Exposure of LETFs and why we don't need cash in the portfolio

Since when did I last wrote financial/stock market stuffs? Probably not since 2008. That doesn't mean I stopped interacting with the market though.

One major change is that ETFs (exchange traded funds) have been much more popular since 10 or 20 years ago instead of ETNs or traditional mutual funds. Simple index funds like SPY and QQQ started around 2000 and we now see similar products all over the world: 2800 (tracker fund) in Hong Kong is nothing new, 0050 (TW top 50 tracker) in Taiwan proved more efficient than most investors, and we even find these products in Tokyo, a traditionally conservative place (they just set up more regulations this year on ETFs!).

But what I wanted to cover today isn't really about index funds, but those leveraged ones (LETFs). No words are needed to describe how they separate themselves from other leveraging methods: they automatically leverages/deleverages themselves on a daily basis. Since their introduction around 2010s, leaders like TQQQ (3x Nasdaq100) has accumulated much popularity, gaining even more attention on every major drawdown.

The ups and downs of LETFs are apparent: you are under leveraged gains (the gains themselves are compounded which magnifies with time), but are also under leveraged risk. Due to volatility decay and the asymmetry of the rise and fall, the return (per unit time) multiplier is always below the leverage ratio and the risk (sd p.a.) is always above the leverage ratio. That gives a lower Sharpe ratio. Of course this is to be expected -- it is impossible for us to leverage a certain portfolio at risk free rate.

Facing these elevated risks are two kinds of people: the yolo apes who are happy to take the risk and went all in TQQQ, and the boogleheads who try to take advantage of the LETFs but also maintaining efficiency in overall. For those who are going all in there isn't much to say -- you probably don't have many alternatives with similar return, and TQQQ is probably one of the best choices. But for those boogleheads the strategy can be extremely diversified.

In light of the classic 60% equity - 40% bond combination, people seem to be applying the same ratio for their leveraged portfolio, and it seemed to work: if you consider TQQQ and TMF (3x 20yr bond) only, max Sharpe occurred at a 55/45 ratio but 60/40 gives almost the same ratio. If you backtest the thing since the beginning of the Nasdaq index, a 60/40 portfolio would beat any other portfolio, including the all in strategy (this is mainly due to 2000 and 2008, admittedly. But who can assure that it won't happen again?). It is also noteworthy that similar conclusion applies if we replace TQQQ by UPRO, the 3x S&P LETF. 

The question is, why? Why does the ratio carries through upon leveraging?

To answer this allow me to introduce exposure, another keyword from the title.

Exposure is a fancy word of leverage ratio, but also applies to a specific component of the portfolio rather than the whole thing. To be precise, a certain portfolio's exposure on a product X is the movement per X's unit price change while leaving all others constant.

For example, QQQ has about 10% of MSFT. That means if MSFT rises by 1% then we would expect a 0.1% corresponding change in QQQ. In a similar way, since TQQQ is leveraged by 3 times, it has a 30% MSFT exposure. A 1% change in MSFT would lead to a 0.3% change in TQQQ.

Amazingly, exposure adds up even among independent tickers. Continuing from the above example let us also introduce SPY which holds approximately 5% of MSFT. If your portfolio consists of 50% TQQQ and 50% SPY, then you will have a 10%*3*0.5+5%*0.5 = 17.5% exposure of MSFT.

And now here is the main principle: portfolio with equal exposure (in all assets) would act identically upon tracking errors due to expense ratio, compounding effect and volatility decays.

The error are all long term tracking errors so the statement holds if it holds in a daily basis which is apparent: if two portfolios have equal exposure in every asset, each of the asset would contribute equal influence to the two portfolios.

Still, the difference between two portfolio with identical exposure due to the above errors are minimal. Given a portfolio suppose we up-leverage part of it and de-leverage another part of it to achieve the same exposure. The elevated risk and decayed return of the up-leveraged part is offset by the opposed effect from the de-leveraged part. 

Let us consider the following two portfolios:

1) 50% TQQQ and 50% QQQ

2) 100% QLD

Both portfolio have 200% QQQ exposure so we expect them to act almost the same, and we now calculate the effect of decay on the two profiles.

Suppose QQQ rises by $x$ one day and drops by $x$ on the next day. Then QQQ drops by $x^2$ in overall. On the other hand, the first portfolio takes a drop of $5x^2$ while the second one takes a drop of $4x^2$. We can see that their decays are really close to each other only differing by $x^2$. Sure this is still the same order as the ordinary decay, but the coefficient is greatly compressed. You can argue the same for expense fees as well as the compounded return.

(Note that the above example shouldn't be interpreted as "the decay of a 2x ETF is 4 or 5 times the underlying ETF" because you are not comparing when underlying ETF is at the original price, so it is natural for the LETF to drop more to reflect such shift. If we alter our method, either by boosting the up-day or to give it another day to rise back to the original price level we will get a much smaller decay. Take the above example again: the first portfolio will suffer a decay of $3x^2+O(x^3)$ and the second is of $2x^2+O(x^3)$ when QQQ is at 100%. This is the truly decayed part.)

Due to convexity and Jensen's inequality, the above always holds when comparing portfolios with identical exposures but distinct leverage irregularities. On the other hand using the same argument, we can see that the compounding effect favors portfolio with higher leverage irregularities. 

In practice however, it is more often to see that portfolios with less leverage irregularities perform better. A possible explanation is the asymmetry of the rise and fall again. A decay lost of $-x^2$ and a compounded gain of $+x^2$ gives a net gain of $-x^4$. Small but accumulate over time.

As a result, even though portfolio of identical exposure behaves almost the same, it's still desirable to reform the portfolio to achieve a homogeneous leverage ratio.

If we treat cash as a 0x etf, then we shouldn't hold cash at all -- instead, spend these cash to deleverage the rest of your portfolio for that extra bit of return.

For example consider a 40% TQQQ 40% SPY 20% cash portfolio. If we are to deleverage it using the 20% cash it could become 60% QLD 40% SPY. Backtest shows that the altered portfolio gives 0.5% more return per year with a drop of 0.6% in risk. Note that TQQQ and QLD has the same expense ratio so the altered portfolio is winning even with higher fees!

Three possible portfolio with identical exposures but different cash levels (0, 20%, 30%). 

Taken from portfolio visualizer.

Now if we operate the other way round: we up-leverage the portfolio to free more cash, what can we do? We are certainly not leaving these cash alone because we know that would leave the portfolio inefficient. Instead we can buy something else using these cash.

Something important that solves our question at the beginning: why does the 60/40 ratio carries through?

(Cont.)