Jefferson Cunha

Newton Da Costa Jr

Sergio Da Silva

English

Research Papers in Economics

Stock Selection Based on Cluster 
Analysis 
 
 
Newton Da Costa, Jr 
Department of Economics, Federal University of Santa Catarina  
 
Jefferson Cunha 
Department of Economics, Federal University of Santa Catarina 
 
Sergio Da Silva 
Department of Economics, Federal University of Santa Catarina 
 
 
 
Abstract 
 
We put forward a technique based on cluster analysis to group stocks in spot markets 
according to a risk-return criterion.  We show how an informed investor will make money 
using the cluster analysis to select stocks of major companies from North and South 
America. 
 
 
Keywords: Stocks, Cluster Analysis 
 
JEL classification: M21, G11, G15  
1.  Introduction 
 
Cluster analysis (Johnson and Wichern 1992) sorts variable sets according to their degree 
of correlation.  One can preset the number of clusters and then let analysis group the 
variables or, alternatively, one can let analysis itself identify natural clusters.  In either 
case, results are displayed in tree diagrams.  Here “fruits” are the variables and “branches” 
show the distance (degree of association) between them.  An advantage of using cluster 
analysis over other regression techniques is its classifying nature. 
We employ the cluster analysis to show that stocks from selected companies of the 
Americas can be categorized according to their degree of integration.  Stock returns are 
likely to be similar in a region thanks to geography and macroeconomic features, just to 
name a few.  So identification of stock clusters allows one to track those with similar 
returns but different risks.  Much literature focuses on integration of international stock 
exchange indices (e.g. Da Costa et al 2005) but there is scarce work on integration of 
individual stocks on an international basis. 
Once stocks are grouped by cluster analysis, an informed investor can use the 
output in his interests.  He will, for instance, look for same-return stocks and then choose 
to minimize risks.  Or else he will pick a cluster of same-risk stocks and high return. 
The rest of the paper is organized as follows.  Section 2 presents data.  Section 3 
analyzes them.  And Section 4 concludes. 
 
2.  Data 
 
Our variables are as follows.  Return, risk, earnings-price ratio, book value-price ratio, 
sales-price ratio, sales-number of stocks ratio, and dividend yield.  To group the stocks, we 
take daily data from two time windows, namely 2 January 1997−31 December 1999 and 2 
January 2000−31 December 2001.  The return and risk variables are calculated between the 
first and last day of either window.  The values for the other variables are those at the last 
day of either window.  We aim to evaluate whether a particular cluster sequence in the first 
window is worth keeping in the second time window. 
From a large set of 1,959 stocks of companies from North and South America listed 
in the Economatica database, we select those with daily average trade volume greater than 
one hundred thousand dollars.  Doing so we leave out low-liquidity stocks with risk 
premiums that are artificially elevated by low trade volumes.  This leaves us with a shorter 
set of 816 company stocks from Argentina, Brazil, Chile, Colombia, Mexico, Peru, 
Venezuela, and the United States.  These still made up 99.96 percent of the overall volume 
traded in such countries’ stock exchanges over the period.  And these also represent 85.8 
percent of the total value of the companies listed in the stock exchanges. 
To reckon return and risk we take closing values of the stocks in US dollar terms.  
These values are corrected for dividends, splits, and other events.  We decide to leave out 
missing-value stocks from the sample.  So the sample ends up with 476 stocks. 
A stock’s total return R is reckoned according to 
1 100
tt
t
PP
R
P
−  −
=⋅ 

                                                                                                      
where Pt is the stock’s closing value at time period t. 
Risk (standard deviation σ) is given by 
[]
2
1
()
n
ii
i
Pr Er σ
=
=− ∑                                                                                                  
where ri is a feasible return, E(r) is expected return, and Pi is the probability related to ri − 
E(r).   
To prevent units from interfering with their relative weight prior to the groupings, 
we take standardized values SVi for all variables Xi, i.e. 
i i
i
XX
SV
σ
−
=                                                                                                               
where the overbar stands for average values. 
The more similar two stocks A and B are, the shorter the Euclidian distance dAB 
between them, i.e. 
[]
2
1
(() () p
ii
AB
i
xA xB
d
p =
−
= ∑                                                                                         
where xi(A) gives the location of stock A compared to plane i’s origin, and p is space size, 
i.e. the number of variables. 
 
3.  Analysis 
 
We take the hierarchical clustering algorithm of Aldenderfer and Blashfield (1984), and 
carry out the cluster analysis with Statistica 5.0.  The stocks’ correlation coefficients are 
used as inputs in a likelihood matrix.  Similar stocks are then merged.  And the resulting 
clusters are displayed in a dendogram, which is a diagram representing the hierarchical 
organization of the stocks’ relationships. 
We employ Sokal and Michener’s (1958) technique to add a stock in an already 
existing cluster.  The technique considers average returns across time to compensate for 
random movements.  After doing the hierarchical clustering, we employ Ward 
methodology to sort the clusters.  Figure 1 shows the resulting dendogram. 
At the arbitrary cut level of 20, we get 10 clusters.  Every stock is assumed to have 
the same weight in a cluster.  So returns are given by the mean.  (Risk and the other 
variables are also given by means.)  Risk-return features are displayed in Figure 2. 
  The preferred clusters in Figure 2 are those with negatively sloped straight lines, 
because these represent higher returns with lower risks.  Cluster 2 is made up of only one 
stock and looks best at first sight.  But it also presents large risk-return.  Then we consider 
it as an outlier and leave it out from analysis.  As result, cluster 6 becomes the best, 
followed by clusters 1, 8, and 10 respectively.  This sequence gives clusters with similar 
returns and increasing risks. 
Once the best cluster is chosen, the investor can perfect his choice by selecting a 
stock within the cluster considering the country where the company is based in.  And here 
he might wish to take country risk into account.  Figure 3 shows the distribution of clusters 
according to place of origin.  (To improve resolution, four stocks with very high returns are 
left out.) 
  Table 1 summarizes the clusters’ features for the first time window.  (Though 
cluster 6 is made up of only one stock we decide not to leave it out because of its good 
fundamental-analysis variables.) 
We then move on to consider the second time window.  Figure 4 displays risk and 
return for the same ten clusters selected for the first time window.  As can be seen, 
preferred clusters continue to be 6, 1, 8, and 10 respectively.  What is more, this cluster 
sequence presents now even higher returns with lower risks.  So the investor will profit (or 
lose less) by keeping the sequence selected for the first time window. 
  Figure 5 shows risk and return of the cluster sequence for the second time window.  
The sequence is located at the bottom left.  This means greater volatility and lower returns 
throughout and happens because the stock exchanges went lower in the second time 
window.  Figure 6 compares risk and return of the two time windows.  As one moves from 
clusters 6 to 1, 8, 10, 4, 7, 9, 5, 3, and 2 respectively, the relationship between risk and  
return gets weaker.  Finally Table 2 shows variable values in the second time window for 
the cluster sequence selected in the analysis of the first time window. 
 
4.  Conclusion 
 
This paper shows how an investor can profit (or lose less) using cluster analysis to select 
stocks.  To illustrate our case we take major company stocks of North and South America 
at two distinct time windows. 
The investor choosing stocks according to the clusters sorted in the first time 
window (2 January 1997−31 December 1999) is found to profit (or to minimize losses) in 
the second time window (2 January 2000−31 December 2001).  
Table 1.  Clusters’ features, 2 January 1997−31 December 1999 
 
Cluster  Number 
of Stocks 
Return 
(%) 
Risk 
(%) 
Earnings-
Price Ratio 
Book-to- 
Price Ratio 
Sales-Price 
Ratio 
Sales-
Number of 
Stocks Ratio 
Dividend 
Yield (%) 
               
44.204 32.8  0.058  0.462 1.258 37.326  2.478 
1 94 
(78.084)  (10.4) (0.045) (0.314) (0.879) (20.996)  (1.931) 
            
15171.324 80.8  0.000  0.011 0.005 1.118 0.000 
2 1 
(0.000)  (0.0) (0.000) (0.000) (0.000) (0.000) (0.000) 
            
751.744 73.3  0.011  0.183 0.245 5.510 0.207 
3 46 
(1211.769)  (29.2) (0.033) (0.196) (0.266) (5.404) (0.529) 
            
43.921 55.4  0.139  1.223 1.112 7.019 8.451 
4 20 
(99.121)  (15.6) (0.171) (0.806) (0.940) (14.636)  (5.563) 
            
−47.605  66.6  −0.445  6.869 3.384 0.038 2.578 
5 5 
(27.483)  (22.3) (0.628) (5.199) (2.470) (0.056) (3.567) 
            
1.300 28.4  0.085  0.590 1.368 77.613  1.057 
6 1 
(0.000)  (0.0) (0.000) (0.000) (0.000) (0.000) (0.000) 
            
106.480 59.3  −0.029  1.050 1.672 8.320 1.669 
7 77 
(731.848)  (17.0) (0.414) (0.660) (1.466) (17.351)  (2.012) 
            
133.144 37.9  0.064  0.512 1.772 69.128  1.580 
8 23 
(284.003)  (15.9) (0.044) (0.330) (1.322) (61.060)  (1.556) 
            
27.980 63.2  −0.126  0.956 3.673 24.096  2.838 
9 13 
(120.861)  (44.4) (0.309) (0.931) (3.382) (22.760)  (4.366) 
            
200.383 41.5  0.032  0.384 0.609 12.149  1.483 
10 196 
(1125.174)  (13.8) (0.083) (0.351) (0.857) (10.411)  (1.836) 
 
Note 
Standard deviations of the averages are in brackets  
Table 2.  Clusters’ features, 2 January 2000−31 December 2001 
 
Cluster  Number 
of Stocks 
Return 
(%) 
Risk 
(%) 
Earnings-
Price Ratio 
Book-to- 
Price Ratio 
Sales-Price 
Ratio 
Sales-Number 
of Stocks 
Ratio 
Dividend 
Yield (%) 
             
15.680 41.1  0.036  0.472 1.587  47.640 2.279 
1 94 
(45.307) (13.5)  (0.074)  (0.349) (1.636)  (34.120) (1.680) 
            
−91.800  93.8  −0.009  0.195 0.071 1.259 0.000 
2 1 
(0.000) (0.0)  (0.000)  (0.000) (0.000) (0.000) (0.000) 
            
−35.875  83.4  −0.170  0.396 0.509 6.663 0.857 
3 46 
(40.223) (23.1)  (0.899)  (0.313) (0.499) (5.776) (2.027) 
            
−8.531  49.2 0.115 1.546 1.835 14.240 10.105 
4 20 
(33.577) (13.3)  (0.126)  (1.020) (1.472)  (36.739) (8.341) 
            
−3.507  64.1  −2.809  10.517 9.994 0.031 0.000 
5 5 
(48.338) (20.9)  (6.118)  (13.053) (17.780)  (0.042)  (0.000) 
            
13.495 31.8  0.164  0.885 2.421  151.229 2.400 
6 1 
(0.000)  (0.0) (0.000) (0.000) (0.000) (0.000) (0.000) 
            
−10.579  57.4 0.037 1.515 2.637 10.459  4.315 
7 77 
(52.838) (20.1)  (0.309)  (2.119) (2.352)  (23.924) (11.977) 
            
13.150 47.1  0.005  0.519 1.913  72.205 1.322 
8 23 
(57.677) (19.5)  (0.135)  (0.359) (1.478)  (62.717) (1.444) 
            
−6.112  69.4  −0.187  1.031  3.484 24.769 13.479 
9 13 
(36.053) (50.4)  (0.542)  (0.901) (3.774)  (27.927) (41.754) 
            
4.538 47.9  −0.008  0.519 0.855  15.449 2.111 
10 196 
(53.312) (17.7)  (0.447)  (0.662) (1.023)  (17.615) (4.128) 
 
Note 
Standard deviations of the averages are in brackets  
Figure 1.  Dendogram using Ward methodology and taking Euclidian distances, 
2 January 1997−31 December 1999 
 
 
 
 
 
Figure 2.  Clusters’ average return and risk, 
2 January 1997−31 December 1999 
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Figure 3.  Return and risk of stocks in a cluster, 
2 January 1997−31 December 1999 
 
-200
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0,0 0,5 1,0 1,5 2,0 2,5
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t
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r
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Figure 4.  Clusters’ average return and risk, 
2 January 2000−31 December 2001 
 
-3.000
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0.000
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Cluster 2
Cluster 3
Cluster 4
Cluster 5
Cluster 6
Cluster 7
Cluster 8
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Figure 5.  Return and risk of stocks in a cluster, 
2 January 2000−31 December 2001 
 
-150
-50
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Figure 6.  Clusters’ average return and risk of a cluster C(n) in time window T(n), 
2 January 1997−31 December 1999 and 2 January 2000−31 December 2001 
 
-200
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References 
 
Aldenderfer, M.S., and R.K. Blashfield (1984) Cluster Analysis. Sage University Paper: 
Quantitative Applications in the Social Sciences, n. 44. 
 
Da Costa, Jr, N., S. Nunes, P. Ceretta, and S. Da Silva (2005) “Stockmarket comovements 
revisited”, Economics Bulletin 7, 1-9. 
 
Johnson, R. A., and D.W. Wichern (1992) Applied Multivariate Statistical Analysis, 
Prentice-Hall International: New Jersey. 
 
Sokal, R., and C.D. Michener (1958) “A statistical method for evaluating systematic 
relationships” University of Kansas Scientific Bulletin 38, 1409-1438. 

A statistical method for evaluating systematic relationships”

Applied Multivariate Statistical Analysis, Prentice-Hall International:

Cluster Analysis. Sage University Paper:

Stockmarket comovements revisited”,

Stock Selection Based on Cluster Analysis

http://128.118.178.162/eps/fin/papers/0509/0509022.pdf

Stock Selection Based on Cluster Analysis

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