Which of the following are considered properties of a ‘coherent’ risk measure:

Which of the following are considered properties of a ‘coherent’ risk measure:

I. Monotonicity

II. Homogeneity

III. Translation Invariance

IV. Sub-additivity
A . II and III
B . II and IV
C . I and III
D . All of the above

View Answer

Answer: B

Explanation:

All of the properties described are the properties of a ‘coherent’ risk measure. Monotonicity means that if a portfolio’s future value is expected to be greater than that of another portfolio, its risk should be lower than that of the other portfolio. For example, if the expected return of an asset (or portfolio) is greater than that of another, the first asset must have a lower risk than the other. Another example: between two options if the first has a strike price lower than the second, then the first option will always have a lower risk if all other parameters are the same. VaR satisfies this property.

Homogeneity is easiest explained by an example: if you double the size of a portfolio, the risk doubles. The linear scaling property of a risk measure is called homogeneity. VaR satisfies this property.

Translation invariance means adding riskless assets to a portfolio reduces total risk. So if cash (which has zero standard deviation and zero correlation with other assets) is added to a portfolio, the risk goes down. A risk measure should satisfy this property, and VaR does. Sub-additivity means that the total risk for a portfolio should be less than the sum of its parts. This is a property that VaR satisfies most of the time, but not always. As an example, VaR may not be sub-additive for portfolios that have assets with discontinuous payoffs close to the VaR cutoff quantile.