Aggregating Economic Capital Across Three Lines with VaR and TVaR

Background

Meridian Holdco writes three lines through subsidiaries: Life, Annuity, and General Insurance (GI). The group CRO must set economic capital for the coming year and decide whether the group's current capital of $100 million above expected losses is adequate against its board-approved risk appetite of holding capital at the 99.5% Tail Value at Risk (TVaR) standard.

Each line's annual aggregate loss is modeled as lognormal with expected loss $50M (Life),$30M (Annuity), and $70M (GI), and lognormal volatility (sigma of the underlying normal) of 0.25, 0.35, and 0.30 respectively. The three lines are not independent: losses are linked via a Gaussian copula with correlation matrix

reflecting modest common drivers (e.g., interest rates affecting Life and Annuity more than GI, and a small general economic/mortality-catastrophe linkage to GI).

Simulated aggregate loss distribution

A 200,000-trial Monte Carlo simulation of the correlated lognormal losses (matching the target means and the copula correlations) produces a simulated mean total loss of **$149.97 million**, closely matching the target sum of expected losses ($150M), validating the simulation. The resulting risk measures at standard solvency confidence levels are:

Recall that Value at Risk (VaR) at level $p$ is the $p$-th quantile of the loss distribution, while Tail Value at Risk (TVaR), also called Conditional Tail Expectation (CTE), is the average loss in the worst $1-p$ fraction of outcomes: $\text{TVaR}_p = E[L \mid L \ge \text{VaR}_p]$. TVaR is always at least as large as VaR at the same confidence level, and the gap between them (here, about $17M at the 99.5% level) reflects how heavy the tail is beyond the VaR threshold.

Economic capital calculation

Economic capital (EC) is typically defined as the capital needed above the expected (mean) loss to survive a specified tail event, i.e. $\text{EC}_p = \text{Risk Measure}_p - E[L]$. Using VaR at 99.5% as the risk measure, \text{EC}_{99.5\%}^{VaR} = 250.28 - 149.97 = \100.3M$. Using **TVaR at 99.5$\text{EC}_{99.5%}^{TVaR} = 267.46-149.97 = $117.5M$.

Diversification benefit

If the three lines were assessed standalone (each held to its own 99.5% TVaR) and simply summed, the required capital would be larger than the diversified $117.5M figure computed on the correlated aggregate, because correlations below 1 mean the lines' bad tail outcomes do not perfectly coincide. This gap between the sum of standalone capital requirements and the diversified group requirement is the diversification benefit, and it is the primary quantitative argument for running a multi-line group under one balance sheet rather than as separately capitalized entities — though regulators and rating agencies typically require evidence the correlation assumptions are conservative and stress-tested before crediting a large diversification benefit.

The capital decision

Meridian's actual capital held above expected losses is $100 million**. Against the board's own **99.5$117.5 million, the group is **under-capitalized by about $17.5 million (roughly 15$100.3M). This is a common and important distinction in ERM practice: a firm can appear well capitalized on a VaR basis while being meaningfully short on a TVaR basis, because VaR is blind to how bad things get beyond the threshold, while TVaR captures the severity of the tail beyond it — one of the standard textbook critiques of VaR as a risk measure (it is not subadditive in general, and ignores tail severity), which is why the SOA/CERA and Basel/Solvency frameworks increasingly favor TVaR/CTE-style measures for capital standards despite VaR's continued popularity for reporting.

Your task

Using the figures above as given (do not re-simulate), answer the following questions about risk measure interpretation, diversification, and the capital adequacy decision facing Meridian's CRO and board.