How to chart a return distribution with a Value-at-Risk cutoff

A histogram of returns shows the shape of the risk you are actually carrying - how often small gains happen, how fat the loss tail is - and a Value-at-Risk line marks the loss you would not expect to exceed on all but the worst few percent of days. Draw the two together and one chart answers both questions at once: what does this return stream look like, and how bad is a bad day.

Build the histogram honestly

Start from a return series - the percentage change from one period to the next, not price levels. Bin the returns and plot frequency. The one decision that changes the story is bin width: too wide and you flatten the tails that matter, too narrow and you get noise. A few dozen bins over a few years of daily returns is a reasonable start. Center the chart on zero so the asymmetry is visible, because equity returns are typically skewed left - the losses cluster further from the mean than the gains.

Overlay a normal to expose the tails

Fit a normal curve with the same mean and standard deviation as your data and draw it over the histogram. The gap between the two is the point. Real return distributions are fat-tailed: the extreme days, in both directions, happen far more often than a normal predicts, and the peak is usually taller and narrower. Seeing the histogram poke out past the normal curve in the left tail is the visual argument for why a risk model built on the normal assumption understates how often you lose a lot.

Mark the Value-at-Risk cutoff

Historical VaR is just a percentile of the loss tail. Sort the returns from worst to best; for a 95% one-day VaR, find the value at the 5th percentile - with 1,000 daily returns that is the 50th-worst observation. If that value is -2.5%, the VaR is reported as 2.5%: the loss you exceed on only 5% of days. Draw it as a vertical line on the histogram and everything to its left is the 5% of days that breach it. VaR is conventionally quoted as a positive loss number, so mind the sign.

Historical versus parametric

The percentile method above is historical VaR - it makes no assumption about the shape of the distribution, only that the past is a fair sample. Parametric VaR instead assumes returns are normal and reads the cutoff off the curve: VaR = -(mean + z times the standard deviation), with z = -1.645 for 95% and -2.326 for 99%. For a mean of zero and a 1% daily standard deviation, 95% parametric VaR is 1.645%. The two methods usually disagree, and the histogram shows why: when the real left tail is fatter than the normal, parametric VaR sits inside the historical one and understates the risk. Report which method you used.

The line is not the whole story

VaR tells you the threshold and nothing about what lies beyond it - a 2.5% VaR is silent on whether the days that breach it lose 3% or 15%. That blind spot is why Expected Shortfall, also called CVaR, is the regulatory preference under Basel III for the trading book: it is the average of the losses past the VaR line. Compute it by taking the mean of every return in the tail beyond the cutoff; by construction it is always at least as large as the VaR. Shade that tail on the chart and label both numbers - the cutoff and the average loss inside it - and you have shown the risk the VaR line alone hides.

[QUADESTO-EMBED: histogram of daily returns with a same-mean-and-std normal overlay, 95% historical VaR vertical line, tail beyond it shaded for Expected Shortfall]

Building it in Quadesto

Feed Quadesto a return series and it plots the distribution with the normal overlay, the VaR line at the percentile you choose, and the Expected Shortfall tail shaded, alongside the underwater drawdown and rolling Sharpe views that describe the same risk through time rather than in aggregate. The free tier embeds it live with a Made with Quadesto credit; Pro (149 pounds a month) removes the attribution and adds branded themes for a risk report.