8. Example

We analyze the daily returns of the exchange rate of the Yen related to the U.S. Dollar from Dec. 1, 1978 to Jan. 31, 1991. Our aim is to fit a generalized Pareto model to the lower tail of the returns to estimate the probability of extreme losses. We start the analysis by loading the finance library and the data set dyr.dat:


  library("finance")

  dyr=read("dyr.dat")

A scatter plot of the data set can be obtained using the command

  plot(1:rows(dyr)~dyr)

In the Academic Version of XploRe the following examples can be executed with the smaller data set dyr1000.dat. Although slightly different results are obtained, one can still recognize that the Hill estimator is unsuited for that data set.

Figure 3: Scatter plot of daily returns of Yen related to U.S. Dollar from Dec. 1978 to Jan. 1991
\includegraphics[scale=0.6]{xtrfig4.ps}

One recognizes from Figure 3 that the distribution of the returns possesses a fat tail. Because our estimators are defined for the upper tail, one must change the sign of the data set with the command


  dyr = -dyr

A suitable threshold can be selected by plotting an estimator diagram. The call

  r=momentgpdiag(dyr,5:500)

  plot(5:500~r)

produces the diagram in Figure 4. We select k=160 extremes (yielding the threshold t=0.00966) and plot the empirical quantile function as well as the estimated parametric one to check the quality of the fit.

Figure 4: Diagram of moment estimator applied to daily returns.
\includegraphics[scale=0.6]{xtrfig2.ps}

This task is performed by the following code (calls to format the graphical output are not shown):


  m=momentgp(dyr,160)



  d=createdisplay(1,1)

  t=aseq(0.965,350,0.0001)

  qf=t~m.mu+m.sigma*qfx("gp",t,m.gamma)

  show(d,1,1,qf)



  empqf=(4284:4444)/4445~sort(dyr)[4284:4444]

  adddata(d,1,1,empqf)

1285xtrm01.xpl

The resulting plot is shown in Figure 5.

Figure 5: GP qf fitted to tail of returns.
\includegraphics[scale=0.6]{xtrfig5.ps}

The Hill estimator yields a similar picture (execute the following lines to add the pertaining Pareto quantile function to the plot).


  h=hillgp1(dyr,160)

  hqf=t~h.sigma*qfx("gp1",t,h.alpha)

  adddata(d,1,1,hqf)

1292xtrm01.xpl

To decide which estimator should be preferred, we employ the mean excess function. Execute the next lines to create the plot of the empirical mean excess function as well as the parametric ones that is shown in Figure 6.

  h=hillgp1(dyr,160)

  m=momentgp(dyr,160)

  d=createdisplay(1,1)

  t=aseq(0.009,210,0.0001)

  ;

  ; plot empirical mean excess function

  ;

  et=sort(dyr)[rows(dyr)-160:rows(dyr)-1]

  eme=et~empme(dyr,et)

  show(d,1,1,eme)

  ;

  hme=t~gp1me(h.alpha,t/h.sigma)*h.sigma

  adddata(d,1,1,hme)

  ;

  mme=t~gpme(m.gamma,(t-m.mu)/m.sigma)*m.sigma

  adddata(d,1,1,mme)

1298xtrm02.xpl

Figure 6: Empirical mean excess function (solid) and GP mean excess function fitted by Hill (dotted) and Moment estimator (dashed).
\includegraphics[scale=0.6]{xtrfig6.ps}

One can recognize that the empirical mean excess function is close to a straight line, which justifies the GP modeling. Yet, the GP mean excess function, based on the Hill estimator, strongly deviates from the empirical mean excess function. This indicates that the Hill estimator is not applicable.



Method and Data Technologies   MD*TECH Method and Data Technologies
  http://www.mdtech.de  mdtech@mdtech.de