Question

You think a certain stock in the gold-mining industry (stock A) is overvalued, so 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 your exposure to the market and to the gold-mining industry. Stock A has a market beta of 1.1, and a gold-mining industry beta of 1.5. Asset B (a gold-miners ETF) has a market beta of 0.8 and a gold-mining industry beta of 1.5 Asset C (SPY) has a market beta of 1 and a gold-mining industry beta of 0.2. If you used assets B and C to get a portfolio that had a market beta and gold-mining industry beta of 0, How many dollars would you put in Asset B? How many dollars would you put in Asset C?

Answer 1

Answer: Dollars to put in Asset B= $9,552.24 , Dollars to put in Asset C= $3,358.21

To solve this question the basic idea is that the betas of the stock that we are shorting will become negative (since we are selling it rather than buying it as is usually done in long only portfolio). Then you can use the formula for Portfolio beta as given below

Where w is the weight & b is the beta of a stock in the portfolio.

Now the problem is reduced to finding the weights Asset b & c such that the above formula yields the value 0.

This can either be done manually by making 2 equations (one for market beta & one for Gold industry beta) in 2 variables (B & C). Then you can solve the system of equations using any method or calculator.

Other way to solve is to use the Excel's solver function add in as shown below. To use Excel solver, you enter any random values for amount of B & C and then use the solver to set value of cell F9 to zero with constraint of G9 also being zero. Cell F9 & G9 should have the above formula of portfolio beta before invoking solver :

Formula view:

Please not that using Excel solver is not necessary to solve this question & you can also solve by making 2 linear equations & solving for the values of variable b & c representing amount to be put in asset B & C respectively.