Arbitrage Pricing Theory
Assume that the returns of individual securities are generated by the following two-factor model:
Rit = E(Rit) + βi1F1t + βi2F2t
Here:
Rit is the return for security i at time t
F1t and F2t are market factors with zero expectation and zero covariance
In addition, assume that there is a capital market for 4 securities, and the capital market for these four assets is perfect in the sense that there are no transaction costs and short sales (i.e., negative positions) are permitted. The characteristics of the four securities follow:
|
Security
|
β1
|
β2
|
E(R)
|
|
1
|
1.0
|
1.5
|
20%
|
|
2
|
0.5
|
2.0
|
20
|
|
3
|
1.0
|
0.5
|
10
|
|
4
|
1.5
|
0.75
|
10
|
a) Construct a portfolio containing (long or short) securities 1 and 2, with a return that does not depend on the market factor, F1t, in any way.
b) Following the procedure in (a), construct a portfolio containing securities 3 and 4 with a return that does not depend on the market factor the F1t. Compute the expected return and β2 coefficient for this portfolio.
c) There is a risk-free asset with expected retrun equal to 4.9%, β1 = 0, and β2 = 0. Describe a possible arbitrage opportunity in such detail that an investor could implement it.
d) What effect would the existence of these kinds of arbitrage opportunities have on the capital markets for these securities in the short and long run? Graph your analysis.