1. Risk: Definition

Intuitively, risk is the possibility that we will not get what we expect. However, if we want to quantify risk, a good starting point would be to define the term more precisely. But here we stumble on a roadblock – even though, living in an uncertain world, we all have an intuitive understanding of risk, there is no single, universally accepted definition.

Definitions of risk are applied in different disciplines such as economics and mathematics; however, the definitions that are used vary across, and even within, those disciplines. To complicate matters further, the definition could be context specific. For example, the Risk and Insurance Management Society (RIMS), a global organization dedicated to risk management, defines risk as “uncertain future outcome(s) that can either improve or worsen one’s position.” In investments, risk can be defined as the likelihood that an investment’s actual return will differ from the one expected. In the context of insurance, risk can be defined as the possibility of loss or injury.

For our purposes, we will use risk and uncertainty interchangeably. However, in his 1921 book Risk, Uncertainty and Profit, Professor Frank Knight distinguishes between risk and uncertainty depending on whether the objective probabilities are known. According to him, risk applies to situations where objective probabilities are known while uncertainty applies to situations where the objective probabilities are unknown. Because, in the words of Prof. Knight, “true uncertainty is not susceptible to measurement,” it is not easy to analyze uncertain situations when the objective probabilities are not known. Consequently, our focus here will be on objective risk, i.e., on quantifying risk using objective probabilities. In Section 5, we will discuss subjective risk whereby probabilities are estimated based on personal beliefs and judgements.

Eminent mathematicians such as Laplace, Jacob Bernoulli and Poincaré believed that we live in a deterministic world where there are definite causes for each effect; everything happens for a reason and there is no such thing as luck. In such a cause-and-effect world, there is no place for games of chance as we would know exactly what outcome will occur when we throw a die or spin the wheel of a roulette. Knowing the causes, it is a simple matter of mathematical calculations to predict the future stock price or the value of a life insurance. However, as we do not always know the underlying laws of nature, we tend to attribute outcomes to chance. In our world of limited knowledge of natural laws, Poincaré notes, “Chance is only the measure of our ignorance.”^[1]

Games of chance have existed since the dawn of civilization. It was the interest in gambling that led to the origin of the probability theory in 17th century Renaissance Europe. A branch of mathematics, the probability theory is a powerful tool for modeling risk and making predictions and decisions. We begin our study of risk by reviewing major concepts from probability theory that will enable us to define several measures of risk.