An alternative to portfolio selection problem beyond Markowitz’s: Log Optimal Growth Portfolio

Muteba Mwamba, John and Suteni, Mwambi (2010): An alternative to portfolio selection problem beyond Markowitz’s: Log Optimal Growth Portfolio.

Abstract

This paper constructs an alternative investment strategy to portfolio optimization model in the framework of the Mean–Variance portfolio selection model. To differentiate it from the ubiquitously applied Mean–Variance model, which is constructed on an assumption that returns are normally distributed, our model makes two assumptions: Firstly, that asset prices follow a Geometric Brownian Motion and that secondly asset prices are Log-normally distributed meaning that continuously compounded returns are normally distributed. The traditional Mean–Variance optimization approach has only one objective, which fails to capture the stochastic nature of asset returns and their correlations. This paper presents an alternative approach to the portfolio selection problem. The proposed optimization model which is an optimal portfolio strategy is produced for investors of various risk tolerance, taking into account the stochastic nature of the returns. Detailed analysis based on log– optimal growth optimization and the application of the model are provided and compared to the standard Mean–Variance approach.

• Item Type: MPRA Paper
• Language: English
• Keywords: Portfolio selection, Kelly criteria, mean variance, optimization
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; G - Financial Economics > G1 - General Financial Markets; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Bachelier, L. (1900). The theory of speculation, (published 1900), Unpublished xx, xx.

Benoulli, D. (1954), ‘Exposition of a new theory on the investment of risk’, Econometrica. 22(1), 23–36.

Black, F. (1993), ‘Estimating expected return’, Financial Analysts Journal, 49, 36–38.

Black, F. & Scholes, M. (1973), ‘The pricing of options and corporate liabilities’, Journal of Political Economy 81, 637–659.

Breiman, L. (1961), ‘Optimal gambling systems for favourable games’, Proceedings of the 4th Berkeley Symposium on mathematical Statistics and Probability. 1, 65–78.

Campbell, J. Y., Lo, A. W. & MacKinlay, A. C. (1997), The Econometrics of Financial Markets, Princeton University Press, United Kingdom.

Christensen, C. M. (2005), ‘On the history of the growth optimal portfolio’, Unpublished. pp. 1–65.

Cremers, Jan-Hein, K. M. & Page, S. (2005), ‘Optimal hedge fund allocations: Do higher moments matter?’, The Journal of Portfolio Management. 5, 1–5.

Elton, E. J. & Gruber, M. J. (1974a), ‘On the optimality of some multi-period selection criteria’, Journal of Business 47, 231–243.

Elton, E. J. & Gruber, M. J. (1974b), ‘On the maximization of the geometric mean with lognormal return distribution’, Management Science 21, 483–488.

Elton, E. J., Gruber, M. J., Brown, S. J. & Goetmann, W. N. (2003), Modern portfolio Theory and Investment Analysis, John Wiley and Sons, New York.

Estrada, J. (2010), ‘Geometric mean maximization: an overlooked portfolio approach’, Unpublished: IESE Business School. pp. 1–31.

Fernholz, R. & Shay, B. (1982), ‘Stochastic portfolio theory and stock market equilibrium’, Journal of Finance 27, 615–624.

Hakansson, N. H. (1971), ‘Multi-period mean-variance analysis: Towards a general theory of portfolio choice’, Journal of Finance. 26, 857–884.

Hakansson, N. & Ziemba, W. (1995), Capital Growth Theory: Handbooks in Operations Research and Management Science, Oxford University Press, New York.

Hunt, B. F. (2000), ‘Growth optimal portfolios: their structure and nature’, Unpublished University of Technology Sydney. pp. 1–24.

Hunt, B. F. (2002), ‘Growth optimal investment strategy efficacy: An application on long run Australian equity data’, Unpublished: University of Technology Sydney. pp. 1–20.

Hunt, B. F. (2005b), ‘Feasible high growth investment strategy’, Journal of Asset Management. 6, 141 – 157.

Kelly, J. (1956), ‘A new interpretation of information rate’, Bell System Techn. Journal 3. 35, 917–926.

Laten, H. (1959), ‘Criteria for choice among risky ventures’, The Journal of political Economy. 38, 145 – 55.

Le, T. & Platen, E. (2006), ‘Approximating the growth optimal portfolio with a diversified world stock index’, Journal of Risk Finance. 7, 559–574.

Lintner, J. (1965), ‘The valuation of risk assets and selection of risky investments in stock portfolios and capital budgets’, Review of Economics and Statistics. 47, 13–37.

Long, J. (1990), ‘The numeraire portfolio’, The Journal of Financial Economics. 26, 29–69.

Luenberger, D. (1998), Investment Science, Oxford University Press, New York.

Markowitz, H. (1952), ‘Mean-variance analysis in portfolio choice and capital markets’, Journal of Finance. 7(1), 77–91.

Markowitz, H. (1959), Portfolio Selection. Efficient Diversification of Investments, John Wiley Sons Ltd, New York.

Markowitz, H. (1976), ‘Investment for the long run: new evidence for an old rule’, Journal of Finance. XXXI, 1279–86.

Merton, R. (1980), ‘On estimating the expected return on the market’, The Journal financial Economics. 8, 323–361.

Mossin, J. (1966), ‘Equilibrium in capital markets’, Econometrica. 35, 768–783.

Norstad, J. (2005), ‘Portfolio optimization unconstrained portfolios’, Unpublished Part1.

Osborne, M. (1959), ‘Brownian motion in stock market’, Operations Research 7, 145–73.

Palczewski, A. (2005), ‘Portfolio optimization a practical approach unconstrained portfolios’, Unpublished: Institute of applied Mathematics Warsaw University Part1.

Platen, E. (2005), ‘On the role of the growth optimal portfolio in finance.’ Australian Economic Papers. 44, 365–388.

Roll, R. (1973), ‘Evidence on the growth optimum model’, Journal of Science XXVIII, 551–566.

Samuelson, P. A. (1971), ‘The fallacy of maximizing the geometric mean in long sequences of investing or gambling’, Proceedings National Acad. Sciences USA 68, 2493–2496.

Sharpe, W. F. (1964), ‘Capital assets prices: a theory of market equilibrium under conditions’, Journal of Finance. 19,, 425–442.

Tim, A. & Kritzman, M. (2007), ‘Mean-variance versus full-scale optimization’, The Journal of Asset Management. 7, 1–5.

Tobin, J. (1958), ‘Liquidity preference as behavior towards risk’, Review of Economic Studies. 25), 65–86.

Weide, J., Peterson, D. & Maier, S. (1977), ‘A strategy which maximizes the geometric mean return on portfolio investments’, Management Science 21.

William, J. B. (1936), ‘Speculation and the carryover.’, The Quarterly Journal of Economics. 50, 436 – 455.

Wilmott, P. (2001), Introduction to Quantitative Finance, John Wiley Sons Ltd, New York.