Risk Measures

Turning "how bad could it get" into a number — and knowing exactly how that number lies to you.

The problem every risk number tries to solve

• A portfolio has thousands of moving parts. A committee has ten minutes. Risk measures compress the first into something that fits the second.
• Every such compression throws information away on purpose — the skill is knowing which information, and whether today is a day when it mattered.
• The three questions the industry actually asks: how much can I lose on an ordinary bad day (VaR), how bad is it when the bad day arrives (expected shortfall), and how much of my result is just the market (beta, tracking error).

Value at Risk: the number that ate risk management

VaR answers one narrow question: over horizon h, what loss is exceeded only (1 − c) of the time? Under a normal distribution it is arithmetic:

• It is a threshold, not a worst case. A 99% one-day VaR of $1m says: on roughly 2–3 trading days a year you lose more than $1m. It says nothing about how much more.
• Three ways to compute it: parametric (the formula above — fast, assumes normality), historical simulation (replay the last N days — no distribution assumed, but blind to anything absent from the window), and Monte Carlo (flexible, slow, only as good as the model inside).
• It is not sub-additive. Combining two books can produce a VaR larger than the sum of the parts — a mathematical embarrassment for a measure used to allocate limits. Expected shortfall fixes this.

Expected shortfall: the average of the bad days

• ES (also CVaR, or tail VaR) is the mean loss given that VaR was breached. It answers the question VaR refuses to.
• It is coherent: sub-additive, so diversification can never look like it adds risk. This is why the Basel market-risk framework moved from 99% VaR to 97.5% expected shortfall.
• Under normality the two are related by a constant: at 99% confidence, ES ≈ 1.15 × VaR. In fat-tailed reality that ratio widens — the gap between them is itself a tail-risk indicator.

Why the tail is always fatter than the model

• Volatility clusters. Big moves follow big moves. Independent-days assumptions understate multi-day disasters — the whole point of a crisis is that it is a sequence, not a draw.
• Correlations converge. The diversification credit in the model evaporates under stress (see diversification & correlation).
• Liquidity vanishes. Historical volatility measures prices you could trade at. In stress the exit price is not the screen price — the model has no term for that gap.
• The window is the whole model. Any measure calibrated on the last two years assumes the next two rhyme. Every case study on this site is a period where they did not: 1998, 2008, 2018, 2020.
• Stress testing exists because of all four. A scenario ("rates +200 bp, equities −25%, spreads +300 bp, correlations to 0.9") makes no distributional claim at all — it just asks what breaks.

Beta: separating the market from the manager

Beta is the sensitivity of a position to a benchmark — the slope of a regression of one on the other. Its practical use is hedging: to neutralise market risk you sell beta-adjusted exposure, not equal exposure.

Tracking error: risk measured against a benchmark

• Tracking error is the volatility of active return (portfolio minus benchmark). An index fund targets ~0.05%; a high-conviction active fund runs 4–8%.
• Information ratio = active return ÷ tracking error — the benchmark-relative cousin of the Sharpe ratio. An IR of 0.5 sustained is genuinely good; an IR of 1.0 sustained is rare enough to be suspicious.
• The uncomfortable arithmetic of proof: statistical significance needs roughly t = IR × √years ≥ 2. At IR 0.5 that is 16 years of data. Most track records end long before the evidence arrives.

Practitioner rules

• Never quote a risk number without its horizon and confidence. "VaR is 2%" is not a statement; "1-day 99% VaR is 2% of NAV" is.
• Backtest the measure, not just the portfolio. Count actual breaches: a 99% VaR breached fifteen times in a year is a broken model, not an unlucky year.
• Pair every statistical measure with a scenario. Distributions describe the market you have seen; scenarios describe the one that ends you.
• Watch the measure become the target. Positions optimised to look small under VaR — selling far out-of-the-money options is the classic — carry enormous risk precisely where VaR cannot see it. See Volmageddon.
• Risk that cannot be hedged must be sized. When there is no instrument, position size is the only risk control left — the link back to Kelly and position sizing.