Final_1.pdf

We now scale each vector by dividing every element in the vector by the
length of the vector. Denoting the unit vectors in small letters, we have

a= (0, 1)

b= (1, 0)

The cosine of the angle between the vectors is given by the inner product of
the unit vectors.

cosq=ab= 0.1 + 1.0 = 0

The value of the cosine is zero, indicating that the angle between the two
vectors is 90 degrees. The two vectors are indeed orthogonal to each other.

Geometric Interpretation

Key to doing the geometric interpretation is the idea of eigenvalue decom-
position of the covariance matrix F. A brief discussion on eigenvalue de-
composition is provided in the appendix. Let the eigenvalue decomposition
ofFbe given as UDUT. If xAandxBare the factor exposure vectors of the
two stocks, let us consider a transformation of the two vectors as shown in
Equations 6.18.

eA=xAUD1/2 (6.18)

eB=xBUD1/2

This is the transformation from the factor exposure space to the factor
return space. Now, using simple matrix manipulations it is easy to verify that

(6.19)

(6.20)

(6.21)

Using Equations 6.19, 6.18, and 6.19, the cosine of the angle between the
vectorseAandeBworks out as follows:

eeABT ==xFxABT cov()r rAB,

length()exFxBBB==T var()rB

length()exFxAAA==T var()rA

length

length

()

()

A

B

=+=

=+=

02 2

30 3

22

22

96 STATISTICAL ARBITRAGE PAIRS