Risk, Uncertainty and Variability defined

Renowned mathematician and statistician Sir David Cox famously said: “Vari- ability is a phenomenon in the physical world to be measured, analysed and where appropriate explained. By contrast, uncertainty is an aspect of knowledge” (Vose, 2008).

Generally, there are two components influencing our ability to predict what the future holds: these are uncertainty and variability.

Uncertainty

Uncertainty is a state of limited information or knowledge (level of ignorance) about the parameters characterising the system or phenomenon being modelled. It is sometimes reducible (usually at an additional cost) through further mea- surement or study, or by consulting more experts. Uncertainty is by definition subjective, as it is a function of the assessor, but there are techniques available to allow one to be ‘objectively subjective’. This essentially amounts to a logical assessment of the information contained in available data about model parame- ters without including any prior, non-quantitative information. The result is an uncertainty analysis that any logical person should agree with, given the available information (Vose, 2008).

Geologists analysing drilling and sampling data often have such discussions and compare notes when they interpret results. In Figure 5.1, deposit A has more uncertainty than deposit B due to the fact that there is a lot less information (drill hole data) available. However, because it contains less variability (i.e. more consistent geological and stratigraphic layering) than B it is nonetheless easier to come up with a reasonable interpretation in terms of the potential lateral extension of the geology. In deposit B however, even though a lot of information is available on the geology, due to its variability it will still be comparatively more difficult to derive a model of the geology.

Variability

Variability is the effect of chance and is a function of the system - it is the ‘nature of the beast’. In the geological context it’s the change in content and quality spatially across the ore body. It is not reducible through either study or further measurement. A simple illustration of variability can be found in tossing a coin. If a coin is tossed, the expectation will be a head (H) or tail (T), each with a probability of 50% if the coin is presumed to be fair. If the coin is tossed twice, four possible outcomes are possible HH, HT, TH, TT, each with a probability of 25% because of the coin’s symmetry. Exactly what the tosses of a coin will produce cannot be predicted upfront with certainty because of the inherent randomness of the coin toss.

In Figure 5.1, deposit A has less variability than B. Total Uncertainty

Total uncertainty is often cited as the combination of uncertainty and variability. These two components act together to erode our ability to be able to predict what the future holds. Uncertainty and variability are philosophically very different, and they are usually kept separate in risk analysis modelling.

Probability

Probability is a numerical measurement of the likelihood of an outcome of some stochastic process. It is thus one of the two components, along with the values of the possible outcomes, that describe the variability of a system. Probability is used to define a probability distribution, which describes the range of values the variable may take, together with the probability (likelihood) that the variable will take any specific value.

Measuring the Center of the Distribution (The First Moment)

The first moment of a distribution measures the expected rate of return on a particular project. It measures the location of the project’s scenarios and possible outcomes on average. The common statistics for the first moment include the mean (average), median (center of a distribution), and mode (most commonly occurring value) - see Figure 5.2.

Figure 5.2: Statistic to demonstrate central tendency - source: Mun (2006)

Measuring the Spread of the Distribution (The Second Moment) The second moment measures the spread of a distribution, which is a measure of risk or uncertainty. The spread or width of a distribution measures the variability of a variable, that is, the potential that the variable can fall into different regions of the distribution. The width or risk of a variable can be measured through several different statistics, including the range, standard deviation (s), variance, coefficient of variation, volatility, and percentiles - see Figure 5.3.

Figure 5.3: Statistic to demonstrate spread - source: Mun (2006)

If we assume we have two projects for which the underlying input uncertainty is different. If likely outcomes for the two projects are generated (e.g. using a Monte Carlo simulation method) then the results for each can be summarised in a statistical distribution. Figure 5.2 and Figure 5.3 illustrate the comparison of

two projects by considering their average (µ1 v.s. µ2) and standard deviation (σ1 v.s. σ2) values.

In Figure 5.2 project 2 has a higher average value than project 1, however both project 1 and 2 have the same spread or standard deviation.

In Figure 5.3 project 1 and 2 have the same average value but project 2 has a greater (wider) spread or standard deviation than project 1.

Coefficient of Variation (CV)

The CV is defined as the ratio of standard deviation to the mean, which means that various risks can be common sized or ‘normalised’ for the sake of compari- son. This measure of risk or dispersion is applicable when a variable’s estimates, measures, magnitudes, or units differ. The CV is useful as a measure of risk per unit of return, or when inverted, can be used as a measure of ‘bang for your buck’ or returns per unit of risk. Thus, in portfolio optimization, one would be inter- ested in minimizing the CV or maximizing the inverse of the CV. In the following sections we will be building on the view that the CV is a suitably representative metric to reflect the risk-return trade off for various options.

A number of statistical concepts required definition here as they will be used extensively in the chapters to come to make sense of outputs and results. The source used in describing these terms is Mun (2006).