Risk Aversion (CVaR)#

By default SDDP minimises the expected total cost, which is the risk-neutral choice. But the average is not always what you care about. The ClearLake tutorial ends with a simulate summary that is cheap on most paths yet has an expensive tail: a median of 0 but a 95th percentile of 1500. When the bad outcomes are what matter, you can train a more conservative policy with Conditional Value-at-Risk (CVaR).

Conditional Value-at-Risk#

For a cost distribution, \(\mathrm{CVaR}_\alpha\) is the average cost over the worst \(\alpha\) fraction of outcomes. With \(\alpha = 0.25\) it is the mean of the costliest quarter of cases, a measure of the tail rather than the centre.

The CVaR risk measure#

Pass a CVaR object to train():

from gamspy.formulations import CVaR

result = sddp.train(n_iter=40, risk=CVaR(tail=0.25, weight=0.5))

CVaR takes two numbers:

tail: the tail probability \(\alpha \in (0, 1]\), the worst fraction CVaR averages over.
weight: the risk weight \(w \in [0, 1]\), how much to lean on the tail versus the average.

At each stage the cost-to-go is aggregated as a blend of the two:

\[(1 - w)\,\mathbb{E}[Z] \;+\; w\,\mathrm{CVaR}_\alpha[Z],\]

where \(Z\) is the stage cost plus future cost. With weight=0 this is the plain expectation (risk-neutral); with weight=1 it is pure CVaR over the tail; values in between trade the two off.

The risk-adjusted bound#

A risk-averse policy guards against the tail, so it costs more on average, and the lower bound reflects a different objective. Training ClearLake under CVaR raises the bound well above the risk-neutral 112.30:

`risk`	lower bound
`None` (risk-neutral)	112.30
`CVaR(tail=0.25, weight=0.5)`	399.17
`CVaR(tail=0.25, weight=1.0)`	916.67

The summary marks the bound as risk-adjusted, because it now bounds the risk measure rather than the expected cost:

========================================================================
  Risk measure         : CVaR(tail=0.25, weight=1.0)
  Lower bound (risk-adj.):    9.166667E+2
  Iterations run       :             40
  Stop reason          :       max_iter
========================================================================

These risk-adjusted bounds are not comparable to the risk-neutral one, since they measure a deliberately more pessimistic objective.

Note

SDDP uses the standard nested, first-stage-neutral convention: the first stage is risk-neutral (an expectation) and CVaR is applied to the later stages. The default risk=None is exactly the risk-neutral expectation, bit-for-bit. CVaR also composes with non-uniform probabilities, so the tail is taken with respect to your scenario weights.