stocks risk portfolio growth that is too low to support inflation adjusted portfolio withdrawals. The question is, what is the right mix? We will answer this question below in the context of all the other related inputs.

Longevity risk is the risk of living longer, or less long, than is budgeted for in the retirement plan. More precisely, longevity risk is the risk associated with not knowing one’s precise age at death. Obviously one’s time until death can not be controlled in any meaningful sense, but the risk of death can be described in terms of the probability of dying at a certain age. For example, Chart 2. illustrates the probability of dying for a non-smoking 65 year old woman at subsequent ages. Observe that the blue line crosses 50% at between age 86 and 87, indicating that 50% of women will die within about 22 years of turning 65.

Chart 2. Probability of dying, female age 65

Source: Statscan (2002)

Once the risk of death can be quantified at each age, it is possible to choose how confident one wishes to be about longevity. Given that a primary goal for many retirement plans is to sustain an income until death, retirees who wish to be very certain about not running out of funds might set their plan to an 85th or 90th percentile survival horizon. Using Chart 2. for females as a guide, a 90th percentile survival horizon might require a plan to age 97, while a 70th percentile plan would require planning to age 91. Clearly, if this theoretical retiree would be willing to accept a 70% longevity confidence rate rather than a 90% rate, she would budget for 6 fewer years of retirement, and this would allow her to withdraw more each year.
Lastly, retirees have some control over the amount of risk they are willing to take with their retirement outcome. This risk is quantified in our analysis by the percentage of periods through history where the assumptions used resulted in failed outcomes. In our example above that assumed a 20 year retirement horizon with a 70/30 portfolio, a 5% annual withdrawal rate, and a 0.5% annual fee, a retiree would be 93.5% confident of success. In our experience, retirees like to target a confidence rate of between 75% and 90%. Obviously, retirees who are willing to accept a higher risk of failure would be able to draw income at a higher rate.
Pulling on the Levers
Now that we have identified the four ways that retirees can control their retirement equation, the next step is to quantify how changes to these levers alter retirement outcomes. We have broken the analysis into two steps: first we will investigate what combinations of asset allocations and withdrawal rates have the highest rates of success. Then we will examine the probability of retirement success at different retirement ages and levels of longevity risk.
Each cell in Chart 3. quantifies the percentage of successful outcomes through history at each equity allocation and withdrawal rate for a 20 year retirement horizon. Each cell represents a run of ~1100 possible retirement experiences over all rolling 240 month periods since 1900 with the specified assumptions. For example, the cell at the intersection of 60% across the top axis and 5% across the left axis quantifies the percentage of all 1100 ‘retirement experiences’ through history where a retirement portfolio finished above zero with a 5% withdrawal rate and a 60/40 stock/bond allocation (92.97%). Note that we also applied 0.5% per year in fees, as this is (quite optimistically) the lowest cost an investor might have paid for access to this portfolio.
Chart 3. Percentage of successful outcomes over 20-year rolling horizons: asset allocation (top axis) vs. withdrawal rates (left axis)

Source: Shiller, Federal Reserve

There are some important takeaways from this matrix. First, note that every asset allocation, even 100% Treasuries, has supported a 3% withdrawal rate, adjusted for inflation. Once withdrawal rates exceed 3%, adding some equities to the portfolio improves success ratios, but only up to a point. Rational investors with low levels of income requirements relative to the size of their retirement portfolio would therefore emphasize a very high bond allocation, as they can achieve the probability of success with less anxiety. This comes with a caveat, however; contemporary interest rates, at 1.8%, are lower than they have been at any time covered in our analysis. Therefore, investors with 100% bond positions today are unlikely to experience successful outcomes with 3% withdrawal rates.

The optimal mix of stocks/bonds for withdrawal rates between 3.5% and 5.5% inclusive seems to be about 70/30 or 80/20. Despite the much higher average return to equities vs. bonds over the past 112 years, retirement portfolios with reasonable withdrawal rates benefitted from some exposure to bonds because they reduced the magnitude and duration of equity bear markets, and reduced overall portfolio volatility.

Once withdrawal rates exceed 5.5%, retirement takes on more of a lottery character whereby successful outcomes are entirely dependent on above-average equity market returns; any allocation to bonds in this withdrawal range reduces the probability of success.

The next step is to determine how sensitive the retirement equation is to longevity risk. To investigate, we generated Chart 4., a matrix of retirement age versus percentile remaining lifespan. The percentile numbers across the top refer to the same concept discussed in reference to Chart 2. For example, from Chart 2. we can see that a woman retiring at age 65 who wants to budget to her 84th percentile lifespan would budget for 30 years of retirement (to age 95). In Chart 4. we have imputed the percentile remaining lifespan corresponding to each retirement age along the y axis.

To interpret Chart 4., simply find your desired retirement age along the y-axis, and find your desired longevity confidence level along the top. The intersecting cell quantifies the percentage of historical ‘retirement experiences’ with successful outcomes with those assumptions. In keeping with our discovery (Chart 3.) that a 70/30 stock/bond allocation was optimal for a 5% withdrawal rate over 20 years, we assumed that allocation for the Chart 4. analysis. We also used female lifespan percentiles to be conservative, and a 5% withdrawal rate with 0.5% fee. You can see that a person retiring at age 62 who wants to hedge 84% of longevity risk would have been successful 78% of the time through history with the assumptions above.

Chart 4. Sensitivity of successful retirement outcomes to percentile of remaining lifespan vs. retirement age (assumes female life table,70% stocks/30% bonds, and 5% withdrawal rate)

Source: Shiller, Federal Reserve

But I Won’t Need as Much Income When I’m Old

One objection to our standard retirement analysis that we hear all the time from clients is, “Why are you planning for me to maintain a stable income through retirement when I surely won’t spend as much in my later years as I will in my early years.” This point requires some discussion.

First of all,


Having said all that, some retirees are willing to live with a much higher risk of humble lifestyle conditions in later life in order to ‘live the adventure’ in their early retirement years. In order to model this, we tested a program of reducing income rates by i percentage points every n years in retirement to see how this impacted retirement outcomes through history.

Chart 5. offers an example. It is the same as Chart 3. except that we model a 1 percentage point reduction every decade through the retirement horizon. In this case, the horizon is just 20 years, so the inflation adjusted income drawn in the second decade is 1 percentage point less than the income drawn in the first decade. If we tested a 30 year horizon, the 3rd decade would be subject to a 2 percentage point decline in income relative to the starting rate.

Chart 5. Percentage of successful outcomes over 20 year horizon with 1 percentage point decrease in withdrawal after 10 years: asset allocation (top axis) vs. withdrawal rates (left axis)

Source: Shiller, Federal Reserve

If we compare results from the first matrix with those of the second matrix, you can see that, on average, retirees who decide to take 1% per year less in income in their second decade of retirement can budget for an extra 0.5 percentage points of income in their first decade with the same success rate. However, this math is not consistent once withdrawal rates exceed 5.5% or 6%; high withdrawal rates receive smaller incremental benefits from reducing income later in life because they are more sensitive to the effects of compounding.

Chart 6. repeats Chart 4. with the same income reduction after year 10. You can see that the income reduction does provide a small reduction in longevity risk.

Chart 6. Sensitivity of successful retirement outcomes to percentile of remaining lifespan (top axis) vs. retirement age with 1 percentage point decrease in withdrawal after 10 years

Source: Shiller, Federal Reserve
Conditional Safe Withdrawal Rates
Until now we have explored the subject of Safe Withdrawal Rates in the context of long-term market averages and percentiles without examining how interest rates and market valuation metrics might figure into the equation. In this section we will investigate the degree to which market valuation and interest rates at the beginning of retirement influence SWRs, and attempt to derive a conditional model to forecast SWRs given current valuation conditions.
The first step was to reverse engineer the ‘retirement experience’ at each month  through history to determine the precise withdrawal rate that would have resulted in total portfolio depletion on the last month of the retirement horizon. In other words, at each month through history we optimized the withdrawal rate using perfect foresight to fully deplete the portfolio on exactly the final month of retirement. (Note that this analysis does not include fees, so the numbers will not align with Charts 3 through 6.)
Chart 7. Optimal withdrawal rate with perfect foresight, 240 month retirement horizon
Source: Shiller, Federal Reserve
Our next step was to determine how we might be able to use contemporaneous market valuation estimates and interest rates to provide a better forecast for optimal withdrawal rates than the long-term average. Dr. Shiller’s database provides Cyclically Adjusted Price to Earnings ratios back to 1881 for U.S. stocks, so we used this valuation metric as another input to our model. Lastly, we used the contemporaneous monthly Treasury rates since bonds are an important part of the overall equation.
For several years we have performed a regular analysis of market valuation that uses a variety of fundamentally diverse valuation metrics to provide a statistical forecast for future market returns over long-term horizons. We are working on a full update to this analysis currently, but we generated the 20-year forecasts for the purpose of discovering how well the model’s return estimates forecast optimal withdrawal rates. Currently our statistical model forecasts 0.9% real annualized total returns to stocks over the next 20 years.
When we regress the realized optimal withdrawal rates each month against the Shiller CAPE, Treasury yields and our own multi-factor valuation model, we are able to forecast safe withdrawal rates with very high accuracy. In fact, the forecast model generates an r-squared value of 0.72, which means that our three inputs explain 72% of the change in safe withdrawal rates through time. This is a very high level of explanatory power for a model of this nature.
Chart 8. Derived optimal withdrawal rate vs. multiple regression estimate from contemporaneous earnings yield and interest rate
Source: Shiller, Federal Reserve
The ANOVA analysis in Table 1. provides us with the factor loadings, which allow us to generate our forecasts from our three inputs. The table also provides factor ranges for the regression which allow us to generate the 95% range around the forecast, which we illustrate graphically in Chart 9.

Table 1. ANOVA regression output

Source: Shiller, Federal Reserve, Doug Short, Chris Turner, Standard & Poor’s

Chart 9. shows the 95th percentile range of estimates for the safe withdrawal rate through time. You can see that the regression model does a fine job of forecasting the safe withdrawal rate given contemporaneous stock market CAPEs, interest rates, and our own multi-factor return estimate model as inputs. You can also see that current estimates for optimal withdrawal rates are near the bottom end of the historical range: 4.21% – 5.38% to be specific. This shouldn’t come as much of a surprise, since interest rates are near 220 year lows and earnings yields are in the top quintile of all monthly periods since 1871.

Chart 9. The black region in the chart is the 95th percentile range of forecasts at each month based on the regression.

Source: Shiller, Federal Reserve, Doug Short, Chris Turner, Standard & Poor’s

Summary and Thoughts

There is a great deal of contention about how to calculate safe withdrawal rates for the tsunami of retiring Boomers that will hit developed economies over the next 15 to 20 years. Many high quality studies have been conducted using Monte Carlo analysis, others through more sophisticated empirical methods, and still more through a purely analytical lens. The most comprehensive studies attempt to model the cointegration between interest rates, stock market returns and inflation, and it is our ambition to add more colour to this particular area of study.

We investigated the three primary sources of risk in retirement, and described a framework for quantifying these risks in a way that allows retirees to take control of their own risk/reward equation. Retirees face longevity risk, which is the risk of living longer (or less long) than expected; failure risk, or the risk of running out of money before death, and; market risk, which is the risk of poor investment returns over the retirement horizon.

We demonstrated how these risks are sensitive to the asset allocation between stocks and bonds, and proposed a model to quantify the optimal allocation given specified withdrawal rates and tolerance for failure. We also offered a model linking retirement age with longevity risk tolerance and tolerance for failure given specified asset allocation and withdrawal rates.

Unfortunately, most retirement planning is conducted under the potentially dangerous assumption of long-term average returns. Such assumptions manifest in amplified risk of plan failure as markets and interest rates deviate from long-term average valuations. We examined the relationship between a priori safe withdrawal rates, and contemporaneous stock market valuations and long-term interest rates. A model was then proposed to estimate a statistically significant range of safe withdrawal rates given stock market valuations and interest rates at the month of plan inception, and it was determined that the model explained over 70% of changes to safe withdrawal rates over the 112 year study.

Investors that look to the above models for guidance must take account of the fact that fees and taxes may substantially reduce SWRs under all assumptions. As such, investors should speak to a qualified Advisor or planner who is well versed in actuarial retirement planning to see how the topics covered above affect their own personal situation. Further, given the sensitivity of safe withdrawal rates to initial valuation conditions, we are of the strong opinion that as markets approach high levels of valuation (e.g. high CAPE, low interest rates), investors would benefit from exploring alternative sources of return to complement, or perhaps even replace traditional investment approaches.

Given that SWRs are especially vulnerable to periods of large or extended portfolio drawdowns, and that these types of drawdowns are much more likely to occur during periods of high market valuations, investors might consider allocating to ‘tactical alpha’ strategies (see Butler, Philbrick, Gordillo, 2012), which have historically managed to deliver stable returns in most market regimes. Examples of this type of approach include dynamic absolute valuation approaches (Butler, Philbrick, Gordillo, 2012), trend following (AQR, 2012), tactical asset allocation (Faber, 2009), robust risk parity, factor risk parity (AQR, 2012), generalized momentum (Keller, 2012), or adaptive asset allocation (Butler, Philbrick, Gordillo, 2012).