S&P500 Index Archives - QUANTITATIVE RESEARCH AND TRADING

June 25, 2022June 26, 2022

Backtest vs. Trading Reality

Kris Sidial, whose Twitter posts are often interesting, recently posted about the reality of trading profitability vs backtest performance, as follows:

While I certainly agree that the latter example is more representative of a typical trader’s P&L, I don’t concur that the first P&L curve is necessarily “99.9% garbage”. There are many strategies that have equity curves that are smoother and more monotonic than those of Kris’s Skeleton Case V2 strategy. Admittedly, most of these lie in the area of high frequency, which is not Kris’s domain expertise. But there are also lower frequency strategies that produce results which are not dissimilar to those shown the first chart.

As a case in point, consider the following strategy for the S&P 500 E-Mini futures contract, described in more detail below. The strategy was developed using 15-minute bar data from 1999 to 2012, and traded live thereafter. The live and backtest performance characteristics are almost indistinguishable, not only in terms of rate of profit, but also in regard to strategy characteristics such as the no. of trades, % win rate and profit factor.

Just in case you think the picture is a little too rosy, I would point out that the average profit factor is 1.25, which means that the strategy is generating only 25% more in profits than losses. There will be big losing trades from time to time and long sequences of losses during which the strategy appears to have broken down. It takes discipline to resist the temptation to “fix” the strategy during extended drawdowns and instead rely on reversion to the mean rate of performance over the long haul. One source of comfort to the trader through such periods is that the 60% win rate means that the majority of trades are profitable.

As you read through the replies to Kris’s post, you will see that several of his readers make the point that strategies with highly attractive equity curves and performance characteristics are typically capital constrained. This is true in the case of this strategy, which I trade with a very modest amount of (my own) capital. Even trading one-lots in the E-Mini futures I occasionally experience missed trades, either on entry or exit, due to limit orders not being filled at the high or low of a bar. In scaling the strategy up to something more meaningful such as a 10-lot, there would be multiple partial fills to deal with. But I think it would be a mistake to abandon a high performing strategy such as this just because of an apparent capacity constraint. There are several approaches one can explore to address the issue, which may be enough to make the strategy scalable.

Where (as here) the issue of scalability relates to the strategy fill rate on limit orders, a good starting point is to compute the extreme hit rate, which is the proportion of trades that take place at the high or low of the bar. As a rule of thumb, for strategies running on typical low frequency infrastructure an extreme hit rate of 10% or less is manageable; anything above that level quickly becomes problematic. If the extreme hit rate is very high, e.g. 25% or more, then you are going to have to pay a great deal of attention to the issues of latency and order priority to make the strategy viable in practise. Ultimately, for a high frequency market making strategy, most orders are filled at the extreme of each “bar”, so almost all of the focus in on minimizing latency and maintaining a high queue priority, with all of the attendant concerns regarding trading hardware, software and infrastructure.

Next, you need a strategy for handling missed trades. You could, for example, decide to skip any entry trades that are missed, while manually entering unfilled exit trades at the market. Or you could post market orders for both entry and exit trades if they are not filled. An extreme solution would be to substitute market-if-touched orders for limit orders in your strategy code. But this would affect all orders generated by the system, not just the 10% at the high or low of the bar and is likely to have a very adverse affect on overall profitability, especially if the average trade is low (because you are paying an extra tick on entry and exit of every trade).

The above suggests that you are monitoring the strategy manually, running simulation and live versions side by side, so that you can pick up any trades that the strategy should have taken, but which have been missed. This may be practical for a strategy that trades during regular market hours, but not for one that also trades the overnight session.

An alternative approach, one that is commonly applied by systematic traders, is to automate the handling of missed trades. Typically the trader will set a parameter that converts a limit order to a market order X seconds after a limit price has been traded but not filled. Of course, this will result in paying up an extra tick (or more) to enter trades that perhaps would have been filled if one had waited longer than X seconds. It will have some negative impact on strategy profitability, but not too much if the extreme hit rate is low. I tend to use this method for exit trades, preferring to skip any entry trades that don’t get filled at the limit price.

Beyond these simple measures, there are several other ways to extend the capacity of the strategy. An obvious place to start is by evaluating strategy performance on different session times and bar lengths. So, in this case, we might look at deploying the strategy on both the day and night sessions. We can also evaluate performance on bars of different length. This will give different entry and exit points for individual trades and trades that are at the extreme of a bar on one timeframe may not be at the high or low of a bar on the other timescale. For example, here is the (simulated) performance of the strategy on 13 minute bars:

There is a reason for choosing a bar interval such as 13 minutes, rather than the more commonplace 5- or 10 minutes, as explained in this post:

Trading Prime Market Cycles

Finally, it is worth exploring whether the strategy can be applied to other related markets such as NQ futures, for example. Typically this will entail some change to the strategy code to reflect the difference in price levels, but the thrust of the strategy logic will be similar. Another approach is to use the signals from the current strategy as inputs – i.e. alpha generators – for a derivative strategy, such as trading the SPY ETF based on signals from the ES strategy. The performance of the derived strategy may not be as good, but in a product like SPY the capacity might be larger.

November 24, 2020

Can Machine Learning Techniques Be Used To Predict Market Direction? The 1,000,000 Model Test.

During the 1990’s the advent of Neural Networks unleashed a torrent of research on their applications in financial markets, accompanied by some rather extravagant claims about their predicative abilities. Sadly, much of the research proved to be sub-standard and the results illusionary, following which the topic was largely relegated to the bleachers, at least in the field of financial market research.

With the advent of new machine learning techniques such as Random Forests, Support Vector Machines and Nearest Neighbor Classification, there has been a resurgence of interest in non-linear modeling techniques and a flood of new research, a fair amount of it supportive of their potential for forecasting financial markets. Once again, however, doubts about the quality of some of the research bring the results into question.

Against this background I and my co-researcher Dan Rico set out to address the question of whether these new techniques really do have predicative power, more specifically the ability to forecast market direction. Using some excellent MatLab toolboxes and a new software package, an Excel Addin called 11Ants, that makes large scale testing of multiple models a snap, we examined over 1,000,000 models and model-ensembles, covering just about every available non-linear technique. The data set for our study comprised daily prices for a selection of US equity securities, together with a large selection of technical indicators for which some other researchers have claimed explanatory power.

In-Sample Equity Curve for Best Performing Nonlinear Model

The answer provided by our research was, without exception, in the negative: not one of the models tested showed any significant ability to predict the direction of any of the securities in our data set. Furthermore, our study found that the best-performing models favored raw price data over technical indicator variables, suggesting that the latter have little explanatory power.

As with Neural Networks, the principal difficulty with non-linear techniques appears to be curve-fitting and a failure to generalize: while it is very easy to find models that provide an excellent fit to in-sample data, the forecasting performance out-of-sample is often very poor.

Out-of-Sample Equity Curve for Best Performing Nonlinear Model

Some caveats about our own research apply. First and foremost, it is of course impossible to prove a hypothesis in the negative. Secondly, it is plausible that some markets are less efficient than others: some studies have claimed success in developing predictive models due to the (relative) inefficiency of the F/X and futures markets, for example. Thirdly, the choice of sample period may be criticized: it could be that the models were over-conditioned on a too- lengthy in-sample data set, which in one case ran from 1993 to 2008, with just two years (2009-2010) of out-of-sample data. The choice of sample was deliberate, however: had we omitted the 2008 period from the “learning” data set, it would be very easy to criticize the study for failing to allow the algorithms to learn about the exceptional behavior of the markets during that turbulent year.

Despite these limitations, our research casts doubt on the findings of some less-extensive studies, that may be the result of sample-selection bias. One characteristic of the most credible studies finding evidence in favor of market predictability, such as those by Pesaran and Timmermann, for instance (see paper for citations), is that the models they employ tend to incorporate independent explanatory variables, such as yield spreads, which do appear to have real explanatory power. The finding of our study suggest that, absent such explanatory factors, the ability to predict markets using sophisticated non-linear techniques applied to price data alone may prove to be as illusionary as it was in the 1990’s.

ONE MILLION MODELS

November 3, 2020November 3, 2020

Range-Based EGARCH Option Pricing Models (REGARCH)

The research in this post and the related paper on Range Based EGARCH Option pricing Models is focused on the innovative range-based volatility models introduced in Alizadeh, Brandt, and Diebold (2002) (hereafter ABD). We develop new option pricing models using multi-factor diffusion approximations couched within this theoretical framework and examine their properties in comparison with the traditional Black-Scholes model.

The two-factor version of the model, which I have applied successfully in various option arbitrage strategies, encapsulates the intuively appealing idea of a trending long term mean volatility process, around which oscillates a mean-reverting, transient volatility process. The option pricing model also incorporates asymmetry/leverage effects and well as correlation effects between the asset return and volatility processes, which results in a volatility skew.

The core concept behind Range-Based Exponential GARCH model is Log-Range estimator discussed in an earlier post on volatility metrics, which contains a lengthy exposition of various volatility estimators and their properties. (Incidentally, for those of you who requested a copy of my paper on Estimating Historical Volatility, I have updated the post to include a link to the pdf).

We assume that the log stock price s follows a drift-less Brownian motion ds = sdW. The volatility of daily log returns, denoted h= s/sqrt(252), is assumed constant within each day, at ht from the beginning to the end of day t, but is allowed to change from one day to the next, from ht at the end of day t to ht+1 at the beginning of day t+1. Under these assumptions, ABD show that the log range, defined as:

is to a very good approximation distributed as

where N[m; v] denotes a Gaussian distribution with mean m and variance v. The above equation demonstrates that the log range is a noisy linear proxy of log volatility ln ht. By contrast, according to the results of Alizadeh, Brandt,and Diebold (2002), the log absolute return has a mean of 0.64 + ln ht and a variance of 1.11. However, the distribution of the log absolute return is far from Gaussian. The fact that both the log range and the log absolute return are linear log volatility proxies (with the same loading of one), but that the standard deviation of the log range is about one-quarter of the standard deviation of the log absolute return, makes clear that the range is a much more informative volatility proxy. It also makes sense of the finding of Andersen and Bollerslev (1998) that the daily range has approximately the same informational content as sampling intra-daily returns every four hours.

Except for the model of Chou (2001), GARCH-type volatility models rely on squared or absolute returns (which have the same information content) to capture variation in the conditional volatility h_t_. Since the range is a more informative volatility proxy, it makes sense to consider range-based GARCH models, in which the range is used in place of squared or absolute returns to capture variation in the conditional volatility. This is particularly true for the EGARCH framework of Nelson (1990), which describes the dynamics of log volatility (of which the log range is a linear proxy).

ABD consider variants of the EGARCH framework introduced by Nelson (1990). In general, an EGARCH(1,1) model performs comparably to the GARCH(1,1) model of Bollerslev (1987). However, for stock indices the in-sample evidence reported by Hentschel (1995) and the forecasting performance presented by Pagan and Schwert (1990) show a slight superiority of the EGARCH specification. One reason for this superiority is that EGARCH models can accommodate asymmetric volatility (often called the “leverage effect,” which refers to one of the explanations of asymmetric volatility), where increases in volatility are associated more often with large negative returns than with equally large positive returns.

The one-factor range-based model (REGARCH 1) takes the form:

where the returns process R_t is conditionally Gaussian: R_t ~ N[0, h_t²]

and the process innovation is defined as the standardized deviation of the log range from its expected value:

Following Engle and Lee (1999), ABD also consider multi-factor volatility models. In particular, for a two-factor range-based EGARCH model (REGARCH2), the conditional volatility dynamics) are as follows:

and

where ln q_t can be interpreted as a slowly-moving stochastic mean around which log volatility ln h_t makes large but transient deviations (with a process determined by the parameters k_h, f_h and d_h).

The parameters q, k_q, f_q and d_q determine the long-run mean, sensitivity of the long run mean to lagged absolute returns, and the asymmetry of absolute return sensitivity respectively.

The intuition is that when the lagged absolute return is large (small) relative to the lagged level of volatility, volatility is likely to have experienced a positive (negative) innovation. Unfortunately, as we explained above, the absolute return is a rather noisy proxy of volatility, suggesting that a substantial part of the volatility variation in GARCH-type models is driven by proxy noise as opposed to true information about volatility. In other words, the noise in the volatility proxy introduces noise in the implied volatility process. In a volatility forecasting context, this noise in the implied volatility process deteriorates the quality of the forecasts through less precise parameter estimates and, more importantly, through less precise estimates of the current level of volatility to which the forecasts are anchored.

On Testing Direction Prediction Accuracy

As regards the question of forecasting accuracy discussed in the paper on Forecasting Volatility in the S&P 500 Index, there are two possible misunderstandings here that need to be cleared up. These arise from remarks by one commentator as follows:

“An above 50% vol direction forecast looks good,.. but “direction” is biased when working with highly skewed distributions! ..so it would be nice if you could benchmark it against a simple naive predictors to get a feel for significance, -or- benchmark it with a trading strategy and see how the risk/return performs.”

(i) The first point is simple, but needs saying: the phrase “skewed distributions” in the context of volatility modeling could easily be misconstrued as referring to the volatility skew. This, of course, is used to describe to the higher implied vols seen in the Black-Scholes prices of OTM options. But in the Black-Scholes framework volatility is constant, not stochastic, and the “skew” referred to arises in the distribution of the asset return process, which has heavier tails than the Normal distribution (excess Kurtosis and/or skewness). I realize that this is probably not what the commentator meant, but nonetheless it’s worth heading that possible misunderstanding off at the pass, before we go on.

(ii) I assume that the commentator was referring to the skewness in the volatility process, which is characterized by the LogNormal distribution. But the forecasting tests referenced in the paper are tests of the ability of the model to predict the direction of volatility, i.e. the sign of the change in the level of volatility from the current period to the next period. Thus we are looking at, not a LogNormal distribution, but the difference in two LogNormal distributions with equal mean – and this, of course, has an expectation of zero. In other words, the expected level of volatility for the next period is the same as the current period and the expected change in the level of volatility is zero. You can test this very easily for yourself by generating a large number of observations from a LogNormal process, taking the difference and counting the number of positive and negative changes in the level of volatility from one period to the next. You will find, on average, half the time the change of direction is positive and half the time it is negative.

For instance, the following chart shows the distribution of the number of positive changes in the level of a LogNormally distributed random variable with mean and standard deviation of 0.5, for a sample of 1,000 simulations, each of 10,000 observations. The sample mean (5,000.4) is very close to the expected value of 5,000.

Distribution Number of Positive Direction Changes

So, a naive predictor will forecast volatility to remain unchanged for the next period and by random chance approximately half the time volatility will turn out to be higher and half the time it will turn out to be lower than in the current period. Hence the default probability estimate for a positive change of direction is 50% and you would expect to be right approximately half of the time. In other words, the direction prediction accuracy of the naive predictor is 50%. This, then, is one of the key benchmarks you use to assess the ability of the model to predict market direction. That is what test statistics like Theil’s-U does – measures the performance relative to the naive predictor. The other benchmark we use is the change of direction predicted by the implied volatility of ATM options.
In this context, the model’s 61% or higher direction prediction accuracy is very significant (at the 4% level in fact) and this is reflected in the Theil’s-U statistic of 0.82 (lower is better). By contrast, Theil’s-U for the Implied Volatility forecast is 1.46, meaning that IV is a much worse predictor of 1-period-ahead changes in volatility than the naive predictor.

On its face, it is because of this exceptional direction prediction accuracy that a simple strategy is able to generate what appear to be abnormal returns using the change of direction forecasts generated by the model, as described in the paper. In fact, the situation is more complicated than that, once you introduce the concept of a market price of volatility risk.

October 6, 2020

Market Timing in the S&P 500 Index Using Volatility Forecasts

There has been a good deal of interest in the market timing ideas discussed in my earlier blog post Using Volatility to Predict Market Direction, which discusses the research of Diebold and Christoffersen into the sign predictability induced by volatility dynamics. The ideas are thoroughly explored in a QuantNotes article from 2006, which you can download here.

There is a follow-up article from 2006 in which Christoffersen, Diebold, Mariano and Tay develop the ideas further to consider the impact of higher moments of the asset return distribution on sign predictability and the potential for market timing in international markets (download here).

Trading Strategy
To illustrate some of the possibilities of this approach, we constructed a simple market timing strategy in which a position was taken in the S&P 500 index or in 90-Day T-Bills, depending on an ex-ante forecast of positive returns from the logit regression model (and using an expanding window to estimate the drift coefficient). We assume that the position is held for 30 days and rebalanced at the end of each period. In this test we make no allowance for market impact, or transaction costs.

Results
Annual returns for the strategy and for the benchmark S&P 500 Index are shown in the figure below. The strategy performs exceptionally well in 1987, 1989 and 1995, when the ratio between expected returns and volatility remains close to optimum levels and the direction of the S&P 500 Index is highly predictable, Of equal interest is that the strategy largely avoids the market downturn of 2000-2002 altogether, a period in which sign probabilities were exceptionally low.

In terms of overall performance, the model enters the market in 113 out of a total of 241 months (47%) and is profitable in 78 of them (69%). The average gain is 7.5% vs. an average loss of –4.11% (ratio 1.83). The compound annual return is 22.63%, with an annual volatility of 17.68%, alpha of 14.9% and Sharpe ratio of 1.10.

The under-performance of the strategy in 2003 is explained by the fact that direction-of-change probabilities were rising from a very low base in Q4 2002 and do not reach trigger levels until the end of the year. Even though the strategy out-performed the Index by a substantial margin of 6% , the performance in 2005 is of concern as market volatility was very low and probabilities overall were on a par with those seen in 1995. Further tests are required to determine whether the failure of the strategy to produce an exceptional performance on par with 1995 was the result of normal statistical variation or due to changes in the underlying structure of the process requiring model recalibration.

Future Research & Development
The obvious next step is to develop the approach described above to formulate trading strategies based on sign forecasting in a universe of several assets, possibly trading binary options. The approach also has potential for asset allocation, portfolio theory and risk management applications.

July 7, 2020July 7, 2020

Trading Prime Market Cycles

Magicicada is the genus of the 13-year and 17-year periodical cicadas of eastern North America. Magicicada species spend most of their 13- and 17-year lives underground feeding on xylem fluids from the roots of deciduous forest trees in the eastern United States. After 13 or 17 years, mature cicada nymphs emerge in the springtime at any given locality, synchronously and in tremendous numbers. Within two months of the original emergence, the lifecycle is complete, the eggs have been laid, and the adult cicadas are gone for another 13 or 17 years.

The emergence period of large prime numbers (13 and 17 years) has been hypothesized to be a predator avoidance strategy adopted to eliminate the possibility of potential predators receiving periodic population boosts by synchronizing their own generations to divisors of the cicada emergence period. If, for example, the cycle length was, say, 12 years, then the species would be exposed to predators regenerating over cycles of 2, 3, 4, or 6 years. Limiting their cycle to a large prime number reduces the variety of predators the species is likely to face.

Prime Cycles in Trading Strategies

What has any of this to do with trading? When building a strategy in a particular market we might start by creating a model that works reasonably well on, say, 5-minute bars. Then, in order to improve the risk-adjusted returns we might try create a second sub-strategy on a different frequency. This will hopefully result in a new series of signals, an increase in the number of trades, and corresponding improvement in the risk-adjusted returns of the overall strategy. This phenomenon is referred to as temporal diversification.

What time frequency should we select for our second sub-strategy? There are many factors to consider, of course, but one of them is that we would like to see as few duplicate signals between the two sub-strategies. Otherwise we will simply be replicating trades, rather than reducing the overall level of strategy risk through temporal diversification. The best way to minimize the overlap in signals generated by multiple sub-strategies is to use prime number bar frequencies (5 minute, 7 minute, 11 minute, etc).

S&P500 Swing Trading Strategy

An example of this approach is our EMini Swing Trading strategy which we operate on our Systematic Algotrading Platform. This strategy is actually a combination of several different sub-strategies that operate on 5-minute, 11-minute, 17-minute and 31-minute bars. Each strategy focuses on a different set of characteristics of the S&P 500 futures market, but the key point here is that the trading signals very rarely overlap and indeed several of the sub-strategies have a low correlation.

The resulting increase in trade frequency and temporal diversification produces very attractive risk-adjusted performance: after an exceptional year in 2017 which saw a 78.58% net return, the strategy is already at +60% YTD in 2018 and showing no sign of slowing down.

Investors can auto-trade the E-Mini Swing Trading strategy and many other strategies in their own account – see the Leaderboard for more details.

March 10, 2020

Capitalizing on the Coming Market Crash

Long-Only Equity Investors

Recently I have been discussing possible areas of collaboration with an RIA contact on LinkedIn, who also happens to be very familiar with the hedge fund world. He outlined the case of a high net worth investor in equities (long only), who wanted to remain invested, but was becoming increasingly concerned about the prospects for a significant market downturn, or even a market crash, similar to those of 2000 or 2008.

I am guessing he is not alone: hardly a day goes by without the publication of yet another article sounding a warning about stretched equity valuations and the dangerously elevated level of the market.

The question put to me was, what could be done to reduce the risk in the investor’s portfolio?

Typically, conservative investors would have simply moved more of their investment portfolio into fixed income securities, but with yields at such low levels this is hardly an attractive option today. Besides, many see the bond market as representing an even more extreme bubble than equities currently.

Hedging Strategies

The problem with traditional hedging mechanisms such as put options, for example, is that they are relatively expensive and can easily reduce annual returns from the overall portfolio by several hundred basis points. Even at current low level of volatility the performance drag is noticeable, since the potential upside in the equity portfolio is also lower than it has been for some time. A further consideration is that many investors are not mandated – or are simply reluctant – to move beyond traditional equity investing into complex ETF products or derivatives.

An equity long/short hedge fund product is one possible solution, but many equity investors are reluctant to consider shorting stocks under any circumstances, even for hedging purposes. And while a short hedge may provide some downside protection it is unlikely to fully safeguard the investor in a crash scenario. Furthermore, the cost of a hedge fund investment is typically greater than for a long-only product, entailing the payment of a performance fee in addition to management fees that are often higher than for standard investment products.

The Ideal Investment Strategy

Given this background, we can say that the ideal investment strategy is one that:

Invests long-only in equities
Is inexpensive to implement (reasonable management fees; no performance fees)
Does not require shorting stocks, or expensive hedging mechanisms such as options
Makes acceptable returns during both bull and bear markets
Is likely to produce positive returns in a market crash scenario

A typical buy-and-hold approach is unlikely to meet only the first three requirements, although an argument could be made that a judicious choice of defensive stocks might enable the investment portfolio to generate returns at an “acceptable” level during a downturn (without being prescriptive as to the precise meaning of that term may be). But no buy-and-hold strategy could ever be expected to prosper during times of severe market stress. A more sophisticated approach is required.

Market Timing

Market timing is regarded as a “holy grail” by some quantitative strategists. The idea, simply, is to increase or reduce risk exposure according to the prospects for the overall market. For a very long time the concept has been dismissed as impossible, by definition, given that markets are mostly efficient. But analysts have persisted in the attempt to develop market timing techniques, motivated by the enormous benefits that a viable market timing strategy would bring. And gradually, over time, evidence has accumulated that the market can be timed successfully and profitably. The rate of progress has accelerated in the last decade by the considerable advances in computing power and the development of machine learning algorithms and application of artificial intelligence to investment finance.

I have written several articles on the subject of market timing that the reader might be interested to review (see below). In this article, however, I want to focus firstly on the work on another investment strategist, Blair Hull.

http://jonathankinlay.com/2014/07/how-to-bulletproof-your-portfolio/

http://jonathankinlay.com/2014/07/enhancing-mutual-fund-returns-with-market-timing/

The Hull Tactical Fund

Blair Hull rose to prominence in the 1980’s and 1990’s as the founder of the highly successful quantitative option market making firm, the Hull Trading Company which at one time moved nearly a quarter of the entire daily market volume on some markets, and executed over 7% of the index options traded in the US. The firm was sold to Goldman Sachs at the peak of the equity market in 1999, for a staggering $531 million.

Blair used the capital to establish the Hull family office, Hull Investments, and in 2013 founded an RIA, Hull Tactical Asset Allocation LLC. The firm’s investment thesis is firmly grounded in the theory of market timing, as described in the paper “A Practitioner’s Defense of Return Predictability”, authored by Blair Hull and Xiao Qiao, in which the issues and opportunities of market timing and return predictability are explored.

In 2015 the firm launched The Hull Tactical Fund (NYSE Arca: HTUS), an actively managed ETF that uses quantitative trading model to take long and short positions in ETFs that seek to track the performance of the S&P 500, as well as leveraged ETFs or inverse ETFs that seek to deliver multiples, or the inverse, of the performance of the S&P 500. The goal to achieve long-term growth from investments in the U.S. equity and Treasury markets, independent of market direction.

How well has the Hull Tactical strategy performed? Since the fund takes the form of an ETF its performance is a matter in the public domain and is published on the firm’s web site. I reproduce the results here, which compare the performance of the HTUS ETF relative to the SPDR S&P 500 ETF (NYSE Arca: SPY):

Although the HTUS ETF has underperformed the benchmark SPY ETF since launching in 2015, it has produced a higher rate of return on a risk-adjusted basis, with a Sharpe ratio of 1.17 vs only 0.77 for SPY, as well as a lower drawdown (-3.94% vs. -13.01%). This means that for the same “risk budget” as required to buy and hold SPY, (i.e. an annual volatility of 13.23%), the investor could have achieved a total return of around 36% by using margin funds to leverage his investment in HTUS by a factor of 2.8x.

How does the Hull Tactical team achieve these results? While the detailed specifics are proprietary, we know from the background description that market timing (and machine learning concepts) are central to the strategy and this is confirmed by the dynamic level of the fund’s equity exposure over time:

A Long-Only, Crash-Resistant Equity Strategy

A couple of years ago I and my colleagues carried out an investigation of long-only equity strategies as part of a research project. Our primary focus was on index replication, but in the course of our research we came up with a methodology for developing long-only strategies that are highly crash-resistant.

The performance of our Long-Only Market Timing strategy is summarized below and compared with the performance of the HTUS ETF and benchmark SPY ETF (all results are net of fees). Over the period from inception of the HTUS ETF, our LOMT strategy produced a higher total return than HTUS (22.43% vs. 13.17%), higher CAGR (10.07% vs. 6.04%), higher risk adjusted returns (Sharpe Ratio 1.34 vs 1.21) and larger annual alpha (6.20% vs 4.25%). In broad terms, over this period the LOMT strategy produced approximately the same overall return as the benchmark SPY ETF, but with a little over half the annual volatility.

Application of Artificial Intelligence to Market Timing

Like the HTUS ETF, our LOMT strategy operates with very low fees, comparable to an ETF product rather than a hedge fund (1% management fee, no performance fees). Again, like the HTUS ETF our LOMT products makes no use of leverage. However, unlike HTUS it avoids complicated (and expensive) inverse or leveraged ETF products and instead invests only in two assets – the SPY ETF and 91-day US Treasury Bills. In other words, the LOMT strategy is a pure market timing strategy, moving capital between the SPY ETF and Treasury Bills depending on its forecast of future market performance. These forecasts are derived from machine learning algorithms that are specifically tuned to minimize the downside risk in the investment portfolio. This not only makes strategy returns less volatile, but also ensures that the strategy is very robust to market downturns.

In fact, even better than that, not only does the LOMT strategy tend to avoid large losses during periods of market stress, it is capable of capitalizing on the opportunities that more volatile market conditions offer. Looking at the compounded returns (net of fees) over the period from 1994 (the inception of the SPY ETF) we see that the LOMT strategy produces almost double the total profit of the SPY ETF, despite several years in which it underperforms the benchmark. The reason is clear from the charts: during the periods 2000-2002 and again in 2008, when the market crashed and returns in the SPY ETF were substantially negative, the LOMT strategy managed to produce positive returns. In fact, the banking crisis of 2008 provided an exceptional opportunity for the LOMT strategy, which in that year managed to produce a return nearing +40% at a time when the SPY ETF fell by almost the same amount!

Long Volatility Strategies

I recall having a conversation with Nassim Taleb, of Black Swan fame, about his Empirica fund around the time of its launch in the early 2000’s. He explained that his analysis had shown that volatility was often underpriced due to an under-estimation of tail risk, which the fund would seek to exploit by purchasing cheap out-of-the-money options. My response was that this struck me a great idea for an insurance product, but not a hedge fund – his investors, I explained, were going to hate seeing month after month of negative returns and would flee the fund. By the time the big event occurred there wouldn’t be sufficient AUM remaining to make up the shortfall. And so it proved.

A similar problem arises from most long-volatility strategies, whether constructed using options, futures or volatility ETFs: the combination of premium decay and/or negative carry typically produces continuing losses that are very difficult for the investor to endure.

Conclusion

What investors have been seeking is a strategy that can yield positive returns during normal market conditions while at the same time offering protection against the kind of market gyrations that typically decimate several years of returns from investment portfolios, such as we saw after the market crashes in 2000 and 2008. With the new breed of long-only strategies now being developed using machine learning algorithms, it appears that investors finally have an opportunity to get what they always wanted, at a reasonable price.

And just in time, if the prognostications of the doom-mongers turn out to be correct.

Contact Hull Tactical

Contact Systematic Strategies

December 3, 2019

Conditional Value at Risk Models

One of the most widely used risk measures is the Value-at-Risk, defined as the expected loss on a portfolio at a specified confidence level. In other words, VaR is a percentile of a loss distribution.
But despite its popularity VaR suffers from well-known limitations: its tendency to underestimate the risk in the (left) tail of the loss distribution and its failure to capture the dynamics of correlation between portfolio components or nonlinearities in the risk characteristics of the underlying assets.

One method of seeking to address these shortcomings is discussed in a previous post Copulas in Risk Management. Another approach known as Conditional Value at Risk (CVaR), which seeks to focus on tail risk, is the subject of this post. We look at how to estimate Conditional Value at Risk in both Gaussian and non-Gaussian frameworks, incorporating loss distributions with heavy tails and show how to apply the concept in the context of nonlinear time series models such as GARCH.

Var, CVaR and Heavy Tails

November 6, 2018

The Lazarus Effect

A perennial favorite with investors, presumably because they are easy to understand and implement, are trades based on a regularly occurring pattern, preferably one that is seasonal in nature. A well-known example is the Christmas effect, wherein equities generally make their highest risk-adjusted returns during the month of December (and equity indices make the greater proportion of their annual gains in the period from November to January).

As we approach the Easter holiday I thought I might join in the fun with a trade of my own. There being not much new under the sun, I can assume that there is some ancient trader’s almanac that documents the effect I am about to describe. If so, I apologize in advance if this is duplicative.

The Pattern of Returns in the S&P 500 Index Around Easter

I want to look at the pattern of pre- and post- Easter returns in the S&P 500 index using weekly data from 1950 (readers can of course substitute the index, ETF or other tradable security in a similar analysis).

The first question is whether there are significant differences (economic and statistical) in index returns in the weeks before and after Easter, compared to a regular week.

It is perhaps not immediately apparent from the smooth histogram plot above, but a whisker plot gives a clearer indication of the disparity in the distributions of returns in the post-Easter week vs. regular weeks.

It is evident that chief distinction is not in the means of the distributions, but in their variances.

A t-test (with unequal variances) confirms that the difference in average returns in the index in the post-Easter week vs. normal weeks is not statistically significant.

It appears that there is nothing special about Index returns in the post-Easter period.

The Lazarus Effect

Hold on – not so fast. Suppose we look at conditional returns: that is to say, we consider returns in the post-Easter week for holiday periods in which the index sold off in the week prior to Easter.

There are 26 such periods in the 65 years since 1950 and when we compare the conditional distribution of index returns for these periods against the unconditional distribution of weekly returns we appear to find significant differences in the distributions. Not only is the variance of the conditional returns much tighter, the mean is clearly higher than the unconditional weekly returns.

The comparison is perhaps best summarized in the following table. Here we can see that the average conditional return is more than twice that of the unconditional return in the post-Easter week and almost 4x as large as the average weekly return in the index. The standard deviation in conditional returns for the post-Easter week is less than half that of the unconditional weekly return, producing and information ratio that is almost 10x larger. Furthermore, of the 26 periods in which the index return in the week prior to Easter was negative, 22 (85%) produced a positive return in the week after Easter (compared to a win rate of only 57% for unconditional weekly returns.

A t-test of conditional vs. unconditional weekly returns confirms that the 58bp difference in conditional vs unconditional (all weeks) average returns is statistically significant at the 0.2% level.

Our initial conclusion, therefore, is that there appears to be a statistically significant pattern in the conditional returns in the S&P 500 index around the post-Easter week. Specifically, the returns in the post-Easter week tend to be much higher than average for periods in which the pre-Easter weekly returns were negative.

More simply, the S&P 500 index tends to rebound strongly in the week after Easter – a kind of “Lazarus” effect.

Lazarus – Or Not?

Hold on – not so fast. What’s so special about Easter? Yes, I realize it’s topical. But isn’t this so-called Lazarus effect just a manifestation of the usual mean-reversion in equity index returns? There is a tendency for weekly returns in the S&P 500 index to “correct” in the week after a downturn. Maybe the Lazarus effect isn’t specific to Easter.

To examine this hypothesis we need to compare two sets of conditional weekly returns in the S&P 500 index:

A: Weeks in which the prior week’s return was negative

B: the subset of A which contains only post-Easter weeks

If the difference in average returns for sets A and B is not statistically significant, we would conclude that the so-called Lazarus effect is just a manifestation of the commonplace mean reversion in weekly returns. Only if the average return for the B data set is significant higher than that for set A would we be able to conclude that, in addition to normal mean reversion at weekly frequency, there is an incremental effect specific to the Easter period – the Lazarus effect.

Let’s begin by establishing that there is a statistically significant mean reversion effect in weekly returns in the S&P 500 Index. Generally, we expect a fall in the index to be followed by a rise (and perhaps vice versa). So we need to compare the returns in the index for weeks in which the preceding week’s return was positive, vs weeks in which the preceding week’s return was negative. The t-test below shows the outcome.

The average return in weeks following a downturn is approximately double that during weeks following a rally and the effect is statistically significant at the 3% level.

Given that result, is there any incremental “Lazarus” effect around Easter? We test that hypothesis by comparing the average returns during the 26 post-Easter weeks which were preceded by a downturn in the index against the average return for all 1,444 weeks which followed a decline in the index.

The t-test shown in the table below confirms that conditional returns in post-Easter weeks are approximately 3x larger on average than returns for all weeks that followed a decline in the index.

Lazarus, it appears, is alive and well.

Happy holidays, all.

October 2, 2018October 2, 2018

Quant Strategies in 2018

Quant Strategies – Performance Summary Sept. 2018

The end of Q3 seems like an appropriate time for an across-the-piste review of how systematic strategies are performing in 2018. I’m using the dozen or more strategies running on the Systematic Algotrading Platform as the basis for the performance review, although results will obviously vary according to the specifics of the strategy. All of the strategies are traded live and performance results are net of subscription fees, as well as slippage and brokerage commissions.

Volatility Strategies

Those waiting for the hammer to fall on option premium collecting strategies will have been disappointed with the way things have turned out so far in 2018. Yes, February saw a long-awaited and rather spectacular explosion in volatility which completely destroyed several major volatility funds, including the VelocityShares Daily Inverse VIX Short-Term ETN (XIV) as well as Chicago-based hedged fund LJM Partners (“our goal is to preserve as much capital as possible”), that got caught on the wrong side of the popular VIX carry trade. But the lack of follow-through has given many volatility strategies time to recover. Indeed, some are positively thriving now that elevated levels in the VIX have finally lifted option premiums from the bargain basement levels they were languishing at prior to February’s carnage. The Option Trader strategy is a stand-out in this regard: not only did the strategy produce exceptional returns during the February melt-down (+27.1%), the strategy has continued to outperform as the year has progressed and YTD returns now total a little over 69%. Nor is the strategy itself exceptionally volatility: the Sharpe ratio has remained consistently above 2 over several years.

Hedged Volatility Trading

Investors’ chief concern with strategies that rely on collecting option premiums is that eventually they may blow up. For those looking for a more nuanced approach to managing tail risk the Hedged Volatility strategy may be the way to go. Like many strategies in the volatility space the strategy looks to generate alpha by trading VIX ETF products; but unlike the great majority of competitor offerings, this strategy also uses ETF options to hedge tail risk exposure. While hedging costs certainly acts as a performance drag, the results over the last few years have been compelling: a CAGR of 52% with a Sharpe Ratio close to 2.

F/X Strategies

One of the common concerns for investors is how to diversify their investment portfolios, especially since the great majority of assets (and strategies) tend to exhibit significant positive correlation to equity indices these days. One of the characteristics we most appreciate about F/X strategies in general and the F/X Momentum strategy in particular is that its correlation to the equity markets over the last several years has been negligible. Other attractive features of the strategy include the exceptionally high win rate – over 90% – and the profit factor of 5.4, which makes life very comfortable for investors. After a moderate performance in 2017, the strategy has rebounded this year and is up 56% YTD, with a CAGR of 64.5% and Sharpe Ratio of 1.89.

Equity Long/Short

Thanks to the Fed’s accommodative stance, equity markets have been generally benign over the last decade to the benefit of most equity long-only and long-short strategies, including our equity long/short Turtle Trader strategy , which is up 31% YTD. This follows a spectacular 2017 (+66%) , and is in line with the 5-year CAGR of 39%. Notably, the correlation with the benchmark S&P500 Index is relatively low (0.16), while the Sharpe Ratio is a respectable 1.47.

Equity ETFs – Market Timing/Swing Trading

One alternative to the traditional equity long/short products is the Tech Momentum strategy. This is a swing trading strategy that exploits short term momentum signals to trade the ProShares UltraPro QQQ (TQQQ) and ProShares UltraPro Short QQQ (SQQQ) leveraged ETFs. The strategy is enjoying a banner year, up 57% YTD, with a four-year CAGR of 47.7% and Sharpe Ratio of 1.77. A standout feature of this equity strategy is its almost zero correlation with the S&P 500 Index. It is worth noting that this strategy also performed very well during the market decline in Feb, recording a gain of over 11% for the month.

Futures Strategies

It’s a little early to assess the performance of the various futures strategies in the Systematic Strategies portfolio, which were launched on the platform only a few months ago (despite being traded live for far longer). For what it is worth, both of the S&P 500 E-Mini strategies, the Daytrader and the Swing Trader, are now firmly in positive territory for 2018. Obviously we are keeping a watchful eye to see if the performance going forward remains in line with past results, but our experience of trading these strategies gives us cause for optimism.

Conclusion: Quant Strategies in 2018

There appear to be ample opportunities for investors in the quant sector across a wide range of asset classes. For investors with equity market exposure, we particularly like strategies with low market correlation that offer significant diversification benefits, such as the F/X Momentum and F/X Momentum strategies. For those investors seeking the highest risk adjusted return, option selling strategies like the Option Trader strategy are the best choice, while for more cautious investors concerned about tail risk the Hedged Volatility strategy offers the security of downside protection. Finally, there are several new strategies in equities and futures coming down the pike, several of which are already showing considerable promise. We will review the performance of these newer strategies at the end of the year.

Go here for more information about the Systematic Algotrading Platform.