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Dynamic Principal Components in Optimal Portfolio Selection

Dynamic Principal Components in Optimal Portfolio Selection. Rodwel Mupambirei, C.G Troskie, R. Guo, N. Hossain Department of Statistical Science University of Cape Town. Agenda. Multiple Index Models Principal Components Principal Components Regression

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Dynamic Principal Components in Optimal Portfolio Selection

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  1. Dynamic Principal Components in Optimal Portfolio Selection Rodwel Mupambirei, C.G Troskie, R. Guo, N. Hossain Department of Statistical Science University of Cape Town

  2. Agenda • Multiple Index Models • Principal Components • Principal Components Regression • Time Series Regression and Model Selection • Portfolio Construction

  3. Multiple Index Models • Sharpe Multiple Index Model • Main Problems with Multi Factor Models • Complexity • Selecting the variables to use as explanatory variables. • Deciding how many factors to include. • Collinearity of the explanatory variables • Principal Components can help with the above problems

  4. Principal Components-Objectives • Reducing the dimension of the data with little loss of information • Identification of the components that explain most of the variance • Construction of orthogonal indices making regression estimates stable • Provides an alternative way of estimating covariance matrices

  5. Principal Components-Applications • Image processing • Biotechnology and chemistry • Computer Science • Finance - Explaining shifts of the yield curve • Finance construction of indices

  6. Principal Components-Methodology 2. Use the SVD decomposition on the mean corrected matrix • X is the data matrix of 8 indices. Alsi, Banks, Findi, Gold, Industrials, Mining, Platinum, Rand 1. Use the SVD decomposition on the mean corrected matrix 3. Compute the score matrix-the orthogonal projection of the indices 4. We now have a set of new indices to use through the multiple index model

  7. Principal Components-Regression Literature Review • Kendal (1957) • Jeffers (1967) • Mansfield et al (1977) • Gunst and Mason (1980)

  8. Principal Components-Regression • Principal Components regression uses the orthogonal projection of the indices to model share returns • There was a misconception for many years that you should only use the principal components associated with the largest variance.- Not True • Hotteling (1957) and Massy (1965) stressed the importance of the smallest principal components in regression • There are numerous examples where even the smallest components are important.

  9. Principal Components-Regression • Unfortunately, despite the evidence, the misconception still exists. • So how should we decide which principal components to use? • We used a model selection procedures to choose the principal components and it turns out the best model was actually, • The principal components that explain the most variation in the indices are not necessarily the ones that explain the most variation in the share returns

  10. Principal Components-Regression An unbiased estimate for the covariance matrix of returns is given by Ωis estimated allowing the residuals from different stocks to be correlated

  11. Dynamic Principal Components • In Dynamic principal components we use time-series models to model the residual variance. • The residuals in the principal components regression may be autocorrelated or heteroskedastic. Gauss-Markoff Theorem • We need to fit AR and/or GARCH models to the residuals in order to invoke the Gauss-Markoff theorem

  12. Principal Components-Results • Model fit-Adjusted R-Squared Statistic • Principal Components Regression explains a greater proportion of the share returns compared to the ordinary least squares.

  13. Principal Components-Results • Model fit- ICOMP • Principal Components Regression provides a much better model fit

  14. Principal Components-Results • Efficient Frontier

  15. Principal Components-Results • Efficient Frontier-Dynamic Models

  16. Principal Components-Results • Minimum Variance Portfolio • The AR-GARCH model results in a significant reduction in the risk

  17. Principal Components-Results • Consider an equally weighted portfolio for proper comparison Portfolio Market Risk Portfolio Residual Risk Specific Risk + extra market risk • The aim is to evaluate the effect of dynamic principal components on the estimates of residual risk

  18. Results-Risk Analysis • Equally weighted portfolio • The Principal Components model results in a reduction the estimates of the residual risk.

  19. Results-Risk Analysis • Equally weighted portfolio • The AR-GARCH model results in a reduction the estimates of the residual risk.

  20. Conclusions • Principal components regression results in a significantly better fit than ordinary least squares. • The smaller principal components can be more important than the larger ones in a regression. • Principal Components leads to a significant reduction in estimates of the residual risk. • The Dynamic Principal Components model performs very when both the autocorrelation an heteroskedasticity are taken in to account.

  21. Further Research • Latent root regression analysis • Questions??

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