In a recent lawsuit, Apex sued Bpex for patent infringement. The jury came back

Question

In a recent lawsuit, Apex sued Bpex for patent infringement. The jury came back with its
decision today. The realized excess rate of return on Apex was 5.4%. The realized
excess rate of return on Bpex was only 4.9%. The market today responded to very
encouraging news about the unemployment rate, and the excess market return was 3%.
The HML portfolio returned 5% and the SMB portfolio returned 4%. The historical
relationship between returns on these stocks and the three factors has been estimated and
this is what we know:
Apex: market beta = 1.4, HML coeff. = 0.6, SMB coeff. = 0.3
Bpex: market beta = 0.6, HML coeff. = 0.4, SMB coeff. = 0.2

a) You believe in the Fama-French 3-factor model. Based on the data
provided in the problem, and assuming that the financial market reacted favorably
towards the winner, which company won the lawsuit?

b.) What other information do you need in order to be certain in your judgment of the outcome?

Explanation / Answer

According to Fama-French 3-factor model, CAPM uses a single factor, beta to compare the excess returns of the portfolio with the excess return on the market as a whole. The equation for this model is: r = Rf + Beta 3 (Km - Rf) + bs. SMB + bv. HML + alpha Substituting the values in the given equation 5.4% = Rf + 1.4 (0.03 - Rf ) + 0.3 (4%) + 0.6 (5%) + 0.0% 0.054 = Rf + 0.042 -1.4 Rf + 0.012 + 0.03 0.054 = Rf (1 - 1.4) + 0.084 -0.03 = -Rf (0.4) Rf = 0.075 or 7.5% Now, calculating the Portfolio return of Apex: r = 0.075 + 1.4 (0.03 - 0.075) + 0.012 + 0.03 = 0.075 + 1.4 (-0.045) + 0.042 = 0.054 or 5.4% Calculating the risk free rate for the Bpex portfolio: r = Rf + Beta 3 (Km - Rf) + bs. SMB + bv. HML + alpha 0.049 = Rf + 0.6 (0.03 - Rf) + 0.2 (4%) + 0.4 (5%) + 0.0% 0.049 = Rf(1-0.6) + 0.0018 + 0.008 + 0.02 0.049 = Rf (0.4) + 0.0298 0.0192 = Rf (0.4) Rf = 0.048 or 4.8% According to the beta values and the return on the individual stocks, the beta for Apex stock is higher for the rate of return than the beta for the Bpex stock which is having relatively low beta . Therefore, the Bpex stock wons the law suit. b) There is no information anything about the risk-free rate which is a major factor in calculating the market risk premium. Also the value of alpha is not given. If we know these two values, we can support our judgement.

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