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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.
In economics, additive utility is a cardinal utility function with the sigma additivity property. [1]: 287–288 ...
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".
You can use a calculator or the simple interest formula for amortizing loans to get the exact difference. For example, a $20,000 loan with a 48-month term at 10 percent APR costs $4,350.
The use of money as compensation can often turn real cases like these into situations where the weak additivity condition is satisfied even if the values are not exactly additive. The value of a type of goods, e.g. chairs, dependent on having some of those goods already is called the marginal utility .
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 particular, the sum of the components X 1 + X 2 + · · · + X n is the riskiest if the joint probability distribution of the random vector (X 1, X 2, . . . , X n ) is comonotonic. [ 2 ] Furthermore, the α - quantile of the sum equals the sum of the α -quantiles of its components, hence comonotonic random variables are quantile-additive.