Consider a single factor APT. Portfolio A has a beta of 2.0 and an expected return of 19%. Portfolio B has a beta of 1.0 and an expected return of 8%. The risk-free rate of return is 3%. You can create a portfolio D which invests % in portfolio A and the rest in the risk-free asset so that it has the same beta as portfolio B, and compare the returns to portfolio D and portfolio B to decide the direction of arbitrage trading.

Invest 50% in portfolio A and the rest 50% in risk-free asset to create Portfolio D, we will have the same systematic risk as that of Portfolio B.

The expected return of Portfolio D = 11%

Portfolio D and Portfolio B have the same beta of 1.0. But, portfolio D has a higher return of 11% as compared to the expected return of Portfolio B of 8%.

Buy Portfolio D, and sell Portfolio B.

Explanation:

A risk free asset is referred to an asset that provides a virtually guaranteed return and no possibility of loss.

Risk-free asset has a beta of 0.

Portfolio D Beta = Wa × Portfolio A Beta + Wb × Risk-free asset beta

1.0 = Wa * 2.0 + Wb * 0

Wa = 1.0/2.0

Wa = 0.50

If we invest 50% in portfolio A and the rest 50% in risk-free asset to create Portfolio D, we will have the same systematic risk as that of Portfolio B.

The expected return of Portfolio D = 0.50 × 0.19 + 0.50 ×0.03

The expected return of Portfolio D = 0.11

The expected return of Portfolio D = 11%

Portfolio D and Portfolio B have the same beta of 1.0. But, portfolio D has a higher return of 11% as compared to the expected return of Portfolio B of 8%.

Buy Portfolio D, and sell Portfolio B.

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