This post is available as a PDF download here.
Summary
- Information does not flow into the market at a constant frequency or with constant magnitude.
- By sampling data using a constant time horizon (e.g. “200-day simple moving average”), we may over-sample during calm market environments and under-sample in chaotic ones.
- As an example, we introduce a highly simplified price model and demonstrate that trend following lookback periods should be a dynamic function of trend and volatility in the time domain.
- By changing the sampling domain slightly, we are able to completely eliminate the need for the dynamic lookback period.
- Finally, we demonstrate a more complicated model that samples market prices based upon cumulative log differences, creating a dynamic moving average in the time domain.
- We believe that there are other interesting applications of this line of thinking, many of which may already be in use today by investors who may not be aware of it (e.g. tracking-error-based rebalancing techniques).
In the 2014 film Interstellar, Earth has been plagued by crop blights and dust storms that threaten the survival of mankind. Unknown, interstellar beings have opened a wormhole near Saturn, creating a path to a distant galaxy and the potential of a new home for humanity.
Twelve volunteers travel into the wormhole to explore twelve potentially hospitable planets, all located near a massive black hole named Gargantua. Of the twelve, only three reported back positive results.
With confirmation in hand, the crew of the spaceship Endurance sets out from Earth with 5,000 frozen human embryos, intent on colonizing the new planets.
After traversing the wormhole, the crew sets down upon the first planet – an ocean world – and quickly discovers that it is actually inhospitable. A gigantic tidal wave kills one member of the crew and severely delays the lander’s departure.
The close proximity of the planet to the gravitational forces of the supermassive black hole invites exponential time dilation effects. The positive beacon that had been tracked had perhaps been triggered just minutes prior on the planet. For the crew, the three hours spent on the planet amounted to over 23 years on Earth. The crew can only watch, devastated, as their loved ones age before their eyes in the video messages received – and never responded to – in their multi-decade absence.
Our lives revolve around the clock, though we do not often stop to reflect upon the nature of time.
Some aspects of time tie to corresponding natural events. A day is simply reckoned from one midnight to the next, reflecting the Earth’s full rotation about its axis. A year, which reflects the length of time it takes for the Earth to make a full revolution around the Sun, will also correspond to a full set of a seasons.
Others, however, are seemingly more arbitrary. The twenty-four-hour day is derived from ancient Egyptians, who divided day-time into 10 hours, bookended by twilight hours. The division of an hour into sixty minutes comes from the Babylonians, who used a sexagesimal counting system.
We impose the governance of the clock upon our financial system as well. Public companies prepare quarterly and annual reports. Economic data is released at a scheduled monthly or quarterly pace. Trading days for U.S. equity markets are defined as between the hours of 9:30am and 4:00pm ET.
In many ways, our imposition of the clock upon markets creates a natural cadence for the flow of information.
Yet, despite our best efforts to impose order, information most certainly does not flow into the market in a constant or steady manner.
New innovations, geopolitical frictions, and errant tweets all represent idiosyncratic events that can reshape our views in an instant. A single event can be of greater import than all the cumulative economic news that came before it; just consider the collapse of Lehman Brothers.
And much like the time dilation experienced by the crew of Endurance, a few, harrowing days of 2008 may have felt longer than the entirety of a tranquil year like 2017.
One way of trying to visualize this concept is by looking at the cumulative variance of returns. Given the clustered nature of volatility, we would expect to see periods where the variance accumulates slowly (“calm markets”) and periods where the variance accumulates rapidly (“chaotic markets”).
When we perform this exercise – by simply summing squared daily returns for the S&P 500 over time – we see precisely this. During market environments that exhibit stable economic growth and little market uncertainty, we see very slow and steady accumulation of variance. During periods when markets are seeking to rapidly reprice risk (e.g. 2008), we see rapid jumps.
Source: CSI Data. Calculations by Newfound Research.
If we believe that information flow is not static and constant, then sampling data on a constant, fixed interval will mean that during calm markets we might be over-sampling our data and during chaotic markets we might be under-sampling.
Let’s make this a bit more concrete.
Below we plot the –adjusted closing price of the S&P 500– and its –200-day simple moving average–. Here, the simple moving average aims to estimate the trend component of price. We can see that during the 2005-2007 period, it estimates the underlying trend well, while in 2008 it dramatically lags price decline.
Source: CSI Data. Calculations by Newfound Research.
The question we might want to ask ourselves is, why are looking at the prior 200 days? Or, more specifically, why is a day a meaningful unit of measure? We already demonstrated above that it very well may not be: one day might be packed with economically-relevant information and another entirely devoid.
Perhaps there are other ways in which we might think about sampling data. We could, for example, sample data based upon cumulative volume intervals. Another might be on a fixed number of cumulative ticks or trades. Yet another might be on a fixed cumulative volatility or variance.
As a firm which makes heavy use of trend-following techniques, we are particularly partial to the latter approach, as the volatility of an asset’s trend versus its price should inform the trend lookback horizon. If we think of trend following as being the trading strategy that replicates the payoff profile of a straddle, increased volatility levels will decrease the delta of the option positions, and therefore decrease our position size. An interpretation of this effect is that the increased volatility decreases our certainty of where price will fall at expiration, and therefore we need to decrease our sensitivity to price movements.
If that all sounds like Greek, consider this simple example. Assume that price follows a highly simplified model as a function of time:
There are two components of this model: the linear trend and the noise.
Now let’s assume we are attempting to identify whether the linear trend is positive or negative by using a simple moving average (“SMA”) of price:
To determine if there is a positive or a negative trend, we simply ask if our current SMA value is greater or less than the prior SMA value. For a positive trend, we require:
Substituting our above definition of the simple moving average:
When we recognize that most of the terms on the left also appear on the right, we can re-write the whole comparison as the new price in the SMA being greater than the old price dropping out of the SMA:
Which, through substitution of our original definition, leaves us with:
Re-arranging a bit, we get:
Here we use the fact that sin(x) is bounded between -1 and 1, meaning that:
Assuming a positive trend (m > 0), we can replace with our worst-case scenario,
To quickly test this result, we can construct a simple time series where we assume a=3 and m=0.5, which implies that our SMA length should be greater than 11. We plot the –time series– and –SMA– below. Note that the –SMA– is always increasing.
Despite being a highly simplified model, it illuminates that our lookback length should be a function of noise versus trend strength. The higher the ratio of noise to trend, the longer the lookback required to smooth out the noise. On the other hand, when the trend is very strong and the noise is weak, the lookback can be quite short.1
Thus, if trend and noise change over time (which we would expect them to), the optimal lookback will be a dynamic function. When trend is much weaker than noise, we our lookback period will be extended; when trend is much stronger than noise, the lookback period shrinks.
But what if we transform the sampling domain? Rather than sampling price every time step, what if we sample price as a function of cumulative noise? For example, using our simple model, we could sample when cumulative noise sums back to zero (which, in this example, will be the equivalent of sampling every 2π time-steps).2
Sampling at that frequency, how many of data points would we need to estimate our trend? We need not even work out the math as before; a bit of analytical logic will suffice. In this case, because we know the cumulative noise equals zero, we know that a point-to-point comparison will be affected only by the trend component. Thus, we only need n=1 in this new domain.
And this is true regardless of the parameterization of trend or noise. Goodbye! dynamic lookback function.
Of course, this is a purely hypothetical – and dramatically over-simplified – model. Nevertheless, it may illuminate why time-based sampling may not be the most efficient practice if we do not believe that information flow is constant.
Below, we again plot the –S&P 500– as well as a standard –200-day simple moving average–.
We also sample prices of the S&P 500 based upon cumulative magnitude of log differences, approximating a cumulative 2.5% volatility move. When the market exhibits low volatility levels, the process samples price less frequently. When the market exhibits high volatility, it samples more frequently. Finally, we plot a –200 period moving average– based upon these samples.
We can see that sampling in a different domain – in this case, the log difference space – we can generate a process that reacts dynamically in the time domain. During the calm markets of 2006 and early 2007, the –200 period moving average– behaves like the –200-day simple moving average–, whereas during the 2008 crisis it adapts to the changing price level far more quickly.
By changing the domain in which we sample, we may be able to create a model that is dynamic in the time domain, avoiding the time-dilation effects of information flow.
Conclusion
Each morning the sun rises and each evening it sets. Every year the Earth travels in orbit around the sun. What occurs during those time spans, however, varies dramatically day-by-day and year-by-year. Yet in finance – and especially quantitative finance – we often find ourselves using time as a measuring stick.
We find the notion of time almost everywhere in portfolio construction. Factors, for example, are often defined by measurements over a certain lookback horizon and reformed based upon the decay speed of the signal.
Even strategic portfolios are often rebalanced based upon the calendar. As we demonstrated in our paper Rebalance Timing Luck: The Difference Between Hired and Fired, fixed-schedule rebalancing can invite tremendous random impact in our portfolios.
Information does not flow into the market at a constant rate. While time may be a convenient measure, it may actually cause us to sample too frequently in some market environments and not frequently enough in others.
One answer may be to transform our measurements into a different domain. Rather than sampling price based upon the market close of each day, we might sample price based upon a fixed amount of cumulative volume, trades, or even variance. In doing so, we might find that our measures now represent a more consistent amount of information flow, despite representing a dynamic amount of data in the time domain.
Taxes and Trend Equity
By Nathan Faber
On April 1, 2019
In Popular, Weekly Commentary
This post is available as a PDF download here.
Summary
Tax season for the year is quickly coming to a close, and while taxes are not a topic we cover frequently in these commentaries, it has a large impact on investor portfolios.
One of the primary reasons we do not cover it more is that it is investor-specific. Actionable insights are difficult to translate across investors without making broad assumptions about state and federal tax rates, security location (tax-exempt, tax deferred, or taxable), purchase time and holding period, losses or gains in other assets, health and family situation, etc.
Some sweeping generalizations can be made, such as that it is better to realize long-term capital gains than short-term ones, that having qualified dividends is better than having non-qualified ones, and that it is better to hold bonds in tax-deferred or tax-exempt accounts. But even these assertions are nuanced and depend on a variety of factors specific to an individual investor.
Trend equity strategies – and tactical strategies, in general – get a bad rap for being tax-inefficient. As assets are sold, capital gains are realized, often with no regard as to whether they are short-term or long-term. Wash sales are often ignored and holding periods may exclude dividends from qualifying status.
However, taxes represent yet another risk in a portfolio, and as you can likely guess if you are a frequent reader of these commentaries, reducing one risk is often done at the expense of increasing another.
The Risk in Taxes
Tax rates have been constant for long periods of time historically, especially in recent years, but they can change very rapidly depending on the overall economic environment.
Source: IRS, U.S. Census Bureau, and Tax Foundation. Calculations by Newfound Research. Series are limited by historical data availability.
The history shows a wide array of scenarios.
Prior to the 1980s, marginal tax rates spanned an extremely wide band, with the lowest tier near 0% and the top rate approaching 95%. However, this range has been much narrower for the past 30 years.
In the late 1980s when tax policy became much less progressive, investors could fall into only two tax brackets.
While the income quantile data history is limited, even prior to the narrowing of the marginal tax range, the bulk of individuals had marginal tax rates under 30%.
Turning to long-term capital gains rates, which apply to asset held for more than a year, we see similar changes over time.
Source: U.S. Department of the Treasury, Office of Tax Analysis and Tax Foundation.
For all earners, these rates are less than their marginal rates, which is currently the tax rate applied to short-term capital gains. While there were times in the 1970s when these long-term rates topped out at 40%, the maximum rate has dipped down as low as 15%. And since the Financial Crisis in 2008, taxpayers in the lower tax brackets pay 0% on long-term capital gains.
It is these large potential shifts in tax rates that introduce risk into the tax-aware investment planning process.
To see this more concretely, consider a hypothetical investment that earns 7% every year. Somehow – how is not relevant for this example – you have the choice of having the gains distributed annually as long-term capital gains or deferred until the sale of the asset.
Which option should you choose?
The natural choice is to have the taxes deferred until the sale of the asset. For a 10-year holding period where long-term capital gains are taxed at 20%, the pre-tax and after-tax values of a $1,000 investment are shown below.
The price return only version had a substantially higher pre-tax value as the full 7% was allowed to compound from year to year without the hinderance of an annual tax hit.
At the end of the 10-year period, the tax basis of the approach that distributed gains annually had increased up to the pre-tax amount, so it owed no additional taxes once the asset was sold. However, the approach that deferred taxes still ended up better after factoring in the tax on the embedded long-term capital gains that were realized upon the sale.
Now let’s consider the same assets but this time invested from 2004 to 2014 when the maximum long-term capital gains rate jumped to 25% in 2013 after being around 15% for the first 8 years.
The pre-tax picture is still the same: the deferred approach easily beat the asset that distributed capital gains annually.
But the after-tax values have changed order. Locking in more of the return when long-term capital gains tax rates were lower was advantageous.
The difference in this case may not be that significant. But consider a more extreme – yet still realistic – example where your tax rate on the gains jumps by more than ten percentage points (e.g. due to a change in employment or family situation or tax law changes), and the decision over which type of asset you prefer is not as clear cut.
Moving beyond this simple thought experiment, we now turn to the tax impacts on trend equity strategies.
Tax Impacts in Trend Equity
We will begin with a simple trend equity strategy that buys the U.S. stock market (the ETF VTI) when it has a positive 9-month return and buys short-term U.S. Treasuries (the ETF SHV) otherwise. Prior to ETF inception, we will rely on data from the Kenneth French Data Library to extend the analysis back to the 1920s. We will evaluate the strategy monthly and, for simplicity, will treat dividends as price returns.
With taxes now in the mix, we must track the individual tax lots as the strategy trades over time based on the tactical model. For deciding which tax lots to sell, we will select the ones with the lowest tax cost, making the assumption that short-term capital gains are taxed 50% higher than long-term capital gains (approximately true for investors with tax rates of 22% and 15%, respectively, in the current tax code).
We must address the question of when an investor purchases the trend equity strategy as a long bull market with a consistent positive trend would have very different tax costs for an investor holding all the way through versus one who bought at end.
To keep the analysis as simple as possible given the already difficult specification, we will look at an investment that is made at the very beginning, assume that taxes are paid at the end of each year, and compare the average annualized pre-tax and after-tax returns. Fortunately, for this type of trend strategy that can move entirely in and out of assets, the tax memory will occasionally reset.
To set some context, first, we need a benchmark.
Obviously, if you purchased VTI and held it for the entire time, you would be sitting on some large embedded capital gains.1
Instead, we will use a more appropriate benchmark for trend equity: a 50%/50% blend of VTI and SHV. We will rebalance this blend annually, which will lead to some capital gains.
The following chart shows the capital gains aggregated by year as a percentage of the end of the year account value.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
As expected with the annual rebalancing, all of the capital gains are long-term. Any short-term gains are an artifact of the rigidity of the rebalancing system where the first business day of subsequent years might be fewer than 365 days apart. In reality, you would likely incorporate some flexibility in the rebalances to ensure all long-term capital gains.
While this strategy incurs some capital gains, they are modest, with none surpassing 10%. Paying taxes on these gains is a small price to pay for maintaining a target allocation, supposing that is the primary goal.
Assuming tax rates of 15% for long-term gains and 25% for short-term gains, the annualized returns of the strategic allocation pre-tax and after-tax are shown below. The difference is minor.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
Now on to the trend equity strategy.
The historical capital gains look very different than those of the strategic portfolio.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
In certain years, the strategy locks in long-term capital gains greater than 50%. The time between these years is interspersed with larger short-term capital losses from whipsaws or year with essentially no realized gains or losses, either short- or long-term.
In fact, 31 of the 91 years had absolute realized gains/losses of less than 1% for both short- and long-term.
Now the difference between pre-tax and after-tax returns is 100 bps per year using the assumed tax rates (15% and 25%). This is significantly higher than with the strategic allocation.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
It would appear that trend equity is far less tax efficient than the strategic benchmark. As with all things taxes, however, there are nuances. As we mentioned in the first section of this commentary, tax rates can change at any time, either from a federal mandate or a change in an individual’s situation. If you are stuck with a considerable unrealized capital gain, it may be too late to adjust the course.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
The median unrealized capital gain for the trend equity strategy is 10%. This, of course, means that you must realize the gains periodically and therefore pay taxes.
But if you are sitting with a 400% unrealized gain in a different strategy, behaviorally, it may be difficult to make a prudent investment decision knowing that a large tax bill will soon follow a sale. And a 10 percentage point increase in the capital gains tax rate can have a much larger impact in dollar terms on the large unrealized gain than missing out on some compounding when rates were lower.
Even so, the thought of paying taxes intermediately and missing out on compound growth can still be irksome. Some small improvement to the trend equity strategy design can prove beneficial.
Improving the Tax Profile Within Trend Equity
This commentary would be incomplete without a further exploration down some of the axes of diversification.
We can take the simple 9-month trend following strategy and diversify it along the “how” axis using a multi-model approach with multiple lookback periods.
Specifically, we will use price versus moving average and moving average cross-overs in addition to the trailing return signal and look at windows of data ranging from 6 to 12 months.2
We can also diversify along the “when” axis by tranching the monthly strategy over 20 days. This has the effect of removing the luck – either good or bad – of rebalancing on a certain day of the month.
Below, we plot the characteristics of the long-term capital gains for the strategies in years in which a long-term gain was realized.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
The single monthly model had about a third of the years with long-term gains. Tranching it took that fraction to over a half. Moving to a multi-model approach brought the fraction to 60%, and tranching that upped it to 2 out of every 3 years.
Source: CSI Data and Kenneth French Data Library. Calculations by Newfound.
From an annualized return perspective, all of these trend equity strategies exhibited similar return differentials between pre-tax and after-tax.
In previous commentaries, we have illustrated how tranching to remove timing luck and utilizing multiple trend following models can remove the potential dispersion in realized terminal wealth. However, in the case of taxes, these embellishments did not yield a reduction in the tax gap.
While these improvements to trend equity strategies reduce specification-based whipsaw, they often result in similar allocations for large periods of time, especially since these strategies only utilize a single asset.
But to assume that simplicity trumps complexity just because the return differentials are not improved misses the point.3
With similar returns among within the trend-following strategies, using an approach that realizes more long-term capital gains could still be beneficial from a tax perspective.
In essence, this can be thought of as akin to dollar-cost averaging.
Dollar-cost averaging to invest a lump sum of capital is often not optimal if the sole goal is to generate the highest return.4 However, it is often beneficial in that it reduces the risk of bad outcomes (i.e. tail events).
Having a strategy – like trend equity – that has the potential to generate strong returns while taking some of those returns as long-term capital gains can be a good hedge against rising tax rates. And having a diversified trend equity strategy that can realize these capital gains in a smoother fashion is icing on the cake.
Conclusion
Taxes are a tricky subject, especially from the asset manager’s perspective. How do you design a strategy that suits all tax needs of its investors?
Rather than trying to develop a one-size-fits-all strategy, we believe that a better approach to the tax question is education. By more thoroughly understanding the tax profile of a strategy, investors can more comfortably deploy it appropriately in their portfolios.
As highly active strategies, trend equity mandates are generally assumed to be highly tax-inefficient. We believe it is more meaningful to represent the tax characteristics an exchange of risks: capital gains are locked in at the current tax rates (most often long-term) while unrealized capital gains are kept below a reasonable level. These strategies have also historically exhibited occasional periods with short-term capital losses.
These strategies can benefit investors who expect to have higher tax rates in the future without the option of having a way to mitigate this risk otherwise (e.g. a large tax-deferred account like a cash balance plan, donations to charity, a step-up in cost basis, etc.).
Of course, the question about the interplay between tax rates and asset returns, which was ignored in this analysis, remains. But in an uncertain future, the best course of investment action is often the one that diversifies away as much uncompensated risk as possible and includes a comprehensive plan for risk management.