Assignment # 1(b)

The paper being reviewed here is titled "Stochastic Linear Programming and Conditional Value at Risk for Water Resources Management", by R. B. Webby, J.Boland, P. G. Howlett, A. V. Metcalfe published in the Australian and New Zealand Industrial and Applied Mathematics Journal (ANZIAM) in 2008.

This paper attempts to illustrates the use of Conditional Value-at-Risk (CVaR) as a decision support tool for water resources managers, focusing on irrigation requirements of a summer crop in a water deficient environment. Water may be available from a number of sources like precipitation, shallow ground aquifers, entitlements of river water and tailwaters (agricultural reuse water).

In financial analysis, Value-at-Risk(VaR) is defined as the maximum loss expected to be incurred over a given time horizon at a specified probability level. VaR gives the specified quantile of the distribution but does not give any info about the upper tail beyond the value. VaR describes the frequency of a sizeable loss butnot the likely severity of such loss. CVaR does contain information about the losses greater than the upper tail as it is the expected value of of the loss, given that a loss greater than or equal to a threshold VaR occurs. A cost model rather than loss is then built and VaR is computed for a given exceedence probability value of the ordered distribution, and CVaR is the mean of the values equal to and beyond the VaR. A stochastic linear programming model is used to optimize the objective function where its possible to have some stochasticity in the constraints of a classic LP problem. The author describes two ways to solve stochastic linear programming problems. One approach is to consider specific values of the variable and solve it deterministically the typical values being the expected value plus or minus multiples of the standard deviation, a full range of values. Another approach is to sample the variables from distributions and then solve deterministically. The author uses the second approach assuming a multivariate normal distribution involving correlated values of rainfall and groundwater. A cost per Ml of water is attributed to each source depending on pumping, storage and application costs and assuming that the same application method is used throughout as well as the environmental costs. No sensitivity tests were carried out.

Simulations were carried out for exceedence probability of 0.9 and time horizon to be the life of the crop throughout. The decision vector represents a course of action taken and the corresponding cost distribution assumed. Each distribution has a CVaR value. Managers for best performance should use the course of action with minimum CVaR. The CVaR value and expected returns on crop is discussed. Value of River water entitlement is also discussed and its shown that with greater valuation of the entitlement the CVaR increases. Finally, a model extension is proposed in order to accommodate multiple crops.

In conclusion, the author would like to quote that his model is able to reveal the exposure to risk pertaining to risky and devastating events. It also quantifies the rate at which supply fails to meet the demand. The stochasticity accounts for the variability in water availability and crop requirements.

My Comments:

I think this paper is a preliminary inquiry into the using the CVaR concept for assessing risk in water resources decision making. The paper introduces the CVaR concept and provides a hypothetical application which requires a lot more detailed analysis before applying to a real problem of water resources decision making. There is scope for including stochasticity in crop prices and climate change to give a realistic model prediction in hand.