Coherent Risk Measures

Overview


Since Markowitz's pioneering paper, much effort has been devoted to identifying measures of financial risk. Some measures that were thought to be intuitive turned out incentive risk taking in certain situations These discoveries have led researcher to identify the types of properties that one should expect from a measure of risk. The axioms of a coherent risk measure have emerged as the top candidate for such a framework.

Axioms


  • Monoticity :
    {% L_1 \leq L_2 \; \rightarrow \; \rho(L_1) \leq \rho(L_2) %}
  • Translation Invariance :
    {% \rho(L + m) = \rho(L) + m %}
  • Subadditivity :
    {% \rho(L_1 + L_2) \leq \rho(L_1) + \rho(L_2) %}
  • Positive Homogeneity : for {% \lambda \geq 0 %}
    {% \rho(\lambda L) = \lambda \rho(L) %}
  • Convexity :
    {% \rho(\lambda L_1 + (1-\lambda)L_2) \leq \lambda \rho(L_1) + (1-\lambda)\rho(L_2) %}

Risk Measures


The following are sample risk measures and whether they comply with the axioms.

  • Standard Deviation - satisfies the axioms
  • Expected Shortfall - satisfies the axioms
  • Value at Risk - falis Subadditivity
(see Roncalli pg 74)

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