Risk Management: VaR, Drawdowns, and Living to Trade Tomorrow

Every earlier lesson computed an edge; this one is about not dying with it. Risk management is the discipline of bounding what one bad day — or one bad model — can do, and its vocabulary (VaR, expected shortfall, drawdown) appears in every trading interview's final act.

Value-at-Risk: useful, gameable, misquoted

VaR at 95%/1-day is the loss threshold exceeded on only 5% of days: "95% 1-day VaR of \1M" means roughly one day a month loses more than \1M. Three ways to compute it — parametric (assume normal: VaR $\approx 1.65\sigma$ at 95%, $2.33\sigma$ at 99%), historical (replay recent days), Monte Carlo (simulate) — and the parametric one inherits every fat-tail sin from the CLT lesson.

The two criticisms you must be able to volunteer:

1. VaR says nothing about how bad the bad 5% is. A book losing at most \1.1M beyond VaR and a book losing \50M beyond VaR can share the same VaR. The fix is expected shortfall (CVaR): the average loss in the tail — the regulatory standard now, precisely for this reason.
2. VaR is gameable: selling far-OTM options (the steamroller trade from the Greeks lesson) produces steady income, tiny VaR — and catastrophic tail. Any risk number computed from a benign sample window blesses exactly the strategies that blow up outside it.

And the structural caveat over all of it: risk numbers are estimated from history, and correlations converge in crises (the portfolio lesson's warning) — the diversification your VaR assumes is weakest when you need it.

Drawdown arithmetic

Drawdown — decline from the running peak (the three-line NumPy computation from the Python lesson) — is the risk measure investors actually feel. The brutal asymmetry every practitioner internalizes: a 50% loss needs a 100% gain to recover. Loss $\ell$ requires gain $\ell/(1-\ell)$ — convex, unforgiving, the volatility-drag/Kelly logic from the other side. Rules of thumb worth having: a strategy's plausible max drawdown over a few years is on the order of 2× its annual volatility if returns were well-behaved — and worse in reality (fat tails, again); and at full-Kelly sizing, deep drawdowns aren't tail risk, they're the expected experience — the practical case for fractional Kelly restated as drawdown tolerance.

Limits: the machinery that enforces humility

Real desks run layered hard limits — position size per name, aggregate exposures (net, gross, per-factor from the CAPM lesson), loss limits with de-risking triggers, concentration and liquidity limits (size vs days-of-volume, tying to the execution lesson: a position you can't exit without massive impact is larger than it looks). Two design principles interviewers probe: limits are set in calm times and obeyed in storms — the moment a limit binds is the moment every incentive argues for an exception, which is why they're hard-coded; and stop-losses don't create expectation (the optional-stopping lesson) — their purpose is truncating tails and, honestly, bounding the damage from broken models rather than unlucky draws. "The stop is for when I'm wrong about the world, not when the world is wrong about me" is the grown-up phrasing.

The interview version

*"Your desk's 99% 1-day VaR is \5M. What does tomorrow's worst case look like?"* — Not \5M: VaR is a threshold, not a maximum; quote expected shortfall, gap risk, and the crisis-correlation caveat. "A strategy made 20% with 5% vol for three years. Risk questions?" — Sharpe ~4 screams short-tail-risk premium (options-selling profile): ask what it loses in a gap, what its worst plausible month is, how correlated it becomes with everything else in stress, and whether its capacity survives the execution lesson. The consistent posture — tails first, averages second — is the risk-management instinct these questions exist to detect.