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

Portfolio E(r) Beta

A 12% 1.2

F 6% 0.0

Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what is the arbitrage strategy?

When beta = 0, there is no risk, so it is risk free

Since beta = 0, the expected return for Portfolio F equals the risk-free rate

For Portfolio A, the ratio of risk premium is (12% 6%)/1.2

.12 - .06/1.2 => .06/1.2 = .05 x 100 = 5%

For Portfolio E, the ratio is lower at (8% 6%)/.6

.08 - .06/.6 => .02/.6 = .0333 x 100 = 3.33%

This implies that an arbitrage opportunity exists

Please explain to me why an arbitrage opportunity exists

Find out weight for Portfolio G:

W₁: Weight in Portfolio A

W₂: Weight in Portfolio G

W₂ = 1 W₁

W₁ B₁ + (1 W₁)B₂ = .6

W₁ 1.2 + (1 W₁)0 = .6

1.2W₁ + 0 = .6

1.2W₁ = .6

W₁ = .6/1.2 = .5

Expected Return and Beta of Portfolio G:

E(r_G) = (.5 x 12%) + (.5 x 6%)

E(r_G) = (.5 x .12) + (.5 x .06)

E(r_G) = .06 + .03 = .09 x 100 = 9%

_G = (.5 x 1.2) + (.5 x 0)

_G = .6 + 0 = .6

Comparing Portfolio G to Portfolio E, G has the same beta, but a higher expected return than E. Therefore, an arbitrage opportunity exists by buying Portfolio G and selling an equal amount of Portfolio E. The profit for this arbitrage will be:

r_G r_E = [9% + (.6 x F)] [8% + (.6 x F)] = 1%

1% of the funds (long or short) in each portfolio

I believe the answer is correct, but I would like to know why an arbitrage opportunity exists. Explain that to me in relation to the problem and please solve this step by step to get 1% r_G r_E = [9% + (.6 x F)] [8% + (.6 x F)] = 1%