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Some New Index-Calculation Methods and Their Mathematical Properties

  • Michael I. C. Nwogugu
Chapter

Abstract

While indices, index tracking funds and ETFs have grown in popularity during then last 10 years, there are many structural problems inherent in the daily trading of indices and ETFs, primarily in the form of Index Arbitrage and ETF Arbitrage (this chapter does not cover the arbitrage of Mutual Funds). The US Congress and many governments have attempted to tackle these problems and have enacted various laws, which have not been effective. Index Arbitrage does not facilitate price discovery, creates excessive and unnecessary volatility and distorts indices and the perceptions of the true values and risks of companies. This chapter contributes to the existing literature by introducing new index calculation methods (for fixed income, equity and commodity indices) that substantially reduce or eliminate Index Arbitrage and ETF Arbitrage.

Keywords

Investor preferences Transferable Utilities Indices Index Arbitrage Price discovery 

Bibliography

  1. Afriat, S. (2012). The index number problem and utility construction. Book manuscript.Google Scholar
  2. Akem, A. (2018). Measuring substitution bias and the CPI—Theory and practice. Australian Bureau of Statistics. https://www.unece.org/fileadmin/DAM/stats/documents/ece/ces/ge.22/2018/Australia_3a.pdf
  3. Alford, A., & Rakhlin, D. (2017). Implementing a smart beta index: The implications of a dual performance objective and limited liquidity. Journal of Portfolio Management, 43(5), 135–146.Google Scholar
  4. Amenc, N., & Goltz, F. (2016). Who cares about purity of factor indexes? A comment on “evaluating the efficiency of ‘smart beta’ indexes”. The Journal of Index Investing, 7(1), 10–13.Google Scholar
  5. Amenc, N., Goltz, F., Martellini, L., & Retkowsky, A. (2010). Efficient indexation: An alternative to cap-weighted indices. Journal of Investment Management, 9(4), 1–23.Google Scholar
  6. Ang, A., Hodrick, R., Xing, Y., & Zhang, X. (2006). The cross-section of volatility and expected returns. Journal of Finance, LXI(1), 259–299.Google Scholar
  7. Arnott, R., Hsu, J., & Moore, P. (2005). Fundamental indexation. Financial Analysts Journal, 61(2), 83–99.Google Scholar
  8. Arnott, R., Kalesnik, V., Moghtader, P., & Scholl, C. (2010, January/February). Beyond cap weight: The empirical evidence for a diversified beta. Journal of Indexes, 16–26.Google Scholar
  9. Aydin, S., & Ozer, G. (2005). National customer satisfaction indices: An implementation in the Turkish mobile telephone market. Marketing Intelligence & Planning, 23(5), 486–504.Google Scholar
  10. Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70, 191–221.Google Scholar
  11. Balk, B. (1995). Axiomatic price index theory: A survey. International Statistical Review, 63, 69–93.Google Scholar
  12. Balk, B. (2016). A review of index number theory. Wiley StatsRef: Statistics Reference Online.Google Scholar
  13. Barnett, W., & Chauvet, M. (2011). Financial aggregation and index number theory. Hackensack: World Scientific.Google Scholar
  14. Bernhard, R. (1971). A comprehensive comparison and critique of discounting indices proposed for capital investment evaluation. The Engineering Economist, 16(3), 157–186.Google Scholar
  15. Birdthistle, W. (2008). The fortunes and foibles of exchange-traded funds: A positive market response to the problems of mutual funds. Delaware Journal of Corporate Law, 33, 69–99. http://works.bepress.com/cgi/viewcontent.cgi?article=1014&context=william_birdthistle
  16. Bleecker, D. (2013). Index theory with applications to mathematics and physics. Bernhelm Books.Google Scholar
  17. Blitz, D., & Van Vliet, P. (2007). The volatility effect: Lower risk without lower returns. Journal of Portfolio Management, 34(1), 102–113.Google Scholar
  18. Blume, M. E., & Edelen, R. M. (2004). S&P 500 indexers, tracking error and liquidity. Journal of Portfolio Management, 30, 37–46.Google Scholar
  19. Bolla, L., Kohler, A., & Wittig, H. (2016). Index-linked investing—A curse for the stability of financial markets around the globe? Journal of Portfolio Management, 42(3), 26–43.Google Scholar
  20. Boudoukh, J., Richardson, M., Subrahmanyam, M., et al. (2002). Stale prices and strategies for trading mutual funds. Financial Analysts Journal, 58(4), 53–71.Google Scholar
  21. Brockett, P., & Garven, J. R. (1998). A re-examination of the relationship between preferences and moment orderings by rational risk averse investors. The GENEVA Papers on Risk and Insurance Theory, 23(2), 127–137.Google Scholar
  22. Carhart, M. M., Carpenter, J. N., Lynch, A. W., & Musto, D. K. (2002). Mutual fund survivorship. Review of Financial Studies, 15, 1439–1463.Google Scholar
  23. Chakravarty, S. (2001). Stealth trading: Which traders’ trades moves stock prices? Journal of Financial Economics, 61, 289–307.Google Scholar
  24. Chan, H., & Howard, P. F. (2002). Additions to and deletions from an open-ended market index: Evidence from the Australian all ordinaries. Australian Journal of Management, 27, 45–74.Google Scholar
  25. Chan, L., Karceski, J., & Lakonishok, J. (1999). On portfolio optimization: Forecasting covariances and choosing the risk model. Review of Financial Studies, 12(5), 937–974.Google Scholar
  26. Che, L. (2018). Investor types and stock return volatility. Journal of Empirical Finance, 47, 139–161.Google Scholar
  27. Chen, H. G., & Singal, N. V. (2006). Index changes and unexpected losses to Investors in S&P 500 and Russell 2000 index funds. Financial Analysts Journal, 62(4), 31–47.Google Scholar
  28. Chen, H., et al. (2006). Index changes and losses to index fund investors. Financial Analysts Journal, 62, 31–34.Google Scholar
  29. Chen, Q., Goldstein, I., & Jiang, W. (2010). Payoff complementarities and financial fragility: Evidence from mutual fund outflows. Journal of Financial Economics, 97(2), 239–262.Google Scholar
  30. Chiappori, P. A., & Ekeland, I. (2006). The micro economics of group behavior: General characterization. Journal of Economic Theory, 130(1), 1–26.Google Scholar
  31. Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification. Journal of Portfolio Management, 35(1), 40–51.Google Scholar
  32. Christie-David, R., & Chaudhury, M. (2001). Coskewness and cokurtosis in futures markets. Journal of Empirical Finance, 8(1), 55–81.Google Scholar
  33. Christoffersen, S., & Musto, D. (2014). Investor flows to asset managers: Causes and consequences. Annual Review of Finance.Google Scholar
  34. Christoffersen, S. E. K., Geczy, C. C., Musto, D. K., et al. (2005). Crossborder dividend taxation and the preferences of taxable and nontaxable investors: Evidence from Canada. Journal of Financial Economics, 78(1), 121–144.Google Scholar
  35. Clarke, R., de Silva, H., & Thorley, S. (2006). Minimum-variance portfolios in the U.S. equity market. Journal of Portfolio Management, 1–14.Google Scholar
  36. Clements, K., Izan, I., & Selvanathan, A. (2006). Stochastic index numbers: A review. International Statistical Review, 74(2), 235–270.Google Scholar
  37. Dash, S. (2002). Price changes associated with S&P 500 deletions: Time variation and effect of size and share prices. New York: Standard & Poor’s.Google Scholar
  38. Dash, S., & Blitzer, D. (2004). Index effect revisited: What happened to the post-addition price pop? New York: Standard & Poor’s.Google Scholar
  39. Dash, S., & Liu, B. (2009). Capturing the index effect via options. Journal of Trading, 4(2), 72–78.Google Scholar
  40. Diewert, E. (1978). Superlative index numbers and consistency in aggregation. Econometrica, 46, 883–900.Google Scholar
  41. Diewert, E. (2005). Index number theory using differences rather than ratios. The American Journal of Economics and Sociology, 64(1), 311–360.Google Scholar
  42. Donnelly, K., & Tower, E. (2007). Time-zone arbitrage in United States mutual funds: Damaging to financial integration between the United States, Asia and Europe? (Working paper). http://www.unescap.org/tid/publication/tipub2563_chap8.pdf
  43. Dopfel, F., & Lester, A. (2018). Optimal blending of smart beta and multifactor portfolios. Journal of Portfolio Management, 44(4), 93–105.Google Scholar
  44. Eggins, J., & Hill, R. J. (2008). Momentum and contrarian stock-market indices (Working paper).Google Scholar
  45. Elliott, D., O’Neill, R., & Winton, J. (2013). Elementary aggregate indices and lower level substitution bias. Statistical Journal of the IAOS, 29, 11–19.Google Scholar
  46. Elton, E. J., Gruber, M. J., & Busse, J. A. (2004). Are investors rational? Choices among index funds. Journal of Finance, 59, 261–288.Google Scholar
  47. Erg, T. (2009). NYSE rule 80A restrictions on index arbitrage and market linkage. Applied Financial Economics, 19(20), 1675–1685.Google Scholar
  48. Falkenstein, E. G. (2009). Risk and return in general: Theory and evidence. Available at SSRN: http://ssrn.com/abstract=1420356
  49. Frino, A., Gallagher, D. R., & Oetomo, T. N. (2005). The index tracking strategies of passive and enhanced index equity funds. Australian Journal of Management, 30, 23–55.Google Scholar
  50. Gastineau, G. L. (2002). Silence is golden: The importance of stealth in pursuit of the perfect fund index. Journal of Indexes, 4, 5–7. Available at: http://www.etfconsultants.com/Silence%20is%20Golden%20Journal%20of%20Indexes.pdf
  51. Goetzmann, W. N., & Massa, M. (2003). Index funds and stock market growth. Journal of Business, 76, 29–38.Google Scholar
  52. Goldstein, M. (2015). Circuit breakers, trading collars, and volatility transmission across markets: Evidence from NYSE Rule 80A (Working paper). http://faculty.babson.edu/goldstein/research/rule80a-2015-04-12-FINAL.pdf
  53. Grant, J., Wolf, A., & Yu, S. (2005). Intraday price reversals in the US stock index futures market: A 15-year study. Journal of Banking & Finance, 29(5), 1311–1327.Google Scholar
  54. Greene, J., & Hodges, C. (2002). The dilution impact of daily fund flows on open-end mutual funds. Journal of Financial Economics, 65(1), 131–158.Google Scholar
  55. Groshen, E., Moyer, B., et al. (2017). How government statistics adjust for potential biases from quality change and new goods in an age of digital technologies: A view from the trenches. Journal of Economic Perspectives, 31(2), 187–210.Google Scholar
  56. Han, L., & Misra, L. (1994). The impact of trading restrictions on the informational relationships between cash, futures, and options markets. International Review of Economics & Finance, 3(4), 429–442.Google Scholar
  57. Hansson, B. (1975). The appropriateness of the expected utility model. Erkenntnis, 9(2), 175–193.Google Scholar
  58. Harris, L. (1997). Circuit breakers and program trading limits: What have we learned ? (USC Working paper). University of Southern California. http://www-bcf.usc.edu/~lharris/ACROBAT/circuit.pdf
  59. Harrison, G., Lau, M., Rutstrom, E., & Sullivan, M. (2004). Eliciting risk and time preferences using field experiments: Some methodological issues. Research in Experimental Economics, 10, 125–218.Google Scholar
  60. Haslem, J. A. (2008, July). Mutual funds: Why have independent directors failed as shareholder watchdogs? (Working paper). SSRN.Google Scholar
  61. Hausman, J. (2003). Sources of bias and solutions to bias in the consumer price index. Journal of Economic Perspectives, 17(1), 23–44.Google Scholar
  62. Heaney, R. (1995). A test of the cost of carry relationship using 90-day Bank accepted bills and the all ordinaries share price index. Australian Journal of Management, 20, 75–104.Google Scholar
  63. Hertzberg, M. (1987). A critique of the dust explosibility index: An alternative for estimating explosion probabilities. U.S. Department of the Interior, Bureau of Mines, Report of Investigations, RI 9095.Google Scholar
  64. Hey, J. D. (2009). Noise and bias in eliciting preferences. Journal of Risk and Uncertainty, 39, 213–235.Google Scholar
  65. Holden, C. (1995). Index arbitrage as cross-sectional market making. Journal of Futures Markets, 15(4), 423–455.Google Scholar
  66. Hosseiny, A. (2017, January). Some correspondences between index number theory in economy and the general theory of relativity in physics. https://arxiv.org/abs/1701.05114
  67. Houge, T., & Wellman, J. (2005). Fallout from the mutual fund trading scandal. Journal of Business Ethics, 62(2), 129–139.Google Scholar
  68. Hurlbert, S. (1971). The non-concept of species diversity: A critique and alternative parameters. Ecology, 52(4), 576–580. http://www.fcnym.unlp.edu.ar/catedras/ecocomunidades/Hurlbert%20-%20The%20Nonconcept%20of%20Species%20Diversity--%20A%20Critique%20and%20Alternative%20Parameters.pdf
  69. IOSCO. (2013). Principles for the regulation of exchange traded funds. www.iosco.org/library/pubdocs/pdf/IOSCOPD414.pdf
  70. Ito, K., & Reguant, M. (2016). Sequential markets, market power and arbitrage. The American Economic Review, 106(7), 1921–1957.Google Scholar
  71. Ivancic, L., Diewert, W., & Fox, K. (2011). Scanner data, time aggregation and the construction of prices indexes. Journal of Econometrics, 161, 24–35.Google Scholar
  72. Jazari, N. (2013). The theory and practice of index numbers. In Encyclopedia of statistical sciences. Wiley.Google Scholar
  73. Jha, R., Murthy, K., & Bhanu, V. (2003, April). A critique of the environmental sustainability index (Working paper). Division of Economics, Australian National University. Available at SSRN: http://ssrn.com/abstract=380160 or  https://doi.org/10.2139/ssrn.380160
  74. Jones, S. K., & Stine, J. B. (2010). Expected utility and the non-normal returns of common portfolio rebalancing strategies. Journal of Asset Management, 10, 406–419.Google Scholar
  75. Kaiser, L., Fleisch, M., & Salcher, L. (2018). Bias and misrepresentation revisited: Perspective on major equity indices. Finance Research Letters, forthcoming.Google Scholar
  76. Kaneko, T. (1976). Note on transferable utility. International Journal of Game Theory, 5(4), 183–185.Google Scholar
  77. Karabatsos, G. (2000). A critique of Rasch residual fit statistics. Journal of Applied Measurement, 1(2), 152–176. http://tigger.uic.edu/~georgek/HomePage/Karabatsos2000JAM.pdf
  78. Keim, D. B. (1999). An analysis of mutual fund design: The case of investing in small-cap stocks. Journal of Financial Economics, 51, 173–194.Google Scholar
  79. Lee, Y. (2003). Indexation of momentum effects. Journal of Investment Management, 1(4).Google Scholar
  80. Lerman, D. (2003, March). Trading the basis: How stock index arbitrage impacts the market (Working paper).Google Scholar
  81. Levell, P. (2015). Is the Carli index flawed? Assessing the case for the new retail price index RPIJ. Journal of the Royal Statistical Society: Series A (Statistics in Society), 178, 303–336.Google Scholar
  82. Lintunen, J., Ropponen, O., & Vartia, Y. (2009, March). Micro meets macro via aggregation (HECER Discussion Papers No. 259).Google Scholar
  83. Locke, P., & Onayev, Z. (2007). Order flow, dealer profitability, and price formation. Journal of Financial Economics, 85(3), 857–887.Google Scholar
  84. Madhavan, A. A., & Ming, K. (2002). The hidden costs of index rebalancing: A case study of the S&P 500 composition changes of July 19, 2002 (ITG working paper).Google Scholar
  85. Maillard, S., Roncalli, T., & Teiletche, J. (2008). On the properties of equally-weighted risk contributions portfolios (Working paper).Google Scholar
  86. Malaney, P. (1996). The index number problem: A differential geometric approach (PhD thesis). Harvard University, Cambridge, MA.Google Scholar
  87. Markham, J. (2006). Mutual fund scandals – A comparative analysis of the role of corporate governance in the regulation of collective investments. Hastings Business Law Journal, 3, 67–81.Google Scholar
  88. McLaughlin, T. (2008). Eyes wide shut: Exchange traded funds, index arbitrage and the need for change. Annual Review of Banking and Financial Law, 27, 597–608.Google Scholar
  89. Murphy, E., & Garvey, E. (2005). Cost of living indices and flexible consumption behaviour: A partial critique (Working paper).Google Scholar
  90. Musto, D. (2011). The economics of mutual funds. Annual Review of Financial Economics, 3(1), 159–172.Google Scholar
  91. Nayebi, H., & Abdollahyan, H. (2006). Social stratification and its indices: A critique. Middle East Critique, 15(3), 249–263.Google Scholar
  92. Neher, D., & Darby, B. (2006). Computation and application of nematode community indices: General guidelines (Working paper). Available at: http://www.uvm.edu/~dneher/Publications/04Neher_Darby%20chapter.pdf
  93. Nwogugu, M. (2005). Towards multifactor models of decision making and risk: Critique of prospect theory and related approaches, part one. Journal of Risk Finance, 6(2), 150–162.Google Scholar
  94. Nwogugu, M. (2006). A further critique of cumulative prospect theory and related approaches. Applied Mathematics and Computation, 179, 451–465.Google Scholar
  95. Nwogugu, M. (2017). Some biases and evolutionary homomorphisms implicit in the calculation of returns. In M. Nwogugu, Anomalies in net present value, returns and polynomials, and regret theory in decision making (Chapter 8). London: Palgrave Macmillan.Google Scholar
  96. Onatski, A. (2009). Testing hypotheses about the number of factors in large factor models. Econometrica, 77(5), 1487–1479.Google Scholar
  97. Overdahl, J., & McMillan, H. (1998). Another day, another collar: An evaluation of the effects of NYSE rule 80A on trading costs and intermarket arbitrage. Journal of Business, 71, 27–53.Google Scholar
  98. Parwada, J., & Yang, J. (2009). Information diffusion among international fund managers: Multicountry evidence. Financial Management, 38(4), 817–835.Google Scholar
  99. Pursiainen, H. (2005). Consistent aggregation methods and index number theory (Doctoral Dissertation). University of Helsinki, Faculty of Social Sciences, Department of Economics. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwj_ve-gnNPdAhXMFsAKHVJ6Ay0QFjAAegQIBBAC&url=https%3A%2F%2Fhelda.helsinki.fi%2Fbitstream%2Fhandle%2F10138%2F21825%2Fconsiste.pdf%3Fsequence%3D2&usg=AOvVaw1AL7J7JfbXdYw7SNo0kZge
  100. Ramsden, J. J. (2009). Impact factors: A critique. Journal of Biological Physics & Chemistry, 9, 139–140.Google Scholar
  101. Sabourin, P. (2012). Measurement bias in the Canadian consumer price index: An update. Bank of Canada Review. Available at: www.bankofcanada.ca
  102. Santoni, G. J., & Liu, T. (1993). Circuit breakers and stock market volatility. Journal of Futures Markets, 13(3), 261–277.Google Scholar
  103. Schultz, M. T. (2001). A critique of EPA’s index of watershed indicators. Journal of Environmental Management, 62(4), 429–442.Google Scholar
  104. Siegel, J. J., Schwartz, J., & Siracusano, L. (2007). The unique risk and return characteristics of dividend-weighted stock indexes (Working paper).Google Scholar
  105. Silver, M. (1982). Appropriate index number formulae for productivity measurement at the plant/organizational level. Managerial and Decision Economics, 3(4), 244–250.Google Scholar
  106. Stoll, H., & Whaley, R. (1997). Expiration-day effects of the all ordinaries share price index futures: Empirical evidence and alternative settlement procedures. Australian Journal of Management, 22, 139–174.Google Scholar
  107. Taylor, N. (2007). A new econometric model of index arbitrage. European Financial Management, 13(1), 159–183.Google Scholar
  108. Törnqvist, L., Vartia, P., & Vartia, Y. (1985). How should relative changes be measured? The American Statistician, 39(1), 43–46.Google Scholar
  109. Turkington, J., & Walsh, D. (1999). Price discovery and causality in the Australian share price index futures market. Australian Journal of Management, 24, 97–113.Google Scholar
  110. Tversky, A. (1975). A critique of expected utility theory: Descriptive and normative considerations. Erkenntnis, 9(2), 163–173.Google Scholar
  111. U.S. Department of Commerce/National Oceanic and Atmospheric Administration. (2003). Report on wind chill temperature and extreme heat indices: Evaluation and improvement projects. http://www.ofcm.gov/jagti/r19-ti-plan/pdf/00_opening.pdf
  112. Van Veelen, M., & Vander Weide, R. (2008). A note on different approaches to index number theory. The American Economic Review, 98(4), 1722–1730.Google Scholar
  113. Vartia, Y. (1976a). Relative changes and index numbers (Doctoral Dissertation in Statistics). The Research Institute of the Finnish Economy, Series A4, Helsinki. https://www.etla.fi/wp-content/uploads/A4.pdf
  114. Vartia, Y. (1976b). Ideal log-change index numbers. Scandinavian Journal of Statistics, 3, 121–126.Google Scholar
  115. Vartia, Y. (1978). Fisher’s five-tined fork and other quantum theories of index numbers. In W. Eichhorn, R. Henn, et al. (Eds.), Theory and applications of economic indices (pp. 271–295). Würzburg: Physica-Verlag.Google Scholar
  116. Vartia, Y. (1983). Efficient methods of measuring welfare change and compensated income in terms of ordinary demand functions. Econometrica, 51(1), 79–98.Google Scholar
  117. Vartia, Y. (2010, January). Principles of defining index numbers and constructing index series (University of Helsinki and HECER discussion paper no. 283).Google Scholar
  118. Vartia, Y., & Vartia, P. (1984). Descriptive index number theory and the Bank of Finland currency index. Scandinavian Journal of Economics, 86(3), 352–364.Google Scholar
  119. Vartia, Y., & Weymark, J. (1981). Four revealed preference tables. Scandinavian Journal of Economics, 83(3), 408–418.Google Scholar
  120. Von Der Lippe, P. (1999). A critique of international recommendations concerning price index formulas. Journal of Economics and Statistics, 218(3+4), 385–414.Google Scholar
  121. Von Der Lippe, P., & Diewert, E. (Eds.). (2010). Index number theory and price statistics. Berlin/Boston: DeGruyter.Google Scholar
  122. Vrijdags, A. (2009). An experimental investigation of transitivity in set ranking. Theory and Decision, 68(1–2), 213–232.Google Scholar
  123. Wagner, N., & Stocker, E. (2009). A new family of equity style indices (Working paper).Google Scholar
  124. White, A. (2007). Biases in consumer price indexes. International Statistical Review, 67(3).Google Scholar
  125. Zhao, X. E. (2008). Determinants of flows into retail international equity funds. Journal of International Business Studies, 39(7), 1169–1177.Google Scholar
  126. Zitzewitz, E. (2003). Who cares about shareholders? Arbitrage-proofing mutual funds. Journal of Law, Economics and Organization, 19(2), 245–280.Google Scholar

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Authors and Affiliations

  • Michael I. C. Nwogugu
    • 1
  1. 1.EnuguNigeria

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