1 VaR Models VaR Models for Energy Commodities Parametric VaR Historical Simulation VaR Monte Carlo VaR VaR based on Volatility Adjusted Quantile Regression Backtesting Value at Risk Models Kupiec Test Christoffersen Test

2 Type of VaR models discussed here: –Parametric VaR –Historical simulation VaR –Monte Carlo VaR –Volatilty Adjusted Quantile Regression It is easy to estimate VaR once we have the return distribution. The only difference between the tree VaR models are due to the manner in which this distribution is constructed Introduction to Value-at-Risk Models

3 Value at Risk and Expected Tail Loss (or Conditional Value at Risk) VaR is simply the quantile on the distribution and Expected Tail Loss is simply the mean to the left/right of the quantile.

4 –Parametric VaR Analytically tractable, but is restricted to linear portfolios and it can only be generalized to a few simple parametric forms (e.g Normally and T distributed data) The well know Risk Metrics method is based on this approach Introduction to Value-at-Risk Models

5 VaR α =Ф -1 (α)*σ σ = Volatility of return of a position/portfolio (assuming 0 mean) Ф -1 = The inverse of a normal density function α = Significance or VaR level Normal VaR

6 Unconditional Volatility

7 Volatility changes dynamically over time and needs to be updated For this reason many institutions use an exponentially weighted moving average (EWMA) methodology for VaR estimation, e.g. using EWMA to estimate volatility in the normal linear VaR formula These estimates take account of volatility clustering so that EWMA VaR estimates are more risk sensitive than equally weighted VaR estimates For example the RiskMetrics TM methodology and supporting data enable the analyst to update the volatility Normal VaR (with volatility adjustment)

8 30 Moving Average Volatility and Dynamic VaR at 1% and 99% Level for EL NP Q contract

9 At portfolio level, we calculate the variance and standard deviation of a portfolio as: Normal VaR Portfolio Level S 2 =wCw’ S=√S 2 w is a column vector of weights. w’ is the transpose, that is a row vector of weights C is the covariance matrix of returns The VaR calculation is as before apart from that we use portfolio standard deviation instead of single position standard deviation

10 Normal VaR Portfolio Level

11 Covariance and correlation changes dynamically over time and needs to be updated For this reason many institutions use an exponentially weighted moving average (EWMA) methodology for VaR estimation, e.g. using EWMA to estimate covariance in the normal linear VaR formula These estimates take account of correlation clustering so that EWMA VaR estimates are more risk sensitive than equally weighted VaR estimates For example the RiskMetrics TM methodology and supporting data enable the analyst to update the covariances Normal VaR (Covariance adjusted)

12 Correlation matrix should be updated continuously with some sort of time-varying correlation model. See previous lecture on how this can be done. Normal VaR (Covariance adjusted)

13 Main method used by banks (in addition to RiskMetrics) No assumption of any parametric distribution for returns Assumes that all possible future variations have been experienced in the past, and that the historically simulated distribution is identical to the returns distribution over the forward looking risk horizon Main constraint is the sample size Volatility clustering not taken into account Scaling is problematic Historical Simulation VaR Historical simulation model

14 Basel Committee recommends a period of 3 and 5 years of daily data be used in the historical simulation model Sample size –VaR are to reflect current market conditions –Historical simulation requires though to actually build the distribution, (and focus on the tails) and thus cannot just rely on recent data –For a 1% VaR we need at lest 2000 daily observations on all assets or risk factors Properties of Historical Value at Risk Sample size and data frequency

15 Historical Simulation VaR

16 Monte Carlo VaR –Choose a stochastic process given stochastic process for prices or return for a given asset or a portfolio of asset and estimate its parameters –Establish a Monte Carlo model for a given stochastic process for prices or return for a given asset or a portfolio of asset (correlated stochastic processes). Run simulations. –Based on simulations, calculate the VaR (or CVaR) for the given distribution the simulation provides

17 Monte Carlo VaR

18 The problems with existing “standard” risk model (that many energy companies have adopted from the bank industry) are: Riskmetrics TM VaR capture time varying volatility but not the conditional return distribution Historical Simulation VaR capture the return distribution but not the time varying volatility Alternatives: GARCH with T, Skew T, or GED CaViaR type models Although these models works fine according to several studies, there is a problem of calibrating these non-linear models and therefore they are only used to a very limited extent in practice Volatility Adjusted Quantile Regression

19 A robust and “easy to implement” approach for Value at Risk estimation based on; First running an exponential weighted moving average volatility model (similar to the adjustment done in Riskmetrics TM ) and then Run a linear quantile regression model based on this conditional volatility as input/explanatory variable The model is easy to implement (can be done in a spreadsheet) This model also shows an excellent fit when backtesting VaR Value at Risk Modelling for Energy Commodities using Volatility Adjusted Quantile Regression

20 1) Calculate Exponentially Weighted Moving Average of Volatility 2) Run a Quantile Regression Regression with the Exponentially Weighted Moving Average of Volatility as the explanatory variable 3) Predict the VaR q t+1 from the model Volatility Adjusted Quantile Regression

21 Volatility Adjusted Quantile Regression

22 Volatility Adjusted Quantile Regression and VaR Calculation - Example El_Nor_Q

23 Returns and EWMA Volatility Example El-Nor_Q Distribution forecast from volatility adjusted quantile regression Todays EWMA vol is 2% on a daily basis. What is 5%, 95% 1 day VaR given our model? Var 5% = *2% = -3.36% Var 95% = *2% = 3.03% Similar equations are found for all the quantiles

24 Backtesting Value at Risk Models Backtesting refers to testing the accuracy of VaR over a historical period when the true outcome is known. The general approach to backtesting VaR for an asset or portfolio is to record the number of occasions over a historical period when the actual loss exceeds the model VaR and compare this number to the pre-specified level

25 Backtesting Value at Risk Models Exceedance: When the true return exceed the VaR predicted by the model (e.g. True model return on day 9 was -2.25% while the VaR10% prediction from the model was -1.96%) The sum of exceedances are calculated over the data sample The sum of exceedances divided by the number of observations in the sample should be close to the VaR quantile we are trying to model

26 Backtesting Value at Risk Models

27 Backtesting Value at Risk Models A proper VaR model has The number of exceedances as close as possible to the number implied by the VaR quantile we are trying to model Exceedances that are randomly distributed over the sample (that is no “clustering” of exceedances). We do not want the model to over/under predict in certain periods

28 Backtesting Value at Risk Models To validate the predictive performance of the models, we consider two types of test: The unconditional test of Kupiec (1995) The conditional coverage test of Christoffersen (1998) The first test check whether the number of exceedances or hits are equal to the predefined VaR level The next test check also whether the exceedances or hit are randomly distributed over the sample

29 Backtesting Value at Risk Models – Kupiec Test The Kupiec (1995) test is a likelihood ratio test designed to reveal whether the model provides the correct unconditional coverage. More precisely, let H t be a indicator sequence where H t takes the value 1 if the observed return, Y t, is below the predicted VaR quantile, Q t, at time t

30 Backtesting Value at Risk Models – Kupiec Test Under the null hypothesis of correct unconditional coverage the test statistic is Where n 1 and n 0 is the number of violations and non- violations respectively, π exp is the expected proportion of exceedances and π obs = n 1 /(n 0 +n 1 ) the observed proportion of exceedances.

31 Backtesting Value at Risk Models – Kupiec Test

32 Backtesting Value at Risk Models – Christoffersen test In the Kupiec (1995) test only the total number of ones in the indicator sequence counts, and the test does not take into account whether several quantile exceedances occur in rapid succession, or whether they tend to be isolated. Christoffersen (1998) provides a joint test for correct coverage and for detecting whether a quantile violating today has influence on the probability of a violating tomorrow.

33 Backtesting Value at Risk Models – Christoffersen test The test statistic is defined as follows: where n ij represents the number of times an observations with value i is followed by an observation with value j (1 is a hit, 0 is no hit). Π 01 =n 01 /(n 00 +n 01 ) and Π 11 =n 11 /(n 11 +n 10 ). Note that the LR cc test is only sensible to one violation immediately followed by other, ignoring all other patterns of clustering

34 Backtesting Value at Risk Models – Christoffersen test

35 Example of analysis: Nordic Rolling Base Front Quarter Contracts

36 High volatility for energy commodities. Annualized values much higher than for stocks and currency markets, specially for short term natural gas and electricity contracts Fat tails for all energy commodities. Some energy commodities have mainly negative skewness (e.g. crude oil), others positive (e.g natural gas) Return distribution and correlation of energy commodities varies a lot over time, hence the evolution of empirical VaR over time Risk Modelling of Energy Commodities Risk modeling of energy commodities will be very challenging. Need models that capture the dynamics of changing return distribution and correlation over time

37 Paper Energy Risk Management

38 Paper Energy Risk Management

39 Paper Energy Risk Management

40 Paper Energy Risk Management

41 –Use the exercise data –Estimate the descriptive statistics and empirical VaR / CVaR at 1%,5% and 95,99% for the whole period for all commodities. Split the dataset in 2 periods and recalculate. Do vol, skewness, kurtosis, empirical var change over time? –Run the following VaR Models Parametric Normal VaR with vol according to historical estimation Parametric Normal VaR with vol according to Exponential Weighted Moving Average (30 days) Historical simulation Monte Carlo VaR from a standard GARCH model Volatility adjusted quantile regression –Backtest the models with Kupiec and Christoffersen’s tests Excel Exercise