Question

In the context of a one factor APT model, you are looking at the following three portfolios:

Portfolio	Expected return	Factor sensitivity
A	6	1.05
B	13	0.76
C	13	1.47

If you construct a composite portfolio "D" from B and C that has the same factor sensitivity as portfolio A, (similar to previous problem) and then go long D and short A (or the other way around) to create a riskless arbitrage profit, what would be your expected return?

Enter return as a percentage.

Hint: Solve for the weights within portfolio D. Then using those weights find the expected return on portfolio D (which is the weighted average of the component assets). Finally when you short one and long the other (you would obviously choose to short the one with the smaller return), your expected return would be the difference between the two returns.

The answer is 7.

Please show how to get to that in excel

Answer 1

Since we want to calculate a portfolio whose factor risk is equivalent to A Lets assume the weight of B in that portfolio is X then the weight of C in that portfolio would be (1-x) 0.76X + (1-X)*1.47 = 1.05 0.76X +1.47 - 1.47X = 1.05 1.47 - 1.05 = 1.47X -0.76X 0.591549296 0.5915 The weight of B is 0.5915 The weight of C would be 1 - 0.5915 0.4085 The expected return from portfolio D would be 0.5915*13 +0.4085*13 13 Now since the factor sensitivity of both the portfolios are same but expected return from the portfolio D is 13% while the expected return from portfolio A is 6% The arbitrage gain would be 7%