QuantConnect ETF Momentum Asset Allocation

Momentum Based ETF Portfolio Rebalancing
Optimizing Portfolio Construction For Optimal Sharpe Ratio
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April-2018 QuantConnect – Momentum Based ETF Portfolio Rebalancing Page 2

Outline
 Basics of Mean Variance Portfolio Construction
 Defining Optimization Function
 LEAN Algorithm Framework
 Implementing Our Model
 Testing and Researching
 Summary

Reduce volatility, increase returns by
calculating optimal weight allocation of
a portfolio for minimum volatility.
Core Idea:
1) Create an estimate of portfolio returns and volatility.
2) Build portfolio of assets; allocating to each by weight.
3) Optimize weights to minimize the volatility in portfolio.
Classic Mean Variance Portfolio Construction

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Jan-18 Feb-18 Mar-18 Apr-18 May-18 Jun-18 Jul-18 Aug-18 Sep-18 Oct-18
Stock A Stock B
Ideal = 50-50

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$2.00
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Jan-18 Feb-18 Mar-18 Apr-18 May-18 Jun-18 Jul-18 Aug-18 Sep-18 Oct-18
Stock A Stock B Stock C
Stock D Stock E Stock F
Stock G Stock H
Most real world applications have portfolios of many assets. We are seeking
to find best balance of hundreds of assets.

Mean Variance Optimization Function
Optimizers experiment with portfolios; seeking to
minimize the objective function.
Classic Mean Variance Optimization the objective
function seeks to minimize expected volatility.

Redefining Our Objective
We will seek to optimize Sharpe Ratio instead of volatility; seeking to
maximize objective function:
𝑥 = m𝑎𝑥(
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑒𝑡𝑢𝑟𝑛
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
)
This target approximates the traditional Sharpe Ratio function.
(We’ll assume the risk free rate of return is zero).

𝑅 𝑝 = (𝑤𝐴, 𝑤 𝐵, 𝑤𝐶)
𝐸(𝑟𝐴)
𝐸(𝑟𝐵)
𝐸(𝑟𝐶)
= 𝑤𝐴 𝐸 𝑟𝐴 + 𝑤 𝐵 𝐸 𝑟𝐵 + 𝑤𝐶 𝐸 𝑟𝐶
With three risk assets in portfolio, the expected return of the portfolio:
We use the variance of each asset, combined with weights to find the variance of the
portfolio:
𝜎 𝑝
2
= 𝑤𝐴, 𝑤 𝐵, 𝑤𝐶
𝑉𝑎𝑟 𝑟𝐴 𝑐𝑜𝑣 𝑟𝐴, 𝑟𝐵 𝑐𝑜𝑣 𝑟𝐴, 𝑟𝐶
𝑐𝑜𝑣 𝑟𝐵, 𝑟𝐴 𝑉𝑎𝑟 𝑟𝐵 𝑐𝑜𝑣 𝑟𝐵, 𝑟𝐶
𝑐𝑜𝑣 𝑟𝐶, 𝑟𝐴 𝑐𝑜𝑣 𝑟𝐶, 𝑟𝐵 𝑉𝑎𝑟 𝑟𝐶
𝑤𝐴
𝑤 𝐵
𝑤𝐶
= 𝑤𝐴
2 𝑉𝑎𝑟 𝑟𝐴
2 + 𝑤 𝐵
2 𝑉𝑎𝑟 𝑟𝐵
2 + 𝑤𝐶
2 𝑉𝑎𝑟 𝑟𝐶
2
+ 2𝑤𝐴 𝑤 𝐵 𝑐𝑜𝑣 𝑟𝐴, 𝑟𝐵 + 2𝑤𝐴 𝑤𝐶 𝑐𝑜𝑣 𝑟𝐴, 𝑟𝐶 + 2𝑤 𝐵 𝑤𝐶 𝑐𝑜𝑣 𝑟𝐵, 𝑟𝐶
Portfolio Variables

Algorithm Framework Modules

Assumptions and Limitations
 Mean Variance Optimization requires expected returns and we provide historical
values. We’re making an assumption these returns will continue in the future.
 Most variance approximations assume a normal distribution.
 Any estimation error in the return prediction magnified.
 The resulting portfolios can be concentrated and nonsensical. In practice its more
common to use Black-Litterman method.

Code and Implementation
Backtest

Summary
 Using portfolio construction techniques we can automatically assign weights
to our portfolio assets.
 This reduces the number of variables we manually define; can improve
returns and lower volatility.
Next Steps – Investigate more robust portfolio construction techniques!
(E.g. Black-Litterman).
Total Trades Drawdown Net Profit Sharpe Ratio
270 12.9% 34% 0.555

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