### Online Portfolio Optimization with Risk Control

#### Abstract

Portfolio selection is undoubtedly one of the most challenging topics in the area of finance. Since Markowitz's initial contribution in 1952, portfolio allocation strategies have been intensely discussed in the literature. With the development of online optimization techniques, dynamic learning algorithms have proven to be an effective approach to building portfolios, although they do not assess the risk related to each investment decision.

In this work, we compared the performance of the Online Gradient Descent (OGD) algorithm and a modification of the method, that takes into account risk metrics controlling for the Beta of the portfolio. In order to control for the Beta, each asset was modeled using the CAPM model and a time-varying Beta that follows a random walk. We compared both the traditional OGD algorithm and the OGD with Beta constraints with the Uniform Constant Rebalanced Portfolio and two different indexes for the Brazilian market, composed of small caps and the assets that belong to the Ibovespa index. Controlling the Beta proved to be an efficient strategy when the investor chooses an appropriate interval for the beta during bull markets or bear markets. Moreover, the time-varying beta was an efficient metric to force the desired correlation with the market and also to reduce the volatility of the portfolio during bear markets.

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PDF#### References

H. Markowitz, “Portfolio selection,” The journal of finance, vol. 7, no. 1, pp. 77–91, 1952.

T. M. Cover, “Universal portfolios,” Mathematical finance, vol. 1, no. 1, pp. 1–29, 1991.

B. Li and S. C. Hoi, “Online portfolio selection: A survey,” ACM Computing Surveys (CSUR), vol. 46, no. 3, p. 35, 2014.

R. Dochow, Online algorithms for the portfolio selection problem. Springer, 2016.

J. L. Kelly, “A new interpretation of information rate,” Bell Labs Technical

Journal, vol. 35, no. 4, pp. 917–926, 1956.

R. M. Bell and T. M. Cover, “Competitive optimality of logarithmic investment,” Mathematics of Operations Research, vol. 5, no. 2, pp. 161–166, 1980.

R. Bell and T. M. Cover, “Game-theoretic optimal portfolios,” Management Science, vol. 34, no. 6, pp. 724–733, 1988.

P. H. Algoet and T. M. Cover, “Asymptotic optimality and asymptotic

equipartition properties of log-optimum investment,” The Annals of Probability, pp. 876–898, 1988.

E. Hazan et al., “Introduction to online convex optimization,” Foundations and Trends in Optimization, vol. 2, no. 3-4, pp. 157–325, 2016.

T. M. Cover and E. Ordentlich, “Universal portfolios with side information,” IEEE Transactions on Information Theory, vol. 42, no. 2, pp. 348–363, 1996.

D. P. Helmbold, R. E. Schapire, Y. Singer, and M. K. Warmuth, “On-line

portfolio selection using multiplicative updates,” Mathematical Finance, vol. 8, no. 4, pp. 325–347, 1998.

A. A. Gaivoronski and F. Stella, “Stochastic nonstationary optimization for finding universal portfolios,” Annals of Operations Research, vol. 100, no. 1, pp. 165–188, 2000.

M. Zinkevich, “Online convex programming and generalized infinitesimal gradient ascent,” in Proceedings of the 20th International Conference on Machine Learning (ICML-03), pp. 928–936, 2003.

A. Agarwal, E. Hazan, S. Kale, and R. E. Schapire, “Algorithms for portfolio management based on the newton method,” in Proceedings of the 23rd international conference on Machine learning, pp. 9–16, ACM, 2006.

E. Hazan, “The convex optimization approach to regret minimization,” Optimization for machine learning, p. 287, 2012.

H. Luo, C.-Y. Wei, and K. Zheng, “Efficient online portfolio with logarithmic regret,” in Advances in Neural Information Processing Systems, pp. 8235–8245, 2018.

P. Das, N. Johnson, and A. Banerjee, “Online lazy updates for portfolio selection with transaction costs,” in Twenty-Seventh AAAI Conference on Artificial Intelligence, 2013.

B. Li, J. Wang, D. Huang, and S. C. Hoi, “Transaction cost optimization for online portfolio selection,” Quantitative Finance, vol. 18, no. 8, pp. 1411–1424, 2018.

Y. Ha and H. Zhang, “Liquidity risks, transaction costs and online portfolio selection,” Transaction costs and Online Portfolio Selection (April 26, 2019), 2019.

W. F. Sharpe, “Capital asset prices: A theory of market equilibrium under conditions of risk,” The journal of finance, vol. 19, no. 3, pp. 425–442, 1964.

J. Lintner, “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets,” The review of economics and statistics, pp. 13–37, 1965.

J. L. Treynor, “How to rate management of investment funds,” Harvard business review, vol. 43, no. 1, pp. 63–75, 1965.

J. Mossin, “Equilibrium in a capital asset market,” Econometrica: Journal of the econometric society, pp. 768–783, 1966.

R. Carmona, Statistical analysis of financial data in R, vol. 2. Springer, 2014.

J. Durbin and S. J. Koopman, Time series analysis by state space methods, vol. 38. OUP Oxford, 2012.

P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical finance, vol. 9, no. 3, pp. 203–228, 1999.

C. Acerbi and D. Tasche, “Expected shortfall: a natural coherent alternative to value at risk,” Economic notes, vol. 31, no. 2, pp. 379–388, 2002.

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