# A Note on CCMV Portfolio Optimization Model with Short Selling and Risk-neutral Interest Rate

### Abstract

In this paper, first we present some drawbacks of the cardinality constrained mean-variance (CCMV) portfolio optimization with short selling and risk-neutral interest rate when the lower and upper bounds of the assets contributions are -1/K and 1/K(K denotes the number of assets in portfolio). Then, we present an improved variant using absolute returns instead of the returns to include short selling in the model. Finally, some numerical results are provided using the data set of the S&P 500 index, Information Technology, and the MIBTEL index in terms of returns and Sharpe ratios to compare the proposed models with those in the literature.### References

D. Bertsimas and R. Shioda, Algorithm for cardinality-constrained quadratic optimization, Computational Optimization and Applications 43 (2009), pp. 1–22.

T.R. Bielecki and I. Jang, Portfolio optimization with a defaultable security, Asia-Paciﬁc Financial Markets 13 (2006), pp. 113–127.

D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming 74 (1996), pp. 121–140.

J.R.Birgeand R.Q.Zhang, Risk-neutral option pricing methods for adjusting constrained cash ﬂows,The Engineering Economist44 (1999), pp. 36–49.

T.J. Chang, N. Meade, J.E. Beasley, and Y.M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers & Operations Research 27 (2000), pp. 1271–1302.

Z. Dai and F. Wen, A generalized approach to sparse and stable portfolio optimization problem, The Journal of Industrial & Management Optimization 14 (2018), pp. 1651–1666.

V. DeMiguel, L. Garlappi, and R. Uppal, Optimal versus naive diversiﬁcation: How inefﬁcient is the 1/n portfolio strategy?, The Review of Financial Studies 22 (2007), pp. 1915–1953.

U. Derigs and N.H. Nickel, Meta-heuristic based decision support for portfolio optimization with a case study on tracking error minimization in passive portfolio management, OR Spectrum 25 (2003), pp. 345–378.

G. Dueck and P. Winker, New concepts and algorithms for portfolio choice, Applied Stochastic Models and Data Analysis 8 (1992), pp. 159–178.

P.A. Frost and J.E. Savarino, For better performance: Constrain portfolio weights, The Journal of Portfolio Management 15 (1988), p. 29.

J. Gao and D. Li, Cardinality constrained linear-quadratic optimal control, IEEE Transactions on Automatic Control 56 (2011), pp. 1936–1941.

J. Gao and D. Li, Optimal cardinality constrained portfolio selection, Operations Research 61 (2013), pp. 745–761.

M. Grant, S. Boyd, and Y. Ye, Cvx: Matlab software for disciplined convex programming, version 2.0 beta (2013).

B.I. Jacobs, K.N. Levy, and H.M. Markowitz, Portfolio optimization with factors, scenarios, and realistic short positions, Operations Research 53 (2005), pp. 586–599.

B.I. Jacobs, K.N. Levy, and H.M. Markowitz, Trimability and fast optimization of long-short portfolios, Financial Analysts Journal 62 (2006), pp. 36–46.

R. Jagannathan and T. Ma, Risk reduction in large portfolios: Why imposing the wrong constraints helps, The Journal of Finance 58 (2003), pp. 1651–1683.

J.D. Jobson and B. Korkie, Estimation for markowitz efﬁcient portfolios, Journal of the American Statistical Association 75 (1980), pp. 544–554.

C.C. Kwan, Portfolio selection under institutional procedures for short selling: Normative and market-equilibrium considerations, The Journal of Banking & Finance 21 (1997), pp. 369–391.

D. Li and W.L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance 10 (2000), pp. 387–406.

D. Li, X. Sun, and J. Wang, Optimal lot solution to cardinality constrained mean variance formulation for portfolio selection, Mathematical Finance 16 (2006), pp. 83–101.

J. Lintner, Security prices, risk, and maximal gains from diversiﬁcation, The Journal of Finance 20 (1965), pp. 587–615.

D. Maringer, Portfolio Management With Heuristic Optimization, Springer Science & Business Media, 2006.

D. Maringer and H. Kellerer, Optimization of cardinality constrained portfolios with a hybrid local search algorithm, OR Spectrum 25 (2003), pp. 481–495.

H. Markowitz, Portfolio selection, The Journal of Finance 7 (1952), pp. 77–91.

A.A. Najafi and H. reza Ghasemi, Portfolio optimization in terms of justifiability short selling and some market practical constraints,Financial Research Journal 14 (2013), pp. 117–132.

N.R. Patel and M.G. Subrahmanyam, A simple algorithm for optimal portfolio selection with fixed transaction costs, Management Science 28 (1982), pp. 303–314.

G.C. Pflug, A. Pichler, and D. Wozabal, The 1/n investment strategy is optimal under high model ambiguity, Journal of Banking & Finance 36 (2012), pp. 410–417.

D.X. Shaw, S. Liu, and L. Kopman, Lagrangian relaxation procedure for cardinality-constrained portfolio optimization, Optimization Methods & Software 23 (2008), pp. 411–420.

J. Xie, S. He, and S. Zhang, Randomized portfolio selection with constraints, Pacific Journal of Optimization 4 (2008), pp. 89–112.

S. Zhang, S. Wang, and X. Deng, Portfolio selection theory with different interest rates for borrowing and leading, Journal of Global Optimization 28 (2004), pp. 67–95.

*Statistics, Optimization & Information Computing*,

*8*(3), 740-748. https://doi.org/10.19139/soic-2310-5070-890

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).