1. Principal Components
The Basics
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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.
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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)
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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.
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5. PCA 2
Under the Statistics Header, Select “Multivariate.” Then “Principal Components.”
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6. *** Principal Components Analysis ***
Standard deviations:
Comp Comp Comp Comp Comp.5
The number of variables is 5 and the number of observations is 351
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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)
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8. Importance of components:
Comp Comp Comp Comp.4
Standard deviation
Proportion of Variance
Cumulative Proportion
Comp.5
Standard deviation
Proportion of Variance
Cumulative Proportion
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9. Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 X90.day
X180.day
X5.Year
X15.Year
X25.Year
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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 * y * y * y * y * y5.
We can use this along with the actual yields to construct the “mimicking portfolio” for the first factor.
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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.
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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 (1.8%) while the variance of the latter is only (0.66%).
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