ModelRisk Optimizer example · Insurance

Reinsurance Programme Chosen by the Optimizer

Which retention, cover and quota share keep the 1-in-200 loss within the capital at the least cost?

Complexity: ★★★★★
Free Excel workbook (IN-09)
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From the model’s saved run

The Optimizer’s plan against the starting plan

The two programmes, and how each does on the same simulated years.

Underwriting result — cumulative probability

S-curve (cumulative probability) of Starting plan, Optimizer’s plan, Worst 1-in-200 result allowed from the saved simulation.

Decision	Unit	Starting plan	Optimizer’s plan	Change
Retention	US$ thousand	2,000	3,000	1,000
Cover above the retention	US$ thousand	23,000	27,000	4,000
Quota share	share ceded	30%	45%	15%

Measure	Unit	Starting plan	Optimizer’s plan	Change
Top of the programme	US$ thousand	25,000	30,000	5,000
Net retention on one claim	US$ thousand	1,400	1,650	250
Mean result	US$m a year	1.65	1.66	0.02	✔ better
1-in-20 result	US$m	−4.60	−4.06	0.53	✔ better
1-in-200 result	US$m	−14.35	−9.83	4.52	✔ better
Standard deviation of the result	US$m	6.06	4.90	−1.15	✔ better
Chance of a loss	of years	29.0%	27.5%	−1.5%	✔ better
Worst 1-in-200 result allowed (− capital)	US$m	−10.00	−10.00
Largest net retention allowed	US$ thousand	2,000	2,000

1-in-20 and 1-in-200: the 5th and 0.5th percentiles of the result. Net retention = retention × (1 − quota share).

Loss exceeded once in T years

Chart: Starting plan, Optimizer’s plan.

The business question

IN-03 compared four reinsurance programmes for a commercial property insurer with about 60 claims a year, most small, a few very large. The insurer holds US$10 million of capital against its underwriting, so a 1-in-200 year must lose no more than that. The broker's programme loses about US$15 million in such a year.

Here the ModelRisk Optimizer chooses the retention, the cover and the quota share together.

Who it is for: Reinsurance buyers, underwriters, capital and pricing actuaries, finance teams of insurers.
Main outputs: The Optimizer's retention, cover and quota share against a starting plan (the broker's programme in IN-03) · for both: the mean, 1-in-20 and 1-in-200 underwriting result, its spread and the chance of a loss · each programme's price and expected cost · the loss exceeded once in 5 to 500 years.

What the Optimizer decides

The Optimizer changes the blue cells. The starting plan is the broker's programme in IN-03: its two layers as one, 23,000 xs 2,000, and a 30% quota share.

Decision	Unit	Starting plan	Optimizer’s plan	Lowest	Highest
Retention	US$ thousand	2,000	3,000	1,000	6,000
Cover above the retention	US$ thousand	23,000	27,000	0	40,000
Quota share	share ceded	30%	45%	0%	50%

Goal: Maximise the mean underwriting result
Limit: 1-in-200 result at least −10.0 (the 0.5th percentile)
Limit: Net retention on one claim: at most the largest net retention, in every plan tried

The Optimizer’s run: Vertex Detector, 124 plans tried, each on 10,000 samples, in 13.6 seconds.

How the model works

Five sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
The expected number of claims, the average claim and how heavy the tail of claim sizes is. The reinsurers' margin on a layer and the quota share's commission; the premium and the expense ratio. The capital the insurer holds and the most of one claim it will keep.
Decisions sheet
The three cells the Optimizer changes — the retention, the cover above it and the quota share — with their bounds and steps, the starting plan (the broker's programme), the goal (the highest mean result) and the limits (the 1-in-200 result within the capital, and the net retention).
Claims sheet
Each iteration draws the year's claims once. VoseAggregateTranche adds up, over every claim, the part that falls in each band of US$0.5 million up to US$50 million, and the part above. A layer from any retention to any top on these steps pays the sum of its bands.
Programme sheet
For each plan: the layer's exact mean and standard deviation (VoseDeductObject) and its price. The quota share's premium and commission; this year's recoveries, the claims kept and the underwriting result, an output.
Results sheet
Both plans on the same simulated years: the mean, 1-in-20 and 1-in-200 result, its spread and the chance of a loss. Each programme's price and expected cost; the loss exceeded once in 5 to 500 years.

ModelRisk techniques

ModelRisk Optimizer: retention, cover and quota share in steps
1-in-200 result as a percentile limit
Net retention as a formula limit
Layer recoveries by claim band VoseAggregateTranche
Layer price from VoseDeductObject

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The results in detail

What each programme costs

Exact, not simulated: the reinsurers' expected margin, by which the programme lowers the mean result.

Measure	Unit	Starting plan	Optimizer’s plan	Change
Layer: expected loss	US$m a year	4.83	3.23	−1.61
Layer: price	US$m a year	6.29	4.58	−1.71
Layer: price ÷ expected loss	×	1.30	1.42	0.12
Quota share: premium less commission	US$m a year	8.03	12.63	4.59
Quota share: expected claims ceded	US$m a year	7.55	12.05	4.50
Expected cost of the programme	US$m a year	1.94	1.93	−0.01	✔ better

Cost = layer price − its expected loss + quota share premium − commission − its expected claims.

How to read these results

The broker's programme, a layer from US$2m to US$25m and a 30% quota share, loses about US$14m in the 1-in-200 year: well over the US$10m of capital the insurer holds.
The Optimizer raises the quota share to 45% and the retention to US$3m. Ceding 45% of every claim keeps the net retention on one claim at about US$1.65m, inside the US$2m allowed. The quota share is cheap here: its commission more than covers the insurer's expenses.
The higher retention drops the busiest part of the layer, from US$2m to US$3m, whose price is mostly the claims it pays back. The saving pays for the larger quota share and a higher top, US$30m. The 1-in-200 loss falls to just inside the capital, about US$9.8m, and the expected cost of the programme stays about the same.
Here the Optimizer’s plan is better on every measure: a slightly higher mean result, less spread, a smaller chance of a loss and a much smaller 1-in-200 loss. Raising the top of the broker's programme would also meet the capital, but at a higher cost than either.
Claims above the top still come back to the insurer, and the limit looks at the 1-in-200 year only: check the 1-in-500 loss on the chart, and the top against the largest sums insured. Change the capital or the largest net retention and run the Optimizer again: the best mix of layer and quota share changes with them.

Run the Optimizer yourself. Open the workbook, open the Optimizer from the ModelRisk ribbon (the saved task opens with it) and press Start; then simulate to compare the plans.

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Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Claims 3 inputs

How many claims a year, and how large (as in IN-03).

Assumption	Unit	Value
Expected number of claimsPolicies in force × claim rate, from the claims history.	claims a year	60
Average claimClaims history, inflated to next year's prices.	US$ thousand	500
Tail of the claim sizes (Pareto shape)Fitted to the large claims (as in IN-02). Must be above 1.	lower = heavier	2.2

Reinsurance prices 2 inputs

What the reinsurers charge for a layer and pay back on a quota share.

Assumption	Unit	Value
Layer price: margin over the expected lossPrice = expected loss + this × standard deviation. 0.25 reproduces the broker's quotes in IN-03.	× std deviation	0.25
Quota share: ceding commissionPaid back by the reinsurer for the insurer's cost of writing the business.	of ceded premium	25%

Premium and expenses 2 inputs

Assumption	Unit	Value
Gross premiumWritten next year.	US$m	42.0
Expense ratioBrokers' commission and administration.	of gross premium	20%

Capital and underwriting policy 3 inputs

The limits the Optimizer must respect.

Assumption	Unit	Value	Uncertainty
Capital held against the underwritingSet by the board: a 1-in-200 year must not lose more.	US$m	10.0	Fixed
Worst 1-in-200 result allowedMinus the capital: the Optimizer's limit on the 0.5th percentile of the result.	US$m	−10.0	Calculated
Largest net retention on one claimThe most of any one claim the insurer keeps: retention × (1 − quota share).	US$ thousand	2,000	Fixed

Units and conventions

Money: US$ million (US$m); claim sizes, retention and cover in US$ thousand.
Excess of loss: A layer '23,000 xs 2,000' pays the part of each claim above US$2m, up to US$23m: the slice of every claim between the retention, US$2m, and the top, US$25m.
Quota share: The reinsurer takes a fixed share of the premium and of the claims left after the layer, and pays back a ceding commission on its share of the premium.
Net retention: The most of one claim the insurer keeps below the top: retention × (1 − quota share).
Result: Underwriting result = premium − expenses − reinsurance premiums + commission − claims kept; before investment income and tax.
1-in-200: The 0.5th percentile of the result: the loss exceeded in one year in 200, against which the capital is held.

Key assumptions and limitations

Claims: Claim numbers are Poisson; claim sizes are independent and Pareto (type 2), heavy-tailed, as in IN-03.
Layer price: Expected loss + 0.25 × the standard deviation of the layer's annual loss. This reproduces the broker's two quotes in IN-03 (1.25 and 2 × expected loss). Priced as one layer, the broker's two cost about US$0.2m less.
Layer: One per-risk layer with unlimited free reinstatements. The part of a claim above the top stays with the insurer. Retention and cover move in steps of US$0.5m, as the broker quotes them.
Quota share: It shares what the insurer keeps after the layer and pays for its share of the layer's price. The commission does not depend on the share, and at most 50% can be placed.
Goal: The capital is fixed for the year, so its cost does not change with the programme: the best programme has the highest mean result whose 1-in-200 loss stays within the capital.
Optimizer: Each plan the Optimizer tries is judged on the same 10,000 simulated years (common random numbers, seed 1) as the simulation on the Results sheet, so plans are compared fairly and the limit the Optimizer met holds on the Results sheet too.
Data: The book and all figures are invented for this example.

How to use it

Enter your claims assumptions, the reinsurers' prices, your premium, the capital you hold and your largest net retention on the Inputs sheet.
On the ModelRisk ribbon, open the Optimizer: the task is saved in this workbook. Press Start. When it finishes, apply the best solution: the retention, cover and quota share on the Decisions sheet change.
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set) to compare the new programme with the starting plan on the Results sheet.
To watch the search from the beginning, copy the starting plan onto the Optimizer’s plan first.

ModelRisk functions used

ModelRisk Optimizer: Chooses the retention, cover and quota share: the highest mean result, the 1-in-200 result at least minus the capital, the net retention within its limit.
VosePoissonObject, VosePareto2Object: The claim number and claim size distributions as objects.
VoseAggregateTranche: Adds up, over every claim of the year, the part that falls in each of 101 bands (an array over 101 cells).
VoseDeductObject, VoseMean, VoseStdev: The layer's payment on one claim, and its exact mean and standard deviation: they set the layer's price.
VoseSimMean, VoseSimStdev: The mean and the spread of both plans' results on the Results sheet.
VoseSimPercentile, VoseSimProbability: The 1-in-20 and 1-in-200 results, the chance of a loss, and the data behind the charts.

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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.