Probability
Measure negative-payoff probability, expected loss, discrete loss VaR, and expected shortfall for a four-leaf decision tree.
TREE DOWNSIDE RISK
This calculator converts a two-branch decision tree into loss probability, probability-weighted expected loss, discrete loss Value at Risk, and expected shortfall. Risk owners can use the measures together to identify frequent modest losses versus less frequent severe losses, while preserving the complete terminal ledger.
TREE DOWNSIDE RISK
Use the downside measures as complementary views and test how probability mass moves at discrete VaR boundaries before setting reserves, controls, or acceptance limits.
| Terminal path | Probability mass (%) | Signed payoff | Expected-payoff contribution | Downside state |
|---|
CURRENT CALCULATION PROCESS
Li = max(0, -xi); EL = sum(pi*Li); VaR_c = inf{l: P(L <= l) >= c}; ES_c = average loss in the worst (1-c) probability mass
Signed terminal payoffs are converted to nonnegative loss magnitudes, leaving gains at zero loss. Expected loss weights every magnitude by path probability. VaR selects the first discrete loss value reaching cumulative confidence, and expected shortfall averages exactly the worst tail mass, including fractional mass at a boundary when required.
HOW TO USE THIS MODEL
TREE DOWNSIDE RISK FUNDAMENTALS
MODEL AND FORMULA
Signed terminal payoffs are converted to nonnegative loss magnitudes, leaving gains at zero loss. Expected loss weights every magnitude by path probability. VaR selects the first discrete loss value reaching cumulative confidence, and expected shortfall averages exactly the worst tail mass, including fractional mass at a boundary when required.
DEEPER ANALYSIS
If a leaf probability is larger than the remaining tail, expected shortfall uses only the portion needed to complete the specified tail mass. Averaging every negative leaf instead answers a different conditional-loss question.
A 95% VaR reports a quantile. Losses worse than VaR can still occur, and a tree with a small catastrophic leaf may show a modest VaR until the confidence level reaches that mass.
Classifying negative values as losses assumes zero is the reference. Budgets, targets, benchmarks, and opportunity costs may require a different shortfall baseline before risk is calculated.
WORKED DECISION CASES
In the default tree, negative leaves total 40% probability. At 95% confidence, the loss VaR and worst-tail expected shortfall both reach the 30-unit A-failure loss because the worst 5% mass lies wholly within that leaf. Controls should address the A-failure mechanism rather than relying on the positive net expected payoff.
A redesigned tree gives one leaf only 1% probability but an extreme loss. Expected loss may remain modest and 95% VaR may not show the catastrophe; expected shortfall at a higher confidence and explicit stress scenarios expose the severity.
TECHNICAL LANGUAGE
EVIDENCE AND DATA LINEAGE
Retain the event-tree version, branch evidence, terminal payoff worksheets, valuation horizon, currency and price date, discounting and netting rules, zero-loss reference, VaR confidence policy, and treatment of ties at discrete boundaries. Stress-test expert probabilities and document dependence or common-cause events that the four fixed leaves omit.
LIMITS AND EXCLUSIONS
RELIABLE SOURCES
FREQUENTLY ASKED QUESTIONS
If non-loss probability already reaches the chosen confidence, the corresponding loss quantile is zero. The rare loss still exists beyond that quantile and should be examined with expected shortfall or stress scenarios.
No. Expected loss averages across the full distribution, while expected shortfall averages only a fixed worst-tail probability mass.
When the entire worst-tail mass fits inside one discrete leaf, every probability slice in that tail has the same loss magnitude.
Only after naming the level and maintaining the same payoff basis, horizon, probability model, and loss reference. A higher level generally probes a smaller, more severe tail.
Not necessarily. It means below the entered zero reference. Define whether payoffs represent cash, net present value, utility, mission score, or another quantity before calling them losses.
No. Reserve and capital rules may require dependence, horizon scaling, scenario stress, parameter uncertainty, liquidity, regulation, and governance beyond this four-leaf calculation.
IMPORTANT NOTE
This page summarizes downside in the entered four-leaf model. It does not validate event probabilities, define risk appetite, or replace regulated capital, safety, actuarial, financial, or enterprise risk methods required for the real decision.