Measuring Risk: Volatility, Drawdown and Value at Risk
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In short
Risk is not one thing, so it does not have one measure.
Scope. This article explains what three common risk measures compute, what each one omits, and how a measure can improve while the danger increases. It sets no acceptable level for any measure and describes no portfolio as risky or safe. All figures use the illustrative teaching parameters fixed on this pillar's hub and are not forecasts.
The three in common use answer different questions, and most confusion in this area comes from quoting one while thinking about another.
Volatility asks how much the value moves. Drawdown asks how far it fell from its high. Value at risk asks how bad a bad period is. A portfolio can look calm on one and alarming on another.
Volatility, and the scaling that gets misused
Volatility is the standard deviation of returns — a measure of dispersion, treating upside and downside identically, which is its first limitation and is why nobody who has lost money finds it a satisfying description of what happened.
It scales with the square root of time, and the same underlying variability therefore looks entirely different depending on the period quoted.
| Period | Standard deviation of the same holding |
|---|---|
| Daily | 1.01% |
| Monthly | 4.62% |
| Quarterly | 8.00% |
| Annual | 16.00% |
Nothing changes between those rows except the period being reported. A volatility figure quoted without its period is uninterpretable, and the square-root scaling that produces the table assumes returns are independent across periods — an assumption that is convenient rather than established.
Drawdown, which is what people actually experience
Drawdown is the fall from a previous peak, and it is the measure that corresponds to how losses are lived through — nobody experiences a standard deviation, but everybody experiences being 30% below where they were.
Simulating 200,000 twenty-year paths on the hub equity parameters, and recording the worst peak-to-trough fall in each:
| Worst drawdown over the twenty years | Share of paths experiencing at least that |
|---|---|
| Deeper than 10% | 94% |
| Deeper than 20% | 67% |
| Deeper than 30% | 35% |
| Deeper than 40% | 14% |
| Median worst drawdown | 25.1% |
Worked example
Worked example — a large fall is the ordinary case, not the exception. Two-thirds of twenty-year paths contain a fall of more than 20% from a previous peak, and the median worst fall is 25.1%. These are paths generated from a positive expected return with no crisis, no regime change and no fat tails — the drawdowns are simply what ordinary variability produces when compounded. A caveat that makes the figures conservative: the simulation uses annual observations, so it cannot see falls that occur and recover within a single year. Real intra-year drawdowns are deeper than this table shows. The relevance to the previous article is direct: a person whose capacity assumes they will not see a 25% fall is assuming something that ordinary arithmetic says will probably happen.
Value at risk, and the question it does not answer
Value at risk states a loss threshold and a probability: a 95% one-year value at risk of 17.32% means that in 5% of years the loss is expected to exceed 17.32%.
It says nothing whatever about how much worse than 17.32% those years are. That is not a subtlety — it is the measure's defining limitation, and it is why expected shortfall exists: the average loss given that the threshold was breached, which on the same parameters is 24.00%.
| Measure, one year, on the hub parameters | Value |
|---|---|
| 95% value at risk | 17.32% |
| 99% value at risk | 28.22% |
| 95% expected shortfall | 24.00% |
The finding that matters most
Both figures above assume returns are normally distributed. Real return distributions have fatter tails, and the effect on the measures is not what most readers would guess.
Comparing the normal assumption against a fat-tailed distribution (a Student-t with four degrees of freedom, rescaled) with exactly the same standard deviation:
| Confidence level | Value at risk under a normal assumption | Value at risk with fat tails, same volatility |
|---|---|---|
| 95% | 17.32% | 15.09% |
| 99% | 28.22% | 33.34% |
| 99.9% | 40.44% | 71.67% |
Worked example — the measure improves while the danger increases. At the 95% level the fat-tailed distribution reports a smaller value at risk than the normal one: 15.09% against 17.32%. A risk report would show the number falling. At the 99.9% level the same distribution reports 71.67% against 40.44% — a loss nearly twice as large. The explanation is that a fat-tailed distribution with a fixed standard deviation has to take mass from somewhere: it moves probability out of the shoulders and into both the centre and the extreme tails. So the ordinary bad year looks milder and the catastrophic year looks far worse, and a measure taken at the 95% level reports only the first. This is the central reason to distrust a single risk number: the confidence level at which a measure is taken determines whether it detects the thing that would actually cause damage, and the level is chosen by whoever produced the report. (The normal figures are closed-form; the fat-tailed figures are simulated and the extreme-tail value moves by a point or two between runs — the pattern does not.)
What none of them capture
Risks that have not occurred in the sample. Every measure here is computed from a distribution — assumed or observed — and neither contains an event outside it.
Illiquidity. A holding that cannot be sold at the quoted price shows normal volatility until the moment it matters.
Correlation change. All portfolio-level measures use a correlation estimate, and correlations move in the direction that hurts.
And the consequence, as opposed to the magnitude. A 25% fall is a number; what it means depends on whether the money was needed.
Frequently asked
8 questions
Why is there more than one risk measure?
Because risk is not one thing. Volatility asks how much the value moves, drawdown asks how far it fell from its high, and value at risk asks how bad a bad period is. A portfolio can look calm on one and alarming on another.
What is wrong with volatility as a measure?
It treats upside and downside identically, and it depends entirely on the period quoted — the same holding shows 1.01% daily, 4.62% monthly and 16.00% annually. The square-root scaling that connects them assumes returns are independent across periods, which is convenient rather than established.
How common are large drawdowns?
On the hub parameters, 94% of twenty-year paths contain a fall deeper than 10% from a previous peak, 67% deeper than 20% and 35% deeper than 30%, with a median worst fall of 25.1% — from ordinary variability, with no crisis assumed. And because the simulation uses annual observations, real intra-year falls are deeper.
What does value at risk tell you?
A loss threshold and a probability: a 95% one-year value at risk of 17.32% means losses exceed that level in about 5% of years. It says nothing about how much worse those years are.
What is expected shortfall?
The average loss given that the threshold was breached — 24.00% on the same parameters against a 95% value at risk of 17.32%. It answers the question value at risk leaves open.
How do fat tails change the numbers?
Counterintuitively. With the same standard deviation, a fat-tailed distribution shows a 95% value at risk of 15.09% against 17.32% for the normal — lower — while showing 71.67% against 40.44% at the 99.9% level. Mass moves out of the shoulders into the centre and the extremes.
Why does that matter?
Because the ordinary bad year looks milder while the catastrophic year looks far worse, and a measure taken at the 95% level reports only the first. The confidence level determines whether the measure detects what would actually cause damage, and it is chosen by whoever produced the report.
What do these measures miss entirely?
Events outside the distribution they were computed from; illiquidity, which looks normal until it matters; changes in correlation; and the consequence of a loss as distinct from its size.
References
- Basel Committee on Banking Supervision — Minimum capital requirements for market risk (the supervisory move from value at risk to expected shortfall) —
- Artzner, Delbaen, Eber and Heath (1999) — Coherent Measures of Risk, Mathematical Finance 9(3) (why value at risk is not coherent and expected shortfall is) —
- Mandelbrot (1963) — The Variation of Certain Speculative Prices, Journal of Business 36(4) (fat tails) —
Educational and informational only — not investment advice, a recommendation, or an offer to buy or sell any security. Investing involves risk, including the possible loss of principal. Worked examples use fictional companies and figures.