Risk Lab · Tail Risk Stress Tester
Tail Risk Stress Tester Guide - VaR and ES
Run a $100M multi-asset book through a correlation breakdown, then compute the Expected Shortfall that VaR leaves out.
What is Value at Risk and Expected Shortfall?
Value at Risk (VaR) is a threshold loss: at a given confidence level (say 99%) and time horizon, VaR is the loss amount that will be exceeded only with the remaining probability (1%). It became the standard risk metric in banking and asset management because it compresses an entire, hard-to-visualize distribution of possible portfolio losses into a single number that fits in a limit, a report, or a regulatory filing.
That compression is also VaR's central weakness. VaR says nothing about how bad things get once the threshold is breached - two portfolios can share an identical 99% VaR while one has a mild, contained tail beyond that point and the other has a catastrophic one that wipes out the firm. VaR is, by construction, silent about everything past its own cutoff. It also fails to be subadditive in general, meaning that combining two portfolios can sometimes produce a VaR larger than the sum of their individual VaRs - a property that makes it possible, in principle, to make a book's risk look smaller than it is just by how it's sliced up.
Expected Shortfall (also called Conditional VaR, or ES) is the standard fix: instead of reporting only the loss at the threshold, it reports the average loss across every outcome worse than that threshold - the expected loss, given that the loss exceeds VaR. Because it averages over the whole tail rather than pointing at its edge, ES at a given confidence level is always at least as large as VaR at that level, and it is subadditive: diversifying can never make ES look worse, which is the property regulators wanted. This is why bank capital rules moved market-risk requirements from VaR toward ES (typically at a lower confidence level, around 97.5%, chosen so the two measures land in a comparable range).
Both measures depend heavily on the assumed shape of the loss distribution, and real markets routinely have fatter tails than a normal distribution predicts - extreme moves happen more often than a bell curve implies. A normal-based VaR or ES will understate true tail risk, which is why risk models are often refit with heavier-tailed distributions for stress scenarios. Correlation assumptions matter just as much: diversification benefits that hold in calm markets can shrink sharply in a crisis, when previously unrelated assets start moving together - a repeatedly observed phenomenon.
How it works
You manage a $100M book with three positions: 50% in an equity index (daily vol 1.2%), 30% in high-yield credit (0.8%), and 20% in exotic derivatives (2.5%). The game runs in four phases. In phase 1 the regime is calm and you compute the portfolio's daily volatility from the given correlation matrix (equity-credit 0.35, equity-derivatives 0.55, credit-derivatives 0.20), entering your answer as a percentage.
In phase 2 the regime flips to stress: every off-diagonal correlation goes to 1.00 while the individual asset vols are unchanged, and you compute the new portfolio volatility. In phase 3 you compute the 99% Expected Shortfall of the stressed book in dollar millions. In phase 4 the game shows what a fat-tailed (Student-t with 4 degrees of freedom) refit does to ES and asks you to pick one of four responses to the situation.
Numeric answers are checked against a tolerance: 0.06 percentage points of volatility for the two vol questions and $0.25M for the ES question. Each question locks after one check, so there are no retries within a run - you can restart the whole stress test at the end.
How scoring works
Points per phase: +2 for the calm portfolio volatility, +3 for the crisis volatility, +3 for the 99% Expected Shortfall in $M, and +2 for choosing the correct hedge, for a maximum of 10 per run.
Volatility answers count as correct within 0.06 percentage points (you type a percent, the game compares it to the exact figure). The ES answer counts within $0.25M. The hedge question has exactly one correct choice: buying deep out-of-the-money index puts.
Each check also records a skill attempt for your readiness profile: the two vol questions record under distributions, the ES question under expected-value, and the hedge choice under optional-stopping.
Portfolio volatility is a double sum you can organize
The formula is sigma_p^2 = sum over i and j of w_i w_j sigma_i sigma_j rho_ij. The clean way to compute it by hand is to first form each position's standalone contribution w_i sigma_i, then square those for the diagonal terms and take pairwise products times correlations for the cross terms. Here the standalone contributions are 0.5 x 1.2% = 0.60%, 0.3 x 0.8% = 0.24%, and 0.2 x 2.5% = 0.50%.
The diagonal terms are 0.60^2 + 0.24^2 + 0.50^2 and each cross pair appears twice, so add 2 x (0.60 x 0.24 x 0.35 + 0.60 x 0.50 x 0.55 + 0.24 x 0.50 x 0.20). Working in percentage units keeps the arithmetic sane; take the square root at the end.
The key sanity check: with correlations below 1, portfolio vol must come out below the plain weighted sum of vols (here 1.34%). The gap between your answer and that weighted sum is the diversification benefit. If your calm answer is at or above 1.34%, you made an arithmetic error.
At rho = 1 the formula collapses - use the shortcut
When every pairwise correlation is 1, the double sum becomes a perfect square: sigma_p = sum of w_i sigma_i, the plain weighted average of the vols. You do not need to redo the matrix computation in phase 2 - just add 0.60% + 0.24% + 0.50% = 1.34%.
The crisis vol jump happens with no change in any single asset's volatility; only the relationships changed. Diversification is a property of the correlation structure, and in a liquidity vacuum that structure is what disappears - everyone sells everything at once.
Knowing the collapse cold also gives you the phase 1 sanity bound for free: the calm vol must sit strictly between the largest single contribution and the weighted sum.
ES is a closed-form multiple of sigma
For a normal loss distribution, 99% VaR is z_0.99 x sigma with z_0.99 = 2.3263, and Expected Shortfall is ES = sigma x phi(z_alpha) / (1 - alpha), where phi is the standard normal density. At the 99% level phi(2.3263) = 0.02665, so ES = sigma x 0.02665 / 0.01 = sigma x 2.665. Notice that is only slightly above the VaR multiplier of 2.3263 - under a normal, the tail beyond VaR is thin.
Convert to dollars at the end: multiply the ES fraction by the $100M NAV. With the crisis sigma of 1.34%, ES = 1.34% x 2.665 = 3.57%, or $3.57M against a VaR of $3.12M.
The final phase makes the conceptual point: ES is always at least VaR, and unlike VaR it is subadditive - merging two books can never make ES look better than the sum of the parts. VaR can, which lets a firm hide risk by splitting a book across desks. That property is why regulators moved to ES.
A worked example
Phase 1 setup: each position's standalone contribution to vol is its weight times its own volatility.
The portfolio variance formula is a double sum over all pairs of positions:
Diagonal terms (i = j, so rho = 1): square each standalone contribution.
Cross terms: each pair appears twice, weighted by its correlation.
Sum the diagonal and cross terms, then take the square root for the calm-regime volatility. Enter 1.07.
Phase 2, crisis: every off-diagonal correlation goes to 1, so the double sum collapses to a perfect square - the plain weighted sum of vols. Enter 1.34.
Phase 3: 99% VaR and Expected Shortfall are both multiples of sigma under a normal model.
Convert ES to dollars against the $100M book. Enter 3.57. Phase 4's t(4) refit then lifts ES roughly 40% higher, to about $5.0M, which is why the deep out-of-the-money index puts are the correct hedge.
Common mistakes
• Forgetting to double the cross terms. Each pair (i, j) appears twice in the double sum. Missing the factor of 2 gives a calm vol that is too low.
• Recomputing the full matrix in the crisis phase. At rho = 1 the answer is just the weighted sum of vols - doing the long computation invites arithmetic slips under time pressure.
• Answering VaR when the game asks for ES. Phase 3 wants sigma x phi(z) / (1 - alpha) in dollars, not z x sigma. The two differ by about 15% at the 99% level, well outside the $0.25M tolerance.
• Mixing units. Vol answers go in as percentages (1.34, not 0.0134); the ES answer goes in as dollar millions (3.57, not a percent). Read the placeholder text.
• Picking further diversification as the hedge. It sounds prudent, but adding more correlated assets when correlations have converged to 1 adds exposure.
Why interviews test this
"What is wrong with VaR?" is asked in essentially every risk interview, and the strong answer is the one this game trains: VaR is silent about everything past its own threshold, and it is not subadditive, so it can be gamed by splitting books. Follow up with why Basel moved to a 97.5% ES and you are ahead of most candidates.
The correlation-breakdown computation is also a live interview exercise: given weights, vols, and a correlation matrix, compute portfolio vol, then recompute at rho = 1. Interviewers use it to check both the double-sum mechanics and whether you know that diversification depends on the correlation structure. The fat-tail phase maps to the standard follow-up about why normal-based risk numbers understate crises.
Play Tail Risk Stress Tester on Outcry
Tail Risk Stress Tester is free to play on Outcry, in the Risk Lab section, and every run is timed and scored. Read the guide once, then play Tail Risk Stress Tester and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.