The Pinball Loss

Ordinary regression trains a model to predict the mean of a target by penalizing over- and under-predictions equally, usually with squared error. If you want to predict the 90th percentile of next-day returns instead of the mean, you need a loss that penalizes the two kinds of mistakes asymmetrically — and that's what the pinball loss (also called quantile loss) does.

For a target quantile $\tau$ (say $\tau = 0.9$ for the 90th percentile), the pinball loss penalizes underestimates by a factor of $\tau$ and overestimates by a factor of $1-\tau$. In words: when you're targeting a high quantile like 0.9, underpredicting is punished nine times as heavily as overpredicting, which pushes the model to sit high enough that only 10% of actual outcomes exceed it — exactly the property a 90th-percentile estimate should have. At $\tau = 0.5$ the two penalties are equal, which reduces the pinball loss to ordinary absolute-error loss and recovers median regression.

A desk building a value-at-risk model might train a gradient-boosted tree with pinball loss at $\tau = 0.05$ to predict the 5th percentile of daily P&L directly, rather than assuming a normal distribution and backing out the quantile from a mean and standard deviation — letting the model learn the actual shape of the tail from data instead.