Diese Projektarbeit stellt zwei dieser Finanzierungstrends vor, welche vermehrt in den Fokus der Finanzierungsproblematik rucken. Read more Read less. Here's how restrictions apply. About the Author Christoph Ruttgers, geb. Grin Publishing June 20, Language: Be the first to review this item Would you like to tell us about a lower price? Start reading Finanzierungstrends im deutschen Mittelstand. Don't have a Kindle? Try the Kindle edition and experience these great reading features: Share your thoughts with other customers. Write a customer review.
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No sector has a positive annual return. Also the standard deviation and the downside range of the mean characterized the risk inherent in this time range. From the density plot in Figure 10 the reader can see that especially in the fat tails the normal distribution underestimates the density. The Student-T distribution has a better overall fit to the empirical distribution even if it underestimates the empirical distribution at the long tails, it is still a better approximation for the VaR model.
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The empirical distribution has a kurtosis of 1. The mean of this portfolio is The p value of the Jarque-Bera-Test is below 2. The three bond indices are price indices clustered by maturity groups and were monthly rebalanced. For this purpose each bond index is weighted equally. The bond index with the ticker EU15PR contains European Government bonds with an average duration of one to five years. Below the reader can see a density plot comparison between the empirical distribution and analytical Normal distribution and Student-T distribution.
The empirical distribution has a kurtosis of 2. A reason for this difference is the unusual high volatility for European Government bonds in the years to caused by the sovereign debt crisis.
In order to test if the performance of a model is independent of the current market situation, the author suggests splitting the observation horizon into two time frames. The main reason for this separation is the evidence of a regime switch after the year concerning the changing behaviour of the credit spreads.
In Figure 15 one can see the development of the credit spreads ofthe Bond Portfolio. In order to keep the analysis clear, the author decided to attach this time frame analysis to each backtesting part of chapter 4. For each model only the unconditional coverage backtest will be performed onto the two time frames. Those two time frames will be later on called pre and post The time frame pre starts at and ends at Post starts at and ends at In this chapter, four different backtests will be performed for each model.
Before the reader continuous to the results of the back tests, the following information is helpful to know how to interpret the results:. The results of the unconditional coverage test and independence test are denoted in p-values. The null hypothesis H0 states that there is no significant difference between expected and realized observations for the unconditional coverage test. In this case the alternative hypothesis Ha concludes that there is a significant difference between expected and realized observations.
A graphical representation can be found in Figure 6. For the independence test, the null hypothesis H0 states that the VaR exceedances are independent. Then, the alternative hypothesis Ha concludes that the dependency of the VaR exceedances is significant. The regulatory test is a daily representation of the VaR exceptions looking only on the last days. If a model is evaluated by this backtest, it is not enough to look at the last days in the time series. An alternative is to move through the time series and make every day a backtest by looking on the last days. In some timeframes more VaR exceptions will occur than in others.
An example of a regulatory backtest time series can be found in Appendix M. For this purpose the author decided to take the worst situation maximum of each time series. The results of the economic test are the annual interest rate costs, relative to its portfolio value. In order to make the results comparable the accrued interest costs are scaled to one year. The last test will incorporate how good a model is in the pre and post time concerning the unconditional coverage test.
For the parametric GARCH model and the FHS the observation period of days will not be covered because the pre data series is smaller than the observation period. Due to that reason, the results for the observation period of days in the pre sample are not that reliable compared to the post sample. The four backtests are performed for each portfolio with the observation period , and days. By looking at the parameter evolvement see Appendix O it can be seen that especially the parameters for an observation period of days is very unstable.
In general it can be said that the parameters are more stable the higher the observation period is. It is possible that the additional parameter degree of freedom that has to be estimated leads to a more unstable estimation result. In the sample set of days, the estimated results are very stable. By looking on the robustness of the estimation results, it can be seen that there occur many convergence problems for lower observation periods like and days.
The t test of every parameter is insufficient for an observation period of days regardless of the model. For the observation period of days the t test is especially for the time span to is insufficient, and the estimation leads in this period to convergence problems. For the observation period of days the t test results in far better results for both models over the whole time range.
In general, one can conclude that an observation period of days is enough to get robust parameter estimates and at the same time good backtesting results. These rejections could be misleading. In both cases the observed exceedances are lower than the expected. Due to the fact that the unconditional coverage test follows a X[2] distribution, model variations with lower exceedances than expected have a lower p-Value. Therefore it is recommended to compare the results with the regulatory backtest. From the results above, there is no trend visible regarding the sample size. The independence test checks if the VaR exceedances are dependent.
From a regulatory point of view all models passed the test. As mentioned before a trend is not visible. Due to the very small sample size, the parameters are more flexible see Appendix O. This leads to a fast adjustment of the volatility. But again, it has to be said that the t-test of many parameter estimates is not significant with this small sample size. Like in the regulatory test, a clear trend cannot be found here. This is the reason because this model reacts to changes in the market very fast and does not constantly overestimate the VaR.
For the confidence level of With a leptokurtic student t distribution the volatile parameters lead to a very high overestimation at this confidence level. More stable parameters are more suitable for higher confidence levels. As already mentioned, the sample size is crucial for the estimation process. Especially for small observation periods, the robustness of the model lacks. Even though the model passes all backtests, it is possible that the calibration process is not stable and therefore from a statistical point of view not valid.
Therefore the author does not recommend choosing a small sample size like days. For some portfolios even sample sizes around can be too small. The analysis of the different time frames results in the conclusion that for the Bond Portfolio the difference between the pre and post backtesting results are higher than for the EFFAS Bond Index Portfolio.
Portfolio, one can see that the results of the backtest are considerable better in the pre time. It is therefore not a valid answer to say that the model delivers a VaR estimation that is equal good for each timeframe. One conclusion from that could be that the regime switch had an unexpected effect on the assumed distribution of returns for the EFFAS Bond Index Portfolio. From the backtests above, the results will be summarized shortly. For a confidence interval of Changes in the credit spread will not be captured.
In Figure 15, the annual credit spread of the Bond Portfolio is plotted. The author wants to evaluate to which extent the two back tests differ. By comparing Figure 3 and Figure 4, one can see that the short-term interest rates in years to could not be replicated fully with only 3 principal components. Therefore the author decided to perform the tests also with 7 principal components. In the tests can be seen that an increase in the principal components does not lead to a more accurate VaR estimate.
In all cases, the test between 3 PC and 7 PC was identical, or the differences were minor. One reason for this is that the most cash flows are mapped to higher maturities. Those higher maturities were possible to replicate even with 3 PC without any loss of information. In general a smaller observation period leads to a more dynamic behaviour of the VaR estimate.
A comparison between advanced Value at Risk models and their backtesting in different portfolios
The higher the observation period the smoother the VaR estimate will be. In the case of days the chance of many exceedances is very high, if the interest rate landscape changes very fast. At a confidence level of It can be seen that the backtest result depends highly on the underlying portfolio. The independence test indicates that for the Bond Portfolio especially longer observation periods are affected of VaR dependency. In those cases, several VaR exceedances are consecutive.
The graphical analysis in Appendix N yields to the conclusion that many clusters appear in the time, where the short term interest rate level and the credit spreads increased especially for longer observation periods. The results of the regulatory test are clear. The higher the observation period, the worse the regulatory backtest is. This result is not fully consistent with the results of the unconditional coverage test. The results suggest that at some point in time, the maximum exceedances of the last days are very high with a sample size of e.
On average, the exceedances are still low otherwise the model would not pass the unconditional coverage test. The economic costs are directly linked to the results of the regulatory test. Due to the reason that the results got worse with a higher observation period, the results are here similar. As already mentioned the results do not differ between 3 PC and 7 PC notable.
For a confidence level of Here, 7 PC yield to the best result for the Bond Portfolio.
The results from the table below are for the Bond Portfolio significant. The unconditional coverage test in the pre time yields to far better results than in the post time. This yields to the conclusion that the model has problems to react appropriate to a regime switch. In Appendix N this circumstance can be seen. By applying the procedure described in chapter 0, the conclusion is that for the Bond Portfolio, no combination at both confidence intervals is appropriate.
For the EFFAS Bond Index Portfolio, one can see that the best results yield from the model with 7 principal components based observation period of days. Both variants yield only to the yellow zone of the regulatory test. The price of each bond with annual coupon payments C and a notional of N is calculated in this way:. The only uncertainty in this equation is the interest ratezt from the zero-coupon yield curve.
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The reader can see in Figure 16 the divergence between the real price and the present value of the portfolio. This graph in combination with Figure 15, suggests that until mid of the credit spread had a minor role. The fact that the credit spread widened does not necessary mean that the returns at high confidence levels differ to a great extent. One reason for this is, that the present value of the bond reacts instantaneous to changes in the zero-coupon yield curve. The observed market prices are duller and do not change immediately to the fair value of the bond. By looking at these results, the reader can see that the regulatory backtest has a very high tolerance.
Models that do not pass the conditional or unconditional coverage test do pass the regulatory test. One reason for that is the very small probability of a type I error at this test. The sample size of days with 7 principal components yields for each portfolio and confidence level to the lowest economic costs. Again, the difference between 3 and 7 principal components is not notable. The unconditional coverage test yields for the pre to better results for both portfolios. In Figure 17, the reader can see the first 3 eigenvectors of the zero-coupon yield curve development over 8 years.
The first three principal components explain approx. These eigenvectors can have different shapes, for different times. The first principal component describes a parallel shift. The second eigenvector shows that an increase in short maturities does not affect longer maturities. This is mainly driven by the strong changes in short-term interest rates between and which can be seen in Figure 2. The third eigenvector describes that a decrease in mid-term maturities yield to an increase in long-term maturities.
The second and the third eigenvector differ to the typical movements described in the literature. The second principal component describes a rotation, where the eigenvector is monotonic decreasing with increasing maturity. The third eigenvector is described in the literature similar, as a quadratic function of the maturity, where the function is highest at short and long maturities and lowest at middle maturities.
The main assumption of this model is that changes in the interest rate level have the greatest proportion of influence on the price change of a bond. Therefore, the author assumed that the results of the backtest based on hypothetical returns lead to better results. This was the reason why this additional test was performed. As already described this assumption did not hold.
In general it makes no difference for these portfolios if 3 or 7 principal components were used for the VaR estimation.
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A further development could be to cluster bonds to different risk categories where the credit spread is highly correlated. This credit spread will be added to the interest rate surface. Out of this compound surface a PCA will be performed in a further step. Like in the models before, for each portfolio the sample size differs between , , , and days. The two different simulations were chosen to evaluate if the simulation size makes a difference.
By looking at the tables below, we can see that the results do not differ to a notable amount at a Due to that reason, it is recommendable to increase the number of simulations to avoid this inaccuracy. This result could be misleading. As already mentioned, models with fewer violations than expected have a lower p-Value. Therefore, one could conclude that depending on the available historical data the sample size should be increased as much as possible, even though the unconditional coverage test leads to worse p-Values.
The conditional coverage test penalizes models that are not able to react on volatility clustering. The model with observations and 10, simulations has to be rejected, because it does not react fast enough to jumps in the volatility. In general, one can say that the independency on a The other model variations are all in the yellow zone.
Due to the fact that for every VaR exceed will be penalized with a factor of 1. One reason for this is, that the penalize factor is in relation to the interest rate level rather high. From the table above, the reader can see that the higher the sample size the lower the economic costs. From the time frame analysis we cannot make any suggestions if the model works better in the pre time than in the post time or vice versa.
Up to the last decision criteria, those two variations are equally good. For the EFFAS Bond Index Portfolio, for both confidence levels, the models with a highest sample size of days deliver the best results after performing all backtests. The choice of a VaR model depends in first consequence on the portfolio and its risk drivers. It seems obvious that a linear bond portfolio should not be treated equally than a non-linear option portfolio. In this thesis, only linear bond portfolios were covered with the interest rate as the main risk driver.
The main question is whether the VaR model is able to capture all risks adequately. The other two models capture only the volatility of the observed market prices. In addition to the market volatility the FHS VaR incorporates also the empirical return distribution. The parametric GARCH models are in practice very easy to implement and yield to the best backtesting results of all three models. Even though the VaR estimation is not stable for all regimes it is recommendable to use this model for linear portfolios.