Default dependence thickens the tail
Economic shocks, sectors, regions, and shared counterparties can make defaults cluster. Independence can then understate the probability of several defaults occurring together even when average PD is correct.
Probability
Portfolio Loss Odds Calculator
Estimate expected defaults, expected loss, loss volatility, and exact binomial odds of exceeding a portfolio loss threshold under a homogeneous independent-account model.
PORTFOLIO LOSS ODDS
Convert account default assumptions into an exact threshold exceedance probability
This calculator models a homogeneous portfolio of independent accounts with one exposure, probability of default, and loss-given-default assumption. Credit and risk analysts can estimate expected loss and the exact binomial chance that aggregate loss reaches a chosen threshold, while keeping concentration and default dependence outside the model clearly visible.
Probability loss reaches threshold-
Expected portfolio loss-
Expected defaults-
Defaults required for threshold-
Modeled loss standard deviation-
Probability of at least one default-
PORTFOLIO LOSS ODDS
Homogeneous portfolio loss ledger
Use threshold exceedance as a transparent benchmark for reserve or limit discussions, then escalate to a heterogeneous correlated portfolio model before relying on it for capital, pricing, or approval decisions.
| Quantity | Exposure / account count | Rate or threshold | Calculated value | Basis |
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CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
loss per default = EAD*LGD; k = ceil(threshold/loss per default); P(loss >= threshold) = sum from x=k to N of C(N,x)*PD^x*(1-PD)^(N-x)
Every account is assigned the same exposure, one-period default probability, and loss fraction. Their independent default count is binomial. The loss threshold is converted to the smallest whole number of equal losses that reaches it, and the exact upper binomial tail is summed without a normal approximation.
HOW TO USE THIS MODEL
Build an exceedance benchmark on one consistent credit basis
- Define the portfolio, default event, and risk horizon before entering the account count and probability of default.
- Use exposure at default and loss given default on matching currency, collateral, recovery-cost, and valuation conventions.
- Set a loss threshold tied to the planning question, such as a reserve trigger, and record whether reaching the threshold counts as exceedance.
- Review the integer defaults required before the tail probability; threshold rounding can materially change the result in a homogeneous model.
- Compare the benchmark with concentration, rating mix, sector, geography, maturity, and default-correlation evidence before using it for a controlled decision.
PORTFOLIO LOSS ODDS FUNDAMENTALS
The homogeneous independent credit model
- Exposure at default
- The modeled amount outstanding when an account defaults, before applying the loss fraction.
- Probability of default
- The chance that one account meets the frozen default definition during the declared horizon.
- Loss given default
- The fraction of exposure lost after modeled recoveries and eligible costs.
- Binomial default count
- With equal independent default probabilities, the number of defaults among N accounts follows a binomial distribution.
- Threshold discretization
- Aggregate loss changes in equal loss-per-default steps, so a monetary threshold maps to the next whole required default count.
MODEL AND FORMULA
Why the loss threshold becomes an exact binomial tail
loss per default = EAD*LGD; k = ceil(threshold/loss per default); P(loss >= threshold) = sum from x=k to N of C(N,x)*PD^x*(1-PD)^(N-x)
Every account is assigned the same exposure, one-period default probability, and loss fraction. Their independent default count is binomial. The loss threshold is converted to the smallest whole number of equal losses that reaches it, and the exact upper binomial tail is summed without a normal approximation.
DEEPER ANALYSIS
Where simplified portfolio loss odds understate real risk
Homogeneity hides concentration
Ten accounts with very different exposures do not have the same threshold behavior as ten equal accounts. A large exposure can cross the threshold with one default, while this model requires equal loss steps.
PD and LGD can move together
Stress conditions can raise both default frequency and loss severity as collateral values and recoveries deteriorate. Treating PD and LGD as fixed independent inputs can miss this wrong-way dependence.
WORKED DECISION CASES
Two portfolio questions with different model adequacy
Homogeneous screening portfolio
For 100 equal accounts at 10,000 exposure, 2% PD, and 60% LGD, each default loses 6,000 and expected portfolio loss is 12,000. A 30,000 threshold requires at least five defaults, so the calculator sums the exact probability of five or more. This is a useful transparent baseline.
Concentrated sector portfolio
A portfolio includes several large borrowers in one cyclical sector. Applying the homogeneous independent result would conceal both exposure concentration and common-shock dependence. The correct action is to move to account-level exposures and a correlated credit model, using this page only as a comparison benchmark.
TECHNICAL LANGUAGE
Credit portfolio loss terminology
- Default event
- The contractual or policy-defined credit event measured over the stated horizon.
- EAD
- Exposure at default, the amount at risk when default occurs.
- PD
- Probability of default for one account over the model horizon.
- LGD
- Loss given default, the share of exposure not recovered under the modeled workout basis.
- Expected loss
- N times PD times EAD times LGD in this homogeneous model.
- Exceedance probability
- The probability that discrete aggregate loss is greater than or equal to the entered threshold.
EVIDENCE AND DATA LINEAGE
Preserve account definition, horizon, EAD, PD, LGD, and recovery basis
Retain the account-level inventory used to justify homogeneity, default definition, observation horizon, rating or underwriting cohort, EAD date, undrawn exposure treatment, collateral and recovery assumptions, workout costs, currency conversion date, and data lineage for PD and LGD. Reconcile account count and total modeled exposure with the source portfolio. Document any exclusions, restructurings, cures, or multiple accounts belonging to one obligor.
LIMITS AND EXCLUSIONS
Material exclusions from the exact binomial benchmark
- Accounts are assumed equal in exposure, PD, and LGD and independent over one common horizon.
- The model excludes obligor, sector, regional, macroeconomic, collateral, and recovery correlation as well as multi-period migration.
- Loss is deterministic once default occurs; LGD uncertainty, timing, discounting variation, and partial default are not simulated.
- The result is not a regulatory capital, allowance, pricing, solvency, or credit approval calculation and may not match governing accounting definitions.
RELIABLE SOURCES
References for this model and its decision limits
FREQUENTLY ASKED QUESTIONS
Questions about homogeneous portfolio loss odds
Why does the threshold default count use a ceiling?
Aggregate loss advances in whole default steps. If four equal defaults fall short of the threshold, the fifth is the first count that reaches or exceeds it.
Is expected defaults a forecast that exactly two accounts will default?
No. It is the long-run mean N times PD and need not be a whole number. Realized defaults are discrete and can differ materially.
Why can at-least-one-default probability be high while expected loss is modest?
Many small independent opportunities can make any default likely, while low EAD or LGD keeps the monetary expectation limited. Frequency and severity answer different questions.
Can I average different account PDs and use this model?
A simple average loses exposure weighting and heterogeneity in the count distribution. Use account-level Poisson-binomial or a richer credit portfolio model when PDs differ materially.
How does default correlation affect the threshold odds?
Positive dependence usually places more probability on joint extremes than an independent model. The direction and size require a defensible dependence model or stress scenarios.
Does the calculation satisfy CECL, IFRS 9, or Basel requirements?
No. Those frameworks include definitions, horizons, scenarios, discounting, segmentation, governance, and validation requirements beyond this homogeneous one-period benchmark.
IMPORTANT NOTE
Do not use an independence benchmark as controlled credit capital
This calculator provides transparent exact-binomial arithmetic for a homogeneous independent portfolio. It does not replace account-level data, correlated default modeling, accounting policy, regulatory methods, model validation, expert credit judgment, or approved governance for reserves, limits, pricing, or capital.