layout: default title: Coherent Risk Measures and Average Value at Risk ——————————————————-

Coherent Risk Measures and Average Value at Risk

Sukrit Mittal Franklin Templeton Investments

Outline

  1. Why coherence?
  2. Axioms of coherent risk measures
  3. Quantiles revisited
  4. Average Value at Risk (AVaR)
  5. Representation of AVaR
  6. AVaR vs VaR
  7. AVaR in the Black–Scholes model
  8. Coherence proofs
  9. Exercises

1. Why Coherence?

VaR answered a practical question.

It failed a theoretical one.

Specifically:

Diversification should not increase risk.

Coherent risk measures enforce this principle.

From Heuristics to Axioms

Risk is not just a number.

It is a functional:

\[\rho : L \to \mathbb{R}\]

mapping random losses to capital requirements.

Axioms discipline this mapping.

2. Axioms of Coherent Risk Measures

A risk measure $\rho(L)$ is coherent if it satisfies:

  1. Monotonicity If $L_1 \le L_2$, then $\rho(L_1) \le \rho(L_2)$

  2. Translation invariance $\rho(L + c) = \rho(L) + c$

  3. Positive homogeneity $\rho(\lambda L) = \lambda \rho(L)$ for $\lambda > 0$

  4. Subadditivity $\rho(L_1 + L_2) \le \rho(L_1) + \rho(L_2)$

These axioms formalize diversification.

Economic Meaning

  • Monotonicity: worse losses imply more risk
  • Translation: cash reduces risk one-for-one
  • Homogeneity: scaling portfolios scales risk
  • Subadditivity: diversification helps

If one axiom fails, interpretation collapses.

3. Quantiles Revisited

Let $L$ denote loss.

Define VaR again:

\[\text{VaR}_\alpha(L) = \inf {x : \mathbb{P}(L \le x) \ge \alpha }\]

VaR is a quantile.

Quantiles are not averages.

4. Average Value at Risk (AVaR)

The Average Value at Risk at level $\alpha$ is defined as:

\[\text{AVaR}*\alpha(L) = \frac{1}{1-\alpha} \int*\alpha^1 \text{VaR}_u(L) , du\]

AVaR averages the worst losses.

This single change fixes VaR.

Interpretation

AVaR answers:

Given that things are bad, how bad are they on average?

It measures tail severity.

This is what risk managers actually want.

5. Representation of AVaR

An equivalent formulation:

\[\text{AVaR}*\alpha(L) = \inf*{m \in \mathbb{R}} \left{ m + \frac{1}{1-\alpha} \mathbb{E}[(L - m)^+] \right}\]

This is a convex optimization problem.

This representation is fundamental.

Why This Representation Matters

  • Convexity becomes explicit
  • Numerical computation becomes tractable
  • Dual representations emerge naturally

VaR has none of these properties.

6. AVaR vs VaR

Key contrasts:

  • VaR: quantile, ignores tail shape
  • AVaR: tail average, tail-sensitive

VaR may penalize diversification.

AVaR never does.

This difference is structural.

7. AVaR in the Black–Scholes Model

Assume log-returns are normal.

Loss $L$ is log-normal.

AVaR admits closed-form expressions.

The math is heavier—but honest.

Normal Loss Approximation

If losses are normal:

\[L \sim \mathcal{N}(\mu_L, \sigma_L^2)\]

Then:

\[\text{AVaR}*\alpha = \mu_L + \sigma_L \frac{\varphi(z*\alpha)}{1-\alpha}\]

where $\varphi$ is the standard normal density.

Interpretation

Compared to VaR:

  • Same quantile cutoff
  • Larger risk value

AVaR prices tail thickness.

Markets ignore it at their peril.

8. Coherence of AVaR

Theorem

AVaR is a coherent risk measure.

Proof outline:

  • Monotonicity: inherited from expectation
  • Translation invariance: linearity
  • Positive homogeneity: scaling inside expectation
  • Subadditivity: convexity of $(x)^+$ and Jensen’s inequality

Each axiom can be proven rigorously.

Why VaR Fails Subadditivity

VaR focuses on a single quantile.

Combining distributions can move mass.

Averages smooth.

Quantiles jump.

This is not a technicality.

9. Exercises

Exercise 1

Show that AVaR is monotone and translation invariant.

Exercise 2

Derive the infimum representation of AVaR.

Explain why convexity matters.

Exercise 3

Compare VaR and AVaR numerically for a heavy-tailed distribution.

Discuss implications for regulation.

Final Takeaways

  • Risk measures must respect diversification
  • VaR fails coherence
  • AVaR repairs this failure
  • Coherence is not philosophy—it is structure

Next: optimization under coherent risk.

Where finance meets convex analysis.