6. Suppose that the Arbitrage Pricing Theory plane is given by:E[rl 6+2.5b,l + 6

ID: 2802483 • Letter: 6

Question

6. Suppose that the Arbitrage Pricing Theory plane is given by:E[rl 6+2.5b,l + 6b2, and that the following well-diversified portfolios are observed. Portfolio Expected Return (%) 15.6 28.8 2.4 1.2 0.6 3.3 Applying financial analysis techniques, you find that portfolio D, with bai-2.04, ba2-1.41, oners E[rd] = 24.96% Referring to the above APT plane, illustrate the arbitrage opportunities that exist for portfolio D. In your answer you must provide a table describing the arbitrage strategy, including the composition of any portfolio involved in this strategy, the sensitivities of the arbitrage portfolio, and the arbitrage profit. In your answer assume a $100 worth of each position taken.

Explanation / Answer

The APT equilibrium expected return of D should be:

E(rd)=6+2.5(2.04)+6(1.41)

=6+5.10+8.46

=19.56

Portfolio D offers a higher return (24.96%) than the long term equilibrium return of 19.56%.

Therefore, an arbitrage opportunity exists. Long Portfolio D and short Portfolio E (which combines portfolio A and B).

To find E we need to solve for xa and xb such that:

e1=xaBa1+xbBb1=Bd1

e2=xaBa2+xbBb2=Bd2

We have 2 equations with 2 unknowns:

2.4xa+0.6xb=2.04…..(1)

1.2xa+3.3xb=1.41…. (2)

Multiplying equation 2 by 2 and subtracting from equation 1 we get:

2.4xa+0.6xb=2.04

-2.4xa-6.6xb=-2.82

xb = 0.13

Plugging in xb=0.13 in equation 1 we get:

2.4xa+0.078=2.04

=2.4xa=1.962

= xa=0.8175

Therefore, the expected return of E is :

E(re)=0.8175(15.60)+0.13(28.80)

=16.497%

The sensitivities of E are:

Be1=0.8175(2.40)+0.13(0.60)=2.04

Be2=0.8175(1.20)+0.13(3.3)=1.41

The arbitrage strategy Long D and Short E:

Initial Investment

End of Period Cash Flow

Bi1

Bi2

Long D

-100

124.96

0.8175

0.13

Short E

100

-116.497

-0.8175

-0.13

Arbitrage Profit

8.463