After a detailed analysis of the oil stock LGCY, you determine that it is overvalued, and you plan on shorting $10,000 of it. You would like to isolate your bet on the alpha of the stock, so you want to hedge out all of your exposure to the market and to the oil industry. LGCY has a market beta of 1.1, and an oil industry beta of 1.5.
XLE (an oil stock ETF) has a market beta of 0.8 and an oil industry beta of 1.6.
SPY has a market beta of 1 and an oil industry beta of 0.1.
How could you use XLE and SPY to reduce your exposure to both risk factors to 0? If your portfolio was made up of just these three assets, what would your final position sizes be?