Principal Components

1
Principal Components

• The Basics

2
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.

3
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)

4
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.

5
PCA 2

• Under the Statistics Header, Select
  Multivariate. Then Principal Components.

6
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

7
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)

8
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

9
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

10
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.

11
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.

12
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).