15.3.1 Historical Simulation (HS)

As a starting point, we estimate VaR using the simple, yet popular, historical simulation (HS) approach. Rather than making an arbitrary parametric assumption about the true (but unknown) return distribution, HS draws on the distribution of historical returns. At any point in time t, the T most recent daily return observations represent the empirical distribution. The next-day estimate of VaR_{t + 1} is given by the q% quantile of this empirical distribution:

15.1

Consistent with Manganelli and Engle (2004), who note that it is common practice to utilize a rolling window of up to 2 years for HS approaches, we assume T = 500 days. If one requires the left-tail (downside) one-day VaR_{t + 1} with α = 1%, one takes the q = α = 1% quantile from the most recent 500 observed daily returns. Conversely, if the right-tail (upside) VaR_{t + 1} is of special interest, one takes the q = 1 − α = 99% quantile from the empirical distribution.

Naturally, the success of the HS approach to VaR estimation depends on the extent to which the prior distribution of returns is an adequate representation of future returns. Even though this assumption is reasonable, HS may still suffer from the fact that it is entirely unconditional – it makes no use of current information in the mean and volatility of the underlying process. The parametric approaches described next propose a variety of ways to incorporate conditioning information into the VaR forecast.

15.3.2 Autoregression with Constant Volatility (AR-ConVol)

Our first parametric approach to estimating VaR utilizes a first-order AR model for daily returns with constant volatility (hereafter denoted AR-ConVol):

15.2

15.3

where r_t is the daily returns and |ρ| < 1 controls the gradual convergence to price equilibrium. The AR-ConVol specification is sufficiently simple to be estimated using the ordinary least-squares method. At each point in time, a rolling window encompassing the previous 5 years' daily returns are utilized to fit the model.⁵ The next-day VaR_{t + 1} is computed as

15.4

where are the parameter estimates from Eqs (15.2) and (15.3), and F⁻¹(q) is the desired q% quantile of the standard normal distribution function.

15.3.3 Auto-regression with Time-Varying Volatility (AR-NGARCH)

In light of extensive empirical evidence refuting constant volatility in asset returns, the second parametric model relaxes the constant volatility assumption defined in Eq. (15.3). Denoted AR-NGARCH, the first-order autoregression is augmented by assuming that the return innovations follow a time-varying conditional variance process:

15.5

15.6

15.7

where Ξ_{t − 1} is the information set at time t − 1 and h_t is the conditional variance process.⁶

The extant literature boasts many possible specifications for the conditional variance process. In modeling oil returns, Marimoutou et al. (2009) adopt a (symmetric) GARCH process. For electricity returns, however, a specification that captures positive leverage effects is desirable. Knittel and Roberts (2005) argue that the convex nature of the marginal costs of electricity generation causes positive demand shocks to have a greater impact on volatility than negative shocks. Accordingly, Eq. (15.7) represents the nonlinear, asymmetric GARCH process introduced by Engle and Ng (1993) and advocated by Christoffersen (2009) for the purpose of forecasting volatility for risk management. The parameter θ captures asymmetric, nonlinear volatility behavior. For θ < 0, a positive shock on day t − 1 increases current volatility by more than a negative shock of the same magnitude.

Given the normality assumption in (15.6), the AR-NGARCH model is estimated by the maximum likelihood approach. The next-day VaR_{t + 1} is computed in a manner similar to Eq. (15.4), with the exception that the constant volatility is replaced with the time-varying estimate :

15.8

15.9

15.3.4 AR-NGARCH with No Distributional Assumption (Filteredhs)

While the HS approach is appealing due to its nonparametric nature, it ignores potentially useful information in the volatility dynamics (Marimoutou et al., 2009). Following Hull and White (1998), Barone-Adesi et al. (1999), and Marimoutou et al. (2009), we attempt to capture the best of both worlds by combining the distribution-free flavor of the HS approach with the conditional variance dynamics in AR-NGARCH.

The AR-NGARCH model given by Eqs. (15.5)–(15.7) is estimated as described above. However, rather than relying on the normal distribution function F⁻¹(q) as in Eq. (15.8), VaR estimation draws on the empirical distribution of filtered residuals (hence, the approach is denoted filteredHS). The residuals from (15.5) are standardized by the conditional volatility estimates from Eq. (15.7):

The next-day VaR_t+1 is then computed as

15.11

where the residuals are standardized residuals. While the AR-NGARCH parameters are estimated using a 5-year rolling window period, to maintain consistency with the HS approach, the tail quantiles are constructed by bootstrapping from the 500 most recent standardized residuals.

15.10

15.3.5 EVT Approaches

Whereas the parametric AR-ConVol and AR-NGARCH approaches model the entire return distribution, EVT focuses on the part that is of primary interest in risk management (i.e., the tail). Explicitly modeling tail behavior seems natural in VaR applications. Following recent literature, the peak over threshold method is used to identify extreme observations that exceed a high threshold u (which is defined below). EVT is then used to specifically model these “exceedences.”

Let x_i denote a sequence of IID random variables from an unknown distribution function. For a chosen threshold u, the magnitude of each exceedence is y_i = x_i − u for i = 1, … , N_y, where N_y is the total number of exceedences. The probability that x exceeds u by an amount no greater than y, given that x > u, is

15.12

Balkema and de Haan (1974) and Pickands (1975) show that F_u(y) can be approximated by the generalized Pareto distribution (GPD)

15.13

where ξ and ν are the shape and scale parameters, respectively. Interestingly, the GPD subsumes various distributions, with ξ > 0 corresponding to heavy-tailed distributions (such as Pareto, Cauchy, and Fréchet), ξ = 0 corresponding to thin-tailed distributions (such as Gumbel, normal, exponential, gamma, and lognormal), and ξ < 0 corresponding to finite distributions (such as uniform and beta distributions). By setting x = y + u and rearranging Eq. (15.12), we obtain

15.14

Noting that the function F_x(u) can be estimated by its empirical counterpart as (T − N_y)/T, where T refers to the total sample observations, and after substituting Eq. (14) into (15.12), we obtain

15.15

The next-day VaR_{t + 1} for a given α% is computed by inverting (15.15):

15.16

Critically, the EVT procedure depends on the threshold u chosen to define exceedences. The choice of u involves a tradeoff. On one hand, u must be set sufficiently high to maintain the asymptotic theory of EVT and generate unbiased estimates of the GPD parameters (particularly the shape variable ξ). On the other hand, if u is set too high, there may be too few exceedences from which to estimate ξ and ν. Guided by Hull (2010) and Christoffersen (2012), we set u such that the most positive (most negative) 5% of observations are used to estimate ξ and ν in the right (left) tail of the distribution. Given u, the parameters (ξ, ν) of the GPD equation (15.13) are estimated by maximum likelihood, and then substituted into (15.16) along with the known values of T, N_y, α, and u to calculate VAR.

Two variations of the EVT approach are explored. First, the approach described above is applied to the raw data. That is, x_t is simply the daily electricity return r_t. This vanilla application is simply denoted as “EVT.” The second approach recognizes that raw electricity returns are unlikely to be IID. With this in mind, McNeil and Frey (2000) advocate applying EVT (as described above) to the “filtered” residuals from a parametric model. Specifically, we fit the AR-NGARCH model Eqs (15.5)–(15.7) as described in Section 15.3.3. Model residuals are then filtered as in Eq. (15.10) and EVT is applied to these standardized residuals. This approach is denoted condEVT. The corresponding next-day VaR_{t + 1} is computed in a manner to Eq. (15.11), with the exception that the normally distributed quantile F⁻¹(q) is replaced by the EVT tail estimator defined in Eq. (15.16).

To summarize, the empirical analysis will explore the relative merits of a number of alternate approaches to estimating VaR for electricity markets. The approaches range from a simple, nonparametric, HS approach through to a sophisticated approach that applies EVT to the filtered residuals of a model that accommodates mean-reverting returns and nonlinear, asymmetric, time-varying volatility (i.e., the condEVT model).

15.4.1 Data

The empirical analysis features two prominent European power markets (European Energy Exchange EEX Phelix in Germany, and PowerNext PWX in France) and one of the major power markets operating in the United States (Pennsylvania, New Jersey, and Maryland; PJM). Daily peak load prices are sourced from DataStream over the period from January 3, 2006 to December 20, 2013 (2010 days). As such, this sample period represents a significant update on the time horizons utilized in prior work (e.g., Chan and Gray, 2006; Huisman et al., 2007; Klüppelberg et al., 2010; Lindström and Regland, 2012).

For each power market, Table 15.1 reports the basic summary statistics that depict daily continuously compounded returns, that is, . Figure 15.1 plots the time series of daily prices (left panels) and continuously compounded returns (right panels). Arguably, if one ignores the scale, the diagrams resemble those commonly observed for traditional assets (or, at least, traditional assets with stationary prices). Prices, while highly volatile, revert to mean. Each market exhibits a number of extreme price movements. Volatility clustering is readily apparent.

This table reports summary statistics for daily returns on the EEX, PWX, and PJM power markets over the period covering January 3, 2006 to December 20, 2013 (2010 days). From daily peak load prices, continuously compounded returns are calculated. All returns are stated on a daily basis in percentage (i.e., multiplied by 100). Q(1) and Q²(1) are the respective Ljung–Box Q-statistics for first-order autocorrelation in returns and squared returns. *** indicates statistical significance at the 1% level.

In the VaR context of this study, the fundamental differences between electricity markets and traditional markets come to the fore when the scale of the diagrams is considered. The magnitude of price movements, and therefore resulting returns, are rarely seen in stock markets. In fact, the electricity movements dwarf even oil markets, despite oil prices being highly vulnerable to economic, political, and military events.⁷ A case in point surrounds the unrelenting US heatwave of July 2013, when temperatures exceeded 100 ºF. PJM peak load price jumped from $98/MW h on July 17, 2013 to $162/MW h the following day. After remaining around this level for a further day, price reverted to $47/MW h.

Figure 15.1 exhibits movements of this magnitude regularly throughout the sample, with obvious consequences for the distribution of returns. While the mean daily return is near zero in each market, the standard deviation is several orders of magnitude higher. The interquartile range and skewness statistics suggest that electricity returns have a near-symmetric distribution. However, the probability mass in each tail far exceeds a normal distribution, with the Jarque–Bera statistic overwhelmingly rejecting the null of normality.⁸ Further, both positive and negative extreme returns are well represented in the distribution of returns. This is relevant to our study of VaR in both left and right tails. Together, Table 15.1 and Figure 15.1 demonstrate the unique characteristics of the power markets: high volatility, volatility clustering, and infrequent extreme movements that result in a fat-tailed return distribution. This casual empiricism provides strong motivation to study various alternative model specifications, particularly the EVT-based models, to forecast VaR.

15.4.2 Parameter Estimates

With the exception of the purely nonparametric HS approach, the alternate approaches to forecasting VaR require an estimate of the parameters of each model. While Section 15.4.3 describes a rolling estimation approach used to generate a series of VaR forecasts, we begin by simply fitting the main model of interest, the condEVT specification, over the entire sample between 2006 and 2013. Table 15.2 presents the parameter estimates and corresponding standard errors (in parentheses) that are computed using the quasi-maximum likelihood (QML) procedure of Bollerslev and Wooldridge (1992).

Panel A reports QML parameter estimates and the corresponding standard errors (in parentheses) for the condEVT model for the EEX, PWX, and PJM markets. The model is estimated over the full sample period covering January 3, 2006 to December 20, 2013. Panel B reports the sample mean for the EVT right- and left-tail quantiles (in absolute values) of the standardized residuals; the sample mean is estimated based on a rolling series of 750 parameter estimates of the EVT corresponding right- and left-tail quantiles from January 4, 2011 to December 20, 2013. For comparison, the tail quantiles F⁻¹(q) (in absolute values) of a normal distribution for q = 95%, 99%, and 99.5% are equal to 1.645, 2.326, and 2.576, respectively.

For each power market, returns are negatively autocorrelated (ρ < 0), suggesting that a large positive (negative) return on a given day is followed by a large negative (positive) return the next day. This is likely to be driven at least in part by the occurrence of large temporary spikes in electricity prices. Each market also exhibits clear GARCH effects (β₁ and β₂ significant). Curiously, the direction of the leverage effect in volatility shocks (θ) differs between markets. In PJM, a positive prior-day shock increases current volatility by more than a negative shock of the same magnitude (θ < 0). For EEX and PWX, the reverse is true.

The final two rows in Panel A reports the corresponding statistics of the shape (ξ) and scale (ν) parameters of the EVT framework, estimated by fitting the GPD distribution to the standardized residuals defined in Eq. (15.10). Estimates of ξ and ν are calculated separately for the left and right tails of the distribution. The left tail is of primary concern to utility- and electricity-provider firms since they have a short position in electricity. On the other hand, when electricity prices increase, electricity buyers such as retailers incur losses because they have a long position with the providers and generators. As such, electricity buyers are more interested with the extreme right tail of the distribution.

Recall that values of ξ > 0 are indicative of heavy-tailed distributions. In the right tail, ξ is positive for all three power markets, corroborating the casual empiricism provided in Figure 15.1 that the distribution of standardized residuals follows a Fréchet distribution. In the left tail, EEX and PWR markets also exhibit fat tails, while ξ is negative for PJM.

Panel B provides an indication of how far the tail behavior deviates from normality. The condEVT model is fitted using a rolling in-sample period (the procedure is described in full in Section 15.4.3). For each day in the out-of-sample period, the right- and left-tail quantiles , defined in Eq. (15.16), (in absolute values) are estimated for α = 5%, 1%, 0.5%. Panel B then reports the time-series mean of the daily quantile estimates.

For comparison, note that the tail quantiles F⁻¹(q) (in absolute values) from a normal distribution for the respective q's are 1.645, 2.326, and 2.576, respectively. The estimates provided in Panel B reveal that the tail quantiles from the condEVT model are higher than those of the normal distribution especially when we move to more extreme quantiles at q = 99% and q = 99.5%, with the empirical right-tail fatness even more apparent for the PJM power market.

15.4.3 VaR Forecasting

In order to compare the various approaches, we generate a series of VaR forecasts over an out-of-sample period. For each parametric approach, this requires an estimate of model parameters.⁹ With a relatively modest time series available, it is a delicate tradeoff between the length of in-sample period used for parameter estimation and the length of the out-of-sample period used to compare performance. For each parametric approach, we employ a rolling in-sample estimation window of approximately 5 years. The initial in-sample period spans January 3, 2006 through December 31, 2010 (1260 days). Parameter estimates from each model are used to forecast next-day VaR_{t + 1} (on January 3, 2011). Rolling forward one day, VaR on January 4, 2011 is based on parameters estimated over January 4, 2006 and January 3, 2011. This procedure is repeated to generate VaR forecasts for a total of 750 out-of-sample days spanning January 3, 2011 through December 20, 2013.

Out-of-sample VaR forecasting performance of the competing models is assessed using a VR. The VR for the right (left) tail is calculated as the percentage of positive (negative) realized returns that exceed (fall below) the VaR predicted by a particular model. If, for example, α = 1% and given that there are a total of 750 observations in the out-of-sample period, violations are expected on approximately 7–8 days. The competing models are then ranked according to how close their actual VRs are to the expected rates.

Consistent with prior studies, we also utilize statistical metrics of forecasting accuracy. First, the unconditional coverage test statistic (LR_uc) assesses whether the actual VR is statistically different from the expected failure rate. The no-difference null is rejected if a model generates either too many or too few violations. Second, following Christoffersen (1998), we employ the conditional coverage test (LR_cc) that jointly tests unconditional coverage and whether VaR violations are independently distributed across time. Under this test, the null hypothesis that the VaR approach is accurate is rejected if the violations are too many, too few, or too clustered in time. A third test, which also follows from Christoffersen (1998), examines the independence property of VaR violations through time (LR_ind).¹⁰ For each of the six competing models, Tables 15.3 and 15.4 report the respective out-of-sample performance relating to the right and left tails of the distribution. For each market, a range of significance levels is examined, that is, α = {5%, 1%, 0.5%}.

The table reports the out-of-sample violation ratios (VR), the model rankings (in parentheses), as well as the likelihood ratio test statistics for unconditional coverage (LR_uc), independence (LR_ind), and conditional coverage (LR_cc) of the right tail of the distribution for all six competing models. Bold test statistics indicate statistical significance at the 5% level or below. The out-of-sample period covers January 3, 2011 to December 20, 2013 (750 days).

The table reports the out-of-sample violation ratios (VR), the model rankings (in parentheses), as well as the likelihood ratio test statistics for unconditional coverage (LR_uc), independence (LR_ind), and conditional coverage (LR_cc) of the left tail of the distribution for all six competing models. Bold test statistics indicate statistical significance at the 5% level or below. The out-of-sample period covers January 3, 2011 to December 20, 2013 (750 days).

Table 15.3 provides strong support for the use of EVT-based approaches to VaR forecasting. The vanilla EVT and condEVT approaches rank in the top two in every case (rankings shown in brackets), with the exception of PJM and EEX for α = 5% and α = 0.5%, respectively. For example, for EEX with α = 5%, VaR forecasts under the conditional EVT approach are violated on 5.47% of the out-of-sample days. Similarly, the vanilla EVT approach has an actual VR of 6.00%. In contrast, forecasts under the HS approach are violated in 8.67% of cases. Comparing the two EVT-based approaches, no clear winner is evident across Table 15.3. Arguably, applying EVT to the distribution of raw returns is of comparable reliability to the more sophisticated condEVT approach that applies EVT to the filtered model residuals.

The success of EVT-based approaches is also evident in the statistical tests. Twenty-seven statistical tests are presented for each approach in Table 15.3 (rejections at the 5% level of significance are indicated in bold). The condEVT approach records no rejections, while the vanilla EVT approach records a single rejection (the independence test in PWX for α = 5%).

With respect to the non-EVT parametric approaches, it is not readily apparent that more sophisticated approaches to modeling volatility are justified. The relative rankings of AR-ConVol, AR-NGARCH, and filteredHS vary widely across Table 15.3. Curiously, the performance of the semiparametric filteredHS, which bootstraps from the filtered model residuals, is underwhelming. This is in sharp contrast to applications in oil markets, where Marimoutou et al. (2009) document that its performance rivals the condEVT approach.

Taken as a whole, the out-of-sample testing of competing approaches to VaR forecasting in the right tail provides strong support for the use of EVT-based approaches. Even a vanilla EVT application to raw (unfiltered) returns generates forecasts of risk exposures that are superior to traditional approaches to risk management.

Table 15.4 presents a similar analysis of left-tail performance, a result which is of particular interest to plant generators and electricity providers since they naturally have a short position in electricity. Examining the rankings of competing approaches, it is readily apparent that no single method dominates across all power markets and significance levels (α). This is in sharp contrast to the right-tail analysis of Table 15.3, where the EVT-based approaches ranked highly in most cases. With the exception of the AR-NGARCH approach, each method ranks first in at least one scenario. While the filteredHS approach performs poorly for α = 5%, it ranks consistently in the top two for higher levels of significance, that is, α = {1%, 0.5%}. In terms of statistical inferences, the two EVT-based approaches generate no significant violations for α = {0.5%, 1%}. However, the same is also true for the HS and filteredHS approaches.

To conclude, it is useful to compare and contrast the findings from Tables 15.3 and 15.4 across the three power markets. With respect to the right tail (Table 15.3), casual empiricism suggests that the PJM market is fundamentally different from EEX and PWX. To illustrate, for α = 5% in EEX, some forecasting approaches underestimate the risk exposure (e.g., HS, 8.67%; EVT, 6.00%), while others overestimate risk (AR-NGARCH, 4.00%; filteredHS, 2.80%). Similar findings occur for the PWX market. In contrast, risk exposures in the PJM market are underestimated in nearly all cases (VR > α). This is true for all levels of α. The possible idiosyncrasies of PJM were highlighted in relation to Table 15.2, where the leverage effect (θ) for PJM differed notably from the other markets. Similarly, the fatness of the tail (ξ) of filtered model residuals was most pronounced for PJM.

In terms of statistical performance, several other idiosyncrasies are observed. The right tail analysis (Table 15.3) suggests that the PWX market behaves differently with respect to the assumption that VaR violations are distributed independently through time. Whereas the other markets display very few rejections of independence, LR_ind is significant for many combinations of forecasting model and α in the PWX market. Curiously, the lack of independence for PWX is confined to the right tail; there are very few significant LR_ind statistics in Table 15.4. Table 15.4 also flags EEX as a difficult market in which to forecast left tail violations. Although not attributable to lack of independence, EEX exhibits many rejections of unconditional coverage (LR_uc). Perhaps not surprisingly, EEX stands out in Table 15.2 as being the only market for which the distribution of returns is negatively skewed.