Question

recall that the definition of arbitrage required the satisfaction of three conditions: one about weights, one about risk, and one about returns. Consider the following scenario in a one factor APT:

E[r] B1 Asset x 3.3% 0.7 Asset y 2.1% 2.3 Asset z 10.1% 3.3

What is the expected return of an arbitrage portfolio composed of all three assets, x, y, and z. Weights will be between +1 and -1. Answer is 5.38. please show how to do.

Answer 1

We see that the if we combine asset x and Asset z we will get a porfolio which will have a higher return than asset y

In order to achieve arbitrage

let the weight of x be =w

and weight of z be =1-w

Hence in order to have a higher return at the same beta the required beta of the portfolio should be equal to asset y's beta

Hence

Beta of X*w+ (1-w)Beta of Z= Beta of Y

0.7w+(1-w)*3.3= 2.3

0.7w+3.3-3.3w= 2.3

hence 3.3-2.3=2.6W

w=1/2.6= 0.384615 or 38.4615%

hence weight of x= 38.4615%

Weight if Z= 1- 0.384615= 0.615385 or 61.5385%

Expected return on the x and z in the above proportion = Weight of x * return of x+Weight of Z * return of z

=38.4615%*3.3+ 61.5385%*10.1%

=1.2692+6.2154= 7.4862

Hence to earn arbitrage profit one should short asset Y and buy Asset x and Asset Z in the above proportion

Weight is of x= 38.4615% or 0.384615

Weight of Y= -100% or -1

Weight of Z=   61.5385% or 0.615385

Expected return =

Weight of x * return of x+Weight of Z * return of z+Weight of y * return of y

=38.4615%*3.3+ 61.5385%*10.1+-100%*2.1

=1.2692+6.2154-2.1= 7.4862-2.1= 5.3862

Since the beta of X and Z will be same as Asset Y hence there will be a portfolio of 0 beta and will result in a gain of 5.3862%