Expected Shortfall (CVaR)

Expected Shortfall (also called CVaR) is VaR’s tougher cousin. VaR tells you the cutoff for “bad days.” Expected Shortfall tells you what the bad days look like on average once you’re already in the tail.

If an asset is fat‑tailed, Expected Shortfall usually looks meaningfully worse than VaR — that gap is the point.

The definition

At level $\alpha$ (we use 5%), Expected Shortfall is the average return in the worst $\alpha$ fraction of days:

$$\text{ES}_{\alpha} = E\left[r \mid r \le \text{VaR}_{\alpha}\right]$$

In practice, for a historical series we sort returns and average the worst $\lceil \alpha \cdot n \rceil$ observations.

How we calculate Expected Shortfall at Gale Finance

1. Historical ES (no distributional assumption). We use the empirical tail.

2. Daily log returns. Like VaR, our tail-risk section uses daily log returns:

$$r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)$$

3. 5% tail. We use $\alpha = 0.05$ for one‑year windows so the estimate isn’t based on 2–3 observations. We also surface how many days are in the tail sample (the “worst N days” count).

4. Native calendars per asset. Crypto includes weekends; trading‑day assets don’t.

How to read it

• VaR answers: “What’s the threshold for a bad day?”
• ES answers: “If we actually hit a bad day, what’s the average damage?”

Both are descriptive. Both can change a lot across regimes. If you want to understand why ES is large (is it frequent small losses or rare crashes?), pair it with skew, kurtosis, and fat tails.