4.13.4 Aggregating Different Risk Types

Basic Prerequisites: VaR & Expected Shortfall and 03 · Systemic Risk Measures.

---

1. Intuition & Practical Objective

This is the single most consequential quantitative problem in the pillar. A bank must report one P&L/QIS capital number, but it runs four different kinds of risk that are shot in different currencies on different clocks:

• Market - daily, mark-to-market, driven by factor returns.
• Credit - default-frequency, long horizon, driven by counterparty health.
• Liquidity - event-driven, worse in stress precisely when it matters.
• Operational - rare, heavy-tailed, almost independent of markets (until it isn’t).

The objective: see why no neutral “add them up” exists, and why copula/scenario methods are the only defensible route. The intuition in one line: expected shortfall is subadditive, so “sum of the marginal ES” is an unattainable upper bound - but treating the risk types as independent is catastrophically wrong in the tail, because the marginals are tail-dependent precisely when they should be independent.

---

2. Mathematical Ground Truth & Derivations

Why the naive sum is “safe-but-wrong”. For any dependence, ${\text{ES}}_{\alpha }$ is subadditive,

$${\text{ES}}_{\alpha }(X_1+X_2)\le {\text{ES}}_{\alpha }(X_1)+{\text{ES}}_{\alpha }(X_2),$$

with equality only in perfect comonotonicity (one’s tail is the other’s). So summing the marginal ES (the regulator’s simple approach) can only overstate - it keeps you solvent at the price of holding too much capital. The Basel formula for operational risk add-ons is a cousin of this: add risk types with a fixed “correlation” coefficient instead of measuring it.

The shared-macro-factor truth. In reality market, credit and liquidity losses are all driven by one macro state $Z$ (a recession is bad for all of them at once). So the naive independence model - VaR_combined = sqrt(VaR_1² + VaR_2²) from summing variances - is the dangerous direction: it treats tails as addable in quadratic form and understates the joint tail, because the covariance term $2\mathrm{Cov}(X_1,X_2)=2(\mathbb{E}[X_1{X}_2]-\mathbb{E}[X_1]\mathbb{E}[X_2])$ is far from zero exactly when $Z$ is extreme.

Copula aggregation (Sklar). A copula $C$ joins marginals and carries only the dependence:

$$F_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=C(F_{X_1}(x_1),\dots ,F_{X_n}(x_n)).$$

The Gaussian copula (one-factor form, Vasicek’s model) is the industry default and encodes zero upper-tail dependence: two Gaussian-copula variables asymptotically never crash together. The Student-t copula (with low df) adds tail dependence - joint extremes are more likely. Choosing between them is not cosmetic: it decides the size of the aggregated tail capital.

The aggregation fallacy in one sentence. Because tail dependence $\lambda ={\lim }_{u\to 1}\mathbb{P}(X_1>F_1^{-1}(u)\mid X_2>F_2^{-1}(u))$ is essentially unobservable (you have ~zero joint-tail observations), the copula - not the data - sets the answer, and every choice errs: sum → overstates; independence → understates; Gaussian copula → assumes no tail dependence; t-copula → assumes a specific one.

---

3. Computational Implementation - two loss streams, Gaussian aggregation + tail dependence

Part A - aggregating by correlation. Two normal loss streams (market, credit). Naive sum = $\text{ES}(\text{market})+\text{ES}(\text{credit})$; then the true aggregate ES for varying correlation $\rho$. Stdlib only, closed form via the normal-ES formula.

Read the output. Part A: the naive sum (63.97) is exactly the $\rho =1$ comonotonic case - you only “achieve” it if the two books crash perfectly together, which never happens. The true aggregate is 45.85 (independence) to 55.65 (corr 0.5), lower than the sum by 8–18 currency units of overstated capital. Part B: co-exceedance at the 1% level is 0.0013 under a Gaussian copula vs 0.0030 under a t-4 copula - more than 2× difference in joint-tail probability. The aggregated capital bearing line sits on that factor of two, and it is chosen by the copula you pick, not by the data. This is precisely the “unmeasurable tail dependence” that makes naive aggregation indefensible.

---

4. Failure Modes & First-Principles Breakdowns

1. Naive sum = assuming perfect comonotonicity. It can only overstate (safe, wasteful). Structural.
2. Naive independence = assuming zero covariance. Quadratic-covariance aggregation ignores the macro factor and understates the joint tail. Dangerous.
3. Gaussian copula = assuming zero tail dependence. Even at corr 0.5 its joint 1% tail is under-represented vs a fat-marginal / t-4 clumping (0.0013 vs 0.0030); Gaussian dependence asymptotically cannot model simultaneous crashes.
4. The copula-freedom illusion. When tail dependence is unobservable, the result is decision-under-model-uncertainty: report a range across copulas (Gaussian vs t vs Clayton) and stress the worst, exactly as Bellini’s scenario-integration does (06 · Advanced Extensions).

---

5. References

• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)
• Bellini, Stress Testing and Risk Integration in Banks (2016)
• Vašíček, Probability of Loss on Loan Portfolio (1987)
• Hull, Options, Futures, and Other Derivatives

---

6. Connected Graph Bridges

• Back: 03 · Systemic Risk Measures · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Base: VaR & Expected Shortfall · Credit Risk & the Merton Model · Operational Risk