Tail Risk and CVaR

Two strategies with identical mean Implementation Shortfall can be completely different objects. The one that occasionally costs you 300 bps is not interchangeable with the one that never costs you more than 20, and a mean cannot tell them apart.

VaR, and why it isn't enough

Value at Risk at 95% is the loss threshold you exceed 5% of the time. It's the industry default and it has a well-known defect: it tells you where the tail starts and nothing about what's inside it. Two strategies with the same VaR can have wildly different worst cases beyond it.

Conditional VaR, also called expected shortfall, fixes that by averaging over the tail instead of pointing at its edge:

$$\text{CVaR}{95} = \mathbb{E}\left[\,X \mid X \ge \text{VaR}\,\right]$$

For execution costs, that's the mean shortfall across your worst 5% of trades. It is also coherent in the technical sense, meaning that combining two positions can't make CVaR worse than the sum of their parts. VaR isn't.

Why it changed what we concluded

In the MoE study, CVaR-95 is the metric that broke the story open. It made visible that an architectural change can move across-seed consistency and per-policy tail exposure in opposite directions. A paper reporting only the first would have claimed the architecture "reduces risk", which would have been half true in the most misleading available way.

Two habits came out of that. Whenever someone says a strategy reduces risk, ask which distribution they took the tail of. And when a result is a variance reduction, ask what the right null is. Averaging $K$ independent things already shrinks variance by $\sqrt{K}$ for free, so "less variance than a single model" is not by itself evidence that anything clever happened.