Introduction

Leverage ETFs are often used as a “booster” vehicle for high-return investment. While chasing high returns can be tempting, it often comes with amplified risk. This micro-study explores a combination of risk parity and minimum portfolio variance using leveraged ETFs across different asset classes to achieve diversification, earn high returns, and mitigate risk. We'll build a portfolio with US Equities, Short Emerging Market Equities, Long-term Bonds, and Gold, aiming for equal risk contribution from each asset.

Background

Risk parity is a portfolio construction method that allocates capital based on the risk of each asset rather than its expected return. This approach aims to minimize portfolio volatility by distributing risk evenly across different asset classes. Leveraged ETFs, with their amplified returns, can enhance the potential gains of a risk parity portfolio while maintaining a balanced risk profile.

We select variance as our risk metric in this study for stability and ease of implementation. Spinu (2013) provided a convex optimization function to implement of risk parity on variance. We modified it a little bit, as shown below:

$$\begin{array}{rcll}\textrm{min}_w & F(w) = \frac{1}{2} w^{T} \Sigma w - b^{T} w\\\textrm{where} & b^T = [\frac{1}{N}, … , \frac{1}{N}]^T\\& \Sigma \in R^{+,N \times N} \textrm{ is the covariance matrix}\\ & N \in N \textrm{ is the # of security}\end{array}$$

The jacobian of the objective function are as below:

$$\begin{array}{rcll}F'(w) = \Sigma w - b^{T}\end{array}$$

You can see that the jacobian is trying to make the risk (variance and covariance) equal to the equal risk budget. We ensure the portfolio is long-only, with the overall weight being 100% of the portfolio value, to ensure the convexity of the objective function.

We choose four leveraged ETFs representing different asset classes:

  • TQQQ: 3x leveraged Nasdaq 100 ETF (US Equities)
  • SVXY: 1x leveraged Short VIX Short-Term Futures ETF
  • VXX: 1x leveraged VIX Short-Term Futures ETF
  • TMF: 3x leveraged 20+ Year Treasury Bond ETF (Long-term Bonds)
  • EDZ: 3x leveraged Short Emerging Market ETF (Short Emerging Market Equities)
  • UGL: 2x leveraged Gold ETF (Gold)

These ETFs are mostly either orthogonal or negatively correlated to one another, offering convexity for risk reduction. The volatility drag of leveraged ETFs will erode your profit, but since we include inversely correlated vehicles, the volatility drag effect will be much lower and we can have the leveraged arbitrage opportunities between them while hedging the excessive, leveraged speculation risk on a single asset. Also, we target large-cap leveraged ETFs in various markets, such that the enormous capital flow would provide better momentum even during market fluctuations when capital reallocates from stock to hedge asset markets like bond and gold, amplifying the compound effect.

 

Implementation

To implement this strategy on QuantConnect, follow these steps:

1. We subscribe to the four ETFs daily data, since we aim to calculate their daily variance and covariance.

self.symbols = [self.add_equity(ticker).symbol for ticker in ["TQQQ", "SVXY", "VXZ", "TMF", "EDZ", "UGL", "UUP"]]

2. Schedule a weekly rebalance function to rebalance the portfolio weekly, such that we can keep the risk exposure stable but still allowing the portfolio to grow compound by not trading too often.

self.schedule.on(self.date_rules.week_start(), self.time_rules.at(8, 0), self.rebalance)

3. In the rebalance function, get historical data and calculate the daily return of each ETF.

ret = self.history(self.symbols, 253, Resolution.DAILY).close.unstack(0).pct_change().dropna()

4. Call the scipy.optimize.minimize function with the objective functions and its derivatives using the SLSQP optimization function.

x0 = [1/ret.shape[1]] * ret.shape[1]
constraints = {"type": "eq", "fun": lambda w: np.sum(w) - 1}
bounds = [(0, 1)] * ret.shape[1]
opt = minimize(lambda w: 0.5 * (w.T @ ret.cov() @ w) - x0 @ w, x0=x0, constraints=constraints, bounds=bounds, tol=1e-8, method="SLSQP")

5. To trade with the calculated weights, call set_holdings with a list of PortfolioTarget objects. 

self.set_holdings([PortfolioTarget(symbol, weight) for symbol, weight in zip(ret.columns, opt.x)])

 

Conclusion

This risk parity strategy using leveraged ETFs offers a potential solution for investors seeking excess risk adjusted return than normal investments, while provide a certain level of portfolio risk management through diversification and risk distribution, making the portfolio all-weather suitable.

Key Advantages:

  • Diversification: The strategy spreads risk across different asset classes, reducing exposure to any single market.
  • Risk Mitigation: By targeting equal risk contribution from each asset, the portfolio aims to minimize overall volatility while equally dissipating the capital fluctuation. The strategy also aims to mitigate individual market risk by allocating funds in various asset class.
  • Leverage: The use of leveraged ETFs enhances potential returns while maintaining a balanced risk profile.

Limitations:

  • Leverage: High leverage amplifies both gains and losses, increasing potential risk. Also, it will incur extra expense costs.
  • Risk Measure: Using a non-coherent risk measure of variance might not be the best choice to calculate a realistic overall risk. It does not obey common risk principles and we only want to avoid downside risk. Instead, using coherent downside risk measures like conditional value-at-risk might be better for considering only the downside risk and the overall portfolio level risk. Also, the covariance matrix is purely based on the historical data, so it may not perfectly demonstrate the forward correlation and fluctuation.
  • Volatility Drag: Some indeterministic direction leveraged ETFs without an inverse counterpart pose extra volatility in value, so you need to pick a negatively-correlated counterpart in most circumstances to minimize the erosion from volatility drag and drops in valuation.

 

References

Spinu, F. (2013). An algorithm for computing risk parity weights. Available at SSRN 2297383.