Risk management has been a crucial part of the daily operations of the financial industry over the past two decades. Value at Risk (VaR), a quantitative measure introduced by JP Morgan in 1995, is the most popular and simplest quantitative measure of risk. VaR has been widely applied to the risk evaluation over all types of financial activities, including portfolio management and asset allocation. This paper uses the implementations of multivariate GARCH models and copula methods to illustrate the performance of a one-day-ahead VaR prediction modeling process for high-dimensional portfolios. Many factors, such as the interaction among included assets, are included in the modeling process. Additionally, empirical data analyses and backtesting results are demonstrated through a rolling analysis, which help capture the instability of parameter estimates. We find that our way of modeling is relatively robust and flexible.
Risk management is an old subject for banking industry, regulators and academia. After the “Black Monday”, financial professionals with quantitative background worried about the firm-wide risk management. Thus, big banks started evaluating risks with quantitative tools. In the early 90s, most banks’ methods were already close to the formal concept of Value at Risk (VaR). In 1994, JP Morgan published its extensively developed quantitative methodology on evaluating the VaR and made a free access, the
This paper presents a hybrid method on the one-day-ahead prediction of high-dimensional portfolios VaR. In time series analysis, a popular way to validate the predictive power of a methodology is to use historical data sets sequentially with moving windows of a certain size. An analysis conducted with moving window is often referred as a rolling analysis. This paper also presents real data rolling analyses with sample portfolios and the corresponding backtesting results, utilizing publicly available historical data sets and different moving windows.
Generally, parametric, nonparametric, and hybrid are three specific approaches to define and calculate VaR. Each approach of VaR calculation requires the forward prediction of the variance, because the variance is the source of fluctuation. To estimate the variance in a sound accuracy level or to even actually estimate it at all, it is inevitable to take the time horizon into account. For instance, with a parametric distribution assumption, a simple percentage based VaR can be calculated with the parametric approach:
where
The rest of the paper is organized as follows. Section 2 introduces the background knowledge of the methodology. Section 3 illustrates simple multi-dimensional examples. Section 4 demonstrates 3-, 5-, and 9-dimensional cases. Section 5 concludes the study and gives remarks.
The Autoregressive-Moving-Average (ARMA) model is a combination of the Autoregressive model and the Moving-Average model. The general form of the ARMA(
where
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model developed by Bollerslev (1986), is the extension of the Autoregressive Conditional Heteroskedasticity (ARCH) model (Engle, 1982). The multivariate version of GARCH has many different specifications mainly to resolve the dependency problem. In this paper, Constant Conditional Correlation (CCC) specification (Bollerslev, 1990), its enhanced version, Dynamic Conditional Correlation (DCC) specification (Engle, 2002), and Generalized Orthogonal GARCH (GO-GARCH) model (Van derWeide, 2002) are used in modeling procedures.
The standard univariate GARCH (
where
The multivariate version of GARCH pushes the edge even further to account for the dependency relationship, or covariance, of the multiple volatility that is included in the multivariate time series model. Assume that we have a time-varying
Note that
where
For the bivariate case, the correlation between variables, or two time series, is presumed to be static over the time. The stochastic relationship does not change between two variables over time. Thus, a simple CCC-GARCH(1, 1) is expressed as:
where
According to CCC, the covariance entirely depends on the prediction of the variance terms, once a fixed correlation estimation is provided with a certain historical time horizon. Therefore, CCC accounts for only a partial effect of the interaction between variables in the system. But a major advantage of this specification is the unrestricted applicability for large systems of time series (Franke
As a successor of CCC, DCC specification is proposed, which bridges the gap of time varying interaction between variables (Engle, 2002). The idea is similar to the CCC specification. The DCC specification estimates the covariance and variance separately, but allows the existence of the time-varying correlations instead of assuming that the correlation remains the same. The correlation matrix
with
where
The generalized orthogonal GARCH or GO-GARCH (Van der Weide, 2002) is a special case of the early BEKK specification (named after Baba, Engle, Kraft, and Kroner), which is one of several ground-breaking specifications of multivariate GARCH models. With the assumption that the errors in the standard GARCH model are dependent on some uncorrelated component and can be modeled by that component through a linear mapping. The logic of GO-GARCH can be expressed as:
where
where
With the above assumption and definition,
where
This specification balances the computational cost and the complexity of covariances over time (Broda and Paolella, 2009).
To help understand the modeling process with the combination of the GARCH and ARMA models, let us take DCC-GARCH(1, 1)-ARMA(1, 1) as an example. With DCC-GARCH(1, 1), the variance and covariance of the multivariate time series can be modeled as:
where
Modeling returns’ dependency of different assets is important, especially for portfolios related to a volatile capital market such as future market and derivatives market. The conventional techniques for modeling the dependence among different variables are multivariate distributions, usually the multivariate normal distribution. However, when modeling the dependency among returns from different assets in a portfolio, the dependency is restricted by a specific theoretical distribution assumption, and the multivariate distribution does not provide a flexible fitting. Furthermore, for higher dimensional portfolios, the joint distribution is difficult to estimate. In addition, if the multivariate normal distribution is used to fit the returns, the tail part is automatically assumed to be not fat and the returns are assumed to be symmetrically distributed. Unfortunately, these assumptions are not verified by the market. A popular alternative for dependency quantification is to utilize copulas, which provides different structures to simplify the estimation of a multivariate joint distribution and also allows different marginal distributions to appear in the structures. For random variables
where
Copula can be defined in different ways for different situations. Commonly used copulas include Gaussian copula,
The function
Statistically, VaR_{1–}
where
For example, if the multivariate GARCH(1, 1) and the ARMA(1, 1) are applied to model the returns, the variance-covariance matrix and the error vector on the right hand side of the above equation can be produced from the multivariate GARCH(1, 1) model with distribution assumption for noises included in the error terms. At the mean time, ARMA(1, 1) can produce the modeling results for mean return vector on the right hand side of the equation.
Inspired by the properties of ARMA models, GARCH models and the concept of copulas, the error terms from ARMA-GARCH models for multivariate cases can be modeled and simulated with copulas and marginal distributions, and a sequential modeling process can be implemented to model and predict VaR:
Step 1: Model the log return time series with multivariate ARMA-GARCH.
Step 2: Fit marginal distributions and a copula using the errors from the multivariate ARMA-GARCH.
Step 3: Get the prediction of mean and variance from the multivariate ARMA-GARCH.
Step 4: Simulate the error terms from the results of Step 2.
Step 5: Acquire a set of predicted log return values with results from Step 3 and Step 4, and determine the value at 100 ×
To avoid any confusion, the error terms from GARCH, or ARMA-GARCH refer to the error terms of the mean process. GARCH models inherit the error terms from the setup of the ARCH model, and the error terms applied to model the variance in the ARCH model ultimately refer back to the error terms in the mean process, which is the ARMA model in our modeling process.
The copula fit of the error terms’ dependency structure is the key to the modeling process. We assume the error distribution to be multivariate normal; however, the choice of copula is flexible depending on the emphasis on the tail distribution of the shocks. The estimation of a copula based multivariate distribution involves both the estimation of the copula parameters
where
However, it costs too much time in practice if the dimension is not moderate. Another popular estimating method is called the Inference for Margins (IFM), unlike the former one step FML method, this method estimates the parameter from the marginal distributions
This work is not limited with the gaussian copula or
To examine the adequacy of the VaR measures produced by our copula-based models, we turn to a conventional “backtesting” validation method, which is used through our entire modeling process. The backtesting procedure evaluates the quality of the forecast of a risk model by comparing the actual results with those generated by the VaR models. For example, consider an event that the loss of a portfolio exceeds its reported VaR, VaR(
Thereupon, the hit function sequence, e.g., (1, 0, 1, 0, 0, 0, 0, 1), tells the times that the loss of the portfolio has exceeded the labeled VaR(
Backtests can often be classified by if they examine an unconditional coverage property or independence property of a VaR measure. However, the test is called an Unconditional Coverage (UC) test if we are only interested in the times of the reported VaRs being violated. According to Christoffersen and Pelletier (2004), the interim between two VaR violations should exhibit no duration dependency, which is specified as the independence property. Therefore, the Conditional Coverage (CC) test is a joint test that checks both the violated times and the independence property.
To examine the stability of the modeling methodology and its accuracy, different example portfolios were established with assets from global financial markets including stocks, index futures, and commodities. A key assumption is that the parameter estimates should be constant over time when analyzing financial time series data using the Multivariate GARCH and Copula models. However, in a real financial market, considerable changes make it unreasonable to make this assumption. Therefore, we compute the parameter estimates over a rolling window of fixed sizes, e.g., 250, 500, through the samples. If the parameters are truly constant over the entire time horizon, then the estimates over the rolling windows should not be too different. If the parameters oscillate too much at some point during the period, then the rolling estimates should capture this volatility. Details of the example portfolios can be found in Table 1. We implemented mainly DCC-GARCH for 3-dimensional cases and GO-GARCH for 5-dimensional and 9-dimensional cases. Different copulas, simulation sizes and moving windows were tested. Details of the modeling specification and corresponding backtesting results can be found in Tables 2
With our 3- and 5-dimensional experimental data sets, the modeling process provides a good and stable performance as indicated with two backtesting results, as well as the exceedance. In addition, it is important to pay attention to the balance of the results in the evaluation of such process. The exceedance should not be too far away from 95%, because capital reserve requirements for financial institutions will be evaluated according to the VaR models and VaR Stress Testing performance. If a model is claimed to be at a 95% confidence level but always delivers results close to 99%, the efficiency of capital allocation will be reduced due to the 4% difference. From this perspective, the 3- and 5-dimensional modeling examples also present a good performance. The exceedance of the modeling results are reasonably close to the 95% target level.
For 9-dimensional examples, especially those examples with skewed-
Copula methods are introduced to decompose the continuous joint distribution into individual marginal distributions so that the dependence structure can be easily studied. This raises concerns on the selection of marginal distribution and fitting of the distributions using the copula structures. Commonly used dispersion structures include: autoregressive of order 1, exchangeable, Toeplitz, and unstructured. Some researchers use one of the correlation matrices (usually autoregressive of order one and simple exchangeable Archimedean (Yan, 2007)) as a direct proxy for multivariate dependency. However, it is widely acknowledged that volatility parameters and variables are not always normally distributed. Many financial related variables often have fat tails and exhibit “tail dependence”. In our study, the skewed-
According to the definition of Normal Mean-Variance Mixture Representation of Skewed-
A DCC-GARCH model is designed to overcome the drawbacks on the covariance estimation that early specifications such as BEKK and CCC models present (Van der Weide, 2002); however, the DCC specification is still not perfect due to the limitation on the possible failure of finding an invertible matrix. DCC assumes the existence of at least one orthogonal matrix that links the observed variables linearly to a set of components similar to the concepts of latent variables. Since the corresponding latent components are assumed to be independent, it is now possible to get dynamically time-varying conditional covariance for each of the multivariate time series systems presenting heteroskedasticity. Nevertheless, it is not guaranteed that a linkage or an invertible matrix will always exist. For example, if the diagonal elements of
Through the investigation on applicability of the proposed modeling methodology, we conclude that the proposed modeling process is a good alternative approach to study and predict VaR. With multivariate time series and copula, the proposed modeling methodology keeps a good balance between the complexity and flexibility. At the meantime, it helps reduce the computational cost. Furthermore, a methodology combining different techniques provides an opportunity to conduct the model selection over a system of models which can be easily constructed. The copula technique plays an important role in the description of the dependency relationship. More importantly, it enables us to directly model the VaR with a regard to the interactions among assets at a high dimensional level. During the study and experimental modeling investigation, we also noticed that the effect of the mean time series on VaR was weak when comparing to the variance effect. The fit of the marginal distributions is also not as important as the fit of GARCH and copula. Additionally, there was no universal solution. Therefore, when modeling with the methodology presented in this paper, we should use a greater amount of caution and patience to adjust different components such as moving window size and copula selection. With the careful adjustment, a model selection process based on backtesting results should be conducted. For future studies on the high-dimension dependency structures, a careful evaluation is highly recommended for the difference between the simulated error terms and the realized error terms. As the dimensionality increases, the difference for fitted multivariate dependency structures can be very vulnerable in terms of precision and accuracy, compared to the realized dependency structure.
Setup of example portfolios
Dimension (portfolio no.)a | Assetsb | Time horizon (moving window) | Prediction coveragec |
---|---|---|---|
3 (A) | AMZN, F, RDS.A | 1000 (250) | 01/01/05–01/25/16 |
3 (B) | BA, SPY, GS | ||
3 (C) | DDD, SPY, GS | ||
3 (D) | BAC, DAL, DG | ||
5 (A) | APPL, NFLX, DDD, BA, TUC | 1750 (500) | 01/01/05–01/25/16 |
5 (B) | AMZN, F, GS, RDS.A, SPY | ||
5 (C) | DDD, TUC, F, BAC, MSFT | ||
5 (D) | WMT, C, WMAR, MS, ACET | ||
5 (E) | RDS.A, C, BAC, MS, ACUR | ||
5 (F) | APPL, DAL, ENZN, SPY, NFLX | ||
5 (G) | AMZN, F, GS, RDS.A, SPY | 2550 (250) | 11/01/05–02/27/15 |
9 (A) | APPL, WMT, NFLX, SPY, C, F, AMZN, MSFT, GS | 01/01/05–01/25/16 | |
9 (B) | ACET, C, ACUR, BAC, MS, DDD, TUC, SH, SPY | ||
9 (C) | WSTL, C, ACUR, BA, MS, DDD, TUC, WMAR, ENZN | ||
9 (D) | AAPL, GOOG, NFLX, AMZN, F, MSFT, GS, RDS.A, SPY, DEXJUP | 12/07/05–02/27/15 |
^{a}The combination of dimension and portfolio number of associated dimension is used as the portfolio name in latter tables.
^{b}Assets included in this table are all from open markets such as NYSE and NASDAQ. Missing values are omitted. Portfolios contain only one share of each asset.
^{c}Date format is mm/dd/yy.
Modeling and validation of 3-dimensional portfolios
Portfolio | GARCH | Marginal distributions | Copula (simulation size) | Unconditional coverage | Conditional coverage | Exceedance (95%) |
---|---|---|---|---|---|---|
3 (A) | DCC-GARCH (1, 1) - ARMA(1, 1) | Skewed- | Gumbel (250) | 0.1386943 | 0.2460887 | 3.87% (29 out of 750) |
Clayton (250) | 0.1027796 | 0.3124458 | 4.0% (30 out of 750) | |||
Gaussian (250) | 0.4333781 | 0.2670521 | 5.06% (38 out of 750) | |||
0.3452114 | 0.1640517 | 3.47% (26 out of 750) | ||||
NAC (250) | 0.6221392 | 0.2290134 | 5.06% (38 out of 750) | |||
3 (B) | Gumbel (250) | 0.0026605 | 0.0035204 | 2.8% (21 out of 750) | ||
Clayton (250) | 0.0037568 | 0.0058276 | 7.47% (56 out of 750) | |||
Gaussian (250) | 0.2673953 | 0.4725123 | 5.46% (41 out of 750) | |||
0.0315854 | 0.0701514 | 6.80% (51 out of 750) | ||||
NAC (250) | 0.0213867 | 0.0691763 | 6.93% (52 out of 750) | |||
3 (C) | Gumbel (250) | 0.0962584 | 0.2504802 | 3.73% (28 out of 750) | ||
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.1386943 | 0.3316343 | 3.87% (29 out of 750) | |||
0.1386943 | 0.3316343 | 3.87% (29 out of 750) | ||||
NAC (250) | 0.1033458 | 0.2134489 | 4.40% (33 out of 750) | |||
3 (D) | Gumbel (250) | 0.1935932 | 0.3367489 | 4.93% (37 out of 750) | ||
Clayton (250) | 0.00011298 | 0.0000013 | 2.26% (17 out of 750) | |||
Gaussian (250) | 0.4760852 | 0.4405354 | 5.60% (42 out of 750) | |||
NA | NA | NA | ||||
NAC (250) | 0.5632115 | 0.5253472 | 5.47% (41 out of 750) |
Modeling and validation of 5-dimensional portfolios
Portfolio | GARCH | Marginal distributions | Copula (simulation size) | Unconditional coverage | Conditional coverage | Exceedance (95%) |
---|---|---|---|---|---|---|
5 (A) | GO-GARCH (1, 1) - AR (1) | Skewed- | Gumbel (250) | 0.3208206 | 0.5854642 | 4.40% (55 out of 1250) |
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.7440254 | 0.8036172 | 4.80% (60 out of 1250) | |||
0.9481972 | 0.7883348 | 4.96% (62 out of 1250) | ||||
NAC-Gumbel (250) | 0.7893542 | 0.8231094 | 4.40% (55 out of 1250) | |||
5 (B) | Gumbel (250) | 0.4063832 | 0.6386581 | 5.52% (69 out of 1250) | ||
Clayton (250) | 0.9481972 | 0.7883348 | 4.96% (62 out of 1250) | |||
Gaussian (250) | 0.8450712 | 0.9801981 | 4.88% (61 out of 1250) | |||
0.4063832 | 0.6386581 | 5.52% (69 out of 1250) | ||||
NAC-Gumbel (250) | 0.8964532 | 0.9354267 | 4.88% (61 out of 1250) | |||
5 (C) | Gumbel (250) | 0.1152822 | 0.2105959 | 6.00% (75 out of 1250) | ||
Clayton (250) | 0.8462265 | 0.7121321 | 5.12% (64 out of 1250) | |||
Gaussian (250) | 0.1386943 | 0.3316343 | 5.60% (70 out of 1250) | |||
0.1386943 | 0.3316343 | 5.68% (71 out of 1250) | ||||
NAC-Gumbel (250) | 0.4418941 | 0.6722767 | 6.00% (75 out of 1250) | |||
5 (D) | Gumbel (250) | 0.0523736 | 0.1324256 | 6.24% (78 out of 1250) | ||
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.2592072 | 0.1463091 | 4.32% (54 out of 1250) | |||
0.0394249 | 0.1073051 | 6.32% (79 out of 1250) | ||||
NAC-Gumbel (250) | 0.9481972 | 0.8661561 | 4.96% (62 out of 1250) | |||
5 (E) | Gumbel (250) | 0.0523736 | 0.1520078 | 6.24% (78 out of 1250) | ||
Clayton (250) | 0.3208206 | 0.0733361 | 4.40% (55 out of 1250) | |||
Gaussian (250) | 0.32082062 | 0.5692731 | 4.40% (55 out of 1250) | |||
0.9483278 | 0.0510023 | 5.04% (63 out of 1250) | ||||
NAC-Gumbel (250) | 0.4690168 | 0.0421323 | 4.56% (57 out of 1250) | |||
5 (F) | Gumbel (250) | 0.0223748 | 0.0420299 | 8.00% (100 out of 1250) | ||
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.0026155 | 0.0051958 | 6.96%(87 out of 1250) | |||
0.1152820 | 0.2235687 | 6.00% (75 out of 1250) | ||||
NAC-Gumbel (250) | 0.6524971 | 0.8679374 | 5.28% (66 out of 1250) | |||
5 (G) | NAC-Clayton (500) | 0.4727282 | 0.03804881 | 5.33% (120 out of 2250) |
Modeling and validation of 9-dimensional portfolios
Portfolio | GARCH | Marginal distributions | Copula (simulation size) | Unconditional coverage | Conditional coverage | Exceedance (95%) |
---|---|---|---|---|---|---|
9 (A) | GO-GARCH (1,1) - AR (1) | Skewed- | NAC-Gumbel (250) | 0.2428564 | 0.4969938 | 4.48% (103 out of 2300) |
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.0 | 0.0 | 1.65% (38 out of 2300) | |||
0.0018221 | 0.0031106 | 1.74% (40 out of 2300) | ||||
NAC-Clayton (250) | 0.0228109 | 0.0031106 | 1.74% (40 out of 2300) | |||
9 (B) | NAC-Gumbel (250) | 0.003 | 0.01 | 3.74% (86 out of 2300) | ||
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.0 | 0.0 | 1.48% (34 out of 2300) | |||
0.0 | 0.0 | 1.52% (35 out of 2300) | ||||
NAC-Clayton (250) | 0.0073 | 0.025 | 3.48% (80 out of 2300) | |||
9 (C) | NAC-Gumbel (250) | 0.02 | 0.005 | 3.74% (86 out of 2300) | ||
Clayton (250) | NA | NA | NA | |||
Gaussian (250) | 0.0 | 0.0 | 1.69% (39 out of 2300) | |||
0.0 | 0.0 | 1.83% (42 out of 2300) | ||||
NAC-Clayton (250) | 0.002 | 0.007 | 3.70% (85 out of 2300) | |||
9 (D) | NAC-Clayton (500) | 0.2428564 | 0.4969938 | 4.48% (103 out of 2300) | ||
Gumbel (500) | 0.7731826 | 0.5368803 | 4.87% (112 out of 2300) |