Hedging Strategy and Electricity Contract Engineering - IFOR

54 Risk management
therefore has a unique solution and only the first order conditions need to
be obtained. The famous diversification principle was here quantified by the
non-perfect correlation between individual assets. The idea of not putting
all your eggs in the same basket was not new, but Markowitz was the first to
formalize and apply it to financial instruments.
It can be shown that no matter what assumptions are made about the distribution
of asset returns, if an investor has a quadratic utility function, utility
is increasing in the mean return and decreasing in the variance and that
only these two first moments matter. Therefore quadratic utility maximizing
investors will only choose portfolios according to the mean-variance portfolio
problem [Arr71]. The quadratic utility however has the unrealistic property of
satiation in the sense that the utility will decrease with wealth beyond a certain
point. Further, the absolute risk aversion will increase with wealth meaning
that the demand for a risky asset will decrease with increased wealth [HL88].
Chamberlain [Cha83] shows that the most general class of distributions that
allow investors to rank portfolios based only on the first two moments is the
family of elliptical distributions. In the electricity market returns are not elliptically
distributed, because of the spikes observed in the prices, as indicated
earlier, why the mean-variance portfolio is very questionable. Actually in any
market a portfolio with options will have asymmetric distributed returns. Further
variance, which is a symmetric measure of risk, where potential upside is
penalized just as much as potential downside is not a preferable property for a
risk measure.
3.4.1.3. Value at Risk
Triggered by the mentioned difficulties with variance, a number of socalled
downside-risk measures have been proposed in the literature
[Roy52, Mar59, Bav78]. Value at Risk (VaR) is the downside risk measure
that is most widely used today. The reason for this is the simple
interpretation of VaR and because it is the risk measure that the Bank for
International Settlements 5 has enforced.
5 An organization of central banks.