*This post is available as a PDF download here.*

# Summary

- While retirement planning is often performed with Monte Carlo simulations, investors only experience a single path.
- Large or prolonged drawdowns early in retirement can have a significant impact upon the probability of success.
- We explore this idea by simulation returns of a 60/40 portfolio and measuring the probability of portfolio failure based upon a quantitative measure of risk called the Ulcer Index.
- We find that a high Ulcer Index reading early in an investor’s retirement can dramatically increase the probability of failure as well as decrease the expected longevity of a portfolio.

**Introduction**

At Newfound we often say, “while other asset managers focus on alpha, our first focus is on risk.”

Not that there is anything wrong with the pursuit of alpha. We’d argue that the pursuit of alpha is actually a necessary component for well-functioning financial markets.

It’s simply that we have never met a financial advisor who has built a financial plan that assumed any sort of alpha. Alpha is great if we can harvest it, but the empirical evidence suggesting how difficult that can be (both for the manager net-of-fees as well as the investor behaviorally) would make the *presumption* of achieving alpha rather bold.

Furthermore, alpha is a zero-sum game: we can’t all plan for it.

Risk, however, is a crucial element of every investor’s plan. Bearing too little risk can lead to a portfolio that “fails slowly,” falling short of achieving the escape velocity required to outpace inflation. Bearing too much risk, however, can lead to sudden and catastrophic ruin: a case of “failing fast.”

When investors hit retirement, the usual portfolio math changes. While we’re taught in Finance 101 that the order of returns does not matter, the introduction of portfolio withdrawals makes the order of returns a large determinant of plan success. This phenomenon is known as “sequence risk” and it peaks in the years just before and after retirement.

Typically, we look at returns through the lens of the investment. In retirement, however, what really matters is the returns of the investor.

We’re often told that our primitive brain, trained on the African veldt, is unsuited for investing. Yet our brain seems to understand quite well that we do not get to live our lives as the average of a Monte Carlo simulation.

If we lose our arm to a lion because we did not flee when we heard a rustle in the bushes, we do not end up with half of an arm because of all the other parallel universes where we did flee. On the timeline we live, the situation is binary.

As investors, the same is true. We live but a single path and there are very real, very permanent knock-out conditions we need to be aware of. Prolonged and significant drawdowns during the first years of retirement rank among the most dangerous.

**Drawdowns and the Risk of Ruin**

A retirement plan typically establishes a safe withdrawal rate. This is the amount of inflation-adjusted money an investor can withdraw from their portfolio every year and still retain a sufficiently high probability that they will not run out of money before they die.

A well-established (albeit controversial) rule is that 4% of an investor’s portfolio level at retirement is usually an appropriate withdrawal amount. For example, if an investor retires with a $1,000,000 portfolio, they can theoretically safely withdraw $40,000 a year. Another way to think of this is that the portfolio reflects 25 years of spending assuming growth matches inflation.

The problem with portfolio drawdowns is that the withdrawal rate now reflects a larger proportion of capital unless it is commensurately adjusted downward. For example, if the portfolio falls to $700,000, a $40,000 withdrawal is now 5.7% of capital and the portfolio reflects just 17.5 years of spending units.

Even shallow, prolonged drawdowns can have a damaging effect. If the portfolio falls to $900,000 and stays stagnant for the next five years, the $40,000 withdrawals grow from representing 4% of the portfolio to nearly 5.5% of the portfolio. If we do not adjust the withdrawal, at five years into retirement we have gone from 25 spending units to 18.5, losing a year and a half of portfolio longevity.

As sudden and steep drawdowns can be just as damaging as shallow and prolonged ones, we prefer to use a quantitative measure known as the Ulcer Index to measure this risk. Specifically, the Ulcer Index is calculated as the root mean square of monthly drawdowns, capturing both severity and duration simultaneously.

In an effort to demonstrate the damaging impact of drawdowns early in retirement, we will run the following experiment:

- Generate 250,000 simulations, each block-bootstrapped from monthly real U.S. equity and real U.S. 5-year Treasury bond returns from 1918 – 2018.
- Assume a 65 year old investor with a $1,000,000 starting portfolio and a fixed real $3,333 withdrawal monthly ($40,000 annual).
- Assume the investor holds a 60/40 portfolio at all times.
- For each simulation:
- Calculate the
*Ulcer Index*of the first five years of portfolio returns (ignoring withdrawals). - Determine how many years until the portfolio runs out of money.

- Calculate the

Based upon this data, below we plot the probability of failure – i.e. the probability we run out of money before we die – given an assumed age of death as well as the Ulcer Index realized by the portfolio in the first five years of retirement.

As an example of how to read this graph, consider the darkest blue line in the middle of the graph, which reflects an assumed age of death of 84. Along the x-axis are different bins of Ulcer Index levels, with lower numbers reflecting fewer and less severe drawdowns, while higher numbers reflect steeper and more frequent ones.

As we trace the line, we can see that the probability of failure – i.e. running out of money before death – increases dramatically as the Ulcer Index increases. While for shallow and infrequent drawdowns the probability of failure is <5%, we can see that the probability approaches 50% for more severe, frequent losses.

Beyond the binary question of failure, it is also important to consider when a portfolio runs out of money relative to when we die. Below we plot how many years prior to death a portfolio runs out of money, on average, based upon the Ulcer Index.

Once again using the darkest blue line as an example, we can see that for most minor-to-moderate Ulcer Index levels, the portfolio would only run out of money a year or two before we die in the case of failure. For more extreme losses, however, the portfolio can run out of money a full decade before we kick the bucket.

It is worth stressing here that these Ulcer Index readings are derived using simulations based upon prior realized U.S. equity and fixed income returns. In other words, while improbable (see the histogram below), extreme readings are not impossible.

It is worth further acknowledging that U.S. assets have experienced some of the highest realized risk premia in the world, and more conservative estimates may put a higher probability mass on more extreme Ulcer Index readings.

**Conclusion**

For early retirees, large or prolonged drawdowns early in retirement can have a significant impact on the probability of success.

In this commentary, we capture both the depth and duration of drawdowns using a single metric known as the Ulcer Index. We simulate 250,000 possible return paths for a 60/40 portfolio and calculate the Ulcer Index in the first five years of returns. We then plot the probability of failure as well as expected portfolio longevity conditional upon the Ulcer Index level realized.

We clearly see a positive relationship between failure and Ulcer Index, with larger and more prolonged drawdowns earlier in retirement leading to a higher probability of failure. This phenomenon is precisely why investors tend to de-risk their portfolios over time.

While the right risk profile and a well-diversified portfolio make for a strong foundation, we believe that investors should also consider expanding their investment palette to include alternative assets and style premia that may be more defensive oriented in nature. For example, defensive equities (e.g. low-volatility and quality approaches) have historically demonstrated an ability to reduce drawdown risk. Diversified, multi-asset style premia also tend to exhibit low correlation to traditional risk factors and a low intrinsic style premia.

Here at Newfound, we focus on trend equity strategies, which seek to overlay trend-following approaches on top of equity exposures in an effort to reduce left-tail risk and create a higher quality of return profile.

However, an investor chooses to build their portfolio, however, it should be risk that is on the forefront of their mind.

## The Path-Dependent Nature of Perfect Withdrawal Rates

By Nathan Faber

On April 22, 2019

In Risk Management, Sequence Risk, Weekly Commentary

This post is available as a PDF download here.## Summary

PWR) is the rate of regular portfolio withdrawals that leads to a zero balance over a given time frame.PWR.PWR, especially as it pertains to the risk of failure in the financial plan.Financial planning for retirement is a combination of art and science. The problem is highly multidimensional, requiring estimates of cash flows, investment returns and risk, taxation, life events, and behavioral effects. Reduction along the dimensions can simplify the analysis, but introduces consequences in the applicability and interpretation of the results. This is especially true for investors who are close to the line between success and failure.

One of the primary simplifying assumptions is the 4% rule. This heuristic was derived using worst-case historical data for portfolio withdrawals under a set of assumptions, such as constant inflation adjusted withdrawals, a fixed mix of stock and bonds, and a set time horizon.

Below we construct a monthly-rebalanced, fixed-mix 60/40 portfolio using the S&P 500 index for U.S. equities and the Dow Jones Corporate Bond index for U.S. bonds. Using historical data from 12/31/1940 through 12/31/2018, we can evaluate the margin for error the 4% rule has historically provided and how much opportunity for higher withdrawal rates was sacrificed in “better” market environments.

Source: Global Financial Data and Shiller Data Library. Calculations by Newfound Research. Returns are backtested and hypothetical. Past performance is not a guarantee of future results. Returns are gross of all fees. Returns assume the reinvestment of all distributions. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.But the history is only a single realization of the world. Risk is hard to gauge.

Perfect Withdrawal RatesThe formula (in plain English) for the perfect withdrawal rate (“

PWR”) in a portfolio, assuming an ending value of zero, is relatively simple since it is just a function of portfolio returns:The portfolio value in the numerator is the final value of the portfolio over the entire period, assuming no withdrawals. The sequence risk in the denominator is a term that accounts for both the order and magnitude of the returns.

Larger negative returns earlier on in the period increase the sequence risk term and therefore reduce the

PWR.From a calculation perspective, the final portfolio value in the equation is typically described (e.g. when using Monte Carlo techniques) as a log-normal random variable, i.e. the log-returns of the portfolio are assumed to be normally distributed. This type of random variable lends itself well to analytic solutions that do not require numerical simulations.

The sequence risk term, however, is not so friendly to closed-form methods. The path-dependent, additive structure of returns within the sequence risk term means that we must rely on numerical simulations.

To get a feel for some features of this equation, we can look at the

PWRin the context of the historical portfolio return and volatility.Source: Global Financial Data and Shiller Data Library. Calculations by Newfound Research. Returns are backtested and hypothetical. Past performance is not a guarantee of future results. Returns are gross of all fees. Returns assume the reinvestment of all distributions. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.The relationship is difficult to pin down.

As we saw in the equation shown before, the –annualized return of the portfolio– does appear to impact the –PWR– (correlation of 0.51), but there are periods (e.g. those starting in the 1940s) that had higher PWRs with lower returns than in the 1960s. Therefore, investors beginning withdrawals in the 1960s must have had higher sequence risk.

Correlation between –annualized volatility– and –PWR– was slightly negative (-0.35).

The Risk in Withdrawal RatesSince our goal is to assess the risk in the historical

PWRwith a focus on the sequence risk, we will use the technique of Brownian Bridges to match the return of all simulation paths to the historical return of the 60/40 portfolio over rolling 30-year periods. We will use the historical full-period volatility of the portfolio over the period for the simulation.This is essentially a conditional

PWRrisk based on assuming we know the full-period return of the path beforehand.To more explicitly describe the process, consider a given 30-year period. We begin by computing the full-period annualized return and volatility of the 60/40 portfolio over that period. We will then generate 10,000 simulations over this 30-year period but using the Brownian Bridge technique to ensure that all of the simulations have the exact same full-period annualized return and intrinsic volatility. In essence, this approach allows us to vary the path of portfolio returns without altering the final return. As PWR is a path-dependent metric, we should gain insight into the distribution of PWRs.

The percentile bands for the simulations using this method are shown below with the actual PWR in each period overlaid.

Source: Global Financial Data and Shiller Data Library. Calculations by Newfound Research. Returns are backtested and hypothetical. Past performance is not a guarantee of future results. Returns are gross of all fees. Returns assume the reinvestment of all distributions. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.From this chart, we see two items of note: The percentile bands in the distribution roughly track the historical return over each of the periods, and the actual PWR fluctuates into the left and right tails of the distribution rather frequently. Below we plot where the

actualPWR actually falls within the simulated PWR distribution.The actual PWR is below the 5

^{th}percentile 12% of the time, below the 1^{st}percentile 4% of the time, above the 95^{th}percentile 11% of the time, and above the 99^{th}percentile 7% of the time. Had our model been more well calibrated, we would expect the percentiles to align; e.g. the PWR should be below the 5^{th}percentile 5% of the time and above the 99^{th}percentile 1% of the time.This seems odd until we realize that our model for the portfolio returns was likely too simplistic. We are assuming Geometric Brownian Motion for the returns. And while we are fixing the return over the entire simulation path to match that of the actual portfolio, the path to get there is assumed to have constant volatility and independent returns from one month to the next.

In reality, returns do not always follow these rules. For example, the skew of the monthly returns over the entire history is -0.36 and the excess kurtosis is 1.30. This tendency toward larger magnitude returns and returns that are skewed to the left can obscure some of the risk that is inherent in the PWRs.

Additionally, returns are not totally independent. While this is good for trend following strategies, it can lead to an understatement of risk as we explored in our previous commentary on Accounting for Autocorrelation in Assessing Drawdown Risk.

Over the full period, monthly returns of lags 1, 4, and 5 exhibit autocorrelation that is significant at the 95% confidence level.

To incorporate some of these effects in our simulations, we must move beyond the simplistic assumption of normally distributed returns.

First, we will fit a skewed normal distribution to the rolling historical data and use that to draw our random variables for each period. This is essentially what was done in the previous section for the normally distributed returns.

Then, to account for some autocorrelation, we will use the same adjustment to volatility as we used in the previously reference commentary on autocorrelation risk. For positive autocorrelations (which we saw in the previous graphs), this results in a higher volatility for the simulations (typically around 10% – 25% higher).

The two graphs below show the same analysis as before under this modified framework.

The historical PWR now fall more within the bounds of our simulated results.

Additionally, the 5

^{th}percentile band now shows that there were periods where a 4% withdrawal rule may not have made the cut.ConclusionHeuristics can be a great way to distill complex data into actionable insights, and the perfect withdrawal rate in retirement portfolios is no exception.

The 4% rule is a classic example where we may not be aware of the risk in using it. It is the commonly accepted lower bound for safe withdrawal rates, but this is only based on one realization of history.

The actual risk investors take on by using this number may be uncertain.

Using simulation techniques, we explored how different assumptions match the historical experience of retirement portfolios.

The simple assumptions (expected return and volatility) commonly used in financial planning Monte Carlo simulations do not seem to reflect as much variation as we have seen in the historical PWR. Therefore, relying on these assumptions can be risky for investors who are close to the “go-no-go” point; they do not have much room for failure and will be more likely to have to make cash flow adjustments in retirement.

Utilizing richer simulation methods (e.g. accounting for negative skew and autocorrelation like we did here or using a downside shocking method like we explored in A Shock to the Covariance System) may be necessary to successfully gauge that risk in a proposed PWR, especially as it pertains to the risk of failure in the financial plan.

Having a number to base planning calculations on makes life easier in the moment, but knowing the risk in using that number makes life easier going forward.