11.4 Real data example - monitoring CAPM

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 $ r_{\textrm{\tiny DAX},t}$ and $ r_{\textrm{\tiny DBK},t}$ the log returns of the DAX and the DBK, respectively, one assumes that the following regression model is valid:

$\displaystyle r_{\textrm{\tiny DBK},t} = \alpha + \beta\,r_{\textrm{\tiny DAX},t} + \varepsilon_t$ (11.14)

Usually, the parameters of the model are estimated by the ordinary least squares method. The parameter $ \beta $ is a very popular measure in applied finance, Elton and Gruber (1991). In order to construct a real portfolio, the $ \beta $ coefficient is frequently taken into account. Research has therefore concentrated on the appropriate estimation of constant and time changing $ \beta $. In the context of SPC it is therefore useful to construct monitoring rules which signal changes in $ \beta $. 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,] $ \;=\beta=0.8838$ a typical value has been obtained. The $ R^2=0.49871$ 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.

Figure 11.5: Partial autocorrelation function of CAPM regression residuals
\includegraphics[width=1\defpicwidth]{capm_pacf_fig.ps}

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 $ \hat\varrho=0.20103$ for the AR(1) model

$\displaystyle \varepsilon_t = \varrho\,\varepsilon_{t-1} + \eta_t$ (11.15)

has been obtained. Eventually, by plugging in the estimates of $ \alpha$, $ \beta $ and $ \varrho$, 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 $ \alpha$ or $ \beta $ in (11.14) or in $ \varrho$ 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 $ \lambda=0.2$ and an in-control ARL of 500, such that the critical value is equal to $ c=2.9623$ (the Markov chain parameter $ r$ was set to 100). Remark, that the computation of $ c$ 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 $ p$ 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).

Figure 11.6: Twosided EWMA chart of the standardized CAPM-AR(1) residuals for the pre-run period (06/01/95 - 03/18/97)
\includegraphics[width=1.2\defpicwidth]{capmewma1.ps}

Figure 11.7: Twosided EWMA chart of the standardized CAPM-AR(1) residuals for the monitoring period (03/19/97 - 04/16/99)
\includegraphics[width=1.2\defpicwidth]{capmewma2.ps}

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.

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