You are asked to consider whether the risk measure [math]\rho ( X ) = E( X ) [/math] is coherent.

Determine which of the following statements is correct.

• [math]\rho ( X )[/math] does NOT possess subadditivity.
• [math]\rho( X ) [/math] does NOT possess monotonicity.
• [math]\rho( X ) [/math] does NOT possess positive homogeneity.
• [math]\rho( X ) [/math] does NOT possess translation invariance.
• [math]\rho ( X )[/math] is a coherent risk measure.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: E

Subadditivity holds: [math]\rho ( X + Y ) = \operatorname{E}( X + Y ) = \operatorname{E}( X ) + \operatorname{E}(Y ) = \rho ( X ) + \rho (Y ) [/math]

Monotonicity holds: If [math]X ≤ Y[/math], then [math]\rho ( X ) = \operatorname{E}( X ) \leq \rho (Y ) = \operatorname{E}(Y ) [/math]

Positive homogeneity holds: [math]\rho (cX ) = \operatorname{E}(cX ) = c\operatorname{E}( X ) = c\rho ( X ) [/math]

Translation invariance holds: [math]\rho ( X + c) = \operatorname{E}( X + c) = \operatorname{E}( X ) + c = \rho ( X ) + c [/math]

Since [math]\rho ( X ) = \operatorname{E}( X )[/math] satisfies all four properties, it is coherent.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.