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However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) . {\textstyle \mu \left(\bigcup _{n=1}^{\infty ...
However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% (= 0.5*0.7 + 0.5*0) since the probability of at least one of the bonds defaulting is 7.84% (= 1 - 0.96*0.96) which exceeds 5%. This violates the sub-additivity property showing that VaR is not a coherent risk measure.
Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2".
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions.
In queueing theory, a discipline within the mathematical theory of probability, the arrival theorem [1] (also referred to as the random observer property, ROP or job observer property [2]) states that "upon arrival at a station, a job observes the system as if in steady state at an arbitrary instant for the system without that job."
Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were applications biased towards a positive result. With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994).
In probability theory, Wald's equation, Wald's identity [1] or Wald's lemma [2] is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities.
In economics, additive utility is a cardinal utility function with the sigma additivity property. [1]: 287–288 ...