Question

APT. Suppose that stock returns are generated from the following linear process:

ri = E(ri) + bi,1X1 + bi,2X2 + εi

where εi and Xk are mean zero processses. The assumptions of Ross’ Arbitrage Pricing Theory hold. Consider three broadly diversified portfolios. Their returns are given approximately1 by

rA =0.06+2X1 +2X2

rB =0.05+3X1 +X2

rC =0.03+3X1

(a) Suppose there is a risk-free asset in the economy. What is its return? 


(b) Determine the factor risk premia. 


(c) Consider a fourth portfolio: 


rD = 0.08+X1 +0.5X2 


Are there arbitrage opportunities? If so, explain how you can exploit them. 


Answer 1

A. ri=E(ri)+bi, 1*1+bi, 2*2+Ei

where, Ei and XK=0

So,

there are three portfolio and it can be divided into 3 parts and form equation as follows

a. bi=1/3(.6)+1/3(.02)+1/3(.04)=.01 and finally we get bi=0.22=0.01 {note: in question itself it is mentioned that return is 1 and it is divided by 100 to simplify it}

b. bi=1/3(.05)+1/3(.03)+1/3(0)=0.01 and finally we get

bi=.18=0.01(0.18 by 1/3(.6/100)+1/3(.02/100)+1/3(.04/100) we get a decimal factor and we round off it) and {0.01 as mentioned above}

c. bi=1/3(0.03)+1/3(.03)=0.01 and we get

bi=0.02=0.01

Therefore,

ri=E(ri)+bi,1*1+bi,2*2+Ei

1=1+.21,1+.01,4+0

1=1

C. In this question we have 4 portfolio and divide into 4 parts.

It can be calculated as follows

3 equation same as above but substitute 1/3 into 1/4 because in this section we divide into 4 parts and forth equation can be derived as

bi=1/4(.08)+1/4(0.1)=0.01

bi=.225=0.01

and balance formulas are

a bi=1/4(.6)+1/4(.02)+1/4(.04)=.01

bi=.165=.01

b bi=1/4(.05)+1/4(.03)+1/4(0)=0.01

bi=0.02=0.01

c bi=1/4(.03)+1/4(.03)=0.01

bi=.015=0.01

finally we get'

ri=E(ri)+bi,1*1+bi,2*2+Ei

.2=0.2