Quantile

A quantile is a cut-off value that splits a set of data, or a probability distribution, at a chosen proportion. The 0.25 quantile, for example, is the value with 25 percent of the observations at or below it and 75 percent above it. Quantiles describe where the observations sit and how widely they are spread, without assuming that the data follow any particular shape.

Several familiar statistics are quantiles under other names. The median is the 0.5 quantile. Quartiles divide data into four equal parts, deciles into ten and percentiles into one hundred. In finance, quantiles are used to rank securities into groups, to describe the tails of return distributions and to define risk measures such as value at risk.

For a random variable with cumulative distribution function $F$, the quantile at probability level $p$ is defined as:

• $Q(p)$ is the quantile, a value on the same scale as the data.
• $p$ is a probability between 0 and 1.
• $F(x)$ is the probability that the variable is less than or equal to $x$.
• $\inf$ means the smallest value of $x$ that satisfies the condition.

In words, the quantile is the smallest value at which the cumulative probability reaches $p$. When $F$ rises smoothly, the quantile function is simply the inverse of $F$. For the standard normal distribution, the 0.05 quantile is about -1.645 and the 0.01 quantile is about -2.326.

With real data the distribution is unknown, so quantiles are estimated from a sample. The observations are sorted from smallest to largest and the quantile is read off at the right position. Statistical software offers several slightly different rules for doing this. A common one places the quantile at position $h$ in the sorted list:

Here $n$ is the number of observations. When $h$ is not a whole number, the result is interpolated between the two neighbouring values.

Suppose nine daily returns, in percent and sorted, are -2.1, -1.4, -0.8, -0.3, 0.1, 0.4, 0.7, 1.2 and 1.9. For the lower quartile, $p = 0.25$ and $h = 8 \times 0.25 + 1 = 3$, so the quartile is the third value, -0.8 percent. The median is the fifth value, 0.1 percent. For the 0.10 quantile, $h = 8 \times 0.10 + 1 = 1.8$, which lies 80 percent of the way from the first value to the second. The gap between them is 0.7, so the quantile is $-2.1 + 0.8 \times 0.7 = -1.54$ percent.

Value at risk is a quantile of the distribution of profit and loss. A one-day 95 percent figure is the loss that is exceeded on only 5 percent of days, which is the 0.05 quantile of daily returns. If a portfolio's daily returns are assumed to follow a normal distribution with a mean of zero and a standard deviation of 1.2 percent, the 0.05 quantile is $-1.645 \times 1.2 = -1.974$ percent. On a $10 million portfolio that is a loss of about $197,400.

The same quantile can instead be taken directly from past returns, an approach known as historical simulation. The two methods often disagree, because market returns have fat tails: extreme quantiles of real data lie further from the centre than the normal curve predicts. A quantile also says nothing about how bad losses are beyond it, which is why risk managers pair it with measures of the average loss in the tail.

• Portfolio sorts. Researchers rank shares on a characteristic and split them into quintiles or deciles, then compare the returns of the top and bottom groups. Studies of cross-sectional momentum and factor investing are built this way.
• Signal thresholds. A strategy may act only when an indicator is above its 90th percentile over a look-back window, a rule that adapts as conditions change. IV rank and similar measures express a current reading as a position within its past range.
• Describing distributions. The interquartile range, the distance between the 0.25 and 0.75 quantiles, measures spread without being distorted by a few extreme values. Quantile-quantile plots compare the quantiles of observed returns with those of a theoretical distribution to reveal skewness and heavy tails.
• Quantile regression. This technique models a chosen quantile of an outcome, such as its 5th percentile, instead of its average.

Sample quantiles are estimates, and extreme ones are unreliable. A 0.01 quantile estimated from one year of daily data rests on only two or three observations. Different interpolation rules give different answers in small samples, so results should state the method used. Quantiles estimated from one period may not hold in the next if market conditions shift, a concern that applies to every statistic drawn from a financial time series.