Breach probabilities and hard loss limits

What you will learn

Optimize a finite exposure grid under separately stated chance and hard-loss constraints without confusing a permitted breach with a guarantee.

Prerequisites: Loss quantiles and tail averages · Risk budgets in a toy model · Unequal outcome weights

Declare exactly what counts as a breach

A chance constraint limits the probability of a specified violation. P(L>20)≤.25 permits up to 25%probability of signed loss strictly greater than 20 under the supplied law. Equality at 20 does not count as a strict breach. A hard loss constraint instead requires L≤20 in every case covered by its uncertainty set. The two statements answer different questions.

The activity has eight equally likely common cases and five available exposures:0 through 4 units. Each unit sends 10 fake chips to provisional escrow. Six cases return 15 per unit, while two return 0. All units share the same case, so scaling exposure scales both gains and losses; these are not independent trials or a diversification argument. Hero begins with 100 and house with 1000 fake chips.

Apply the constraint before optimizing the objective

At k units, net outcome is +5 k in six cases and −10 k in two. Expected net is (6×5 k−2×10 k)/8=1.25 k. Larger available exposures have higher means, but a constraint can exclude them. With threshold 20 and breach allowance.25, four units are feasible: the 40-loss failure occurs in two of eight cases, exactly the permitted probability. Feasible does not mean that the 40 loss is harmless or impossible.

Under an all-cases cap of 20 with the same outcomes, only exposures up to two units qualify. Two units have maximum loss 20 and expected net 2.5. Four units have expected net 5 but violate the hard cap. Keep the objective and constraint separate: a positive mean cannot make an infeasible choice feasible. The exercise chooses the best mean only within its declared five-point grid and model.

Reconcile actual transfers and model scope

Each hypothetical outcome restarts the same hero 100/house 1000 baseline. At four units, provisional reserve 60 plus escrow 40 plus house 1000 conserves 1100 chips. Success returns 60 gross, so hero ends 120 and house 980: net gain 20. Failure returns 0, so hero ends 60 and house 1040: net loss 40. Gross receipt includes the returned stake and is not net profit.

The second model sets threshold 10 and allowance.125. Any exposure of two or more has breach probability.25, above the allowance; one unit has maximum loss 10 and no strict breaches. Both criteria then choose one unit. A zero modeled breach probability in this finite positive-probability support still says nothing about omitted real-world cases. Probabilities, support and payoff assumptions require their own evidence; no financial recommendation or future guarantee follows.

Hands-on: Fund exposures under probability and support constraints

Construct strict breach cases, maximize a finite expected-net objective under each criterion and settle conserved success/failure counterfactuals.

Reading limit workshop…

Common mistakes

• Counting equality as a strict exceedance.
• Treating an allowed breach probability as a guarantee.
• Using positive expected net to ignore a binding constraint.
• Adding counterfactual payouts or mistaking gross receipt for net gain.

Keep these ideas

• Specify the inequality, threshold and probability law.
• Construct all strict breach cases before evaluating the chance rule.
• A hard cap covers every case in its stated support.
• Conserve chips and separate mean, probability and worst loss.

Further reading

• John Duchi — Optimization with uncertain data (2018), section 6 ↗

  Stanford EE364b notes, printed pages15–16: all-scenario protection versus a probability-of-violation constraint, with an explicit quantile-convention caution. Only this bounded section reviewed; no full-notes or video claim. Our fake-chip finite-exposure model is original and independently enumerated. Checked 2026-10-21.