# Risk Analysis in Optimization

Risk is often argued to be a subjective phenomenon involving exposure and uncertainty. Risk is qualified as an asymmetric phenomenon – it is related to loss only. Instead, uncertainty is a symmetric notion; it relates to possible positive and negative deviations from the expected price or return.

Risk analysis helps the decision maker recognize the difference between the expected value of a decision alternative and the payoff that may actually occur. A decision alternative and a state of nature combine to generate the payoff associated with a decision. The risk profile for a decision alternative shows the possible payoffs along with their associated probabilities.

Risk analysis is the systematic study of uncertainties and risks we encounter in business, engineering, public policy, and many other areas. Risk analysts seek to identify the risks faced by an institution or business unit, understand how and when they arise, and estimate the impact (financial or otherwise) of adverse outcomes. Risk managers start with risk analysis, then seek to take actions that will mitigate or hedge these risks.

Some institutions, such as banks and investment management firms, are in the business of taking risks every day. Risk analysis and management is clearly crucial for these institutions. One of the roles of risk management in these firms is to quantify the financial risks involved in each investment, trading, or other business activity, and allocate a risk budget across these activities. Banks in particular are required by their regulators to identify and quantify their risks, often computing measures such as Value at Risk (VaR), and ensure that they have adequate capital to maintain solvency should the worst (or near-worst) outcomes occur.

Quantitative Risk Analysis

Quantitative risk analysis is the practice of creating a mathematical model of a project or process that explicitly includes uncertain parameters that we cannot control, and also decision variables or parameters that we can control. A quantitative risk model calculates the impact of the uncertain parameters and the decisions we make on outcomes that we care about — such as profit and loss, investment returns, environmental consequences, and the like. Such a model can help business decision makers and public policy makers understand the impact of uncertainty and the consequences of different decisions.

One way to learn how to deal with uncertainty is to perform an experiment. But often, it is too dangerous or expensive to perform an experiment in the “real world” — so we resort to a model, such as a scale model of an airplane in a wind tunnel. With a model, we can simulate what would happen in the real world, and perform many experiments — for example, subjecting our model airplane to various air currents and forces — and learn how it behaves. We can introduce uncertainty into our experiments using devices such as a coin toss, dice roll, or roulette wheel. A single experiment that involves a coin toss may not tell us very much, but if we perform a simulation that consists of many experiments or trials, and collect statistics about the results, we can learn quite a lot.

If we have the skills and software tools needed to create a mathematical model of a project or process on a computer, we can perform a simulation with many trials in a very short time, and at very low cost. With such advantages over experiments in the real world, it’s no wonder that computer-based simulation has become so popular.

There are 4 ways to respond to risk, which are

• Transfer: The responsibilities of bearing the risk are transferred to another entity, usually in the form of insurance.
• Avoidance: The organization does all it can to ensure that the risk does not occur.
• Mitigation: The organization reduces the chances of the risk occurring and also identifies alternatives for reducing the consequences.
• Acceptance: When there’s no way to avoid, transfer or mitigate a risk, the organization accepts that there is nothing that can be done and makes no effort to deal with it.

Generally, a risk model consists of two parts:

• Probabilistic models are constructed for the underlying sources of risk, such as market or credit risk factors, and the portfolio loss distribution is described by means of the probabilistic models.
• Risk is quantified by means of a risk measure that associates a real number to the portfolio loss distribution.

Measures of Dispersion

Measures of dispersion represent how observations in a dataset are distributed and whether the variability around the mean of the distribution is high or low. Intuitively, if a quantity is nonrandom, then it is equal to its mean with probability one and there is no fluctuation whatsoever around the mean. We look at three measures widely used in practice: the standard deviation and the mean absolute deviation.

Markowitz (1952) was the first to recognize the relationship between risk and reward and introduced the standard deviation as a measure of risk. He was also the first to suggest the semi-standard deviation as an alternative to deal with the standard deviation’s symmetric nature. We saw, however, that that one cannot be a true risk measure either.