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5. The expected return for Portfolio F equals the risk-free rate since its
beta equals 0.
For Portfolio A, the ratio of risk premium to beta is: (12 6)/1.2 = 5
For 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 a Portfolio G with beta equal to 0.6 (the same as E’s) by combining Portfolio A and Portfolio F in equal weights. The expected
return and beta for Portfolio G are then:
E(r G ) = (0.5 12%) + (0.5 6%) = 9%
G = (0.5 1.2) + (0.5 0) = 0.6
Comparing Portfolio G to Portfolio E, G has the same beta and higher
return. 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% + (0.6 F)] [8% + (0.6 F)] = 1%
That is, 1% of the funds (long or short) in each portfolio.
6. Substituting the portfolio returns and betas in the expected return-
beta relationship, we obtain two equations with two unknowns, the risk-free rate (r f ) and the factor risk premium (RP):
12 = r f + (1.2 RP)
9 = r f + (0.8 RP)
Solving these equations, we obtain:
r f = 3% and RP = 7.5%
7. a. Shorting an equally-weighted portfolio of the ten negative-alpha
stocks and investing the proceeds in an equally-weighted portfolio
of the ten positive-alpha stocks eliminates the market exposure and
creates a zero-investment portfolio. Denoting the systematic market
factor as R M , the expected dollar return is (noting that the
expectation of non-systematic risk, e, is zero):
$1,000,000 [0.02 + (1.0 R M )] $1,000,000 [(–0.02)
+ (1.0 R M )]
= $1,000,000 0.04 = $40,000
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