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Short-Term Santa Claus Rally Strategy

12 月 28, 2013 によってアンドリュー・セルビー Leave a Comment

This is the time of year when it seems like everyone is talking about a “Santa Claus Rally.” According to Investopedia, the concept of the Santa Claus Rally describes a spike in stock prices over the week between the Christmas and New Year’s holidays.

There are a number of factors that get credit for this spike, including tax purposes, people investing their Christmas bonuses, and bears taking the week off. Investopedia also suggests that the spike in stock prices is likely due to investors moving into stocks because they are anticipating a January rally.

定量的なトレーダーとして, we are able to see that many of these conversations are based on selectively bullish memories that are dramatically skewed towards recent years. Are there really any significant numbers to support the idea of a Santa Claus Rally?

Jeff from System Trader Success published an interesting article that takes a strictly quantitative look at how the S&P 500 has traded over the ten days before and after Christmas since 1964.

Jeff’s Christmas Trade Strategy

In order to first test the concept of a Santa Claus Rally, Jeff wrote a simple system that started with $50,000 and would risk 2% of its capital on a long SPX trade every Christmas season. He defined the risk on a trade as three times the 10-day 平均該当範囲 of the SPX.

Jeff from System Trader Success did some research and produced a very simple Santa Claus Rally Strategy.

To determine the best entry date for his strategy, Jeff tested the performance of SPX based on purchases made between one and ten days before Christmas. He used the same idea to determine an exit date by testing the performance of SPX between one and ten days after Christmas.

The performance before Christmas was positive regardless of the entry date, but the performance improved with longer dates. The post-Christmas results were very similar with longer hold times generally performing better. Jeff elected to go with five days before Christmas as his entry and five days after Christmas as his exit.

Backtesting the Strategy

Backtesting Jeff’s strategy over 48 potential Santa Claus Rallies produced a net profit of $23,964 on his $50,000 capital. The strategy recorded a win rate of 79% and an average profit of $499.

In a simple attempt to improve the strategy’s performance, Jeff also tested implementing a trend filter using the 200-day simple moving average. Only taking the Christmas trades that occurred during bull markets eliminated 16 取引. On the remaining 32 取引, the win rate jumped to 81% and the average profit jumped to $566.

意外にも, the results of the 16 trades that occurred during bear markets were not terrible. Those trades recorded a win rate of 75% and an average profit of $365.

What About This Year?

How would trading Jeff’s Christmas strategy have worked this year?

With three days to go, it looks like Jeff’s Christmas Trade is going to be profitable again this year!

あなたが見ることができます。, five days before Christmas the SPX had a big up day. If the system would have been able to buy at the open on that day, it would be sitting on more than a 3% 利益 with three days left until it exits.

One of the most interesting aspects of this strategy is that it only employs your capital for 10 days every year. This means that it won’t make a great strategy on its own, but could be very possibly be worked into another strategy.

What Quantitative Value Do Stops Actually Have?

12 月 20, 2013 によってアンドリュー・セルビー Leave a Comment

One of the questions that every quantitative trader must address is whether adding a stop-loss component to their system will help or hinder its performance. I have written a good deal lately about the pros and cons of different types of stops, but haven’t had much actual backtesting data to work with.

When I wrote a post about about Cesar Alvarez’s S&P Rotational Strategy a few weeks ago, I suggested that adding a stop-loss might lower the maximum drawdowns. This would give the strategy a way to exit losing positions during the month, rather than waiting for the monthly redistribution. Theoretically, this would have reduced some of the big losses that the strategy suffered in 2008.

We assume that adding a stop loss component has the quantitative value of a safety net, but that isn’t always the case.

In addition to writing about that idea here, I also commented on Alvarez’s post. In response to that, he has written a follow-up post addressing my suggestion to implement stops:

Continuing from the post, we are adding a maximum stop loss. The stop is evaluated at the close each day with the exit happening at the close. The tested stops are 5%, 10% と 15%.

興味深いことに, Alvarez finds that adding stops can be helpful in some situations and terrible for performance in other situations. While adding stops may always seem like a logical idea in theory, Alvarez shows that actual performance can prove otherwise.

Best Performing Stocks

The version of the best performing stocks strategy that we looked at in the previous post utilized a market timing indicator and a six month look-back period. That strategy produced a CAR of 10.48% 最大ドローダウンと 42.22%. Here are the numbers when 5, 10, と 15 percent stops were added:

5% 停止: 10.51% CAR, 26.30% MDD
10% 停止: 10.85% CAR, 38.05% MDD
15% 停止: 10.84% CAR, 39.48% MDD

あなたが見ることができます。, adding the stop loss doesn’t do much for the CAR, but it does a great job of reducing the maximum drawdown. When the stops were applied to the version of the strategy with a 12 month look-back period, the impact on maximum drawdown was similar, but the CAR saw a bit more of an increase. When the stops were applied to each of the two versions without the market timing indicator, we saw a slightly less impact on drawdown and a much greater impact on CAR.

Alvarez also commented that in almost all cases, 、 5% stop was the best performer, which he thought was unusual:

Normally close stops tend to be the worst but the 5% stop tends to be the best.

Worst Performing Stocks

The worst performing stocks version of the strategy that we looked at used the market timing indicator and a six month look-back period. The strategy without stops had a CAR of 13.05% 最大ドローダウン 27.88%. Here are what the numbers look like when the different levels of stops were applied:

5% 停止: 5.11% CAR, 28.26% MDD
10% 停止: 8.36 CAR, 30.90% MDD
15% 停止: 10.47% CAR, 30.87% MDD

この場合, adding the stops has really hurt the strategy. While there was some improvement in the maximum drawdowns of some of the versions, adding stops basically crippled the CAR of all of the worst performing stocks strategies.

Alvarez notes that this is the result he expected:

For the worst N-month ranking, stops appear to hurt the all results. These results support previous research that stops on short-term mean reversion hurt results.

取引のボラティリティ: 基本的な VXX 戦略

12 月 12, 2013 によってアンドリュー・セルビー Leave a Comment

絶えず進化する ETF ・ ETN 市場は信じられないほど簡単プロのトレーダーのために予約するために使用するさまざまな市場の多くへのアクセスを持っている平均小売業者にしました。. それと新しい可用性戦略と定量的アプローチのためのアイデアの大量流入してきたこれらの市場.

VXX ETN の作成は、市場のボラティリティを取引する能力の小売業者を提供しています。. 単に一般的な市場の指標として、VIX を使用するために使用トレーダーが実際にそのインジケーターを交換することができます。.

小売業者のオプションに取引のボラティリティ、VXX なります. ジェイビアガーデンが取引戦略を私たちのユニークなアイデアを与える.

公開取引のボラティリティからジェイビアガーデン、 VXX の取引のための戦略 VXX 毎週ロール収量からの信号をに基づいてください。 (意地の悪い) 10 日間の簡単な移動平均.

この考えのための規則は非常に単純です:

2 つあります。 “半分” この取引戦略の:
1) 短い VXX/UVXY をされています。 (または長い 14 世/SVXY) いつでも、WRY が移動平均を下回る, と
2) 長い VXX/UVXY をされています。 (または短い XIV) いつでも、WRY が移動平均を上回る.

彼は彼のバックテストパラメーターを提供してくれます:

私は別の戦略を backtested “短い VXX のみ” と “長い VXX のみ” VXX の当初から (1/30/2009) を通じて 12/10/2013.
決定点は日の決算データを使って作られています (分析で個々の取引を表示することができます。ここで).

ジェイの短い側のリターンは非常に有望に見える:

短い VXX のみの:
# 益: 59
# 損失の: 47
平均リターン: +4.1%
最大ゲイン: +40.4%
最大損失: -19.2%
利益の合計 & 損失: +438.6%

彼の長い側のリターンで収益性の高いはないです。:

長い VXX のみの:
# 益: 32
# 損失の: 74
平均リターン: -0.8%
最大ゲイン: +60.6%
最大損失: -17.4%
利益の合計 & 損失: -81.9%

あなたが見ることができます。, この戦略の短辺がエッジを持っているよう, 長い側はかなりスタックが、. ジェイは、個々の取引のリターンを報告するヒストグラムを提供する続いてください。.

彼も収益性の高いに多く長辺取引の開始を示唆しています。, しかし、その利益で返したし、わずかな損失になって. 彼の理論は、トレーリングストップを実装するだろうそれらの利益を保護および長い側に成功戦略をレンダリング.

トレーリングストップを追加することに加えて, 私も長期的トレンドフィルターを追加する、リターンにどのような影響が見て興味があります。. どのように彼の負けトレードの多くの反対側で行われたのだろうか、 100 または 200 日単純移動平均.

もう 1 回お願いします, 我々は、ユニークで有益な何かに開発される可能性が戦略を持っています。. しかし, この時点で、それは面白いアイデアだけでは基本的に.

あなたは、データスヌーピングは、あなたのバックテストに影響を与えるまかせています?

12 月 11, 2013 によってアンドリュー・セルビー Leave a Comment

It is important to eliminate as much bias as possible from your backtesting process. This will ensure that any positive results are more likely to continue moving forward. しかし, eliminating those biases can be more difficult that you might think.

Data snooping is one of the more wide-ranging biases that you might be exposing your backtesting to. It can range from obvious things like limiting your data sample to a bull market period. It can also show up more discretely in areas like optimizing parameters for different indicators your strategy might be using.

Being overly selective about the data or parameters you use for your backtest can expose your results to data snooping bias.

CXO Advisory group took a look at snooping bias in a recent post that is actually a preview of a chapter in their upcoming book. The name of the chapter, Avoiding or Mitigating Snooping Bias, suggests that there is likely to be some form of snooping bias in any backtest. Our goal is to reduce its affect on the performance numbers as much as possible.

The chapter starts with a list of the different forms that snooping bias can take on:

Snooping bias, also called mining bias and more loosely benefit of hindsight, is a notorious artificial booster of backtest performance. It takes multiple forms:
Picking the best of many rules/indicators (戦略, models) for a given data sample
Optimizing rule parameters for a given data sample
Restricting a data sample to find favorable performance of a given rule
Running an investment contest among many individuals

The chapter then continues with a legal analogy:

A sentiment shared among researchers in stochastic fields is: “If you torture the data long enough, it will confess to anything.”
Because returns are noisy (substantially random), trying many combinations of rules, parameter settings and data samples will generate strategies that outperform benchmarks by extreme good luck.
A prosecutor (an investor) satisfied with false confessions is likely to lose in court (the market).

This analogy does a good job of providing a broad overview of snooping bias. While it is almost impossible to completely remove biases from your backtesting, using some simple common sense in your attempt to optimize a strategy can do a great deal to reduce exposure to snooping bias.

転送最適化を歩く: A More Detailed Explanation

12 月 10, 2013 によってアンドリュー・セルビー Leave a Comment

Two weeks ago, we looked at an example of using walk forward optimization by VBO Systems that tested a volatility breakout system. While this article was interesting from a nuts and bolts aspect, it left a lot on the table in terms of explanation.

The author has since expanded that post to include a more detailed explanation of exactly what we are trying to do with walk forward optimization. This new introduction to the article provides us with some details and background on why walk forward optimization is so effective.

Walk forward optimization allows up to test how a strategy would have traded in a live environment and evaluate which parameters would have performed best.

The article starts by listing some of the reasons that systems can lose their edge:

The system is not based on a valid premise
Market conditions have changed in a dramatic way that invalids the theoretical premises on which the system was developed
The system has not been developed and tested with a sound methodology. 例えば, (、) lack of robustness in a system due to improper parameters, と (b) inconsistent rules and improper testing of the system using out-of-sample and in-sample data

It continues by explaining how a basic walk forward optimization is conducted:

Walk forward analysis is the process of optimizing a trading system using a limited set of parameters, and then testing the best optimized parameter set on out-of-sample data.
This process is similar to how a trader would use an automated trading system in real live trading. The in sample time window is shifted forward by the period covered by the out-of-sample test, and the process is repeated.
At the end of the test, all of the recorded results are used to assess the trading strategy.

In order to make sure the concept is understood, it is also explained another way:

他の言葉で, walk forward analysis does optimization on a training set; tests on a period after the set and then rolls it all forward and repeats the process.
We have multiple out-of-sample periods and look at these results combined. Walk forward testing is a specific application of a technique known as Cross-validation.
It means taking a segment of data to optimize a system, and another segment of data to validate. This gives a larger out-of-sample period and allows the system developer to see how stable the system is over time.

As we covered in the previous post, there are three main aspects of this process:

Define in-sample and out-of-sample periods
Define a robust parameters area
Execute the walk forward

あなたが見ることができます。, performing a walk forward optimization on a system that you are developing will help you to gain an understanding about how a system will perform in real time, while at the same time finding the optimal parameters for the strategy.

Should Your Equities Strategy Focus on Relative Strength or Weakness?

12 月 8, 2013 によってアンドリュー・セルビー 2 コメント

One of the long-running debate for designers of equities-based quantitative strategies can be boiled down to three words: strength vs. weakness.

The strength side of the argument maintains that stocks that are breaking into new highs are statistically more likely to continue higher. 反対側に, the weakness argument advocates buying stocks at their lows because they are priced at a discount to their actual value.

There are investing and trading icons on both side of this argument going all the way back to guys like Jesse Livermore and Benjamin Graham. There have also been plenty of successful examples of both strategies over the past hundred years.

Cesar Alvarez put together a simple rotational system that allows us to compare 相対的な強さ to weakness.

In one of his recent posts, Cesar Alvarez decided to look into designing a simple rotational strategy that would use the S&P 500 stocks as its universe and rebalance its positions on a monthly basis. The interesting debate that he faced was whether to focus on holding the best performing stocks or the worst performing stocks.

もちろんです, the best way to determine the answer to that debate was to actually test both strategies. Here are the rules and parameters as he defined them:

Test Parameters

差出人 1/1/2001 宛先 10/31/2013
$.01/share for commission
3 month T-bill interest rate used for cash

Best Performing Stocks

It is the first trading day of the month
Stock is a member of the S&P500
Rank stocks from the highest to lowest by (6,12) month return
Buy the top 20 ranked stocks at the close
Optional market timing rule: SPX close > 200 day moving average of SPX closes on the entry day
Hold until next month

Worst Performing Stocks

It is the first trading day of the month
Stock is a member of the S&P500
Rank stocks from the lowest to highest by (6,12) month return
Buy the top 20 ranked stocks at the close
Optional market timing rule: SPX close > 200 day moving average of SPX closes on the entry day
Hold until next month

バックテスト結果

Alvarez lists the stats for each version of this simple system, but we can boil those down with a few quick observations. 最初, the systems that utilize the optional market timing rule always outperformed the systems that did not utilize the rule. 2 番目, in each instance, using a 6-month return produced better returns than using a 12-month return.

After processing those observations, it is clear that we just need to look at two systems. The system that held the best performing stocks each month produced a CAR of 13.05% 最大ドローダウンと 27.88%. The system that held the worst performing stocks each month produced a CAR of 10.48% 最大ドローダウンと 42.22%.

While the numbers indicate that holding the best stocks would have produced a better and less volatile return than holding the worst stocks, both strategies dramatically outperformed a simple buy and hold approach.

Looking at the equity chart that Alvarez provides, you can see that the Worst Stocks Strategy performs just as well (if not better) than the Best Stocks Strategy for most of the twelve year period. The key differences are that the Worst Stocks Strategy got crushed around 2008, and the Best Stocks Strategy have performed phenomenally well this past year.

Alvarez suggests that he would also repeat the test changing the number of stocks held and trying different lookback periods to rank the stocks. I would be curious to see how adding a stop-loss that allowed the strategy to exit positions mid-month would affect the drawdowns.

転送最適化を歩く: Explained in Plain English

11 月 29, 2013 によってアンドリュー・セルビー Leave a Comment

Traders that gravitate towards quantitative strategies are typically nerds.

I don’t mean that in a negative way, because I consider myself one as well. しかし, nerds have a tendency to speak and write using far more complex words and sentences than actually necessary.

Because of that tendency, entry-level explanations can often be confusing for beginners to understand.

This comes across especially well on the topic of walk forward optimization. Most of the articles about the topic are very complex and involve some high level math. This can be extremely discouraging for someone just looking into systematic trading strategies.

Many quantitative traders have a hard time explaining walk forward optimization in simple terms.

VBO Systems posted a very helpful case study using walk forward optimization this week. They started by briefly explaining the three main steps of their walk forward optimization process:

Define in-sample and out-of-sample periods
Define a robust parameters area
Execute the walk forward

That’s simple enough. 次, they specified the system and data that they would be using for the case study:

For this test we will use the FDAX and a volatility breakout (VBO) intraday trading system.
We will use NinjaTrader and CQG historical 1-minute data, assuming 3 points of slippage for each R/T trade to cover trading frictions.

The first step in their process was to identify the in-sample and out-of-sample periods. Here is how they explained it:

We will choose as in-sample 1/1/2001 宛先 12/31/2009 for system design and in-sample optimization and 1/1/2010 宛先 12/31/2012 as out-of-sample period to evaluate the in-sample optimization robustness and execute the walk forward. We will then use a 3:1 ratio for the WFO (walk forward optimization):
Optimize 2007 宛先 2009 and verify performance out-of-sample in 2010
Optimize 2008 宛先 2010 and verify performance out-of-sample in 2011
Optimize 2009 宛先 2011 and verify performance out-of-sample in 2012

The next step is to define the parameters that they are looking to optimize. Here are the three that they listed:

Lookback period of the fast average
Lookback period of the slow average
Volatility filter

これまでのところ, this has been a pretty simple process, and VBO Systems does a great job of keeping their explanations simple. In order to define the robust area for each of these parameters, the article uses a 3D chart to identify the moving average lookback periods that perform reasonably well over the course of the in-sample data. The same process is applied on a standard chart to get the volatility filter parameters.

The final step is to perform the walk forward using the identified data parameters on the defined data periods. 基本的には, they just see which moving averages and volatility filter would have worked best on each of the in-sample data periods, and then test those parameters on the out-of-sample data periods to see if the returns are in-line with expectations.

The result of this case study is that each of the walk forward optimization periods produce similar returns to the overall system returns for the entire in-sample data set. This gives reassurances that the strategy has a certain level of robustness.

AAPL の短期ボリンジャーバンドのブレイクアウト戦略アイデア

11 月 27, 2013 によってアンドリュー・セルビー Leave a Comment

人気の先物と外国為替市場に加えて, テストし、液体とアクティブな個々の株式の戦略を貿易のような多くの定量的なトレーダー.

AAPL はいずれかの世界で最も活発な株式, 出来高とニュース報道. それはまた過去の 10 年にわたって信じられないほど強気にされて. AAPL があり、人気と報道をされている在庫はそれ用に設計されたいくつかの戦略のカスタム行き.

Paststat.com は、トレーダーが取るし、潜在的な戦略の開発を奨励するさまざまな定量的なアイデアに彼らのウェブサイトの全体のセクションを捧げる. 彼らの最近の記事の 1 つはのためのアイデアを紹介します。短期ブレイクアウトシステム AAPL の貿易のために特別に設計されています.

戦略の基本的な考え方は、時間がかかる AAPL のロングポジションの在庫が勃発し、その上のボリンジャーバンドの上で閉まる. 戦略は、間の在庫を保持します。 1 と 5 販売する前に日. 日足のボリンジャーバンドの設定とに基づいて戦略 20 移動平均期間と 2 標準偏差.

この単純なボリンジャーバンドのブレイクアウト戦略 AAPL のために設計されています、 82% 率を獲得し、 1.48 過去数年間で利益率.

それは非常に簡単に含まれているチャートについて説明します:

今後の $AAPL ロングのため取引の確率 1/2/3/4/5 時まで期間日取引 $AAPL すぐ上のボリンジャーバンドの上で交差します。

記事 backtests 12 月のブレイクアウトからこの戦略 2009 11 月まで 2013. 戦略のシンプルさを考慮しました。, 結果は、実際にはかなり印象的です.

1 日の保留期間を使用してください。, 戦略生成 22 受賞者のうち 28 取引. 勝ちトレードの平均利益します。 1.04% 負けトレードの平均損失が、 0.57%.

保持期間は 5 日間に増加したとき, 増資戦略 23 wins. これらの勝ちトレードの平均利益 2.69% 負けトレードの平均損失だったと 1.82%.

記事は、5 日間の開催期間のアプローチについてバックテスト期間にわたって記録された取引のそれぞれの結果を投稿するに行く. これは本当に勝ちトレード数が多いを強調し、負けトレードよりも大きいが、勝ちトレードという事実.

これは、サイズの小さいサンプルでは確かに, 初期のバックテスト結果は、さらにいくつか実際の戦略にこれを開発する時間を投資する価値があることを示す. リターンがトレンドフィルターまたはいくつかの他の確認のインジケーターを実装することによって上改善される可能性があります。.

非常に取引 AAPL のこの戦略のさらなるテストを見るは興味深いだろう. それはまた他の個別銘柄にそれをテストするのには興味深いだろう, さらにそれを改良しようとして.

さらにバックテストの結果が将来のパフォーマンスを保証しないことを証明します。

11 月 18, 2013 によってアンドリュー・セルビー Leave a Comment

理論的には, 時間の経過とともに市場の根本的変化が不採算になること端を持っていたシステムを原因完璧な理にかなって. しかし, 我々は現在の時間まで一般的にバックテストシステム, このような劇的な変化にシステムが発生するは一般的ではないです。.

マットはありません恐怖クマ最近からシステムに出くわしたそれは 2 つの異なる期間をテストする機会を与えた. 彼は書かれていた発見したシステムとの backtested 2009. その時点できれいに撮れました, しかし、それはケースのように継続しませんでしたが表示されます。. マットは前の期間にわたって試験以前のバックテストの結果を複製することが, but the results moving forward were drastically different.

Matt shows us a system that performed one way up through 2009 and then produced inverse results ever since.

Matt shows us a system that performed one way up through 2009 and has produced inverse results ever since.

Here is how Matt came across those posts:

I was browsing my twitter feed this week and saw a couple old Quantifiable Edges posts (ここでとここで) linked to by @PsychTrader.
The two posts were written in mid-2009 and detail a simple weekly strategy that uses the relative performance of the S&P 500 and Nasdaq indexes to time the market.
They showed how investing in the SP 500 or Nasdaq when Nasdaq has been outperforming (based on 10 week relative performance) has generally beat out buy and hold.

Matt decided to test how a simple system that held either QQQ or cash depending on whether QQQ was either outperforming or underperforming the SPY would fare. He tested two different versions. The first version held QQQ when it was outperforming the SPY and cash when it was not. The second version held QQQ when it was underperforming the SPY and cash when it was not.

Here is what Matt found:

First let’s look at the period from 1999 (inception of QQQ) to the end of 2008.
The strategy to invest in QQQ when it underperformed SPY got crushed during the bear markets and just treaded water during the bull market.
The inverse strategy did much better and held its ground through bull and bear period alike.

興味深いことに, the strategies flip-flopped starting in 2009:

前に大きな時間を吸い込ま戦略がされて着実に急上昇高く以前より良い戦略は、水を踏みされている中.
私は市場の気まぐれで自分の頭の上になっている戦略の一例ですね. これは私だろうことを光らせて今後.

あなたが見ることができます。, この毎週の QQQ の戦略論理にかなってし、交換することはかなり容易になります。. 唯一の問題は前進貿易にどのバージョンを知っているだろう. これはさらにバックテストの結果は、将来のパフォーマンスを保証しません、ポイントを強調します。!

どのように正確な Etf を活用?

11 月 17, 2013 によってアンドリュー・セルビー Leave a Comment

One of the main reasons that many quantitative traders are attracted to the futures markets is the amount of leverage that those products offer. Traders seeking ways to leverage their systems can also make use of leveraged ETFs, but how well do those ETFs actually produce leveraged results?

If an ETF is supposed to replicate two or three times the returns of a particular market or index, any variation could severely impact a trading system. If the ETF doesn’t achieve its target leverage, your system will underperform. If it is overleveraged, the system will be exposed to a greater risk of ruin.

Using leveraged ETFs can be very appealing, but how well do those ETFs actually track the markets they represent?

CXO Advisory group published an article that breaks down the daily and long term variations between leveraged ETFs and their targets. They profiled 46 different 2X and -2X ETFs and 10 3X and -3X ETFs in order to determine exactly how accurate these leveraged ETFs actually are in real trading.

Here is how they got started:

We measure achieved average daily leverage by comparing the average daily return of each leveraged ETF to the average daily return of a 1X ETF designed to track the same index.
We measure achieved long-term leverage by comparing the terminal return of each leveraged ETF to the terminal return of a 1X ETF designed to track the same index.

These are the results they discovered for the 2X and -2X ETFs:

The 2X (-2X) ETFs modestly underachieve (achieve or slightly overchieve) targeted daily leverage on average, perhaps due to leveraged fund expenses and the contribution of dividends to the underlying 1X ETFs.
The 2X ETFs generally do not deliver the targeted daily leverage over a long period. Terminal leverages range from -1.8 宛先 35.1, with the latter outlier (real estate sector) relating to a 1X underlying ETF with a small negative terminal return.
The -2X ETFs are generally more consistent than the 2X ETFs over a long period, with all these funds losing most of their value. しかし, two of the -2X funds deliver positive leverage. The outlier with leverage 3.0 relates to an underlying 1X ETF with negative terminal return (financial sector). The outlier with leverage 46.6 relates to the same underlying ETF as the above 2X outlier (real estate sector).

Here are the results they discovered for the 3X and -3X ETFs:

As above, the 3X (-3X) ETFs slightly underachieve (overchieve) targeted daily leverage on average, perhaps due to leveraged fund expenses and the contribution of dividends to the underlying 1X ETFs.
Over the generally bullish sample period, the 3X ETFs are generally effective at delivering terminal leverage.
しかし, the -3X funds do not (cannot) deliver -3X leverage, which would require negative fund values. This result reflects the asymmetry of gains and losses based on simple returns (the largest possible loss is -100%).