# On the use of conditional expectation in portfolio selection problems

- 94 Downloads

## Abstract

In this paper, we examine the use of conditional expectation, either to reduce the dimensionality of large-scale portfolio problems or to propose alternative reward–risk performance measures. In particular, we focus on two financial problems. In the first part, we discuss and examine correlation measures (based on a conditional expectation) used to approximate the returns in large-scale portfolio problems. Then, we compare the impact of alternative return approximation methodologies on the ex-post wealth of a classic portfolio strategy. In this context, we show that correlation measures that use the conditional expectation perform better than the classic measures do. Moreover, the correlation measure typically used for returns in the domain of attraction of a stable law works better than the classic Pearson correlation does. In the second part, we propose new performance measures based on a conditional expectation that take into account the heavy tails of the return distributions. Then, we examine portfolio strategies based on optimizing the proposed performance measures. In particular, we compare the ex-post wealth obtained from applying the portfolio strategies, which use alternative performance measures based on a conditional expectation. In doing so, we propose an alternative use of conditional expectation in various portfolio problems.

## Keywords

Conditional expectation Large-scale portfolio selection Performance measures Return approximation Heavy tailed distribution## Notes

### Acknowledgements

This paper was supported by the Italian funds MURST 2016/2017 and by STARS Supporting Talented Research—Action 1—2017. The research was also supported by the Czech Science Foundation (GACR) under Project 17-19981S, and by VSB-TU Ostrava under the SGS Project SP2018/34.

## References

- Ait-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices.
*The Journal of Finance*,*53*(2), 499–547.CrossRefGoogle Scholar - Angelelli, E., & Ortobelli, S. (2009). American and European portfolio selection strategies: The Markovian approach. In P. N. Catlere (Ed.),
*Financial hedging*(pp. 119–152). New York: Nova Science.Google Scholar - Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk.
*Mathematical Finance*,*9*(3), 203–228.CrossRefGoogle Scholar - Backus, D., Foresi, S., & Zin, S. (1998). Arbitrage opportunities in arbitrage-free models of bond pricing.
*Journal of Business and Economic Statistics*,*16*, 13–24.Google Scholar - Bertocchi, M., Consigli, G., D’Ecclesia, R., Giacometti, R., Moriggia, V., & Ortobelli, L. S. (2013).
*Euro bonds: Markets, infrastructure and trends*. Singapore: World Scientific.CrossRefGoogle Scholar - Biglova, A., Ortobelli, S., & Fabozzi, F. (2014). Portfolio selection in the presence of systemic risk.
*The Journal of Asset Management*,*15*, 285–299.CrossRefGoogle Scholar - Biglova, A., Ortobelli, S., Rachev, S., & Stoyanov, S. (2004). Different approaches to risk estimation in portfolio theory.
*Journal of Portfolio Management*,*31*(1), 103–112.CrossRefGoogle Scholar - Boente, G., & Fraiman, R. (1989). Robust nonparametric regression estimation.
*Journal of Multivariate Analysis*,*29*, 180–198.CrossRefGoogle Scholar - Bowman, A. W., & Azzalini, A. (1997).
*Applied smoothing techniques for data analysis*. London: Oxford University Press.Google Scholar - Brockwell, P., & Davis, R. (1998).
*Time series: Theory and methods*. New York: Springer.Google Scholar - Chamberlain, G. (1983). A characterization of the distributions that imply mean-variance utility functions.
*Journal of Economics Theory*,*29*, 185–201.CrossRefGoogle Scholar - Cherubini, U., Luciano, E., & Vecchiato, W. (2004).
*Copula methods in finance*. Chichester: Wiley.CrossRefGoogle Scholar - Coqueret, G., & Milhau, V. (2014).
*Estimating covariance matrices for portfolio optimization*. ERI Scientific Beta White Paper.Google Scholar - Daly, J., Crane, M., & Ruskin, H. (2008). Random matrix theory filters in portfolio optimization: A stability and risk assessment.
*Physica A: Statistical Mechanics and its Applications*,*387*(16–17), 4248–4260.CrossRefGoogle Scholar - Davidson, R., & Jean-Yves, D. (2000). Statistical inference for stochastic dominance and for the measurement of poverty and inequality.
*Econometrica*,*68*(6), 1435–1464.CrossRefGoogle Scholar - Dehong, L., Hongmei, G., & Tiancai, X. (2016). The meltdown of the Chinese equity market in the summer of 2015.
*International Review of Economics & Finance*,*45*, 504–517.CrossRefGoogle Scholar - DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?
*Review of Financial Studies*,*22*, 1915–1953.CrossRefGoogle Scholar - Fabozzi, F. J., Dashan, H., & Guofu, Z. (2010). Robust portfolios: Contributions from operations research and finance.
*Annals of Operations Research*,*176*(1), 191–220.CrossRefGoogle Scholar - Fama, E. (1965). The behavior of stock market prices.
*Journal of Business*,*38*, 34–105.CrossRefGoogle Scholar - Fan, J., Hu, T. C., & Truong, Y. K. (1994). Robust non-parametric function estimation.
*Scandinavian Journal of statistics*,*21*(4), 433–446.Google Scholar - Farinelli, S., Ferreira, M., & Rossello, D. (2008). Beyond Sharpe ratio: Optimal asset allocation using different performance ratios.
*Journal of Banking & Finance*,*32*(10), 2057–2063.CrossRefGoogle Scholar - Georgiev, K., Kim, Y. S., & Stoyanov, S. (2015). Periodic portfolio revision with transaction costs.
*Mathematical Methods of Operations Research*,*81*, 337–359.CrossRefGoogle Scholar - Härdle, W., & Müller, M. (2000). Multivariate and semiparametric kernel regression. In M. G. Schimek (Ed.),
*Smoothing and regression: Approaches, computation, and application*(pp. 357–392). New York: Wiley.Google Scholar - Ingersoll, J. E. (1987).
*Theory of financial decision making*. Rowman & Littlefield Studies in Financial Economics.Google Scholar - Jones, M. C., Marron, J. S., & Sheather, J. S. (1996). A brief survey of bandwidth selection for density estimation.
*Journal of the American Statistical Association*,*91*(433), 401–407.CrossRefGoogle Scholar - Kan, R., & Zhou, G. (2007). Optimal portfolio choice with parameter uncertainty.
*Journal of Financial and Quantitative Analysis*,*42*(3), 621–656.CrossRefGoogle Scholar - Kondor, I., Pafka, S., & Nagy, G. (2007). Noise sensitivity of portfolio selection under various risk measures.
*Journal of Banking and Finance*,*31*, 1545–1573.CrossRefGoogle Scholar - Ledoit, O., & Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection.
*Journal of Empirical Finance*,*10*, 603–621.CrossRefGoogle Scholar - Ledoit, O., Wolf, M., & Honey, I. (2004). shrunk the covariance matrix.
*Journal of Portfolio Management*,*30*(4), 110–119.CrossRefGoogle Scholar - Levy, H., & Markowitz, H. M. (1979). Approximating expected utility by a function of mean and variance.
*American Economic Review*,*69*, 308–317.Google Scholar - Mandelbrot, B. (1963). The variation of certain speculative prices.
*Journal of Business*,*26*, 394–419.CrossRefGoogle Scholar - Markowitz, H. M. (1952). The utility of wealth.
*Journal of Political Economy*,*60*, 151–158.CrossRefGoogle Scholar - Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006).
*Robust statistics: Theory and methods*., Wiley series in probability and statistics Chichester: Wiley.CrossRefGoogle Scholar - Martin, D., Rachev, S., & Siboulet, F. (2003). Phi-alpha optimal portfolios and extreme risk management.
*Wilmott Magazine of Finance*,*2003*(6), 70–83.CrossRefGoogle Scholar - Müller, A., & Stoyan, D. (2002).
*Comparison methods for stochastic models and risks*. New York: Wiley.Google Scholar - Nadaraya, E. A. (1964). On estimating regression.
*Theory of Probability and its Applications*,*9*(1), 141–142.CrossRefGoogle Scholar - Nolan, J. P., & Ojeda-Revah, D. (2013). Linear and nonlinear regression with stable errors.
*Journal of Econometrics*,*172*(2), 186–194.CrossRefGoogle Scholar - Ortobelli, S., & Lando, T. (2015). Independence tests based on the conditional expectation.
*WSEAS Transactions on Mathematics*,*14*, 335–344.Google Scholar - Ortobelli, S., Petronio, F., & Lando, T. (2017). A portfolio return definition coherent with the investors preferences.
*IMA Journal of Management Mathematics*,*28*(3), 451–466.Google Scholar - Ortobelli, S., & Tichý, T. (2015). On the impact of semidefinite positive correlation measures in portfolio theory.
*Annals of Operations Research*,*235*(1), 625–652.CrossRefGoogle Scholar - Owen, J., & Rabinovitch, R. (1983). On the class of elliptical distributions and their applications to the theory of portfolio choice.
*Journal of Finance*,*38*, 745–752.CrossRefGoogle Scholar - Papp, G., Pafka, S., Nowak, M. A., & Kondor, I. (2005). Random matrix filtering in portfolio optimization.
*ACTA Physica Polonica B*,*36*, 2757–2765.Google Scholar - Rachev, S. T., Menn, C., & Fabozzi, F. J. (2005).
*Fat-tailed and skewed asset return distributions: Implications for risk management, portfolio selection, and option pricing*. New York: Wiley.Google Scholar - Rachev, S. T., & Mittnik, S. (2000).
*Stable paretian models in finance*. Chichester: Wiley.Google Scholar - Rachev, S., Ortobelli, S., Stoyanov, S., Fabozzi, F., & Biglova, A. (2008). Desirable properties of an ideal risk measure in portfolio theory.
*International Journal of Theoretical and Applied Finance*,*11*(1), 19–54.CrossRefGoogle Scholar - Ross, S. (1978). Mutual fund separation in financial theory-the separating distributions.
*Journal of Economic Theory*,*17*, 254–286.CrossRefGoogle Scholar - Ruppert, D., & Wand, M. P. (1994). Multivariate locally weighted last squares regression.
*The Annals of Statistics*,*22*(3), 1346–1370.CrossRefGoogle Scholar - Samorodnitsky, G., & Taqqu, M. S. (1994).
*Stable non-Gaussian random processes: Stochastic models with infinite variance*(Vol. 1). Boca Raton: CRC Press.Google Scholar - Schoutens, W. (2003).
*Levy processes in finance*. New York: Wiley.CrossRefGoogle Scholar - Scott, D. W. (2015).
*Multivariate density estimation: Theory, practice, and visualization*. New York: Wiley.Google Scholar - Sharpe, W. F. (1994). The sharpe ratio.
*Journal of Portfolio Management, Fall**21*, 45–58.Google Scholar - Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk.
*The Journal of Finance*,*52*(5), 1973–2002.CrossRefGoogle Scholar - Statman, M. (2004). The diversification puzzle.
*Financial Analysts Journal*,*60*(4), 44–53.CrossRefGoogle Scholar - Stoyanov, S., Rachev, S., & Fabozzi, F. (2007). Optimal financial portfolios.
*Applied Mathematical Finance*,*14*(5), 401–436.CrossRefGoogle Scholar - Szegö, G. (2004).
*Risk measures for the 21st century*. Chichester: Wiley.Google Scholar - Tobin, J. (1958). Liquidity preference as behavior towards risk.
*Review of Economic Studies*,*25*, 65–86.CrossRefGoogle Scholar