Finance
Portfolio Loss Distribution Calculator
Estimate an analytic portfolio loss distribution, value at risk, expected shortfall, and threshold exceedance probability over a chosen horizon.
ANALYTIC LOSS MAP
Turn return and volatility assumptions into a decision-ready loss range
Use this page when the decision is about the shape of possible losses over one horizon. A positive loss means capital lost; a negative loss represents a gain. The model reports both a percentile boundary and the average severity beyond that boundary.
CURRENT DECISION RECORD
Loss percentile ledger
Every row is generated from the current inputs and reused by Copy, TXT, and the page-specific PDF.
| Percentile | z score | Loss amount | Loss as % of portfolio |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
Loss ~ Normal(meanLoss, sdLoss); meanLoss = -V x r x h/252; sdLoss = V x sigma x sqrt(h/252); VaR(c) = meanLoss + z(c) x sdLoss
Waiting for valid inputs.
HOW TO USE THE DISTRIBUTION
Five steps from assumptions to a risk conversation
- Enter the capital currently exposed, not an aspirational future balance.
- Use a return and volatility estimate measured on a basis consistent with the portfolio.
- Choose the decision horizon, such as a 20-day liquidity window or 63-day review cycle.
- Set a concrete loss threshold tied to a covenant, spending need, or governance trigger.
- Compare VaR, expected shortfall, and threshold probability; preserve the inputs with the exported record.
FOUNDATIONS
Five ideas that keep the output interpretable
Loss sign convention
The calculator defines loss as minus portfolio return. Positive numbers consume capital; negative numbers add capital.
Time scaling
Mean return scales with time, while standard deviation scales with the square root of time under independent, stable increments.
Percentile, not maximum
A 95% VaR is a boundary in the modeled distribution. It does not cap the remaining 5% of outcomes.
Tail average
Expected shortfall answers how severe losses are on average after the VaR boundary has been crossed.
Threshold probability
A policy threshold asks a different question from VaR: how often a specific loss amount is modeled to be exceeded.
DEEP ANALYSIS
Three decisions this model can inform
Capital buffer sizing
Use VaR as a percentile reference and expected shortfall as the more conservative tail severity. If available capital covers VaR but not expected shortfall, governance should explicitly document that residual exposure.
Horizon mismatch
A one-day risk number is not a substitute for a 20-day liquidation window. Increasing the horizon changes both drift and dispersion, so the decision window belongs in the model rather than in a footnote.
Tail-model challenge
Compare the analytic result with historical stress losses. A large gap is evidence that skew, jumps, concentration, or correlation changes are material and that a normal model should not stand alone.
DECISION CASES
Two ways the same output leads to different action
Treasury reserve before a payment
A foundation expects a $4 million grant payment in 20 trading days. It models the liquid portfolio over that exact horizon and compares the 95% tail measures with cash already ring-fenced. The decision is whether to sell risk assets now, not whether the long-run allocation is attractive.
Committee review after volatility rises
A pension portfolio keeps the same expected return but raises annual volatility from 12% to 22% after a regime shift. Mean loss changes little, while VaR and expected shortfall widen sharply. The committee records the assumption change and tests whether its drawdown policy still holds.
TERMS
Portfolio distribution glossary
- Arithmetic expected return
- The average one-period return used as the center of the normal return model.
- Annual volatility
- The annualized standard deviation of returns, used here as the dispersion input.
- Loss distribution
- A probability description of possible dollar losses over the selected horizon.
- Value at risk
- The loss percentile associated with a chosen confidence level, not the worst possible loss.
- Expected shortfall
- The conditional average loss among outcomes at or beyond the VaR cutoff.
- Exceedance probability
- The modeled chance that loss is greater than a user-specified threshold.
EVIDENCE RECORD
What to preserve with the result
Retain the valuation date, portfolio holdings or benchmark, return and volatility estimation window, rebalancing convention, selected horizon, confidence level, threshold rationale, and any independent stress result. The PDF records current values and the illustration used to explain the tail decision.
LIMITS
Model boundaries
- Normal returns are symmetric and thin-tailed; real losses may be skewed, clustered, or discontinuous.
- Square-root-of-time scaling assumes stable volatility and weak dependence across increments.
- The model treats the portfolio as a linear exposure without options, margin calls, taxes, or forced sales.
- Expected return and volatility are assumptions, not observations about the future.
Disclaimer: Use the result as one governed scenario alongside stress tests, liquidity analysis, and professional judgment.
SOURCES
Methods and investor-risk references
FAQ
Questions specific to analytic portfolio loss distributions
Why can mean loss be negative?
A positive expected return becomes a negative expected loss under the page's sign convention. It means an expected gain, even though adverse outcomes remain possible.
Is 95% VaR the loss in the worst 5% of cases?
No. It is the boundary where the upper 5% tail begins. Expected shortfall summarizes the average loss inside that tail.
Can I compare a 20-day result with a one-year budget?
Only after aligning horizons and assumptions. A short-horizon normal model does not automatically describe a full-year path with rebalancing, cash flows, and regime changes.
What volatility should I enter?
Use a defensible estimate for the whole portfolio, including correlations among holdings. A weighted average of individual volatilities generally misses diversification effects.
Why does a higher expected return reduce modeled loss?
The return assumption shifts the center of the distribution. It does not reduce dispersion or guarantee that tail losses will be smaller in observed markets.
When should I use simulation instead?
Use simulation when nonlinear payoffs, changing correlations, discrete events, fat tails, or path dependence materially affect the decision.