Flirting with Models

The Research Library of Newfound Research

Month: February 2019

Three Applications of Trend Equity

By

On February 25, 2019

In Portfolio Construction, Risk & Style Premia, Trend, Weekly Commentary

This post is available as a PDF download here.

What is Trend Equity?

Trend equity strategies seek to meaningfully participate with equity market growth while side-stepping significant and prolonged drawdowns.  These strategies aim to achieve this goal by dynamically adjusting market exposure based upon trend-following signals.

A naïve example of such a strategy would be a portfolio that invests in U.S. equities when the prior 1-year return for U.S. equities is positive and divests entirely into short-term U.S. Treasuries when it is negative.

The Theory

This category of strategies relies upon the empirical evidence that performance tends to persist in the short-run: positive performance tends to beget further positive performance and negative performance tends to beget further negative performance. The theory behind the evidence is that behavioral biases exhibited by investors lead to the emergence of trends.

In an efficient market, changes in the underlying value of an investment should be met by an immediate, commensurate change in the price of that investment.  The empirical evidence of trends suggests that investors may not be entirely efficient at processing new information.  Behavioral theory (Figure I) suggests that investors anchor their views on prior beliefs, causing price to underreact to new information.  As price continues to drift towards fair value, herding behavior occurs, causing price to overreact and extend beyond fair value.  Combined, these effects cause a trend.

Trend equity strategies seek to capture this potential inefficiency by systematically investing in equities when they are exhibiting positively trending characteristics and divesting when they exhibit negative trends.  The potential benefit of this approach is that it can try to exploit two sources of return: (1) the expected long-term risk premium associated with equities, and (2) the convex payoff structure typically associated with trend-following strategies.

The Positive Convexity of Trend Following

As shown in Figure II, we can see that a hypothetical implementation of this strategy on large-cap U.S. equities has historically exhibited a convex return profile with respect to the underlying U.S. equity index, meaningfully participating in positive return years while reducing exposure to significant loss years.

“Risk Cannot Be Destroyed, Only Transformed.”

While the flexibility of trend equity strategies gives them the opportunity to both protect and participate, it also creates the potential for losses due to “whipsaw.”  Whipsaws occur when the strategy changes positioning due to what appears to be a change in trend, only to have the market rapidly reverse course.  Such a scenario can lead to ”buy high, sell low” and “sell low, buy high” scenarios.  These scenarios can be exacerbated by the fact that trend equity strategies may go several years without experiencing whipsaw to only then suddenly experience multiple back-to-back whipsaw events at once.

As Defensive Equity

The most obvious implementation of trend equity strategies is within a defensive equity sleeve.  In this approach, an allocation for the strategy is funded by selling strategic equity exposure (see Figure III).  Typically combined with other defensive styles (e.g. minimum volatility, quality, et cetera), the goal of a defensive equity sleeve is to provide meaningful upside exposure to equity market growth while reducing downside risk.

This implementation approach has the greatest potential to reduce a policy portfolio’s exposure to downside equity risk and therefore may be most appropriate for investors for whom ”failing fast” is a critical threat.  For example, pre-retirees, early retirees, and institutions making consistent withdrawals are highly subject to sequence risk and large drawdowns within their portfolios can create significant impacts on portfolio sustainability.

The drawback of a defensive equity implementation is that vanilla trend equity strategies can, at best, keep up with their underlying index during strong bull markets (see Figure IV).  Given the historical evidence that markets tend to be up more frequently than they are down, this can make this approach a frustrating one to stick with for investors.  Furthermore, up-capture during bull markets can be volatile on a year-to-year basis, with low up-capture during whipsaw periods and strong up-capture during years with strong trends.  Therefore, investors should only allocate in this manner if they plan to do so over a full market cycle.

Implementation within a Defensive Equity sleeve may also be a prudent approach with investors for whom their risk appetite is far below their risk capacity (or even need); i.e. investors who are chronically under-allocated to equity exposure.  Implementation of a strategy that has the ability to pro-actively de-risk may allow investors to feel more comfortable with a larger exposure.

Finally, this approach may also be useful for investors seeking to put a significant amount of capital to work at once.  While evidence suggests that lump-sum investing (“LSI”) almost always out-performs dollar cost averaging (”DCA”), investors may feel uncomfortable with the significant timing luck from LSI.  One potential solution is to utilize trend equity as a middle ground; for example, investors could DCA but hold trend equity rather than cash.

Pros

  • Maintains overall strategic allocation policy.
  • May help risk-averse investors more confidently maintain an appropriate risk profile.
  • May provide meaningful reduction in exposure to significant and prolonged equity losses.

Cons

  • High year-to-year tracking error relative to traditional equity benchmarks.
  • Typically under-performs equities during prolonged bull markets (see Figure IV).

As a Tactical Pivot

One creative way of implementing a trend equity strategy is as a tactical pivot within a portfolio.  In this implementation, an allocation to trend equity is funded by selling both stocks and bonds, typically in equal amounts (see Figure V).  By implementing in this manner, the investor’s portfolio will pivot around the policy benchmark, being more aggressively allocated when trend equity is fully invested, and more defensively allocated when trend equity de-risks.

This approach is often appealing because it offers a highly intuitive allocation sizing policy.  The size of the tactical pivot sleeve as well as the mixture of stocks and bonds used to fund the sleeve defines the tactical range around the strategic policy portfolio (see Figure VI).

One benefit of this implementation is that trend equity no longer needs to out-perform an equity benchmark to add value.  Rather, so long as the strategy outperforms the mixture of stocks and bonds used to fund the allocation (e.g. a 50/50 mix), the strategy can add value to the holistic portfolio design.  For example, assume a trend equity strategy only achieves an 80% upside capture to an equity benchmark during a given year.  Implemented as a defensive equity allocation, this up-capture would create a drag on portfolio returns relative to the policy benchmark.  If, however, trend equity is implemented as a tactical pivot – funded, for example, from a 50/50 mixture of stocks and bonds – then so long as it outperformed the funding mixture, the portfolio return is improved due to its tilt towards equities.

Implementation as a tactical pivot can also add potential value during environments where stocks and bonds exhibit positive correlations and negative returns (e.g. the 1970s).

One potential drawback of this approach is that the portfolio can be more aggressively allocated than the policy benchmark during periods of sudden and large declines.  How great a risk this represents will be dictated both by the size of the tactical pivot as well as the ratio of stocks and bonds in the funding mixture.  For example, the potential overweight towards equities is significantly lower using a 70/30 stock/bond funding mix than a 30/70 mixture.  A larger allocation to bonds in the funding mixture creates a higher downside hurdle rate for trend equity to add value during a negative equity market environment.

Pros

  • Lower hurdle rate for strategy to add value to portfolio during positive equity environments.
  • Intuitive allocation policy based on desired level of tactical tilts within the portfolio.
  • May provide cushion in environments where stocks and bonds are positively correlated.

Cons

  • Portfolio may be allocated above benchmark policy to risky assets during a sudden market decline.
  • Higher hurdle rate for strategy to add value to portfolio during negative equity environments.

As a Liquid Alternative

Due to its historically convex return profile and potentially high level of tracking error exhibited over short measurement horizons, trend equity may also be a natural fit within a portfolio’s alternative sleeve.  Indeed, when analyzed more thoroughly, trend equity shares many common traits with other traditionally alternative strategies.

For example, a vanilla trend equity implementation can be decomposed into two component sources of returns: a strategic portfolio and a long/short trend-following overlay.  Trend following can also be directly linked to the dynamic trading strategy required to replicate a long option position.

There are even strong correlations to traditional alternative categories.  For example, a significant driver of returns in equity hedge and long/short equity categories is dynamic market beta exposure, particularly during significant market declines (see Figure VII).  Trend equity strategies that are implemented with factor-based equity exposures or with the flexibility to make sector and geographic tilts may even more closely approximate these categories.

One potential benefit of this approach is that trend equity can be implemented in a highly liquid, highly transparent, and cost-effective manner when compared against many traditional alternatives.  Furthermore, by implementing trend equity within an alternatives sleeve, investors may give it a wider berth in their mental accounting of tracking error, allowing for a more sustainable allocation versus implementation as a defensive equity solution.

A drawback of this implementation, however, is that trend equity will increase a portfolio’s exposure to equity beta.  Therefore, more traditional alternatives may offer better correlation- and pay-off-based diversification, especially during sudden and large negative equity shocks.  Furthermore, trend equity may lead to overlapping exposures with existing alternative exposures such as equity long/short or managed futures.  Investors must therefore carefully consider how trend equity may fit into an already existing alternative sleeve.

Pros

  • Highly transparent, easy-to-understand investment process.
  • Implemented with highly liquid underlying exposures.
  • Investors often given alternatives a wider berth of allowable tracking error than more traditional allocations.

Cons

  • May be more highly correlated with existing portfolio exposures than other alternatives.
  • Potentially overlapping exposure with existing alternatives such as equity long/short or managed futures.

G̷̖̱̓́̀litch

By

On February 19, 2019

In Risk & Style Premia, Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary­

  • Trend following’s simple, systematic, and transparent approach does not make it any less frustrating to allocate to during periods of rapid market reversals.
  • With most trend equity strategies exhibiting whipsaws in 2010, 2011, 2015-2016, and early 2018, it is tempting to ask, “is this something we can fix?”
  • We argue that there are three historically-salient features that make trend following attractive: (1) positive skew, (2) convexity, and (3) a positive premium.
  • We demonstrate that the convexity exhibited by trend equity strategies is both a function of the strategy itself (i.e. a fast- or slow-paced trend model) as well as the horizon we measure returns over.
  • We suggest that it may be more consistent to think of convexity as an element than can provide crisis beta, where the nature of the crisis is defined by the speed of the trend following system.
  • The failure of a long-term trend strategy to de-allocate in Q4 2018 or meaningfully re-allocate in Q1 2019 is not a glitch; it is encoded in the DNA of the strategy itself.

There’s an old saying in Tennessee – I know it’s in Texas, probably in Tennessee – that says, fool me once, shame on – shame on you.  Fool me – you can’t get fooled again!  — George W. Bush

It feels like we’ve seen this play before.  It happened in 2010.  Then again in 2011.  More recently in 2015-2016.  And who can forget early 2018?  To quote Yogi Berra, “It’s déjà vu all over again.”  We’re starting to think it is a glitch in the matrix.

Markets begin to deteriorate, losses begin to more rapidly accelerate, and then suddenly everything turns on a dime and market’s go on to recover almost all their losses within a few short weeks.

Trend following – like the trend equity mandates we manage here at Newfound – requires trends.  If the market completely reverses course and regains almost all of its prior quarter’s losses within a few short weeks, it’s hard to argue that trend following should be successful.  Indeed, it is the prototypical environment that we explicitly warn trend following will do quite poorly in.

That does not mean, however, that changing our approach in these environments would be a warranted course of action.  We embrace a systematic approach to explicitly avoid contamination via emotion, particularly during these scenarios.  Plus, as we like to say, “risk cannot be destroyed, only transformed.”  Trying to eliminate the risk of whipsaw not only risks style pollution, but it likely introduces risk in unforeseen scenarios.

So, we have to scratch our heads a bit when clients ask us for an explanation as to our current positioning.  After all, trend following is fairly transparent.  You can probably pull up a chart, stand a few feet back, squint, and guess with a reasonable degree of accuracy as to how most trend models would be positioned.

When 12-month, 6-month, and 3-month returns for the S&P 500 were all negative at the end of December, it is a safe guess that we’re probably fairly defensively positioned in our domestic trend equity mandates.  Despite January’s record-breaking returns, not a whole lot changed.  12-, 6-, and 3-month returns were negative, negative, and just slightly positive, respectively, entering February.

To be anything but defensively positioned would be a complete abandonment of trend following.

It is worth acknowledging that this may all just be Act I.  Back when this show was screening in 2011 and 2015-2016, markets posted violent reversals – with the percent of stocks above their 50-day moving average climbing from less than 5% to more than 90% – only to roll over again and retest the lows.

Or this will be February 2018 part deux.  We won’t know until well after the fact.  And that can be frustrating depending upon your perspective of markets.

If you take a deterministic view, incorrect positioning implies an error in judgement.  You should have known to abandon trend following and buy the low on December 24th.  If you take a probabilistic view, then it is possible to be correctly positioned for the higher probability event and still be wrong.  The odds were tilted strongly towards continued negative market pressure and a defensive stance was warranted at the time.

We would argue that there is a third model as well: sustainability (or, more morbidly, survivability).  It does not matter if you have a 99% chance of success while playing Russian Roulette: play long enough and you’re eventually going to lose.  Permanently.  Sustainability argues that the low-probability bet may be the one worth taking if the payoff is sufficient enough or it protects us from ruin.

Thus, for investors for whom failing fast is a priority risk, a partially defensive allocation in January and February may be well warranted, even if the intrinsic probabilities have reversed course (which, based on trends, they largely had not).

But sustainability also needs to be a discussion about being able to stick with a strategy.  It does not matter if the strategy survives over the long run if the investor does not participate.

That is why we believe transparency and continued education are so critical.  If we do not know what we are invested in, we cannot set correct expectations.  Without correct expectations, everything feels unexpected.  And when everything feels unexpected, we have no way to determine if a strategy is behaving correctly or not.

Which brings us back to trend equity strategies in Q4 2018 and January 2019.  Did trend equity behave as expected?

Trend following has empirically exhibited three attractive characteristics:

  • Positive Skew: The return distribution is asymmetric, with a larger right tail than left tail (i.e. greater frequency of larger, positive returns than large, negative returns).
  • Convex Payoff Profile: As a function of the underlying asset the trend following strategy is applied on, upside potential tends to be greater than downside risk.
  • Positive Premium: The strategy has a positive expected excess return.

While the first two features can be achieved by other means (e.g. option strategies), the third feature is downright anomalous, as we discussed in our recent commentary Trend: Convexity & Premium.  Positive skew and convexity create and insurance-like payoff profile and therefore together tend to imply a negative premium.

The first two characteristics make trend following a potentially interesting portfolio diversifier.  The last element, if it persists, makes it very interesting.

Yet while we may talk about these features as historically intrinsic properties of trend following, the nature of the trend-following strategy will significantly impact the horizon over which these features are observed.  What is most important to acknowledge here is that skew and convexity are more akin to beta than they are alpha; they are byproducts of the trading strategy itself.  While it can be hard to say things about alpha, we often can say quite a bit more about beta.

For example, a fast trend following system (typically characterized by a short lookback horizon) would be expected to rapidly adapt to changing market dynamics.  This allows the system to quickly position itself for emerging trends, but also potentially makes the strategy more susceptible to losses from short-term reversals.

A slow trend following system (characterized by a long lookback period), on the other hand, would be less likely to change positioning due to short-term market noise, but is also therefore likely to adapt to changing trend dynamics more slowly.

Thus, we might suspect that a fast-paced trend system might be able to exhibit convexity over a shorter measurement period, whereas a slow-paced system will not be able to adapt rapidly.  On the other hand, a fast trend following system may have less average exposure to the underlying asset over time and may compound trading losses due to whipsaw more frequently.

To get a better sense of these tradeoffs, we will construct prototype trend equity strategies which will invest either in broad U.S. equities or risk-free bonds.  The strategies will be re-evaluated on a daily basis and are assumed to be traded at the close of the day following a signal change.  Trend signals will be based upon prior total returns; e.g. a 252-day system will have a positive (negative) signal if prior 252-day total returns in U.S. equity markets are positive (negative).

Below we plot the monthly returns of a ­-short-term trend equity system (21 day)- and a -long-term trend equity system (252 day)- versus U.S. equity returns.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not a guarantee of future results.  All returns are hypothetical and backtested.  Returns are gross of all fees.  For the avoidance of doubt, neither the Short-Term nor Long-Term Trend Equity strategy reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary.  It is not possible to invest in an index.

We can see that the fast-paced system exhibits convexity over the monthly measurement horizon, while the slower system exhibits a more linear return profile.

As mentioned above, however, the more rapid adaptation in the short-term system might cause more frequent realization of whipsaw due to price reversals and therefore an erosion in long-term convexity.  Furthermore, more frequent changes might also reduce long-term participation.

We now plot annual returns versus U.S. equities below.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not a guarantee of future results.  All returns are hypothetical and backtested.  Returns are gross of all fees.  For the avoidance of doubt, neither the Short-Term nor Long-Term Trend Equity strategy reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary.  It is not possible to invest in an index.

We can see that while the convexity of the short-term system remains intact, the long-term system exhibits greater upside participation.

To get a better sense of these trade-offs, we will follow Sepp (2018)1 and use the following model to deconstruct our prototype long/flat trend equity strategies:

By comparing daily, weekly, monthly, quarterly, and annual returns, we can extract the linear and convexity exposure fast- and slow-paced systems have historically exhibited over a given horizon.

Below we plot the regression coefficients (“betas”) for a fast-paced system.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not a guarantee of future results.  All returns are hypothetical and backtested.  Returns are gross of all fees.  For the avoidance of doubt, the Short-Term Trend Equity strategy does not reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary.  It is not possible to invest in an index.

We can see that the linear exposure remains fairly constant (and in line with decompositions we’ve performed in the past which demonstrate that long/flat trend equity can be thought of as a 50/50 stock/cash strategic portfolio plus a long/short overlay2).  The convexity profile, however, is most significant when measured over weekly or monthly horizons.

Long-term trend following systems, on the other hand, exhibit negative or insignificant convexity profiles over these horizons.  Even over a quarterly horizon we see insignificant convexity.  It is not until we evaluate returns on an annual horizon that a meaningful convexity profile is established.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not a guarantee of future results.  All returns are hypothetical and backtested.  Returns are gross of all fees.  For the avoidance of doubt, the Long-Term Trend Equity strategy does not reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary.  It is not possible to invest in an index. 

These results have very important implications for investors in trend following strategies.

We can see that long-term trend following, for example, is unlikely to be successful as a tail risk hedge for short-term events.  Short-term trend following may have a higher probability of success in such a scenario, but only so long as the crisis occurs over a weekly or monthly horizon.3

Short-term trend following, however, appears to exhibit less convexity with annual returns and has lower linear exposure.  This implies less upside capture to the underlying asset.

Neither approach is likely to be particularly successful at hedging against daily crises (e.g. a 1987-type event), as the period is meaningfully shorter than the adaptation speed of either of the strategies.

These results are neither feature nor glitch.  They are simply the characteristics we select when we choose either a fast or slow trend-following strategy.  While trend-following strategies are often pitched as crisis alpha, we believe that skew and convexity components are more akin to crisis beta.  And this is a good thing.  While alpha is often ephemeral and unpredictable, we can more consistently plan around beta.

Thus, when we look back on Q4 2018 and January 2019, we need to acknowledge that we are evaluating results over a monthly / quarterly horizon.  This is fine if we are evaluating the results of fast-paced trend-following strategies, but we certainly should not expect any convexity benefits from slower trend models.  Quite simply, it all happened too fast.

Conclusion

When markets rapidly reverse course, trend following can be a frustrating style to allocate to.  With trend equity styles exhibiting whipsaws in 2010, 2011, 2015-2016, and early 2018, the most recent bout of volatility may have investors rolling their eyes and thinking, “again?”

“Where’s the crisis alpha?” investors cry.  “Where’s the crisis?” managers respond back.

Yet as we demonstrated in our last commentary, two of the three salient features of trend following – namely positive skew and positive convexity – may be byproducts of the trading strategy and not an anomaly.  Rather, the historically positive premium that trend following has generated has been the anomaly.

While the potential to harvest alpha is all well and good, we should probably think more in the context of crisis beta than crisis alpha when setting expectations.  And that beta will be largely defined by the speed of the trend following strategy.

But it will also be defined by the period we are measuring the crisis over.

For example, we found that fast-paced trend equity strategies exhibit positive convexity when measured over weekly and monthly time horizons, but that the convexity decays when measured over annual horizons.

Strategies that employ longer-term trend models, on the other hand, fail to exhibit positive convexity over shorter time horizons, but exhibit meaningful convexity over longer-horizons.  The failure of long-term trend strategies to meaningfully de-allocate in Q4 2018 or rapidly re-allocate in Q1 2019 is not a glitch: it is encoded into the DNA of the strategy.

Put more simply: if we expect long-term trend models to protect against short-term sell-offs, we should prepare to be disappointed.  On the other hand, the rapid adaptation of short-term models comes at a cost, which can materialize as lower up-capture over longer horizons.

Thus, when it comes to these types of models, we have to ask ourselves about the risks we are trying to manage and the trade-offs we are willing to make.  After all, “risk cannot be destroyed, only transformed.”

 

 

Trend: Convexity & Premium

By

On February 11, 2019

In Trend, Weekly Commentary

Available as a PDF download here.

Summary­

  • Trend following is unique among style premia in that it has historically exhibited a convex payoff profile with positive skew.
  • While the historical premium is anomalous, the convexity makes sense when we use options to replicate trend following strategies.
  • We explore reasons why frequent rebalancing in trend following strategies is necessary and decompose the return contributions from different portions of the option replication model.
  • Most of the historical premium associated with trend following comes from the trading impact that is tied to the asset’s Sharpe ratio rather than the convexity.
  • By separating the impacts of convexity and trading impact, we can gain a deeper understanding of the types of risk exchanges that come with investing using trend following strategies.

Introduction

Unlike many of the other style premia, trend following has historically exhibited a convex payoff profile with positive skew.  In less mathematical terms, that means it tends to harvest many small losses (due to reversals) and just a few large gains (when trends take off).

This is unique, as most risk and style premia exhibit the opposite: concave payoffs with negative skew.

To simplify, we can think of concave, negative-skew trades as akin to selling insurance, while convex, positive-skew trades are akin to buying it.  With respect to traditional financial literature, the mental model of selling insurance makes quite a bit of sense, as we can think of expected excess returns as being the reward earned for being willing to bear the risk others wish to transfer away.

Trend following, then, is a bit of an anomaly.  Not because it exhibits a convex, positive skew profile, but because it does so and has historically exhibited a positive premium.  That is not supposed to happen – you don’t expect to profit when you buy insurance – and it leaves many scratching their head asking, “could this be an actual market anomaly?”

Unfortunately, a discussion of why trend following works often conflates the convexity of the strategy with the oddness of the historically positive premium.  The latter is, for sure, anomalous.  But what we hope to show in this commentary is that the former is just a byproduct of the trading strategy itself and does not require any investor misbehavior.

How will we do this?  Whenever we talk about buying or selling insurance, a very natural language to use is that of put and call options.  Thus, our goal in this commentary is to approach trend following through the lens of options and demonstrate that simple trend-following strategies can be thought of as naively replicating the pay-off of a straddle.

In doing so, our goal is to differentiate between two key elements of trend following: the convexity of returns it exhibits and the historically positive premium it has generated.

Please note, for all the option geeks out there, that this will be a highly simplified interpretation with lots of hand-waving.  For example, we will generally assume that interest rates are zero, dividends do not exist, that price is continuous (i.e. no jumps) and there are no trading frictions.  None of this is true, of course, but we do not think it meaningfully takes away from the intuition established.

Pricing is Replication

A foundational principle in financial engineering is the Law of One Price, which states that any two securities with identical future payouts, no matter how the future turns out, must have an identical price.  Otherwise, we could construct an arbitrage.

Thus, to price an option, we only need to replicate its payoff.  This is, of course, easier said than done, as options have non-linear payoffs.  A call option, for example, pays nothing when price is below the strike at maturity and pays the difference between price and the strike otherwise.

Source: Wikipedia

Given access to a risk-free bond and the underlying stock, we cannot easily replicate this payoff with some sort of static portfolio.  We can, however, attempt to replicate it using a dynamic trading strategy that adjusts our mixture of the stock and bond over time.

Consider the following example: a stock is priced at $80 and will be worth either $100 or $60 in one year.  We have a call option with a strike of $90.  This means if the stock ends up at $100, the payoff will be $10 while if the stock ends up at $60, the payoff will be zero.

How can we replicate this?

We need to solve the simultaneous equations:

$100Δ – B(1+r) = 10
$60Δ – B(1+r) = 0

Here, Δ is the number of shares of stock to buy, B is how much to borrow, and r is the risk-free rate.  Given r, we can solve the equations for the replicating portfolio.   If we assume r=0%, we find that Δ=0.25 and therefore B=$15.

Thus, to replicate the call option, we need to borrow $15 and buy 0.25 shares at $80, for a total cost of $5.  Since this portfolio replicates the option payout exactly, this must also be the price of the option!

Of course, these are highly simplified assumptions.  But if we collapse the time period down from 1-year to an infinitesimally small unit of time, we can repeat this exercise over and over such that we have a dynamic trading strategy that will replicate the option’s final payoff, and therefore the option’s value over time.

(For a more thorough – but still highly accessible – introduction to this concept, we recommend Emanuel Derman’s The Boy’s Guide to Pricing & Hedging.)

Trend Following is “Long Gamma”

Let us now connect trend following to options.

Consider the following case where the underlying stock price follows a binomial tree and we want to replicate the payoff of a call option with a strike of $80.  Again, to simplify things, we will just assume that our risk-free rate is 0%.

Let’s start when price is at $90.  At this point, to replicate the final payoff, we have the following two equations:

$100Δ – B = 20
$80Δ – B = 0

Again, solving simultaneously, we find that we need to buy 1 share (at $90) and therefore B=$80.  The option, therefore, is equal to $10.

Now let’s consider the bottom case when the stock price is $70,

$80Δ – B = 0
$60Δ – B = 0

In this case, we find that the number of shares is equal to zero and therefore B=$0.

Finally, using these prices for the option in those two states, we can step back to the starting case, where we now know:

$90Δ – B = 10
$70Δ – B = 0

Here we find that the number of shares is equal to 0.5 and B is equal to $35, making the option therefore worth $5.

This highly simplified model tells us that:

  • The number of shares held in the replicating portfolio informs how sensitive the option price is to movement in the underlying stock. Note that at the initial step the option value was $5 and the number of shares held was 0.5.  As price changed by +/- $10, the value of the option changed by +/- $5.  The more shares held, the more sensitive the option price is to the stock price change.  In option’s parlance, this is known as the option’s “delta.”
  • The number of shares required by the replicating portfolio also changed based upon changes in the underlying stock price. In option’s parlance, this is known as an option’s “gamma.” This measures how the options delta changes with changes in the stock price.

Note that as the stock price increased, the number of shares required to replicate the option increased.  This implies that a call option has positive gamma.

If we repeated this whole exercise but used a put option instead, we would similarly find that a put option has positive gamma: the more price depreciates below the strike, the more shares we need to short to replicate.

Which means that these options can be roughly replicated using a very naïve trend-following strategy.

If we buy a put and a call at the same strike and same maturity, we have constructed a trade known as a “straddle”.  We plot an example straddle payoff profile below.

Source: theoptionsguide.com

Since a long straddle is simply the combination of a put and a call, we can replicate its payoff by just replicating both positions independently and summing up our total exposure.

It should come as no surprise that the replication of this straddle is, in essence, a trend following strategy.  As the underlying stock price increases, we buy shares, and as underlying stock price decreases, we sell shares.

With this naïve model, we can already see a few interesting trade-offs:

  • The convexity of trend following may have nothing to do with any sort of market “anomaly,” but rather is a function of the trading strategy employed.
  • Purchasers of a straddle will realize the payoff minus the up-front cost of the options, which will be a function of implied volatility. The replicating trend-following strategy will realize the same payoff minus the trading costs, which will be a path-dependent function of returns (and, therefore, realized volatility).

For a more nuanced dive into deriving this relationship, we recommend the paper “Tail protection for long investors: Trend convexity at work” by Dao, Nguyen, Deremble, Lempérière, Bouchaud, and Potters (2016).

Straddle-ish

While we have demonstrated that a straddle can replicated with a (continuous) trend-following strategy, it is not the traditional trend-following archetype by any means.  Furthermore, trend-following strategies are continuous in nature, while straddles have a defined expiration date.

So, let us consider a more realistic (albeit, still a toy) trend-following implementation.  We will go long the S&P 500 when its prior 12-month return is positive and short when it is negative.  We will rebalance the strategy at the end of each month.  For simplicity, we will assume any available capital is invested in risk-free bonds that return 0%.

How might we translate this strategy into a semi-equivalent straddle replication?  One interpretation may be that at the end of each month, we use the price from 12 months ago to set the strike of our straddle.  To compute the delta, we will lean on the Black-Scholes equation, where we will assume that the time until expiration is one month, we’ll assume no dividend payment, and we’ll use prior short-term realized volatility as our input for implied volatility.1  Since the delta will vary between -1 and 1, we will use it as our allocation to the S&P 500, investing remaining capital in risk-free bonds that return 0%.

It is important to note that in the prior section, the simplified trend following strategy replicated the straddle payoff because we were able to delta hedge over infinitesimally small time horizons at zero cost.  Here, we are rebalancing monthly, applying a much more static mode of replication.

Below we plot the growth of $1 in each strategy.  The correlation in monthly log-returns between the two strategies is 95.8%.

Source: CSI Data.  Calculations by Newfound Research.  Returns are backtested and hypothetical.  Returns assume the reinvestment of all distributions.  Returns are gross of all fees except for underlying ETF expense ratios.  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. 

We can see that a trend following strategy looks incredibly similar to a strategy that replicates a straddle.  Why?  Let’s look at the trend response function versus delta.

Note that the x-axis is measured in standard deviations from the strike.  This is important.  A 10% move in one asset class may be highly significant while a 10% move in another may not.  Furthermore, a 10% move in one market environment may be significant while a 10% move in another may not.  The delta function captures this as implied volatility is an input to the measure, while the trend signal does not.

We can see that the binary trend signal and the straddle delta are nearly identical when price is greater than 0.6 standard deviations from the strike in either direction. In this model, this can be interpreted as a strong trend where the trend strategy allocation and delta straddle allocation will coincide.

Within that range, however, we can see that the trend signal over-estimates the delta.  Therefore, in cases where price continues away from the strike, the binary signal will out-perform and in cases where price reverts back towards the strike, the binary signal will under-perform.

We can use this insight to explore a few questions.  For example, why do trend following strategies have to be traded frequently?  Let’s consider the case where we only rebalance quarterly.  Note what happens to the delta function of the straddle:

We can see that the trend signal and the straddle delta only meet when price is 1.2 standard deviations from strike.  The delta is taking into account how long there is until expiration and therefore adjusting itself downward in magnitude in acknowledgement that price might revert back towards the strike.  The binary signal does not.

So, is the answer just to rebalance as frequently as possible?  After all, as the ratio of the rebalance period to the lookback period goes to zero, the shape of the delta function approaches the binary trend signal.  Conversely, as the ratio goes to infinity (i.e. the holding period length far exceeds the lookback), the shape of the delta function approaches y=0 at all points.

The answer in the real world, where prices are not continuous, we are not trading infinitesimally small horizons, and there are trading costs, is “no.”

But if we just statically replicate the remaining time until option expiration, doesn’t that remove the entire long gamma aspect?  Have we not lost the convexity created by the trading strategy?

In an attempt to answer this question, we can ask a slightly different one: “how different is the change in delta from rolling into the new straddle versus replicating the original straddle?”  We derive the math in the appendix, but under some general assumptions we can say that the deltas will converge when the ratio of the rebalance time-step and the time until option expiration goes to zero.

dt/(T-t) → 0

As this ratio becomes smaller, therefore, the delta change from rolling into the new straddles will approximate the delta change of replicating the prior straddle, and thus conserve the replicating strategy’s natural convexity.

There are two important drivers for this limit here: we want dt to be as small as possible and T-t to be as large as possible.

Unfortunately, this implies that we are delta hedging over an infinitesimally small time period at the beginning of the life of the straddle, a time at which the delta is approximately zero because it is struck at-the-money!  On the other hand, if we go towards expiration, T-t goes to zero!

The dilemma at hand, then, is for a simple binary long/short trend strategy to reflect the delta of a straddle, it needs to be close to expiration when the delta function looks more like a step change.  However, for the changes in delta from rolling the straddle position to reflect changes in delta from replication, the roll must occur near inception!

Using this information, we can attempt to get a rough approximation of how much a binary trend strategy’s return comes from: (1) the replication of a straddle, (2) excess delta exposure from rolling straddle exposure, and (3) excess delta exposure from using a binary signal.

Specifically, rebalancing our portfolios once a month we will:

  • Calculate the delta of a 24-month straddle with 12-months left until expiration. This allows the straddle to reflect the same strike price as the rolling and binary trend signals, but with enough life until expiration that dt/(T-t) is small enough that the delta may closely reflect the delta from replication.
  • Calculate the delta of a 13-month straddle with 1-month until expiration. Calculate the difference in delta exposure from this step to the last and label this the excess delta from taking a rolling approach.
  • Finally, calculate the binary trend signal and calculate the difference in exposure versus the rolling approach. This will reflect the excess exposure from our binary approximation.

Below we plot the total return from each of these three series.

Source: CSI Data.  Calculations by Newfound Research.  Returns are backtested and hypothetical.  Returns assume the reinvestment of all distributions.  Returns are gross of all fees except for underlying ETF expense ratios.  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.

While we can see that replication has had a meaningful positive return, we can also see that the rolling model has a non-insignificant positive impact on returns.  The trend model, despite being a net contributor through 2012, was more-or-less insignificant over the full period.

However, there is a bit of a head-scratcher here worth discussing: why would we expect the replication of a straddle to have a positive return?  Positive convexity, sure.  But a positive return?

Convexity versus Premium

Bruder and Gaussel (2011)2 suggest that any single-asset trading strategy can be broken down into two component pieces: an option profile and trading impact.  A simple constant stop-loss level, for example, can be thought of as a perpetual call option payoff with trading costs that alter exposure between long and flat, creating frictions due to non-continuous prices and delays in rebalancing.

For trend-following strategies, they suggest a framework whereby the trend is measured as an exponential moving average of daily returns and exposure is proportional to this measured trend, the asset’s variance, and the investor’s risk tolerance (a suggestion that is theoretically consistent with an optimal Markowitz/Merton strategy).

With these assumptions in hand, they introduce the following proposition:

Using this proposition, we can decompose long/short trend equity strategies into these two component pieces.  Using this model, Jusselin, Malongo, Roncalli, Lezmi, Masselin, and Dao (2017)3 demonstrate that theoretical model returns can be decomposed as:

Following this model, we can extract the option payoff (G) and trading impact (g) of a trend following S&P 500 strategy.

What is clear from this analysis is that trend following strategies have two different components driving returns.

The first is the underlying option payoff.  In the Bruder and Gaussell (2011) model, the option is similar to a straddle struck on the trend of the underlying asset.  We can see in the graph above that this component has a short memory and provides much of the convexity often associated with trend-following returns.

The second component is the trading impact.  We can see that this component is low frequency and the driving factor is the realized Sharpe ratio.  If we squint our eyes a bit at the return formula, it even resembles the gamma gain minus gamma loss of a traditional straddle delta-hedging strategy.4

We can see that the long-term driver of returns in trend-following strategies, then, is not the convexity, but rather the trading impact.  Given that trend following has positive gamma, we would expect the trading impact to be positive for returns that exhibit autocorrelation (i.e. trending returns).  Interestingly, Jusselin, Malongo, Roncalli, Lezmi, Masselin, and Dao (2017) demonstrate that autocorrelation may not even be a necessary component for positive returns; rather, for this particular trend following model, the trading impact will have positive profit or loss based upon the underlying asset’s Sharpe ratio.

It is worth acknowledging that this analysis is based upon a particular exposure model that is driven by the underlying asset’s realized trend, volatility, and the investor’s risk tolerance.  Thus, the delta-based model and binary signal model explored in prior sections will not match as neatly.  Nevertheless, it serves as further evidence for trend following’s inherent return convexity with respect to the underlying asset.

Conclusion

Trend following is unique among style premia in that it has historically exhibited a convex payoff profile with positive skew. By replicating example trend following strategies using straddle options, we demonstrated how convexity is inherent to trend following strategies outside of any historical premium.

While the historical premium is anomalous, the convexity makes sense.

By replicating the payoff of a rolling straddle strategy, we saw potential reasons why frequent rebalancing in trend following strategies is necessary and were able to decompose the return contributions of the replication, the rolling model, and a binary trend following approach under more realistic assumptions.

In the simplified model we found that most of the historical premium associated with trend following comes from the trading impact that is tied to the asset’s Sharpe ratio.

A key step in sticking with any trading strategy is an understanding of why it may work in the future. A good historical backtest is nice to see, but there has to be a reason – be it behavioral, economic, or structural – for the backtest to have any reliability.

Using options to isolate the return sources in trend following strategies is a way to separate the impacts of convexity and trading impact while gaining a deeper understanding the types of risk exchanges that come with investing using trend following strategies.

 

 

No Pain, No Premium

By

On February 4, 2019

In Popular, Risk & Style Premia, Risk Management, Weekly Commentary

Summary

  • In this commentary, we discuss what we mean by the phrase, “no pain, no premium.”
  • We re-frame the discussion of portfolio construction from one about returns to one about risk and argue that without risk, there should be no expectation of return.
  • With a risk-based framework, we argue that investors inherently act as insurance companies, earning a premium for bearing risk.  This risk often manifests as significant negative skew and kurtosis in the distribution of asset returns.
  • We introduce the philosophical limits of diversification, arguing that we should not be able to eliminate risk from the portfolio without eliminating return as well.
  • Therefore, we should seek to eliminate uncompensated risks while diversifying across compensated ones.
  • We explore the three axes of diversification – what, how, and when – and demonstrate how thinking in a correlation-driven, payoff-driven, and opportunity-driven framework may help investors find better diversification.

1. Is it About Risk or Return?

For graduate school, I pursued my Masters of Science in Computational Finance at Carnegie Mellon University.  One of the first degrees of its kind in the late 1990s, this financial engineering program is a cross-disciplinary collaboration between the finance, mathematics, statistics, and computer-science departments.

In practice, it was an intensive year-and-a-half study on the theoretical and practical considerations of pricing financial derivatives.

I do not recall quite when it struck me, but at some point I recognized a broader pattern at play in every assignment.  The instruments we were pricing were always about the transference of risk in some capacity.  Our goal was to identify that risk, figure out how to isolate and extract it, package it into the appropriate product type, and then price it for sale.

Risk was driving the entire equation.  Pricing was all about understanding distribution of the potential payoffs and trying to identify “fair compensation” for the variety of risks and assumptions we were making.

For every buyer, there is a seller and vice versa and, at the end of the day, sellers who did not want risk and would have to compensate buyers to bear it.

1.1 Stocks for the Long Run

The idea that reward is compensation for risk is certainly not a new one.  It is, more or less, the entire foundation of modern finance.

But sometimes, it seems, we forget it.

We are often presented with a return-based lens through which to evaluate the world of finance.  Commonly reprinted are graphs like the one below, demonstrating century-long returns for stocks, bonds, and cash and accompanied by broad, sweeping generalizations like, “stocks for the long run.”

The truth is, if you plot anything on a log-axis over a long enough time horizon and draw it with a thick enough crayon, the line will eventually look pretty straight.

But if we zoom in to a horizon far more relevant to the lifecycle of most individual investors, we see a very different picture.

What we see is the realization of risk.  We have to remember that the excess returns we expect to earn over the long run are compensation for bearing risk.  And that risk needs to manifest, from time-to-time.  Otherwise, if the probability of the risk being realized goes down, then so should the excess premium we expect to earn.

From a quantitative perspective, risk is often measured as volatility.  In our opinion, that’s not quite right.  We believe, given a long enough return history with enough realized risk events, risk can be better measured in a return’s distribution symmetry and fat-tailed-ness (i.e. “skew” and “kurtosis” respectively).

Below we plot the annualized excess real return distribution for U.S. equities over the last 100 years.  We can see that the distribution is “leaning” to the right, indicating that large losses are more frequent than large gains.

We would argue that when we buy equities, what we are really buying is a risk.  In particular, we are buying an uncertain stream of cash flows.

Now, this might seem a little weird.  Why would we ever pay someone to bear their risk?

The answer is because, in many ways, we can think of equities as a swap of cashflows: one up-front bullet payment for the rights to an uncertain stream of future cash flows generated by the underlying business.

In theory, the price we pay today should be less than the net present value of all those future cash flows, with the difference representing the premium we expect to earn over time.

Uncertainty is the wedge between the values.  Without uncertainty, no rational seller would give up their future cash flows for less than they are worth (or, if we do have an irrational seller, we would expect buyers to compete over those cashflows to the point they are fairly valued).

Thus, the premium will be driven both by certainty about the future cash flows (growth rate and duration) as well as the market’s appetite for bearing risk.

The more certain we are of those future cash flows or the higher the market’s appetite to bear risk, the smaller the expected premium should be.

1.2 “Funding Secured”

To get a better sense of the play between certainty and premium in the market, we can explore an example where we effectively collapse price into a binary “yes or no” event.

On August 7th, 2018, Elon Musk sent out the following tweets:

At the time he sent the tweet, Tesla shares were trading around approximately $365.  The stock had opened around $340 that day and had jumped on news reporting that the Saudi sovereign fund had built a $2b stake in Tesla and some speculation about a potential buy-out.

Now let’s assume, for a moment, that Elon’s tweet said, “Deal struck to take Tesla private at $420, effectively immediately”  What should the price of Tesla’s stock jump to?  $420, of course.

Now Elon’s tweet merely said he was considering it.  He also did not specify a timeline.  But let’s consider two cases:

  • The market believes a deal will be struck to take Tesla at $420 in the near future.
  • The market does not believe Tesla will be taken private.

In the former case, the right price is approximately $420.  In the latter case, the appropriate price is whatever the shares were trading at before the announcement.1

Thus, where price trades between the two points can be interpreted as to the market’s confidence in the deal being done.

Hence, I tweeted the following:

(Note that when I sent out the first tweet, I hadn’t realized trading had been halted in Tesla.)

Assuming the entire day’s move was attributed to the buyout news, a price change from $340 to $380 only represents a 50% move towards the buy-out price of $420.  The market was basically saying, “we give this coin-flip odds.”

1.2 Well ‘Skews Me

While modern portfolio theory uses volatility as the measure of risk, the connection between excess realized premia and volatility is tenuous at best.  It certainty falls apart in highly skewed, fat-tailed return distributions.

Rather, skewness appears to be a much better measure of risk for most financial assets.  And when we look at equity markets around the globe, we see the same fact pattern emerge: return distributions with negative skew indicating that losses tend to be (much) bigger than gains.

2. You’re An Insurance Company

What this type of risk-based thinking all boils down to is that you – and your portfolio – are really acting as an insurance company of sorts.

When we purchase insurance, we are really transferring our associated risk to the insurance company.  To incentivize them to bear the risk, we have to pay an annual premium.

Similarly, when we buy stocks, we are really trading a certain cashflow today (the price) for a stream of uncertain cash flows in the future.  The discount between the price we pay and the net present value of future cash flows is the premium we expect to earn.  And when we sell stocks, we are effectively paying that premium.

So in building our portfolios, we should think like an insurance company.

Like an insurance company, we want to diversify the premiums we earn.  Not only do we want to diversify within a given type of insurance, but we probably also want to diversify the type of insurance we offer.  And, in an ideal world, the type of insurance would be uncorrelated!

2.1 Diversifying with Bonds

Enter the most traditional portfolio diversifier: bonds.  Typically considered to be a “safe” asset, if we look at them through the lens of real excess returns, we can see that bond returns also exhibit negative skew and fat tails.

This makes sense, as when we buy a bond we are still bearing all sorts of risks.  Not only do we bear the risk of a default, but we also bear inflation risk and interest rate path-dependency risk.

With U.S. Treasuries, default risk is likely minimized (depending on your perspective), and the other two risks might be less correlated than the traditional risks (e.g. economic growth) we see with equities.  So combining stocks and bonds should help us control skew, right?

Well, not quite.  Below we plot the annualized excess real returns for a 60/40 portfolio.

We see that skew and kurtosis remain.  What gives?

Well, one answer is that while a 60/40 portfolio might be close to balanced in the terms of notional dollar exposure to each asset, it is completely unbalanced from the perspective of residual volatility.

Below we plot the relative contribution to risk of stocks and bonds over time in a 60/40 portfolio.

Because the payout for bonds is far more certain than the payout for stocks, not only is the expected excess premium much lower, but volatility tends to be much lower as well.  This means that the premium earned from holding bonds is not large enough to offset the losses realized in equities.

Savvy readers will recognize this as the driving thesis behind risk parity.  To strike a balance, we need to allocate to stocks and bonds in such a manner that they provide equal contribution to portfolio risk.

Below, we plot the annual excess real return distribution for a stock/bond risk parity portfolio that is levered to a constant volatility target of 8%.

What do we see?  Skew and fat tails remain.  Perhaps the answer is simply that we need more diversification.  While in practice this might mean buying different assets, in theory it means exposing ourselves to different types of risk sources that lead to uncertainty in the value of future cash flows.  We enumerate a few below.

In traditional asset allocation, trying to isolate and add these different exposures is very difficult.

First, it is worth acknowledging that not every type of risk necessarily deserves to earn compensation.  In theory, we should only be compensated for un-diversifiable risks.

Furthermore, many of these risks have time-varying correlations and magnitudes, and often collapse towards a single risk factor during crisis states of the world.

Yet we would argue that there is a deeper, philosophical limit we should consider.

3. The Philosophical Limits of Diversification

What we keep running up against is what we call the “philosophical limit of diversification.”

The simplest way to think about the limit is this: If we can diversify away all of our risk, we should not expect to earn any reward.

After all, if we found some magical combination of assets that eliminated downside risk in all future states of the world, we would have constructed an arbitrage.  We could simply borrow at the risk-free rate, invest in the appropriate blend of assets, and reap our risk-free reward.

That is why years like 2018, when 90% of assets lose money, have to occur from time to time.  Without the eventual realization of risk, there is no reason to expect return.

3.1 The Frustrating Law of Active Management

A corollary of this philosophical limit is what we like to call “The Frustrating Law of Active Management.”

We go further in depth into this idea in another commentary, but the basic idea follows: if an investment strategy is perceived both to have alpha and to be easy, investors will allocate to it and erode the associated premium.

How can a strategy be “hard”?  Well, a manager might have a substantial informational or analytical edge.  Or, the manager might have a structural moat, accessing trades others do not have the opportunity to pursue.

But for most well-known edges (e.g. most major style premia), “hard” is going to be behavioral.  The strategy has to be hard enough to hold on to that it does not get arbitraged away.  Which implies that,

For any disciplined investment approach to outperform over the long run, it must experience periods of underperformance in the short run.

This also implies that,

For any disciplined investment approach to underperform over the long run, it must experience periods of outperformance in the short run.

For active managers, the frustration is that not only does their investment approach have to under-perform from time-to-time, but bad strategies will have to out-perform.  The latter may seem confusing until we consider that a purposefully bad strategy could simply be inverted to create a purposefully good one.2

And, as above, we cannot simply diversify our way out of the problem.  After all, if there were a magic combination of active strategies that earned the same expected alpha but reduced the risk, everybody would pursue that combination.

4. Investment versus Investor Returns

So is the answer here to just, “suck it up?”  Do we simply look at periods like 2000-2010 and say, “it’s the price we pay for the opportunity to earn long-run returns?”

We would argue both “yes” and “no.”

It all depends upon where an investor falls within their lifecycle.  Young investors who are pursuing growth mandates may simply need to accept that skew and fat tails are the cost of earning the long-run premium.  Too much diversification may lead to “failing slow.”

For investors in the later stages of their lifecycle, however, the math changes.  Indeed, this is true for any individual or institution where withdrawals are concerned.  When we have a withdrawal-driven mandate, it is the risk of “failing fast” that we need to concern ourselves with.

The problem is that investment-centric thinking often makes diversification seem foolish.  To quote Brian Portnoy, “diversification means always having to say you’re sorry.”

Not only do we have to contend with the fact that the relative performance of the investments in our portfolio will vary wildly from one another year-to-year, but evidence suggests that so will the investor’s utility function.

Consider the graphic below, where the investor’s utility oscillates between relative (“I didn’t do as well as my peers!”) and absolute returns (“I lost money!”), making the diversified profile a consistent loser.

Source: BlackRock.

(3/14/2019 Update: It was pointed out to me that based upon the numbers in the table above, the total return reported the Diversified Portfolio is actually understated.  Total return should be 202.4%, with $100K turning into $302,420.) 

However, if we actually think about investor returns, rather than investment returns, the picture changes.  Below we plot the growth of $1,000,000 since 2000 with a fixed $40,000 withdrawal.  In this highly simplified example, we can begin to see the benefits of increased diversification.

Despite the philosophical limits of diversification, we clearly should not forgo it entirely.  But what is the right framework to think about diversification and how it can be introduced into a portfolio?

5. The Three Axes of Diversification

At Newfound, we talk about three potential axes of diversification that investors can try to exploit.

We call these axes the what, the how, and the when axes, and they aim to capture what we invest in (“correlation driven”), how we make the decisions (“pay-off driven”), and when we make those decisions (“opportunity driven”).

Below, we explore each axis individually and how to might be able to contribute to a portfolio’s overall diversification profile.

5.1 What Axis (“Correlation Diversification”)

The “what” axis asks the question, “what are we investing in?”  It captures the traditional notions of asset class and geographic diversification.  As we have explored in this commentary, it also implicitly captures risk-based diversification.

We can also think of this axis as being responsible for “correlation-driven” diversification.  As we will see, however, the empirical evidence of the effectiveness of this type of diversification is limited.

5.1.1 It’s Hard to Allocate Our Way Out of a Bear Market

Empirical evidence suggests that correlation-driven diversification is not tremendously effective at limited losses in crisis events.  Consider the returns plotted below for a number of asset classes during 2008.  We can see that by the end of the year, almost all had fallen between -20% to -50%.


As it turns out, most of the risk reduction benefits seen in a traditional asset allocation are not actually due to diversification benefits, but rather simply due to outright de-risking.

In their 2016 paper The Free Lunch Effect: The Value of Decoupling Diversification and Risk, Croce, Guinn and Robinson demonstrate that most of the risk reduction seen in moving from and all-stock portfolio to a balanced portfolio is simply due to the fact that bonds are less volatile than stocks.

That is not to say that de-risking is without its own merits.  Outright de-risking a portfolio is simple way to reduce total loss potential and is one of the driving forces behind the benefits of glide-path investing’s ability to control sequence risk.

Investors looking to maintain a return profile while reducing risk through the benefits of diversification, however, may be disappointed.

In When Diversification Fails, Page and Panariello demonstrate that asset correlations tend to be bi-modal in nature.  Unfortunately, the dynamics exhibited are the exact opposite of what we would like to see: diversification opportunity is ample in positive market states, but correlations tend to crash towards one during equity crises.

This does not make traditional diversification outright worthless, however, for growth-oriented investors.

Consider the table below from a paper titled, The Risk of Premiums, in which the author summarizes his findings about the statistical significance of different realized equity risk premia around the globe over different time horizons.

The five countries with stars on the left-hand side of the table have historically exhibited statistically significant risk premia across rolling 1-, 5-, 10-, and 20-year periods.  Those with stars on the right did not exhibit statistically significant risk premia across any of the rolling periods.

It is important to remember that risk premia are expected, but by no means guaranteed.  It is entirely possible that markets mis-estimate the frequency or magnitude with which risks manifest and fail to demand an adequately compensating premium.

Things have worked out exceptionally well for U.S. investors, but the same cannot be said for investors around the globe.

With the exception of explicit de-risking, what diversification may not necessarily provide much support in managing the left-tails of systematic risk factors.  Nevertheless, what diversification may be critical in helping reduce exposure to idiosyncratic risks associated with a specific geographic region or asset class.

5.2 The How Axis – Payoff Diversification

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

How need not be complex.  Low-cost, tax-efficient passive asset allocation is a legitimate how.

But this axis also captures the variety of other active investment styles that can create their own, and often independent, return streams.

One might go so far as to call them “synthetic assets,” but most popular literature simply refers to them as “styles.”  Popular categories include: value, momentum, carry, defensive (quality / low-volatility), trend, and event-driven.

The how axis is able to take the same what and create what are potentially unique return streams.  The return profile of a currency momentum portfolio may be inherently different than a commodity value portfolio, both of which may offer diversification from traditional, economic risk factors that drive currency and commodity beta.

If the what axis captures correlation driven diversification, we would argue that the how axis captures pay-off driven diversification.

5.2.1 Style Diversification

In When Diversification Fails, Page and Panariello also found that correlations for many styles are bi-modal, but some may offer significant diversification in equity crisis states.

2018, however, once again proved that there are philosophical limits to the benefits of diversification.  For styles to work over the long run, not only do there have to be periods where they fail individually, but there have to be periods where they fail simultaneously.

If we want to keep earning reward, we have to bear some risk in some potential state of the world.

It is no surprise, then, that it appears that most major styles appear to offer compensation for their own negative skew.  In their 2014 paper Risk Premia: Asymmetric Tail Risks and Excess Returns, Lemperiere, Deremble, Nguyen, Seager, Potters and Bouchaud find that not only do most styles exhibit negative skew, but that there appears to be a positive relationship with skew and the style’s Sharpe ratio.

As with asset classes, return appears to be a compensation for bearing asymmetric risk.

The two exceptions in the graph are trend and equity value (Fama-French HML).

The authors of the paper note that the positive skew of equity value is somewhat problematic, as it implies it is an anomaly rather than a risk compensation.  However, using monthly returns to recreate the above graph shifts the skew of equity value back to negative, implying perhaps that there is a somewhat regime-driven nature to value that needs to be further explored.

Trend, on the other hand, has long-been established to exhibit positive skew.  Indeed, it may very well be a mathematical byproduct of the trading strategy itself rather than an anomaly.

5.2.2 Payoff Diversification

While the findings of Lemperiere, Deremble, Nguyen, Seager, Potters and Bouchaud (2016) imply that style premia are not exceptions to the “no pain, no premium” rule, we should not be dissuaded from considering the potential benefits of their incorporation within a portfolio.

After all, not only might we potentially benefit from the fact that their negative states might be somewhat independent of economic risk factors (acknowledging, as always, the philosophical limits of diversification), but the trading strategies themselves create varying payoff profiles that differ from one another.

By combining different asset classes and payoff functions, we may be able to create a higher quality of portfolio return.

For example, when we overlay a naive trend strategy on top of U.S. equities, the result converges towards a distribution where we simply miss the best and worst years.  However, because the worst years tend to be worse than the best years are good, it leads to a less skewed distribution.

In effect, we’ve fought negative skew with positive skew.

At Newfound, we often say that “risk cannot be destroyed, but only transformed.”  We tend to think of risk as a blob that is spread across future states of the world.  When we push down on that blob in one future state, in effect “reducing risk,” it simply displaces to another state.

Trend may have historically helped offset losses during crisis events, but it can create drawdowns during reversal markets.  Similarly, style / alternative premia may be able to harvest returns when traditional economic factors are going sideways, but may suffer during coincidental drawdowns like 2018.

Source: PIMCO

That is why we repeat ad nauseam “diversify your diversifiers.”

5.2.3 Specification Risk

While the above discussion of how pertained to style risks, there is another form of risk worth briefly discussing: specification risk.

Specification risk acknowledges that two investors implementing two identical styles in theory may end up with very different results in practice.  Style risk tells us that equity value managers struggled as a category in 2016; specification risk tells us how each manager did individually.

Whether we are compensated for bearing specification risk is up for debate and largely depends upon your personal view of a manager’s skill.

In the absence of a view of skill, what we find is that combining multiple managers tends to do little for a reduction in traditional portfolio volatility (except in highly heterogenous categories), but can tremendously help reduce portfolio skew as well as the dispersion in terminal wealth.

For example, below we generate a number of random 30-stock portfolios and plot their returns over the last decade.

We can see that while the results are highly correlated, the terminal wealth achieved varies dramatically.

If instead of just picking one manager we pick several – say 3 or 4 – we find that the potential dispersion in terminal wealth drops dramatically and our achieved outcome is far more certain.

You can read more on this topic in our past commentary Is Multi-Manager Diversification Worth It?

5.3 When Axis

We believe that the when axis may be one of the most important, yet overlooked opportunities for diversification in portfolio construction.  So much so, we wrote a paper about it titled Rebalance Timing Luck: The Difference Between Hired and Fired.

The basic intuition behind this axis is that our realized portfolio results will be driven by the opportunities presented to us at the time we rebalance.

In many ways, diversification along the when axis can be thought of as opportunity-diversification.

For example, Blitz, van der Grient, and van Vliet demonstrated in their 2010 paper Fundamental Indexation: Rebalancing Assumptions and Performance that the quarter in which an annually-rebalanced fundamental index is reconstituted can lead to significant performance disparity.  For example, the choice to rebalance the portfolio in March versus September would have lead to a 1,000 basis point performance difference in 2009.

This difference was largely driven by the opportunities perceived by the systematic strategy at the time of rebalancing.

This risk is not limited to active portfolios.  In the graph below we plot rolling 1-year return differences between two 60/40 portfolios, one of which is rebalanced at the end of each February and one that is rebalanced at the end of each August.

We can see that the rebalance in early 2009 lead to a 700 basis point gap in performance by spring 2010.

While we believe this has important implications for how research is conducted, benchmarks are constructed, and managers build portfolios, the more practical takeaway for investors is that they might benefit from choosing managers who rebalance on different schedules.

6. Summary

Investors often focus on returns, but it is important to keep in mind why we expect to earn those returns in the first place.  We believe a risk-based mindset can help remind us that we expect to earn excess returns because we are willing to bear risk.

In many ways, we can think of ourselves and our portfolios as insurance companies: we collect premiums for bearing risk.  Yet while we can we can seek to diversify the risks we insure, there are few truly independent risk factor and the premiums aren’t often large enough to offset large losses.

We also believe that there exist theoretical limits to diversification.  If we eliminate risk through diversification, we also eliminate reward.  In other words: no pain, no premium.

This does not inherently mean, however, we should just “suck it up.”  The implications of risk-based thinking is dependent upon where we are in our investment lifecycle.

The primary risk of investors with growth mandates (e.g. investors early in their lifecycle) is “failing slow,” which is the failure to growth their capital sufficiently to outpace inflation or meet future liabilities.  In this case, our aim should be to diversify as much as possible without overly de-risking the portfolio.  With a risk-based mindset, it becomes clear why approaches like risk parity, when targeting an adequate volatility, may be philosophically superior to traditional asset allocation.

For investors taking withdrawals (e.g. those late in their lifecycle or endowments/pensions), the primary risk is “failing fast” from large drawdowns.  Diversification is likely insufficient on its own and de-risking may be prudent.  Diversifying payoff types and introducing positive skew styles – e.g. trend – may also benefit the investment plan by creating a more consistent return stream.

Yet we should acknowledge that even return opportunities available along the how axis appear to be driven largely by skew, re-emphasizing that without potential pain, there should be no premium.

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