Introduction

Value at Risk (VaR) is a widely used metric for assessing financial risk by estimating the maximum potential loss under specified confidence levels. While VaR provides valuable insights into risk exposure, it falls short in meeting the criteria of a coherent risk measure. Coherent risk measures possess certain properties that ensure consistency and soundness in risk assessment. In this chapter, we delve into the reasons why VaR does not meet these coherence properties and examine its implications for risk management.

Properties of a Coherent Risk Measure

A coherent risk measure is characterized by four properties: subadditivity, positive homogeneity, translation invariance, and monotonicity. These properties ensure that the measure aligns with the principles of risk assessment and decision-making.

1) Subadditivity: A coherent risk measure should satisfy the principle that the risk of a combined position is not greater than the sum of the risks of its individual components. Mathematically, for any two portfolios A and B, their combined risk should be less than or equal to the sum of their individual risks:

$$\text{Coherent Measure (A+B)} \le \text{Coherent Measure (A)}+\text{Coherent Measure (B)}$$

2) Positive Homogeneity: This property ensures that increasing the size of a position proportionally increases its risk measure. Mathematically, for any portfolio A and any positive constant $c$:

$$\text{Coherent Measure (cA)}= c \text{Coherent Measure(A)}$$

3) Translation Invariance: A coherent risk measure remains unchanged when the portfolio’s underlying value is shifted by a constant amount. Mathematically, for any portfolio A and any constant $c$:

$$\text{Coherent Measure (A+c)} = \text{Coherent Measure (A)} + c$$

4) Monotonicity: Monotonicity asserts that a portfolio with higher expected losses should have a higher risk measure. Mathematically, for portfolios A and B, if the expected loss of A is greater than the expected loss of B:

$$\text{Coherent Measure (A)} \ge \text{Coherent Measure (B)}$$

VaR and its Incoherence

Value at Risk (VaR) fails to satisfy the subadditivity property. It violates the principle that the combined risk should not exceed the sum of individual risks. In certain situations, diversification among assets does not lead to proportional risk reduction. This inconsistency arises due to VaR’s focus on the worst-case scenario for each individual position, which doesn’t guarantee a conservative estimate of the combined risk.

Example: Consider two portfolios A and B with VaR(A) = 100 and VaR(B) = 80. According to subadditivity, VaR(A + B) should be less than or equal to 180. However, due to VaR’s nature, the combined VaR(A + B) might exceed 180, violating the subadditivity property.

Implications

The incoherence of VaR can lead to inaccurate risk assessments and hinder effective risk management strategies. Investors might be misled into believing that diversification provides more risk reduction than it actually does, potentially exposing them to higher-than-expected losses.