Aggregate Risk and Capital

Tail Risk, Dependence, and Capital

Calculate VaR and TVaR, model common shocks, and preserve the boundary between risk theory and regulation.

Tail Risk, Dependence, and Capital

Expected loss supports pricing and budgeting; capital is concerned with adverse outcomes around and beyond selected tolerances. First define the loss variable $L$ —gross claims, net claims, underwriting loss, or change in own funds—and its horizon.

1. VaR and TVaR

At confidence level $\alpha$ ,

\operatorname{VaR}_\alpha(L) =\inf\{l:P(L\le l)\ge\alpha\}.

For a continuous loss distribution, VaR is its $\alpha$ quantile. Tail Value at Risk is

\operatorname{TVaR}_\alpha(L) =\frac{1}{1-\alpha}\int_\alpha^1\operatorname{VaR}_u(L)\,du.

For continuous distributions it equals $E[L\mid L>\operatorname{VaR}_\alpha(L)]$ .

Measure	Reads as	Main limitation
VaR 99%	threshold exceeded in about 1% of modelled periods	says nothing about how bad exceedances are; not generally subadditive
TVaR 99%	mean loss in the worst 1% under a continuous model	more tail-sensitive and data/model hungry

Neither is “capital” until premiums, expenses, assets, management actions, horizon, and decision rules are specified.

2. Empirical estimation

For simulated losses $L_1,\ldots,L_M$ :

sort the losses;
use a documented quantile convention for VaR;
average losses beyond the selected threshold for a simple continuous-sample TVaR estimate;
quantify Monte Carlo error—especially when $(1-\alpha)M$ is small.

At 99.5%, 10,000 simulations leave only about 50 tail observations. That is usually weak evidence for a stable TVaR.

3. Dependence can erase apparent diversification

Two regions can have identical marginal count and severity models but share catastrophe states. The next experiment compares independent regional states with one shared state while preserving each region's marginal distribution.

Same marginals, different dependence

The marginal model for each region is unchanged. Only dependence changes, yet aggregate tail risk should increase materially under the shared state.

4. Reinsurance and capital

Calculate risk measures on the correctly transformed net loss:

L^{\text{net}} =L^{\text{gross}}-R(L^{\text{gross}}) +\text{reinsurance premium and costs},

where $R$ must implement the treaty at its contract unit. Then examine:

gross and net expected result;
gross and net VaR/TVaR;
attachment and exhaustion probability;
basis and counterparty scenarios;
model sensitivity near the layer.

Ignoring reinsurance price makes every ceded layer look beneficial.

5. Parameter and model risk

Tail results are functions of estimated inputs:

\rho(L;\widehat\theta,\widehat M),

where $\theta$ are parameters and $M$ is model structure. A professional analysis varies:

frequency trend and catastrophe rate;
severity tail family and threshold;
inflation and portfolio mix;
dependence/common-shock structure;
treaty wording, exhaustion, and default;
reserve and premium uncertainty.

Report model spread alongside simulation error. Running more simulations reduces only Monte Carlo error.

6. Current catastrophe evidence

Swiss Re Institute's 2025 catastrophe estimate—USD 220bn economic losses, 49% insured, and 92% of insured losses from secondary perils—motivates repeated-event and regional-dependence scenarios. It does not supply Harbour Mutual's catastrophe probability, severity tail, or correlation.

Use event catalogues and insurer exposure to calibrate; use current publications to challenge scenario completeness.

7. Regulatory boundary

Classical aggregate models and VaR/TVaR are analytical building blocks. They are not, by themselves:

an IFRS 17 insurance-contract measurement;
a Solvency UK Solvency Capital Requirement;
a complete internal model or Own Risk and Solvency Assessment.

Solvency UK reporting applies for reporting dates from 31 December 2024, and the PRA's current supervisory statements define requirements within a broader legal and governance framework. Always verify the current rulebook and firm permissions; do not infer compliance from a classroom percentile.

Practice

A loss sample has VaR $_{99}=£10$ m and TVaR $_{99}=£18$ m. Explain the difference in one sentence.
Why does doubling simulations not solve tail-model uncertainty?
Two portfolios have zero linear correlation. Does that prove tail independence?

Answers

£10m is the modelled 99th-percentile threshold, while £18m is the mean loss in the corresponding worst tail under the stated convention.
It reduces Monte Carlo error conditional on the fitted model, not error from the wrong tail family, parameters, or dependence structure.
No. Non-linear or tail dependence can remain even when covariance is zero.

Sources and further reading

Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999), “Coherent Measures of Risk,” Mathematical Finance, 9(3), 203–228.
Bank of England — Solvency II/UK capital requirement supervisory statement
Bank of England — Solvency UK insurance reporting
Swiss Re Institute — natural catastrophes in 2025

Next, move from a one-period loss distribution to surplus paths and ruin.

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