To understand financial market dynamics, it is helpful to distinguish beliefs, attitudes towards risk, and attitudes towards ambiguity. Beliefs are subjective evaluations of future cash flows. Risk refers to uncertainty within a model of the asset’s return. And ambiguity means uncertainty about the model and probability distributions. Accordingly, one can separate price dynamics into three effects: changes in beliefs, changes in risk premia and changes in ambiguity premia. Ambiguity premia seem to be dominant, particularly when investors have little information about the nature of a particular risk. Traditional risk premia seem to be much less significant. Belief effects are negligible when ambiguity is high but increase as information accumulates. Often trading opportunities arise from the mean reversion of ambiguity premia and the “under-adjustment” of beliefs.

The below are mostly quotes from:
Li, Wenhui and Christian Wilde (2021), “Separating the effects of beliefs and attitudes on pricing under ambiguity”, SAFE Working Paper No. 311.
with some additional quotes whose sources are linked separately.

The post ties up with this site’s summary on risk management.

Three pricing factors

“The pricing of an ambiguous asset, whose cash flow stream is uncertain, may be affected by three factors: the belief regarding the realization likelihood of cash flows, the subjective attitude towards risk, and the attitude towards ambiguity…In theory, these three factors are mutually independent concepts, and thus their effects on decisions are separable.”

“Price deviation can be decomposed into three components: the belief effect (i.e. price deviation ascribed to a subject’s personal belief deviating from the neutral benchmark belief), the risk premium (i.e. price deviation arising from a subject’s non-neutral attitude towards risk), and the ambiguity premium (i.e. price deviation arising from a subject’s non-neutral attitude towards ambiguity)…All three components are relative terms, relative to the benchmark certainty equivalent. The dynamic of the total price deviation…is driven by the dynamics of the three components combined.”

Belief

Beliefs can be defined as subjective evaluations about the ambiguous environment. It describes, in a decision maker’s opinion, how likely a possible scenario is to occur. [This is different from] attitude towards risk and attitude towards ambiguity [which] capture a decision maker’s preference.”

“The belief effect is defined as the price deviation arising from the difference between the all-neutral benchmark belief characterization and a subject’s personal belief characterization…A positive belief effect indicates that [the person] prices the asset lower than the benchmark belief implies, as she holds a more unfavorable belief than the benchmark belief…The belief effect can also be understood as the difference between two certainty equivalents of the ambiguous asset: the benchmark certainty equivalent based on the all-neutral belief and the [person’s] certainty equivalent based on the personal belief, assuming that [he/she] is risk- and ambiguity-neutral…From this point of view, it is also clear that belief effects are only belief-relevant and independent from attitudes, since all attitudes are assumed to be neutral.”

“When a [person] updates her current belief, two factors may determine where the updated belief locates: the stickiness to her current belief, and her pessimistic/optimistic inclination.”

Risk

“The risk premium [is] the price deviation arising from a subject’s non-neutral risk attitude.”

“Aversion towards risk describes that a decision maker prefers a less risky situation to a more risky situation…A positive risk premium indicates risk aversion, a negative premium indicates risk loving, and zero indicates risk neutrality. The higher the value is, the more risk averse is [the person].”

Ambiguity

“A sophisticated investor has to account for different sources of uncertainty…The first one is denoted as ‘risk’ or ‘uncertainty ‘within the model’ and it reflects the stochastic nature of the problem. A second source, denoted ‘model uncertainty” or ‘ambiguity’, reflects the limits of an agent to quantify the uncertainty related to potential outcomes. Not being able to identify the data generating process behind some observable variables, she evaluates a set of possible models and she attaches to them a subjective probability.” [Girardi]

Ambiguity describes a situation in which some outcomes of an event have unknown [or] non-singular probability measurements. Ubiquitous in financial markets, the presence of ambiguity is shown to affect individuals’ behaviors such as market participation, asset pricing, and portfolio choices.”

“Aversion towards ambiguity describes that a decision maker prefers a less ambiguous situation to a more ambiguous situation…The ambiguity premium [is] the price deviation arising from non-neutral ambiguity attitude…The ambiguity premium is a clean measurement of the price deviation arising from ambiguity attitude.”

“Experimental evidence supports the idea that individual decision makers exhibit a strong preference towards prudent choices both under risk and model uncertainty…A prudent attitude towards ambiguity captures an aversion towards model uncertainty which strengthens the more the investor believes that unfavourable events are likely to realize, such as during financial crises or contractions of the business cycle.” [Girardi]

Separating the pricing factors

Belief, attitude towards risk, and attitude towards ambiguity are…separable factors which influence individuals’ decisions independently. The focus of this paper is to cleanly measure the effect of each individual factor. The effects are denoted as belief effect, risk premium, and ambiguity premium, respectively…The clean measurements allow us to investigate the qualitative feature (sign) and the quantitative feature (magnitude) of each individual effect.”

“This paper gathers data from a series of laboratory experiments…Subjects are told that there is a binary lottery whose outcome is either winning or losing. Subjects are also told that the lottery pays out a positive financial reward in case of winning, and that the lottery pays out zero in case of losing. However, neither the winning probability nor the losing probability of the lottery is known to any subject. In this way, an ambiguous lottery is set up. To elicit beliefs, we use guess games to investigate a subject’s belief regarding the winning probability of the ambiguous lottery. To elicit ambiguity attitude and risk attitude, we use choice lists to elicit a subject’s certainty equivalents of the ambiguous lottery and her certainty equivalents of some purely risky lotteries, respectively.”

“We apply both non-parametric and parametric methods to cleanly separate the belief effects, the risk premiums, and the ambiguity premiums from each other.

  • As a first analysis, we disentangle the three price deviation components with a non- parametric method. The results are directly derived from the experiment data…For each subject we can disentangle the three price deviation components, directly computed from the responses of the three ‘parallel’ games…The non-parametric method does not require any model specification or parameter estimation, which is beneficial for the precision of this analysis.
  • As a second analysis, we apply a parametric method [that] estimates the parameters governing risk attitude and ambiguity attitude, respectively. The model provides a theoretical foundation for the separation between beliefs and attitudes: risk attitude and ambiguity attitude are governed by different utility functions; Beliefs stay outside of the utility functions, and thus stay separated from attitudes.”

“Failing to disentangle beliefs from attitudes may lead to incorrect attitude characterizations and subsequently confound the estimation of the effect of attitude.”

Findings

“This paper reaches the following findings:

  • Belief effects on average tend to be zero when the degree of ambiguity is high, indicating that neither substantial pessimism nor substantial optimism is found in initial beliefs. As new information accumulates, belief effects emerge. This is driven by the under-adjustment of beliefs in updating compared with what Bayes’ rule implies…
  • From an individual’s point of view…a subject is more likely to price the asset lower than the benchmark belief implies, before any information is revealed, but more likely to price the asset as the benchmark belief implies, after observing a certain amount of information. It also means that a subject may bias towards the unfavorable situations in belief under completely ambiguity, and may turn to evaluating the situations neutrally as more and more information is revealed
  • On average subjects display risk neutrality independent of draw implementation. On an individual level, however, there exists rather large heterogeneity…
  • When the degree of ambiguity is high, ambiguity premiums are on average significantly positive. Overall, the ambiguity premiums are evidently larger than the risk premiums and the belief effects in a completely ambiguous environment…In a completely ambiguous environment, overall subjects tend to price the ambiguous asset lower than the all-neutral benchmark, which leads to a positive total price deviation. As information accumulates, on average the total price deviation decreases.”
  • There exists heterogeneity [differences across participants in the experiments] in all three price deviation components…The belief effect and the ambiguity premium tend to display decreasing heterogeneity as [information of the return-generating process] increases, while the risk premium displays increasing heterogeneity. It is understandable since the belief effect and the ambiguity premium are closely affected by [information]. As ambiguity gradually resolves, more and more subjects slide to the benchmark in belief and require less compensation for exposure to ambiguity. Thus, both distributions have the tendency compressing towards zero. A [person’s] risk attitude is, in theory, independent from the draws, and thus the risk premium is not systematically affected by [information].

Lessons for macro trading (editor’s comments)

Ambiguity seems to be the dominant factor of market dynamics in times of shocks and crises. Hence, for assessing the severity of a shock to markets, it is important to judge if it brings to investors’ minds a new type of uncertainty for which are not prepared (view post here). Unpreparedness means high ambiguity and, hence, high premia. When a market faces rising ambiguity risk assets may underperform. When ambiguity is stabilizing or declining, as information and experience are growing, risk assets may outperform, even if the underlying risk is not actually decreasing.

Volatility alone may be more important for risk management than for end investors. An asset with higher price fluctuation may not necessarily command a proportionately higher premium, particularly if the nature of volatility and the underlying stochastic process is well understood and accepted. This may explain part of the historic success of low-risk strategies that prefer leveraged low-risk assets over high-risk assets (view post here), as leverage may command a higher premium than volatility.

Rapid and efficient use of information about dominant risk factors should be profitable, particularly under high ambiguity. The trader that works faster with relevant information would benefit from popular under-adjustment of beliefs as much as from the ability to predict the dynamics of ambiguity premia. Put simply, if new significant risk factors arise, investors should quickly re-focus their research resources on those less well-understood factors.

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Ralph Sueppel is founder and director of SRSV, a project dedicated to socially responsible macro trading strategies. He has worked in economics and finance for over 25 years for investment banks, the European Central Bank and leading hedge funds. At present, he is head of research and quantitative strategies at Macrosynergy Partners.