\section{Value-at-Risk}


To quantify the risk associated with the models applied, it was decided to use the Value-at-Risk\index{value-at-risk} (VaR) metric. The VaR metric\index{VaR metric}, for the next day, is defined for a probability $\alpha$ as the value the loss will not exceed with probability $\alpha$. For a series of stochastic variables $X_t$, which would here be the future price of electricity, the VaR will be defined as,
\begin{equation}
\Prob(X_t \le \mathrm{VaR}_{\alpha}) = \alpha.
\label{eq:var1}
\end{equation}
So, the VaR is directly related to the quantiles of the distribution of electricity prices.
For the models used in this chapter, the VaR is straight-forward to calculate. For the POT method, the VaR is simply obtained by inverting the GP distribution for a given probability. This gives the VaR as, cf. Eq.~\eqref{eq:pot.4},
\begin{equation}
\mathrm{VaR}_{\alpha} =\, u \,+\, \frac{\sigma}{\gamma} \big((1 - \alpha)^{-\gamma} - 1 \big),
\label{eq:var2}
\end{equation}
where $u$ denotes the chosen threshold value, $\sigma$ and $\gamma$ are the scale and shape parameters of the GP distribution of exceedances of $u$, respectively. For the ACER method, the VaR is found just as easily by inverting the estimated ACER function for the desired exceedance level $\alpha$, cf. Eq.~\eqref{eq:gev3}. The VaR then becomes, 
\begin{equation}\label{eq:var3}
\mathrm{VaR}_{\alpha} = \, b\, + \,\Bigg(\frac{1}{\tilde{a}} \bigg( \Big(\frac{q}{1-\alpha}\Big)^{\xi}-1 \bigg) \Bigg)^{1/c},
\end{equation}
where $\tilde{a}$, $b$, $c$, $q$ and $\xi$ are the parameters estimated by fitting the assumed parametric ACER function to the observed data. In the derivation of Eq.~\eqref{eq:var3}, the approximation $\exp(-x) \approx 1 - x$ for small $x$ is used. It should be noted that when using the conditional approach, the VaR estimations need to be inserted into the equation for the conditional quantiles. As a last reminder, it should be noted that for the VaR to be interpreted as the unlikely loss, these two methods need to be fitted to the left tail of the return distribution. So, using the VaR, a simple and straight-forward method to quantify the risk in the Nordic Power Exchange spot market is at hand. 



\section{Application to Simulated Time Series of Electricity Prices}\index{electricity prices!simulated time series}

Before using the method developed on real time series, it was decided to try to motivate the use of the ACER method instead of the POT method by applying the two to simulated time series with known properties. By this approach, it should be possible to get an idea of the difference in performance for the two methods, and if the ACER method performs better, argue that it should be used when estimating extreme quantiles. The initial step was simply simulating iid innovations from a Student's $t$ distribution and comparing the estimated quantiles with the ones from the actual distribution. Data sets were simulated consisting of 10 time series with 3650 independent outcomes of the $t$ variable in each, which then provides us with data sets of 36500 observations. This was repeated 5 times, resulting in 5 data sets for comparing the performance of the ACER and POT methods. For the data sets simulated, a $t$ variable with $\nu=4$ degrees of freedom was used. Applying the ACER and POT methods to the simulated data sets, and estimating the 100 time series return level, which is the level that is expected to be exceeded once per 100 time series, the results presented in Table~\ref{tab:acerpott1} were obtained. In this table, the estimated 100 time series return level is presented along with the percentage deviation from the real return level obtained from the Student's $t$ distribution, and $95\%$ confidence intervals for the estimated return level. From this table it is seen that while the ACER method estimates a return level closer to the real return level 4 out of 5 times, the width of the confidence intervals is far greater when using the POT method. The reason for the difference in the confidence intervals is likely due to the fact that the ACER method uses approximately $48\%$ of all observations, compared to approximately $10\%$ of all observations used by the POT method.

\begin{table}
	 \caption{$100$ time series return level estimated with ACER and POT for the $t$-model with $\nu=4$.}
	\label{tab:acerpott1}
  \centering
  \begin{tabular}{c c | c c}
  \hline
    $\eta_{ACER}^{100\mathrm{ts}}$ & $\mathrm{CI}_{ACER}$ & $\eta_{POT}^{100\mathrm{ts}}$ & $\mathrm{CI}_{POT} $\\
\hline
32.99 \: (2.1\%) & (29.32, 37.80) & 33.82\: (4.7\%) & (23.75, 43.88)\\
27.78 \:(14.0\%) & (24.17, 31.59) & 24.56\:(24.0\%) & (18.35, 31.05)\\ 
31.68 \: (1.9\%) & (28.43, 34.72) & 30.24\: (6.4\%) & (21.87, 38.61)\\
29.76 \: (7.7\%) & (26.88, 33.31) & 31.83\: (1.5\%) & (22.46, 41.20)\\
31.80 \: (1.5\%) & (26.29, 34.25) & 29.05\:(10.1\%) & (21.15, 36.96)\\
\hline
  \end{tabular}
\end{table}

