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

# Summary

- Portfolio risk is often measured as the variance of returns over time. Another form of risk is the variance of terminal wealth that can arise from small variations in strategy inputs or asset returns.
- Strategies or portfolios that are more sensitive to small changes in inputs are inherently “fragile.”
- Fragile strategy design makes it difficult to rely upon backtests or historical results in setting forward expectations.
- We explore how diversification across the “what,” “how,” and “when,” axes of portfolio construction can help reduce strategy fragility.

**Introduction**

At Newfound, we spend a lot less time trying to figure out how to be *more right *than we spend trying to figure out how to be *less wrong.* One area of particular interest for us is the idea of unintended bets: the exposures in a portfolio we may not even be aware of. And if we knew we had the exposure, we might not even want it.

For example, consider a portfolio that invests in either broad U.S., broad international, or broad emerging market equities based upon valuations. A significant tilt towards non-U.S. assets may be a valuation-driven decision, but for U.S. investors it creates significant exposure to fluctuations in the U.S. dollar versus foreign currencies.

Of course, exposures are not limited only to assets. Exposures may be broader macro-economic, stylistic, thematic, geographic, or even political factors.

These unintended bets can go far beyond explicit and implicit exposures. In our example, the choice of how to measure value may lead to meaningfully different portfolios, despite the same overarching thesis. For example, a naïve CAPE ratio versus adjusting for differences in relative sector composition dramatically alters the view of whether international equities are significantly cheaper than U.S. equities. These potential differences capture what we like to call “model specification risk.”

Finally, we can be subject to unintended bets based upon when the portfolio is re-evaluated and reconstituted. Evaluating valuations in January, for example, may lead to a different decision versus evaluating them in July.

How can we avoid these unintended bets? At Newfound, we believe that the answer falls back to diversification: not only in the traditional sense of *what *we invest in, but also across *how *we make decisions and *when *we make them.

When left uncontrolled, unintended bets can make a strategy incredibly fragile.

What, precisely, does it mean for a strategy to be fragile? A strategy is fragile when small variations of strategy inputs – be it asset returns or other measures – lead to meaningful dispersion in realized results.

Now we want to distinguish between volatility and fragility. Volatility is the dispersion of strategy returns across time, while fragility is the dispersion in end-of-period wealth across variations of the strategy.

As an example, a portfolio that invests only in the S&P 500 is very volatile but not particularly fragile. Given the last ten years of returns for the S&P 500, slight variations in annual returns would not lead to significant dispersion in end-of-period wealth. On the other hand, a strategy that flips a coin every December and invests for the next year in the S&P 500 when it lands on heads or short-term U.S. Treasuries when it lands on tails would have *lower expected *volatility than the S&P 500 but would be much more fragile. We need simply consider a few scenarios (e.g. all heads or all tails) to understand the potential dispersion such a strategy is subject to.

In the remainder of this commentary, we will demonstrate how diversification across the *what*, *how, *and *when *axes can reduce strategy fragility.

**The Experiment Setup**

Since a large degree of our focus at Newfound is on managing trend equity mandates, we will explore fragility through the lens of the style of measuring trends. For those unfamiliar with the approach, trend equity strategies aim to capture a significant portion of equity market growth while avoiding substantial and prolonged drawdowns through the application of trend following. A naïve implementation of such an idea would be to invest in the S&P 500 when its prior 12-month return has been positive and invest in short-term U.S. Treasuries otherwise.

To learn something about the fragility of a strategy, we are going to have to inject some randomness. After all, no amount of history will tell us about the fragility of a teacup that has spent its entire life sitting on a shelf; we will need to see it fall on the floor to actually learn something.

As with our recent commentary *When Simplicity Met Fragility*, we will inject randomness by adding white noise to asset returns. Specifically, we will add to daily returns a draw from a random normal distribution with mean 0% and standard deviation 0.025%. Using this slightly altered history, we will then run our investment strategy.

By performing this process a large number of times (10,000 in this commentary), we can explore how the outcome of the strategy is impacted by these slight variations in return history. The greater the dispersion in results, the more fragile the strategy is.

To demonstrate how diversification across the three different axes can affect fragility, we will start with a naïve trend equity strategy – investing in broad U.S. equities using a single trend model that is rebalanced on a monthly basis – and vary the three components in isolation.

** The What**

The “what” axis simply asks, “what are we invested in?”

How can our choice of “what” affect fragility? Consider a slight variation to our coin-flip strategy from before. Instead of flipping a single coin, we will now flip two coins. The first coin determines whether we invest 50% of the portfolio in either the S&P 500 or short-term U.S. Treasuries, while the second coin determines whether we invest the other 50% of the portfolio in either the Russell 1000 or short-term U.S. Treasuries.

In our single coin example, each year we expected to invest in the S&P 500 50% of the time and in short-term U.S. Treasuries 50% of the time. With two coins, we now expect to be fully invested 25% of the time, partially invested 50% of the time, and divested 25% of the time.

Let’s take this notion to further limits. Consider now flipping 100 coins where each determines the allocation decision for 1% of our portfolio, where heads leads to an investment in a large-cap U.S. equity portfolio and tails means invest in short-term U.S. Treasuries. Now being fully invested or divested is an infinitesimally small probability event; in fact, for a given year there is a 95% chance that your allocation to equities falls between 40-60%.^{1}

Even though we’ve applied the exact same process to each investment, diversifying across more investments has dramatically reduced the fragility of our coin-flipping strategy.

Now let’s translate this from the theoretical to the practical. We will begin with a simple trend following strategy that invests in the underlying asset when prior 12-1 month returns have been positive or invests in the risk-free rate, re-evaluating the trend at the end of each month.

To explore the impact of diversifying our *what*, we will implement this strategy five different ways:

- A single in-or-out decision on broad U.S. equities.
- Applied across 5 equally-weighted U.S. equity industry groups.
- Applied across 12 equally-weighted U.S. equity industry groups.
- Applied across 30 equally-weighted U.S. equity industry groups.
- Applied across 48 equally-weighted U.S. equity industry groups.

The graph below plots the distribution of log difference in terminal wealth against the median outcome for each of these five approaches. Lines within each “violin” show the 25^{th}, 50^{th}, and 75^{th}percentiles.

The graph clearly demonstrates that by increasing our exposure across the “what” axis, the dispersion in terminal wealth is dramatically reduced.

*Source: Kenneth French Data Library. Calculations by Newfound Research.*

But why is reduced dispersion in terminal wealth necessarily better?

It implies a greater consistency in outcome, which is not only important for setting forward expectations, but is also important for evaluating past performance (whether backtested or live). This evidence tells us that if we are evaluating a trend equity strategy that employs a single model to make in-or-out decisions on broad U.S. equities on a monthly basis, it will be nearly impossible to tell whether the realized results are in line with reasonable expectations or overly optimistic (we can probably guess that they aren’t overly pessimistic, as those sorts of returns typically aren’t marketed).

To justify a concentration in the “what” axis, we would have to demonstrate that the worst-case scenarios would still represent a meaningful improvement in expected terminal wealth versus a more diversified approach.

It should be noted that our experiment design prohibits dispersion from every being *fully *reduced, as we are injecting randomness into past returns. Even if no strategy is applied, there will be some inherent dispersion in final wealth. For example, below we plot the dispersion that occurs simply from adding randomness to past returns with a buy-and-hold approach.

Increasing the number of assets in the portfolio inherently reduces dispersion for buy-and-hold because diversification helps drive the expected impact of the injected randomness towards its mean: zero. With only one asset, on the other hand, outlier events are free to wreak havoc on results.

*Source: Kenneth French Data Library. Calculations by Newfound Research.*

Note that adding a strategy on top of buy-and-hold can exacerbate the fragility issue, making diversification that much more important.

**The How**

The “how” axis asks, “how are we making investment decisions.”

Many investors are already somewhat familiar with diversification along the “how” axis, often diversifying their active exposures across multiple managers who might have similar investment mandates but slightly different processes.

We like to call this “process diversification” and think of it as akin to the parable of the blind men and the elephant. Each blind man touches a different part of the elephant and pronounces his belief in what he is touching based upon his isolated view. The blind man touching the leg, for example, might think he is touching a sturdy tree while the blind man touching the tail might believe he is grabbing a rope.

None is correct in isolation but taken together we may gain a more well-rounded picture.

Similarly, two managers may claim to invest based upon valuations, but the manner in which they do so gives them a very different picture of where value can be found.

The idea of process diversification was explored in the 1999 paper “Do You Need More than One Manager for a Given Equity Style?” by Franklin Fant and Edward O’Neal. Fant and O’Neal found that while a multi-manager approach does very little for return variability across time (i.e. portfolio volatility), it does a *lot *for end-of-period wealth variability. They find this to be true across almost all equity style box categories. In other words: taking a multi-manager approach can reduce fragility.

Let us return to our prior coin flip example. Instead of making a choice to invest in the S&P 500 based upon a coin-flip, however, we will combine a number of different signals. For example, we might flip a coin, roll a die, measure the weather, and look at the second hand of a clock. Each signal gives us some sort of in-or-out decision, and we average these decisions together to get our allocation. As with before, as we incorporate more signals, we decrease the probability that we end up with extreme allocations, leading to a more consistent terminal wealth distribution.

Again, we should stress here that the objective is not just outright elimination of dispersion in terminal wealth. After all, if that were our sole pursuit, we could simply stuff our money under our mattress. Rather, assuming we will be implementing some active investment strategy that we hope has a positive long-term expected return, our aim should be to reduce the dispersion in terminal wealth for that strategy.

Of course, in investing we would not expect the processes to be entirely independent. With trend following, for example, most popular models are actually mathematically linked to one another, and therefore generate signals that are highly correlated. Nevertheless, even modest diversification can have meaningful benefits with respect to strategy fragility.

To explore the impact of diversification along the how axis, we implement our trend following strategy six different ways. Each invests in broad U.S. equities and rebalances monthly but differs in the number of trend-following models employed.^{2}

The results are plotted below.

*Source: Kenneth French Data Library. Calculations by Newfound Research.*

Again, we can see that increased diversification across the how axis dramatically reduces dispersion in terminal wealth. Our takeaway is largely the same: without an ex-ante view as to which particular model (or group of models) is best (i.e. a view of how to be *more right*), diversification can lead to greater consistency in results. We will be less wrong.

A subtler conclusion of this analysis is that it should be very, very difficult to necessarily conclude that one model is better than another. We can see that if we risk selecting just one model to govern our process, seemingly minor variations in historical returns leads can lead to dramatically different terminal wealth results, as evidenced by the bulging distribution. Inverting this line of thinking, we should also be suspect of any backtest that seeks to demonstrate the superiority of a given model using a single backtest. For example, just because a 12-1 month total return model performs better than a 10-month moving average model on historical S&P 500 returns, we should be highly skeptical as to the robustness of the conclusion that the 12-1 model is best.

**The When**

Then “when” axis asks, “when are we making our investment decision?”

This is an oft overlooked question in public markets, but it is commonly addressed in the world of private equity and venture capital. Due to the illiquid nature of those markets, investors will often attempt to diversify their business cycle risk by establishing positions in multiple funds over time, giving them exposure to different “vintages.” The idea here is simple: the opportunity set available at different points in time can vary and if we allocate all of our earmarked capital to a particular year, we may miss out on later opportunities.

Consider our original coin-flipping example where we flipped a single coin every December to determine whether we would buy the S&P 500 or hold our capital in short-term Treasuries. But why was it necessary that we make the decision in December? Why not July? Or January? Or September?

While we would not expect there to be point-in-time risk for coin flipping, we can still consider the net effect of a vintage-based allocation methodology. Here we will assume that we flip a coin each month and rebalance 1/12^{th}of our capital based upon the result.

Again, the probability of allocating to the extremes (100% invested or 100% divested) is dramatically reduced (each has approximately a 0.02% chance of occurring) and we reduce strategy fragility to any specific coin flip.

But just how impactful is this notion? Below we plot the rolling 1-year total return difference between two 60% S&P 500 / 40% 5-year U.S. Treasury fixed-mix portfolios, with one being rebalanced in February and one in August. Even for this highly simplified example, we can see that the total return spread between the two portfolios blows out to over 700 basis points in March 2010 due to the fact that the February portfolio rebalanced back into equities at nearly the exact bottom of the crisis.

*Source: Global Financial Data. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.*

To increase diversification across the “when” axis, we want to increase the number of vintages we deploy. For our trend following example, we will assume that the portfolio allocates between broad U.S. equities and the risk-free rate based upon a single model, but with an increasing number of evenly-spaced vintages. Again, we will run 10,000 simulations that each slightly perturb historical U.S. equity market returns and compare the terminal wealth variation for approaches that employ a different number of vintages.

We can see in the graph below that, as with the other axes of diversification, as we increase the number of vintages employed, the variance decreases. While the 25^{th}and 75^{th}percentiles do not decrease as dramatically as for the other axes, we can see that the extreme variations are reined in substantially when we move from 1 monthly tranche to 4 weekly tranches.

*Source: Kenneth French Data Library. Calculations by Newfound Research.*

**Conclusion**

We see two critical conclusions from this analysis:

- To develop confidence in achieving our objective we have to consider our sensitivity to unintended bets that may be included within the portfolio.

- Fragility makes it incredibly difficult to distinguish between luck and skill, particularly as strategy fragility increases. This is true for both backtested and live performance.

To conclude our analysis, below we present a graph that combines diversification across all three axes. We again run 10,000 samples, randomly perturbing returns. For each sample, we then run four variations:

- A single, randomly selected model run in broad U.S. equities that is rebalanced monthly.
- A random selection of 3 models run on 5 industry groups in 2 bi-weekly tranches.
- A random selection of 6 models run on 12 industry groups in 4 weekly tranches.
- A random selection of 9 models run on 30 industry groups in 20 daily tranches.

It should come as no surprise that as we increase the amount of diversification across all three axes, the dispersion in terminal wealth is dramatically reduced.^{3}

*Source: Kenneth French Data Library. Calculations by Newfound Research.*

It is also important to note that while our analysis focused on trend following strategies, this same line of thinking applies across all investment approaches. As an example, consider a quantitative value manager who buys the top five cheapest stocks, as measured by price-to-book, in the S&P 500 each December and then holds them for the next year. Questions worth pondering are:

- What does it say about our conviction when the 6
^{th}stock in the list is incredibly close to the 5^{th}stock? - What happens if some of our measures of book value are incorrect (or even just outdated)?
- How different would the portfolio look if we ranked on another value measure (e.g. price-to-earnings)?
- How different would the opportunity set be if we ranked every June versus every December?

While low levels of diversification across the *what, **how*, and *when *axes are not necessarily an indicator that a model is inherently fragile, it should be a red flag that more effort is required to disprove that it is not fragile.

## Dart-Throwing Monkeys and Process Diversification

By Corey Hoffstein

On December 24, 2018

In Portfolio Construction, Risk Management, Weekly Commentary

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

IntroductionIn last week’s commentary (

What do portfolios and teacups have in common?), we explored at great length the potential benefits of diversification in the domains ofwhat,how, andwhen.The crux of our argument is that for investors, return dispersions across time (i.e. “volatility”) can be a potentially misleading risk characteristic and that it is important to consider the potential dispersion in terminal wealth as well.

These are by no means original or unique thoughts. Often the advisors and institutions we work with intuitively understand them: they just have not been presented with the math to justify them.

Therefore, in contrast to last week’s rather expansive note, we aim to keep this week’s note short, simple, and punchy in an effort to drive how manager / process diversification can help deliver more consistent outcomes.

Dart-Throwing MonkeysConsider the following experiment.

We begin with thousands and thousands of dart-throwing monkeys. Every month, the monkeys throw their darts at a board that determines how they will be invested for the next month. In this hypothetical scenario, we will assume that the monkeys are investing in different industry groups.

^{1}Some monkeys are “concentrated managers,” throwing just a single dart and holding that pick for the next month. Other monkeys are more diversified, throwing up to 30 darts each month and equally allocating their portfolio across their investments. Portfolio sizes can be either 1, 5, 10, 15, 20, 25, or 30 equally-allocated investments.

It is our job, as an allocator, to choose different monkeys to invest with. Do we invest with just 1 concentrated monkey manager? Five different diversified managers? How much difference does it really make at the end of the day?

We learn in Finance 101 that once we diversify our portfolio sufficiently, we have eliminated nonsystematic risk. But does that mean we expect the portfolios to necessarily end up in the same place?

As an example, if we pick 10 dart-throwing monkeys who each pick 10 investments per month, how different would we expect our final wealth level to be from another allocator who picks 10 different dart-throwing monkeys who each pick 10 investments per month?

Process Diversification and Terminal Wealth DispersionBelow we plot the dispersion in terminal wealth

^{2}as a function of (1) the number of securities picked by each monkey manager and (2) the number of monkey managers we allocate to.As an example of how to read this graph, the orange line tells us about portfolios comprised of monkey managers who pick five investments each. As we move from left to right, we learn about the dispersion in terminal wealth based upon the number of managers we allocate to.

We can think of this two ways. First, we can think of it as potential dispersion in results among our peers who make the same type of decision (e.g. picking 5 managers who pick 5 investments each) but different specific choices (e.g. might pick different managers). Second, we can think of this as the dispersion in possible results if we were able to live across infinite universes simultaneously.

Source: Kenneth French Data Library. Calculations by Newfound Research.Unfortunately, we cannot live across infinite universes and this graph tells us that choosing a single, highly concentrated manager can lead to wildly different outcomes depending upon the manager we select.

As the managers further diversify and we further diversify among managers, this dispersion in potential outcomes decreases.

^{3}ConclusionThe intuition behind these results is simple:

It is equally valid to interpret this analysis as saying there is greater opportunity for out-performance in taking concentrated bets in highly concentrated managers. We would argue this is

more rightthinking: the win condition requires both that we pick the right managers and the managers pick the right stocks. While a little bit of diversification can go a long way here in clipping outlier events, the dispersion can still far exceed a more diversified approach.At Newfound, we prefer the

less wrongapproach. Allocations to a few diversified managers each taking a different approach can lead to significantly less dispersion in outcomes and, therefore, allow for better financial planning.