Risk mathematics
Formulas, units, and worked calculations. Every result depends on a clearly defined population, period, and set of assumptions.
| Concept | Formula | Worked example |
|---|---|---|
| Basis points | Rate × 10,000 | 0.002 = 20 basis points = 0.2%. |
| Count loss rate | Affected transactions ÷ eligible transactions | 20 ÷ 10,000 = 0.2%. Use a defined cohort and outcome. |
| Value loss rate | Loss value ÷ eligible payment value | $8,000 ÷ $400,000 = 2%. State whether loss is gross or net. |
| Expected credit loss, simplified | PD × LGD × EAD | 0.04 × 0.50 × $10,000 = $200 over the stated horizon. |
| Expected transaction loss, simplified | Loss probability × loss severity in dollars | 0.01 × $200 = $2. The probability must have a validated meaning. |
| Contribution after expected loss | Contribution before risk − expected loss | $6 − $2 = $4 before other omitted costs. |
| Precision | True positives ÷ (true positives + false positives) | 80 ÷ (80 + 120) = 40%. |
| Recall | True positives ÷ (true positives + false negatives) | 80 ÷ (80 + 20) = 80%. |
| False-positive rate | False positives ÷ actual negatives | 120 ÷ 9,900 ≈ 1.21%. This denominator differs from precision. |
| Reserve accrual | Eligible volume × retention rate | $50,000 × 10% = $5,000. The rate is illustrative. |
| Reserve closing balance | Opening + accruals − uses − releases | $12,000 + $5,000 − $2,000 − $3,000 = $12,000. |
| Uncovered exposure | Gross obligation − eligible available cover | $120,000 − $30,000 = $90,000 in the stated stress case. |
| Currency conversion | Source amount × destination units per source unit | $100 × 0.90 EUR/USD = €90 before fees. The rate is fictional. |
| Cash conversion cycle | Inventory days + receivable days − payable days | 40 + 30 − 20 = 50 days. A simplified average, not a full cash forecast. |
| Debt-service coverage, simplified | Defined cash available for debt service ÷ debt service | $15,000 ÷ $12,000 = 1.25×. Define the cash measure and period. |
| Little’s Law | Average work in progress = arrival rate × average system time | 60 cases/hour × 2 hours = 120 cases under suitable stable-system assumptions. |
| Net recovery | Gross recovery − relevant recovery costs | $5,000 − $800 = $4,200 under this metric definition. |
| Available balance, simplified | Eligible settled balance − applicable reservations | $500 − $100 = $400 when no other restrictions or adjustments apply. |
The denominator is part of the result
A 1% rate is incomplete until the reader knows 1% of what. Transaction count, value, approved traffic, attempts, accounts, and original exposure answer different questions. Keep numerator and denominator in the same cohort and observation period.
Expected value does not describe every outcome
A $2 expected loss on one payment does not mean the payment will lose exactly $2. It describes an average under the stated probability and severity assumptions. Correlated losses and extreme outcomes need a separate stress analysis.
Keep money and time explicit
Store monetary values with their currency and a suitable precision convention. Do not add different currencies without a defined conversion. Match hours with hours and months with months. Round for display after the calculation, using the product’s approved monetary rules where actual funds are involved.
Read the calculations in context
Credit fundamentals explains PD, LGD, and EAD. Risk models explains prediction metrics and calibration. Case operations explains queue timing. Reserves and limits explains why recorded reserves and eligible cover can differ.