Hedging Strategy and Electricity Contract Engineering - IFOR

7.1 Efficient frontier 177
lG x 6 9 6 y jH 39 336x 40
336
1 iF
l
G S i5 j6 D i5 jH x D i5 j
8
g7
D i5 j
336
iF 1
S i5 j
30
lF 1
4
l
9
1
0
8
70
30
lF 1
4
l
9
g4l
G S i5 j6 D i5 jH
(7.14)
The marginal costs for production and pumping are, as seen, not present. They
are typically very low and are negligible in this analysis.
7.1. Efficient frontier
The tightest risk constraint for which there exists a feasible solution to
(7.3)-(7.12) is C min 5
8
657 million CHF. This means that the polyeder of
feasible solutions is empty for risk constraints lower than C min , which hence is
the minimum achievable risk.
A number of optimizations were performed for different constraints on the risk,
varying from C min to 2 million CHF, where the latter corresponds to a pure
profit maximization without any regards to the riskiness of the portfolio. The
maximum expected profit plotted against the risk constraint gives us the utility
specific efficient frontier, which is shown in Figure 7.3. Any portfolio under
the efficient frontier is inefficient, since there exists a portfolio with a higher
expected profit at the same risk. This corresponds to the efficient frontier in the
Markowitz portfolio optimization approach, where however risk is measured
as variance [Mar52]. The efficient frontier is shown to be piecewise linear
and concave, which is consistent with Corollary 6.13. The expected profit
as a function of the maximum risk is strictly increasing for risk constraints
tighter than 1325000 CHF. For looser risk constraints the expected profit is
constant, as seen in Figure 7.3. The reason for this is that the risk constraint
is inactive for C 1325000 CHF. The future position is then already at its
maximum, x 10000 MW, as can be seen in Figure 7.4, and the production
cannot be changed towards a more profit oriented dispatch. This would in the
mean-variance case correspond to a full investment in the single stock with the
highest expected profit and with short positions prohibited. The marginal value