● portfolio risk — VaR / CVaR & marginal-VaR
How much could this book lose — and which name is the culprit?
Portfolio risk in one view: historical + Monte-Carlo Value-at-Risk and Expected Shortfall (CVaR) for your basket, the marginal-VaR that pins the risk on a single holding, and notes on what each number means. Computed on ~2 years of real daily returns.
AAPLMSFTNVDA
⚠ The culpritNVDA drives 51% of this book's Value-at-Risk — trimming it cuts your downside most efficiently.
Value-at-Risk & Expected Shortfall
| Horizon | VaR 95% | ES 95% | VaR 99% | ES 99% |
|---|---|---|---|---|
| Monte-Carlo · 1d | 2.73% | 3.43% | 3.88% | 4.38% |
| Monte-Carlo · 10d | 8.64% | 10.83% | 12.26% | 13.86% |
| Historical · 1d | 2.65% | 3.95% | 4.88% | 5.61% |
| Historical · 10d | 8.37% | 12.49% | 15.42% | 17.74% |
| Parametric · 1d | 2.78% | — | 3.93% | — |
Method agreement (1-day VaR 95%): 3 independent methods span 2.65%–2.78% — tight — methodology-robust.
VaR = loss threshold (19/20 days). ES/CVaR = average loss on the bad 1-in-20 day. Monte-Carlo = correlated simulation; Historical = empirical replay; Parametric = sigma-based (VaR only). Agreement = confidence.
Marginal VaR — which holding drives the risk
AAPL24.20%
MSFT24.43%
NVDA · top driver51.37%
Marginal (component) VaR splits TOTAL VaR into each holding's share — the slices sum to the total. Trim the biggest to cut risk most efficiently.
19 days out of 20 this book should not lose more than 2.73% in a day; but on the bad 1-in-20 day the average loss is about 3.43%. Computed on ~2 years of real daily-return data — your decisions, your risk.
Below, the engine walks the tail one concept at a time — VaR, Expected Shortfall, historical vs Monte-Carlo, who owns the tail, and how it scales — so you can read your own risk.
What is VaR, in one sentence?
Historical 1-day 95% VaR is 2.65% of the book; the Monte-Carlo estimate is 2.73%.
Value-at-Risk (VaR) at 95% over 1 day means: "on 19 out of 20 normal days you should not lose more than this." It is a loss threshold, not a worst case — 1 day in 20 you can lose MORE.
Why Expected Shortfall (ES / CVaR) matters more
On a 95% breach day you would lose about 3.95% on average (historical ES) — that is 1.49× the VaR threshold.
VaR tells you the threshold; it says nothing about HOW bad the bad days are. Expected Shortfall is the AVERAGE loss on the days you do breach VaR — the true "if it goes wrong, how wrong" number. ES is always at least as large as VaR, and regulators (FRTB) now prefer it.
Historical vs Monte-Carlo — two honest lenses
Here the two methods land within 0.08% of each other at 95% — close agreement.
Historical simulation replays exactly what happened to your book, so it captures real fat tails but can only show losses it has already seen. Monte-Carlo estimates the covariance between your assets, Cholesky-factorises it, and generates thousands of NEW correlated scenarios — smoother, and able to imagine combinations that have not occurred yet. Agreement between the two is a confidence signal.
Who owns the tail? (marginal VaR)
The largest single contributor to tail risk is asset #2 (51.37% of total VaR).
Total risk is not the sum of each position's standalone risk — correlations matter. Marginal (component) VaR splits the TOTAL VaR into the slice each asset is actually responsible for, and the slices sum exactly to the total. Trim the biggest slice to cut risk most efficiently.
Scaling to longer horizons
Your 10-day 95% VaR (8.64%) is the 1-day figure × √10.
A 1-day risk number is scaled to h days with the "square-root-of-time" rule (multiply by √h). It assumes returns are independent day-to-day; real markets trend or mean-revert, so treat multi-day figures as an approximation, not gospel.
Risk glossary
- VaR
- Value-at-Risk — a loss threshold you expect to breach only (1 − confidence) of the time.
- Expected Shortfall (CVaR)
- The average loss GIVEN that you have already breached VaR — the mean of the worst tail.
- Monte-Carlo VaR
- Simulating thousands of correlated scenarios from the estimated covariance (via Cholesky) and reading the tail.
- Cholesky
- A way to factor the covariance matrix (L·Lᵀ = Σ) so independent random draws come out correlated like your real assets.
- Marginal / Component VaR
- Each asset's share of total VaR (Euler allocation); the shares sum to 100% of VaR.
- √h rule
- Multiply a 1-day risk figure by the square root of the horizon in days to scale it.
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