Principal Components The Basics Principal Components 1. 11/30/2018

Dimensionality Reduction One of the most fundamental questions in managing a portfolio is what risk is it exposed to, and what do these sources of risk look like. If we have a portfolio with 8,000 different securities, it may well be that there are just 20 sources of fundamental risk that it is exposed to. Principal Components 1. 11/30/2018

Dimensionality Reduction In the Treasury Complex, for example, it is long observed that only 3 factors are needed to account for all the risks in the entire yield curve. Litterman and Scheinkman use Principal Components on the bond yields, and identify the 3 factors: Level Slope Curvature (or Volatility) Principal Components 1. 11/30/2018

Principal Components Analysis We can use S-Plus to analyze the data and provide factor loadings, as well as report the relative importance of each factor. First set up the data so that it is as you want to analyze in Excel. Next launch S-plus, and under the “File” heading, select “Import Data” and then “From File.” Identify your excel file here. Principal Components 1. 11/30/2018

PCA 2 Under the Statistics Header, Select “Multivariate.” Then “Principal Components.” Principal Components 1. 11/30/2018

*** Principal Components Analysis *** Standard deviations: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 0.02774643 0.009870496 0.001872219 0.0009389359 0.0004868663 The number of variables is 5 and the number of observations is 351 Principal Components 1. 11/30/2018

Component names: "sdev" "loadings" "correlations" "scores" "center" "scale" "n.obs" "terms" "call" "factor.sdev" "coef" Call: princomp(x = ~ ., data = kfbyo, scores = T, cor = F, na.action = na.exclude) Principal Components 1. 11/30/2018

Importance of components: Comp.1 Comp.2 Comp.3 Comp.4 Standard deviation 0.02774643 0.009870496 0.001872219 0.0009389359 Proportion of Variance 0.88295817 0.111738742 0.004020121 0.0010111083 Cumulative Proportion 0.88295817 0.994696910 0.998717032 0.9997281400 Comp.5 Standard deviation 0.0004868663 Proportion of Variance 0.0002718600 Cumulative Proportion 1.0000000000 Principal Components 1. 11/30/2018

Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 X90.day 0.619 -0.348 0.447 0.398 0.370 X180.day 0.612 -0.292 -0.222 -0.529 -0.460 X5.Year 0.388 0.403 -0.713 0.207 0.368 X15.Year 0.261 0.615 0.289 0.353 -0.588 X25.Year 0.152 0.503 0.398 -0.628 0.413 Principal Components 1. 11/30/2018

The “Factors” We can use this analysis to construct the historical realizations of each of our factors. In this example, the first PC is equal to: .69 * y1 + .612 * y2 + .388 * y3 + .261 * y4 + .152 * y5. We can use this along with the actual yields to construct the “mimicking portfolio” for the first factor. Principal Components 1. 11/30/2018

Factor Regressions Next, when we regress each of the yields on the factor, we note that indeed the regression coefficient corresponds to the weight we put on that yield in constructing the factor. We also see the importance of each factor in explaining each yield. For example, we know that the first factor explains 88.3% of the total variance of the entire yield curve. Principal Components 1. 11/30/2018

Variance Decomposition This includes 95.8% of the 90-Day Bill yield, but only 41.0% of the 25-Year PO Strip yield. But note that during this sample, the variance (standard deviation) of the former is .0003 (1.8%) while the variance of the latter is only .00004 (0.66%). Principal Components 1. 11/30/2018