Vose SoftwareModelRisk example · Insurance
Life Portfolio and Lapse
What is a block of life policies worth, and what makes it uncertain — chance, or not knowing the rates?
From the model’s saved run
What is the block worth?
Present value of ten years of premiums less death claims and expenses (US$m).
| Projection | Mean | 5th pct | 95th pct | 1-in-200 0.5th pct | Loss chance |
|---|---|---|---|---|---|
| Expected (best estimate, no randomness) | 10.29 | ||||
| Chance only (best-estimate rates) | 10.20 | 3.77 | 16.56 | 0.12 | 0.5% |
| Uncertain rates | 8.91 | −1.64 | 18.95 | −7.85 | 8.4% |
| Rate uncertainty widens the 5th–95th range by | 61% | loss chance × | 18.2× |
S-curve (cumulative probability) of Chance only, Uncertain rates from the saved simulation.
The business question
A life insurer's block of 20,000 ten-year term policies pays US$250,000 on death. Policyholders pay US$650 a year — unless they become disabled, when the premium is waived but the cover continues. A quarter of the disabled recover each year and start paying again. About 7% a year let their policy lapse.
The block looks profitable on the best-estimate rates, but those rates come from limited experience, and lapses in particular move with the economy. The model follows every policyholder between the four states each year, first with the best-estimate rates and then with rates that are themselves uncertain, and values the block's premiums, death claims and expenses.
- Who it is for
- Life actuaries, product and pricing teams, risk managers of life insurers; anyone modelling movements between states.
- Main outputs
- The present value of the block's profits (best estimate, with chance only, and with uncertain rates) and the chance of a loss · deaths over ten years · where today's policyholders will be in ten years · which rate drives the value.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The number of policies, the premium, the sum assured and expenses. The yearly chances of becoming disabled, lapsing, dying and recovering; PERT ranges for how far each rate could be from its best estimate.
Projection sheet
The transition matrix holds a year's chance of each move. Its ten-year power (VoseMarkovMatrix) shows where today's policyholders will be. The block is projected three ways: the expected numbers (matrix multiplication), random moves with the best-estimate rates (VoseMarkovSample), and random moves with this iteration's uncertain rates. Premiums, death claims and expenses follow, and their present value.
Results sheet
The present value of each projection — mean, percentiles and the chance of a loss; deaths over ten years; the rates that drive the value.
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The results in detail
Deaths over ten years
The number of death claims: chance alone, and with uncertain mortality.
| Projection | Mean | Std dev | 5th pct | 95th pct | Spread std dev ÷ mean |
|---|---|---|---|---|---|
| Expected (best estimate) | 305 | ||||
| Chance only | 306 | 17.5 | 277 | 335 | 5.7% |
| Uncertain rates | 309 | 32.7 | 257 | 364 | 10.6% |
| Rate uncertainty multiplies the spread by | 1.9× |
What drives the value?
Lowest and highest mean value over ten equal groups of each uncertain rate (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 |
|---|---|---|---|
| Mortality level | 0.05 | 16.58 | 1 |
| Lapse level | 7.89 | 9.87 | 2 |
| Disability level | 8.02 | 10.00 | 3 |
| Recovery level | 8.56 | 9.27 | 4 |
How to read these results
- On the best-estimate rates the block is worth about US$10.3 million over ten years. Chance alone — which of the 20,000 policyholders die, lapse or fall ill — moves that by a few million either way, and a loss is very unlikely (about 0.5%).
With 20,000 lives the law of large numbers works: deaths over ten years vary by only about 6% from chance. Uncertainty about the rates themselves nearly doubles that spread (× 1.85), widens the range of the value by about 60% and makes a loss about eighteen times as likely (8.4%) — the rows under each table.
Better experience data would remove more risk than a larger block would.
- The average value with uncertain rates is lower (about US$8.9 million), mainly because the ranges on the Inputs sheet lean towards more deaths, more disability and more lapses than the best estimates.
- Mortality drives the value: the mean value is about US$16.6 million in the tenth of futures with the lightest mortality and about zero in the tenth with the heaviest. Lapses matter little here because every policy is profitable at the same rate. For a product priced on the assumption that many policyholders will lapse, lapse uncertainty could dominate.
- The Markov chain also shows where the block is heading: after ten years about 47% of today's policyholders are still paying, 51% have lapsed and 1.5% have died — the ten-year matrix on the Projection sheet.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
The block 5 inputs
Ten-year term policies with a premium waiver on disability.
| Assumption | Unit | Value |
|---|---|---|
| Policies in force todayAll active today. | policies | 20,000 |
| Premium per policyLevel, paid at the start of each year by active policyholders. | US$ a year | 650 |
| Sum assuredPaid at the end of the year of death. | US$ | 250,000 |
| Expenses per policyFor active and disabled policies. | US$ a year | 60 |
| Discount rateRisk-free rate for ten years. | a year | 3.0% |
Best-estimate rates 5 inputs
A year's chance of each move, from the company's experience.
| Assumption | Unit | Value |
|---|---|---|
| Active → disabledDisability: premiums are then waived. | a year | 0.40% |
| Active → lapsedPolicyholders who stop paying and leave. | a year | 7.0% |
| Active → deadAverage over the block's ages for ten years. | a year | 0.20% |
| Disabled → active (recovery)They start paying again. | a year | 25% |
| Disabled → deadDisabled lives die more often. | a year | 1.0% |
How sure are the rates? 4 inputs
Multipliers on the best-estimate rates, one value for all ten years.
| Assumption | Unit | Uncertainty | Low | Most likely | High |
|---|---|---|---|---|---|
| Mortality levelExperience studies rest on few deaths. | × best estimate | PERT | 0.85 | 1.00 | 1.25 |
| Lapse levelLapses move with the economy and competitors' prices. | × best estimate | PERT | 0.60 | 1.00 | 1.60 |
| Disability level | × best estimate | PERT | 0.70 | 1.00 | 1.50 |
| Recovery level | × best estimate | PERT | 0.70 | 1.00 | 1.20 |
Units and conventions
- Money
- Premiums, sums assured and expenses per policy in US$; portfolio amounts in US$ million (US$m).
- States
- Active: paying premiums. Disabled: premiums waived, still covered. Lapsed and dead: left the block for good.
- Transition matrix
- Row = the state at the start of a year, column = the state at its end; each row adds up to 100%.
- Value
- Present value of premiums less death claims and expenses over ten years.
Key assumptions and limitations
- Rates
- The same yearly rates for all ten years (an average over the block's ages); one set of uncertain multipliers for the whole projection.
- Lives
- Policyholders move independently of each other, so a year's moves are multinomial.
- Timing
- Premiums and expenses at the start of a year, death claims at its end.
- Data
- The block and all figures are invented for this example.
How to use it
- Enter your block's size, premium, sum assured and expenses.
- Enter your best-estimate rates and how uncertain each one is.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Compare 'chance only' with 'uncertain rates': the gap is what better experience data would buy.
The tornado chart fills in after a simulation with ModelRisk’s classic engine.
ModelRisk functions used
- VoseMarkovSample
- Moves a whole population between states for a number of periods, at random (an array).
- VoseMarkovMatrix
- The transition matrix for several periods: where a policyholder will be after t years.
- VosePERT
- How far each rate could be from its best estimate.
- VoseSimMean, VoseSimPercentile, VoseSimProbability
- Statistics of the simulation on the Results sheet.
- VoseSimSensitivityMean
- The tornado chart: every VoseInput ranked by how far the mean value moves across its range.
Further reading
- Sensitivity analysis in Excel: a tornado chart step by step (Vose guide)
- Haberman, S. & Pitacco, E. (1999). Actuarial Models for Disability Insurance. Chapman & Hall/CRC.
- Dickson, D. C. M., Hardy, M. R. & Waters, H. R. (2020). Actuarial Mathematics for Life Contingent Risks, 3rd edition. Cambridge University Press.
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley — Markov chains.
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.