3.3 Semiparametric VaR
3.3.4 Empirical results
Jan04 Jan06 Jan08 Jan10
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04
0.06 Elliptical VaR 10%, 400 STI stocks, Shrink to Identity
Period: 2003 − 2011
Profit and Loss
Figure 12: Elliptical VaR for 400-stocks portfolio based on shrinkage to Identity;
red - 10% VaR.
10%
Shrink to Identity 85 Shrink to Diagonal 90 Shrink to Market 84 Shrink to Two param 60
FFL 91
SIM 94
Table 6: Number of exceedances for elliptical VaR for 400-stocks portfolios based on di↵erent covariance matrix estimators.
3.4 Comparison of estimation methods
Despite of strikingly di↵erent distributional characteristics of portfolio returns the results of the VaR estimation do not di↵er so drastically. The results for the histori-cal simulation are of similar magnitude for all portfolios. This is an expected result, since the value-at-risk depends only on the values of the returns themselves.
For delta-normal VaR one can see a certain positive bias for the two estimators for which portfolio returns were more Gaussian-distributed. The same holds true for the Monte-Carlo simulations based on normal distribution. However, for other returns series the number of exceedances are drastically high which makes these methods certainly impossible to apply in practice.
This justifies introduction of a new method which takes into account leptokurtic tails of distribution, namely, semi-parametric VaR considered in the last subsection.
It should be noted that the numerical procedure for integration and quantile search is quite lengthy, therefore, the estimation period was reduced up to the end of 2011 and only 10
4 Conclusion
In the following thesis several critical issues of the modern finance were addressed:
first, several covariance matrix estimators for high-dimensional data were analyzed;
secondly, the issue of leptokurtic tails was addressed in value-of-risk computation.
This analysis based on empirical data allows to draw certain conclusions with respect to the di↵erent methods of high-dimensional covariance matrix estimation as well as with respect to the value-at-risk calculation.
In the theoretical literature there exist three main approaches to covariance ma-trix estimation when the dimension is greater than the sample size: factor-based approach, shrinkage approach and direct operations on sample covariance matrix.
Although based on di↵erent methodology all of them aim at obtaining a lower di-mensional representation of high-didi-mensional data. Moreover, some of them despite of being contrasted to each other seem to arrive at the similar result. Thus, for example, shrinkage estimators when the number of parameters to be estimated is high (in this case, p parameters are considered to be a ’high’ number to estimate)
produce similar portfolio volatility as covariance matrix estimators based on factor models. Therefore, one can conclude that the true distinction is not in the method itself, but in the magnitude with which the high-dimensional data is reduced to a lower-dimensional space - the more the data is ’squeezed’ into lower subspace, the greater are the di↵erences between the estimators.
Given that one can certainly think of a situation when common factors (suggested by Fama and French, for example) are indeed not relevant for the market of interest.
In such a situation the shrinkage estimators developed by Ledoit and Wol↵can serve as substitutes for the factor-based models.
It is important to note that di↵erent methods perform di↵erently in calm and crisis periods. Although the maximum shrinkage result in general in lower volatil-ity a portfolio, in certain periods less dimension reduction perform as good sa high amount of shrinkage or there are even periods when estimators based on less dimen-sion reduction perform better. For an investor this means that it can be beneficial to switch between various estimators depending on the expected situation on the market. High amount of shrinkage can be also viewed as a hedge against increased volatility .
Moreover, as a result of optimization with di↵erent covariance estimators one obtains portfolios have di↵erent values which are distributed di↵erently. Portfolio returns with more shrinkage are more ’normally’ distributed than the portfolio re-turns with less amount of shrinkage or factor-based models. This is explained by the fact that the latter estimators are more responsive to the market, thus, the volatility of the market is reflected in the volatility of a portfolio. Therefore, the methodology of value-at-risk calculations should be adjusted respectively.
For the future research it is interesting to consider di↵erent reduction methods which will incorporate the information available in the market. For example, if one knows that certain stocks are positively correlated, then shrinkage to identity can be substituted for the shrinkage to constant variance and certain predefined positive covariances. Moreover, multivariate shrinkage targets should be considered.
Valuation of risk should be adjusted for di↵erent estimators, otherwise, the value-of-risk can be over- or underestimated.
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6 Appendix
Jan05 Jul07 Jan10 Jul12
0 0.2 0.4 0.6 0.8 1 1.2
Shrinkage intensities for different estimators based on daily returns of 200 STI stocks
Period: 2003 − 2013
Shrinkage intensity
Figure 13: Shrinkage intensities for di↵erent covariance estimators (400 stocks); red:
shrinkage to constant correlation, blue: shrinkage to identity, green: shrinkage to diagonal m., cyan: shrinkage to market, magenta: shrinkage to two parameteres.
Jan05 Jul07 Jan10 Jul12
Standard deviations of GMV portfolios based on different covariance matrix estimators, STI index
Period: 2003 − 2013
Average returns of GMV portfolios based on different covariance matrix estimators, STI index
Period: 2003 − 2013
Return
Figure 14: Standard deviations and average returns of 200 STI-stocks GMV portfolio with di↵erent covariance estimators; blue: shrinkage to identity, green: shrinkage to diagonal m., cyan: shrinkage to market, magenta: shrinkage to two parameteres, yellow: FFL, coral: SIM.
Jan05 Jul07 Jan10 Jul12
−0.1
−0.05 0 0.05 0.1 0.15
Sharpe ratios for 95 samples based on different covariance matrix estimators, STI index
Period: 2003 − 2013
Sharpe ratio
Figure 15: Sharpe ratios for 200 STI-stocks GMV portfolio with di↵erent covari-ance estimators; blue: shrinkage to identity, green: shrinkage to diagonal m., cyan:
shrinkage to market, magenta: shrinkage to two parameteres, yellow: FFL, coral:
SIM.
0 0.02 0.04 0.06 0.08 0.1
1 2 3 4 5 6
Boxplot for standard deviations for GMV portfolio based on different covariance estimators
Standard deviation
−1.5
−1
−0.5 0 0.5 1 1.5
1 2 3 4 5 6
Boxplot for returns for GMV portfolio based on different covariance estimators
Returns
Figure 16: Boxplot for monthly standard deviations and returns of a 200 STI-stocks GMV portfolio with di↵erent covariance estimators, 1: shrinkage to identity, 2:
shrinkage to diagonal m., 3: shrinkage to market, 4: shrinkage to two parameteres, 5: FFL, 6: SIM
Declaration of Authorship
I hereby certify that this master thesis has been composed by me and is based on my own work, unless stated otherwise. No other person’s work has been used without due acknowledgement in this master thesis. All references and verbatim extracts have been quoted, and all sources of information, including graphs and data sets, have been specifically acknowledged.
Natalia Sirotko-Sibirskaya
Berlin, November 19, 2013 Signature: