ModelRisk example · General techniques
Envelope Correlation: Interest and Mortgage Rates
How do you make one variable follow another when you only know its range at a few points? The envelope method.
From the model’s saved run
Interest and mortgage rates
Percentiles of the two rates and of the spread between them.
| Measure | Unit | 10th pct | Median | 90th pct | Mean |
|---|---|---|---|---|---|
| Interest rate | rate | 3.94% | 5.14% | 6.46% | 5.17% |
| Mortgage rate | rate | 8.68% | 10.39% | 12.29% | 10.44% |
| Spread (mortgage − interest) | rate | 3.99% | 5.27% | 6.56% | 5.27% |
| Measure | Unit | Value |
|---|---|---|
| Rank correlation of the two rates | 0.70 |
S-curve (cumulative probability) of Interest rate, Mortgage rate from the saved simulation.
The business question
A bank's treasury models next year's base interest rate as a PERT distribution from 3% to 8%, most likely 5%. Mortgage rates follow interest rates, but loosely: the margin changes with competition and with the rate itself.
The experts give three statements: if the interest rate is 3%, the mortgage rate will be between 6% and 10%, most likely 8%; at 5% between 7.7% and 12.8%, most likely 10.3%; at 8% between 10% and 15.9%, most likely 13.3%.
The envelope method draws a curve through each set of three points, so that every interest rate has a minimum, a most likely and a maximum mortgage rate, and draws the mortgage rate from a PERT between them.
- Who it is for
- Analysts who need to correlate two variables from expert judgement rather than data.
- Main outputs
- The mortgage rate and the spread over the interest rate at P10 / median / P90 · their rank correlation · the mortgage rate when interest rates are low or high · a chart of the envelope and of 1,000 simulated pairs.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The base interest rate as a PERT, and the limits below and above which an interest rate counts as low or high on the Results sheet.
Rates sheet
The envelope table gives the mortgage rate's minimum, most likely and maximum at three interest rates. Lagrange's formula gives the parabola through each set of three points, which gives the range at this iteration's interest rate; VosePERT draws the mortgage rate in it. The chart shows the three curves and 1,000 simulated pairs.
Results sheet
Percentiles of the interest rate, the mortgage rate and the spread; their rank correlation; the mean mortgage rate in iterations with a low or a high interest rate.
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The results in detail
The mortgage rate when interest rates are low or high
Mean mortgage rate over the iterations in which the interest rate is below or above the limits on the Inputs sheet.
| Measure | Unit | Value |
|---|---|---|
| Chance of a low interest rate | probability | 11.5% |
| Mean of mortgage rate × (low) | rate | 1.01% |
| Mean mortgage rate when the interest rate is low | rate | 8.84% |
| Chance of a high interest rate | probability | 20.7% |
| Mean of mortgage rate × (high) | rate | 2.45% |
| Mean mortgage rate when the interest rate is high | rate | 11.82% |
| Mean mortgage rate over all iterations | rate | 10.44% |
How to read these results
- The envelope makes the mortgage rate's whole range move with the interest rate, so the two are strongly but not perfectly related: a rank correlation of about 0.7. The chart on the Rates sheet shows every simulated pair inside the envelope.
- When the interest rate is low the mortgage rate averages about 8.8%; when it is high, about 11.8%. A single correlation coefficient would not tell you that the range itself widens as rates rise.
- The envelope is the experts' knowledge written down: three ranges are easier to give than a correlation coefficient, and they keep impossible pairs (a 6% mortgage with an 8% interest rate) out of the model.
- The curves are parabolas through three points. Keep the interest rate's range inside the experts' three rates, where they are interpolated rather than extrapolated.
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.
Interest rate 1 input
Next year's base interest rate.
| Assumption | Unit | Uncertainty | Low | Most likely | High |
|---|---|---|---|---|---|
| Base interest rateKeep it inside the envelope's three rates. | rate | PERT | 3.0% | 5.0% | 8.0% |
Low and high interest rates 2 inputs
Used only on the Results sheet, to compare the mortgage rate when interest rates are low or high.
| Assumption | Unit | Value |
|---|---|---|
| An interest rate below this is low | rate | 4.0% |
| An interest rate above this is high | rate | 6.0% |
Units and conventions
- Rates
- Annual rates, in percent.
- Spread
- Mortgage rate minus interest rate.
- Envelope
- The minimum and maximum curves within which the second variable is drawn.
Key assumptions and limitations
- Quadratic curves
- Each curve is the parabola through the experts' three points, as in the original example. Outside the three interest rates it is an extrapolation: keep the interest rate's range inside them (a model check does).
- PERT
- Within its range the mortgage rate is a PERT; the interest rate is the only link between the two.
- Data
- The rates are those of ModelRisk's earlier Correlating Interest and Mortgage Rates example.
How to use it
- Set the interest rate's range on the Inputs sheet.
- Replace the envelope on the Rates sheet with your experts' ranges at three interest rates.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Check the chart: every simulated pair should sit inside the envelope, and the envelope should look plausible.
ModelRisk functions used
- VosePERT
- The interest rate, and the mortgage rate within this iteration's envelope.
- Lagrange's formula
- The parabola through the experts' three points, written as plain arithmetic.
- VoseInput, VoseOutput
- Name the two rates and the spread.
- VoseSimValue
- The chart's simulated pairs: the values of the first 1,000 iterations.
- VoseSimCorrelation
- The rank correlation between the two rates.
- VoseSimMean, VoseSimPercentile
- Statistics of the simulation.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: modelling dependencies with the envelope method.
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.