Portfolio Optimization

Implementation & modification of a set of papers of portfolio optimization problem.

Course project for OIT 604 Spring19 of Graduate School of Business@Stanford.

The codes are on the basis of Yihao Kao's RPCA code.

Problem Formulation

Consider an optimization problem:

Here could be considered as centered stock returns, are stock holding decisions for stocks and are combinations of expected stock returns and other costs. The term is the total risk and the term is the expected revenue. By maximizing the weighted sum (the weight could be merged into , so it is ignored), we can get a decision .

With history data , the problem could be solved via efficient data driven decision making approaches such as PPCA ¹, RPCA²and DPCA³. These methods could be considered as sample average approximation (SAA) approaches.

Furthermore, inspired by kernel methods⁴⁵⁶, suppose we can access a set of additional history feature data along with s, given a new feature observation , we try to improve the out-of-sample performance with additional information.

References

1. Tipping, M. E. and Bishop, C. M. (1999), Probabilistic Principal Component Analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61: 611-622. (link)

2. Kao, Y.H. and Van Roy, B., 2013. Learning a Factor Model via Regularized PCA. Machine Learning, Volume 91, Number 3, pp. 279-303. (link)

3. Kao, Y.H. and Van Roy, B., 2014. Directed principal component analysis. Operations Research, 62(4), pp.957-972. (link)

4. Bertsimas, D. and Kallus, N., 2014. From predictive to prescriptive analytics. arXiv preprint arXiv:1402.5481. (link)

5. Nadaraya, Elizbar., 1964. On estimating regression. Theory Probab. Appl. 9(1) 141-142.

6. Watson, Geoffery., 1964. Smooth regression analysis. Sankhya A 359-372.