Risk Measures and Capital

Risk measures translate a loss distribution into a single number used for pricing, reserving margins, and capital setting. The choice of measure matters: it determines what gets capitalized, how capital is allocated across business units, and whether the resulting incentives are sound.

Value at Risk

$\text{VaR}_\alpha(X) = \inf\{x : F_X(x) \ge \alpha\}$

is the $\alpha$-quantile of the loss distribution $X$ — "the loss we expect to exceed only $(1-\alpha)$ of the time." VaR is easy to compute and communicate but says nothing about the severity of losses beyond the threshold, and — critically — is not subadditive in general.

Tail VaR, CTE, and Expected Shortfall

$\text{TVaR}_\alpha(X) = \frac{1}{1-\alpha}\int_\alpha^1 \text{VaR}_u(X)\,du = E[X \mid X \ge \text{VaR}_\alpha(X)]$

for continuous distributions (Conditional Tail Expectation, CTE, and Expected Shortfall, ES, coincide with TVaR in the continuous case; they can differ subtly at mass points in discrete distributions). TVaR averages all the tail outcomes beyond the quantile, so it is sensitive to tail thickness in a way VaR is not — a key reason regulators (Solvency II's economic underpinnings, Swiss Solvency Test) and rating agencies favor it for tail-risk capital.

Coherence and VaR's subadditivity failure

Artzner, Delbaen, Eber & Heath (1999) define a coherent risk measure as one satisfying:

1. Monotonicity: $X \le Y \Rightarrow \rho(X) \le \rho(Y)$
2. Translation invariance: $\rho(X+c) = \rho(X) + c$
3. Positive homogeneity: $\rho(\lambda X) = \lambda \rho(X)$ for $\lambda \ge 0$
4. Subadditivity: $\rho(X+Y) \le \rho(X) + \rho(Y)$

Subadditivity formalizes "diversification cannot increase risk" — a portfolio's risk measure should not exceed the sum of its parts' risk measures. TVaR is coherent; VaR is not, and the standard counterexample makes this concrete.

Counterexample: VaR is not subadditive

Consider two independent risks, each defaulting with probability 4% for a loss of 100 (else 0), so each risk's 95% VaR is 0 (since $P(\text{loss}=0)=96\% \ge 95\%$). For the sum $X+Y$:

$P(X+Y = 0) = 0.9216 < 0.95$, so the 95th percentile of $X+Y$ falls in the "exactly one defaults" outcome: $\text{VaR}_{0.95}(X+Y) = 100$.

But $\text{VaR}_{0.95}(X) + \text{VaR}_{0.95}(Y) = 0 + 0 = 0 < 100 = \text{VaR}_{0.95}(X+Y)$ — subadditivity is violated: diversifying two risks increased the measured VaR from 0 to 100. This is exactly the pathology that motivates TVaR/ES for tail capital.

Distortion and spectral risk measures

A distortion risk measure reweights the survival function through a concave distortion function $g:[0,1]\to[0,1]$ with $g(0)=0, g(1)=1$:

$\rho_g(X) = \int_0^\infty g(S_X(x))\,dx.$

The Wang transform uses $g(u) = \Phi(\Phi^{-1}(u) + \lambda)$, shifting the standard normal quantile by a market price of risk $\lambda$; the proportional hazard (PH) transform uses $g(u) = u^{1/\gamma}$ for $\gamma \ge 1$. Both are coherent when $g$ is concave, and both generalize VaR/TVaR as special (degenerate) cases. A spectral risk measure generalizes further by weighting each quantile of the loss distribution by an explicit risk-aversion function $\phi(u)$ integrating to 1: $\rho_\phi(X) = \int_0^1 \phi(u)\,\text{VaR}_u(X)\,du$, and is coherent whenever $\phi$ is non-decreasing (more weight on worse outcomes) — TVaR is the spectral measure with $\phi(u) = \mathbb 1\{u > \alpha\}/(1-\alpha)$.

Euler (gradient) capital allocation

Given total capital $\rho(X)$ for a portfolio $X = \sum_i X_i$, the Euler allocation splits capital in a way that is exactly additive (allocations sum to total capital) and reflects each unit's marginal contribution to tail risk, exploiting the homogeneity of $\rho$:

$\rho_i = X_i \cdot \frac{\partial \rho(X)}{\partial X_i}\bigg|_{\text{at actual weights}}.$

For $\rho$ = standard deviation, this reduces to $\rho_i = \dfrac{\text{Cov}(X_i, X)}{\sigma(X)}\sigma(X)$ — i.e., allocate in proportion to each unit's covariance with the total. For TVaR, the Euler allocation is $\rho_i = E[X_i \mid X \ge \text{VaR}_\alpha(X)]$, the expected loss from unit $i$ conditional on the portfolio being in its tail scenario — capturing exactly how much each line contributes when things go badly, not merely its stand-alone size.

Worked two-line Euler example

Two lines, simulated (or modeled) jointly over 10 equally-likely scenarios, total losses $X = X_1 + X_2$:

At $\alpha = 0.80$ (worst 20% = top 2 of 10 scenarios: 9 and 10), $\text{TVaR}_{0.80}(X) = (90+120)/2 = 105$.

Euler allocation to line 1: $E[X_1 \mid \text{tail}] = (20+30)/2 = 25$. Euler allocation to line 2: $E[X_2 \mid \text{tail}] = (70+90)/2 = 80$. Check: $25+80=105$ ✓ — capital is allocated in proportion to each line's actual tail behavior (line 2 dominates the tail even though it is not always the larger line), not its stand-alone volatility.

RBC and Solvency II SCR

The NAIC Risk-Based Capital (RBC) formula for P&C insurers combines charges for asset risk (R0–R2), credit risk (R3), underwriting risk — reserve and premium risk (R4, R5) — via a covariance adjustment (square-root formula, treating major risk categories as independent) to produce Total Adjusted Capital vs. Authorized Control Level RBC, triggering regulatory action at specified ratios (Company Action Level, Regulatory Action Level, etc.).

Solvency II's SCR (Solvency Capital Requirement) is calibrated as the 99.5% one-year VaR of Basic Own Funds, computed either via a standard formula (modular capital charges for market, counterparty default, life/health/non-life underwriting risk, combined via prescribed correlation matrices) or an approved internal model. Both RBC and SCR are ultimately risk measures applied to an insurer's own balance sheet, differing chiefly in calibration philosophy (formula factors and correlations vs. VaR at a specified confidence level) and in whether diversification benefits are recognized explicitly.

ORSA (Own Risk and Solvency Assessment) requires insurers to conduct their own forward-looking assessment of risk and capital adequacy — beyond the formulaic SCR/RBC — incorporating the insurer's own risk profile, business strategy, and stress scenarios, and is a qualitative/quantitative complement rather than a single number.

Pitfalls

• Using VaR for capital and diversification credit simultaneously — since VaR is not subadditive, "diversified VaR" can understate risk in ways TVaR would not.
• Ignoring estimation error in tail risk measures — TVaR at extreme percentiles is highly sensitive to tail model choice (see Extreme Value Theory).
• Allocating capital by stand-alone volatility rather than tail contribution, understating the true capital need for a line that is small on average but dominant in tail scenarios.

Exam relevance

Coherent risk measures, TVaR, and capital allocation are tested on CAS Exam 9 (previously Exam 8/ERM material) and referenced throughout SOA ERM/QFI tracks.

Further reading

• Artzner, Delbaen, Eber & Heath (1999), Coherent Measures of Risk
• Wang (2000), A Class of Distortion Operators for Pricing Financial and Insurance Risks
• NAIC, Risk-Based Capital Forecasting & Instructions