The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 562 Friday, March 12, 2004 12:40 PM

562 The Mathematics of Financial Modeling and Investment Management

Discriminant analysis divides a set into two parts. However, one might

want to classify stocks in several groups. In this case, the problem is one of

clustering. Clustering means forming groups so that objects in each group

are similar while objects in different groups are dissimilar. For instance,

classification in several different styles is an example of clustering. To per-

form clustering one needs a distance function that gives the distance

between any two objects. Clustering will find groups, i.e., clusters, that

have the minimum possible distance. A popular way of classifying stocks is

through hierarchical clustering based on correlation distance.^6

Style Classification Systems

Now that we have a general idea of the two main style categories, growth

and value, and the further refinement by size, let’s see how a portfolio man-

ager goes about classifying stocks that fall into the categories. We call the

methodology for classifying stocks into style categories as a style classifica-

tion system. Vendors of style indices have provided direction for developing a

style classification system. However, managers will develop their own system.

Developing such a system is not a simple task. To see why, let’s take

a simple style classification system where we just categorize stocks into

value and growth using one measure, the price-to-book value ratio. The

lower the P/B ratio the more the stock looks like a value stock. The style

classification system would then be as follows:

■ Step 1. Select a universe of stocks.

■ Step 2. Calculate the total market capitalization of all the stocks in the

universe.

■ Step 3. Calculate the P/B ratio for each stock in the universe.

■ Step 4. Sort the stocks from the lowest P/B ratio to the highest P/B

ratio.

■ Step 5. Calculate the accumulated market capitalization starting from

the lowest P/B ratio stock to the highest P/B ratio stock.

■ Step 6. Select the lowest P/B stocks up to the point where one-half the

total market capitalization computed in Step 2 is found.

■ Step 7. Classify the stocks found in Step 6 as value stocks.

■ Step 8. Classify the remaining stocks from the universe as growth

stocks.

While this style classification system is simple, it has both theoreti-

cal and practical problems. First, from a theoretical point of view, in

(^6) Clustering is broad topic. An excellent reference is Richard O. Duda, Peter E.
Heart, and David G. Stork, Pattern Classification (New York: John Wiley & Sons,
2001).