Two Types of Risk

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Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queue, 2006, 1, pp. 109 - 140
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The risk encountered in many environmental problems appears to exhibit special two-sided characteristics. For instance, in a given area and in a given period, farmers do not want to see too much or too little rainfall. They hope for rainfall that is in some given interval. We formulate and solve this problem with the help of a two-sided loss function that depends on the above range. Even in financial portfolio optimization a loss and a gain are two sides of a coin, so it is desirable to deal with them in a manner that reflects an investors relative concern. Consequently, in this paper, we define Type I risk: the loss is too big and Type II risk: the gain is too small. Ideally, we would want to minimize the two risks simultaneously. However, this may be impossible and hence we try to balance these two kinds of risk. Namely, we tolerate certain amount of one risk when minimizing the other. The latter problem is formulated as a suitable optimization problem and illustrated with a numerical example.
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