Risk
Hornby (1985) defines ‘risk’ as: “(instance of) possibility or chance of meeting danger, suffering loss, injury, etc.” Also: “at the ~ risk of / at ~ of, with the possibility of (loss etc.)”.
Thus, if there are possible outcomes O = {o1, o2, ..., on}, then the situation is risky if at least one of the o’s represents a loss. The risks are the oi that are losses, thus Risks[O] = {oi
O | oi is a loss}. The risk factors are the positions or index numbers of the risky outcomes, the i’s, or the dimensions (the causes that make such positions to be filled).
We will use the term ‘valued risk’ when a risk is valued with money or utility. When all risks have been made comparable by valuing them, then we can add them, and we will use the term expected risk value for the expected value of the ‘valued risks’. Then, crucially, once these definitions are well understood, then we may also use ‘the risk’ for the expected risk value. [110]
With such understanding, risk will be r = -Ex<0[x] [111] or for short r = -E[x < 0]. [112]
Valued risk deals with the cases when probabilities are known or when unknowns are assumed to be uniformly distributed over known categories. It is not customary to use the term ‘risk’ for unknown categories. For example, it is uncommon to say, or write economics papers about this, that “all our lives are at risk of a suddenly imploding universe, or black hole hitting Earth, or waking up as a cockroaches”. Such real ‘Acts of God’ are commonly neglected. Note though that it still remains possible to say that a situation is risky even though one cannot put a number to it. Above expectation may be indeterminate since one may lack knowledge about the probability distribution or even the categories.