Abstract
This paper provides a plausible explanation for why the optimum number of stocks in a portfolio is elusive, and suggests a way to determine this optimal number. Diversification has a lot to do with the number of stocks in a portfolio. Adding stocks to a portfolio increases the level of diversification, and consequently leads to risk reduction up to a certain number of stocks, beyond which additional stocks are of no benefit, in terms of risk reduction. To explain this phenomenon, this paper investigates the relationship between portfolio diversification and concentration using a genetic algorithm. To quantify diversification, we use the portfolio Diversification Index (PDI). In the case of concentration, we introduce a new quantification method. Concentration is quantified as complexity of the correlation matrix. The proposed method quantifies the level of dependency (or redundancy) between stocks in a portfolio. By contrasting the two methods it is shown that the optimal number of stocks that optimizes diversification depends on both number of stocks and average correlation. Our result shows that, for a given universe, there is a set of Pareto optimal portfolios containing a different number of stocks that simultaneously maximizes diversification and minimizes concentration. The choice portfolio among the Pareto set will depend on the preference of the investor. Our result also suggests that an ideal condition for the optimal number of stocks is when variance reduction benefit of diversification is off-set by the variance contribution of complexity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It is worth mentioning that Kirchner and Zunckel (2011) claim portfolio weight can be accommodated under PDI by considering weighted returns.
If the covariance matrix of a stock portfolio is an identity matrix then all covariances are zero, meaning that the stock returns are statistically independent and all the variances have the same magnitude. Another possibility is when portfolio contain only one stock.
Woerheide and Persson (1993) use a variant of the HHI as a diversification measure i.e. 1- HHI.
i.e. Ratio of arithmetic and geometric mean equals 1 if and only if all values are equal.
The suggestion of Kirchner and Zunckel (2011) to incorporate weights into PDI can be accommodated by ICOMP so that we have a concentration measure that accounts for both correlation structure and weights.
The magnitude of the increase will depend on the correlation value, as we will show later.
This is difficult to see on the graphs because of the magnitude of complexity compared with the other axis, as noted earlier, complexity is a monotonic function of number of stocks.
Note that this marginal is calculated as \(\frac{\hbox {PDI}\left[ {\hbox {i}+1} \right] -\hbox {PDI}\left[ \hbox {i} \right] }{1}\hbox {where\,i}=1,2\ldots 83\). Therefore they represent change in PDI or ICOMP for a change in the number of stocks on average.
To estimate the covariance or correlation matrix, the portfolio must contain at least 2 stocks. The single objective GA is implemented with the “genalg” package in R (Willighagen et al. 2015) while the multi-objective one is implemented with the “nsga2R” package, also in R. Details of the parameters are supplied in subsequent sections.
As noted earlier, one may want to put more weight on ICOMP when the market is volatile (or when there is high correlation between stocks).
This result is because more than one combination of stocks can achieve a difference of 9 between PDI and ICOMP. Stocks or combination of stocks that are more correlated can be used interchangeable to form portfolios with similar diversification potential.
References
Bozdogan, H. (2000). Akaike’s information criterion and recent developments in information complexity. Journal of Mathematical Psychology, 44(1), 62–91.
Bozdogan, H. (2004). Statistical data mining and knowledge discovery. Boca Raton: CRC Press.
Choueifaty, Y., Froidure, T., & Reynier, J. (2013). Properties of the most diversified portfolio. Journal of Investment Strategies, 2(2), 49–70.
Cooper, B. (2000). Modelling research and development: How do firms solve design problems? Journal of Evolutionary Economics, 10(4), 395–413.
Cremers, K. J. M., & Petajisto, A. (2009). How active is your fund manager? A new measure that predicts performance. Review of Financial Studies, 22(9), 3329–3365.
Crezée, D. P., & Swinkels, L. A. P. (2011). High-conviction equity portfolio optimization. Journal of Risk, 13(2), 57.
Czarnitzki, D., & Doherr, T. (2002). Genetic algorithms: A tool for optimization in econometrics-basic concept and an example for empirical applications. ZEW discussion Paper, Mannheim
Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. A. M. T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182–197.
Divecha, A. B., Drach, J., & Stefek, D. (1992). Emerging markets: A quantitative perspective. The Journal of Portfolio Management, 19(1), 41–50.
Diyarbakırlıoğlu, E., & Satman, M. H. (2013). The Maximum Diversification Index. Journal of Asset Management, 14(6), 400–409.
Evans, J. L., & Archer, S. H. (1968). Diversification and the reduction of dispersion: an empirical analysis. The Journal of Finance, 23(5), 761–767.
Frahm, G., & Wiechers, C. (2011). On the diversification of portfolios of risky assets. Tech. rep., Discussion papers in statistics and econometrics.
Holland, J. H. (1975). Adaptation in natural and artificial system: An introduction with application to biology, control and artificial intelligence. Ann Arbor: University of Michigan Press.
Hovenkamp, H. J. (1985). Economics and federal antitrust law.
Hwang, C.-L., & Masud, A. S. M. (2012). Multiple objective decision making–methods and applications: A state-of-the-art survey (Vol. 164). New York: Springer.
Kirchner, U., & Zunckel, C. (2011). Measuring portfolio diversification. arXiv:1102.4722.
Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics (JSTOR), pp. 79–86.
Lin, D., Li, X., & Li, M. (2005). A genetic algorithm for solving portfolio optimization problems with transaction costs and minimum transaction lots. In Advances in Natural Computation, (pp. 808–811), New York: Springer.
Magnus, J. R., & Neudecker, H. (1995). Matrix differential calculus with applications in statistics and econometrics. New York: Wiley.
Markowitz, H. (1952). Portfolio selection*. The journal of finance, 7(1), 77–91.
Maschek, M. K. (2015). Economic modeling using evolutionary algorithms: The influence of mutation on the premature convergence effect. Computational Economics, pp. 1–23.
Meucci, A. (2010). Managing diversification.
Oyenubi, A. (2010). Information theoretic measure of complexity and stock market analysis: Using the JSE as a case study. http://open.uct.ac.za/handle/11427/10967.
Rudin, A. M., & Morgan, J. S. (2006). A portfolio diversification index. The Journal of Portfolio Management, 32(2), 81–89.
Scrucca, L. (2012). GA: A package for genetic algorithms in R. Journal of Statistical Software, 53(4), 1–37.
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379.
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 623.
Statman, M. (1987). How many stocks make a diversified portfolio? Journal of Financial and Quantitative Analysis, 22(03), 353–363.
Tang, G. Y. N. (2004). How efficient is naive portfolio diversification? An educational note. Omega, 32(2), 155–160.
Van Emden, M. H. (1971). An analysis of complexity. MC Tracts (pp. 1–86). Amsterdam: Centrum Voor Wiskunde en Informatica.
Van Heerden, J. D., & Saunderson, S. (2008). The effect of the South African market concentration on portfolio performance. Corporate Ownership & Control, p. 99.
Willighagen, E., Ballings, M., & Ballings, M. M. (2015). Package genalg.
Woerheide, W., & Persson, D. (1993). An index of portfolio diversification. Financial Services Review, 2(2), 73–85.
Author information
Authors and Affiliations
Corresponding author
Additional information
This measure was originally introduced in the author’s Master’s thesis under the supervision of A.E Clark and C.G Troskie [see Oyenubi (2010) for details].
Appendix
Appendix
Symbol | Company | Symbol | Company | ||
|---|---|---|---|---|---|
1 | AAPL | Apple Inc. | 43 | KLAC | KLA-Tencor Corp. |
2 | ADBE | Adobe Systems | 44 | LBTYA | Liberty Global plc |
3 | ADI | Analog Devices | 45 | LLTC | Linear Technology Corp. |
4 | ADP | Automatic Data Processing Inc. | 46 | MAT | Mattel Inc. |
5 | ADSK | Autodesk Inc. | 47 | MCHP | Microchip Technology |
6 | AKAM | Akamai Technologies Inc | 48 | MDLZ | Mondelez International, Inc. |
7 | ALTR | Altera Corp. | 49 | MNST | Monster Beverage Corporation |
8 | ALXN | Alexion Pharmaceuticals, Inc | 50 | MSFT | Microsoft Corp. |
9 | AMAT | Applied Materials | 51 | MU | Micron Technology |
10 | AMGN | Amgen | 52 | MXIM | Maxim Integrated Products, Inc |
11 | AMZN | Amazon Corp. | 53 | MYL | Mylan Inc. |
12 | ATVI | Activision Blizzard, Inc. | 54 | NFLX | Netflix, Inc. |
13 | BRCM | Broadcom Corporation | 55 | NTAP | NetApp |
14 | CA | CA, Inc. | 56 | NUAN | Nuance Communications, Inc |
15 | CELG | Celgene Corp. | 57 | NVDA | Nvidia Corporation |
16 | CERN | Cerner Corporation | 58 | ORLY | O’Reilly Auto Parts |
17 | CHKP | Check Point Software Technologies Ltd | 59 | PAYX | Paychex Inc. |
18 | CHRW | C. H. Robinson Worldwide | 60 | PCAR | PACCAR Inc. |
19 | CMCSA | Comcast Corp. | 61 | PCLN | The Priceline Group Inc |
20 | COST | Costco Co. | 62 | QCOM | QUALCOMM Inc. |
21 | CSCO | Cisco Systems | 63 | REGN | Regeneron Pharmaceuticals, Inc. |
22 | CTSH | Cognizant Technology Solutions | 64 | ROST | Ross Stores Inc |
23 | CTXS | Citrix Systems | 65 | SBAC | SBA Communications Corp. |
24 | DLTR | Dollar Tree, Inc. | 66 | SBUX | Starbucks Corp. |
25 | DTV | DIRECTV Group Inc. | 67 | SHLD | Sears Holdings Corporation |
26 | EBAY | eBay Inc. | 68 | SIAL | Sigma-Aldrich |
27 | EQIX | Equinix, Inc | 69 | SIRI | Sirius XM Holdings Inc. |
28 | ESRX | Express Scripts | 70 | SNDK | SanDisk Corporation |
29 | EXPD | Expeditors Int’l | 71 | SPLS | Staples Inc. |
30 | FAST | Fastenal Co | 72 | SRCL | Stericycle Inc |
31 | FFIV | F5 Networks, Inc. | 73 | STX | Seagate Technology Public Limited Company |
32 | FISV | FIserv Inc. | 74 | SYMC | Symantec Corp. |
33 | FOSL | Fossil Group, Inc. | 75 | TXN | Texas Instruments |
34 | FOXA | Twenty-First Century Fox, Inc. | 76 | VOD | Vodafone Group Plc |
35 | GILD | Gilead Sciences | 77 | VRTX | Vertex Pharmaceuticals Incorporated |
36 | GMCR | Keurig Green Mountain, Inc. | 78 | WDC | Western Digital |
37 | GOLD | Randgold Resources Limited | 79 | WFM | Whole Foods Market, Inc |
38 | GOOG | Google Inc. | 80 | WYNN | Wynn Resorts Ltd |
39 | GRMN | Garmin Ltd. | 81 | XLNX | Xilinx Inc |
40 | HSIC | Henry Schein, Inc. | 82 | XRAY | Dentsply Intl |
41 | INTU | Intuit Inc. | 83 | YHOO | Yahoo Inc. |
42 | ISRG | Intuitive Surgical Inc. |
Rights and permissions
About this article
Cite this article
Oyenubi, A. Diversification Measures and the Optimal Number of Stocks in a Portfolio: An Information Theoretic Explanation. Comput Econ 54, 1443–1471 (2019). https://doi.org/10.1007/s10614-016-9600-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-016-9600-5