Question

Consider an economy where a one-factor APT holds. All portfolios are well diversified. Portfolio A has an expected return of 10% and its factor beta is 1. Portfolio F has an expected return of 8% and its factor beta is 0.6. Suppose another portfolio E is well diversified with a beta of 0.8 and an expected return of 10%. Does an arbitrage opportunity exist in this case?

Answer 1

Expected return

Beta

Portfolio A

10%

1

Portfolio F

8%

0.6

Portfolio E

10%

0.8

To find: Arbitrage opportunity exists

Step 1:-

We need to choose the portfolio with highest and lowest beta.

In this case,

Highest Beta

Portfolio A

Lowest Beta

Portfolio F

Step 2:-

Construct a hypothetical portfolio say Portfolio B in this case, combining Portfolio A and F with Beta equal to that of Portfolio E

Let the weight of Portfolio A be Wa

so weight of portfolio F= (1-Wa)

We know,

Beta of portfolio= Weighted average

Wa*1+(1-Wa)*0.6= 0.8

Wa+0.6-0.6Wa= 0.8

0.4Wa= 0.2

Wa= 0.5

Weight of portfolio A= 0.5

Weight of portfolio F= (1-0.5)=0.5

Step 3:-

Expected return from the hypothetical portfolio i.e Portfolio B= Weighted average

0.5*10+0.5*8

9%

Step 4:-

Comparing Portfolio B so created with Portfolio E we get,

Potfolio

Beta

Return

B

0.8

9%

E

0.8

10%

The above results violate the law of one price i.e. stocks with equal risk should provide equal return. Hence there is an arbitrage opportunity:-

Short sell Portfolio B and invest the proceeds in Portfolio E

Arbitrage profit= (10-9)%

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