Leveraged Funds, Cash, and QQQ/SPY Portfolios: Risk and Return monte_carlo_sim_cagr

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Analysis of Monte Carlo Simulation Results for Asset Classes and Portfolios

Preface

Past history is only one path within a random process. Monte Carlo simulation is a useful tool for studying the possible outcomes of many paths for a random event.

The simulation sample begins at the start of 2008 because it is limited by the launch date of the 2x leveraged fund QLD. It ends at the end of 2022, rather than the end of 2023, to avoid overly optimistic results.

The focus below is compound annual growth rate (CAGR) and maximum drawdown (MDD), without inflation adjustment. The long-term risk of investing is a CAGR that is too low. If performance is poor or even trails long-term inflation, real purchasing power shrinks and social class can slide toward poverty. Short-term risk comes from excessive MDD. A large drawdown can damage confidence and lead investors to sell, creating permanent loss. If cash flow depends on invested assets, a decline creates psychological pressure; if securities-backed borrowing is used, a large decline may also cause forced liquidation.

This article organizes Portfolio Visualizer simulation results and presents them as box charts for easier comparison. The bottom and top of each box are the first quartile (Q1) and third quartile (Q3), and the middle line is the median. The lower and upper whiskers mark the 10th and 90th percentiles. Intuitively, the five points from bottom to top represent the outcomes of the unluckiest investor, a less lucky investor, an ordinary investor, a lucky investor, and the luckiest investor.

Box charts are useful for comparing assets and holding periods, but they do not show the full probability distribution of simulation results. Still, they help reveal features such as interquartile range (IQR), which reflects variability. Whether the median is centered in the box and whether the whiskers are balanced also indicates whether the distribution is skewed.

Finally, Monte Carlo simulation cannot know every future outcome. It only shows many possible future paths, so the exact value at each box-chart point is not the key issue. The key is comparing asset classes or combinations and different holding periods. Because simulation output is strongly affected by the original sample distribution, we first compare SPY historical data with simulation results to assess reliability.

SPY vs TLT vs CASH. First, look at JPMorgan's 1950-2020 historical data. The light-green stock-market return distributions for rolling 5-year, 10-year, and 20-year periods are basically covered by the SPY CAGR box chart and whiskers in Figure 1(a). Therefore, using data from the start of 2008 to the end of 2022 as simulation input is broadly useful and has reference value.

For CASH, the most distinctive feature is that MDD is always zero regardless of holding period. Compared with the uncertainty of potential stock-market declines, this explains why many people prefer holding cash instead of investing. But risk-averse investors holding large cash positions often ignore the unbearable long-term risk of low CAGR. Avoiding short-term risk also means giving up returns and indirectly pushing oneself toward poverty.

For SPY, once the holding period exceeds 15 years, even the worst 10% of simulated CAGR outcomes remain positive. The variability of outcomes narrows as the holding period lengthens: the best outcomes gradually come down, and the worst outcomes gradually move up.

This shows that investing is fair over long periods. The gap between the luckiest and unluckiest investors can become small enough to satisfy most people. SPY's MDD can become larger as the holding period increases, but investors are rewarded for bearing that decline. Over a 25-year SPY holding period, a bad market may produce MDD over -60%, yet CAGR can still be around 4%. A decline alone is never a reason to sell stocks.

The surprising result is TLT. Even over 25 years, the long-term Treasury fund can still produce negative returns. Its CAGR median is only around 4%, and its MDD distribution is wider and lower than SPY's, meaning larger possible losses. TLT exposes holders to both long-term and short-term risks without corresponding returns, so it is not an ideal investment.

SPY vs QQQ vs QLD. Here we compare SPY, QQQ, and the daily 2x leveraged ETF QLD. If someone believes in long-term market growth, should they go all in on leveraged ETFs?

In Figure 2, QQQ's CAGR boxes sit above SPY's, showing higher returns, but QQQ's MDD boxes are slightly lower, showing deeper drawdowns. Both are diversified and avoid idiosyncratic risk, but SPY's additional 400 stocks provide somewhat better downside protection, while removing those 400 stocks improves CAGR. QQQ's boxes are only slightly longer than SPY's, meaning luck affects QQQ only slightly more than SPY. Even the unluckiest long-term QQQ investor does better on average than an SPY investor.

QLD's five-year CAGR upper box edge reaches an extreme 58%, attracting many investors, but its five-year lower whisker is also an extreme -16%, which repels others. QLD's CAGR box width is roughly twice QQQ's. Leverage raises median return but also raises variability. Lucky investors who buy near lows can earn very high average annual returns, while unlucky investors who buy at bubble peaks may still fail to earn positive returns after 20 years. QLD's MDD can reach about -98% over a 25-year period. If that happens early, recovery may be possible; if late, it can be extremely painful.

If single-year returns are independent random variables, then by the central limit theorem, long-term average geometric annual returns are log-normally distributed. Daily 2x leverage magnifies volatility and the standard deviation of average annual returns, increasing skewness. High skew means a few paths have very high returns, while most paths are far below them, similar to lottery payoff distributions.

QQQ vs Case 1 vs 50/50. The real function of leveraged ETFs is to complement cash: cash reduces QLD's severe MDD, while QLD offsets cash drag and weak CAGR. Suppose an investor must keep cash for emergency reserves or collateral borrowing. If the same proportion is invested in a 2x leveraged ETF and the leveraged ETF and cash are rebalanced back to 1:1 every year, the cash drag problem can be reduced.

The red box chart in Figure 3 is QQQ:QLD:CASH = 60:20:20. Its expected beta is equivalent to 100% QQQ. CAGR box width is almost the same as 100% QQQ and slightly higher overall, suggesting slightly better average annual return. But MDD inherits some QLD characteristics and is slightly deeper. After daily market moves, QQQ:QLD:CASH weights drift. When QLD rises, its weight increases, increasing future drawdown risk until rebalancing removes the problem.

If cash is increased further, the limit under this rule is QLD:CASH = 50:50. CAGR remains almost as wide as 100% QQQ and slightly higher, while MDD becomes less variable and somewhat smaller than 100% QQQ. This appears to offer lower short-term risk and better long-term return. But one must ask why such a high cash ratio is needed. High cash can create behavioral errors: chasing after rises, trying to bottom-fish after declines, failing to rebalance after deeper declines, or incorrectly taking profits or stop-losses because of leveraged ETF volatility. Even with discipline, high cash and high leveraged ETF positions require rebalancing, which creates transaction costs and possibly capital gains tax. For pledged borrowing, lenders may not accept leveraged ETFs as collateral, while cash and unlevered index funds usually create less concern.

QQQ vs beta 1.2 vs 60/40. The Kelly formula tells gamblers how much to bet to maximize long-term average return when expected return, risk-free rate, and standard deviation are known. Past research applying this principle to the S&P 500 suggested an allocation of 117%, meaning all-in was not enough; borrowing an extra 17% of principal would produce the expected optimal average annual return. Borrowing, however, has interest costs and requires collateral or good credit. Can the same effect be achieved another way?

The red CAGR boxes in Figure 4 use QQQ:QLD:CASH = 60:30:10, with expected volatility at 120% of QQQ. The median CAGR is about 2% higher than QQQ's. For holding periods above ten years, the lower whisker is almost aligned with QQQ's, while the upper whisker and upper box edge are higher. This suggests that using cash and 2x leveraged funds to raise volatility can improve upside without lowering expected minimum average annual return. There is no free lunch, however: MDD boxes are lower than QQQ's, so the price of higher return is larger drawdown.

The prior section's effect of reducing MDD by raising cash also appears here. At 120% expected volatility, maximizing cash leads to QLD:CASH = 60:40. The yellow CAGR box is roughly similar to the red box, while the yellow MDD box sits higher, meaning higher cash lowers maximum drawdown without reducing return.

A preliminary conclusion: leveraged funds are powerful tools. They can be combined with unlevered index funds and cash to adjust leverage without the complexity of borrowing or derivatives. If unlevered index funds are steak, cash and leveraged funds are seasoning. Properly used, they improve the meal, but no one orders only the seasoning.

QQQ vs 60/40 vs 60/40(TLT). TLT is not always negatively correlated with stocks, but because it has low long-term correlation with stocks, it is popular in traditional stock-bond allocations. What happens if TLT replaces cash?

In the CAGR box chart, the yellow box again shows a familiar effect: its lower whisker aligns with or exceeds QQQ's lower whisker, while the upper whisker and upper box edge are higher. Replacing cash with TLT slightly improves expected minimum average annual return and raises expected maximum average annual return. But again there is no free lunch. The cost is higher MDD: yellow MDD boxes are lower than blue and red. Whether comparing QQQ to SPY, 120% volatility portfolios to QQQ, or TLT replacing cash, higher CAGR comes with larger MDD. Higher return means higher volatility; this is a beta effect, not alpha.

Returning to the steak analogy, replacing salt, or cash, with seasoned salt, TLT, may make the flavor better, but not everyone can accept it.

Conclusion: this article first shows the risks of cash, TLT, and QLD used alone, such as low growth, low returns, or high variability that makes outcomes depend heavily on luck. But when these asset types are combined in appropriate ratios, interesting characteristics appear. Still, investment cannot be driven by the rearview mirror. The variables needed by the Kelly formula are unknown, so the optimal leverage level cannot truly be known. Monte Carlo simulation is objective, but it uses data from a limited period. Black swan events are often not visible in the historical sample.

Investing has many possibilities, and adding leveraged funds creates even more. This article offers a perspective and background knowledge. In investing, the greatest danger is not decline itself, but ignorance: taking large risks without knowing it, or not understanding the cost of refusing to take risk.

Once an asset allocation exists, rebalancing is needed. Rebalancing does not have to be only periodic and mechanical; tactical methods are possible. But those more complex operations are harder to simulate and are beyond this article's scope.

Peta Hung, 2024.10.19