Value at Risk Models in Indian Markets: A Predictive Ability Evaluation Study

Value at risk (VaR) is a method of measuring the potential loss in portfolio value for a given distribution of historical returns over a given time period. Measurement of risk therefore becomes essential for a corporate decision. This study attempts to rank the overall predictive ability of select value at risk models in estimating market risks of Indian financial markets. This study estimates the respective predictive ability by employing numerical and graphical measures. The findings plug the gaps in the literature and estimate the best method to be used in the industry. The results evidentially prove that parametric model using normal distribution with GARCH (1,1) fits best for estimating value at risk.


Introduction
Risk is an important element when evaluating the effectiveness of business operations. A Risk management plan can help a firm identify future losses, operational inefficiencies, reduce uncertainty and ultimately provide a healthier bottom line. Due to increased global competition, increasing regulations, financial engineering leading to development of complicated securitisation and derivative product, risk management is gaining huge importance. One of the most important steps in risk management is risk measurement. Risk is measured using some common tools such as standard deviation, beta, value at risk and conditional value at risk or sophisticated risk models can also be developed for better results.

Value at Risk (VaR)
Value at risk (VaR) is a method of measuring the potential loss in portfolio value for a given distribution of historical returns over a given time period.
The maximum possible periodic loss under normal circumstances at a certain confidence level is represented by VaR i.e. Value at Risk. If the loss exceeds VaR, the expected loss is known as Expected Shortfall (ES) or Conditional Value at Risk (CVaR). Like volatility, with respect to a long term basis, VaR may be extrapolated by multiplying it by the square root of the number of days (i.e. the square root rule). Illustratively, if the daily VaR is to be converted to annual VaR, multiply the daily VaR by the square root of 252 (considering 252 trading days in a year).

Scope
The study aims to identify, measure and predict market risk for positions in three types of markets viz, Stock Indices, Commodities and Exchange rates. The research intends to use various parametric and nonparametric models to forecast the Value at Risk and also answers why market risk assessment and management are essential for financial institutions.

1) Analysis and forecasting of one-day value at risk for positions in Nifty 50
Index, INR/USD and gold bullion.
2) Determining the accuracy of four VaR models viz. SMA, EWMA, GARCH and Historical Simulation, in predicting one-day Value at Risk.
3) To rank the overall predictive ability of select value at risk models in estimating Market Risks of Indian Financial Markets.
The above objectives were researched by extensive study performed at NMIMS University, Mumbai.

Review of Literature
Economic and financial activities globally have been impacting the NASDAQ composite since the last one decade. The paper by [1] KeithKuester, (2006) compared the performance of various existing approaches and some new models for predicting value-at-risk (VaR) in a univariate context. A hybrid method, combining a heavy-tailed generalized autoregressive conditionally heteroskedastic (GARCH) filter with an extreme value theory-based approach exhibited the best results when applied on 30 years of the daily return data on the NASDAQ Composite Index. Also, a new model based on heteroskedastic mixture distributions showed suitable performance. An extension to a particular Conditional autoregressive VaR (CAVaR) model was provided as most CAVaR models perform inadequately.
The paper by [2] Jimmy Skoglund, Donald Erdman, Wei Chen, (2010) evaluates risk models by backtesting the performance of VaR models in predicting The results show that extreme value theory (EVT) performs better than the parametric models as it focuses on the fat tail risk as well.
[10] Timotheos. A, (2010), in the research, analyze the behaviour of the risk management techniques and models for both long and short VaR trading positions. The study investigates three markets form the period of January 3rd 1989 to June 30th 2003. No single model seems to provide statistically acceptable value at risk (VaR) estimate for all securities. However, the forecasting ability of parametric model under GARCH produces comparatively better results.
The seminal work by [11] Alfred Lehar, Christian Schittenkopf and Martin Scheicher, (2002) compares the Black-Scholes framework, namely GARCH and Stochastic Volatility (SV) option pricing model performances in order to eventually estimate the Value at Risk (VaR) for an options portfolio. The models are applied to UK's Financial Times-Stock Exchange 100 Index (FTSE 100) option prices. GARCH clearly dominates stochastic volatility and shows significant overall improvements in pricing performance.
[12] Walsh M. David and Tsou Yu-Gen Glenn, (1998) compare the historical volatility model, an improved extreme-value method (IEV), ARCH/GARCH class of models and an EWMA model of volatility. The data used included the three price indices collected every five minutes from 1 January 1993 to 31 December 1995. The hourly data analysis showed the EWMA and GARCH (1,1) techniques to be the best predictors, depending upon the loss function used, though the difference between them being very slight. The results for the daily data were also same, thereby creating difficulties in identifying the better one between EWMA and GARCH. The weekly data results indicated that EWMA was the best predictor for weekly volatility.
A comparative study by [13] Yu Jun, (2002) evaluates the performance of 9 alternative models for predicting stock price volatility using daily New Zealand data from the period 1980 to 1998. Evaluation method: root mean square error (RMSE), the mean absolute error (MAE), the Theil-U statistics and the LINEX loss function. The RMSE statistics indicated that the SV model provided the most accurate forecast. Despite its simplicity, the random walk model was not found to be a very good forecaster according to RMSE. The MAE statistics favoured the exponential smoothing model, while the SV model came as the second-ranked. Under the Theil-U statistics, SV model again was the best and only one model (i.e. ARCH model) performs worse than the random walk model. When "a" was assigned a value of 10 the LINEX loss function identified SV as the best. With a value of −10 and −20 also LINEX loss function again selected SV

Data Period
The study investigates three markets: equity, currency and commodity for a period of 20 years so as to incorporate different economic conditions. There are a total of 4981, 5281 and 5081 observations for Nifty 50 returns, INR/USD returns and gold returns respectively ( Table 1).
The daily close price for each of the asset is considered as the input to the various models. The close prices are then converted into lognormal returns using the following equation: Ln = log normal.

Sources of Data
The secondary sources of data used in this study are obtained from NSE, BSE, RBI and Bloomberg database amongst others.

Various Approaches for Estimating VaR
The VaR Estimation Methodologies have two types of classification ( Figure 1). The first classification is based on the type of portfolio position or exposure. A portfolio may have linear exposures like stocks, forwards and futures or non-linear exposures such as options, embedded options and structured products.  The Delta Normal Valuation Method is to be used in case of linear derivative such as forwards, Futures and swaps.
The Delta Gamma Valuation Method is to be used in case of non-linear derivatives which are well behaved like options and non-option embedded bonds.
The Full Revaluation Method is to be used in case of non-linear misbehaved series like option embedded bonds i.e. callable bonds or puttable bonds and for structured products like Mortgage Backed Securities (MBS). It is also to be used when there exist cross partial effects ( Figure 2). The second type of classification is based on the type of data used for calculating VaR. Historical based approaches use historical data to calculate the volatility for VaR while the implied volatility approach uses option price and Black and Scholes Model (BSM) to back calculate the forward looking implied volatility.

Parametric Approaches
Parametric Approaches are based on the assumption that the asset returns follow a certain distribution, say normal distribution. It is also known as Delta Normal Method, Analytical Approach or Variance Co-Variance (VCV) Approach.
1) Why volatility is required for calculating VaR: Suppose we have a portfolio whose returns are normally distributed with a daily mean = 0 and daily volatility = 1.5%. The portfolio is currently worth $1000 million. The 1-day VaR at 99% confidence interval is as follows: For an area of 1 % in a tail z = 2.33.
VaR (in $ terms) = 3.5% of $1000 million = $ 35 million. Thus, maximum possible loss that can take place in a day is $ 35 million with 99% confidence i.e. with just 1% chance of being exceeded. In 1% of the worst cases, loss would be at least $ 35 million.  The above example goes to show that the standard deviation is a necessary input for estimating the Value at Risk.
2) Observed properties of volatility A volatility estimation engine must capture the following three properties of volatility: a) Adaptability: Volatility is dynamic or regime switching so the engine should keep on forecasting new volatility based on innovation. b) Persistence: Volatility is persistent or sticky, i.e. it tends to cluster around the current value. c) Mean Reverting: Volatility tends to revert back to a particular mean reversion level. 3

) Parameter Estimation (MLE) and choice of window size Parameter Estimation Techniques
The technique is designed in such a manner that we try to maximize the probability of the observations occurring. The Maximum Likelihood Estimation technique is used in the study for parameter calibration of the various probability distributions considered in the VaR models.

Choice of window size
There is a trade off between statistical accuracy i.e. precision and adaptability when selecting a window size for a volatility estimation engine.
The study uses 1000 days window size to incorporate persistency. The VaR is estimated using approximately past 4 years data.

4) Method 1: VaR using SMA
The simple moving average is also known as the historical standard deviation approach. The unbiased formula for calculation variance under the SMA approach is as follows: where, μ = Past Returns k-window size. The beauty of EWMA is the requirement of less storage. We just need to store the previous day's estimate of volatility i.e. σn−1 and the recent innovation σn−1.

5) Method 2: VaR using EWMA
The RiskMetrics approach is just an EWMA model that uses a pre-specified decay factor for daily data, which is equal to 0.94 and a factor equal to 0.97 for monthly data. 6) Method 3: VaR using GARCH One of the most popular methods of estimating volatility is the generalized autoregressive conditional heteroskedastic (GARCH) (1,1) model. The best way to describe GARCH (1,1) model is to take a look at the formula: where: α = weight on the previous period's return β = weight on the previous volatility estimate ω = weighted long-run variance = ƔVL. VL = long-run average variance = ω/(1 -α − β)α+ β + Ɣ = 1α + β < 1 for stability so that Ɣ is not negative.
If α + β > 1 then Ɣ < 0, then GARCH model becomes unstable, as the variance becomes mean fleeing then compared to mean reverting. 7) Analogy between EWMA with GARCH  EWMA (Exponential weighted moving average) is a special case of GARCH when α + β = 1.
 Conceptually GARCH is better than EWMA as it captures mean reverting tendency, which EWMA approach doesn't.
 EWMA is considered better than GARCH, as it is parsimonious. Requires just one parameter estimation i.e. λ. Whereas GARCH requires three parameters α, β and ω.
 However, empirically it was found that GARCH performed better than EWMA for each asset class under consideration (Figure 3).

Non Parametric Approaches 1) Historical Simulation
Historical Simulation does not assume any particular distribution. It also does not suffer from the problem of fat tails as it considers the extreme values of the returns as well.
The following steps were performed to estimate VaR under HS approach: Step 1: Collected past returns of each asset class.
Step 2: Arranged them from worst to best.
Step 3: Sliced 1%, 5% and 10% from the worst side. Theoretical Economics Letters 2) Historical Simulation with Bootstrapping Historical Simulation with Bootstrapping is an improved approach of Historical Simulation approach. It involves random sampling from past observations with replacement. We then arrange the data from worst to best. To get the VaR, we slice the significance level percentage from the worst side.

Analysis and Findings
This study employs both numerical measures and graphical measures to evaluate the predictive ability of the VaR models as stated in the previous section. Multiple analytical techniques as mentioned in the figure below may be employed ( Figure 4).

Descriptive Analysis
Descriptive analysis involves describing the characteristics of data i.e. summarising the data. a) Numerical Measures (Table 2 The above figure reflects the INR/USD log returns. The stationarity and volatility clustering in the market is visible (Figure 8).
The daily gold close price from 1 st Jan 1999 to 31 st Dec 2018 is chalked in the figure above (Figure 9). The horizontal axis corresponds to time while the vertical axis displays gold close price.    Even ingold returns, the stationarity and volatility clustering in the market areee visible (Figure 10).

Unit Root Analysis
Unit root analysis is tested using the Augmented Dickey-Fuller (ADF) test. To test for stationary, we conducted the ADF test. It is used for larger and more complicated set of time series models. We can perform GARCH forecasting only on stationary data. Thus, the data needs to be made stationary.       When having a unit root ρ1 = 1 in the following equation This is called the augmented Dickey-Fuller (ADF) test and implemented in many statistical and econometric software packages. We have conducted ADF test on Excel through Real Stat software package.

Back-Testing Results with Exceedances
Back testing is the process of comparing losses predicted by the value at risk (VaR) model to those actually experienced over the sample testing period.
If a model were completely accurate, we would expect VaR to be exceeded (also called as an exception) with the same frequency predicted by the confidence level used in the VaR model. In other words, the probability of observing a loss amount greater than VaR is equal to the significance level. When using the above back testing approach, the following three desirable attributes of VaR estimates are to be evaluated: VaR estimate should be unbiased, adaptable and robust.

Inferences
 The results show that parametric model using normal distribution with GARCH (1,1) fits best for estimating value at risk at 99% confidence interval in all the three markets under consideration. Theoretical Economics Letters

Limitations
 The statistical probability distributions assumed for testing purposes are limited to normal and student-t distributions only.
 The calculations done are to estimate only 1-day VaR. However, the study can be extended to estimate 30-day VaR, 50-day VaR, 100 day-VaR according to the investors' need and risk exposure.
 The research studies only the market risk in the Indian markets. The VaR concept can be applied to estimate the credit risk, operational risk and liquidity risk.
 The rolling window size considered for backtesting is of 1000 days. Rolling window sizes play a crucial role in determining the VaR as it leads to a choice between persistence and adaptability.
 The study considers three markets viz. index, exchange rate and commodities. The models can be used for individual stocks as well for derivative instruments like futures and options.