References

[1]
D. Cajas. Entropy Pooling with CVaR and EVaR Views. Available at SSRN 7120258 (2026). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
[2]
G. P. Brinson and N. Fachler. Measuring non-US. equity portfolio performance. The Journal of Portfolio Management 11, 73–76 (1985). 1 2 3
[3]
[4]
[5]
N. J. Higham. Computing the nearest correlation matrix—a problem from finance. IMA Journal of Numerical Analysis 22, 329–343 (2002). 1
[6]
H. Qi and D. Sun. A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM Journal on Matrix Analysis and Applications 28, 360–385 (2006). 1
[7]
M. M. De Prado. Machine learning for asset managers (Cambridge University Press, 2020). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
[8]
V. A. Marčenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik 1, 457 (1967). 1 2 3 4 5 6 7 8 9 10 11
[9]
D. H. Bailey and M. Lopez de Prado. The Sharpe ratio efficient frontier. Journal of Risk 15, 3–44 (2012). 1 2
[10]
H. Markowitz. Modern portfolio theory. Journal of Finance 7, 77–91 (1952). 1 2 3 4 5 6 7 8 9 10
[11]
S. Gerber, H. Markowitz, P. Ernst, Y. Miao, P. Sargen and others. The Gerber statistic: A robust co-movement measure for portfolio optimization. Available at SSRN 3880054 (2021). 1 2 3 4 5 6 7 8 9 10 11 12
[12]
E. Flint and D. Polakow. Deconstructing the Gerber statistic. Finance Research Letters 56, 104144 (2023). 1 2
[13]
W. Smyth and D. Broby. An enhanced Gerber statistic for portfolio optimization. Finance Research Letters 49, 103229 (2022). 1 2 3 4 5 6 7 8 9 10 11 12
[14]
G. J. Székely, M. L. Rizzo and N. K. Bakirov. Measuring and testing dependence by correlation of distances. The Annals of Statistics 35, 2769–2794 (2007). 1 2
[15]
M. Sibuya. Bivariate extreme statistics, I. Annals of the Institute of Statistical Mathematics 11, 195–210 (1960). 1
[16]
G. De Luca and P. Zuccolotto. A tail dependence-based dissimilarity measure for financial time series clustering. Advances in Data Analysis and Classification 5, 323–340 (2011). 1 2
[17]
K. H. Knuth. Optimal data-based binning for histograms and histogram-based probability density models. Digital Signal Processing 95, 102581 (2019). 1 2 3
[18]
D. Freedman and P. Diaconis. On the histogram as a density estimator: L2 theory. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 453–476 (1981). 1 2 3
[19]
D. W. Scott. On optimal and data-based histograms. Biometrika 66, 605–610 (1979). 1 2 3
[20]
A. Hacine-Gharbi, P. Ravier, R. Harba and T. Mohamadi. Low bias histogram-based estimation of mutual information for feature selection. Pattern Recognition Letters 33, 1302–1308 (2012). 1
[21]
A. Hacine-Gharbi and P. Ravier. A binning formula of bi-histogram for joint entropy estimation using mean square error minimization. Pattern Recognition Letters 101, 21–28 (2018). 1
[22]
C. E. Shannon. A mathematical theory of communication. The Bell System Technical Journal 27, 379–423 (1948). 1 2
[23]
A. Meucci. Risk and Asset Allocation (Springer Berlin Heidelberg, 2005). 1 2 3 4 5 6 7
[24]
Y. Feng and D. P. Palomar. A signal processing perspective of financial engineering. Foundations and Trends in Signal Processing 9, 1–231 (2016). 1 2 3 4 5 6 7
[25]
P. Jorion. Bayes-Stein estimation for portfolio analysis. The Journal of Financial and Quantitative Analysis 21, 279–292 (1986). 1
[26]
T. Bodnar, O. Okhrin and N. Parolya. Optimal shrinkage estimator for high-dimensional mean vector. Journal of Multivariate Analysis 170, 63–79 (2019). 1
[27]
F. Black and R. Litterman. Global portfolio optimization. Financial Analysts Journal 48, 28–43 (1992). 1 2 3 4
[28]
D. Cajas. Convex Optimization of Portfolio Kurtosis. Available at SSRN 4202967 (2022). 1 2 3 4 5 6 7
[29]
D. Cajas. On the Spectral Decomposition of Portfolio Skewness and its Application to Portfolio Optimization. Available at SSRN 4540021 (2023). 1 2
[30]
J. A. Nelder and R. W. Wedderburn. Generalized linear models. Journal of the Royal Statistical Society: Series A (General) 135, 370–384 (1972). 1
[31]
H. Akaike. A new look at the statistical model identification. IEEE Transactions on Automatic Control 19, 716–723 (1974). 1
[32]
C. M. Hurvich and C.-L. Tsai. Regression and time series model selection in small samples. Biometrika 76, 297–307 (1989). 1
[33]
G. Schwarz. Estimating the dimension of a model. The Annals of Statistics 6, 461–464 (1978). 1
[34]
R. R. Hocking. The analysis and selection of variables in linear regression. Biometrics 32, 1–49 (1976). 1 2 3
[35]
H. Theil. Economic Forecasts and Policy. 2 Edition (North-Holland, 1961). 1
[36]
M. A. Efroymson. Multiple regression analysis. In: Mathematical Methods for Digital Computers, edited by A. Ralston and H. S. Wilf (John Wiley & Sons, 1960); pp. 191–203. 1 2 3 4 5
[37]
K. Pearson. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2, 559–572 (1901). 1
[38]
H. Hotelling. Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 24, 417–441 (1933). 1
[39]
M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61, 611–622 (1999). 1
[40]
B. D. Fekedulegn, J. J. Colbert, R. R. Hicks Jr. and M. E. Schuckers. Coping with multicollinearity: an example on application of principal components regression in dendroecology. Technical Report NE-RP-721 (U.S. Department of Agriculture, Forest Service, Northeastern Research Station, 2002). 1 2 3 4
[41]
T. G. Andersen, T. Bollerslev, P. F. Christoffersen and F. X. Diebold. Volatility and correlation forecasting. In: Handbook of Economic Forecasting, Vol. 1, edited by G. Elliott, C. W. Granger and A. Timmermann (North-Holland, Amsterdam, 2006); Chapter 15, pp. 777–878. 1 2 3
[42]
B. J. Christensen and N. R. Prabhala. The relation between implied and realized volatility. Journal of Financial Economics 50, 125–150 (1998). 1
[43]
B. J. Christensen and C. S. Hansen. New evidence on the implied-realized volatility relation. The European Journal of Finance 8, 187–205 (2002). 1
[44]
T. Egbers and L. Swinkels. Can implied volatility predict returns on the currency carry trade? Journal of Banking & Finance 59, 14–26 (2015). 1
[45]
S. Gerber, W. Smyth, H. Markowitz, Y. Miao, P. Ernst and P. Sargen. Squeezing financial noise: A novel approach to covariance matrix estimation. Available at SSRN 4986939 (2025). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
[46]
S. Van Dongen and A. J. Enright. Metric distances derived from cosine similarity and Pearson and Spearman correlations, arXiv preprint arXiv:1208.3145 (2012). 1 2
[47]
S. Yue, X. Wang and M. Wei. Application of two-order difference to gap statistic. Transactions of Tianjin University 14, 217–221 (2008). 1
[48]
R. Tibshirani, G. Walther and T. Hastie. Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63, 411–423 (2001). 1
[49]
P. J. Rousseeuw. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics 20, 53–65 (1987). 1
[50]
M. López de Prado and M. J. Lewis. Detection of false investment strategies using unsupervised learning methods. Quantitative Finance 19, 1555–1565 (2019). 1
[51]
D. Müllner. Modern hierarchical, agglomerative clustering algorithms, arXiv preprint arXiv:1109.2378 (2011). 1
[52]
P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa and P. van Mulbregt. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods 17, 261–272 (2020). 1 2
[53]
[54]
W.-M. Song, T. Di Matteo and T. Aste. Hierarchical information clustering by means of topologically embedded graphs. PloS one 7, e31929 (2012). 1 2 3 4
[55]
[56]
[57]
M. Tumminello, T. Aste, T. Di Matteo and R. N. Mantegna. A tool for filtering information in complex systems. Proceedings of the National Academy of Sciences 102, 10421–10426 (2005). 1 2
[58]
S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 129–137 (1982). 1
[59]
L. C. Freeman. A set of measures of centrality based on betweenness. Sociometry 40, 35–41 (1977). 1
[60]
U. Brandes. A faster algorithm for betweenness centrality. The Journal of Mathematical Sociology 25, 163–177 (2001). 1
[61]
L. C. Freeman. Centrality in social networks conceptual clarification. Social Networks 1, 215–239 (1979). 1 2
[62]
P. Bonacich. Power and centrality: a family of measures. American Journal of Sociology 92, 1170–1182 (1987). 1
[63]
L. Katz. A new status index derived from sociometric analysis. Psychometrika 18, 39–43 (1953). 1
[64]
S. Brin and L. Page. The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30, 107–117 (1998). 1
[65]
T. W. Valente and R. K. Foreman. Integration and radiality: measuring the extent of an individual's connectedness and reachability in a network. Social Networks 20, 89–105 (1998). 1
[66]
A. Shimbel. Structural parameters of communication networks. The Bulletin of Mathematical Biophysics 15, 501–507 (1953). 1
[67]
J. B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society 7, 48–50 (1956). 1
[68]
O. Borůvka. O jistém problému minimálním. Práce Moravské Přírodovědecké Společnosti 3, 37–58 (1926). 1
[69]
R. C. Prim. Shortest connection networks and some generalizations. The Bell System Technical Journal 36, 1389–1401 (1957). 1
[70]
R. N. Mantegna. Hierarchical structure in financial markets. The European Physical Journal B 11, 193–197 (1999). 1 2
[71]
E. Estrada. The Structure of Complex Networks: Theory and Applications (Oxford University Press, 2011). 1
[72]
D. Cajas. A Graph Theory Approach to Portfolio Optimization. Available at SSRN 4602019 (2023). 1 2 3 4 5
[73]
D. Cajas. A Graph Theory Approach to Portfolio Optimization Part II. Available at SSRN 4667426 (2023). 1 2 3 4
[74]
F. Ricca and A. Scozzari. Portfolio optimization through a network approach: network assortative mixing and portfolio diversification. European Journal of Operational Research 312, 700–717 (2024). 1 2
[75]
J. Fan, Y. Fan and J. Lv. High dimensional covariance matrix estimation using a factor model. Journal of Econometrics 147, 186–197 (2008). 1
[76]
J. Walters. The Black-Litterman model in detail. SSRN Electronic Journal (2011). 1
[77]
T. Idzorek. A step-by-step guide to the Black-Litterman model: incorporating user-specified confidence levels. In: Forecasting Expected Returns in the Financial Markets (Academic Press, 2007); pp. 17–38. 1
[78]
P. N. Kolm and G. Ritter. On the Bayesian interpretation of Black-Litterman. European Journal of Operational Research 258, 564–572 (2017). 1
[79]
W. Cheung. The augmented Black-Litterman model: a ranking-free approach to factor-based portfolio construction and beyond. Quantitative Finance 13, 301–316 (2013). 1
[80]
A. Meucci. Fully flexible views: theory and practice. Risk 21, 97–102 (2008). 1 2 3 4 5 6 7 8 9 10 11 12
[81]
A. Vorobets. Sequential entropy pooling heuristics. Available at SSRN 3936392 (2021). 1 2 3 4 5
[82]
A. Meucci, D. Ardia and S. Keel. Fully flexible extreme views. The Journal of Risk 14, 39–49 (2011). 1 2 3
[83]
F. Dietrich and C. List. Probabilistic opinion pooling generalized. Part one: general agendas. Social Choice and Welfare 48, 747–786 (2017). 1 2 3 4
[84]
[85]
C. Martini and J. Sprenger. Opinion Aggregation and Individual Expertise. In: Scientific Collaboration and Collective Knowledge (Oxford University Press, 2017). 1 2
[86]
K. Boudt, W. Lu and B. Peeters. Higher order comoments of multifactor models and asset allocation. Finance Research Letters 13, 225–233 (2015). 1
[87]
L. Martellini and V. Ziemann. Improved estimates of higher-order comoments and implications for portfolio selection. The Review of Financial Studies 23, 1467–1502 (2010). 1
[88]
M. Sousa Lobo and S. Boyd. The Worst-Case Risk of a Portfolio (Stanford University, 2000). 1
[89]
F. J. Fabozzi, P. N. Kolm, D. A. Pachamanova and S. M. Focardi. Robust Portfolio Optimization and Management (John Wiley & Sons, Hoboken, NJ, 2007). 1 2 3 4
[90]
D. N. Politis and J. P. Romano. The stationary bootstrap. Journal of the American Statistical Association 89, 1303–1313 (1994). 1 2 3
[91]
D. N. Politis and J. P. Romano. A circular block-resampling procedure for stationary data. In: Exploring the Limits of Bootstrap (John Wiley & Sons, 1992); pp. 263–270. 1 2 3
[92]
H. R. Künsch. The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17, 1217–1241 (1989). 1 2 3
[93]
S. Boyd and L. Vandenberghe. Convex Optimization (Cambridge University Press, Cambridge, UK, 2004). 1 2 3 4 5
[94]
S. Diamond and S. Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research 17, 1–5 (2016). 1 2 3 4
[95]
R. H. Tütüncü and M. Koenig. Robust asset allocation. Annals of Operations Research 132, 157–187 (2004). 1 2
[96]
P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977). 1 2 3 4 5
[97]
H. Konno and H. Yamazaki. Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science 37, 519–531 (1991). 1 2
[98]
D. Cajas. Portfolio Optimization of Even Moments using Power Cone Programming. Available at SSRN 6518258 (2026). 1 2
[99]
D. Cajas. Approximation of Portfolio Kurtosis through Sum of Squared Quadratic Forms. Available at SSRN 4472793 (2023). 1 2
[100]
A. Chekhlov, S. Uryasev and M. Zabarankin. Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance 8, 13–58 (2005). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
[101]
R. T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk. Journal of Risk 2, 21–41 (2000). 1 2 3 4 5 6 7 8 9 10
[102]
P. Mohajerin Esfahani and D. Kuhn. Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Mathematical Programming 171, 115–166 (2018). 1 2 3 4 5
[103]
A. Ahmadi-Javid. Entropic value-at-risk: A new coherent risk measure. Journal of Optimization Theory and Applications 155, 1105–1123 (2012). 1 2 3 4
[104]
D. Cajas. Portfolio Optimization of Relativistic Value at Risk. Available at SSRN 4378498 (2023). 1 2 3 4
[105]
D. Cajas. OWA portfolio optimization: A disciplined convex programming framework. Available at SSRN 3988927 (2021). 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[106]
D. Cajas. Efficient Gini Mean Difference and Tail Gini Portfolio Optimization based on P-Norms. Available at SSRN 4711326 (2024). 1 2 3 4 5 6 7
[107]
D. Cajas. Higher order moment portfolio optimization with L-moments. Available at SSRN 4393155 (2023). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
[108]
W. Ogryczak and T. Śliwiński. On solving linear programs with the ordered weighted averaging objective. European Journal of Operational Research 148, 80–91 (2003). 1 2 3 4 5
[109]
W. Ogryczak and A. Ruszczyński. Dual stochastic dominance and quantile risk measures. International Transactions in Operational Research 9, 661–680 (2002). 1 2 3
[110]
S. Yitzhaki. Stochastic dominance, mean variance, and Gini's mean difference. The American Economic Review 72, 178–185 (1982). 1
[111]
P. G. Martin and B. B. McCann. The Investor's Guide to Fidelity Funds (John Wiley & Sons, 1989). 1 2
[112]
D. Cajas. Portfolio Optimization of Brownian Distance Variance. Available at SSRN 4561293 (2023). 1
[113]
M. R. Young. A minimax portfolio selection rule with linear programming solution. Management Science 44, 673–683 (1998). 1
[114]
D. P. Palomar. Portfolio Optimization: Theory and Application (Cambridge University Press, 2025). 1 2 3 4
[115]
P. A. Krokhmal. Higher moment coherent risk measures. Quantitative Finance 7, 373–387 (2007). 1 2 3 4 5
[116]
D. Cajas. Semidefinite Relaxation of Higher Portfolio Moments. Available at SSRN 5284483 (2025). 1 2 3 4 5
[117]
P. J. Rousseeuw and C. Croux. Alternatives to the median absolute deviation. Journal of the American Statistical Association 88, 1273–1283 (1993). 1
[118]
M. López de Prado. Building diversified portfolios that outperform out of sample. The Journal of Portfolio Management 42, 59–69 (2016). 1
[119]
[120]
T. Raffinot. Hierarchical clustering-based asset allocation. The Journal of Portfolio Management 44, 89–99 (2017). 1
[121]
T. Raffinot. The hierarchical equal risk contribution portfolio. SSRN Electronic Journal (2018). 1
[122]
T. Roncalli and G. Weisang. Risk Parity Portfolios with Risk Factors. Available at SSRN 2155159 (2012). 1 2 3
[123]
A. Meucci. Risk Contributions from Generic User-Defined Factors. Available at SSRN 930034 (2007). 1
[124]
D. Cajas. Robust Portfolio Selection with Near Optimal Centering. Available at SSRN 3572435 (2019). 1
[125]
T. de Graaf. Robust Mean-Variance Optimization. Master's thesis, Leiden University (2016). 1
[126]
S. Maillard, T. Roncalli and J. Teiletche. On the Properties of Equally-Weighted Risk Contributions Portfolios. Available at SSRN 1271972 (2008). 1 2 3
[127]
B. Bruder and T. Roncalli. Managing Risk Exposures Using the Risk Budgeting Approach. Available at SSRN 2009778 (2012). 1 2 3
[128]
MOSEK ApS. MOSEK Portfolio Optimization Cookbook, https://docs.mosek.com/portfolio-cookbook/index.html (2023). 1
[129]
J.-C. Richard and T. Roncalli. Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications & Puzzles. Technical Report 1902.05710 (arXiv, 2019). 1 2 3 4 5
[130]
V. Gambeta and R. Kwon. Risk Return Trade-Off in Relaxed Risk Parity Portfolio Optimization. Journal of Risk and Financial Management 13, 237 (2020). 1 2
[131]
H. Mausser and O. Romanko. Computing Equal Risk Contribution Portfolios. IBM Journal of Research and Development 58, 5:1–5:12 (2014). 1
[132]
M. López de Prado. A robust estimator of the efficient frontier. SSRN Electronic Journal (2019). 1 2
[133]
D. H. Wolpert. Stacked generalization. Neural Networks 5, 241–259 (1992). 1 2 3
[134]
W. Shen and J. Wang. Portfolio selection via subset resampling. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (2017); pp. 1517–1523. 1 2 3
[135]
R. A. Martin. PyPortfolioOpt: portfolio optimization in Python. Journal of Open Source Software 6, 3066 (2021). 1 2 3 4 5
[136]
M. López de Prado. Advances in Financial Machine Learning (John Wiley & Sons, Hoboken, NJ, 2018). 1 2 3 4
[137]
J. Bergstra and Y. Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research 13, 281–305 (2012). 1 2
[138]
J. L. Kelly. A New Interpretation of Information Rate. Bell System Technical Journal 35, 917–926 (1956). 1
[139]
E. O. Thorp. The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. In: Handbook of Asset and Liability Management, Vol. 1 (North-Holland, 2008); pp. 385–428. 1
[140]
R. Chares. Cones and Interior-Point Algorithms for Structured Convex Optimization Involving Powers and Exponentials. Ph.D. Thesis, Université catholique de Louvain (Louvain-la-Neuve, Belgium, 2009). 1
[141]
W. F. Sharpe. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance 19, 425–442 (1964). 1
[142]
S. Schaible and T. Ibaraki. Fractional Programming. European Journal of Operational Research 12, 325–338 (1983). 1
[143]
A. Charnes and W. W. Cooper. Programming with Linear Fractional Functionals. Naval Research Logistics Quarterly 9, 181–186 (1962). 1
[144]
R. C. Grinold and R. N. Kahn. Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk. 2 Edition (McGraw-Hill, New York, 1999). 1
[145]
B. Tóth, Y. Lémperière, C. Deremble, J. de Lataillade, J. Kockelkoren and J.-P. Bouchaud. Anomalous Price Impact and the Critical Nature of Liquidity in Financial Markets. Physical Review X 1, 021006 (2011). 1
[146]
V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal. A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms. Management Science 55, 798–812 (2009). 1 2 3

Contributors

Index