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TZA Basics and Its Relationship with TNA

TZA (Direxion Daily Small Cap Bear 3X Shares) is an inverse ETF targeting -3x the daily return of the Russell 2000 index. With a 1.01% expense ratio, managed by Direxion. It is the 'mirror image' of TNA (Direxion Daily Small Cap Bull 3X), profiting when the Russell 2000 declines.

The Russell 2000 comprises 2,000 U.S. small-cap stocks with higher volatility than the S&P 500 or NASDAQ100. Daily volatility is approximately 1.3% (annualized ~21%), well above the S&P 500's 1.0% (annualized 16%). This elevated volatility accelerates TZA's decay.

TNA and TZA are 3x bull and -3x bear on the same index, yet their long-term returns are completely asymmetric. TNA can potentially generate long-term profits, while TZA is guaranteed to decay. The mathematical basis for this asymmetry follows.

Bull vs Bear Asymmetry - Mathematical Proof

A leveraged ETF's long-term return can be approximated from the base index's return (mu) and volatility (sigma). The 3x bull's expected log return is 3*mu - 9*sigma^2/2; the -3x bear's expected log return is -3*mu - 9*sigma^2/2.

The critical difference is clear. The 3x bull has 3*mu (trend-direction profit) minus 9*sigma^2/2 (volatility decay). When mu > 0 (long-term uptrend), the bull can produce positive returns as long as 3*mu > 9*sigma^2/2.

The -3x bear, however, has -3*mu (loss from fighting the trend) AND -9*sigma^2/2 (volatility decay), both negative. As long as mu > 0, the bear's expected return is always negative. The double punishment of trend loss plus decay makes long-term holding of bear products mathematically impossible.

Why Russell 2000's High Volatility Is Especially Punishing for the Bear Side

Russell 2000's annualized volatility of 21% is lower than NASDAQ100's 24% but higher than the S&P 500's 16%. However, Russell 2000's long-term average return is approximately +7% annually, below NASDAQ100's +12% and S&P 500's +10%.

For the 3x bull, the return-to-volatility ratio matters. Russell 2000's mu/sigma^2 = 0.07/0.044 = 1.59, far below the S&P 500's 0.10/0.026 = 3.85. This means TNA has less return cushion against volatility decay compared to UPRO (S&P 500 Bull 3X).

For the -3x bear TZA, higher volatility means greater decay. With Russell 2000's sigma^2 = 0.044, the term 9*sigma^2/2 = 0.198, meaning approximately -20% annual volatility decay. Adding trend loss of -3*mu = -21%, TZA's annual expected return is approximately -41%.

Concrete Calculation - Russell 2000 at +10% Annual Return

Consider a year where Russell 2000 returns +10%. TNA (bull 3x) theoretical return: 3 x 10% minus volatility decay = +30% - approximately 20% = roughly +10%. Despite decay, positive returns are maintained.

TZA (bear 3x) theoretical return: -3 x 10% minus volatility decay = -30% - approximately 20% = -50%. A mere +10% Russell 2000 rise halves TZA's value. Two consecutive such years produce -75%; three years, -87.5%.

Actual data confirms this. TZA's cumulative return since inception (2008) exceeds -99.99%. One million yen became less than 100 yen. Meanwhile, TNA returned over +500% during the same period (though with a maximum drawdown of -99% along the way). This asymmetry is the fundamental difference between bull and bear.

Correct Use of TZA - Ultra-Short-Term Only

TZA generates profits only when the Russell 2000 crashes sharply over a brief period. Small caps tend to fall harder than large caps during crashes; in March 2020, the Russell 2000 dropped over -40%. During this period, TZA returned over +100%.

However, small-cap recoveries are also rapid. From the March 2020 bottom through March 2021, the Russell 2000 rebounded over +100%, and TZA suffered -95% losses. Even if you profit from TZA during a crash, missing the exit timing means losing everything.

TZA holding periods should be limited to 1-2 weeks maximum. Due to Russell 2000's high volatility, decay is even faster than SQQQ or SPXS. Just one month of holding can produce over -10% decay. Quantitative investing books that cover the relationship between compounding and volatility are prerequisites for leveraged ETF investing.

What Happens When You Hold Both Bull and Bear Simultaneously

If you hold equal amounts of TNA and TZA, intuition suggests a net-zero position with no gain or loss. In reality, both decay from volatility, so combined assets steadily decline.

Concrete calculation: on a day Russell 2000 gains +2%, TNA rises +6% and TZA falls -6%. With 1 million yen each, TNA = 1.06M, TZA = 0.94M, total 2M unchanged. Next day Russell 2000 falls -2%: TNA = 1.06 x 0.94 = 0.9964M, TZA = 0.94 x 1.06 = 0.9964M, total 1.9928M. Two days, -0.36% loss.

This loss accumulates daily. Annually, approximately -20% of combined assets vanishes. Holding bull and bear simultaneously is not 'hedging' but 'guaranteed asset destruction.' If hedging is needed, reduce position size or move to cash.

TZA Investment Conclusions and Lessons

TZA is the most extreme embodiment of bull-bear asymmetry in leveraged ETFs. Despite both being 3x on the Russell 2000, TNA can potentially profit long-term while TZA is mathematically incapable of long-term holding.

The root of this asymmetry is the fact that equity markets rise over time. Bull products riding the uptrend can potentially earn returns exceeding volatility decay. Bear products fighting the trend suffer the double burden of trend loss and decay.

The only rational scenario for TZA is a short-term trade when you are confident small caps will crash sharply and can exit within days. Every other use case violates the mathematics of compounding.