How do banks aggregate risk across trading desks, and what are the challenges with recognizing diversification benefits?
I'm studying market risk for FRM Part II. Each desk has its own VaR, but how does a bank compute firm-wide VaR? My textbook mentions that diversification should reduce total risk, but regulators are skeptical. What's the issue?
Risk aggregation is one of the most practically important and theoretically challenging problems in market risk management. The challenge: individual desk VaRs don't simply add up, and the diversification benefit depends critically on correlation assumptions that may fail during crises.
The Basic Aggregation Problem:
Suppose Westbrook Bank has three trading desks:
- Rates desk: VaR = $15M
- Equities desk: VaR = $20M
- FX desk: VaR = $10M
Simple sum (no diversification): $15M + $20M + $10M = $45M
With correlations (assuming rho = 0.3 between all pairs):
Firm VaR = sqrt(15^2 + 20^2 + 10^2 + 2(0.3)(15)(20) + 2(0.3)(15)(10) + 2(0.3)(20)(10))
= sqrt(225 + 400 + 100 + 180 + 90 + 120) = sqrt(1115) = $33.4M
Diversification benefit = $45M - $33.4M = $11.6M (26% reduction)
Aggregation Methods:
| Method | Description | Pros | Cons |
|---|---|---|---|
| Variance-covariance | Sum using correlation matrix | Simple, closed-form | Assumes normal, linear |
| Full revaluation | Single Monte Carlo for all desks | Most accurate | Computationally expensive |
| Copula-based | Model dependency structure directly | Flexible correlation | Complex to implement |
| Simple sum | No diversification | Conservative | Overstates risk |
Why Regulators Are Skeptical:
- Correlations break during crises: The 0.3 correlation assumed above might jump to 0.8+ during a market crash, eliminating most diversification benefit
- Tail correlation is higher: Even if average correlation is low, extreme events tend to be more correlated
- Model risk: Correlation estimation is noisy and unstable
- Incentive effects: Banks have incentives to overstate diversification to reduce capital
Basel FRTB Approach:
The Fundamental Review of the Trading Book uses Expected Shortfall (not VaR) and recognizes limited diversification:
- Calculate ES for each risk class (GIRR, CSR, EQ, Commodity, FX)
- Aggregate using prescribed correlation parameters
- Apply a floor: aggregated ES cannot be less than a specified fraction of the simple sum
FRM Key Points:
- Diversification benefits are real in normal markets but can vanish in crises
- Sub-additivity: ES is sub-additive (portfolio ES <= sum of individual ES); VaR is NOT always sub-additive
- Banks should stress test the correlation assumptions used in aggregation
- The simple sum provides an upper bound that's useful for stress scenarios
Explore risk aggregation techniques in our FRM Part II Market Risk course.
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