Insurer Capital and Diversification Benefit

Dependence	Std dev	Capital VaR 99.5%	TVaR 99.5%	Diversif. benefit	Capital ÷ premium
Independent	32.4	103.2	173.6	55%	40%
Normal copula	39.1	118.1	181.1	48%	45%
t copula	40.1	128.6	189.3	43%	49%
Square-root formula	–	136.6	–	40%	53%
Sum of the standalone capitals		227.0

The business question

A general insurer writes US$260 million of premium a year in motor, property and liability insurance, holds US$300 million of reserves for claims from earlier years, and is exposed to storms and floods. Its regulator requires enough capital to survive a 1-in-200 year. Capitalising each risk on its own would need about US$234 million.

But the risks do not all go wrong at once, so the company needs much less — how much less depends on how they move together. Ordinary correlation says little about the worst years: in a crisis, inflation lifts motor and liability claims and reserves at the same time.

The model computes the capital three ways — risks independent, joined by a Normal copula, and by a t copula that makes bad years cluster — and compares them with the regulator's square-root formula. It then shares the capital out between the risks.

Who it is for: Capital and risk actuaries, chief risk officers, finance teams of insurers; anyone building an internal model.
Main outputs: The capital at the 1-in-200 (VaR 99.5%) and the average bad year beyond it (TVaR) for independent risks, a Normal copula and a t copula · the square-root formula for comparison · the diversification benefit · each risk's standalone capital, and the capital allocated to it by covariance and by its share of the bad years.

How the model works

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

Inputs sheet
For each line the premium, the expected loss ratio and its variability. The reserves and the variability of their run-off; the frequency and size of catastrophes; six correlations; the degrees of freedom of the t copula. The threshold of a bad year for the allocation.
Capital sheet
Each iteration draws the year's loss — claims above or below their expected value — for every risk, three times: independently, through a Normal copula and through a t copula with the same correlations. Catastrophes are independent of the rest in all three. The totals, and each risk's loss in the bad years, are recorded.
Results sheet
The capital (99.5th percentile of the total loss) and the TVaR under each model, against the square-root formula and the sum of standalone capitals. The capital allocated to each risk by covariance and by its share of the bad years.

ModelRisk techniques

Copula-driven loss ratios VoseCopulaMultiT
Catastrophe losses VoseAggregateMC
VaR and TVaR VoseSimCVARp
Square-root formula
Capital allocation VoseSimCorrelation

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The results in detail

Capital by risk

Standalone: each risk capitalised alone. Allocated: the t-copula capital shared out by covariance and by the bad years.

Total loss, t copula — distribution

Histogram of t copula from the saved simulation.

Risk	Standalone VaR 99.5%	Std dev	Correl. with total	By covar. allocated	By bad yrs allocated
Motor	19.6	6.9	0.69	15.3	10.5
Property	22.0	7.2	0.24	5.6	5.6
Liability	28.5	9.1	0.70	20.2	16.3
Reserves	68.4	24.1	0.84	64.6	42.2
Catastrophe	88.6	16.7	0.43	22.8	54.0
Total	227.0			128.6	128.6
Share of the t-copula capital allocated				100.0%	100.0%

Loss by risk in the bad years	All years mean, US$m	Bad years mean, US$m
Motor	0.16	11.4
Property	0.08	6.0
Liability	0.25	17.6
Reserves	0.65	45.6
Catastrophe	0.83	58.4
Share of years that are bad (threshold on the Inputs sheet)	1.41%

The square-root formula ρ × capital × capital (US$m²)	Motor	Property	Liability	Reserves
Motor	385	108	279	335
Property	108	484	156	376
Liability	279	156	809	972
Reserves	335	376	972	4,672
Catastrophes (independent: their capital²)	7,853

How to read these results

Capitalising each risk on its own would take about US$227 million. Together they need US$103 to 129 million, depending on how they move together: diversification saves 43% to 55% of the capital.
The model of dependence matters even with the same correlations. The t copula, which makes the worst years cluster, needs about a quarter more capital than independent risks and about 9% more than the Normal copula: ignoring dependence, or modelling it with a Normal copula alone, understates the capital.
The regulator's square-root formula, applied to the simulated standalone capitals with the same correlations, gives about US$137 million here, a little above the t copula. It is not reliable in general: it treats the risks as if they combined like Normal ones, and with heavy tails it can fall on either side of the simulated capital.
The TVaR — the average of the years worse than the 1-in-200 — is about US$190 million under the t copula, far above the VaR, because the few catastrophe years beyond it are extreme.
Sharing the capital out depends on the method. By covariance, the reserves take about half, because they vary most, and catastrophes about a sixth. In the bad years — here a total loss above US$100 million, about one year in 70 — catastrophes bring over two fifths of the loss, about US$58 million in an average bad year.
Prices and return-on-capital targets by line inherit that choice.
Simulation error: the model ships with 50,000 samples, so a 1-in-200 rests on about 250 simulated years rather than 50. The catastrophes' heavy tail still leaves the capitals uncertain by about ±5% — at 10,000 samples two seeds gave t-copula capitals of 137 and 121. The gap between the Normal and t copulas, about 9%, is only just clear of it.

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.

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Inputs and assumptions

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

Lines of business 9 inputs

Premium written next year, and the claims it is expected to bring.

Assumption	Unit	Value
Motor: premiumMany small claims: a steady line.	US$m	120
Motor: expected loss ratioClaims ÷ premium, on average.	of premium	72%
Motor: variability of the loss ratioHow much a year's claims vary around their expected value.	std dev ÷ mean	8%
Property: premiumFires and storms below the catastrophe level.	US$m	80
Property: expected loss ratioClaims ÷ premium, on average.	of premium	60%
Property: variability of the loss ratioHow much a year's claims vary around their expected value.	std dev ÷ mean	15%
Liability: premiumLong-tail claims, hard to predict.	US$m	60
Liability: expected loss ratioClaims ÷ premium, on average.	of premium	75%
Liability: variability of the loss ratioHow much a year's claims vary around their expected value.	std dev ÷ mean	20%

Reserves 2 inputs

Claims from earlier years, not yet paid.

Assumption	Unit	Value
Claims reservesBest estimate (see IN-05).	US$m	300
Variability of the run-offPrediction error of the reserves over one year.	std dev ÷ mean	8%

Catastrophes 3 inputs

Storms and floods hitting the property book.

Assumption	Unit	Value
Catastrophes a yearAbout one every three years.	events	0.3
Pareto scaleSets the size of an event.	US$m	20
Pareto shapeAverage event = scale ÷ (shape − 1).	lower = heavier	2.5

Dependence 7 inputs

Between the lines and the reserves; catastrophes are independent.

Assumption	Unit	Value
Correlation: motor and property		0.25
Correlation: motor and liabilityInflation drives both.		0.50
Correlation: motor and reserves		0.25
Correlation: property and liability		0.25
Correlation: property and reserves		0.25
Correlation: liability and reservesLiability reserves move with new liability claims.		0.50
t copula: degrees of freedomThe lower, the more strongly the worst years come together.		4

Capital allocation 1 input

Assumption	Unit	Value
A bad year: total loss aboveAbout a 1-in-80 year: the capital is shared by each risk's part in these years.	US$m	100

Units and conventions

Money: US$ million (US$m).
Loss: Claims above their expected value in the year: positive is bad. Expected losses are already in the premium.
Capital: The 99.5th percentile of the total loss: what survives a 1-in-200 year.
TVaR: The average loss in the years worse than the 1-in-200: how bad the very bad years are.
Diversification benefit: 1 − capital ÷ the sum of the standalone capitals.

Key assumptions and limitations

Lines: Each line's loss ratio is Lognormal with the given mean and coefficient of variation.
Reserves: The run-off of the reserves is Lognormal around their best estimate (see IN-05).
Catastrophes: Poisson events with Pareto-distributed losses, independent of the other risks.
Dependence: The same correlations in the Normal and the t copula; the t copula adds tail dependence.
Scope: Underwriting and reserve risk only: no market, credit or operational risk; one year.
Data: The insurer and all figures are invented for this example.

How to use it

Enter your lines' premiums and loss ratios, your reserves and your catastrophe exposure.
Set the correlations and the t copula's degrees of freedom (lower = more clustering of bad years).
Run a simulation from the ModelRisk ribbon (50,000 samples with a fixed seed are set: a 1-in-200 needs more than the usual 10,000, and the run takes several minutes).
Compare the capital under the three models and the square-root formula; read where the capital goes.

ModelRisk functions used

VoseCopulaMultiNormal, VoseCopulaMultiT: Four correlated U values each iteration: Normal, and Student (bad years cluster).
VoseLognormal: Each risk's loss from its U value: the same distribution under every dependence model.
VoseAggregateMC, VosePoissonObject, VosePareto2Object: The year's catastrophe losses.
VoseSimPercentile, VoseSimCVARp: The 1-in-200 loss (VaR) and the average beyond it (TVaR).
VoseSimStdev, VoseSimCorrelation, VoseSimMean: The statistics behind the capital allocation. The covariance method needs the Pearson correlation (type 1): the shares then add up to the capital.

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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.

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