Market Risk Measurement and Management
The core toolkit
Market risk questions revolve around quantifying tail losses and then testing whether the number is credible. Be fluent in the three VaR engines - variance-covariance (delta-normal), historical simulation, and Monte Carlo - and know when each fails. Delta-normal misses gamma and fat tails; historical simulation is hostage to its sample window and cannot produce a loss worse than the worst observed day unless volatility scaling or filtering is applied; Monte Carlo is only as good as the assumed joint distribution.
Expected shortfall (ES) averages losses beyond the VaR quantile. It is subadditive and captures tail shape, which is why FRTB replaced 99%/10-day VaR with 97.5% ES. VaR itself is not subadditive, so a combined portfolio can show a higher VaR than the sum of its parts.
Volatility and dependence
EWMA (lambda around 0.94 for daily RiskMetrics) is a GARCH(1,1) special case with no mean reversion; GARCH adds reversion to long-run variance. For dependence, linear correlation is insufficient - tail dependence and copulas matter, and the Gaussian copula has zero tail dependence by construction.
Traps
Square-root-of-time scaling is valid only for i.i.d. returns; positive autocorrelation makes it understate risk. Backtesting a 99% VaR over 250 days should average 2.5 exceptions, and the Basel green zone allows 4 - 5 exceptions is amber, not automatic rejection. A 99%/10-day VaR is not ten times the 1-day figure. Also watch delta-normal vs delta-gamma for option books, key rate durations for curve risk, and the low statistical power of backtests.
Worked example
1-day 99% VaR is $10m. Under i.i.d. normal returns, 10-day VaR = 10m x sqrt(10) = about $31.6m. If returns are positively autocorrelated, this understates true risk.