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This article is the first to provide a detailed method for range-based CARR model to estimate the VaR and its out-of-sample prediction. In this paper, we use GARCH and CARR volatility models to compare the VaR’s out-of-sample forecasting performance. Using the historical simulation method as benchmark for VaR estimation, we found that the historical simulation approach for VaR measurement is more conservative than GARCH and CARR methods. The mean violation rate for the CARR VaRs is lower than that of the GARCH VaRs. Meanwhile, the CARR VaR is able to deliver lower required capital levels without producing bigger violations. This paper argued that the CARR VaR valuation approach is suitable as an internal model method for financial institution in VaR forecasting.

There is no consistent approach for forecasting market risk. Value-at-Risk (VaR) has become a standard benchmark for measuring financial risk. The formula for determining the market risk capital was set by 1996 Market Risk Amendment to the 1988 Basel I Accords. An international financial institution experiencing an excess number of violations based on their required VaR will be penalized by having to set aside a larger amount of risk capital. The VaR models are able to be broadly defined as a quantitative tool whose goal is to assess the possible loss that can be incurred by an international financial institution over a given time period and for a given portfolio of assets. A common, though not very good, assumption is that the change in the asset value over the time horizon is normally distributed. The mean change in the asset value is usually assumed to be zero. Also see Hull [^{−1} is the inverse cumulative normal distribution. Namely, regardless of the time period, the measurement of VaR for a specific confidence level is proportional to the volatility of asset’s value change.

The volatility of portfolio return is time-varying and predictable, but forecasting the future level of volatility is not easily. One of the main reasons is the volatility forecasts are sensitive to the specification of the volatility model. Conventionally, the returned based GARCH family models are popularized for describing volatility process. Surveying from a bundle of related literatures, the GARCH family models have provided effective tools in estimating the volatility process. Also see Engle [

Giot and Laurent [^{1} They proposed that the skewed Student APARCH model performs best in all six commodity markets. Giot and Laurent [

The range variable is much more efficient volatility proxy, a fact known since the paper of Parkinson [

Above all, Chou [

In this paper, we aim to address the VaR topic by comparing the performance of a return-based daily GARCH model with the performance of a CARR model based on the daily range volatility model when the one-day-ahead VaR measure is to be calculated. Chen, Gerlach, Hwang and McAleer [^{2} The CSI 300 index covers more than 90% of the total market value in Chinese stock markets. All the data in this paper are collected from the dataset of Bloomberg. For a better asset allocation, it is essential to realize the VaR measurements for China financial markets with various volatility models for investors. The VaR modeling is an intuitive application of volatility models as VaR measures are directly related to the expected volatility over the specific time horizon given some confidence level.

The rest of this paper is organized as follows. Section 2 reviews the GARCH and CARR models for estimating volatility process and its application for VaR measurement. Section 3 describes the data and empirical results. Finally, the conclusion is organized in Section 4.

An important feature of GARCH model is the asset value’s conditional variance h_{t} which is determined by model parameters and recent return observations. Let daily return for asset is r_{t} at time t. The simple GARCH(1,1) model can be expressed as Equation (1):

r t = ε t

ε t | I t − 1 ~ N ( 0 , h t ) (1)

h t = α 0 + α 1 ε t − 1 2 + β 1 h t − 1

where α 0 , α 1 and β 1 are parameters to be estimated. I_{t}_{−1} denotes the available information set until time t−1. The GARCH parameters can be estimated with data available up to time t.

Basically, the 99 percent VaR prediction at time t is given by

V a R t + 1 | I t = r t + 1 − 2.33 σ t + 1 2 (2)

The symbol of r t + 1 denotes asset’s return at time t+1. The symbol of σ t + 1 2 denotes the conditional variance of the asset.

We collect the daily data of CSI 300 index spot and futures series from 24 May 2010 to 16 December 2016. There are 1597 daily data for this empirical study. To obtain stable estimates for the initial period, estimates for days 1 through 1,000 are in-sample to construct the GARCH model.^{3} While comparing the performance for VaR prediction based on the GARCH model, we introduced the historical simulation (HS, hereafter) for VaR estimation with the 1000 sample size as the criticalvalue. Days when the actual loss exceeds VaR are referred to as exceptions or violations. If exceptions happen on about 1% of the days, we can feel reasonably comfortable with the GARCH model for computing VaR. Rolling out-of-sample forecasts begin after day 1000. Out-of-sample estimates are updated daily. Given parameter estimates, we get the next day’s 99 percent VaR assuming normality of the model’s disturbance term.

Based on GARCH(1,1) model, we can take the expectation value for the conditional variance equation for one-step-ahead. Hence,

h t + 1 = α 0 + α 1 ε t 2 + β 1 h t ,

E t ( h t + 1 ) = E ( α 0 + α 1 ε t 2 + β 1 h t ) = α 0 + ( α 1 + β 1 ) σ 2 + α 1 ( ε t 2 − σ 2 ) + β 1 ( h t − σ 2 ) = σ 2 + α 1 ( ε t 2 − σ 2 ) + β 1 ( h t − σ 2 ) = h ^ t + 1 (3)

Here the E t ( h t + 1 ) indicates the conditional variance on time t+1 given information set on time t. σ 2 is the unconditional variance for this GARCH model. The unconditional variance can be regarded as the long term average variance for this series. Meanwhile, it represents that the GARCH prediction can be thought of as a weighted average of unconditional variance. It is easily to obtain the unconditional variance ( σ 2 ) by taking expectation from the variance equation of GARCH model, namely E t ( h t ) = E t ( α 0 + α 1 ε ( t − 1 ) 2 + β 1 h t − 1 ) . In other words, σ 2 = α 0 + α 1 σ 2 + β 1 σ 2 . So, the unconditional variance is given by σ 2 = α 0 / 1 − α 1 − β 1 .

The GARCH parameters are estimated with the 1000 daily data. To get stable estimates for the initial period, forecasts for 1 through 1000 are in-sample. Next the GARCH parameter estimated are fitted from the sample 2 through 1001. Rolling out-of-sample forecasts begin after day 1000. One-day-ahead estimate is updated daily. We collect 596 VaR estimates from the GARCH-VaR approach. Then, comparing to the 99 percent VaR benchmark value which is inferred by historical simulation method based on the same in sample period. Finally, we are able to compare the targeted violation rate to the predicted violation rate.

The range based models are also suitable to delineate the process of asset’s volatility. The related studies indicated that range variable has relative efficient in describing volatility process, but did not many empirical support. Chou [

R t = λ t μ t

μ t | I t − 1 ∼ exp ( 1 ; . ) (4)

λ t = ω + α R t − 1 + β λ t − 1 .

The variable R t is the difference between ln(H_{t}) and ln(L_{t}), and H_{t} and L_{t} are the highest and lowest intraday price in a trading day respectively. The variable λ t is the conditional mean of the range during the trading day t. The symbol μ t is the innovation assumed to follow the exponential distribution with a unit mean. Finally, the ω , α and β are parameters to be estimated. Similar to the out-of-sample prediction for GARCH model, we can obtain the one-step-ahead prediction that is derived from CARR model.

λ ^ t + 1 = ( E λ t + 1 | I t ) = ω ^ + α ^ R t + β ^ λ t (5)

The same sample size is used to estimate the coefficients of CARR model. Following the Equation (5), it is easily to get the one-day-ahead volatility prediction. Again, the CARR parameter estimates are obtained from the sample 2 through 1001. Similar to the GARCH model, the 596 range-based volatility estimates for the one-day-ahead prediction could be obtained from the CARR model.

Due to the range-based volatility scale for CARR model is different from the return-based scale for GARCH model. For the comparative purpose in VaR estimation with these two volatility models, the range volatility indicators from CARR model has to be adjusted.

h ^ t = ψ λ ^ t + e t , t = 1001 , 1002 , ⋯ , 1597 (6)

where h ^ t is conditional variance estimate from the GARCH model and e_{t} is the residual term. Let the adjusted forecasted volatility from CARR model at time t is AFV_{t}. We can transfer the range volatility indicator as AFV_{t} series in Equation (7) which have the same scale with return-based GARCH model.

A F V t = φ ^ λ ^ t (7)

Thus, we can use the concept of Equation (2) to express the one-day-ahead out-of-sample VaR forecast for CARR model.^{4}

We collect the daily observations of CSI-300 stock index and its futures index from 24 May 2010 to 16 December 2016.^{5} There are 1597 daily data for our VaR study. All the data are collected from Bloomberg.

The coefficient estimates of GARCH and CARR models for in-sample data, namely, day 1 to day 1000 are demonstrated in

The mean value of VaR represents the average for all of the one-day-ahead out-of-sample VaR forecasting based on the historical simulation method, GARCH model and CARR model and displays their results in column 3, 4 and 7 respectively. The number of out-of-sample size is 597. In

The historical simulation approach for VaR measurement is appropriate as a benchmark in analyzing the VaR violation rate for GARCH and CARR models. Judging from the column 5 and 8 in

GARCH | CARR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

r t = ε t | R t = λ t μ t | |||||||||

ε t | I t − 1 ∼ N ( 0 , h t ) | μ t | I t − 1 ∼ exp ( 1 ; . ) | |||||||||

h t = α 0 + α 1 ε t − 1 2 + β 1 h t − 1 | λ t = ω + α R t − 1 + β λ t − 1 . | |||||||||

CSI-300 Spot | α ^ 0 | α ^ 1 | β ^ 1 | Q(12) | LLF | ω ^ | α ^ | β ^ | Q(12) | LLF |

1.063 (1.987) | 0.133 (3.765) | 0.814 (15.537) | 21.319 [0.445] | −2293.301 | 0.381 (3.415) | 0.262 (8.374) | 0.676 (16.655) | 7.320 [0.831] | −1835.309 | |

CSI-300 Futures | α ^ 0 | α ^ 1 | β ^ 1 | Q(12) | LLF | ω ^ | α ^ | β ^ | Q(12) | LLF |

1.021 (2.432) | 0.211 (4.153) | 0.712 (16.021) | 23.551 [0.512] | −2107.547 | 0.372 (2.827) | 0.237 (6.593) | 0.612 (12.613) | 7.412 [0.845] | −1722.232 |

Notes: 1) Sample period is from 24 May 2010 to 16 December 2016. All the daily data numbers are 1597. 2) Numbers in parentheses are t values. Numbers in square bracket are p values. 3) All the original data are collected from Bloomberg.

Historical Simulation | GARCH VaRs | CARR VaRs | ||||||
---|---|---|---|---|---|---|---|---|

Obs. | Mean VaR | Mean VaR | Number Violation | Mean Violation | Mean VaR | Number Violation | Mean Violation | |

CSI-300 Spot | 597 | −3.27 | −3.12 | 7 | −3.35 | −3.01 | 5 | −3.32 |

CSI-300 Futures | 597 | −3.56 | −3.34 | 8 | −3.47 | −3.23 | 4 | −3.38 |

Notes: This table displays back testing of GARCH and CARR Value-at-Risk forecasts. All of the markettrading data are collected from Bloomberg. The GARCH VaR forecast is based on a simple GARCH(1,1) with conditionally normal innovations and a CARR(1,1) with conditionally exponent innovations. All of these mean VaRs are calculated by daily data.

Additionally, the number violation in

This paper has presented the empirical study on the performance of GARCH and CARR VaR models for CSI-300 spot and futures index. The results show that the VaR out-of-sample prediction for the range-based CARR volatility model is better than the traditional return-based GARCH model. Historical simulation approach in VaR estimation appears more conservative phenomena that imply higher levels of capital coverage for trading risk. In other words, the predictive VaR by historical simulation approach is less useful as a measure of actual financial risk. Despite the insightful information employed by GARCH model, its VaR prediction did not outperform the CARR model for VaR estimation. Even though the time series models of range variable cannot account for position’s sensitives to current risk shocks. But their flexibility and parsimony are precise and convenient for modelling the VaR information. The range-based CARR model may have advantages in forecasting and as an alternative method for identifying the shortcomings of other VaR valuation approaches.

Wu, C.C. (2018) Measuring and Comparing the Value-at-Risk Using GARCH and CARR Models for CSI 300 Index. Theoretical Economics Letters, 8, 1179-1187. https://doi.org/10.4236/tel.2018.86078