Introduction

In the realm of risk management, coherent risk measures play a pivotal role in assessing and quantifying the uncertainty associated with financial assets and portfolios. Coherent risk measures exhibit certain properties that make them reliable tools for risk assessment. In this chapter, we will delve into the properties that define a coherent risk measure, highlighting the significance of each property in capturing the essence of risk in financial contexts.

Properties of a Coherent Risk Measure:

1) Subadditivity: Subadditivity, also known as the diversification property, asserts that the risk of a combined portfolio is never greater than the sum of individual risks. Mathematically, for any set of risky positions or portfolios {$X_1$, $X_2$, …, $X_n$}, the following holds:

$$Risk(X_1 + X_2 + … + X_n) \le Risk(X_1) + Risk(X_2) + … + Risk(X_n)$$

This property aligns with the idea that diversification can help mitigate risk. For example, if an investor holds multiple stocks, the combined risk of the portfolio should be lower than the sum of individual stock risks due to potential offsetting movements.

2) Monotonicity: Monotonicity states that if one portfolio has higher expected returns and lower risk than another, the risk measure should assign a lower risk to the former. In other words, risk and expected return should move in opposite directions. Formally, if Portfolio A has higher expected return and lower risk compared to Portfolio B, then:

$$Risk(A) \le Risk(B)$$

3) Positive Homogeneity: Positive Homogeneity implies that scaling a portfolio by a positive factor also scales its risk measure by the same factor. Mathematically, for any portfolio $X$ and a positive constant $c$:

$$Risk(cX) = cRisk(X)$$

This property ensures that risk measures remain consistent regardless of the scale of the portfolio.

4) Translation Invariance: Translation Invariance asserts that adding a fixed amount of capital to a portfolio does not change the relative risk. Mathematically, for any portfolio $X$ and a constant amount $c$:

$$Risk(X + c) = Risk(X)$$

This property reflects that the absolute level of wealth does not affect the risk measure.