Some limitations of frequency as a component of risk: an expository note

Students of risk analysis are often taught that "risk is frequency times consequence" or, more generally, that risk is determined by the frequency and severity of adverse consequences. But is it? This expository note reviews the concepts of frequency as average annual occurrence rate and as the reciprocal of mean time to failure (MTTF) or mean time between failures (MTBF) in a renewal process. It points out that if two risks (represented as two (frequency, severity) pairs for adverse consequences) have identical values for severity but different values of frequency, then it is not necessarily true that the one with the smaller value of frequency is preferable-and this is true no matter how frequency is defined. In general, there is not necessarily an increasing relation between the reciprocal of the mean time until an event occurs, its long-run average occurrences per year, and other criteria, such as the probability or expected number of times that it will happen over a specific interval of interest, such as the design life of a system. Risk depends on more than frequency and severity of consequences. It also depends on other information about the probability distribution for the time of a risk event that can become lost in simple measures of event "frequency." More flexible descriptions of risky processes, such as point process models can avoid these limitations.