The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 142 Wednesday, February 4, 2004 12:49 PM

142 The Mathematics of Financial Modeling and Investment Management

and only if they have the same dimensions n = m and the same compo-

nents:

x = y ⇔ xi = yi, i = 1 , ..., n

Vectors can be row vectors or column vectors. If the vector compo-

nents appear in a horizontal row, then the vector is called a row vector,

as for instance the vector

x = [1 2 8 7]

Here are two examples. Suppose that we let wn be a risky asset’s

weight in a portfolio. Assume that there are N risky assets. Then the fol-

lowing vector, w, is a row vector that represents a portfolio’s holdings of

the N risky assets:

w = w 1 w 2 ......... wN

As a second example of a row vector, suppose that we let rn be the

excess return for a risky asset. (The excess return is the difference

between the return on a risky asset and the risk-free rate.) Then the fol-

lowing row vector is the excess return vector:

r = r 1 r 2 ......... rN

If the vector components are arranged in a column, then the vector

is called a column vector as, for instance, the vector

x =

1

2

8

7

For example, as explained in Chapter 19, a portfolio’s excess return

will be affected by what can be different characteristics or attributes that

affect all asset prices. A few examples would be the price-earnings ratio,

market capitalization, and industry. We can denote for a particular

attribute a column vector, a, that shows the exposure of each risky asset

to that attribute: