Question

. Consider the following data:

Portfolio	Expected return	beta
a	10%	1.1
b	14%	1.4
c	18%	2.0

Portfolios a, b, and c are well diversified. If there is an arbitrage opportunity, how will it be done?

A. Buy stocks a and short stocks b and c,

B. buy stocks a and b and short c,

C. buy stock b and short a and c.

D.  There is no arbitrage opportunity.

Please can somebody answer this? I think there is no arbitrage opportunity since 1 of the betas has to be equal to 0. Please correct me if I'm wrong

Answer 1

For an arbitrage opportunity, Beta has to be equal for two different portfoliod, and if expected return is different, then we can buy the one with higher return and sell the one with lower expected return.

In this case, Beta of b is 1.4

So, we need to build a portfolio with a and c such that weighted beta becomes 1.4

Therefore, let weight of portfolio a be "x" therefore weight of portfolio c will be "1-x"

Weighted Beta = (Beta of A × Weight of A) + (Beta of C × Weight of C)

1.4 = (1.1*x) + {2*(1-x)}

1.4 = 1.1x + 2-2x

0.9x = 0.6

Therefore, X = Weight of A = 0.666 = 2/3rd and therefore, weight of C will be 1/3rd

Cross verification: [(2×1.1) + (1×2)]/2 = 1.4

Expected Return of above portfolio will be

(Expected return of A × Weight of A) + (Expected return of C × Weight of C)

[(10×2)+(18×1)]/3 = 12.67

Portfolio 1: 2 units of A and 1 unit of C, Beta = 1.4, Expected Return = 12.67%

Portfolio 2: 3 units of B, Beta = 1.4, Expected Return = 14%

Therefore, there is an arbitrate opportunity.

Buy Portfolio 2 and Sell Portfolio 1

Its Beta will become zero and expected return will be 14-12.67=1.33% which will be risk free return i.e. arbitrage gain.

Therefore, Correct Option is C. Buy stock b and short a and c.

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