Vose SoftwareModelRisk example · Corporate finance
Portfolio Value at Risk and Expected Shortfall
How much could the reserve portfolio lose in a bad year? Value at risk and expected shortfall, with normal and fat tails.
From the model’s saved run
How much could the portfolio lose in a year?
Loss over one year (US$m). Same returns, volatilities and correlations; different tails.
| Measure | Normal model | Fat-tailed model | Difference |
|---|---|---|---|
| Mean loss (negative = expected gain) | −27.2 | −27.1 | |
| Value at risk, 95% | 45.8 | 42.1 | −8% |
| Value at risk, 99% | 79.0 | 88.6 | +12% |
| Expected shortfall, 97.5% | 80.2 | 92.5 | +15% |
| Expected shortfall, 99% | 97.3 | 121.9 | +25% |
| Chance of a large loss (Inputs sheet) | 0.3% | 0.7% | +109% |
| Within the 95% VaR limit? (Inputs sheet) | ✔ within | ✔ within |
| Measure | Unit | Value |
|---|---|---|
| Amount invested | US$m | 500 |
| Limit on the one-year 95% value at risk | US$m | 50 |
S-curve (cumulative probability) of Normal model, Fat-tailed model from the saved simulation.
The business question
A company holds a US$500 million reserve: 40% in global equities, 30% in government bonds, 20% in corporate bonds and 10% in listed real estate. Its risk policy caps the one-year 95% value at risk. The standard way to measure it treats returns as Normal and joins them with their correlations.
Market history says otherwise: extreme returns happen more often than a Normal distribution allows, and in a crisis the assets fall together. The model measures the same portfolio twice — once the standard way, and once with fat-tailed returns joined by a Student copula, which makes joint extremes more likely — with identical expected returns, volatilities and correlations, and compares their tails.
- Who it is for
- Treasurers and investment committees of corporate reserves, pension and insurance portfolios; risk managers.
- Main outputs
- One-year loss: value at risk at 95% and 99%, expected shortfall at 97.5% and 99%, and the chance of a large loss, under a Normal model and a fat-tailed model · whether each keeps the 95% VaR within the policy's limit · value at risk from 90% to 99.9% · what drives the losses.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The amount invested and the weights. The expected one-year return and volatility of each asset class; the six correlations; the degrees of freedom of the fat-tailed returns and of the Student copula.
Portfolio sheet
The Normal model draws four correlated U values from VoseCopulaMultiNormal and turns them into Normal returns. The fat-tailed model draws them from VoseCopulaMultiT and turns them into Student returns (VoseStudent3) with the same mean and volatility. The loss is the fall in the portfolio's value.
Results sheet
Value at risk (a percentile of the loss) and expected shortfall (the average loss beyond it, VoseSimCVARp) under both models. The chance of losing more than a threshold; the drivers of the loss.
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The results in detail
Value at risk at each confidence
The loss exceeded in only 10%, 5%, … of years (US$m): the further into the tail, the more the models part.
Histogram of Fat-tailed model from the saved simulation.
| Confidence | Normal model | Fat-tailed model | Difference |
|---|---|---|---|
| 90% | 29.8 | 23.2 | −22% |
| 95% | 45.8 | 42.1 | −8% |
| 97.5% | 60.6 | 61.4 | +1% |
| 99% | 79.0 | 88.6 | +12% |
| 99.5% | 90.1 | 110.2 | +22% |
| 99.9% | 122.1 | 165.6 | +36% |
What drives the losses
Lowest and highest mean loss over ten equal groups of each asset class's return (VoseSimSensitivityMean), drawn from the overall mean.
Tornado chart: the inputs ranked by how far the mean output moves across each input’s range.
The ranking as a table 4 inputs
| Input (ranked by influence) | Lowest mean | Highest mean | Rank |
|---|---|---|---|
| Global equities return | −101.7 | 46.4 | 1 |
| Corporate bonds return | −91.1 | 36.0 | 2 |
| Listed real estate return | −91.4 | 33.5 | 3 |
| Government bonds return | −39.1 | −16.7 | 4 |
How to read these results
- Both models give the portfolio the same expected return, volatility and correlations. They differ only in the tails: how often extreme returns happen, and whether the assets suffer them together.
- At 95% the two models agree — the fat-tailed model's VaR is even a little lower — and both keep it within the policy's limit. Further out they part: the fat-tailed model's 99% VaR and, above all, its expected shortfall are much larger. A limit set on 95% VaR alone passes both and does not see the difference.
- Expected shortfall averages the losses beyond the VaR, so it measures how bad the bad years are, not just where they start. Bank regulation moved from 99% VaR to 97.5% expected shortfall for this reason.
- Equities drive the losses, with real estate and corporate bonds falling alongside them; government bonds, which tend to move the other way, cushion them.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Portfolio 2 inputs
How much is invested, and the risk policy's limit.
| Assumption | Unit | Value |
|---|---|---|
| Amount invested | US$m | 500 |
| Limit on the one-year 95% value at riskRisk policy: 10% of the reserve. | US$m | 50 |
Tails 3 inputs
What makes the second model fat-tailed.
| Assumption | Unit | Value |
|---|---|---|
| Degrees of freedom of the returnsLower = fatter tails; 5 fits monthly equity data. | 5 | |
| Degrees of freedom of the Student copulaLower = more joint extremes. | 4 | |
| A large loss20% of the portfolio. | US$m | 100 |
Asset classes 5 inputs
Weight, and expected return and volatility over one year — the same in both models.
| Asset class | Weight of the total | Uncertainty | Expected return a year | Volatility a year |
|---|---|---|---|---|
| Global equities | 40% | Fixed | 7.0% | 16.0% |
| Government bonds | 30% | Fixed | 3.5% | 6.0% |
| Corporate bonds | 20% | Fixed | 4.5% | 8.0% |
| Listed real estate | 10% | Fixed | 6.0% | 18.0% |
| Total | 100% | must total 100% |
Correlations 4 inputs
Between the one-year returns — the same in both models. Enter the yellow cells above the diagonal; the cells below it mirror them.
| Equities | Gov. bonds | Corp. bonds | Real estate | ||
|---|---|---|---|---|---|
| Global equities | 1.00 | −0.20 | 0.60 | 0.70 | Equities and government bonds tend to move apart. |
| Government bonds | −0.20 | 1.00 | 0.40 | 0.00 | |
| Corporate bonds | 0.60 | 0.40 | 1.00 | 0.50 | |
| Listed real estate | 0.70 | 0.00 | 0.50 | 1.00 |
Units and conventions
- Money
- US$ million (US$m); a loss is positive.
- Value at risk
- The loss that is exceeded with the stated small probability in a year (VaR 99% = 99th percentile of the loss).
- Expected shortfall
- The average loss in the worst cases beyond the VaR (ES 97.5% = mean of the worst 2.5%).
- Horizon
- One year; no rebalancing during the year.
Key assumptions and limitations
- Returns
- Each asset's return has the stated mean and volatility in both models; only the tails differ.
- Dependence
- The same correlation matrix in both models; the Student copula adds tail dependence.
- Data
- The portfolio and all figures are invented for this example.
How to use it
- Enter your portfolio's weights, return and volatility assumptions and correlations.
- Set the tail parameters from market data: lower degrees of freedom mean fatter tails.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Set limits on expected shortfall as well as VaR, and test how the weights change both.
The tornado chart fills in after a simulation with ModelRisk’s classic engine.
ModelRisk functions used
- VoseEigenValues
- The eigenvalues of the correlation matrix: all above 0 for a valid matrix.
- VoseCopulaMultiNormal, VoseCopulaMultiT
- Four correlated U values each iteration: Normal, and Student (tail dependence).
- VoseNormal, VoseStudent3
- Asset returns from the U values: Normal, and Student with the same mean and standard deviation.
- VoseSimCVARp
- Expected shortfall: the mean of the loss beyond a percentile.
- VoseInput, VoseOutput, VoseSimPercentile, VoseSimMean
- Drivers, outputs and the statistics on the Results sheet.
- VoseSimSensitivityMean
- The tornado chart: every VoseInput ranked by how far the mean loss moves across its range.
Further reading
- Sensitivity analysis in Excel: a tornado chart step by step (Vose guide)
- McNeil, A. J., Frey, R. & Embrechts, P. (2015). Quantitative Risk Management, revised edition. Princeton University Press.
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley — copulas.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.