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

Question

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(rG) = (.5 x 12%) + (.5 x 6%)

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

E(rG) = .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:

rG – rE = [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% rG – rE = [9% + (.6 x F)] – [8% + (.6 x F)] = 1%

Explanation / Answer

Answer:

The expected return for PortfolioFequals the risk-free rate since its beta equals 0.For Portfolio A, the ratio of risk premium to beta is: (12 - 6)/1.2 = 5For Portfolio E, the ratio is lower at: (8 – 6)/0.6 = 3.33

This implies that an arbitrage opportunity exists. For instance, you can create aPortfolio Gwith beta equal to 0.6 (the same as E’s) by combining PortfolioAandPortfolioFin equal weights. The expected return and beta for PortfolioGare then:

E(rG) = (0.5 × 12%) + (0.5 × 6%) = 9%G= (0.5 × 1.2) + (0.5 × 0) = 0.6

Comparing PortfolioGto PortfolioE,Ghas the same beta and higher return.Therefore, an arbitrage opportunity exists by buying Portfolio G and selling an equalamount of PortfolioE. The profit for this arbitrage will be

rG– rE=[9% + (0.6 × F)][8% + (0.6 × F)] = 1%

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

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