Estimating VaR with Copula Function and Empirical Research

Estimating VaR of portfolio by conditional copula-GARCH method

Contents

1.    Introduction.. 1

2.    Theory of copula.. 1

2.1      Introduction to copula.. 1

2.2      Copula family.. 2

2.3      Estimation method.. 3

2.4      Estimation of VaR.. 3

3.    Empirical results.. 3

3.1      The data and the marginal distribution.. 3

3.2      Copula modeling.. 5

3.3      Estimation of VaR.. 5

4.    Conclusion.. 7

References.. 8

1.     Introduction

Value at Risk (VaR) has become the standard measure used by financial institutions to quantify the market risk of an asset or a portfolio. Estimating VaR with one asset is not difficult, but it becomes complex when the portfolio contains more than one asset. The main difficulties are to model the joint multivariate distribution and the dependence structure.

The traditional approaches for estimating VaR are based on the assumption of multivariate normal distribution and the linear correlation. However, in reality, the distribution of financial data usually has fatter tails than normal distributions. Hence, it is shown in many empirical works that such multivariate distributions do not provide adequate results due to the presence of asymmetry.

In order to overcome these problems, we combine copula and the forecast function of the GARCH model, and then propose the conditional copula-GARCH method to compute the VaR of portfolios. Copula in particular has recently become one of the most significant new tools, it is generally applied in the financial field, such as risk management, portfolio allocation, derivative asset pricing etc. In our work we focus on portfolio risk management, especially in estimating VaR. Copula theory allows us to construct a flexible multivariate distribution with different margins and different dependent structures.

In the empirical analysis, we construct a portfolio composed of NASDAQ and STI indices, and apply various copulas completely with different marginal distribution to estimate VaR for the portfolio. In addition, compared with traditional methods, including the historical simulation method, variance–covariance method and EWMA model, we prove that the Student-t copula-GARCH model captures the VaR of the portfolio more successfully.

The rest of the paper is organized as follows. Section 2 presents GARCH models for marginal distributions. Section 3 presents Sklar’s theorem and the copula families. In addition, we introduce the estimation procedures of VaR. Section 4 presents the empirical analysis and results, followed by a conclusion in Section 5.

2.      Theory of copula

In our project, we use the joint distribution function of returns of individual assets in a portfolio to compute the portfolio’s VaR. We first find marginal distribution of individual returns by GARCH model and then derive the joint distribution of the portfolio by using copula.

2.1   Introduction to copula

This part illustrates the statistical background of how we get the joint distribution of a random vector when its marginal distributions are all known, which is called copula method.

Let 

 denote a random vector with marginal cumulative distribution function

respectively. Applying the probability integral transform:

Then copula is defined as the joint cumulative distribution function of:

When an appropriate copula is found, Sklar’s theorem help us to compute the joint cumulative distribution function of X.

Sklar’s theorem. Every multivariate cumulative distribution function F of a random vector

can be expressed in terms of its marginals and a copula C:

By taking derivatives, we get the result which will be used in our computation of VaR:  

Where c is the copula density function.

So how can we find an appropriate copula? Does it satisfy any conditions? A lemma below tells us an important property of copula.

Lemma. Assume ξ has continuous cumulative distribution function F(x), then θ = F(ξ) follows uniform distribution on [0,1].

Property. The copula’s marginal probability distribution of each variable is uniform.

This property tells us which type of cdf can be selected into the copula family.

2.2   Copula family

       In practice, we need to find the best copula fit for our data. We choose copula from a copula family and compute the model AIC and BIC to determine the best one.

Copula family:

Gaussian copula, the Student-t copula, Clayton copula, Rotated-Clayton copula, Frank copula, Plackett copula, Gumbel copula, the Rotated-Gumbel copula and so on.

       For our project, Gaussian copula and Student-t copula perform very well.

Gaussian copula:

where Φ is the cumulative distribution function of a standard normal and Φρ is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix ρ.
Student-t copula:

where ρ is the correlation matrix of multivariate Student-t distribution and v is the degree of freedom.

2.3   Estimation method

Denote θ the set of all parameters of both the marginal distribution and copula. We use maximum likelihood method (MLE) to estimate this θ. According to the previous result:

the log-likelihood function is:

By maximization, we obtain the maximum likelihood estimator:

2.4   Estimation of VaR

  Suppose our portfolio has two assets X_1,t and X_2,t. The weight of X_1,t is 0.5 and the confidence level is 0.05. We can calculate the VaR by definition:

  Where c is the copula function and f is the marginal distribution.

  We also calculate four other different methods of VaR: Historical simulation; Variance-covariance method; Exponentially weighted moving average(EWMA); and the traditional GARCH-VaR model.

3.     Empirical results

3.1   The data and the marginal distribution

In order to see the performance of copula-GARCH method, we conduct model estimation and analysis based on real-world data. We simply choose two major stock indices, STI and SPX (Singapore Straits Time Index, and S&P500 Index), from Oct 19, 2010 to Oct 17, 2019 with 2199 daily observations. Price data comes from Bloomberg database, and we transfer them into return data by:

Where Pt is the value of index at time t. In addition, to eliminate the spurious correlation caused by holiday divergence, we eliminate those observations which coincide with holidays of either Singapore or the U.S. The market returns and absolute returns of STI and SPX are shown in Fig.1.

Fig.1. Daily returns and absolute returns of STI and SPX

In Fig.1, you can find volatility significant clustering in both historical data. Hence, we use ARCH test to examine the serial correlation of squared return, i.e. ARCH effects (see Table 1)

Table 1. Descriptive statistics and ARCH effects

Table 1 exhibits descriptive statistics on returns and the result of ARCH tests. We can find that both STI and SPX has a negative skewness. The LM(lag=k) results indicate that ARCH effects can be found in both SPI and SPX returns.

Hence, we consider the GARCH model as the univariate marginal model, to fit for the return series as the initial models with normal and student-t distribution. Table 2 exhibits the MLE results, Ljung-box test and ARCH test for model adequacy, as well as the AIC and BIC criterion for model selection.Table 2. Parameter estimates of GARCH model and statistical t

From Table 2, we find that model parameters are significant at 5% significant level, and the result of Ljung-box tests and ARCH tests shows no anymore residual autocorrelations and ARCH effects. Therefore, we conclude that both GARCH-n and GARCH-t models are adequate.

3.2  Copula modeling

We use the copula method based on the marginal distribution discussed above. We use MLE method, and the copula functions will be fitted to the residuals series. The result of copula-GARCH modeling is shown in Table 3.

Table 3. Parameter estimates for Gaussian and Student-t copula and model selection statistics

In Table 3, we can obviously find that the best fitting copula function is student-t copula, whose AIC and BIC are generally smaller compared to Gaussian copula. This means that, student-t copula performs better in describing the dependence structure of the bivariate return series. In particular, the best fitting model among the four model we discuss is the student-t copula GARCH-n model.
3.3 Estimation of VaR

We initially uses the sample-in data, which contains 1699 return observations, to estimate VaR1700, and at each new observation we re-estimate VaR, until the sample-out observations we have updated are used up and we will get 500 tests for VaR.

Figure 2. Estimated VaR with T copula GARCH-n method

The numbers of violations in Table 5 are the numbers of sample observations being located out of the critical value. The mean error shows for each copula function, the total absolute discrepancy per marginal model between the observed and expected number of violations at α = 0.05 and 0.01. From the results in Table 5, the Student-t copula GARCH-n method shows the minimum mean error with a 95% and 99% level of confidence. That is, the Student-t copula GARCH-n is the most accurate model for estimating VaRs of the portfolio. Besides, Fig. 2 shows the VaR plot we estimate using the Student-t copula with a marginal distribution, the GARCH-n model at α = 0.05 and 0.01. In this Figure the VaR of portfolio is located almost below the portfolio returns, and describes the expectation of investment loss well.

Figure 3. Comparison of VaR using various methods at α = 0.05

It is also obvious to see the VaR of the historical simulation method and variance-covariance method are underestimated and this represents the highest mean error at α = 0.05 and 0.01. The univariate GARCH-VaR method is better than the HS and VC methods, but it still underestimates VaR, and the mean error is still high. The EWMA method is also not as good as the method of conditional copula-GARCH. The various VaRs we estimate are shown in Fig. 3 with α = 0.05. In this figure we can see that trends of the HS and VC methods are more flat and these methods can not reflect the risk of portfolio returns with time-varying. The methods of GARCH reflect too much while the estimation of EWMA is always not enough. The t-copula-GARCH-n method captures the extremes most successfully compared with others.

4.  conclusion
Table 5. Number of violations of VaR estimation.
the correlation among the price or volatility behaviors of the financial assets within a portfolio is crucial for the proper estimation of the VaR amount. However, restrictions on the joint distributions of the financial assets within the portfolio might decrease the performance of the VaR estimation. We describe a model for estimating portfolio VaR by the conditional copula-GARCH model, which allows for a very flexible joint distribution by splitting the marginal behaviors from the dependence relation, and which is lack of in traditional methods. We estimates different copulas with different univariate marginal distributions, and traditional methods to compare the results. The Student-t copula describes the dependence structure of the portfolio return series quite well, in which we choose it by the AIC and BIC of the model criterion, producing the best results of the reliable VaR limit. The copula method has the feature of flexibility in distribution, which is more appropriate in studying highly volatile financial markets compared with traditional methods.

References

1. Jondeau, E. & Rockinger, M. (2006). The copula-garch model of conditional dependencies:

An international stock market application. Journal of international money and finance, 25 (5), 827-853.

2. Hotta, L. K., Lucas, E. C. & Palaro, H. P. (2008). Estimation of VaR using copula and extreme value theory. Multinational Finance Journal, 12 (3/4), 205-218.

3. Lu, X. F., Lai, K. K. & Liang, L. (2014). Portfolio value-at-risk estimation in energy futures markets with time-varying copula-GARCH model. Annals of Operations Research, 219 (1), 333-357.