Question

We observe three well-diversified portfolios A, B and C with a return of 15%, 6%, and 12% respectively in the market. We also know that portfolio A, B and C has a beta of 1.5, 0.5, and 1.0 respectively. Construct an arbitrage strategy, and how much is the arbitrage profit?

Answer 1

Given,
Portfolio	Expected return	Beta
A	15%	1.5
B	6%	0.5
C	12%	1
Step 1: Choose the portfolio with highest beta and lowest beta i.e Portfolio A and Portfolio B
Step 2: We need to construct a hypothetical portfolio 'D' combining Portfolio A & B with beta equal to that of Portfolio C
We know,
Portfolio Beta= Weighted average
Let the Weight of portfolio A be Wa than weight of Portfolio B will be (1-Wa)
Wa*1.5+(1-Wa)*0.5= 1
1.5Wa+0.5-0.5Wa=1
1Wa=0.5
Wa= 0.5
Therefore,
Weight of Portfolio A= 0.5
Weight of Portfolio B= 0.5
Step 3: We need to calculate the expected return of the hypothetical portfolio so created in Step 2
Expected return= Weighted average
	0.5*15+0.5*6
	10.50%
Step 4: We need to calculate expected return of portfolio C with hypothetical portfolio D
Portfolio	Expected return	Beta
C	12%	1
D	10.50%	1
The above data violates the law of one price i.e. stocks with equal risk should provide equal return.
Hence, there is an arbitrage opportunity
Short sell Portfolio D and invests the proceeds in Portfolio C
Arbitrage profit= (12-10.50)%
	1.50%