# Generating stress scenarios: null correlation is not enough

December 28, 2010
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(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

In a recent post (here, by @teramonagi), Teramonagi mentioned the use of PCA to model yield curve, i.e. to obtain the three factor,parallel shift“, “twist” and “butterfly“. As in Nelson & Siegel, if m is maturity, $yleft( m right)$ is the yield of the curve at maturity m, assume that

where β0, β1, β2 and τ, are parameters to be fitted

• β0 is interpreted as the long run levels of interest rates (the loading is 1, it is a constant that does not decay)
• β1 is the short-term component (it starts at 1, and decays monotonically and quickly to 0);
• β2 is the medium-term component (it starts at 0, increases, then decays to zero);
• τ is the decay factor: small values
produce slow decay and can better fit the curve at long maturities,
while large values produce fast decay and can better fit the curve at
short maturities; τ also governs where β2 achieves its maximum.

(see e.g. here). Those factors can be obtained using PCA,

`term.structure = read.csv("C:\tmp\FRB_H15.csv",stringsAsFactors=FALSE)term.structure = tail(term.structure,1000)term.structure = term.structure[,-1]label.term = c("1M","3M","6M","1Y","2Y","3Y","5Y"                      ,"7Y","10Y","20Y","30Y")colnames(term.structure) = label.termterm.structure = subset(term.structure,term.structure\$'1M' != "ND")term.structure = apply(term.structure,2,as.numeric)term.structure.diff = diff(term.structure)term.structure.princomp = princomp(term.structure.diff)factor.loadings = term.structure.princomp\$loadings[,1:3]legend.loadings = c("First principal component","Second principal component","Third principal component")par(xaxt="n")matplot(factor.loadings,type="l",  lwd=3,lty=1,xlab = "Term", ylab = "Factor loadings")legend(4,max(factor.loadings),legend=legend.loadings,col=1:3,lty=1,lwd=3)par(xaxt="s")axis(1,1:length(label.term),label.term)> summary(term.structure.princomp)Importance of components:                          Comp.1    Comp.2     Comp.3     Comp.4     Comp.5      Comp.6      Comp.7      Comp.8Standard deviation     0.2028719 0.1381839 0.06938957 0.05234510 0.03430404 0.022611518 0.016081738 0.013068448Proportion of Variance 0.5862010 0.2719681 0.06857903 0.03902608 0.01676075 0.007282195 0.003683570 0.002432489Cumulative Proportion  0.5862010 0.8581690 0.92674803 0.96577411 0.98253486 0.989817052 0.993500621 0.995933111`

using Teramonagi’s
R code, When
generating stress scenarios, the idea is to generate independently
those factors (or components) and then to aggregate them (using the expression given above). With Principal Component Analysis, PCA, we get orthogonal components, while with Independent Component Analysis, ICA, we get independent components. And independence and null correlation can be rather different. We recently discussed that idea in a paper with Christophe Villa (available soon here).

Consider the following sample

`ns=10000X=runif(ns)Y=runif(ns)I=(Y<.25)*(Y<3*X)*(Y>X/3) +    (Y>.75)*(Y<X/3+3/4-1/12)*(Y>3*X-2)+   (Y>.25)*(Y<.75)*(Y<3*X)*(Y>3*X-2) FACT1=X[I==1]FACT2=Y[I==1]DATA=data.frame(FACT1,FACT2)PCA<-princomp(DATA)op <- par(mfrow = c(1, 2))plot(FACT1[1:2000],FACT2[1:2000],main="Principal component analysis",col="black",cex=.2,xlab="",ylab="",xaxt="n",yaxt="n")arrows(.5,.5,.8,.8,type=2,col="red",lwd=2)arrows(.5,.5,.2,.8,type=2,col="red",lwd=2)plot(PCA\$scores,cex=.2,main="Principal component analysis",xaxt="n",yaxt="n")`

The PCA obtain the following projections on the two components (drawn in red, below)

`> X=PCA\$scores[,1];> Y=PCA\$scores[,2];> n=length(FACT1)> x=X[sample(1:n,size=n,replace=TRUE)]> y=Y[sample(1:n,size=n,replace=TRUE)]> PCA\$loadings Loadings:      Comp.1 Comp.2FACT1 -0.707  0.707FACT2 -0.707 -0.707                Comp.1 Comp.2SS loadings       1.0    1.0Proportion Var    0.5    0.5Cumulative Var    0.5    1.0 > F1=0.707*x-.707*y> F2=0.707*x+.707*y`

Hence, with PCA, we have two components, orthogonal, with a triangular distribution, so if we generate them independently, we obtain

which is quite different, compared with the original sample. On the other hand, with ICA,we obtain factors that are really independent….

`library(fastICA)nt=10000ICA<-fastICA(DATA,2)m=ICA\$Kx=ICA\$S[,1]y=ICA\$S[,2]plot(ICA\$S,cex=.2,main="Independent component analysis",xlab="",ylab="",xaxt="n",yaxt="n")`

see below for the graphs, and here for more details,

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