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

Let wA, wB and wC be the weight of Asset B and Asset C ,

wA+wB+wC =1.............................(1)

Since Beta of a portfolio is the weighted average beta

For market beta

wA*1.1+wB*0.8+wC*1 =0.......................(2)

For gold mining beta

wA*1.5+wB*1.5+wC*0.2 = 0....................(3)

Solving , we get

wA= - 3.4359

wB= 3.282051

and wC =1.153846

So, if Asset A is shorted for $10000 , Asset B and C has to be bought and

amount invested in Asset B = 3.282051/3.4359*10000 =$9252.24

amount invested in Asset C = 1.153846/3.4359*10000 =$3358.21