Question

Consider the following data for a one-factor economy. All portfolios are well diversified.

Suppose another portfolio E is well diversified with a beta of 2/3 and expected return of 9%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be?

Short answer

The correct answer would be:

a. The arbitrage opportunity exists

b. Portfolio His an arbitrage portfolio.

Step-by-step solution

Given Information

r_A = 10% and r_F = 4%

Beta for Portfolio F = 0

So expected return = risk free rate

Beta for Portfolio E = 2/3 (Given) and

Expected rate of return for Portfolio E= 9% (Given)

Calculation of ratio of risk premium to beta of Portfolio A and E

So, forPortfolio A, the ratio of risk premium to beta is: E[(r_A) – (r_F)]/ Beta

= (10% - 4%)/1 = 6%

But the ratio for Portfolio E is higher: (9% - 4%) / (2/3) = 7.5%

From the above scenario, it appears that the arbitrage opportunity exists.

Identification of arbitrage strategy

To check the arbitrage strategy, let’s create Portfolio G withβ = 0 by taking a long position on Portfolio E and short position in Portfolio F.

Forthe beta of G to equal 1.0, the proportion of funds invested in E must be: 3/2 = 1.5

The expected return of G is then:

E(r_G) = [(-0.50) x r_F] + (1.5 x r_E) [(-0.50) x 4%] + (1.5 x 9%) = 11.5%

β_G = 1.5 x (2/3) = 1.0

On comparison, it is noted that Portfolio G has the same Beta and a higher expected return.

Now, let’s consider Portfolio H, which is a short position in Portfolio A with the proceeds invested in Portfolio G:

β_H = 1β_G + (-1)β_A = (1 x 1) + [(-1) x 1] = 0

E(r_H) = (1 x r_G) + [(-1) x r_A] = (1 x 11.5%) + [(- 1) x 10%] = 1.5%

Since there is zero risk (β =0) and a positive return of 1.5%, Portfolio H is an arbitrage portfolio.