Expectiles as Risk Measures

Value-at-Risk asks "what loss is exceeded only $\alpha\%$ of the time," and Expected Shortfall asks "what's the average loss given that you're already in that bad tail." Expectiles ask a related but different question: instead of counting how often a threshold is exceeded, they weight how far outcomes are above versus below a candidate value, using squared distances rather than a count.

Formally, the $\tau$-expectile $\mu_\tau$ of a loss distribution $L$ is the value solving:

$$\tau \, \mathbb{E}\big[(L - \mu_\tau)^+\big] = (1-\tau) \, \mathbb{E}\big[(L - \mu_\tau)^-\big],$$

where $(L-\mu_\tau)^+$ is the shortfall above $\mu_\tau$ and $(L-\mu_\tau)^-$ is the excess below it. In plain English: pick $\mu_\tau$ so that the "outsized loss" side and the "smaller than expected" side balance, but weighted $\tau$ against $1-\tau$ rather than counted equally — for $\tau$ close to 1, big losses above $\mu_\tau$ are weighted heavily, pulling $\mu_\tau$ up toward the tail, similar in spirit to a high-confidence VaR.

The reason expectiles matter beyond being a curiosity is that, unlike VaR, an expectile-based risk measure at any $\tau$ is coherent (it correctly rewards diversification, i.e. it's subadditive), and unlike Expected Shortfall, it's also elicitable — meaning there's a well-defined scoring rule for backtesting it against realized data, which Expected Shortfall famously lacks.

Worked example. For a simple symmetric loss distribution, the $\tau=0.5$ expectile equals the ordinary mean. Moving $\tau$ to 0.95 shifts $\mu_\tau$ into the upper tail of the distribution, functioning much like a 95% VaR or ES level, but computed by balancing squared deviations rather than by a quantile count.