Hedging Strategy and Electricity Contract Engineering - IFOR

3.5 The need for a new risk measure 61
the mean CVaR problem [BLS00].
Pflug [Pfl99] has shown that CVaR under certain assumptions is a coherent risk
measure, which implies that it encourages diversification. Furthermore, CVaR
has because of its convexity in the portfolio positions very tractable properties
in terms of optimization [RU01]. A global minimum is relatively easy to find,
as only the first order conditions has to be fulfilled. For more information on
coherent risk measures in general and CVaR in particular, see [Del00].
3.5.1. CVaR as an appropriate risk measure in the electricity market
We believe that CVaR is an appropriate risk measure in the electricity market.
The skewness in the distribution of profit and loss caused by the price spikes
and the options built into basically any power contract makes an asymmetric
risk measure that can really penalize extreme events, such as CVaR interesting.
The tractable computational features of CVaR do indeed strengthen its position.
Further the connection between a utility maximizing agent and CVaR is strong
enough to motivate this risk measure over others also from a utility theory point
of view. Hence we will use CVaR as risk measure in this work.
3.5.2. Calculation of CVaR
There are several ways to calculate CVaR. It can actually be calculated
analytically for certain distributions. As with VaR, CVaR can be calculated
using the variance/covariance method, where the assumption of normally
distributed returns is made. For that case it can be easily shown that CVaR and
VaR are simply given by a scaling of the standard deviation [RU01]. CVaR as
risk measure is however motivated by heavy tailed distributions, as opposed to
the normal distribution, and CVaR can actually be calculated analytically for
heavy-tailed distributions with extreme value theory [EKM97]. This approach,
called peaks over threshold (POT), is however not straight-forward and certain
assumptions have to be made to derive the analytical CVaR. Most important is
probably that this method currently only handles one-dimensional problems,
implying that one has to have a historical data set, to estimate the parameters
of the extreme value distribution, of the whole portfolio. This data is seldom