Question

Consider a one factor economy where the risk free rate is 5%, and portfolios A and B are well diversified portfolios. Portfolio A has a beta of 0.6 and an expected return of 8%, while Portfolio B has a beta of 0.8 and an expected return of 10%. Is there an arbitrage opportunity in this economy? If yes, how could you exploit it?

Answer 1

In order to test an arbitrage opportunity, we need to check what will be the expected return of the portfolio when the beta of the portfolio is zero. Hence then we can compare the portfolio return with risk free position which are now comparable as both the things have a beta of zero.

Portfolio Beta is the weighted average beta of the stock where weight is the investment in each stock.

Lets assume investment in portfolio A is X. hence investment in portfolio B shall be 1-X

We need to find the weight where portfolio beta is zero

i.e Portfolio Beta = Weight A * Beta A + Weight B * Beta B

0 = X * 0.8 + (1-X)* 0.6

0 = 0.8X + 0.6- 0.6X

-0.6= 0.2X

X = -3

Hence Weight of A = -3 and B = (1-(-3)) = 4

That means we need to sell portfolio A 3 times and Buy B 4 times

Expected Return of above position = Weight * Return A + Weight * Return B

= (-3) * 10 + 4 * 8 = -30 + 32 = 2%

At beta of 0, through portfolio we can earn a risk free return of 2% , however given risk free rate is 5%. Hence arbitrage opportunity exists.

How to exploit?

We will buy the asset with high return and sell the asset with lower return.

Step 1: Buy Risk Free Asset and Sell the portfolio

Buy Risk free asset with return of 5%

Sell the effective portfolio with return of 2%( i.e Buy A upto 3 times and Sell B upto 4 times)

Step 2: At maturity

We will get return of 5% on risk free and will have to effective pay 2% on the portfolio of A&B

Net Return = 3%

Close the position by liquidating risk free asset and reversing the portfolio position.

Hence here we have earned a risk free arbitrage return of 3%.