Risk

Financial risk

In finance, an elementary measure of risk for financial asset prices is with volatility, the degree of variation of a trading price over time, usually measured by the standard deviation of logarithmic returns.

Portfolio theory

Modern portfolio theory measures the riskiness of a portfolio using the variance (or standard deviation) of the portfolio. If we denote return by ${\displaystyle R(w)}$ of a portfolio with weight vector ${\displaystyle w=(w_1,\dots ,w_n{)}^{\prime }}$ then the risk, as measured by variance of the portfolio is given by

${\displaystyle \text{Risk}=\mathrm{V}\mathrm{a}\mathrm{r}(R(w))=\sum _{i=1,j=1}^n{w}_j{w}_j\mathrm{C}\mathrm{o}\mathrm{v}(R_i,R_j)}$

where ${\displaystyle R_i}$ denotes the return of asset ${\displaystyle i=1,\dots ,n}$. Modern portfolio theory tells us an optimal combination of weights creates an optimal portfolio - known as the tangency portfolio - that still has undiversifiable risk. The model implies this 'systematic' source of risk should be the only factor considered, as all other sources of risk can be diversified away. An extension of this is the Capital asset pricing model (CAPM), where this optimal portfolio becomes known as the market portfolio.

The beta coefficient measures the sensitivity of an individual asset to overall market changes, and is defined as the linear projection coefficient of asset ${\displaystyle i=1,\dots ,n}$ returns on the returns of a market portfolio,

${\displaystyle {\beta }_i=\frac{\mathrm{C}\mathrm{o}\mathrm{v}(R_i,R_{mkt})}{\mathrm{V}\mathrm{a}\mathrm{r}(R_i)}}$

In a CAPM world, ${\displaystyle {\beta }_i}$ can be interpreted as the contribution of systemic risk to the risk of asset ${\displaystyle i}$.

Risk-neutral measure

In mathematical finance, a risk-neutral measure is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under the measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing.

Let ${\displaystyle S}$ be a d-dimensional market representing the price processes of the risky assets, ${\displaystyle B}$ the risk-free bond and ${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$the underlying probability space. Then a measure ${\displaystyle Q}$ is a risk-neutral measure if

1. ${\displaystyle Q\approx P}$, i.e., ${\displaystyle Q}$ is equivalent to ${\displaystyle P}$,
2. the processes ${\displaystyle {\left(\frac{S_t^i}{B_t}\right)}_t}$ are (local) martingales w.r.t. ${\displaystyle Q}$ ${\displaystyle \mathrm{\forall }i=1,\dots ,d}$.

Mandelbrot's mild and wild theory

Benoit Mandelbrot distinguished between "mild" and "wild" risk and argued that risk assessment and analysis must be fundamentally different for the two types of risk. Mild risk follows normal or near-normal probability distributions, is subject to regression to the mean and the law of large numbers, and is therefore relatively predictable. Wild risk follows fat-tailed distributions, e.g., Pareto or power-law distributions, is subject to regression to the tail (infinite mean or variance, rendering the law of large numbers invalid or ineffective), and is therefore difficult or impossible to predict. A common error in risk assessment and analysis is to underestimate the wildness of risk, assuming risk to be mild when in fact it is wild, which must be avoided if risk assessment and analysis are to be valid and reliable, according to Mandelbrot.

Estimation

• Proxy or analogue data from other contexts, presumed to be similar in some aspects of risk.
• Theoretical models, such as Monte Carlo simulation and Quantitative risk assessment software.
• Logical models, such as Bayesian networks, fault tree analysis and event tree analysis
• Expert judgement, such as absolute probability judgement or the Delphi method.

Management

Risk management is the set of actions that organisations take to avoid and mitigate downside risks, taking into account factors such as the possibility of upside risk opportunities, innovation, the environment, safety, scientific evidence, culture, politics, and law. Risk management operates at the strategic, operational, and individual level, and may form part of an overarching governance, risk, and compliance strategy. It comprises the assessment of risk as regards an organisation's objectives and strategies, as well as risk mitigation options, such as risk transformation, risk transfer, risk avoidance, risk reduction, and risk retention.

Assessment

Risk assessment is a systematic approach to recognising and characterising risks, and evaluating their significance, in order to support decisions about how to manage them. ISO 31000 defines it in terms of its components as "the overall process of risk identification, risk analysis and risk evaluation":

• Risk identification is "the process of finding, recognizing and recording risks". It "involves the identification of risk sources, events, their causes and their potential consequences." ISO 31000 describes it as the first step in a risk assessment process, preceding risk analysis and risk evaluation. In safety contexts, where risk sources are known as hazards, this step is known as "hazard identification".
• The ISO defines risk analysis as "the process to comprehend the nature of risk and to determine the level of risk". In the ISO 31000 risk assessment process, risk analysis follows risk identification and precedes risk evaluation. Risk analysis often uses data on the probabilities and consequences of previous events.
• Risk evaluation involves comparing estimated levels of risk against risk criteria to determine the significance of the risk and make decisions about risk treatment actions. In most activities, risks can be reduced by adding further controls or other treatment options, but typically this increases cost or inconvenience. It is rarely possible to eliminate risks altogether without discontinuing the activity. Sometimes it is desirable to increase risks to secure valued benefits. Risk criteria are intended to guide decisions on these issues.

For example, the tolerability of risk framework, developed by the UK Health and Safety Executive, divides risks into three bands:

• Unacceptable risks – only permitted in exceptional circumstances.
• Tolerable risks – to be kept as low as reasonably practicable (ALARP), taking into account the costs and benefits of further risk reduction.
• Broadly acceptable risks – not normally requiring further reduction.

Attitude, appetite and tolerance

The terms risk appetite, attitude, and tolerance are often used similarly to describe an organisation's or individual's attitude towards risk-taking. One's attitude may be described as risk-averse, risk-neutral, or risk-seeking.

Mitigation

• Risk transformation describes the process of not only mitigating risks but also employing risk factors into advantages.
• Risk transfer is the shifting of risks from one party to another, typically an insurer.

Psychology of risk

Risk perception

Risk perception is the subjective judgement that people make about the characteristics and severity of a risk. At its most basic, the perception of risk is an intuitive form of risk analysis.

Adults have an intuitive understanding of risk, which may not be exclusive to humans. Many ancient societies believed in divinely determined fates, and attempts to influence the gods can be seen as early forms of risk management. Early uses of the word 'risk' coincided with an erosion of belief in divinely ordained fate. Notwithstanding, intuitive perceptions of risk are often inaccurate owing to reliance on psychological heuristics, which are subject to systematic cognitive biases. In particular, the accuracy of risk perception can be adversely affected by the affect heuristic, which relies on emotion to make decisions.

The availability heuristic is the process of judging the probability of an event by the ease with which instances come to mind. In general, rare but dramatic causes of death are over-estimated while common unspectacular causes are under-estimated; an "availability cascade" is a self-reinforcing cycle in which public concern about relatively minor events is amplified by media coverage until the issue becomes politically important. Despite the difficulty of thinking statistically, people are typically subject to the overconfidence effect in their judgements, tending to overestimate their understanding of the world and underestimate the role of chance, with even experts subject to this effect. Other biases that affect the perception of risk include ambiguity aversion.

Paul Slovic's "psychometric paradigm" assumes that risk is subjectively defined by individuals, influenced by factors such as lack of control, catastrophic potential, and severity of consequences, such that risk perception can be psychometrically measured by surveys. Slovic argues that intuitive emotional reactions are the predominant method by which humans evaluate risk, and that a purely statistical approach to disasters lacks emotion and thus fails to convey the true meaning of disasters and fails to motivate proper action to prevent them. This theory has received support from retrospective studies and evolutionary psychology. Hazards with high perceived risk are therefore, in general, seen as less acceptable and more in need of reduction.

Cultural theory of risk views risk perception as a collective phenomenon by which different cultures select some risks for attention and ignore others, with the aim of maintaining their particular way of life. Hence risk perception varies according to the preoccupations of the culture. The theory outlines two categories, the degree of binding to social groups, the degree of social regulation. Cultural theory can be used to explain why it can be difficult for people with different world-views to agree about whether a hazard is acceptable, and why risk assessments may be more persuasive for some people than others. However, there is little quantitative evidence that shows cultural biases are strongly predictive of risk perception.

Decision theory

In decision theory, regret (and anticipation of regret) can play a significant part in decision-making, distinct from risk aversion. Framing is also a fundamental problem with all forms of risk assessment. In particular, because of bounded rationality, the risk of extreme events is discounted because the probability is too low to evaluate intuitively. As an example, one of the leading causes of death is road accidents caused by drunk driving – partly because any given driver frames the problem by largely or totally ignoring the risk of a serious or fatal accident. The right prefrontal cortex has been shown to take a more global perspective, while greater left prefrontal activity relates to local or focal processing. Reference class forecasting is a forecasting method by which biases associated with risks can be mitigated.

Risk taking

Researchers may categorize some humans as risk-takers ("chancers")

Psychologists have run randomised experiments with a treatment and control group to ascertain the effect of different psychological factors that may be associated with risk taking, finding that positive and negative feedback about past risk-aking can affect future risk-taking. For example, one experiment showed that belief in competence correlated with risk-taking behavior. The theory of risk compensation suggests that people typically adjust their behavior in response to the perceived level of risk, becoming more careful where they sense greater risk and less careful if they feel more protected. People also show risk aversion, such that they reject fair risky offers because of the perception of loss. Further, intuitive responses appear less risk-averse than subsequent reflective response.

Sex differences

Sex differences in financial decision making are relevant and significant. Numerous studies have found that women tend to be financially more risk-averse than men and hold safer portfolios. Scholarly research has documented systematic differences in financial decisions such as buying investments versus insurance, donating to ingroups versus outgroups (such as terrorism victims in Iraq versus the United States), spending in stores, and the endowment effect-or asking price for goods people have.

Philosophy of risk

Peter L. Bernstein (2012) showed that people used risk estimates before statistics and probability calculations were developed. Instead of relying on numbers, people used narratives and letters. Captains and merchants shared voyage stories at coffeehouses, comparing notes about hazards on new routes and seasonal patterns. Through a web of correspondents, letters became increasingly important as people could update their beliefs about weather, wars, or piracy over long distances. These qualitative data helped investors and underwriters judge how dicey a proposed voyage felt.

This kind of evidence has led philosophers to think there is more to (objective) risk than the likelihood of an undesirable outcome. Ebert et al. (2020) suggest distinguishing risk monists from risk pluralists: risk monists argue that there is just one correct way to understand risk. Tversky and Kahneman can be considered monists in this sense; probability judgments that diverged from the probability calculus were deemed incorrect or biased. By contrast, pluralists claim that there are different, valid notions of risk. On this view, people who lived before statistics were developed may have been doing something legitimate when they estimated risks—even if those estimates conflict with a statistical notion. Without statistics, what else could they have done?

According to the modal account of risk, a situation is risky when nearby possible worlds—differing only slightly from the actual one—contain serious harm. Risk tracks the closeness of such bad outcomes rather than their probability; hence a low-chance disaster may still count as high risk if only a small change would have led to it. On the normic account of risk, a situation is risky when the bad outcome would be normal or unsurprising. Risk is assessed through system functions and norms rather than bare probability. A harm counts as high risk when it would occur in the most normal continuations of the present setup; the less departure from normality needed for the harm, the greater the risk. Especially in domains where we lack predictive power, such approaches allow us to consider risk without relying on unknown probabilities, as illustrated by the normic account of suicide risk.

Society and culture

Risk and autonomy

The experience of many people who rely on human services for support is that 'risk' is often used as a reason to prevent them from gaining further independence or fully accessing the community, and that these services are often unnecessarily risk averse. "People's autonomy used to be compromised by institution walls, now it's too often our risk management practices", according to John O'Brien. Michael Fischer and Ewan Ferlie (2013) find that contradictions between formal risk controls and the role of subjective factors in human services (such as the role of emotions and ideology) can undermine service values, so producing tensions and even intractable and 'heated' conflict.

Risk society

Anthony Giddens and Ulrich Beck argued that whilst humans have always been subjected to a level of risk – such as natural disasters – these have usually been perceived as produced by non-human forces. Modern societies, however, are exposed to risks such as pollution, that are the result of the modernization process itself. Giddens defines these two types of risks as external risks and manufactured risks. The term Risk society was coined in the 1980s and its popularity during the 1990s was both as a consequence of its links to trends in thinking about wider modernity, and also to its links to popular discourse, in particular the growing environmental concerns during the period.