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

# Summary

- Even when an investment has a positive expected average growth rate, the experience of most individuals may be catastrophic.
- By focusing on the compound average growth rate, we can see the median realizations – which account for risk – are often more crucial decision points than ensemble averages, which are the focal point of Monte Carlo analysis.
- These arguments also provide a simple explanation for investor behavior that avoids the need for utility theory concepts that have been used for the past 200+ years.
- Since we can neither average our results with other investors nor average our results with potential copies of ourselves in infinite states of the world, the best we can do is try to average over time.
- Because we all live in a multi-period world where we have a single investment portfolio that compounds over time, managing risk can help us maximize our long-term growth rate even if it seems foolish in hindsight.

Pretend we come to you offering a new investment strategy. Each week, you earn 0.65% (such that over a year you earn 40%), but there is a 1-in-200 chance that you lose -95%. Would you invest?

If we simulate out a single trial, we can see that within a year, we may lose most of our money.

Of course, just because things went wrong in our singular example does not mean that this is necessarily a bad investment. In fact, if we evaluate the prospects of this investment by looking at the average experience, we end up with something far more attractive (the “Ensemble,” which is essentially a Monte-Carlo simulation of the strategy).

The math here is simple: 99.5% of the time we make 1.0065x our money and 0.5% of the time, we end up with 0.05x our money. On average, then, we end up with 1.0017x, or 1.092x annualized. While the average experience is not the 40% annualized we sought, the 9.2% return after a year is still nothing to scoff at.

Of course, the average is not actually achievable. There are not infinite variations of this investment strategy for you to allocate your capital across, nor, we suspect, do you have access to infinite versions of you living in parallel universes who can pool their risk.

Rather, you are forced to diversify your risk *over time*. Here we end up with a different picture.

Another series of unfortunate events?

Not so fast. You see, when we move to diversifying over time, we need to look at a *time-weighted *average. It is not the arithmetic mean we are after, but rather the *geometric* mean which will account for the effects of compounding. Calculating the geometric mean – 1.0065^{99.5%} x 0.05^{0.5%} – leaves us with a value of 0.9915, i.e. our wealth is expected to *decay* over time.

Wait.

How is it possible that *on average *the strategy is a winner if each and every path is expected to decay over time?

The simple answer: A few fortunate outliers make up for all decaying paths.

The slightly more complex answer: In this investment, our wealth can never go below $0 but we can theoretically make an infinite amount of money. Thus, over time, the average is dragged up.

**The Misleading Mean**

In many cases, the *average experience* can be entirely misleading for the experience you can expect. In the world of bell-curves and normal distributions, we typically expect experiences to be clustered around the average. For example, there are more people close to the average height than there are far away.

However, when other distributions apply, the average can be unlikely. Wealth distribution is a perfect example of this. In 2013 in the United States, the top 10% of families held 76% of the wealth while the bottom 50% held 1%. Using 2017 figures, if we divided net worth among the U.S. population – i.e. the “average” household wealth – it would come out to around $760,000 per family. The bottom 50%, however, have a net worth closer to $11,000 per family.

In other words, if you pick a random person off the street, their experience is likely much closer to $11,000 than $760,000. It’s the wealthy outliers that are pulling the average up.

A more applicable metric, in this case, might be the *median*, which will say, “50% of experiences are below this level and 50% are above.”

**The Role of Risk**

As it turns out, the median is important for those of us diversifying over time as well. If we consider our hypothetical investment strategy above, our intuition is that the median result is probably not great. *Eventually*, it feels like, everyone goes practically bankrupt. If we plot the median result, we see almost exactly that.

(As a side note, if you’re wondering why the median result exhibits a sawtooth pattern rather than the smoother results of the mean, the answer is the median is the actual result that sits at the 50^{th} percentile. Knowing that the probability of losing 95% of our wealth is 1-in-200, it takes time for enough individuals to experience a poor result for the median to drop.)

In fact, if we model investment wealth as a Geometric Brownian Motion (a commonly used stochastic process for modeling stock prices), then over the long run an investor’s compound growth rate approaches the *median*, not the mean.[1] The important difference between the two is that while volatility does not affect the expected level of wealth, it does drive the mean and median further apart. In fact, the median growth rate is the mean growth rate minus half the volatility squared (which you might recognize as being the common approximation for – drum roll please – the geometric growth rate).

In other words: volatility matters.

Most investors we speak with have an intuitive grasp of this concept. They know that when you lose 10% of your wealth, you need to gain 11.11% back to get to break even.

And when you lose 50%, and you need to earn 100% to get back to break even. Under compound results, feeling twice the pain from losses than the pleasure from gain makes complete sense. There are no individual and independent trials: results have consequences.

This is why taking less risk can actually lead to greater growth in wealth in the long run. If we take too little risk, we will will not participate, but too much risk can lead to ruin. For example, below we plot final wealth results after a 50% drop in market value and a 100% recovery depending on your *capture ratio*.

As an example of reading this graph, if we start with $1 and experience a 50% loss and a 100% gain, but are only 50% exposed to each of those movements (i.e. we lose 25% and then gain 50%), we end up with $1.125. At the far right of the graph, we can see that at 2x exposure, the first 50% move completely wipes out our capital.

**Common Sense Utility Theory**

What economists have found, however, is that even if we offer our investment as a one-off event – where the expected return is definitively positive – most would still forego it. To resolve this conundrum, economists have proposed utility theory.

The argument is that investors do not actually try to maximize their expected change in wealth, but rather try to maximize the expected *utility* of that change. The earliest formalization of this concept was in a paper written by Daniel Bernoulli in 1738, where he proposed a mathematical function that would correct the expected return to account for risk aversion.

Bernoulli’s originally proposed function was *log-utility.* And under log-utility, our investment strategy offering is no longer appealing: log(1.0065) x 99.5% + log(0.05) x 0.50% is a negative value. What’s interesting about log utility is that, due to the property of logarithms, it ends up creating the *identical* decision axiom as had we used our compound growth rate model.

log(1.0065) x 99.5% + log(0.05) x 0.50% = log(1.0065^{99.5%}) + log(0.05^{0.5%}) = log(1.0065^{99.5%} x 0.05^{0.5%})

So while utility theory is supposed to correct for behavioral foibles like “risk aversion,” what it really does is take a single-period bet and turn it into a multi-period, compound bet.

Under the context of multi-period, compounding results, “risk aversion” is not so foolish. If we have our arm mauled off by a lion on the African veldt, we cannot simply “average” our experience with others in the tribe and end up with 97% of an arm. We cannot “average” our experience across the infinite universes of other potential outcomes where we were not necessarily mauled. Rather, our state is permanently altered for life.

Similarly, if we lose 50% of our money, we cannot just “average” our results with other investors. Nor can we average our results with all the potential infinite alternate universes where we did not lose 50%. The best we can do is try to average over time, which means that our compound growth rate matters. And, as we demonstrated above, so does risk.

**Conclusion**

Ex-post, managing risk can often feel foolish. Almost exactly 9 years after the bottom of the 2008-2009 bear market, the S&P 500 has returned more than 380%. Asset class, geographic, and process diversification largely proved foolish relative to simple buy-and-hold.

Ex-ante, however, few would forgo risk management. Ask yourself this: would you sell everything today to buy only U.S. large-cap stocks? If not, then there is little to regret about not having done it in the past. While the narratives we spin often make realized results seem obvious in hindsight, the reality is that our collective crystal balls were just as cloudy back then as they are today.

Few lament that their house did not burn down when they buy fire insurance. We buy insurance “in case,” not because we want the risk to materialize.

We all live in a multi-period world where we have a single investment portfolio that compounds over time. In such a world, risk matters tremendously. A single, large loss can take us permanently off plan. Even small losses can put us off course when compounded in a streak of bad luck. While a focus on risk aversion may seem foolish in hindsight when risk does not materialize, going forward we know that managing risk can help us maximize our long-term growth rate.

[1] Derivations for this result can be found in our commentary *Growth Optimal Portfolios*

## Dynamic Spending in Retirement Monte Carlo

By Nathan Faber

On July 15, 2019

In Risk Management, Sequence Risk, Weekly Commentary

Thispost isavailable as a PDF download here.## Summary

Monte Carlo simulations are a prevalent tool in financial planning, especially pertaining to retirement success calculations.

Under a typical framework of normally distributed portfolio returns and constant inflation-adjusted withdrawals, calculating the success of a given retirement portfolio is straightforward. But as with most tools in finance, the art lies both in the assumptions that go into the calculation and in the proper interpretation of the result.

If a client is told they have a 10% chance of running out of money over their projected retirement horizon, what does that mean for them?

They cannot make 9 copies of themselves to live out separate lives, with one copy (hopefully not the original) unfortunately burning through the account prematurely.

They also cannot create 9 parallel universes and ensure they do not choose whichever one does not work out.

We wrote previously how investors follow a single path (You Are Not a Monte-Carlo Simulation). If that path hits zero, the other hypothetical simulation paths don’t mean a thing.

A simulation path is only as valuable as the assumptions that go into creating it, and fortunately, we can make our simulations align more closely with investor behavior.

The best way to interpret the 10% failure rate is to think of it as a 10% chance of

havingto make an adjustment before it hits zero. Rarely would an investor stand by while their account went to zero. There are circumstances that are entirely out of investor control, but to the extent that there was something they could do to prevent that event, they would most likely do it.Derek Tharp, on Michael Kitces’ blog, wrote a post a few years ago weighing the relative benefit of implementing small but permanent adjustments vs. large but temporary adjustments to retirement withdrawals and found that making small adjustments and leaving them in place led to greater likelihoods of success over retirement horizons (Dynamic Retirement Spending Adjustments: Small-But-Permanent Vs Large-But-Temporary).

In this week’s commentary, we want to dig a little deeper into some simple path dependent modifications that we can make to retirement Monte-Carlo simulations with the hope of creating a more robust toolset for financial planning.

The Initial PlanSuppose an investor is 65 and holds a moderate portfolio of 60% U.S. stocks and 40% U.S. Treasuries. From 1871 until mid-2019, this portfolio would have returned an inflation-adjusted 5.1% per year with 10.6% volatility according to Global Financial Data.

Sticking with the rule-of-thumb 4% annual withdrawal of the initial portfolio balance and assuming a 30-year retirement horizon, this yields a predicted failure rate of 8% (plus or minus about 50 bps).

The financial plan is complete.

If you start with $1,000,000, simply withdraw $3,333/month and you should be fine 92% of the time.

But what if the portfolio drops 5% in the first month? (It almost did that in October 2018).

The projected failure rate over the next 29 years and 11 months has gone up to 11%. That violates a 10% threshold that may have been a target in the planning process.

Or what if it drops 30% in the first 6 months, like it would have in the second half of 1931?

Now the project failure rate is a staggering 46%. Retirement success has been reduced to a coin flip.

Admittedly, these are trying scenarios, but these numbers are a key driver for financial planning. If we can better understand the risks and spell out a course of action beforehand, then the risk of making a rash emotion-driven decision can be mitigated.

Aligning the Plan with RealityWhen the market environment is challenging, investors can benefit by being flexible. The initial financial plan does not have to be jettisoned; agreed upon actions within it are implemented.

One of the simplest – and most impactful – modifications to make is an adjustment to spending. For instance, an investor might decide at the outset to scale back spending by a set amount when the probably of failure crosses a threshold.

Source:Global Financial Data. Calculations by Newfound.This reduction in spending would increase the probability of success going forward through the remainder of the retirement horizon.

And if we knew that this spending cut would likely happen if it was necessary, then we can quantify it as a rule in the initial Monte Carlo simulation used for financial planning.

Graphically, we can visualize this process by looking at the probabilities of failure for varying asset levels over time. For example, at 10 years after retirement, the orange line indicates that a portfolio value ~80% of the initial value would have about a 5% failure rate.

Source:Global Financial Data. Calculations by Newfound.As long as the portfolio value remains above a given line, no adjustment would be needed based on a standard Monte Carlo analysis. Once a line is crossed, the probability of success is below that threshold.

This chart presents a good illustration of sequence risk: the lines are flatter initially after retirement and the slope progressively steepens as the time progresses. A large drawdown initially puts the portfolio below the threshold for making and adjustment.

For instance, at 5 years, the portfolio has more than a 10% failure rate if the value is below 86%. Assuming zero real returns, withdrawals alone would have reduced the value to 80%. Positive returns over this short time period would be necessary to feel secure in the plan.

Looking under the hood along the individual paths used for the Monte Carlo simulation, at 5 years, a quarter of them would be in a state requiring an adjustment to spending at this 10% failure level.

Source:Global Financial Data. Calculations by Newfound.This belies the fact that some of the paths that would have crossed this 10% failure threshold prior to the 5-year mark improved before the 5-year mark was hit. 75% of the paths were below this 10% failure rate at some point prior to the 5-year mark. Without more appropriate expectations of a what these simulations mean, under this model, most investors would have felt like their plan’s failure rate was uncomfortable at some point in the first 5 years after retirement!

Dynamic Spending RulesIf the goal is ultimately not to run out of funds in retirement, the first spending adjustment case can substantially improve those chances (aside from a large negative return in the final periods prior to the last withdrawals).

Each month, we will compare the portfolio value to the 90% success value. If the portfolio is below that cutoff, we will size the withdrawal to hit improve the odds of success back to that level, if possible.

The benefit of this approach is greatly improved success along the different paths. The cost is forgone income.

But this can mean forgoing a lot of income over the life of the portfolio in a particularly bad state of the world. The worst case in terms of this total forgone income is shown below.

Source:Global Financial Data. Calculations by Newfound.The portfolio gives up withdrawals totaling 74%, nearly

19 years’ worth. Most of this is given up in consecutive periods during the prolonged drawdown that occurs shortly after retirement.This is an extreme case that illustrates how large of income adjustments could be required to ensure success under a Monte Carlo framework.

The median case foregoes 9 months of total income over the portfolio horizon, and the worst 5% of cases all give up 30% (7.5 years) of income based off the initial portfolio value.

That is still a bit extreme in terms of potential cutbacks.

As a more realistic scenario that is easier on the pocketbook, we will limit the total annual cutback to 30% of the withdrawal in the following manner:

These rules still increase the success rate to 99% but substantially reduce the amount of reductions in income.

Looking again at the worst-case scenario, we see that this case still “fails” (even though it lasts another 4.5 years) but that its reduction in come is now less than half of what it was in the extreme cutback case. This pattern is in line with the “lower for longer” reductions that Derek had looked at in the blog post.

Source:Global Financial Data. Calculations by Newfound.On the 66% of sample paths where there was a cut in spending at some point, the average total cut amounted to 5% of the portfolio (a little over a year of withdrawals spread over the life of the portfolio).

Even moving to an even less extreme reduction regime where only 10% cuts are ever made if the probability of failure increases above 10%, the average reduction in the 66% of cases that required cuts was about 9 months of withdrawals over the 30-year period.

In these scenarios, the failure rate is reduced to 5% (from 8% with no dynamic spending rules).

Source:Global Financial Data. Calculations by Newfound.ConclusionRetirement simulations can be a powerful planning tool, but they are only as good as their inputs and assumptions. Making them align as closes with reality as possible can be a way to quantify the impact of dynamic spending rules in retirement.

While the magnitude of spending reductions necessary to guarantee success of a retirement plan in all potential states of the world is prohibitive. However, small modifications to spending can have a large impact on success.

For example, reducing withdrawal by 10% when the forecasted failure rate increases above 10% nearly cut the failure rate of the entire plan in half.

But dynamic spending rules do not exist in a vacuum; they can be paired with other marginal improvements to boost the likelihood of success:

While failure is certainly possible for investors, a “too big to fail” mentality is much more in line with the reality of retirement.

Even if absolute failure is unlikely, adjustments will likely be a requirement. These can be built into the retirement planning process and can shed light on stress testing scenarios and sensitivity.

From a retirement planning perspective, flexibility is simply another form of risk management.