Brief Notes on Asymmetric Dependencies in Risk Analysis
“All right,” said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest had gone. “Well! I’ve often seen a cat without a grin,” thought Alice; ” but a grin without a cat!” It’s the most curious thing I ever saw in all my life!“ — Alice’s Adventures in Wonderland
Abstract
Understanding and measuring dependencies between risks is essential for risk management. In real life, most dependencies are not symmetrical, i.e. the predictive power between the variables is not symmetric. Nevertheless, in enterprise risk management it is frequently assumed - mostly tacitly - that there are only symmetrical relationships. We would like to shed light on this fact and list possible solutions.
Why are dependencies important in risk management?
There are in fact many reasons for this, the most important of which we would like to highlight shortly:
Risk Aggregation
Dependencies between risks can have a heavy impact on risk aggregation. Usually, risk diversification is over estimated when risk dependencies are neglected.Risk Mitigation
A deep knowledge of risk dependencies can be essential in defining efficient mitigation strategies.Allocation of Risk Capital
When determining an appropriate proportion of risk capital for a risk or a bundle of risks, it is necessary to understand the dependencies with other risks with which the company is confronted.
Dependencies are considered in many risk models with little differentiation. In most cases, linear correlations or rank correlations are estimated and then used in the Monte-Carlo Simulation in order to perform risk aggregation and quantification of mitigation strategies.
As the determination of dependencies between risks can sometimes only be carried out very inadequately, it is often advisable not to exceed the granularity when creating risk registers.
≫In analyzing the risk landscape as part of ERM
or as part of project risk management
the quantification of risk dependencies
is both:
Important and Difficult.≪
(In-)Dependence in Probability Theory
Often Pearson’s linear correlation or rank correlations (Spearman’s 𝛒 and Kendall’s 𝜏 are employed to analyse dependencies and are also used as an input for Monte- Carlo Simulation especially in proceeding the risk aggregation.
Although linear correlations or rank correlations are technically relatively easy to handle, they must be used with caution and their limits must be observed.
Two events A and B are called independent if and onley if the probability of both events occurring together (the intersection of A and B) is equal to the product of their individual probabilities:
P(A ∩ B) = P(A) * P(B)
Sets of events A1, A2, …, An are mutually independent if every event is independent of any intersection of the other events. This means that for any subset of events, the probability of their intersection equals the product of their probabilities.
In real life independence is certainly the exception rather than the rule. Dependent events are also called correlated1.
If two events A and B are correlated either (i) or (ii) applies
(i) positive correlation : P(A ∩ B) > P(A) * P(B)
(ii) negative correlation: P(A ∩ B) < P(A) * P(B)
If A,B are possible realizations of risks, it follows that dependent risks are realized either systematically less or systematically more in the link than would be the case with independent risks. This is why understanding dependencies is important in risk analysis.
The structure of a dependency of a pair of variabels (v,w) is given by it’s joined distribution Pv,w. Based on this one can try to define a measure of the strength of dependency Dep as a function with, cf. Rényi2
0 ≤ Dep(Pv,w) ≤ 1
Dep(Pv,w) = 0 if and only if v,w are independent
Dep(Pv,w) = 1 if and only if w = f(v) or v = g(w),
(where f and g are Borel measurable functions)
This has some resemblance to correlation coefficients. In fact correlation coefficients are functions to [-1,1] instead of [0,1]. But, if we would like to measure only the strength of dependency via correlation we could simply take absolute amount of the coefficient which would result into a function into [0,1]. Moreover, linear correlation equals 1 exactly if and only if there is a functions f of the form f: x |- a*x+c almost surely with a>0 and w= f(v) which implies of course also the existence of a corresponding g. In the case of rank correlation, on the other hand, f only has to be positive mononotone and not necessarily linear.
Correlation coefficients are symmetric, i.e. cor(v,w) = cor(w,v) for each pair of variables and it does not follow from cor(v,w) = 0 that v and w are independent. In other words:
≫Correlation coefficients does not capture all forms of dependence
and are as such not helpful
in detecting asymmetries in Dependencies.≪
Asymmetric Dependencies
Even if there is no direct causal link between a pair of risks, according to Reichenbach’s principle of common cause, correlations are often indicators for a third hidden causal factor. The Common Cause Principle was introduced by Hans Reichenbach3, in The Direction of Time, which was published posthumously in 1956.
Suppose that two events A and B are positively correlated: p(A∩B)>p(A)p(B) . If we assume additionally , that neither event is a cause of the other. Then, Reichenbach’s Common Cause Principle (RCCP) states:
A and B will have a common cause that renders them conditionally independent.
Reichenbach incorporated his RCCP into a new probablistic theory of causation, and used it to describe a (purported) macrostatistical temporal asymmetry in analogy with the second law of thermodynamics which is often used to explain why there is a direction of time.
Causation -as part of its definition- has a direction it is a vector from cause to effect. Classical dependency measures like linear correlation are symmetric and therefore cannot give any indication of causation simply because they do not include a direction.
Asymmetrical dependencies can indicate causal dependencies.4 However, this will not be discussed further here.
Even when copula methods are used, symmetric copulae are usually the tools of choice, which are sometimes parameterized on the basis of an assumed or calculated rank correlation.
The figure above displays 4 examples of quantile-risk pairs whose dependency structure is defined by different types of copulae. The parameters are chosen so that a rank correlation of + 0.75 results. All diagrams show 5,000 simulated points respectively. The upper two copulae allow for asymmetry in relation to the (black)diagonal between (0,1) to (1,0) but all are symmetrical to the (red) diagonal between (0,0) to (1,1).
With the help of such copulae it is possible to represent different types of dependencies, but we still assume symmetry (related to the red line) and the closeness of the dependency is measured with just one number without an arrow from cause to effect.
The adaptation of a so-called checkerboard copula to empirical values can lead to the copula being asymmetrical in relation to both diagonals and thus offering more flexibility.
Our intention is to draw the attention to the question of symmetry to the (red) diagonal between (0,0) to (1,1).
Dependencies can have a parabolic or sinus type shape for instance.
If the relation is a clear parabola (or sinus curve) a perfect functional dependency between the risks is given in the sense that the knowledge of the value of risk x delivers a unique precise value for risk y.
But nevertheless, linear correlation and rank correlation are zero in this case. Thus, by no means correlation zero implies non-existence of any link between the risks.
According to Rényi a dependency measure should obviously result in 1 in this simple case because we can transform x to y simply by squaring.
Moreover, there is an important difference between the two risks in regard to their relation to each other:
While we can determine the value of y perfectly if we know x, this is not possible in the opposite case: With a single exception (vertex of the parabola) we always get two x values from one y value and those pairs of x values can be show a wide distant between the two points
A similar observation can be made when there is a relation shaped like a sinus curve and for a multitude of further dependency structures.
The next figure shows a scatter plot of a pair of pseudo data with a nearly parabolic dependency in the lower left corner and the same pair of values next to it, but with random redistribution of the pairing. In both cases, the correlation coefficients are very close to zero. So when we look at these two pairs of related distributions, we see no difference only through the lens of the correlation coefficients and distributions of the single risks. Nevertheless, the aggregation leads to very different distributions, as shown in the upper part of the figure. This results into large differences in the VaRs.
Real Life Examples of U-Shaped Dependencies
Frequently correlation coefficients are understood as measures of information one variable inherits regarding another one. While there is truth in this, this can also be very misleading, as just explained there are examples showing exceptionally low or even zero correlation but high level of inter risk information.
Moreover, in certain examples we see high or even full information in one direction, but lower information in the other direction. Such an asymmetry does not proof a causal link but can be understood as an indication of such a relation.
It is intuitively clear that examples similar to our parabola case contain more information in one variable (in the example the variable of the x axis) than in the reverse risk. Thus (in the exact case) a value on the y-axis is assigned two different values on the x-axis that are far apart from each other, while conversely there is a clear assignment.
Such a dependency between variables can be observed in practice, for example when it comes to the reliability of complex technical structures such as power plants depending on their age. It is known that the number of failures is relatively high shortly after commissioning, only to fall again once the initial weak points have been eliminated. As the power plant gets older, the number of failures increases again, for example due to material fatigue.
In motor insurance we see U-shapes in age related accident frequencies.
The degree of utilization of some machines often has a U-shaped dependency on the susceptibility to faults. The same can be observed in many areas with regard to the workload of employees and the quality of work.
We see very similar dependency structures in many areas. A completely different example would be the error rate and processing speed of routine tasks depending on the intensity of supervision by superiors. If there is no or very little supervision, this can lead to poorer performance. If, on the other hand, too much supervision is exercised, the associated frustration of employees can also lead to a deterioration.
It is fair to say that we often find ourselves in this or a similar situation when the target value cannot be achieved simply by maximizing or minimizing a few variables, but an optimization process is necessary.
Interestingly,on the metalevel of risk perceiption which of course can have retroactive effects on the risk level itself we alo find such dependencies: Inverted U-shaped model: How frequent repetition affects perceived risk5
Entropy Measures & Kernel Methods
Entropic measures of similarity of probability distribution are well known since quite some time. Let us only mention here the Kullback-Leibler-Entropy. In fact, there are a lot of different versions of these measuring of distribution similarities.6 More recently, kernel or copula methods have also been used for this purpose.
Based on Kernel regression methods H. D. Vinod employs in the R package generalCorr a generalized correlation coefficient leading to asymmetric generalized correlation matrices.
Using pseudo data, we demonstrate the application for the two-dimensional case:
Another method to detect asymmetric and directed dependence in bivariate samples is used in the package qad. where a copula based method is used. 7
Empiric bivariate data can first be approximated by checkerboard copulae, in order to then be examined for any asymmetric dependencies.
Modeling Causality
If a causal chain is known as such, it should also be modelled as such and not only with the help of correlation coefficients or similar instruments. In this way, we implement a directional arrow in our model, which is lost if we use symmetric dependency structures only. This approach is also useful when the causal chain is not known exactly, but only with a wide corridor of uncertainty.
If causal dependencies are modelled explicitly, this implies that any asymmetries are automatically taken into account.
A suitable method of analysing causal dependencies while taking uncertainties into account is the use of Bayesian networks.
The use of risk drivers explicates Rechenbach’s common cause principle and is often useful when external factors, such as general economic factors or climatic contingencies, need to be taken into account.
It should be ensured that the simulation models used in the ERM are open to such extended techniques.
Footnotes
Correlation is sometimes used as an abbreviation for “having a non-zero linear correlation coefficient” or “having a non-zero rank correlation”, but sometimes also in the wider sense to indicate that variables are not independent. In this paper as long we do not talk explocitly about linear or rank correlation the word correlation refers to correlation in the wider sense.↩︎
A. Rényi,On Measure of Dependence, Acta Mathematica Academiae Scientiarum Hungarica Volume10,pages 441–451, (1959) WebLink↩︎
Hans Reichenbach (* September 26, 1891 in Hamburg; † April 9, 1953 in Los Angeles, California) was a German physicist, philosopher and logician. At Einstein’s suggestion, Reichenbach was appointed associate professor of philosophy of physics at the University of Berlin in 1926. Together with Rudolf Carnap, Reichenbach founded the journal Erkenntnis, the organ of logical positivism, in 1930. Reichenbach was among the first lecturers to be dismissed from the university when the National Socialists seized power in 1933.↩︎
H. D. Vinod, Generalized Correlations and Kernel Causality Using R Package generalCorr October 3, 2017(Vignette to the R package generalCorr.)↩︎
Xi Lu, Xiaofei Xie and Lu Liu, Inverted U-shaped model: How frequent repetition affects perceived risk Published online by Cambridge University Press: 01 January 2023 WebLink↩︎
Arthur Gretton, Ralf Herbrich, Alexander Smola, Olivier Bousquet and Bernhard Schölkopf,Kernel Methods for Measuring Independence, Journal of Machine Learning Research 6 (2005) 2075–2129↩︎
Florian Griessenberger, Wolfgang Trutschnig, Robert R. Junker, qad: An R-package to detect asymmetric and directed dependence in bivariate samples, British Ecological Society Volume 13, Issue 10, Pages: 2089-2302,October 2022 WebLink↩︎