There are different ways of applying SPC to financial data. Here, we use a
twosided EWMA chart for monitoring the Deutsche Bank (DBK) share. More
precisely, a capital asset pricing model (CAPM) is fitted for DBK and the DAX
which is used as proxy of the efficient market portfolio. That is, denoting with
and
the log returns of the DAX
and the DBK, respectively, one assumes that the following regression model is
valid:
Usually, the parameters of the model are estimated by the ordinary least squares
method. The parameter is a very popular measure in applied
finance, Elton and Gruber (1991).
In order to construct a
real portfolio, the
coefficient
is frequently taken
into account.
Research has therefore concentrated on
the appropriate estimation of
constant and time changing
.
In the context of SPC it is therefore
useful to construct monitoring
rules which signal
changes in
.
Contrary to standard SPC application in
industry there is no obvious state of the process which one can call
''in-control'', i.e. there is no target process. Therefore, pre-run time
series of both quotes (DBK, DAX) are exploited for building the in-control state. The
daily quotes and log returns, respectively, from january, 6th, 1995 to march,
18th, 1997 (about 450 observations) are used for fitting (11.14):
A N O V A SS df MSS F-test P-value _________________________________________________________________________ Regression 0.025 1 0.025 445.686 0.0000 Residuals 0.025 448 0.000 Total Variation 0.050 449 0.000 Multiple R = 0.70619 R^2 = 0.49871 Adjusted R^2 = 0.49759 Standard Error = 0.00746 PARAMETERS Beta SE StandB t-test P-value ________________________________________________________________________ b[ 0,]= -0.0003 0.0004 -0.0000 -0.789 0.4307 b[ 1,]= 0.8838 0.0419 0.7062 21.111 0.0000
With b[1,]
a typical value has been obtained. The
is not very large. However, the simple linear regression is
considered in the sequel. The (empirical) residuals of the above
model are correlated (see Figure 11.5). The SPC
application should therefore be performed with the (standardized) residuals of an AR(1)
fit to the regression residuals.
For an application of the
XploRe
quantlet armacls
(quantlib
times
) the regression residuals were standardized. By using the
conditional least squares method an estimate of
for the
AR(1) model
has been obtained. Eventually, by plugging in the estimates of ,
and
, and standardizing with the sample standard deviation of the
pre-run residuals series (see (11.15)) one gets a series of
uncorrelated data with expectation 0 and variance 1, i.e. our in-control
state. If the fitted model (CAPM with AR(1) noise) remains valid after the
pre-run, the related standardized residuals behave like in the in-control state.
Now, the application of SPC, more precisely of a twosided EWMA chart, allows to
monitor the series in order to get signals, if the original model was changed.
Changes in
or
in (11.14) or in
in
(11.15) or in the residual variance of both models lead to shifts or
scale changes in the empirical residuals series. Hence, the probability of an
alarm signaled by the EWMA chart increases (with one exception only - decreased
variances). In this way a possible user of SPC in finance is able to monitor an
estimated and presumed CAPM.
In our example we use the parameter
and an in-control
ARL of 500, such that the critical value is equal to
(the Markov
chain parameter
was set to 100). Remark, that the computation of
is
based on the normality assumption, which is seldom fulfilled for financial data.
In our example the hypothesis of normality is rejected as well with a very small
value (Jarque-Bera test with quantlet
jarber
). The estimates of
skewness 0.136805 and kurtosis 6.64844 contradict normality too. The fat tails
of the distribution are a typical pattern of financial data. Usually, the fat
tails lead to a higher false alarm rate. However, it would be much more
complicated to fit an appropriate distribution to the residuals and use these
results for the ''correct'' critical value.
The Figures 11.6 and 11.7 present the EWMA graphs of the pre-run and the monitoring period (from march, 19th, 1997 to april, 16th, 1999).
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In the pre-run period the EWMA chart signals 4 times. The first 3 alarms seem to be outliers, while the last points on a longer change. Nevertheless, the chart performs quite typical for the pre-run period. The first signal in the monitoring period was obtained at the 64th observation (i.e. 06/24/97). Then, we observe more frequently signals than in the pre-run period, the changes are more persistent and so one has to assume, that the pre-run model is no longer valid. A new CAPM has therefore to be fitted and, if necessary, the considered portfolio has to be reweighted. Naturally, a new pre-run can be used for the new monitoring period.