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

**Safe Withdrawal Rate**

**Aftcasting**

*, 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*

**Jan 1***, 1900 and track the outcome. Next we simulate the experience for a person retiring on*

**Feb 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.*

**March 1****The 4 Levers of Retirement**

*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)

**Pulling on the Levers**

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,

Secondly, XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

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*

**Conditional Safe Withdrawal Rates**

*real annualized total returns*to stocks over the next 20 years.

Table 1. ANOVA regression output

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.

**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).