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,

1）How many dollars would you put in Asset B?

2）How many dollars would you put in Asset C?

Answer 1

We know that the Asset A has been shorted which means the current outstanding:

Market Beta = -1.1  

Industry Beta = -1.5  

(Please note, the negative sign as the stock has been shorted)

Now, if we need to neutralize the market and industry risk, then we need to purchase Asset B & Asset C to make the overall Market and Industry Beta = 0

Let us assume:

We purchase ”X” times $10,000 of Asset B

We purchase ”Y” times $10,000 of Asset C

The market and industry beta of Asset A, B & C are given below:

Asset	Market Beta	Industry Beta	Action
A	1.1	1.5	Short
B	0.8	1.5	Long
C	1.0	0.2	Long

Hence, after purchasing 'X times Asset B' and 'Y times Asset C':

Overall Market Beta => 0.8x + 1.0y – 1.1

Overall Industry Beta => 1.5x + 0.2y -1.5

We need to make overall market and industry beta 0. Hence, equating the above two equations to 0 and solving for x & y.

0.8x + 1.0y - 1.1 = 0 ……Eq1

1.5x + 0.2y - 1.5 = 0 ……Eq2

Solving for X & Y

Step 1: Eq2 * 5 – Eq1

7.5x – 0.8x = 7.5 – 1.1

6.7x = 6.4

X = (6.4/6.7)

X = 0.95522

Step2: Use value of X in Eq1 to calculate Y

0.8 * (0.95522) + y = 1.1

Y = 1.1 – 0.76417

Y = 0.33582

Amount of Asset B to be bought is = 0.95522 * 10000 = $9,552

Amount of Asset C to be bought is = 0.33582 * 10000 = $3,358