Skip to main content
SpringerLink
Log in

Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

  • Chapter
Book cover Handbook of Quantitative Finance and Risk Management

Abstract

In this chapter, we first introduce utility function and indifference curve. Based on utility theory, we derive the Markowitz’s model and the efficient frontier through the creation of efficient portfolios of varying risk and return. We also include methods of solving for the efficient frontier both graphically and mathematically, with and without explicitly incorporating short selling.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 869.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,099.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A cardinal utility implies that a consumer is capable of assigning to every commodity or combination of commodities a number representing the amount or degree of utility associated with it. An ordinary utility implies that a consumer needs not be able to assign numbers that represent (in arbitrary unit) the degree or amount of utility associated with commodity or combination of commodity. The consumer can only rank and order the amount or degree of utility associated with commodity.

  2. 2.

    Technically, these conditions can be represented mathematically by ∂U ∕ ∂μ w > 0 and ∂U ∕ ∂σ w < 0.

  3. 3.

    Following Fama (1970) and Alexander and Francis (1986:177), a fair game means that the expected returns, given information set θ, equal the expected returns without the information set. Note that this does not mean the expected returns are zero or positive – they could be negative.

  4. 4.

    By definition, an indifference curve shows all combinations of products (investments) A and B that will yield same level of satisfaction or utility to consume. This kind of analysis is based upon ordinal rather than cardinal utility theory.

  5. 5.

    Most texts do not identify the Markowitz model with restrictions on short sale. Markowitz (1952), in fact, excluded short sales.

  6. 6.

    See Baumol (1963).

  7. 7.

    The process of finding the efficient frontier graphically described in this section was originally developed by Markowitz (1952). Francis and Archer (1979) have discussed this subject in detail.

  8. 8.

    In Fig. 5.15a and b the vertical axis is defined as \(({\overline{R}}_{p} - {R}_{f})/{\sigma}_{p}\), which is the Sharpe performance measure as defined in Chap. 4. \(({\overline{R}}_{p} - {R}_{f})/{\sigma}_{p}\) is the objective function for portfolio optimization; it will be explored in Chap. 8.

References

  • Alexander, G. J. and J. C. Francis. 1986. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Baumol, W. J. 1963. “An expected gain-confidence limit criterion for portfolio selection.” Management Science 10, 171–182.

    Article  Google Scholar 

  • Bertsekas, D. 1974. “Necessary and sufficient conditions for existence of an optimal portfolio.” Journal of Economic Theory 8, 235–247.

    Article  Google Scholar 

  • Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–455.

    Article  Google Scholar 

  • Blume, M. 1970. “Portfolio theory: a step toward its practical application.” Journal of Business 43, 152–173.

    Article  Google Scholar 

  • Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York.

    Google Scholar 

  • Brealey, R. A. and S. D. Hodges. 1975. “Playing with portfolios.” Journal of Finance 30, 125–134.

    Article  Google Scholar 

  • Breen, W. and R. Jackson. 1971. “An efficient algorithm for solving large-scale portfolio problems.” Journal of Financial and Quantitative Analysis 6, 627–637.

    Article  Google Scholar 

  • Brennan, M. J. 1975. “The optimal number of securities in a risky asset portfolio where there are fixed costs of transaction: theory and some empirical results.” Journal of Financial and Quantitative Analysis 10, 483–496.

    Article  Google Scholar 

  • Cohen, K. and J. Pogue. 1967. “An empirical evaluation of alternative portfolio-selection models.” Journal of Business 46, 166–193.

    Google Scholar 

  • Dyl, E. A. 1975. “Negative betas: the attractions of selling short.” Journal of Portfolio Management 1, 74–76.

    Article  Google Scholar 

  • Elton, E. J. and M. Gruber. 1974. “Portfolio theory when investment relatives are log normally distributed.” Journal of Finance 29, 1265–1273.

    Article  Google Scholar 

  • Elton, E. J. and M. Gruber. 1978. “Simple criteria for optimal portfolio selection: tracing out the efficient frontier.” Journal of Finance 13, 296–302.

    Article  Google Scholar 

  • Elton, E. J., M. Gruber, and M. E. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 11, 1341–1357.

    Article  Google Scholar 

  • Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York.

    Google Scholar 

  • Evans, J. and S. Archer. 1968. “Diversification and the reduction of dispersion: an empirical analysis.” Journal of Finance 3, 761–767.

    Google Scholar 

  • Fama, E. F. 1970. “Efficient capital markets: a review of theory and empirical work.” Journal of Finance 25, 383–417.

    Article  Google Scholar 

  • Feller, W. 1968. An introduction to probability theory and its application, Vol. 1, Wiley, New York.

    Google Scholar 

  • Francis, J. C. and S. H. Archer. 1979. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Gressis, N., G. Philiippatos, and J. Hayya. 1976. “Multiperiod portfolio analysis and the inefficiencies of the market portfolio.” Journal of Finance 31, 1115–1126.

    Article  Google Scholar 

  • Henderson, J. and R. Quandt. 1980. Microeconomic theory: a mathematical approach, 3d Edition, McGraw-Hill, New York.

    Google Scholar 

  • Lee, C. F. and A. C. Lee 2006. Encyclopedia of finance, Springer, New York.

    Book  Google Scholar 

  • Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore.

    Google Scholar 

  • Levy, H. and M. Sarnat. 1971. “A note on portfolio selection and investors’ wealth.” Journal of Financial and Quantitative Analysis 6, 639–642.

    Article  Google Scholar 

  • Lewis, A. L. 1988. “A simple algorithm for the portfolio selection problem.” Journal of Finance 43, 71–82.

    Article  Google Scholar 

  • Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolio and capital budgets.” Review of Economics and Statistics 47, 13–27.

    Article  Google Scholar 

  • Mao, J. C. F. 1969. Quantitative analysis of financial decisions, Macmillan, New York.

    Google Scholar 

  • Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd ed., Wiley, New Jersey.

    Google Scholar 

  • Markowitz, H. M. 1952. “Portfolio selection.” Journal of Finance 1, 77–91.

    Google Scholar 

  • Markowitz, H. M. 1959. Portfolio selection. Cowles Foundation Monograph 16, Wiley, New York.

    Google Scholar 

  • Markowitz, H. M.1976. “Markowitz revisited.” Financial Analysts Journal 32, 47–52.

    Article  Google Scholar 

  • Markowitz, H. M. 1987. Mean-variance analysis in portfolio choice and capital markets. Blackwell, Oxford.

    Google Scholar 

  • Martin, A. D., Jr. 1955. “Mathematical programming of portfolio selections.” Management Science 1, 152–166.

    Article  Google Scholar 

  • Merton, R. 1972. “An analytical derivation of efficient portfolio frontier.” Journal of Financial and Quantitative Analysis 7, 1851–1872.

    Article  Google Scholar 

  • Mossin, J. 1968. “Optimal multiperiod portfolio policies.” Journal of Business 41, 215–229.

    Article  Google Scholar 

  • Ross, S. A. 1970. “On the general validity of the mean-variance approach in large markets,” in Financial economics: essays in honor of Paul Cootner, W. F. Sharpe and C. M. Cootner (Eds.). Prentice Hall, Englewood Cliffs, NJ, pp. 52–84.

    Google Scholar 

  • Sharpe, W. F. 1970. Portfolio theory and capital markets, McGraw-Hill, New York.

    Google Scholar 

  • Von Neumann, J. and O. Morgenstern. 1947. Theory of games and economic behavior, 2nd Edition, Princeton University Press, Princeton, NJ.

    Google Scholar 

  • Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rutgers University, New Brunswick, NJ, USA

    Cheng-Few Lee & Hong-Yi Chen

  2. University of Illinois, Urbana-Champaign, IL, USA

    Joseph E. Finnerty

Authors
  1. Cheng-Few Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph E. Finnerty
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong-Yi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-Yi Chen .

Editor information

Editors and Affiliations

  1. Department of Finance and Economics, Rutgers University, 94 Rockafeller Road, Janice H. Levin Bldg., New Brunswick, NJ, 08854-8054, USA

    Cheng-Few Lee

  2. State Street Corp., Boston, MA, 94132, USA

    Alice C. Lee

  3. Center for PBBEF Research, North Brunswick, NJ, USA

    John Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Lee, CF., Finnerty, J.E., Chen, HY. (2010). Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation