Show how different risk factors can act on one another, and outline why combining risk exposures into a total is hard.
The idea
Risks do not act alone. Rising interest rates can raise credit losses on floating-rate borrowers. A fall in asset prices can force fire sales that turn funding trouble into market losses. Wrong-way risk occurs when exposure to a counterparty is largest just when its default probability is highest. In crises, correlations between risk factors tend to rise, so diversification shrinks. Combining risks into one total is hard because market, credit and operational risk are measured with different models, horizons and confidence levels. Some risks, such as reputation, are hard to quantify. Simply adding stand-alone figures assumes perfect correlation and so overstates the total. Assuming zero correlation understates it. With correlation ρ, the total of two figures A and B is Total = √(A² + B² + 2 × ρ × A × B). Correlations estimated in calm periods are unreliable under stress.
Formulas
Total of two risk figures with correlation
total = √(A² + B² + 2 × ρ × A × B)
- A, B
- the two stand-alone risk figures, on the same horizon and confidence level
- ρ
- the correlation between the two risks
Put both figures on a common horizon and confidence level first. At ρ = 1 it is the simple sum and at ρ = 0 the square root of the sum of squares; calm-period correlations understate crisis totals.
A practice question
During a severe market crisis, correlations between risk factors typically:
- fall toward zero, because markets become segmented.Why not: The opposite of the observed pattern: stress tends to bring assets and risk types closer together.
- stay constant, because correlations are fixed structural parameters.Why not: Correlations are unstable and are estimated from data that change with market conditions.
- rise, so diversification benefits assumed in normal times shrink. - correct
- turn negative, because safe assets rise while risky assets fall.Why not: That happens for only a few asset pairs; across most risk factors correlations rise in stress.
Correlations tend to increase in crises, so a total built with calm-period correlations understates risk when it matters most.
A harder one
A bank's stand-alone economic capital is USD 60 million for market risk and USD 80 million for credit risk. Assuming the aggregate follows the square-root rule with a correlation of 0.5 between them, the total is closest to:
- USD 121.7 million. - correct
- USD 140.0 million.Why not: Adds the two figures, which assumes a correlation of 1 and gives no diversification benefit.
- USD 100.0 million.Why not: Uses a correlation of zero and ignores the 0.5 interaction term.
- USD 111.4 million.Why not: Omits the factor of 2 on the covariance term, giving 2 x rho x A x B = 4,800 divided by 2.
Total = sqrt(60² + 80² + 2 x 0.5 x 60 x 80) = sqrt(3,600 + 6,400 + 4,800) = sqrt(14,800) = about 121.7. It lies between the zero-correlation figure (100) and the simple sum (140).
The hardest tier
An analyst reports firm-wide risk as the sum of a market risk VaR (10-day, 99%) of 30, a credit risk figure (one-year, 99.9%) of 200 and an operational risk figure (one-year, 99.9%) of 80. What is the most important flaw to fix before aggregating?
- None: the sum is conservative, so any error can only overstate risk and is harmless.Why not: Ignores that mixing horizons and confidence levels makes the numbers inconsistent, so the sum has no clear meaning.
- The figures use different horizons and confidence levels and must first be put on a common basis. - correct
- Market risk VaR should be excluded because it is not expressed in currency units.Why not: VaR is expressed in currency; the problem is comparability, not units.
- The sum should be replaced with the average of the three, because averaging removes model differences.Why not: Averaging does not create a common horizon or confidence level and gives a meaningless number.
Aggregation requires comparable inputs. Converting a 10-day, 99% figure and one-year, 99.9% figures to a common horizon and confidence level comes first; correlation assumptions are the next issue.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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