# TQQQ: Leveraged ETF Decay Costs February 2022 Update

## Summary

- As expected, LETF Decay costs have continued to increase.
- LETF leakage costs still remain very low compared to March 2020 COVID correction.
- However, unlike March 2020, LETF Decay costs may not be overshadowed by the stratospheric April 2020 and subsequent bounce and bull run.
- LETF decay costs will bite hard in sideways markets.

## Key points carried forward from January 2022

- Leveraged ETF decay costs have doubled in the last 3 months and the trend is expected to continue.
- 2X ProShares Ultra S&P 500 (ARCX:SSO) will leak only 1.5% per year relative to holding 1X SPDR S&P 500 ETF (SPY) or Vanguard S&P 500 ETF (VOO). SSO still represents a very inexpensive form of gearing/leverage.
- However, 3X Direxion Daily Semiconductor Bull (ARCX:SOXL) LETFs at current levels, all things being held equal, is set to lose 41% of its investment value to decay costs every year.
- Similarly, the short LETFs ProShares UltraPro Short QQQ (SQQQ) and ProShares UltraPro Short S&P500 (SPXU) are very expensive and are expected to become a lot more expensive in market correction conditions. LETF costs will increase in multiples.
- Investors should migrate from 3x to 2x LETFS, avoid LETFs on less liquid indices other than on the S&P500 and Nasdaq100, and avoid short LETFs in current market conditions.

### Decay costs continue to increase

On 4 February I wrote an article "Time To De-risk And Optimize Your Riskier Leveraged ETF Positions" I concluded that "decay costs will increase considerably in the coming weeks. This is a good time to de-risk: Move from 3X to 2X LETFs, and avoid or liquidate positions in high leakage, high decay, high cost LETFs - including esoteric LETFs on less liquid, difficult, costly to hedge indices (including gold, minerals, semiconductors, and financial LETFs). Short LETFs will also become significantly more expensive moving forward."

The article generated some debate which critical analysis is very welcome to stress-test the theories and modelling.

### Decay costs are important

I believe that it is important for investment decision making purposes to understand and quantify (and be able to compare) LETF decay costs relative to lower cost LETFs or investing in the underlying index. In my previous article I mentioned that fortunes have been made by Investment Banks, Private Equity Players, Hedge Funds and Issuers benefiting from *inter alia* a low interest rate environment and buoyant stock prices ("positive leverage effect"), particularly where these counterparties are able to control the narrative. Financial products (including LETFs), can be a zero sum gain where somebody's loss is somebody else's gain. I want to know exactly how much I am paying away for my LETF investments (particularly in sideways markets).

As expected LETF Decay costs have in fact continued to increase from my 04 February 2022 article.

Leveraged ETF decay costs have continued to increase as projected, I submit, for 2 primary reasons: LETF hedging costs (including option hedging costs) increase in times of uncertainty and higher volatility, and moreover, and secondly I hold a theory that there is a fixed cost component to LETFs carried forward in SWAP contracts (numerator to LETF cost calculation) and that these full costs must now be absorbed by a far diminished LETF asset base (denominator to LETF cost calculation).

The following bar chart references the same decay data as the above line chart but ranks the covered LETFs from lowest to highest decay cost:

The above bar chart ranks the leading LETFs with the lowest cost S&P and Nasdaq 2x LETFs on the left and higher cost 3X, and Short, esoteric LETFs on difficult, costly to hedge indices (including gold, minerals, semiconductors, and financial LETFs) on the right.

I have previously used the term "Liquidity" within the LETF decay context to mean difficulty/ relative illiquidity / relative costliness of hedging the LETF and daily tracking errors relative to the 2X S&P SOO. (I was not trying to imply that it may be difficult to liquidate your LETF position).

Swap counterparties may withdraw from the market during extreme market March 2020 COVID selloff conditions creating Swap "illiquidity". 3X LETFs have 2 parts leverage covered by 1 part equity and would sail considerably closer to breaching their collateral covenants during March 2020 conditions than 2X LETFs which have one part leverage covered by 1 part equity. Gearing coverage ratios are twice has healthy on 2X LETFs than 3X LETFs. Direxion converted the following LETFs from 3X to 2X in May 2020: (presumably due to difficulties in hedging / hedging "illiquidity" during the March 2020 COVID selloff):

Direxion Gold Miners Bull&Bear 3X converted to 2X May 2020,

Direxion Junior Gold Miners Bull&Bear 3X converted to 2X May 2020 ,

Direxion Energy Bull&Bear 3X converted to 2X May 2020,

Direxion S&P Oil&Gas Prod Bull&Bear 3X converted to 2X May 2020,

Direxion MCSI Brazil Bull 3X converted to 2X May 2020,

Direxion Russia Bull 3X converted to 2X May 2020.

Note that should the current market conditions persist, that while 2X S&P LETF will only cost you ~3.2% per year for 2x leverage, 3X semiconductor LETFs are set to lose ~41.7% relative to the underlying semiconductor SOXX or SOXL should that SOXX remain flat for the next year.

3X LETF data analysis:

- ProShares UltraPro QQQ 3X (NASDAQ:TQQQ)
- Direxion Daily Semiconductor Bull 3x SOXL
- Direxion Daily S&P 500 Bull 3X (SPXL)
- ProShares UltraPro S&P500 (UPRO)
- ProShares UltraPro Short QQQ SQQQ
- Direxion Daily Small Cap Bull 3X (TNA)
- ProShares UltraPro Short S&P500 SPXU

2X LETF data analysis:

- ProShares Ultra QQQ QLD
- ProShares Ultra S&P 500 SSO

### Decay costs appear unrealistically high

**Question from my 4 February 2022 article:** "So let me get this correct if I invest $1,000 in SOXL for one year based on the way I understand it coming from you I'm going to pay $417* for fees correct me if I'm wrong it just doesn't sound right - thank you": (*updated for 25 Feb decay data)

**Answer:** Good Question... Please have a look at the following chart which illustrates the leakage of Semiconductor Bull 3X SOXL (blue chart) relative to the Semiconductor index 1X SOXX (pink chart) during the period March 2020. You will observe that in the pink chart 1X SOXL recovered from the March 2020 crash by Mid-June 2020 (while in June 2020 the 3X SOXL chart was still underwater by 45% relative to SOXX). So in those 3 months 3X SOXL underperformed SOXX by 45%!!! (refer red arrows). An investor who bought SOXL on 1 March 2020 would be 45% down relative to an investor who bought SOXX on 1 March 2020 and broke even on 15 June 2020. In the absence of any leakage or decay, both SOXX and SOXL would have both broken even on 15 June 2020.

### Sideways markets allow us to visualize decay

Sideways and correction markets give us a rare glimpse, opportunity for visual corroboration of the true decay costs of LETFs (which decay costs can be extremely material, difficult to quantify correctly and masked and hidden in the continued bull market of the last 10 years). If (by zooming in and out of price charts) you search for a period in which the index returns to its original levels (zero growth) (as has occurred at the beginning of 2022), and plot and compare the index to its LETFs, then, during this zero growth period, you will have eliminated two very significant distortions to LETF decay data: 1. Growth (which masks LETF costs) and 2. you will have eliminated the compounding effect. (compounding, unless eliminated, significantly distorts decay data). By virtue of the product of all index price movements (i.e. price movement factors) reverting back to their original level, compounding effects, and index growth masking effects are eliminated from your analysis, and you can visually observe whether the LETF too will return to exactly its original levels. (Observe that LETFs practically never do return to their original levels completely. LETFs always leak value).

Fortunately, the extremely high decay costs of March to June 2020 have reduced in multiples but decay costs are notwithstanding very material to investment decisions in current market conditions. Investors fixate of the disclosed 0.95% LETF fees, while the actual leakage or decay costs of LETF can be 30 times higher (largely due to the costs of hedging and leveraging these less liquid indices e.g. the Semiconductor index SOXX). The same visual decay observations can be made for smaller subsequent corrections. Please also note that the 45% decay above occurred over only 3 months and that the annualized decay rate of SOXL relative to SOXX was higher than 45% (refer Figures 1-3 above) but these costs were absorbed and hidden by the bull

### Visual Leakage corroboration

The following 2 charts give us a visual corroboration of the leakage of TQQQ and SQQQ relative to the underlying index QQQ. You will notice monthly decay in February 2022 of roughly 5% for the leveraged ETFs. Note that this leakage occurred in only one month (February) and annual decay *ceteris paribus* (current status quo remaining constant) would be 12 times greater (which will put a significant dent in your investment portfolio relative to lower cost LETFs or the underlying index).

### Daily LETF tracking errors

Let's have a look at daily variances between achieved LETF multiples and how these correlate to volatility:

### Daily LETF tracking error observations

For Volatility (left hand column) Green cells show low VIX volatility. (Red: high VIX volatility).

For LETF achieved multiplier analysis (right hand columns): Green cells show outperformance or overachievement of the 2X or 3X LETF multiples. (Red: underperformance the 2X or 3X LETF multiples indicating higher decay).

All LETFs achieved multiples match their prospectus objectives with remarkable accuracy on a daily basis. The dark red and dark green cells show the highest variances which differ from the benchmark by a factor of 0.0001 for 2X S&P and Nasdaq LETFs ((but considerably higher (but still infinitesimally small)) daily variances for higher decay LETFs.

Generally there is a loose correlation between VIX volatility red periods, and LETF red periods of higher decay. However, you will observe that during high volatility periods, there are days of underperformance but also days of considerable outperformance (green cells). Tracking errors (positive and negative) (although infinitesimally small), when measured on a daily basis are more prevalent during periods of higher volatility.

Then there are periods (Orange circles) when volatility is average and yet variances are very high. In my previous articles I have mathematically analyzed these correlations and i am yet to be convinced that there is an absolute pattern between Volatility (VIX) and Decay (as evidenced by the M&M random scattering of varied colors in the above diagram (although there is admittedly, at certain times, a loose pattern). My views on the volatility vs decay debate go against most academic articles which promote Black Scholes type volatility formulae to generate a one fits all decay solution. (I use a fixed income security bootstrapping model: ("Bootstrapping the Decay Strip") to quantify and isolate 1. index growth and 2. compounding effects, and I submit that that daily analysis produces a better fit model that caters for the Orange anomalies and the M&M random scatter colored pattern of variances.

### Compounding dramatically distorts decay data over time

These infinitesimally small daily variance factors do compound up to very significant monthly and annual decay leakages.

For the quant / propeller-heads out there: The proper derivation of the above achieved multiple is properly calculated (using Excel formula notation) as: LOG(LETFDailyPriceMoveFactor,UnderlyingIndexDailyPriceMoveFactor). The method of simply dividing LETFDailyPriceMoveFactor by UnderlyingIndexDailyPriceMoveFactor yields a small compounding / calculation error which snowballs considerably when extrapolating (annualizing) these calculations. Refer my first LETF article Figure 6 and Annexure B to that Article if you are really struggling to fall asleep at night.

### Simple interest vs Compound interest comparison

The very dramatic effects of compounding (and the extent to which compounding and price movements can distort LETF decay calculations) are well illustrated by the **following question:** "The reality is that TQQQ has 5,000% 10-year return, while QQQ has 457% 10-year return. TQQQ return relative to QQQ return is greater than 10X (not 3X as expected). All these "professional" investors are talking about decay and volatility. The reality is that TQQQ has outperformed QQQ significantly over the past 10 years as a long-term investment. Can an investment professional explain that one to me?"

**ANSWER:** Great Question and Great Observation. The truth is that TQQQ did not actually outperform QQQ by more than 10 times over the last 10 years if performance is measured properly. The apparent 10 times TQQQ outperformance relative to QQQ is severely distorted by not eliminating the compounding effect. Believe me that there are no windfall gains or free lunches of this magnitude in the financial markets. In fact, TQQQ has actually underperformed Proshares' 3X objective over the last 10 years. The actual multiple achieved by Proshares for TQQQ over the last 10 years is in reality 2.3 times QQQ if you run the comparison analysis on a daily basis, or continuous basis over the last 10 years. More recently (and particularly if you exclude the March 2020 - June 2020 COVID crash period), TQQQ has performed very close to its 3X objective. During its initial years when TQQQ volumes were much lower, Proshares struggled to achieve 3X performance and TQQQ also performed well below its 3X objective during the COVID crash.

The following diagram shows a monthly analysis of the actual 3X multiples achieved by TQQQ which illustrates a monthly return much closer to the 3X TQQQ objective. The monthly analysis is still distorted by not eliminating the compounding effects: intra-monthly compounding and growth effects, but the compounding distortion effects are far less prevalent in the monthly analysis than the 10-year analysis.

#### Conversion from 10 year simple rate ("NAC10") to Daily compounded rate ("NACD")

The Proshares Prospectus states on page 364 that the use of the simple formula of Drift = (LETF Return - (IndexReturn x LETF Multiplier)) does not work for periods longer than 1 day. I agree with this statement. Curiously none of the Issuers provide the relatively simple math to solve this compounding problem (which is the key to solving the long term decay analysis problem).

The formula to convert a 10 year simple rate to a daily rate is as follows:

**NACDailyCompoundedRate = (((NAC10yrSimpleRate + 1) ^ ( 1 / (10years*365days) -1 ) * 365days.**

So applying this NACD formula to the above **question:** * "The reality is that TQQQ has 5,000% 10-year return, while QQQ has 457% 10-year return. TQQQ return relative to QQQ return is greater than 10X (not 3X as expected)."* we get the following solution:

(((NAC10yrSimpleRate + 1) ^ ( 1 / (10years*365days) -1 ) * 365days

=(((TQQQ5000%+1)^(1/(10years*365days)-1)*365days

=((51)^(1/3650)-1)*365days [plugging this into Excel or a calculator]

=39.339% TQQQ return per year on a daily compounded basis (stated as "Nominal Annual Compounded Daily" or "NACD" in bond trading lingo.

In the same way (comparing apples with apples) we can solve the underlying index QQQ daily compounded rate ("NACD" rate) using the same formula:

(((NAC10yrSimpleRate + 1) ^ ( 1 / (10years*365days) -1 ) * 365days

=(((QQQ457%+1)^(1/(10years*365days)-1)*365days

=((5.57)^(1/3650)-1)*365days [plugging this into Excel or a calculator]

=17.178% QQQ index return per year on a daily compounded basis (stated as "Nominal Annual Compounded Daily" or "NACD" in bond trading lingo.

Now that we have our daily compounded annualised rates for both TQQQ of 39.399% nacd and QQQ of 17.178% nacd we can easily calculate the achieved multiple rate using Proshares' guidance/method as follows

i.e. 39.339%TQQQreturn / 17.178%QQQreturn = 2.3X multiple achieved for TQQQ over the 10 year period. This 2.3X is below the 3X target TQQQ objective due to decay and costs. (TQQQ's growth was not 10X of QQQ's over the 10 years as it would initially appear).

The overwhelming majority of analysts fail to consider the above (relatively straight-forward) NACD conversion when analyzing drift or decay and you will observe from the above calculation that this omission results in staggeringly incorrect results for Drift analysis periods greater than 1 day. (This calculation error as is stated, acknowledged in the Proshares prospectus). My interest rate conversion calculator in Figure 12 below (downloadable with formulas : follow the link below) allows you to convert from any (e.g. 10 year) simple rate back to any compounded (e.g. daily) rate and visa versa. Take a look at the formulae in the model which correspond to the above calculations. An understanding of these formulae is critical in analyzing LETF decay.

### Compounding errors tend to zero as time scales shorten

If you run a daily or minute-by-minute, or continuous analysis, the compounding effects and distortions are for all intents and purposes eliminated completely.

The following is a model that illustrates the dramatic impact (and performance distortion effects) of compounding. I have updated the model to allow you to convert your 10-year growth factors (without compounding) to daily, and continuous rates (which NACD rates are remarkably different from the NAC10 rates). You will observe that the 10-year "simple" rate (without compounding) differs dramatically from the daily rate or the continuously compounded rate. You will also note that the daily rates and the continuously compounded rates are extremely close and for daily comparison purposes only can be assumed to be identical.

Polynomial math and conversion from a 10-year simple (no compounding) rate to a daily (NACD) or continuously compounded rate and understanding the conversion formulae is the key to unlocking the compounding problem, and proper, accurate performance measurement (as used in bond and fixed income security analysis), and solving the decay / leakage calculations for LETFs.

## Conclusion:

Based on severely diminished LETF prices which diminished prices seem to be persisting, I predict costs to increase further ((by at least ~20% (i.e. by a factor of at least 1.2 (1+20%) above current LETF levels)) and I don't expect costs to come down to November 2021 levels until (if ever) the bull market returns and market recovers to its original highs (where legacy Swap costs will then be absorbed by a much higher LETF asset base). So investors should brace themselves for escalating LETF costs with a gloomy sideway or downward index outlook: As always LETFs are not for the feint at heart, and particularly in these market conditions. Only dabble into LETFs with small capital you can afford to lose.

I am (perhaps controversially) not a believer in trying to time the market and as in March 2020, am remaining long LETF market exposure, but I have migrated from more costly 3X LETFs to lower cost 2X LETFs (grossed up to achieve the same market exposure) and now only hold S&P SSO and Nasdaq QLD LETFs. I intend to ride out the storm (which might be around for quite some time) because I do not want to miss out on the next April-June 2020 type bounce. Perhaps this is wishful thinking, but in the very long term I am happy to ride the following wave:

### Caveat:

As stated in my previous article, the charts in this article build on the (somewhat different) leveraged ETF decay pricing methodology that I have tried to explain in my previous articles (see here and here). Please note that I am not expressing any views whatsoever on the likely price movements or weightings in the underlying indices (including SPY, VOO, VB, and semiconductor indices). I am specifically only looking at the investment decision relating to the holding costs of LETFs relative to other lower-cost LETF (less expensive) alternatives that provide an equivalent market exposure. The objective is only to reduce LETF holding or decay costs in times of a market correction, not to time the market or suggest sector weightings (other than to highlight that certain illiquid, difficult to hedge sectors incur significantly higher LETF decay costs than SPY, VOO and QQQ). I therefore continue to be more bullish on SSO and QLD relative to other (more expensive) LETF investments, but I express no view whatsoever on the short term, coin flip, random walk direction of the underlying indices.

This article was written by

**Analyst’s Disclosure:** I/we have a beneficial long position in the shares of QLD, SSO either through stock ownership, options, or other derivatives. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

**Seeking Alpha's Disclosure:** Past performance is no guarantee of future results. No recommendation or advice is being given as to whether any investment is suitable for a particular investor. Any views or opinions expressed above may not reflect those of Seeking Alpha as a whole. Seeking Alpha is not a licensed securities dealer, broker or US investment adviser or investment bank. Our analysts are third party authors that include both professional investors and individual investors who may not be licensed or certified by any institute or regulatory body.

#### Recommended For You

#### Comments (28)

QQQ has returned 17.178% per year andTQQQ has returned 39.339% per year.Agreed that TQQQ rate of return is not 3X of QQQ rate of return, but it's still significant !Historically, semiconductor industry has been cyclical and I don't think it's going to be different this time either. Whereas QQQ & TQQQ track the top 100 companies listed on NASDAQ. Hence, in my opinion QQQ / TQQQ are likely to trace a better upward trajectory.Best wishes,- A

Keep in mind, that the daily rebalancing is what leads to the decay... and the compounding. If QQQ averaged 17% for the next 5 years, TQQQ could do 30%, 40%, 16%, -10%. We have no idea, because the amount of the decay is a function of 2 things, frequency of directional changes in the index, and the severity of those directional changes. Read Warwicks responses to a few other posts where he explains the difficulty in projecting long-term results.I personally would not be using leveraged ETF's as the increased volatility that I think we have thru the FED normalization process will yield volatility that will increase the decay too much. But who knows... Ive been wrong before.

SP500 is about 107%With that data, let us assume that 5 years ago, I invested:1k in UPRO and 1k in SP500Question: Is the following math correct?UPRO: $1,000 * 290% = $2,900, so do: $1,000 (principle) + $2,900 (return) = $3,900 is total $ amount you have 5 years later.SP500: $1,000 * 107% = $1,070, so do: 1,000 (principle) + $1,070 (return) = $2,070 is total $ amount you have 5 years later.

UPRO: $1,000 grows to $3,900 5 years later.

SP500: $1,000 grows to $2,070 5 years later.These factors however need further manipulation: UPRO 5 year growth rate of 290% isn't directly comparable to the SP 5yr growth rate of 107% without some compounding adjustments. This is the same as saying that a $1000 5 year 20% roll-up loan will roll up to $2488 (whereas a $1000 5 year 10% roll-up loan will only grow to $1610). The $1488 interest roll-up on the 20% loan isn't directly comparable to twice the $610 interest roll-up on the 10% loan without some mathematical adjustment i.e. reducing both to a v short term compounding rate. This compounding conversion is one of the critical steps in calculating LETF decay or leakage or beta slippage for periods longer than 1 day without which results for comparison periods longer than one day will always be materially misstated. Comparison between various LETFs is also not possible without adjusting for compounding. The Proshares Prospectus states on page 364 that the use of the simple formula of Drift = (LETF Return Factor [3.90 per yr example] - (IndexReturn Factor [2.07 per yr example] x LETF Multiplier {2X])) does not work for periods longer than 1 day. I agree with this Proshares statement. drive.google.com/...To solve the relationship between SP and UPRO: You need to calculate the daily (or continuously) compounded rate for both either by using polynomial math or by using my model here: docs.google.com/...The tricky math makes the analysis of LETFs and their costs difficult and can mask the cost of investing into LETFs for unknowing/unaware investors.

decay is a generic term in the investment industry... essentially its the slow loss of value to factors other than simple price movement.

In the case of leveraged ETF's, (LETF), the decay comes from what is more commonly referred to as "tracking error", or "beta slippage." Google "leveraged ETF tracking error", or "leveraged ETFbeta slippage" and you will get the explanation.In a nutshell, it occurs because LETF's apply their leverage DAILY, rather than leverage across a period of time. Simplke example, if the NASDAQ went up 1% today, and down 1% tomorrow a triple leveraged fund would go up 3% and then down 3%. In the index If we started with $100, on day 1 we'd have $100 x 1.01 = $101 as expected. But when we lose 1% the next day, we are losing 1% of $101, not 1% of $100. So now we have $101x .99= $99.99. Now look at the 3x leverage fund. If we started with $100, on day 1 we'd have $100 x 1.03 = $103 as expected. But when we lose 3% the next day, we are losing 3% of $103, not 3% of $100. so now .97x $103= $99.91. 8 cents less than the underlying index. Thats what causes the decay. The higher the leverage, the higher the market volatility, and the more frequent the directional changes, the higher the decay will be.

"A fund holds $200 of stock and $100 of margin loan. During Day 1, the value of the underlying stock falls by 5%, so at the end of the day the fund has $190 of stock and $100 of debt, netting to $90 of equity. The fund must rebalance its assets to maintain its leverage ratio. So, it pays down $10 of its excess debt by selling $10 of stock, and now its balance sheet includes $180 of stock, and $90 of debt, netting to $90 of equity. During Day 2, the stock rises by 5.263%, which nearly exactly cancels out its previous day loss of 5%. The fund's $180 of stock is now worth $189.47. To maintain the leverage ratio, the amount of debt in the fund is raised from $90 to $94.74. The fund has $189.47 in stock and $94.74 in debt, netting to $94.73 in equity. What was the net result? An investor who owned $100 of stock would still have $100 of stock at the end of Day 2, but an investor who owned $100 of equity in the 2x levered fund would only own $94.73 in equity. Sounds like a terrible deal - over 2 days, the leveraged investor lost 5.27% of their equity!"My response to the above apparent disparity between TQQQ and QQQ is as follows:

@Stephen: In further support of your findings the corollary of the 5% example must also hold true: Assume that there is an equal probability that the market will rise or fall: Assuming that the market rises, then the corollary 5% "Volatility Drag" example applies as follows: (the maths below does not give an accurate long-term projection for leveraged ETFs btw. - more on that below):A fund buys $200 of stock at 9.30am on Day 3 and holds $100 of margin loan. During Day 3, the value of the underlying stock rises by 5.263%, so at the end of the day the fund has $210.52 of stock and $100 of debt, netting to $110.52 of equity. The fund must rebalance its assets to maintain its leverage ratio. So, it raises $10.52 of extra debt, and buys further equity and now its balance sheet includes $221.05 of stock, and $110.53 of debt, netting to $110.52 of equity. During Day 4, the stock falls by 5%, which exactly cancels out its previous day’s gain of 5.263%. The fund's $221.05 of stock is now worth $210.00. To maintain the leverage ratio, the amount of debt in the fund is reduced from $110.52 to $105.00. So, it pays down $5.52 of its excess debt by selling $5.52 of stock, and now its balance sheet includes $210.00 of stock, and $105.00 of debt, netting to $105.00 of equity. What was the net result? An investor who owned $100 of stock would still have $100 of stock at the end of Day 2, but an investor who owned $100 of equity in the 2x levered fund would now own $105.00 in equity. Sounds like an amazing deal - over 2 days, the leveraged investor is up 5% of their equity! [if you correct the flawed maths logic this $5 “windfall gain” on days 3 and 4 matches and offsets the $5.27 “unlucky” losses on Days 1 and 2. So the net effect of the "Volatility Drag" per the example above (ie arising out of some mischievous mathematical anomaly) should offset and cancel each other over time.Leveraged ETFs make use of option strategies and Total Return Equity Swaps to fine tune their hedging and prevent calamitous losses, and in so doing pay away "Volitility Spreads" but Volatility spreads (on options) are not what is being described in the (mathematically flawed) example above.We know that the effects of compounding increase exponentially with time and also increase exponentially as returns increase. Using a “Polynomial Method” to include compounding concepts in our modelling work very well when applied to pricing long term leveraged ETF movements:Returning to the two WindfallGain vs Unlucky Funds example above, the correct maths (I believe) should be described as follows: A 2X leveraged ETF investor holds $100 of stock (roughly comprising $200 of synthetic and actual equity positions) and roughly $100 of synthetic and actual debt (including Swaps). During Day 1, the value of the underlying stock falls by 5%. The value of the 2X leveraged ETF is calculated using the compounding method as follows: $100* ((1-5%)^2) = $90.25. [In Excel type: “=100*((1-5%)^2)”]During Day 2, the stock rises by the reciprocal of 5%, which exactly cancels out its previous day loss of 5%. The value of the 2X leveraged ETF is calculated using the compounding method as follows: $90.25* ((1-5%)^-2) = $100.00. [In Excel type: “=90.25*((1-5%)^2)”]And a 3X ETF would be calculated as follows: Day 1 (5% index drop) $100* ((1-5%)^3) = $85.73. Day 2 (reciprocal $5 index increase): $85.73* ((1-5%)^-3) = $100. [In Excel type: “=100*((1-5%)^3)” and “=85.73*((1-5%)^-3)” into cells A1 and A2.Compounding can be explained in polynomial expressions as follows:

"(1+0.01) is 1.01 or 1 plus 1%. If you make 1% per period -- let's say per week -- then you multiply your account times 1.01 each period. If you start out with 1000, then after a week it's 1000 * 1.01= 1010. After 2 weeks you multiply by 1.01 again, so after 2 weeks it's 1000 * 1.01 * 1.01 = 1020.1. After three weeks it's 1000 * 1.01 * 1.01 * 1.01. Multiplying the $1000 account by 1.01 3 times is represented as $1000 * 1.01^3. After N periods, you raise the periodic growth to the Nth power. So if you made 1% per week, after a year your $1000 account would grow to $1000 * 1.01^52. Which is actually 67.7% growth, not 52%, because the earnings compound.seekingalpha.com/...

good luck with your trading to all.

https://scrible.com/s/sC5M0

.